A N A L Y T I C A L A N D EXPERIMENTAL EQUIPMENT VIBRATION STUDIES O F T H E BEHAVIOUR O F ISOLATORS UNDER SEISMIC CONDITIONS by FRANK C. F. L A M B.A.Sc, The University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T O F T H E REQUIREMENTS MASTER FOR T H E D E G R E E O F O F APPLIED SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH COLUMBIA April H85 e Frank C. F. Lam,/985 In presenting degree at freely this the of Department publication this or in partial for reference thesis by his and study. for scholarly or her of purposes requirements for an Department of Civil Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 agree that permission may It of this thesis for financial gain shall April 85 the I further representatives. permission. Date: fulfilment University of British Columbia, 1 agree that the Library shall available copying thesis be is granted understood by the that advanced make it for extensive Head of my the copying or not be allowed without my written ABSTRACT Analytical isolators effect and experimental studies of the under seismic conditions are presented. of equipment-structure equipment-structure conditions which of equipment vibration A preliminary parametric study of the interaction on the ultimate equipment response of general systems is considered under behaviour a first non-interactive The results of approach can this study yield indicate adequate the ultimate equipment response estimates. A model of a prototype air handling unit mounted on vibration isolators constructed for elastomeric isolators tested under characteristics indicate and experimental open and spring dynamic studies. isolators conditions. Two types with The of vibration uni-directional frequency isolators - restraint - seismic response and were of these vibration isolated systems were obtained. The experimental results the values. substantially in the static that damping use was vibration The higher isolators results level also of base of the have show nonlinear that excitation the stiffness elastomeric than the characteristics and isolators survive open spring can isolators high a with uni- directional restraint Analytical identification series vibration test results, have been analysis, agreement models was between used the to solve experimental isolated systems, based on the model formulated. A numerical procedure, ultilizing time the equations results and of the motion analytical of the results systems. was Good observed. This study indicates that the analytical procedure can be used to accurately predict the response characteristics of vibration isolated equipment systems subjected excitation inputs. ii to known base Table of Contents ABSTRACT ii LIST OF FIGURES vi LIST OF TABLES xiii ACKNOWLEDGMENTS 1. xiv INTRODUCTION 1 1.1 GENERAL : 1.2 LITERATURE REVIEW 2. 1 3 1.2.1 The Seismic Response of Linear Equipment Systems 3 1.2.2 The Seismic Response of Nonlinear Equipment Systems 6 1.2.3 Dynamic Testing Techniques and Seismic Qualification Practice 7 1.3 OBJECTIVES AND SCOPE 7 ANALYTICAL STUDY ON THE EFFECT OF EQUIPMENT-STRUCTURE INTERACTION 9 2.1 INTRODUCTION 9 2.2 STEADY STATE RESPONSE '..9 2.2.1 Coupled Differential Equations of Motion 10 2.2.2 Uncoupled Differential Equations of Mouon 12 2.2.3 Comparsion of the Coupled and Uncoupled Steady State Response ....14 2.3 SEISMIC RESPONSE OF EQUIPMENT-STRUCTURE SYSTEM 3. 15 2.3.1 System Parameters Studied 16 2.3.2 Solution Method 21 2.3.3 Response of Linear Equipment 22 2.3.4 Method of Shake Table Testing 23 TESTING FACILITIES AND TEST MODEL PARAMETERS 42 3.1 INTRODUCTION 42 3.2 TESTING FACILITIES 42 3.3 TEST MODEL CHARACTERISTICS 44 iii 4. 5. E X P E R I M E N T A L STUDY 53 4.1 INTRODUCTION 53 4.2 STATIC TESTING 4.3 D Y N A M I C TESTING 7. 53 PROGRAM 55 4.3.1 Sinusoidal Testing 55 4.3.2 Sinusoidal Decay Testing 56 4.3.3 Random White Noise Testing 57 4.3.4 Seismic Motion Testing 58 A N A L Y T I C A L STUDY 62 5.1 INTRODUCTION 62 5.2 M O D E L IDEALIZATIONS 62 5.3 DEVELOPMENT OF THE MATHEMATICAL MODEL 65 5.3.1. The Structural Model 65 5.3.2 67 5.4 6. PROGRAM The Equipment Model 5.3.3 The Combined Equipment Structure Model 77 M E T H O D O F SOLUTION 80 E X P E R I M E N T A L A N D A N A L Y T I C A L RESULTS 85 6.1 INTRODUCTION 85 6.2 M O D E L INDENTIFICATION TEST RESULTS (STATIC) 85 6.3 M O D E L INDENTIFICATION TEST RESULTS (DYNAMIC) 89 6.4 RANDOM MOTION TEST RESULTS 91 CONCLUSIONS 125 7.1 S U M M A R Y A N D CONCLUSIONS 125 7.2 F U T U R E STUDIES 127 BIBLIOGRAPHY 128 APPENDIX A 131 APPENDIX B 134 iv APPENDIX C ! 139 v LIST O F FIGURES Figure Page 2.1 Two degree of freedom equipment-structure system 2.2 Uncoupled and coupled equipment response transfer function. e=0.01, $. = 0.01 2.3 26 27 Uncoupled and coupled equipment response transfer function. e=0.001, $ . = 0.01 28 i 2.4 Uncoupled and coupled equipment e =0.0001, $ . = 0.01 2.5 response transfer function. '. Uncoupled and coupled equipment response transfer function. 6=0.01, £ . = 0.02 2.6 30 Uncoupled and coupled equipment response transfer function. 6=0.001, $. = 0.02 2.7 31 Uncoupled and coupled equipment response transfer function. e =0.0001, $. = 0.02 2.8 32 Uncoupled and coupled equipment response transfer function. 6=0.01, $. = 0.05 2.9 33 Uncoupled and coupled equipment response transfer function. e=0.001, $j = 0.05 2.10 34 Uncoupled and coupled equipment response transfer function. e=0.0001, $ . = 0.05 2.11 29 35 Uncoupled and coupled equipment e=0.01, $ . = 0.10 response transfer function. 36 ° I vi Figure 2.12 Page Uncoupled and coupled equipment response transfer function. e =0.001, $. = 0.10 37 o 1 2.13 Uncoupled and coupled equipment response transfer function. e =0.0001. 5 . = 0.10 38 2.14 Three degree of freedom structural system 39 2.15 Single degree of freedom equipment system 39 2.16 Combined equipment-structure 39 2.17 Coupled and uncoupled peak response of tuned secondary system to El Centra earthquake system N - S component. 2% damping in structure, 5% damping in equipment. 2.18 40 Coupled and uncoupled peak response of tuned secondary system to El Centro earthquake N-S component 2% damping in structure, 2% damping in equipment 41 3.1 Overall function of U.B.C. shaking table 48 3.2 Experimental model showing instrument locations 49 3.3 Prototype air handling system 50 3.4 Experimental model 51 3.5 Open spring isolator with uni- directional restraint 52 3.6 Elastomeric isolator 52 4.1 Schematic respresentation 4.2 Static test set up 4.3 Close up view of bi-directional static test 61 4.4 Typical measured load-deflection curve 61 5.1 Idealized model of the vibration isolated equipment system 82 of bi-directional static test . vii 60 60 Figure 5.2 Page Nonlinear load-deflection curve (static test) and piecewise linear load-deflection curve (analytical model) 5.3 Schematic representation 83 of Coulomb friction between isolator and snubber device 5.4 83 Equilibrium forces in a displaced vibration isolated equipment system 84 6.1 Axial load-deflection curves of an elastomeric vibration isolator. 6.2 Lateral load-deflection curves of an elastomeric vibration isolator. 95 6.3 Axial load-deflection curves of an open spring isolator with uni-direcuonal restraint. 6.4 ..94 96 Lateral load-deflection curves of.an open spring isolator with uni-directional restraint. 97 st 6.5 Decaying sinusoidal response time history at 1 natural frequency (1.8 Hz): open spring isolator with uni-direcuonal restraint 6.6 98 Decaying sinusoidal response u'me history at 3r<^ natural frequency (4.9 Hz): open spring isolator with uni-directional restraint 6.7 .' 99 Horizontal acceleration frequency response function: open spring isolator with uni-directional restraint 6.8 6.9 100 Rotational acceleration frequency response function: open spring isolator with uni-directional restraint 101 Predicted and measured mode shapes: open spring isolator with uni-directional restraint 102 viii Figure Page st 6.10 Decaying sinusoidal response time history at 1 natural frequency (4.6 Hz): elastomeric isolator 6.11 103 Decaying sinusoidal response time history at 3TC^ natural frequency (15 Hz): elastomeric isolator 6.12 Horizontal acceleration frequency response function: 104 elastomeric isolator 6.13 105 Rotational acceleration frequency response function: elastomeric isolator 106 6.14 Predicted and measured mode shapes: elastomeric isolator 107 6.15 Predicted and measured horizontal acceleration response time history: open spring isolator with uni- directional restraint Base motion: El Centro 1940 N-S. Peak acceleration scaled to O.lg 6.16 ;..108 Predicted and measured rotational acceleration response time history: open spring isolator with uni-directional restraint Base motion: El Centro 1940 N-S. Peak acceleration scaled to O.lg 6.17 109 Predicted and measured horizontal acceleration response time history: open spring isolator with uni-directional restraint Base motion: San Fernando 1971 N21E. Peak acceleration scaled to 0.18g 6.18 110 Predicted and measured rotational acceleration response time history: open spring isolator with uni-directional restraint Base motion: San Fernando 1971 N21E. Peak acceleration scaled to 0.18g ix Ill Figure 6.19 Page Predicted and measured horizontal acceleration response time history: open spring isolator with uni-directional restraint. Base motion: Taft 1950. Peak acceleration scaled to O.llg 6.20 112 Predicted and measured rotational acceleration response time history: open spring isolator with uni-directional restraint. Base motion: Taft 1950. Peak acceleration scaled to O.llg 6.21 113 Predicted and measured horizontal acceleration response time history: open spring isolator with uni-directional restraint Base motion: Band limited (0 - 25 Hz) white noise signal. Peak acceleration scaled to O.lg 6.22 114 Predicted and measured rotational acceleration response time history: open spring isolator with uni-directional restraint Base motion: Band limited (0 - 25 Hz) white noise signal. Peak acceleration scaled to O.lg 6.23 115 Predicted and measured horizontal acceleration response time history: elastomeric isolator. Base motion: El Centro 1940 N-S. Peak acceleration scaled to O.lg 6.24 116 Predicted and measured rotational acceleration response time history: elastomeric isolator. Base motion: El Centro 1940 N-S. Peak acceleration scaled to O.lg x 117 Figure 6.25 Page Predicted and measured horizontal acceleration response time history: elastomeric isolator. Base motion: San Fernando 1971 N21E. Peak acceleration scaled to 0.18g 6.26 118 Predicted and measured rotational acceleration response time history: elastomeric isolator. Base motion: San Fernando 1971 N21E. Peak acceleration scaled to 0.18g 6.27 119 Predicted and measured horizontal acceleration response time history: elastomeric isolator. Base motion: Taft 1950. Peak acceleration scaled to 0.1 lg 6.28 120 Predicted and measured rotational acceleration response time history: elastomeric isolator. Base motion: Taft 1950. Peak acceleration scaled to 0.1 lg 6.29 121 Predicted and measured horizontal acceleration response time history: elastomeric isolator. Base motion: Band limited (0 - 25 Hz) white noise signal. Peak acceleration scaled to O.lg 6.30 122 Predicted and measured rotational acceleration response time history: elastomeric isolator. Base motion: Band limited (0 Peak acceleration scaled to O.lg xi 25 Hz) white noise signal. 123 Figure Page 6.31 Failed open spring isolator with uni-directional restraint. 124 6.32 Failed open spring isolator with uni-directional restraint. 124 A.1 Power spectral density Function of a band limited white noise signal 133 A. 2 Power spectral density function B. l Single input / mutiple output system xii 133 135 LIST OF TABLES Table 6.1 Page Model and prototype inertia properties. xiii 86 ACKNOWLEDGMENTS 1 wish to express my sincere gratitude and appreciation to my three advisors, Dr. S. Cherry, Dr. N. D. Nathan, and Dr. D. L. Anderson, for their encouragement and guidance received from throughout the research and preparation of this thesis. Mr. R. A. Strachan of Brown Strachan Associates The cooperation was very helpful in this investigation and it is sincerely appreciated. My thanks are also extended to fellow graduate students, Mr. Bill Lipsett, Mr. Ken Tarn, and Mr. Bill Barwig, for their valuable advice during the course of this thesis. This thesis was made possible by the financial assistance Research Council of Canada in the form of an research assistantship. the vibration isolators appreciated. 1 also by Airmaster Sales Ltd. and wish to thank Genstar Structures Air System Ltd. for of the National The donation of supplies Ltd. is also partially donating the experimental model. Finally 1 wish to express my thanks to the faculty and technical staff Civil Engineering Department at U . B . C , fellow graduate their encouragement, advice, and assistance. xiv students, of the and my parents for Chapter 1 INTRODUCTION 1.1 G E N E R A L The interest. protection Essential sprinkling of equipment equipment systems, control such from as seismic pressure equipment, and hazards vessels, piping is pumps, are vital of wide power to of extreme importance. Within these lifeline and emergency generators, fire certain emergency systems. The operability of these systems during and after engineering lifeline and an earthquake systems there are is some items of equipment, such as air handling units and chillers, which are not vital to the operation of these systems. Compared with essential equipment, it is not as important that these non-essential items remain operative during or after it is dangerous earthquake an earthquake. However, to suggest that their seismic restraint criteria can be relaxed; during an such units could be displaced from their foundations, causing damage to neighboring essential equipment Past earthquakes have shown that the failure of equipment systems which lead to the breakdown of emergency services can produce loss of human life and extensive property damage. Also, the failure of lifeline systems such as transportation and energy transmission can facilities, cause extreme various secondary can be significant the value further disorder may in turn increase disasters. In addition, the economic consequences Specialized development, to replace compounds California in a city, which equipment of their housing structures engineering required and the disruption of communication and water supply networks Division the of and specialized initial Mines be construction much more costs. expensive investment Furthermore, equipment may involve a revenue financial loss and Geology 1 caused by an estimated earthquake. that, for of equipment failure because of the considerable manufacturing, damaged systems may the potential in In than in their the time loss which 1973, California the alone, 2 twenty-one billion dollar worth of damage could be expected and 2000 as equipment a design result of equipment principles developing criteria for were failure not due improved to between the years earthquakes [1]. Thus if the the seismic protection of equipment are 1970 the then current main motivations for improvement in safety and protection of capital investment. Although the consequences systems, with defence of equipment failure can be severe, most equipment the exception of nuclear power plants, installations, are rarely designed with the power transmission degree of care stations, and actually warranted. The difficulties in providing for the proper seismic restraint of equipment systems arise from three sources: 1. Most equipment systems and their housing structures as separate primary units. This structural practice system and neglects the the effect secondary are of analysed and designed interaction equipment system; between the therefore, the design may be inadequate. 2. The seismic response of an equipment system depends on the characteristics of the equipment, of the housing structure, and of the ground motion, and also on the location of the equipment within a building. Therefore, the designer very difficult task when attempting to represent the seismic faces a environment accurately. 3. Most equipment experimentally, systems to have withstand the not been seismic fully qualified, environment The either risk analytically of or catastrophic failure of these systems during an earthquake may therefore be high. Vibration isolated equipment is particularly vulnerable to seismic forces. Typically, this type of equipment fs mounted on special spring units, or other resilient supports, in order to significantly reduce any tranmission of shock or periodic forces into the primary system as a result of normal equipment operation. Motion restraining devices are sometimes used to prevent the development of excessive response in the 3 resiliently mounted equipment resilienfjy destroyed such which can may damage with without result due anything motion including to the situated motion restraining earthquakes program, equipment even though their housing structures cases, failure mountings actual mounted Experience from past earthquakes or in a simulated a consistent may suffer dislodgement near devices restraining the and of the be isolator totally spring units, Moreover, some extensively environment design may that relatively minor damage. In not been seismic analysis devices equipment have has demonstrated A tested reliable procedure, resiliently mounted equipment is to be protected from earthquake is resilient either in qualification needed if the damage. 1.2 L I T E R A T U R E REVIEW During the past decade a great deal of research has led to a considerable advance in our understanding of the response of equipment systems under earthquake excitation. These studies can be broadly categorized into three major groups: 1. The analytical study of seismic response of linear equipment systems. 2. The analytical study of seismic response of nonlinear equipment systems. 3. Techniques of dynamic testing and seismic qualification. 1.2.1 T H E SEISMIC RESPONSE O F LINEAR EQUIPMENT The earliest non-interactive separately methods approach, from the of in seismic which secondary the analysis of primary equipment system. SYSTEMS equipment systems adopt structural system is analysed The time history response of the primary system at the point of attachment of the secondary system obtained from a time computed time history secondary system. Since step analysis. record, the is A then generation response used of a a spectrum, as time the generated input history from excitation record is is first to the the usually lengthy and costly procedure, various authors have developed simpler approaches. a 4 Bigg and Roessei [2], Amin et al [3], and Kapur and Shao [4] empirical rules for generating floor response spectra directly from ground response spectra and the modal properties and Vanmarke [6], Singh proposed [7], of the primary structure. and Vanmarke [8] based on random vibration theory; Scanlan Singh [5], Chakravarti developed and Sachs [9] alternate approaches suggested a technique involving the Fourier transform method. All of these conventional important drawbacks. secondary They consistently system is not negligible system and when a natural natural frequency Kapur and interaction and of Shao between significant the point the primary developed Asfura • [11] if pseudo static part the problems that system introduced Crandall under and if is artificially also state that the several when the mass of the Mark [10], conditions secondary uncoupled suggest that proper have system coincides and these the an methods with the mass of the primary of the secondary system. out response They experience spectrum in comparision primary are response frequency [4] errors Kiureghian and be floor modal Singh the system analysis with a [5], effect is is of important used. Der combination rules cannot separated into and effect non-classical of a dynamic a damping in the combined system, and the cross correlation between closely coupled modes in the primary system and the secondary system, is often neglected or improperly treated using a conventional floor response spectrum method of analysis. Various researchers have investigated to take it into account the important primary systems. motion and of secondary both the primary Crandall and Mark [10] excitation of a In and effect these the different analytical methods of dynamic interaction methods, secondary the combined systems must between the equations of be have found the root mean square response to two degree of freedom motion. Penzien and Chopra [12] designed considered. stationary system using the combined equations of proposed an innovative method which reduces 5 an N degree of secondary system excitation for freedom primary system into N , two degree of the above method requires not a freedom usually a response spectrum, which Newmark [13] introduced the concept of first is and single degree systems. two degree available. In effective of freedom However, the input of freedom another ground development, mass ratio to arrive at approximate solutions to simple equipment-structure interaction problems. Nakahata et al [14] later improved upon this concept to approximate modal properties and modal responses with various rules for modal combination. Sackman and Kelly [15] recognized that the perturbation method can be used to formulate equations of motion for a general primary system and a light single degree of freedom secondary system. They arrived at closed form solutions of the perturbed Perturbation differential equations methods are equations parameters of mathematical containing et al [18] using tools Laplace which parameters. transform can In and damping values be this used study techniques. to solve the small of the secondary Villaverde and Newmark [17], system. and Der Kiureghian also used perturbation methods and modal combination rules to arrive at approximate modal properties [16] Fourier used a equations. Villaverde multiple attachment and Nour-Omid a floor transform and and Asfura [11] approach Newmark [17] studied a single response spectrum interaction Kiureghian and Igusa [19] secondary of the combined system. points and non-classical [18] equipment-structure supported small are the mass, stiffness, Robinson and Ruzicka [16], derived motion by to solve considered perturbed secondary differential systems with damping. Der Kiureghian, Sackman, degree of which using the Robinson and Ruzicka freedom can random secondary include vibration the system and effect techniques. of Der extended this method to multiply tuned and arbitrarily systems with non-classical damping. Finally, Der Kiureghian develped a new floor response spectrum method which includes 6 the effect of equipment-structure systems. Good analytical solutions interaction to the for multiply problem of supported linear secondary equipment-structure interaction were obtained by these researchers. 1.2.2 T H E SEISMIC RESPONSE O F NONLINEAR EQUIPMENT SYSTEMS The analysis of the seismic response of a nonlinear equipment system is much more difficult than that of a linear equipment system. A popular analytical approach to the nonlinear problem makes use of numerical time step analysis which accounts for a proper modelling of the equipment nonlinearities. A case of good example resiliency supported nonlinear characteristics the stiff of a nonlinear equipment system equipment with motion is illustrated limiting by the constraints. The arise when the equipment system comes into contact with motion limiting device. Iwan [20, by replacing the set of nonlinear springs 21] studied by a set this problem analytically of equilvalent linear springs whose stiffness depends on the displacement amplitude. From a conventional floor response spectrum, the maximum kinetic energy of the can then be obtained as a function of natural frequency. states that, in any one mode of a conservative energy, which kinetic Iwan energy, arrived natural depends which at a frequency of equilvalent linear system system, Rayleigh's principle the maximum potential on the displacement amplitude, must equal the maximum depends function the on which equipment. the natural relates frequency. the He obtained Using displacement another this principle, amplitude function to the relating the displacement amplitude to the natural frequency of the equilvalent linear support stiffness, which depends on displacement amplitude. The transcendental equation obtained by combining these relations is solved by an iterative procedure which yields an estimate of the seismic response of the equipment system. This method offers an equilvalent linear, non-interacting seismic analysis for isolated equipment 7 systems with motion restraint devices. DYNAMIC 1.2.3 TESTING TECHNIQUES AND SEISMIC QUALIFICATION PRACTICE Rigorous guidelines standards generating [25]. for seismic stations Rainer [26] for seismic qualification tests are qualification [22], and by of Class IE Silva [23], McGavin and Nielson [27] given by the equipment [24], for nuclear power and Wilson et al have suggested procedures for testing of structures. Bendat and Piersol [28, 29] and Newland [30] techniques of spectral provided insights these analysis for dealing with into structural experimental techniques random data. modelling and experimental are used in the testing IEEE the dynamic improved the Sabnis et al techniques. program [31] Some of in the present study. 1.3 OBJECTIVES AND S C O P E The main objectives of this research are: 1. to investigate the seismic response of vibration isolated equipment systems 2. to study the effect of equipment-structure interaction 3. to evaluate the seismic performance of two typical equipment vibration isolators. The parts. detailed Chapter investigation two involves a presented parametric in this study of thesis can be divided into three the of equipment-structure effect interaction on the ultimate equipment response of a general equipment-structure system. It is found that the effect of equipment-stucture interaction can significantly influence the dynamic response of some equipment-structure systems. Next, consideration is given to the experimental investigation of the structural dynamic properties (natural frequencies, mode shapes, and damping values) and the seismic response of a model of an equipment system mounted on commercially available vibration isolator units. The 8 modelling techniques and experimental methods are presented in Chapter three and four. The final stage of the study pertains to the development of an analytical model of vibration isolated means equipment units and the solution of the theoretical of a time series analysis method. The analytical study is discussed five and the analytical and experimental results are presented problem by in Chapter in Chapter six. Chapter seven contains the summary and conclusions of the results of the investigation. Chapter 2 ANALYTICAL STUDY O N T H E E F F E C T OI EQUIPMENT-STRUCTURE INTERACTION 2.1 INTRODUCTION As stated in the literature review, the effect if improperly' treated, can introduce significant of equipment-structure interaction, errors in the dynamic analysis of the equipment system. The effect of equipment-structure interaction is most significant when the mass of the secondary system is not negligible in comparision with the mass of the primary system or when a natural frequency of the secondary system is tuned to a natural the frequency effect of equipment-structure equipment-structure affect their of the primary system. systems response. response of an attached and examines the seismic can The This of to first interaction by a reviewing section equipment single by of this having degree the of a of this chapter is to study analysing the important chapter single freedom deals degree parameters with of structure. response of such an unit when attached Finally, the last section discusses shake table response the steady can state which second to a three testing tuned which freedom The of is section degree of methods which include the effect of equipment-structure interaction. 2.2 STEADY and item tuned freedom structure. The and The purpose STATE RESPONSE study of equipment-structure interaction is best introduced with the simplest most fundamental two degree of freedom simple system contains all the essential assemblage, such as shown in Fig. 2.1. properties which characterize the more general multi- degree of freedom equipment-structure system; therefore, the results from this study can easily be extended to more complicated equipment-structure systems. 9 10 2.2.1 C O U P L E D DIFFERENTIAL The coupled system are single and formulated degree subsystems equations of are stiffness EQUATIONS of motion of the simple two degree of freedom in terms of the parameters of the individual fixed based freedom subsystems. mass influence influence coefficents M , C, and K, The coefficients be m., k^, where and i = 2 refers to the secondary Let OF MOTION physical properties damping influence i= l refers to of the two coefficents c., the primary system system. the mass, damping, and stiffness -c2 kj + k2 matrix of the combined system respectively. Then: 0 m, M = C equation of K = m2 _ _0 The Ci + C2 _-C 2 motion the of two = (2.1) _-k2 C2 degree of -k: freedom k2 _ system horizontal base motion x (t) is g Mx + Cx + K x = - M I where x = [xj, displacement, motion and I T x2] velocity, =[1, 1] , (2.2) x (t) g x and = [x,, x2] acceleration T , X vectors is the influence =[X J ( T X] , 2 measured are relative the to horizontal the base vector coupling the input base motion to the individual degrees of freedom of the system. Now consider harmonic base motion. The base motion is given in complex form as X (t)=X e gv' g ILOt (2.3) 11 where X g is the base acceleration amplitude and CJ is the base motion driving frequency. The steady state response of the two subsystems can be expressed in complex form as xI(t) = B e I x,(t) = B, e "* = i ( w t i ( u ^ t where A, and A 2 are well to as a e ) ) A l = A ! complex; i w t (2.4) e (2.5) iu)t they contain information on the amplitudes as the phases of the steady state damped response of the subsystems due harmonic derivatives of base excitation. equations (2.4) Substituting and (2.5) equations (2.1), into equation (2.2) (2.3), yields and the the following matrix equation (k 1 +k 2 - m 1 a> 2 )+ia>(c l + c2) - ( k 2 + iwc 2 ) -niiX A - ( k 2 + ia>c2) (k m cj )+icL)c - 2 (2.6) = 2 2 v 2 6 -m2x T Equation subsequent subsystem (2.6) can be substitution solved of A to find into responses. The absolute the response equation acceleration (2.4) vector, and A = [Alt equation of the secondary Aj (2.5) system , and gives the is given by (2.7) x 2 a (t)=x I (t) + X (t) The complex transfer H2C(w), response represents of the function of the secondary system from a coupled analysis, the amplitude and the phase of the steady secondary system due to a unit harmonic state damped base excitation. 12 Therefore, by setting x (t) = l e 1 £ J ^ , the amplitude of the complex equipment c transfer function is: | H 2 1 n ((A2 + B 2 ) i / Z ) / C ( « ) | = c (2.8) where A = D2 + E2-(DF+EG)u2 B=(EF-DG>J ; C = D 2 + E2 D=k 3 2 -(k, + k 2 -m 1 oj 3 Xk 2 -m ; cj 2 ) + ((ci + c 2 )c 2 -2c 2 2 >j 2 E=-aj((c 1 + c2)k2 + c 2 k 1 + k 2 - 2k2c2) + (m2(ci + c2) + m!C 2 Xj 3 F=m 2 (k, + k2) + mik2-mimjO)2 G = (m2Ci + c2 + miC2)cj From equation frequency for (2.8), various |H 2 C (<^) | systems, can having be obtained damping as ratios, a function $ j, and of driving mass ratios, e, where $ . = C./(2CL> .m.) (i = l , 2) and e = m I / m 1 . 2.2.2 U N C O U P L E D same DIFFERENTIAL EQUATIONS For comparative purposes, two degree freedom of now system. consider OF MOTION an Assuming uncoupled that the analysis of the response of the uncoupled subsystems is harmonic, and using the same approach illustrated in the previous section, the absolute acceleration response of the uncoupled single degree of freedom structure, X ^ a (0. is: X l a ( t ) = ( l + ((Q-Ri)m 1 a; J )/(Q 2 + R 2 ) X g e l u t (2.9) 13 where Q = k1-wJm1 R=wc, Using x 2a(0 ^ a s moiion D a s e applied to the uncoupled equipment, and repeating the above procedure, the absolute acceleration response of the uncoupled single degree of freedom equipment system, ^ a ^ ' x2a(t)=(l + S)X g e 's: (2.10) l c J t where S = (T-Ui)mjul/V T = ( Q Y - RZ)u'm, + ( Q U=-((Q + R 2 2 J +R ) Z + ( Q Z + Y = (Q + R )(Y 2 2 2 +Z 2 2 )Y YR)m,ca 2 ) ) Y = kj-w 2 m 2 Z=CJC The 2 amplitude of the complex equipment transfer function , l ^ u c ^ ^ ' ^ r o m the uncoupled analysis then becomes: H Again,| H 2 u c ( C J ) | =((V + Tm 2 c; 2 ) 2 + ( U m 3 a ; 2 ) 2 ) 1 / 2 ) / V (^) damping ratios, | as a function of driving frequency $ ., and mass ratios, (2.11) for systems with e, is easily obtained. different 14 2.2.3 COMPARSION OF T H E COUPLED AND UNCOUPLED STEADY STATE RESPONSE Fig. 2.2 - driving frequency systems with Fig. 2.13 obtained different show equipment transfer from the parameters. coupled The results analysis, in some cases, are very conservative a steady state underestimates coupled the significantly equipment depends equipment-structure analysis; in response. on' the system, but the from uncoupled a steady analysis state for uncoupled as compared with the results from other The cases, accuracy mass ratio the driving and and functions as a function of and the uncoupled analysis of. the uncoupled analysis the damping frequency of ratio the of the harmonic base motion. In general, when the driving frequency coincides with or is close to the tuned natural overestimates frequencies analysis frequencies the on of the subsystems, equipment response. either side of the the uncoupled However, there tuning yields nonconservative results. frequency exists a for analysis range which grossly of the driving uncoupled Moreover, as the level of damping of the subsystems increases, or as the mass of the secondary system becomes negligible as compared to the mass of the primary system, the accuracy of the uncoupled analysis increases. For combined tuning a tuned coupled two degree system has frequency, i.e. there of two is a freedom natural frequency equipment-structure frequencies shift on either system, side of in the combined system from the tuning frequency of the uncoupled system. This frequency shift on that the a various band of system high parameters amplification and it is occurs. around these shifted Therefore, if the analysis may be unsafe. involves energy exchanges The response of the tuned coupled the away depends tuning poles forcing happens to lie in this region, depending on the system parameters, the frequency the uncoupled combined system between the equipment and its housing structure at a 15 beat frequency which is much lower than the tuned frequency of the subsystems. It is this classic beating phenomenon which into consideration. magnitude The coupling of the damping and mass ratios the secondary system the subsystems is weak from coupled tuning pole using an between heavily analysis obtained uncoupled transmitted an analysis of the may some very depends subsystems. If much uncoupled analysis. be justified. takes time and systems, between subsystems to take on the the mass of and the location of the tuning poles obtained does not differ from the subsystems damped the fails is negligible compared to the primary system, the coupling between a between the uncoupled analysis of the the Therefore, location of the in this Futhermore, the requires energy from many cycles may be situation, energy of oscillation. In dissipated the subsystems. As a result, in systems with transfer before it is high damping the error of an uncoupled analysis is usually not significant. 2.3 SEISMIC RESPONSE O F E Q U I P M E N T - S T R U C T U R E The seismic response characteristics SYSTEM and the steady state response of the tuned equipment-structure system can be quite different characteristics Earthquake motions are not purely harmonic in nature, since they contain more than one predominant forcing frequency. can be Moreover, the duration of the ground motion is such considered as a transient disturbance. In the that the following sections, disturbance the seismic response of a tuned, linear single degree of freedom equipment system attached to the top floor analysis. of a three degree An uncoupled analysis of freedom structure is obtained of the equipment-structure system from is a time step considered first Then a coupled analysis of the equipment-structure system is" presented. Various system parameters corresponding to different tuning conditions and different damping and mass ratios are also studied. 16 2.3.1 S Y S T E M The frame PARAMETERS structural building with STUDIED system under consideration masses lumped at each is a simple story shear floor as shown in Fig. 2.14. mass of each floor is 175,000 kg and the interstory stiffness The damping of the structure three is assumed to be classical The is 350,000,000 N / m . and the damping ratio is taken as 2% for each mode ($ g . = 2% i = l , 2, 3). The equation of motion of the structural system subjected to base motion X (t) is: M X + C x +K x = - M I X (t) s s s s s s s g (2.12) T where x g =[xsj. 2> T 's ^ X S relative displacement, x g T e is the relative velocity, and X g = [ x s | . each floor measured with x s 2> =txs^ 2> X S * j\ s * s the relative acceleration of respect to the base. The influence vector 1 =[1, 1, T 1] couples the base input motion to the individual floor degrees of freedom. M , C , K are the mass, damping, and stiffness matrix of the structure. M v s s s s ° and K s are as follows: 175 175 xlO 3 175 (2.13) K s = 7.0 -3.5 -3.5 7.0 -3.5 -3.5 3.5 xlO 8 N/m The structural damping ..matrix can be found from the structural mass matrix M 17 and the structural stiffness damping, the structural matrix Ks- By considering mode shape matrix $ vector T can be found from the eigenvalue the structure and the structural without natural period solution to the following undamped free vibration problem: M X +K x =0 s s s s (2.14) where 0 is a zero vector. Assuming the free vibration response is harmonic, the equation of motion can be expressed as [K - u 2 M ] $ r=0 1 s r s s where $ cJ r s r * is the mode shape for the r * is the corresponding natural (2.15) mode of the structure's vibration and frequency. A nontrival solution is possible only when the determinant of equation (2.15) vanishes, yielding the frequency equation of the system: |Ks- w2Ms||=0 (2.16) Expanding the determinant yields an N the degree of freedom N natural frequencies first element in the r the natural system. th frequencies degree algebraic The N roots of the system. mode shape, $ s for structural modal matrix $ each r mode equation in u | of this equation are the square of Assuming an arbitrary value can be found by substituting into equation (2.15). and the structural natural periods T s for a The for the in turn resulting are as follows: s 18 * = s s L s .0780 -.1762 -.1413 .1413 -.0784 0.1762 x l O - 2 .1762 0.1413 -.0784 m (2.17) T = S [T,. T,, TJ = [.316. We assume the modal matrix $ s the classical .113, .078] sec can uncouple the damping matrix, as occurs in approach. Thus, pre- and post- multiplying the damping matrix by the modal matrix yields a diagonal matrix of generalized damping C *= s * T C * s s = s 5 2 u> m 2 2 -T s s is: -1 * s Instead of inverting <i> (2.18) 2 $3 cj 3 m 3 Then the structural damping matrix C s coefficients: (2.19) s s one can make use of the relationship $ -1 = s $ T s M s to yield the symmetrical structural damping matrix as: 424 C =M * s C s s s $ s M s = SYM. -124 - 270 397 -151 273 xKT NS/M (2.20) 19 Now equipment consider item the shown characteristic in Fig. of 2.15. the linear single Assuming the degree secondary of freedom system's natural period is tuned to one of the natural periods of the structure, and defining the ratio, e, as the mass ratio between the mass of the secondary system, M f , and the mass of equipment a floor stiffness of the building, characteristics condition. For an equipment for system a m^ = 175,000 given mass kg, one ratio and having a mass ratio, can a select the given tuning e and tuned to the til i mode of the structure, the equipment stiffness damping of the damping ratio equation of equipment is varied motion system from of the is 2% to assumed is K e = ((27r)/Tsj)2 to be classical and the modal 5% ($ e = C £ / ( 2 M e w ) = 2% or uncoupled equipment system subjected enij. The 5%). The to a base motion (x, + x ) is: M X + 2$ ( K e M £ ) 1 / 2 x e + K£ x e = - M (X g (t)+x,(t)) e e e (2.21) e where X , which is obtained from an uncoupled analysis of the building alone, is 3 the acceleration of the third floor relative to the ground which is undergoing an acceleration X . g Consider next the equations of a single degree mass ratio attached of freedom to the top of the coupled system. The combined system equipment floor of with the a three given tuning condition and story structure is shown in Fig. 2.16. The equation of motion of the coupled system is: M X + Cx + K x = - M I where x =[xi, X (t) x , x , xj 2 T 3 is (2.22) the displacement, x =[% u x , x , xj 2 3 T is the T velocity, and X =[Xi, X , X , x j 2 3 is the acceleration of each floor relative to 20 j the base. Here I =[1, 1, 1, 1] is the influence vector coupling the base input motion to the individual degrees of freedom. The mass matrix and the stiffness matrix of the combined system are as follows: 175 175 M = x l O 3 kg (2.23) 175 el75 K = 7.0x10' •3.5x10° -3.5x10* 7.0x10 8 -3.5x10 N/m 7.0x10 8 + K ( -3.5x10' -K where K £ dealing parameters, such as with large tuned subsystems differences out by Hurty and Rubinstein [32], different, the even though because the mode phase relationship system undergoes of non-classical The combined the with in mass or combined system needs to be treated damped K depends on the choice of the mass ratio e and the tuning condition. When are (2.24) •K with extra large stiffness, care. the system can no longer individual subsystems damping of the considered classical damping complex matrix of the individual subsystems C s and C^. C is built classically This is and there degree of freedom mode. Failure matrix as damping. is a when the to include the damping in the analysis can introduce significant system system In such cases, as pointed be have the movement of each free vibration in a given combined in if the damping ratios between the subsystems shapes of the combined system are between differences effect errors. from the damping 21 424x10 -124x10 3 C = 3 -270xl0 J -124x10' -270x10 397x10' -151x10 3 J Ns/m -c 273xl0 3 + C -151x10" (2.25) -C These system parameters will be used in the solution to the equations of motion; the method of solution is examined in the next section. 2.3.2 SOLUTION The coupled method analysis specified interval. METHOD is variation The of the in solution well the differential used known change equation here for Newmark of of both Beta acceleration motion the uncoupled method, which response in incremental over form a and assumes a small for time history base acceleration record is given by Clough and Penzien [33] time a given as: MAX(t) + CAx(t) + KAx(t) =-MIAX (t) % The application differential of the numerical step by the (2.26) step integration technique to this equation of motion yields the following results for each time step: X(t+At) = X(t)+Ax(t)/(/3At 2 )-X(t)/(0At)-X(t)/(2/3) x(t+At) = x(t)+X(t)At(l-a/(20)hx(t)a/0 +Ax(t)a/(0At) x(t+At) = x(t) + K * _ 1 F * K * = M / ( | 3 A t 2 ) + aAx(t)/(/3At) + K F*=-M(x (t+At)-X (t)) + M(X(t)/2+x(t)/At)//3 +C(-X(t)(l-a /(20 ))At+ x(t)a /0) (2.27) 22 The values of a and 0 depend on the assumption of the variation of the response acceleration within each time step. If a constant variation in acceleration, taken as the average value between the acceleration at the start and at the end of a time interval, within linear variation in each time step is assumed, a = .5 acceleration within each time step is and /3 = .25. assumed, a = .5 If a and 0 = .16667. The average acceleration method is unconditionally stable, i.e. the solution is stable for any chosen length of time step, and the accuracy will, in general, increase as the chosen time step interval is decreased. The linear acceleration method, however, is not unconditionally stable, i.e. the solution may oscillate with increasing for the amplitude linear if too large a time step is chosen. The stablility criterion acceleration method is that the chosen time step must be less than 0.551 of the smallest natural period of the system. Moreover, both methods generally yield good results if the chosen time step interval is less the one-tenth of the smallest natural period of the system. 2.3.3 RESPONSE O F LINEAR A comparison of EQUIPMENT the results from coupled time step analyses uncoupled time step analyses for the linear single degree of freedom with various mass ratios and tuning building is shown in Fig. 2.17 difference between the seismic response equipment systems is that overestimate the equipment analyses may sometimes attached to the equipment, three story Fig. 2.18. These figures indicate that the major response of peak conditions, and of the response, underestimate the equipment systems uncoupled seismic whereas peak the and analyses generally uncoupled equipment steady state steady state response. However, there are some similarities between the seismic response analyses and the steadystate response analyses. 23 1. The errors from the uncoupled analyses are most severe when the mass the uncoupled analyses. In ratios are high. 2. Tuning conditions also affect the accuracy of general, the error from an uncoupled analysis is most significant when the equipment is tuned to the fundamental natural frequency of the structure. 3. Damping also level of influences damping of the accuracy the of subsystems the uncoupled analyses. increases, the As the uncoupled seismic analyses become more accurate. These similarities arise because the mass ratios, the damping ratios, and the tuning conditions determine the strength of coupling between the subsystems. 2.3.4 M E T H O D O F S H A K E T A B L E In conventional methods subjected to a prescribed predicted building floor of simulated motion TESTING shake table seismic testing.the equipment motion which corresponding to the system closely resembles floor response is the spectrum supplied by the design requirement. This response spectrum is called the required response spectrum of the housing (RRS) and it is often obtained from a non-interactive study structure alone; therefore, the effect of equipment-structure interaction is usually neglected in conventional practice. Shake interaction. combined mass table testing This is can diTectly include the effect achieved by equipment-structure and stiffness of the building and testing system. The scaled actual system. of a parameters However, this equipment-structure scaled are model of the the dimensions, approach has several major drawbacks. First, the physical size of the shake table dictates the allowable physical parameters of the testing model; this limits applicable equipment-structure systems. Second, fragility testing the is of use of interest scaled In models such tests presents the added difficulties maximum capacity when of the 24 equipment system is determined for both single and multiple frequency waveforms that until comply specimen with fails future requirements. and the failure operational type failure mechanism is not obtainable Moreover, the monitored structural model and the Fragility actual equipment tests are conducted may be operational the or structural. But from dynamic testing of scaled models. failure may not be valid because the system may have different ultimate scaled strengths since such models cannot accurately represent all aspects of the prototype. Finally, perhaps the most important drawback of scaled model testing originates difficultes actual in accurately system. modelling the Without an accurate nonlinearities and the from the damping of the model, the results of shake table testing are not valid. A more approach. analysis. as practical method uses of a combined analytical the combined system A mathematical model is This involves the determination of the various system and experimental needed for the parameters, such the mass, damping, and stiffness of the individual uncoupled subsystems. The system parameters physical characteristics upgraded system can from the be obtained performance identification. resembles static testing or from the known of the subsystems. These parameters can be checked and Knowing model of the combined system closely from the of the dynamic tesung. system parameters, This an process accurate is known as mathematical can be built. For a given ground motion, which predicted ground motion producing the response spectrum provided by the design requirements, a floor motion response record at the point of equipment attachment can be combined equipment-structure the effect of equipment-structure generated system. This from a time step analysis floor motion record, which interaction, can then be used the shake table to test the actual equipment system. as of the includes the input to 25 Another similar approach entails the determination of a floor new response spectrum as defined by some of the authors mentioned in section such as Sackman and Kelly [15], Sackman floor and Nour-Omid response spectrum, interaction, can than a motion time step are One can and Der which be generated spectrum spectrum. [18], Villaverde and Newmark [17], also The again needed then the directly from the analysis. use system the motions corresponding to the to the actual equipment system. effect and to obtain seismic of This new equipment-structure ground response spectrum parameters in order simulated Der Kiureghian, Kiureghian and Asfura [11]. includes motions, a^ prescribed the 1.2.1 new floor which closely rather ground response resemble new response spectrum, as a shake table input 26 - Uncoupled - Coupled Analysis Analysis o •—to zo; (X /A 0.0 1 0.2 Figure 1 0.4 2.2 1 0.6 r— 0.6 r 1.0 FREQUENCY (HZ) — 1.2 1.4 1.6 1.6 2.0 Plot or uncoupled and coupled equipment response transfer function, e = 0.01 5. = 0.01 Uncoupled Coupled Analysis Analysis t 0.0 1 0.3 Figure 2.3 1 0.4 1 0.6 0.8 1 1.0 = * V FREQUENCY (HZ) T— 1.2 J 1.4 —I 1.6 1.6 2.0 Plot of uncoupled and coupled equipment response transfer function. e = 0.001 5j = 0.01 T i Uncoupled Peak of coupled analysis — — Coupled Analysis Analysis X" a: £si- a 0.0 —i 0 2 1 0.4 1 0.6 f» • 0.8 P 1.0 |'2 FREQUENCY (HZ) ' Figure 2.4 ~ l — 1.4 —I— 1.6 ~l 1.8 2.0 Plot of uncoupled and coupled equipment response transfer function. e = 0.0001 5. = o.oi Uncoupled Coupled Analysis Analysis V 0.0 1 0.2 Figure 2.5 1 0.4 —f-^— 0.S I 0.8 1 1.0 1 FREQUENCY (HZ) 1.2 ^ f— 1.4 1.6 ~1— t.e 2.0 Plot of uncoupled and coupled equipment response transfer function. e = 0.01 $. = 0.02 T Uncoupled — Coupled Analysis Analysis i i 3 o !. a"" i! UJ it fM R ii R P- i \ V 0.0 —|— 0.2 Figure 2.6 i 0.4 • •• i 0.6 i 0.8 1 1.0 1 FREQUENCY IHZ) 1.2 ~-r~1.4 \- 1.6 1.8 2.0 Plot of uncoupled and coupled equipment response transfer function, e = 0.001 5 j "= 0.02 i Peak o f coupled t "1 analysis . . . ._ U n c o u p l e d A n a l y s i s — j I! H _ Coupled Analysis M l! o !! I— u tl_«l a " (jj u. tn i 1 11 1 1 1 I 1 1 1 / / j 1 0 0.2 Figure 2.7 • 0.4 I 1 0.6 0.6 / 1 I / 1 1.0 FREQUENCY •v. (HZ) 1 1.2 •-•-»t— —• 14 1.6 r-— 1.6 2.0 Plot of uncoupled and coupled equipment response transfer function, e = 0.0001 $. = 0.02 t-J _ — Uncoupled — Coupled Analysis Analysis <_> LU i u. <n z \ a" v. a /»' I! / I 0.0 0.2 Figure 2.8 I 1 0.4 0.6 • T — 0.6 1 1 1 I 1.0 1.2 t.4 1.6 FREQUENCY IHZ) ~ 1 1.6 2.0 Plot of uncoupled and coupled equipment response transfer function. e = 0.01 5j = 0.05 Uncoupled — Coupled Analysis Analysis p- (_> I i LJ ; ! 01 Iw- w V 0.0 1 0.2 Figure 2.9 1 0.4 1 0.6 1 0.8 1 1.0 1 FREQUENCY (HZ) 1.2 1 1.4 i"~ ——r— 1.6 2.0 1.8 Plot of uncoupled and coupled equipment response transfer function. f = 0.001 5 = 0.05 4i Peak of coupled analysis 11 Uncoupled Coupled Analysis Analysis / *3o t (_) z =3o a: u. i \ l i I*? 01 • / 0.0 —i— 0.2 Figure 2.10 —1 0.4 w i 1 1 0.8 0.8 1.0 r— -i I 2 FREQUENCY (HZ) ' f——• 1.8 1 1.8 Plot or uncoupled and coupled equipment response transfer function, e = 0.0001 "J = o.05 , 2.0 Uncoupled Coupled n Analysis Analysis \ I; H V Ii w I! n i > i • I i \ \ \ 0.0 N 1 1 1 1 1 1 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 2.11 FREQUENCY (HZ) 1— 1.6 i— 1.8 2.0 Plot of uncoupled and coupled equipment response transfer function. e = 0.01 5j = 0.10 Uncoupled Coupled i! AnalyBiB Analysis \ Ii \ /' * I; i; /•' i I, \- t W Ii V I! W I, Ii V // 0.0 I— 0.2 Figure 2.12 -1 0.4 1 1 1 0.6 0.8 1.0 1— FREQUENCY [HZ) 1.2 i 1.4 I— 1.6 I— 1.8 2.0 Plot of uncoupled and coupled equipment response transfer function. e = 0.001 $j = 0.10 Uncoupled Coupled 'A Analysis Analysis i, /; /• I I, V. V. l'< I! i I. \ \ w I i \ 0.0 - 1 — 0.2 Figure 2.13 -1 0.4 1 1 1 0.6 0.8 1.0 1— FREQUENCY (HZ) 1.2 1.4 —i— 1.6 - 1 — 1.8 2.0 Plot of uncoupled and coupled equipment response transfer function, c = 0.0001 $i = 0.10 39 s3 m m s3 s2 sl k k s2 sl M K e C e e 7 7 ^ Figure 2.14 Three D.O.F. structural system. Figure 2.15 e e e s3 m s2 m . si S.D.O.F. equipment system. s3 k s2 k . si Figure 2.16 Combined equipment structure system. Analysis type — — -— - Structure tuning mode Uncoupled Coupled First Uncoupled Coupled Second Uncoupled Coupled Third 2 % Damping in structure 5 % Damping in equipment r— , 0.00001 0.0001 Figure 2.17 , 0.001 , — 0.01 Coupled and uncoupled peak response of 1 0.1 tuned secondary system to El Centro earthquake N-S component Mass Ratio G Analysis type 15 — — \ 10 \ Structure tuning mode Uncoupled Coupled First Uncoupled Coupled Second Uncoupled Coupled Third 2 % Damping in all modes \ \ . . . . \ . ... \ \ \ \ 000001 0.0001 Figure 2.18 0.001 0.01 0.1 Coupled and uncoupled peak response of tuned secondary system to El Centro earthquake N-S component Mass Ratio Chapter 3 TESTING FACILITIES A N D T E S T M O D E L PARAMETERS 3.1 INTRODUCTION One of the main objectives vibration The isolated equipment systems subjected tests were This chapter in this study requires the testing and modelling of divided into two to single axis horizontal base excitation. categories a) discusses the facilities at static testing and b) dynamic the University of British Columbia which were used to undertake these experiments. The properties of the model chosen to an actual air handling equipment testing. system, and the properties represent of the chosen vibration isolation spring units, are also discussed. 3.2 TESTING The Laboratory FACILITIES static of tesung the program Department Columbia. The static testing which is capable biaxial kip mounted hydraulic and ram. displacement can (MTS Civil Engineering program to MTS The 202.01) each model input to other. each the at Structural the Each actuator respectively) hydraulic ram Engineering University of British piston system in two perpendicular directions. This hydraulic actuator 661.22 in employed a hydraulically actuated consists of a 11 kip capacity model perpendicular 661.21A-02 performed of applying loads simultaneously hydraulic system capacity of was (MTS model 204.61) and a 50 with corresponding has a incorporated can either load into be cell loading rigs (MTS model the load arm of the controlled or controlled by an MTS control console (MTS model 483.02). This console also monitor the ram displacements and the applied loads. The results can be displayed on X - Y plotters. The Laboratory dynamic of the testing program Department of was Civil performed Engineering 42 in at the the Earthquake University Engineering of British 43 Columbia. This test facility consists of a 3.3 m x 3.3 m earthquake simulation shaking table. It has a capacity to drive a payload of 16000 kg with a maximum acceleration of 2.5 g and a maximum displacement cellular aluminum structure of 130 mm peak to peak. The table is a weighing 2090 kg. It has a grid system of threaded steel inserts into which the test specimens are anchored. The shaking table is driven by an MTS 16000 kg hydraulic actuator Corporation attainable electronic servo and the feedback valve system. Only is controlled by an MTS Systems one-directional horizontal motion is by the table; all other movements are restrained by hydrostatic slip bearings and by supporting columns. A PDP 11/04 minicomputer commands the table by mean of digitized prescribed motion records stored on floppy disks. The records can be actual recorded seismic motions or synthesized motions to comply with the relevant requirements. The minicomputer conversion module. system and The real also provides time clock analog signals data acquisition functions received are from capability. controlled the data by Analog/digital an AR-11 acquisition signal real time instruments are preconditioned by amplifiers. The signals are reconditioned by variable gain buffers and variable cut-off filters before conversion into digital form by the AR-11 real time module. Finally, the digitized records are stored on floppy disks. Up to 16 channels of input signal can be handled by the present system. The sampling rate depends on the total number of channels sampled. The maximun sampling rate is 2 kHz in continuous mode and 35 kHz in burst 470 V8 main (High-level powerful frame mode. A communication link computer and the PDP 11/04 is between the U B C Amdahl possible through Data Link Control). This connection allows the user to make Amdahl computer for data storage and analysis. Fig. 3.1 a HDLC use of the shows the overall function of the shaking table system. The differential test instrumentation transducers (LVDT). consisted Five of accelerometers accelerometers were and used linear to voltage monitor the 44 acceleration of the specimen and the table. A KisfJer servo accelerometer with a range of ± 5 0 g was mounted on the table. Equipment mounted accelerometers were of the strain-gauge order to type with monitor a range the horizontal, centre of mass. Three isolator springs. vibration with a They isolator fourth of LVDTs LVDT. the They were located vertical, and the on the specimen in rotational acceleration of were connected to the specimen at measured springs ± 2 0 g. the relative Fig. 3.2 horizontal to the table. and the The table the its level of the vertical displacements of the displacement was measured shows the mounting position of the instruments. The instrumentation was calibrated periodically to ensure accurate and consistent results. 3.3 T E S T An MODEL air CHARACTERISTICS handling arrangement 9 with a system, 405 US Chicago Vane Axial Fan, Design 34 size 54 1/4 100 bhp motor frame and inertia base, mounted on vibration isolators was modelled for the purpose of the expermental study. In order to accurately represent the prototype system, the model characteristics must be chosen carefully. In the present study there were several idealizations adopted. 1. The actual air handling fan unit was assumed to be very rigid as compared to the relatively soft vibration isolators, i.e., the fan unit was assumed to undergo rigid body motion under seismic excitation. 2. The mass of the fan unit was assumed to be evenly distributed over its volume. 3. The response of the air handling system under its normal operation was considered to be negligible compared with its seismic response. 4. The vibration isolators were assumed to meet standard specifications for noise and vibration control. Based on the above assumptions, characteristics such as mass, mass moment of inertia, dimensions, location of centre of mass, location of vibration vibration isolators were chosen to model the actual system. isolators, and types of 45 From the specification of the Chicago vane axial fan, the mass of the actual system without an inertia base was estimated centre of mass was shows the location of the basic components of the air handling system. The isolator spacing along Standard the estimated horizontal axes specifications distance to be for noise between the adjacent 1.367 to be 1123 kg and the location of the xx m above and and zz was vibration corner isolators the vibration isolators. 1.55 m control and specify 0.9 m that respectively. the should at least be equal Fig. 3.3 minimum to the height of the centre of gravity of the equipment Since this specification cannot be met with the normal arrangement an inertia base with a mass of 730 kg was constructed for the air handling system. The addition of an inertia base lowers the centre of gravity of the during system equipment vibration there and which are isolators major without reduces start up. the rocking motion and However, with the are needed, which results differences inertia bases. between This the point eliminates additional in a different seismic will not mass, a different set of dynamic system. Therefore, response of be excessive movement examined equipment further systems with in the present study. The total mass of the air handling system with an inertia base was to be 1853 kg and the location of the centre of mass was estimated estimated to be 0.95 m above the vibration isolators. The isolator spacing along the horizontal axes xx and zz was the same as before, 1.55 m and 0.9 m respectively. The mass moment of inertia about the horizontal axes xx and zz was estimated kg.m.m respectively. The above characteristics to be 1772.4 kg.m.m and 1507.7 of the actual system had to be modelled in the study. An inverted concrete box mounted on an inertia base was constructed to model the actual system. This model was designed to withstand an acceleration of 2 g's in all directions. The concrete box, shown in Fig. 3.4, has five components: a top piece and four side pieces. The perimeter of each component is encased by steel angles. 46 The top panel is 0.152 m x 1.054 1.499 m. m x 1.499 The five m x 0.953 m x 0.889 m. Two of the side panels are 0.076 m and the other two side panels are 0.051 pieces were panels. Two A325 M10 threaded thick side panel at a distance assembled rods are by welding along the m x 0.737 m x perimeters of the implanted longitudinally into each 0.076 m of 0.08 m from the base and 0.06 m apart. They are used to connect the box to its supporting angles and inertia base. The two 1.1 m long supporting angles are 200 x 200 x 200 mm. The inertia base consists of two 2 m long W410 x 39 supporting I beams and a 0.11 m x 0.88 m x 1.91 m concrete base. The I beam was needed to raise the centre of mass of the system to the correct height final model. The characteristics system; These components are bolted together to form the of the model were estimated to be close to the actual this was confirmed later in the system identification part of the experimental study. The selection of the vibration isolators of 886 rpm under normal operation. was based on the estimated Assuming the damping to be fan negligible speed and defining the transmissibilty T^ as the ratio between the maximum force transmitted to the floor and the maximum applied force due to the rotating fan, the vertical stiffness required for the vibration isolators was: K= (27rf(l/Tr+l)1/2/60)2m/4 ' (3.1) where f= driving frequency m= Assuming =886 rpm total mass of the system T r can vary between 5% and 20%, the vertical stiffness of the vibration isolators can vary between 209 kN/m and 730 kN/m. Based on the above results, two 47 types of isolators uni-directional were restraint, selected as shown for in testing: Fig. an 3.5 open spring (Airmaster isolator ASL-2-208) with and a an elastomeric isolator as shown in Fig. 3.6 (Korfund series F F D D 1800). These isolators are commerically available and their respective and 350 kN/m. rated vertical stiffness were 630 kN/m Input Teletype Output C A L C O M P Plotter l< 1 , 1 ! i Floppy AMDAHL Disk i I if HDLC 170 VB Computer i Data A c q u i s i t i o n Do to Actuator and Foundation Restraint Figure 3.1 Overall function of U.B.C. shaking table. Processing _ G e n e r a t i o n of Command S i g n a l End view Side view Accelerometers B • Accelerometer Accelerometer O 5 o ± 0 1.03 m 77IT 0.89 m LVDTs 0.78 m „ 2.00 m Figure 3.2 Experimental model showing instrument locations. 0.78 m LVDTs Figure 3.3 Prototype air handling system (Design H Chicago vane axial Tan sl;e 54 1/4. arrangement 9). o Figure 3.4 Experimental model. 52 Figure 3.6 FJastomeric isolator Chapter 4 EXPERIMENTAL STUDY 4.1 INTRODUCTION The experimental program was divided into two parts: i) static testing and ii) dynamic testing. The static testing program was conducted prior to the dynamic testing program to confirm and explore model. The model characteristics experimentally the characteristics the mass of the model and the model. Without the system, based solely inertia properties on information, an the commercially of the system, the experimental of interest were the axial and lateral stiffness of the chosen isolators, above of may be location of centre analytical rated model isolator of the stiffness erroneously constructed. of mass of the vibration and the the vibration behaviour accurate of the isolated model equipment under model seismic and excitation. analytical model of the vibration isolated to predict its response in a seismic environment to investigate Based on characteristics experimentally the equipment system The present calculated The dynamic testing program was subsequently performed to further identify experimentally the of isolated above results, the an can be developed chapter describes the testing methods and procedures adopted in the experimental program. 4.2 STATIC TESTING Two isolators PROGRAM types of isolators were tested in the static testing and ii) open spring isolators are commercially available. Only the isolator stiffness is available from the manufacturers; and the lateral directions are i) with uni-directional motion restraints. of isolators axial program: however, the needed to stiffness construct Both types in the axial direction characteristics the elastomeric analytical in both the model. The bi-directional hydraulic system described in Chapter three was used in the static testing program. Each type of isolator tested was mounted horizontally in pairs in an inverted 53 54 series as shown schematically in Fig. 4.1; the actual test arrangement is illustrated in Figs. 4.2 and 4.3. The isolator assemblage was initially loaded to a specific axial load using the hydraulic load control system, and then laterally displaced using the hydraulic displacement control system. This procedure was repeated so that levels the of lateral axial load-deflecton preloads, were relationships obtained. for The for a range of axial preloads each lateral pair of stiffness isolators, of the determined from these lateral load-deflection curves. Also, the effect at various isolators was of axial load on the lateral stiffness of these isolators was found. With the same experimental set up, the isolator assemblage was brought to an initial lateral preload using the load controlled hydraulic system. Maintaining the lateral preload, the isolators were displaced axially by the displacement controlled hydraulic ram, which yielded an axial load-deflection curve having the characteristics indicated in Fig. 4.4. axial Repeating load-deflection the above curves for procedure the for isolator a range of lateral assemblage at preloads various resulted in levels of lateral preloads. Again, from the load-deflection curves, the axial stiffness characteristics of the isolators were obtained for comparison with the commercially rated values. Moreover, the influence of lateral load on the axial stiffness characteristics was obtained. The mass of the model was measured from model a 10 kip capacity was by suspending its individual components MTS load cell. The location of the centre of mass of the determined by suspending the model along one of its top edges from a crane. A ' plumb line was dropped from the point of suspension. repeated This procedure was by suspending the model along the opposite top edge and the intersection of the plumb lines yielded the location of centre of mass. of the model. Finally, based on these results, was calculated. the mass moment of inertia of the model about the centroidal axis 55 4.3 D Y N A M I C The tesung TESTING PROGRAM dynamic testing program was conducted after program. The first series of dynamic tests the completion of the static were exploratory in nature. They consisted of low-level sinusoidal testing, low-level sinusoidal decay testing, and random white noise frequencies, testing. the From mode these tests, the shapes, and the system modal characteristics damping ratios such as the natural were' found. The next series of dynamic tests were seismic motion tests. The seismic response and the failure modes of the model were obtained from these tests. 4.3.1 SINUSOIDAL TESTING The displacement the sinusoidal tests amplitudes. possibility of isolators damage to the with performed The magnitude base motion displacement spring were of model amplitudes uni-directional relatively the amplitude at for at system was were from the static testing, a range mm of forcing frequencies chosen to excite the model. Approximately 25 frequencies range. Low pass filters with frequencies frequencies below above instrumentation results. 10 noise Hz, 10 a cut off and Hz. which 40 and 1.3 The (0 - low low might otherwise pass pass have affected of the were used within this filters filters mm 15 Hz) was frequency of 10 Hz were used Hz the and the open respectively. Based on a preliminary estimation of the natural frequencies model base As a result, isolators 1.8 fixed limited to avoid resonance. the elastomeric restraints small were used eliminated the at test at test extraneous accuracy of the A sampling duration of 10 seconds was chosen to ensure that the model responded in a steady state vibration. Digitized time history records of the model response and of the table motion were recorded and stored on floppy disks discussed in Chapter three. as 56 The freedom ratio of between the the model and steady the response amplitude forcing motion amplitude for each of the degree of table can be obtained by examining the response records at each forcing frequency. A plot of this response frequency amplitude response ratio curve at for the each forcing system. The frequency natural then frequencies yielded and the mode shapes of the system were obtained from these results. 4.3.2 SINUSOIDAL DECAY TESTING Damping is perhaps dynamic characteristic sinusoidal From results of natural frequencies was vibrating in a vibrating system. The purpose in damped the static and sinusoidal testing, a good of estimate of of the system was obtained. In the sinusoidal excited at each the suddenly cut off, free measure the decay tests was to determine the modal damping ratios of the system. the system to the most important and very often the most difficult of its corresponding the system vibration. The natural natural responded resulting frequencies mode. at its decaying so When natural that the response was decay lest, the it responded table frequency the motion by was in a mode of recorded and the modal damping ratios were determined by the well known logarithmic decrement method. Several assumptions are made in using the logarithmic decrement method: i) the damping of the system is viscous, ii) the system is underdamped, and iii) the different modes of vibration in a multi-degree of freedom system are well separated. The percentage of critical damping in each D^ ln(x,/x mode of vibration is given as: )/(J2TT) (4.1) 57 where D= percentage of critical damping in a mode Xj= acceleration amplitude of the j th This approximate cycle of decay method of determining the damping is valid for systems having a percentage of critical damping in each mode which is less than 20%, which is valid in our case. 4.3.3 R A N D O M W H I T E NOISE TESTING Another method of system identification the system response due to white noise excitation. 25 Hz) white noise signals can be generated sinusoids (see defined as for frequencies such a system Random, band analysis of limited (0 - having a constant 25 Hz) white noise signal power spectral density is function from 0 to 25 Hz. The frequency band of the white noise signal to span record within the spectral by the technique of summation of Appendix A). A band limited (0 - a random variable was chosen involves is the estimated used as an natural input to frequency the system, the range of 0 to 25 Hz are resulting input/output relationships range of the model. When excited can be analysed all the with frequencies equal of the intensity. The by the techniques of spectral analysis to yield the frequency response functions for the system of interest The theory of system identification Appendix been using B. A computer program, written to analyse the time spectral FOUR.S, series analysis techniques is presented based on the above principles records. The computer listing for in has this program is provided in Appendix C. In the random white noise generated band limited white noise tests, the model was signal. The sampling excited by a randomly- duration of each test was 26 seconds, with a recording interval of 0.006 second. The recording interval 58 was kept as short as Ideally, choice the sampling of sampling possible to ensure good resolution and to avoid aliasing. duration should be kept as duration was chosen to long as possible. minimize possible A practical bias errors or truncation errors in the spectral analysis. The input base motion was scaled to a peak aceleration of 0.1 g on the MTS system, which generated strong model responses without causing damage to the model. Low pass filters with a 40 Hz cut off frequency were used. The model responses and the table motions were recorded and stored on floppy disks. The recorded data were transferred to the main frame computer for analysis to yield the frequency response curves and the phase and relationships the resulting for the system. frequency The experiments response curves were repeated and phase angle three times relationships were averaged to minimize possible random errors. 4.3.4 SEISMIC M O T I O N The acceleration as input motions to TESTING records of actual the shaking table earthquake to ground motions were used generate seismic responses of the model. The choice of these input motion records was based on their frequency content in comparison to the natural chosen records were: frequencies i) El Centro N - S of the equipment model. The (1940), ii) San Fernando N21E (1971), and iii) Taft (1950). The input acceleration interval was 0.02 second and the sampling interval was 0.006 second. The sampling duration was the same as the input duration of the recorded acceleration earthquakes. The low pass filters were set of the input motion was scaled at to appropriate 10 Hz. The peak levels ranging from 0.1 g to 0.2 g, so that strong model responses were recorded without damaging the model. The model responses were, recorded on floppy transferred onto the main frame computer for further analysis. disks and later 59 After the completion of were the response time histories histories from an analytical study. compared the next chapter. Once satisfactory responses motion. was obtained The chosen an floor seismic ground with the motion tests, predicted model The method of analysis will the equipment actual floor motion record was motion record equipment and were the response time be discussed in agreement between the predicted and measured was the San used Fernando Blvd. roof response N61W (1971). Since the natural frequencies and the resulting well separated, and was assumed structure the mass it the input 2500 Wilshire of the structure ratio small, as between follows the that equipment-structure interaction was not significant in this case. The peak input floor acceleration was gradually increased until the isolators failed and the failure mode was observed and recorded. This represented the end point of all the experimental experimental studies are presented analytical method used to equipment-structure systems. work in this thesis. The results of the in Chapter six. The next chapter describes the predict the seismic response of vibration isolated 60 Fixed Cydic axial axial load Cyclic lateral load load Fixed lateral load Figure 4.1 Schematic representation of -0- bi-directional static test 61 Figure 4.4 Typical measured load - deflection curve Chapter 5 ANALYTICAL STUDY 5.1 I N T R O D U C T I O N The subjected dynamic response to strong motion of vibration isolated earthquakes may equipment also be housed investigated in a by an structure analytical approach. This approach requires the formulation of an accurate mathematical model of the dynamic system of interest, for which a set of equations that approximately represent the dynamic characteristics of the system can be written. The solution to this set of equations will yield a prediction of the dynamic response of the system to prescribed ground motions. In- this chapter, the assumptions of A vibration isolated fairly general incorporate equipment systems and their housing structures mathematical typical and idealizations in the mathematical modelling model nonlinear properties of of equipment-structure vibration isolators, will be systems, will also presented. which be can developed. Finally, a time step numerical integration scheme will be employed to solve the set of equations, yielding the dynamic response of vibration isolated equipment systems. 5.2 M O D E L For isolated and IDEALIZATIONS the purpose of equipment-structure idealizations developing systems, regarding the it a is dynamic general mathematical necessary to properties make of the model of simplifying system. vibration assumptions The dynamic properties of interest are the mass, stiffness, and damping parameters of the system. There models and are ii) lumped-parameter continuous two broad categories continuous-parameter models parameters are models more are of models. common more system Both models than accurate. 62 modelling: In are i) lumped-parameter idealized; continuous-parameter the present study, in general, models the but simpler 63 lumped-parameter system modelling method is characteristics are dissipate the seismic are the energy spring respectively. into In lumped-parameter representative elements energy input of the system. Examples mass and idealized lumped adopted. elements In contrast, lumped-parameter which dampers elements store are are the which either various store or of energy storage elements kinetic energy typical energy assembled models, and dissipating mathematically the potential elements. to form These the final analytical model. Let mathematical us now examine modelling of the the various housing idealized structure. lumped-parameter It is commonly elements assumed in the that the housing structure of interest is a multistory simple shear building frame. Each floor of the building is assumed to act as a rigid diaphragm in its own plane. In addition, seismic motion is assumed to act in a single horizontal direction normal to a vertical face of the shear building frame. This idealization allows each rigid floor diaphragm to translate in a single horizontal direction only. The second idealization is that the mass of each floor experienced by is entirely the direction. Interstory rigid stiffness lumped floor into the diaphragm rigid are diaphragm. the inertia may by pre- and post- damping ratios less than about two, the assumption of classical widely in of the structure multiplying the matrix of the structure. In general, this assumption and forces such that a completely diagonal matrix of generalized be obtained critical only varying inertia material damping damping matrix However, as stiffness are assumed to coefficients by the modal is valid for structures twenty percent forces the horizontal of the building is provided by the elastic lateral of the interstory columns. The damping characteristics be classical The which have stated in Chapter damping fails for systems which have tuned frequencies properties, such stiffness or mass. With the above assumptions, as large differences in the interstory the idealized lumped parameters of the mathematical model of the housing structure are the mass of each floor, the interstory stiffness, and the damping of the building. 64 The formulation of the mathematical model of the vibration isolated also involves many assumptions. First, the inertia properties, such as equipment the mass or the mass moment of inertia of the vibration isolated machinery are assumed to be lumped entirely at the inertia forces centre of mass of the system. This experienced by the equipment system assumption dictates that all the act through the centre of mass of the system. Moreover, the machinery is assumed to be infinitely rigid as compared to the vibration isolators; on vibration isolators forcing therefore, the machinery is with the proper modelled as inertia properties base motion is assumed to act in a single vertical plane of a pair of isolators. a rigid unit mounted at the centre of gravity. The horizontal direction normal to the The response of the system to such base motion can be completely described by the three components of rigid body motion: the lateral movement, the rotation, and the vertical motion at the centre of gravity of the system. This system has three degrees of freedom as shown in Fig. 5.1. The and vibration isolators rotational nonlinear spring stiffness stiffness. characteristics. nonlinearity of the isolators material nonlineaT system, nonlinear effect geometrically lies provide and the the equipment system Typically, vibration These nonlinearities ii) the geometric displacements isolators arise and strains are its exhibit nonlinearity of in the nonlinear load deformation nonlinear systems, the can with some from the considered relationship lateral, vertical, i) form of materially system. For a small and the of the system. In nonlinearities are the results of large deformations in the system. The analytical model used in this study attempts to model the material nonlinearity of the vibration isolators only. group of discontinuous linear elastic springs The as nonlinear isolator shown in Fig. 5.1. is idealized as a Depending on the nonlinear stiffness characteristic of the vibration isolators, each of the discontinous linear elastic springs in the model has a specific spring constant Therefore, a piecewise linear stiffness Fig. characteristic is assumed to model the nonlinear isolator stiffness as shown in 5.2. The damping characteristic of the equipment system is assumed to be 65 provided solely by the damping in the vibration isolators. assumed. Modal damping ratios of the equipment system mathematical model of the equipment system. These Classical damping is again are needed to formulate the system parameters were obtained from the experimental tests. 5.3 D E V E L O P M E N T O F T H E M A T H E M A T I C A L M O D E L The idealizations in the equipment-structure systems have those a idealizations, mathematical of the modelling been presented mathematical model mathematical model of housing in the the vibration last section. system structure of will will be be isolated Now, based on developed. presented. First, a Secondly, a mathematical model of the equipment system, which can incorporate the nonlinearity of the vibration isolators, will be studied. Finally, a mathematical model combining the equipment and the structure will be examined. 5.3.1 T H E S T R U C T U R A L The story each structural model is building frame studied shear however, the floor MODEL is system is denoted very similar expanded in to a bv m . (i = l,2,..,r) to the model of the simple Chapter two. maximum of where In ten the present floors. three model, The mass of r ^ l O denotes the number of si floors in the structure. structural The interstory stiffness is given by k s . (i = l,2,...,r). This system can have a maximum of ten modes of vibration resulting in a maximum of ten modal damping ratios denoted by $ . (i = l,2,...,r). The equation g of motion of the structural system when excited by base motion x (t) is: M where x + C x +K x = - M I ss ss ss s x s 1 (t) g is (5.1) w the relative displacement vector, x s 66 = [k s j,x ^ relative x s r l is the relative acceleration vector of velocity vector, and x g the floors with = respect txsj'Xs2-—XSP to the moving 's ^ base. I T = [1,1, ,1] is the influence vector coupling individual floor. The structural mass matrix M the base motion input to the and the structural stiffness matrix s K s are: m si m s2 M m sr (5.2) k s l + k s2 -k s2 ~ k s2 k -+k , s2 s3 . K -k -k Finally, matrix damping the structural M , s the matrix damping matrix structural Cg as stiffness illustrated C s matrix in is obtained K , s Chapter sr from k sr sr the structural mass and the structural generalized two. The structural generalized damping matrix and the structural damping matrix are as follows: 67 5 C = si u slmsl 2 K J sr u> m sr sr (5.3) * T * M C =M * C 5 S S s s s where = structural mode shape matrix <> i th u> = sr the r *sr = natural frequency of the structure the percentage of critical damping of the r^ 1 mode of the structure Knowing the mass, stiffness, and damping matrix, the equations of motion of the structural system can be formulated. 5.3.2 T H E EQUIPMENT The equipment formulation will the mathematical First model consider supported lumped at the centre by four of the vibration the idealized shown in Fig. 5.1. The system inertia properties mass is of now be presented. equipment system its MODEL isolated vibration isolated consists of a ; rigid mass with of gravity of the model. The rigid sets of springs which provide the system with its lateral, rotational, and vertical stiffness. Each restoring set of springs force equilibrium to the consists of a system when linear the position. The development of elastic spring rigid mass is excessive equipment which provides a displaced seismic from its motion is 68 prevented Such the provision of devices system to by provide an abrupt when the displacement model the snubbers characteristics which increase in act as motion restraint restoring force to the of the unit exceeds a prescribed of these devices, a discontinuous devices. equipment value. In order linear spring is added to the assemblage, which provides the required addtional stiffness (see Fig. 5.1). This is a simple spring In order to model a system discontinuous system linear softens spring when assemblage modelling a bilinear hardening system. which exhibits degrading stiffness can the be assigned displaced characteristics, the negative stiffness, such that the a equipment system exceeds a prescribed displacement as shown in Fig. 5.2. From the static testing program, it was discovered that when the horizontal snubber units in the open spring isolators with uni-directional motion restraints were the engaged at some Fixed lateral load, axial suffness of the springs increased dramatically. However, under a continuously increasing axial load, the vibration isolators eventually regained their original axial stiffness as shown in Fig. 5.3. This phenomenon is the result of coulomb damping. With the horizontal snubbers engaged, the springs were held axially by friction between the snubber and the spring assemblage. An increased axial load was needed to overcome this friction force before slippage between the snubber and the spring assemblge could occur in the axial stiffness. To model vertically mounted becomes active besides the direction; at this point, the spring regained effect, springs in only effect vibration isolators this when a the the friction spring mathematical horizontal is included model and snubber of coulomb damping, the effect units its normal axial in the are of material each friction engaged. set of spring Finally, damping of the needs to be considered. Material damping is modelled by the provision of linear viscous dashpot dampers in the mathematical model. 69 The equations of motion of the model to base excitation x b (t) at the point of equipment support is formulated about the centre of mass of the model and with reference to the static equilibrium position of the dynamic system. The equations of motion are given as: M x +C x +K'(x e e e e e v )X = - M IX. (t) e b c' e (5.4) v w where x = [x .,x _,x ,] is the relative displacement vector, x = [x ,,x _,x ,] v e el e2 e3 e el e2 e3 is the relative velocitv vector, X = [x ,,X ~,x ,] is the relative acceleration e el e2 e3 vector of the degrees of equipment freedom x ^, x ^ , rotation, and the vertical system. The motion to sign with reference to the and x ^ are respectively movement of the convention for the right, clockwise positive rotation, base motion X^(t). the lateral The three motion, the centre of gravity of the equipment movements and is defined as downward motion. The the input horizontal vector I 7 = [1,1,1] is the influence vector coupling base motion to the individual degrees of freedom of the equipment The stiffness matrix is nonlinear because it depends on the equipment displacement response, x . The equipment mass, damping, and the following manner. First, x £ , assumed' to equations of sin(xe2) = x £ 2 be small. motion Based can and cos(xe.j) = l matrices the relative displacements on be stiffness this assumption, neglected. are assumed Also, formulated in of the equipment are second the are order terms linearization in the assumption to apply. Summing all the forces in the direction of each degree of freedom of the system and setting the result to zero results in the desired equations of motion. So, assuming the system is undamped and excited by base motion X^(t), the inertia and spring forces of the displaced system are shown in Fig. 5.4. A compressive spring force is defined as positive by our sign convention. Equilibruim in the direction of the horizontal, ' 70 rotational, and vertical degrees of freedom of the equipment results in the following equations of motion: Mx . + C + D = - M x . ( tv ) el b ' Jx £ 2 -A(arm Mx a) + b(arm b)-C(arm c)-D(arm d) = 0 (5.5) ,+A+B=0 e3 where M J = = mass of the equipment system mass moment of inerua of the equipment system about the centroidal axis A = total vertical spring force on the left side B = total vertical spring force on the right side C = total horizontal spring force on the right side D = total horizontal spring force on the left side arm a = horizontal distance between the eg. and the vertical spring on the left side arm b = horizontal distance between the e.g. and the vertical spring on the right side arm c = vertical distance between the eg. and the horizontal spring on the right side arm d = vertical distance between the eg. and the horizontal spring on the left side The various displaced system. distances between system is determined Assuming small rotations the springs and from the geometry and displacements, the centre of mass and the displacements the various of the of the distances are as 71 follows: arm a = b. + ax . 1 e2 arm b = b -ax 0 r e2 arm c = arm d = a-b,xe2 a + b rxe2 where a = vertical distance between the eg. and the undisplaced vibration isolators bj = horizontal distance between the e.g. and the undisplaced vibration isolators on the left br = horizontal distance between the eg. and the undisplaced vibration isolators on the right At this point, let us examine the restoring spring forces in the equations of motion. These spring forces may be nonlinear because of the discontinuous springs. First, let us assume that the system remains in its linear state. And let us denote K-^^, as the initial lateral stiffness of the vibration isolator on the left and the right side of the system respectively, and K - j y L » ^ J V R A S the initial vertical stiffness of the vibration isolator on the left and right side of the system respectively. displacements From the of the vibration isolators geometry of the system, due to the movements at gravity of the system are given by equation (5.7). xc = horizontal displacement of the vibration isolators = x .-ax . el e2 the various the centre of 72 Vj V = vertical displacement of the left vibration isolator = " b l x e 2 + x e3 = vertical displacement of the right vibration isolator = From The b ( 5 J ) rxe2+xe3 the sign convention, positive displacements mean shortening of the springs. linear restoring spring forces as a result of the various displacements are: A = B K 1VLV1 = = K lVRVr C = K lHLXc D = K lHRXc Substituting equations K lVL(-bIXe2+ = K lVR(brxe2+ = K lHL (5.6) e r a x e3) e3) x e2 (5.8) ) lHR(xeraXe2) K = ( x X and (5.8) into equation (5.5) yields the equation of motion for the linear three degree of freedom system. Let us now examine spring units are engaged. the situation Let the clearance discontinuous spring units be denoted discontinuous springs by K ^ H ' " ^ n e s e by in which the horizontal discontinuous between the mass and the horizontal c and the stiffness of the horizontal discontinuous springs are engaged when the absolute value of the horizontal displacement of the vibration isolators x £ exceeds the clearance units are c. The Final ^2HL horizontal spring ~ K 1HL forces of horizontal stiffness + K 2H the a n d system K 2HR with of the left = the K 1HR + K and the right spring 2H r e s Pecuvel>- T n e horizontal discontinuous spring units engaged are: C D = K = 2HRXc-cK2H(sgn(xc)) K 2HLXc-cK2H(sgn(Xc)) ( 5 9 ) 73 where sgn(xc) = The expressions the algebraic sign of for the vertical xc restoring spring forces remain unchanged since they are still linear. Again, the expressions for the spring forces and the spring distances can be substituted into equation (5.5) to yield .the equations for the case where only the horizontal discontinuous springs are The situation when the cases: left i) side engaged. in which only the vertical discontinuous spring on the left side of the model is engaged will two possible of motion the of now be studied. discontinuous the model spring has In this situation, there is lifted engaged beyond in a the tensile are mode static equilibrium position, and ii) the discontinuous spring is engaged in a compressive mode when the left side of the model has dropped beyond the static equilibrium position. Some vibration characteristics. equilibrium isolators Since the position, and have equations the very of isolators different motion are are in compressive may not compressive stiffness dead when load of provide stiffness any the the motion lifting written initially in dead weight of the machinery, such vibration isolators change tensile vibration may increase the due may experience isolator devices compressive about compression overcomes machinery. Moreover, some restraining and against open abruptly when the spacing to the an abrupt the spring uplift, static initial isolators whereas their between the spring coils is closed in a compressive mode. In order from the static to model the above situations, equilibrium position, between the the vertical vertical mass during an uplift is allowed to take on a different a downward movement. Also the tensile and spring and measured the rigid value from that during compressive discontinuous vertical springs may be assigned different clearance, values. stiffness of the 74 Let us the denote the rigid mass by D j that a positive vertical uplift . clearance and it is engaged in denote the compressive denote the springs on tensile displacement the *MVL+^2V1 left m a n is suffness side defined as spring and by D^- Keeping in mind downward, the discontinuous is engaged in tension when V j is less than - D j compression stiffness the discontinuous the downward clearance vertical spring on the leftside and between of of when Vj is greater of the discontinuous vertical these springs. The total the denoted model P^'ft mode and ^2YL+^2V2 u new vertical restoring spring forces is than m D^. Let ^2\2 springs and Kjyi vertical by stiffness K^vx, w m c n of the equals downward mode. The a for the case where the vertical discontinuous spring units on the left side are engaged in a downward mode and in an uplift mode are given by equation (5.10) and (5.11) respectively. A = V,K A = V j K ^ - f D j K ^ Repeating discontinuous When is 2 V L -D2K the above spring = 2 V 2 ( - b j X ^ + x ^ X K j ^ + K ^ h D ^ ^ (5.10) = (-b1xe2 + x e 3 X K 1 V L + K 2 V i ) + D 1 K 2 v l ( 5 . 1 1 ) procedure on the for right side is the situation engaged, when only yields the the vertical following results. is less than ~^>y the vertical discontinuous spring on the right side engaged in tension. When V r is greater than D 2 , the vertical discontinuous spring on the right side is engaged in compression. The total vertical stiffness of the springs on the right side of the model is denoted by K- 2 v -^, which equals F£ 1 V R + ^ 2 V 1 The in an uplifting mode and new restoring spring forces for the ^•JYR case + ^2V2 where m the a n downward vertical m °de. discontinuous spring unit on the right side is engaged in a downward mode and in an uplift mode are given by equation (5.12) and (5.13) respectively. 75 B = B = V rK2VR-D2K2V2 V KK r 2VR+ Depending D lK2Vl on which = < b r X e2+ X e3)(KlVR + 5 J = < b r X e2+ X e3)(KlVR+K2V2>+DlK2Vl ^ 1 nonlinear springs are equation (5.13) and equation (5.6) can be substituted the equations system. of motion Finally equation for the nonlinear three (5.5) can be rearranged K 2V2>"D2K2V2 activated, ( equation 2 3 (5.9) ) ) - into equation (5.5) to yield degree of freedom into matrix equipment form to yield the mass and stiffness matrices in equation (5.4). Let us equipment now formulate mass matrix equipment generalized equipment natural matrix $ are obtained for the undamped M e > damping frequencies free the equipment damping matrix C the equipment stiffness matrix K£, and the in Chapter two. The matrix u . C g as illustrated (i = 1,2,3) and the equipment from the g mode from an eigenvalue solution of the frequency vibrating system. equipment The equipment shape equation generalized damping matrix and the equipment damping matrix are as follows: C = 2 e *e2we2Me2 S e3we3Me3 (5.14) C = e M * C *<i> T M e e e e e where th $ £T=percentage of critical damping of the r Knowing the equipment mass, stiffness mode of the equipment and damping motion of the equipment system can be formulated. matrix, the equation of 76 So analysis. springs. far, Let the us treatment focus our of the attention friction on the spring has been mathematical omitted formulation in the of these The laws of Coulomb are assumed to govern and the friction force is assumed to be proportional to the normal force surfaces. In the case of the open spring isolators acting between the with uni-directional sliding restraints, the normal force acting between the sliding surfaces is the lateral load acting on the snubber units when the snubber units are engaged. The friction force always acts in a direction opposite to the velocity of the system, as in the case of the dashpot damper. independent However, of the the magnitude velocity. The frictional of the resistance is frictional resistance formulated as an is extra vertical spring load, which is proportional to the lateral spring forces C and D. The constant The vertical horizontal of proportionality u spring forces discontinuous is on the left springs are determined and engaged, from the experimental results. right side of the model, when the are given by equation (5.15) and equation (5.16) respectively A = K ^ V j+sgnKMAx^^Vj) (5.15) B = K1VRVr+sgn((uAxcK2H),AVr) (5.16) where Ax£ = ^xc^2H AVj AVr = x c (t+At)-x c (t) = = s n u '3^er s Pnn§ change in the lateral spring displacement f°rce Vjd+AO-Vjd) = change in the vertical left spring displacement = V r (t+At)-V r (t) = change in the vertical right spring displacement 77 Substituting (5.5) equation (5.9) for the appropriate - equation (5.16) and equation (5.6) into equation cases yields the complete system of nonlinear equations of motion for the vibration isolated equipment system. 5.3.3 T H E C O M B I N E D The will now mathematical be presented. EQUIPMENT modelling of The equations STRUCTURE the MODEL combined of motion for equipment-structure such a system system under a ground excitation Xg(t)is as follows: Mx + Cx+K(x)x=-MIx (t) (5.17) Again x, x, X are respectively the relative displacement, velocity, and acceleration of each degree of freedom to the ground and I is the influence vector coupling the input ground motion to each degree of will be considered to have r stories (r ^ 10); freedom. The structure of interest the overall system matrices depend upon the location of the equipment system in the building. Assume that the vibration isolated equipment is attached to the i^ floor of an T story building. The various system 1 matrices can be found in the usual fashion, with the mass matrix of the system given as: 78 m, m i+1 M = m.i + 2 (5.18) m. i+ 3 m where .th mass of the i m. = 1 m. + j m The i + 2 — ni. + 2 = m a s s m = o m floor mass of the equipment system e n t 0I " i n e r u a ° f the equipment system suffness matrix of the combined system is formulated by the same method and is given by: k, + k K = 2 . k. + k„-kn -k -k n kn kn k -k 21 k k„ k k k 2a k. 3 32 12 — (5.19) -k -k r k r 79 where k. k.j = i ^ 1 interstory suffness of the structure = the equipment restoring force corresponding to the i 1 * 1 equipment degree of freedom due to unit displacement of the j ^ 1 equipment degree of freedom (ij The various equipment stiffness entries kjj = 1,2,3) in the combined stiffness matrix can be obtained directly from the suffness of the vibration isolated equipment system formulated in the previous section. Finally, the combined equipment-structure damping matrix can be formulated in a similar fashion and is given by: c C = +cn - C n "Cn -c -Cn Cn Cn Cn -C21 Cn Cn c 23 -c31 CJI c32 c 33 (5.20) rr M _ entries from the structural damping matrix Cy The = technique entries from the equipment damping matrix used to motion will be discussed solve the resulting in the next section. nonlinear system of equations of 80 5.4 M E T H O D O F SOLUTION The method of solution of the nonlinear system of equations of motion is the well known Newmark-Beta method. The solution scheme is similar to that discussed in Chapter two. The major difference between the present and the former scheme treatment of the nonlinear stiffness equations of motion by this terms. step-by-step In order to solve integration is the the nonlinear system method, the equations of must be expressed in incremental form: MAX(t)+CAx(t) + K(x)Ax(t) where A represents an =-MIAx (t) s incremental (5.21) change in value within a time step At. The nonlinearity in equation (5.21) exists because the stiffness matrix is dependent upon the displacement solution vector x. However, within a time step At each nonlinear spring can be assumed to have a constant stiffness. The stiffness of each nonlinear spring is assumed to increment at the be the tangent stiffness Therefore, the stiffness of the spring at the matrix only needs to be conclusion of each time increment Then beginning examined of a time for modification the incremental equations of motion within a time increment can be considered to be linear. Either a constant or a linear variation adopted in the change of acceleration response over a small time interval can be in the Newmark-Beta method. The resulting expressions of the response from the application of the numerical integration scheme are given by equation (2.27). The above time step integration scheme may lead to erroneous results if the chosen time step is too laTge. In addition, the nonlinear springs are modelled by a piecewise can occur abruptly. When they linear scheme an abrupt stiffness have been which means change computed on that a change in stiffness occurs, the calculated responses are incorrect because the stiffness basis of linear Moreover, dynamic equilibrium will not generally be satisfied within a time step. at the conclusion of the 81 lime step. The solution scheme adopts two strategies to reduce caused by these conditions: i) equilibrium correction and ii) the possible error event to event integration scheme. In the equilibrium correction scheme, the dynamic equilibrium is checked at the conclusion and of are each added time to the interval. Any unbalanced loads are load next increment for the treated time step. increments must be used for this method, otherwise substantial load-displacement increments By path is possible. into sub-increments using an iterative The alternative when an abrupt from small an loads time from the real subdivides in stiffness, technique, the solution can proceed external Very departure scheme change as the event, time occurs. event to event. There is never an unbalanced load in this solution scheme and the solution follows the exact event to event method has the disadvantage load deformation relationship. However, the of being computationally In the present increment The study much more expensive the equilibrium event to than the equilibrium correction scheme. correction event method is scheme only is used employed during when the every time horizontal discontinuous springs are engaged. This concludes the discussion on the analytical study of the seismic response of the vibration isolated on the above equipment-structure principles has been system. written to A computer analyse the program, seismic EVIS.S, based response of systems. The computer listing for this program is provided in Appendix C. such Discontinuous spring Discontinuous spring \ '2H \ 1HL T L a -KNH -mm 2V1 (tens.) K.JY2 (comp.) 1VL Friction spring I 1HR - r 1VR Discontinuous springs H4 Figure 5.1 Idealized model of the vibration isolated equipment system. D 2 83 Experimental Figure 5.2 M o d e l Nonlinear load- deflection curve (static test) and piecewise linear load- deflection curve (analytical model) Figure 5.3 Schematic representation of coulomb friction between isolator and snubber device. Figure 5.4 Equilibruim forces in a displaced vibration isolated equipment system. Chapter 6 E X P E R I M E N T A L A N D ANALYTICAL RESULTS 6.1 INTRODUCTION The this experimental and analytical results chapter. results and mathematical physical The results are ii) excitation random model parameters of the obtained in this study are presented in divided into two categories: test physical results. system of the mathematical As of i) described interest model are model identification test in has been Chapter five, developed. a The based on the model identification test results obtained from the static and the dynamic testing program. Also, a computer program utilizing time step analysis has been developed to generate numerical solutions of the system of dynamic equations of motion. In this chapter, the model identification test results presented and first. the physical Then the parameters random adopted excitation numerical solutions of the model equations base motion inputs. systems are Finally in test the results mathematical are of motion subjected the performance of the compared model are with the to the same random two vibration isolated equipment discussed. 6.2 M O D E L INDENTIFICATION T E S T R E S U L T S (STATIC) The static test program yielded the inertia properties of the experimental model and the stiffness characteristics of the vibration isolators. Table 6.1 shows the measured inertia properties of the model and the calculated equipment system. Design 34 size 54 The prototype 1/4 equipment arrangement 9 with inertia properties system a 405 is the US of the prototype Chicago Vane Axial 100 bhp motor frame Fan and inertia base. The idealizations and the characteristics of this system have been discussed in Chapter four. 85 86 Mass Model 2136.6 kg 1510.8 kg.m.m .965 m Prototype 1852.9 kg 1772.4 kg.m.m .950 m Table 6.1 A Location of eg. above base Mass moment of inertia w.r.t z-z axis Model and prototype inertia properties. fairly close match can be seen between the model and prototype inertia properties. Exact matching of the model inertia properties with the prototype inertia properties proved to be a difficult task. However, this difference is not significant as the aim of the present study is to observe the seismic response of typical vibration isolated equipment systems which include a wide range of inertia characteristics. The prototype equipment system serves to provide the inertia characteristics As long as the model inertia properties properties, and they do not exceed the of one of these systems. are reasonably close dead load capacity to the prototype inertia of the selected vibration isolators, the model is considered acceptable. Instead of modelling exactly the particular prototype equipment system, our experiment models a similar vibration isolated equipment system which possesses the measured inertia properties summarized in Table 6.1. The lateral and axial stiffness characteristics of the elastomeric isolators and the open spring program. under isolators Fig. 6.1 three with shows different motion the axial constant restraint were load-deflection lateral preloads obtained curves (0, of 1.1, from the 2.2 the static elastomeric kN). The testing isolators following conclusions can be drawn from these results: 1. The elastomeric isolators have nonlinear axial stiffness characteristics. The isolator tends to stiffen under an increasing axial load and the loading path differs 87 substantially from the unloading path. This indicates that high hysteretic damping characteristics can be expected from the elastomeric 2. The effect of significant, at lateral least preload in the on the preload axial range isolators. stiffness examined. characteristics The axial is not load-deflection curves are similar and consistent under the various lateral preloads imposed. 3. The measured rated axial stiffness stiffness approximate the axial value. value differs Under an substantially axial load of from 5240 the commercially N (1200 lb), the dead load of the equipment model supported by a vibration isolator, measured tangent stiffness was 1350 kN/m (7700 lb/in); the commercially rated axial stiffness is 635 kN/m (3600 lb/in). The lateral stiffness characteristics of an elastomeric isolator under various constant axial preloads are shown in Fig. 6.2 For sake of clarity the complete unloading branch of only one test is shown in this figure. The results indicate the following: 1. 2. The loading path differs from the unloading path; therefore, the lateral stiffness characteristics are nonlinear. Hysteretic damping characteristics can be expected . The characteristics influence elastomeric increasing of axial isolator axial is preloads significant preload. on the The The lateral lateral lateral stiffness stiffness stiffness tends during loading to of increase of one the with elastomeric isolator varied from 232 kN/m (1333 lb/in) to 800 kN/m (4570 lb/in) when the axial lb) preloads were 2.2 kN (500 in compression respectively. The lateral stiffness isolator tension to 18 kN (4000 during unloading of one varied from 455 kN/m (2600 lb/in) to 1750 lb) in elastomeric kN/m (10000 lb/in) under axial preloads of 2.2 kN (500 lb) in tension to 18 kN (4000 lb) in compression. Fig. 6.3 shows the axial load-deflection uni-directional restraint under two characteristics lateral preload of an open spring isolator with conditions. (0 and 4.4 kN). Two conclusions are drawn from these results: 1. UndeT low lateral preloads, the axial stiffness characteristics of the vibration 88 isolator are tensile 394 trilinear. The isolator stiffness is zero. Under kN/m (2251 these springs spacings cannot carry any low compressive tensile load; therefore, load, a linear axial the stiffness of lb/in) was measured. The commercially rated axial stiffness of is 350 kN/m (2000 between the spring coils lb/in). are Under closed; high thus, compressive the isolator loads, axial the stiffness increases abruptly to 3500 kN/m (20000 lb/in). 2. As explained in Chapter five, under a high lateral load, when the horizontal snubber unit is engaged, the effect of Coulomb damping can be observed to act between snubber which the spring relates the unit extra and the axial frictional unit. load to The coefficient the lateral of friction normal load u was measured as 0.08. The lateral load-deflection characteristics of the open spring isolators with uni-directional restraint under various axial preloads are shown in Fig. 6.4. The results indicate the following: 1. The lateral suffness stiffness characteristics is 98 kN/m (558 of the isolator lb/in) and the second (4784 lb/in). The abrupt increase in suffness are bilinear. The initial lateral stiffness lateral is 839 kN/m occurs when the gap between the snubber unit and the spring unit is closed. The measured gap is 50 mm (.2 in). 2. The influence of axial preloads on the lateral stiffness characteristics is not and the significant. Based suffness stiffness the characteristics equipment isolator on systems with can results from to be used now be the noted testing in the analytical defined. uni-directional restraint, characteristics static above For the are the measured incorporated program, models system the inertia of these vibration isolated containing an piecewise linear axial into the analytical open spring and lateral model, which also involves friction springs with a friction coefficient of 0.08. In the analytical model of the elastomeric isolator, a linear axial stiffness of 2700 kN/m (7700 lb/in) and a 89 linear lateral stiffnesses is stiffness of to represent strictly incorrect; 3500 kN/m (9990 the nonlinear suffness however, the dynamic lb/in) was used. characteristics tesung The use of of the elastomeric results reveal that this linear isolators analytical model provides reasonable response predictions for various base motion inputs. Also, the lateral stiffness of the elastomeric isolator adopted high. The choice is justified in the analytical model seems to be however by the dynamic model identification test results presented in the next section. 6.3 M O D E L INDENTIFICATION T E S T R E S U L T S (DYNAMIC) The dynamic model identification tests yielded decaying sinusoidal curves and frequency response curves for the two vibration isolated equipment systems. Fig. 6.5 - Fig. 6.9 show the dynamic model identification test results of the system incorporating an open spring isolator with uni-directional restraint By exciting the system at each of its natural frequencies (1.8 Hz and 4.9 Hz) until a steady state response was reached, and then suddenly stopping the table motion, the decaying sinusoidal response of the centre of gravity of the system at each natural frequency was obtained as shown in Fig. 6.5 and Fig. 6.6. Based on the logarithmatic decrement method, the modal damping ratios were found to be 0.103 quite high; springs In the vibration Based close for examination these two tests. These of the on experiment springs the to rub experimental when the system against results, their 10% casings seemed revealed to be thai the the model to lean slightly toward was damping values vibration isolators were not uniform in height, which caused one side. the however, a and 0.105 excited, and in this misalignment created all extra modes caused damping was used load. in the analytical model. Fig. response gravity 6.7 and Fig. 6.8 functions of the of system. this show the horizontal and rotational acceleration system. The results These were responses were measured obtained experimental from at the frequency centre sinusoidal of tests, 90 experimental white noise tests, analytical sinusoidal tests, and analytical white noise tests. The experimental sinusoidal and white noise tests have been described in Chapter four. The analytical counterparts; sinusoidal however, and these solution of the mathematical four types amplitudes. of tests show white noise results tests were were generated model for the system. good correlation in natural their from experimental the computer In general, the results from these the natural frequencies and response However, the experimental white noise test results do not clearly show the caused spectral to analytically response peaks in the frequency range of 4.3 Hz system similar the spectral analysis analysis is strictly frequencies approach valid for 4.9 Hz. The nonlinearity in the to perform poorly since the technique of linear systems only. Based on these results, the of this system are 1.8 Hz, 4.3 Hz and 4.9 Hz. The predicted and measured mode shapes are analytical approach is shown a pure in Fig. 6.9. vertical experimentally by exciting the system The second translational mode mode, which with a lateral predicted cannot be by the duplicated base motion. As predicted by the analytical model, the first and the third modes of vibration are out of phase by 180 degrees. Fig. 6.10 elastomeric (4.6 Fig. 6.14 show the dynamic model identification test results for the isolator Hz and 15.0 system. By exciting the Hz) until system at each a steady state response was of its natural frequencies reached, and then suddenly stopping the table motion, the decaying sinusoidal response of the centre of gravity of the elastomeric shown isolator in Fig. 6.10 and logarithmatic decrement mode. These demonstrated damping that vibration isolator system high Fig. 6.11. each of values are hysteretic the The modal method were 0.17 system. These the analytical model. at frequencies damping ratios as was damping high, can but be were adopted the static expected as test from the modal obtained, calculated' by for the first mode and 0.18 fairly values natural as the for the third results an have elastomeric damping ratios in 91 Fig. 6.12 and Fig. 6.13 show the horizontal and rotational acceleration frequency- response functions of the elastomeric isolator. Again these responses were monitored at the centre of gravity of the system. Based on the static test results, which yielded the nonlinear stiffness characteristics of the elastomeric isolators, it was decided not to test this system for with the white noise since the spectral linear systems only. The experimental between the measured natural frequencies shapes of this system and analytical results and predicted natural of the system 6.4 R A N D O M M O T I O N RESULTS and response amplitudes. The TEST response of the analytical models of the two vibration isolated systems were compared with the results The agreement are shown in Fig. 6.14. The predicted mode shapes agree well results. conditions. frequencies show good were 4.6 Hz, 8 Hz and 15.0 Hz. The three mode with the corresponding measured The analysis on the response is valid experimental models equipment obtained experimentally under random motion were excited by four different random base motions: 1. El Centro 1940 N - S (maximum acceleration scaled to 0.1 g), 2. San Fernando 1971 N21E (maximum acceleration scaled to 0.18 g), 3. Taft 1950 (maximum acceleration scaled to 0.11 g), 4. Band limited (0 - 25 Hz) white noise record (maximum acceleration scaled to 0.11 g). The peak model base motion acceleration was kept within 0.1 g and 0.18 response could be generated without damaging the g so that strong experimental model. These same base motion records were used as the input excitations for the analytical models of the two vibration isolated equipment systems, thereby allowing a comparision to be made between the numerical solutions and the measured responses. 92 Fig. 6.15 acceleration Fig. 6.22 show the predicted and measured horizontal and rotational responses uni-direcuonal at restraint the For centre this of gravity system, good of the open agreement is spring system observed with between the responses of the analytical and the experimental models. The results also show that, in some cases, values; the analytical this may be which were, in fact Fig. 6.23 acceleration system. responses. experimental overestimate damping the actual values for measured response the analytical model slightly lower than those existing in the actual model. Fig. 6.30 show the predicted and measured horizontal and rotational at the centre of is again observed agreement The results the result of assuming responses Good model analytical results. The that of the measured model gravity predictions frequency of the between are elastomeric the slightly isolator predicted more content of the analytical equipment and measured conservative response is than the higher than response, which indicates that this analytical model is stiffer than the experimental model. This may be > due to modelling the nonlinear equipment by a linear analytical model, which isolator may have softened is strictly incorrect the system when subjected The nonlinearity of the actual to dynamic excitation. The final stage of the experimental program involved the determination of the failure modes of the two vibration isolated equipment systems. The open spring system was excited with an actual floor motion record: San Fernando 2500 Wilshire Blvd roof response N61W (1971). The base motion acceleration isolator failed. springs were about to dislodge was 1.2 g; Fig. 6.31 however, and the Fig. 6.32 the failed until the vibration vibration isolators; from their casing. The recorded peak exact determined since the isolators show was increased base motion acceleration failed at some acceleration at failure the open base acceleration could level during a spike not be in the input base motion. The elastomeric isolator equipment system was excited by the same input base motion and it survived with no apparent not experience an acceleration spike in this damage; case. In order however, the table did to determine the failure 93 mode of the elastomeric isolators, the system was excited with a sinusoidal input at the fundamental natural frequency of the system (4.6 Hz). The base motion acceleration amplitude suffered vibration damping was then only increased to 2.5 g. The elastomeric minor isolators. damage: These characteristic environment there results and was some indicate nonlinearity, that tensile isolators tearing along the base the elastomeric survived remained intact and the isolator, extreme with simulated of the its high seismic Axial load (lb) Lateral deflection (in) Lateral load-deflection curves of an elastomeric vibration isolator. TIME (SECI Ct Figure 6.5 Decaying sinusoidal response time history at 1 natural frequency (1.8 Hz). oo a TIME ( S E C ) Figure 6.6 Decaying sinusoidal response time history at 3 r d natural frequency (4.9 Hz). "i 4.0 Figure 6.7 1 r 50 6.0 FREQUENCY frequency IHZI Horizontal acceleration response function. p Experimental sinusoidal results -e- Analytical sinusoidal results Experimental White Noise results p 00. Analytical White Noise results Open spring isolator with uni-direcuonal restraint. 1 4.0 5 0 FREQUENCY (HZ) Figure 6.8 1— 6.0 " i — 70 ~~i— Rotational acceleration frequency response function. 8.0 — i — 9.0 10.0 Open spring isolator with uni-directional restraint. Figure 6.9 Predicted and measured mode shapes. •© - Experimental sinusoidal results — Analytical sinusoidal results FREQUENCY (HZ) Figure 6.13 Rotational acceleration frequency response function Elastomeric isolator. Predicted Measured 4 First mode Second mode Third mode Predicted and measured mode shapes. o Experimental response ( , Analytical response Open spring isolator with uni-directional restraint. Base motion: El Centro 1940 N-S. i j 3.0 40 50 6.0 7.0 8.0 9.0 TIME ( S E C ) Predicted and measured horizontal acceleration response time history. 10.0 Experimental response o Analytical response Open spring isolator with uni-directional restraint. o Base motion: El Centro 1940 N-S. Peak acceleration scaled to O.lg. in m Q (X z o o <M_ 0.0 "I— 1.0 -1— 2.0 Figure 6.16 ~i— 3.0 ~i 40 1 50 TIME ( S E C ) i— 6.0 7.0 8 0 i— 9.0 Predicted and measured rotational acceleration response time history 10. Experimental response Analytical response Open spring isolator with uni-directional restraint Base motion: San Fernando 1971 N21E Peak acceleration scaled to 0.18g. o.o -1— i.o Figure 6.17 — 2.0 i— 3.0 3.0 4.0 SO TIME ( S E C ) 6.0 ~I— 7.0 7.0 — I 8.0 8.0 Predicted and measured horizontal acceleration response time i— 9.0 history. 10. Experimental response Analytical response Open spring isolator with uni-directional restraint. Base motion: San Fernando 1971 N21E Peak acceleration scaled to 0.18g. 0.0 to Figure 6.18 i— 2.0 I 1 1 3.0 4.0 5.0 TIME ( S E C ) 1— 6.0 —|— I— 8.0 7.0 Predicted and measured rotational acceleration response time history 9.0 10.0 Experimental response m °~ ?- - Analytical response Base motion: Taft 1950. Peak acceleration scaled to O.llg. INI ?H 0.0 1 1.5 Figure 6.19 1 3.0 1 45 ! 6.0 , 7.5 TIME ( S E C ) 1 9.0 , 10.S , 12.0 Predicted and measured horizontal acceleration response lime history , 13.S 15 o Experimental response Analytical response » V C o' 1 tnrM a?- Ii Open spring isolator with uni-directional restraint. P Base motion: Taft 1950. Peak acceleration scaled lo 0.1 lg. 0.0 1 1.5 Figure 6.20 3.0 i " 5 6.0 6.0 TIME 1 75 7.5 (SECI r— 9.0 9.0 10.5 Predicted and measured rotational acceleration response ume ~I 12.0 history. 13.5 15.0 CM Experimental response Analytical response Open spring isolator with uni-directional restraint. Base motion: Band limited (0 - 25 Hz) white noise signal. Peak acceleration scaled to O.lg. 0.0 I— 2.5 Figure 6.21 -1— s.o 7.5 -i 10.0 1 12.5 TIME ISECI 1 15.0 1 17.5 1 20.0 7~T 22 5 Predicted and measured horizontal acceleration response time history. 25 o Experimental response Analytical response Open spring isolator with uni-directional restraint. Base motion: Band limited (0 - 25 Hi) white noise signal. Peak acceleration scaled to O.lg. 0.0 1 2.S Figure 6.22 1 5.0 1 7.5 1 10.0 TIME 1 12.5 CSEC) 1 15.0 17.5 20.0 Predicted and measured rotational acceleration response lime history. 22.S 25.0 Experimental response Analytical response Elastomeric isolator. Base motion: El Centro 1940 N - S Peak acceleration scaled to O.lg. o.o ~~i— 0.7 Figure 6.23 1.4 i— 2.1 I 1 2.0 3.5 1— 4.2 ~I— 4.9 —1— 5.6 TIME ISEC) Predicted and measured horizontal acceleration response time history. ~~i— 6 3 7.0 Experimental response Analytical response Elastomeric isolator. Base motion: El Centro 1940 N-S. Peak acceleration scaled to O.lg. o.o i— 0.7 Figure 6.24 ~T— 1.4 -1 2.1 1 2.8 TIME 1 3.5 ISEC) 1 4.2 1 4.9 I 5.6 Predicted and measured rotational acceleration response time history. Experimental response Analytical response Elastomeric isolator. Base motion: San Fernando 1971 N21E Peak acceleration scaled to 0.18g. o.o —j— 0.7 1.4 Figure 6.25 2.1 2.a 35 TIME ISEC) 4.2 -1— 4.9 4.9 5.6 i 6.3 Predicted and measured horizontal acceleration response lime history. 7.0 Experimental response Analytical response Elastomeric isolator. Base motion: San Fernando 1971 N21E. Peak acceleration scaled lo 0.18g. — i — 0.0 0.7 Figure 6.26 ~ i — 1.4 1.4 2.1 i 1 2.a 3.5 TIME ISECI 1— 4.2 -1— 4.9 I— 5.6 Predicted and measured rotational acceleration response lime history -l— 6 3 7.0 Experimental response Analytical response Flasiomcric isolator. Base motion: Taft 1950. Peak acceleration scaled to 0.1 lg. o.o —i— IS Figure 6.27 3.0 4.5 6.0 7.5 TIME ( S E C ) 9.0 10.S ~ i — 12.0 12.0 13.5 Predicted and measured horizontal acceleration response lime history. 15. Experimental response 11 0.0 1 1 1 1 I.S 3.0 4.5 6.0 Figure 6.28 TIME 1 i 7.5 9.0 (SECI i 10.5 i 12.0 Predicted and measured rotational acceleration response lime i 13.3 history. ISO Experimental response Analytical response Base motion: Band limited (0 - 23 Hz) white noise1 signal. Peak acceleration scaled to O.lg. o.o i 0.5 Figure 6.29 1.0 I 1 5 I.S I 2.0 2.0 1 2.5 2.5 TIME ( S E C ) 1— 3.0 3.0 l— 3.5 3.5 7 — I— 4.0 4.0 Predicted and measured horizontal acceleration response lime history. 4.5 5.0 « o" Experimental response Analytical response s Elastomeric isolator. i- Base motion: Band limited (0 - I Peak acceleration scaled to 0.1g> o.o -1— o.s Figure 6.30 -t.o T— -i I 5 signal. 25 H i ) while noise 1 1 1 2.0 2.5 3.0 I 3.5 I— -1— 4.0 4.5 TIME (SEC) Predicted and measured rotational acceleration response time history s.o to 124 Figure 6.32 Failed open spring isolator with uni-directional restraint Chapter 7 CONCLUSIONS 7.1 S U M M A R Y AND CONCLUSIONS Analytical and experimental studies on the behaviour of equipment isolators under seismic conditions have been investigated in three stages as follows: 1. A parametric study was conducted to investigate interaction systems. on the The maximum equipment steady state equipment-structure system first study Then the characteristics of a response subjected was general response of a of general general two equipment-structure degree to harmonic base excitation extended four the effect of equipment-structure to degree of study, one examine freedom the of freedom was considered transient response equipment-structure system subjected to seismic excitation. From the results of this can ascertain the validity of the conventional practice of non-interactive dynamic analysis and testing of equipment The effect of equipment-structure interaction depends on the parameters of the system of interest and the characteristic of the base excitation. In heavily damped equipment-structure systems, the ultimate response predictions. equipment non-interactive The approach non-interactive yields satisfactory approach is also . adequate for ultimate equipment response predictions when the equipment is very light in harmonic equipment comparision base with motion response its inputs, when the supporting the floor non-interactive driving frequency frequency of the subsystems and underestimate driving frequency is close in the housing results coincides structure. For overestimate the with the tuning the equipment response when the to the tuning frequency of the subsystems. The error in a non-interactive analysis is greatest when the driving frequency coincides with the tuning frequency of the subsystems. In the study with seismic 125 base motion 126 inputs, the non-interactive results are generally conservative. 2. An experimental study vibration equipment isolated of the seismic units was response conducted. handling unit mounted on vibration isolators The vibration isolators investigated characteristics of two A model of a was constructed different prototype for testing air purposes. in this study were of the elastomeric type and the open spring type with uni-directional restraint. The two isolator models were tested both statically and dynamically. The stiffness and inertia properties of the systems were obtained in the static testing program. The dynamic testing program yielded the damping and dynamic response characteristics systems. Finally, the failure modes of the of the vibration isolated vibration isolated equipment systems were obtained by subjecting the systems to simulated seismic conditions. The static and dynamic tests provided information on the nonlinear axial and lateral isolators. stiffnesses The characteristics, isolators and on the high damping characteristics elastomeric isolator, with its nonlinearity of these vibration and high damping survived the extreme simulated seismic conditions. The open spring with uni-directional restraint failed in a brittle manner under a substantially lower level of base excitation. 3. Based on vibration the system isolated identification equipment systems test results, were analytical formulated. A models of computer the two program, utilizing time step analysis and incorporating the nonlinearity of the systems, was developed to solve the equations of motion of the systems. The analytical results, such as the frequency response curves, mode shapes, and seismic responses, were compared with the experimental results to verify the analytical model. The predicted response values response using obtained the analytical model agreed experimentally. It can be concluded that well with the the analytical model is adequate for the seismic analysis of general vibration isolated equipment systems. 127 7.2 F U T U R E STUDIES The work presented vibration isolated in this thesis has shown that the current design rules for equipment systems may be inadequate to develop a rational seismic design method for and therefore there is a need vibration isolated equipment systems. In developing such a procedure the following items should be addressed: 1. The effect of equipment-structure interaction on the ultimate equipment response. 2. The nonlinearities and the damping characteristics of the vibration isolators. Studies should also be undertaken to examine the seismic response of vibration isolated equipment systems subjected can include the effect to oblique base of excitation. And shake equipment-structure interaction should table be tests which implemented in future experimental programs. Since there exists a large variety of commericial vibration isolators, it is not feasible isolators which through the are capable experimental response characteristics futher investigation. to qualify each type of vibration isolator; however, vibration of Such and of resisting analytical expected seismic approach. Finally, vibration isolated studies are equipment important in forces the effect without an order vibration isolated equipment systems from seismic hazards. to can be developed on the inertia adequately seismic base protect needs the BIBLIOGRAPHY 1) California Division of Mines and Geology Bulletin 198, 2) J. 1973. M . Biggs and J. M . Roesset, "Seismic Analysis of Equipment Mounted on a Massive Structure," in Seismic Design of Nuclear Power Plants, Ed. R. J. Hansen, pp 319-343, M.I.T. Press, 1970. 3) M . Amin, W. J. Hall, N. M . Newmark and R. P. 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Singh, " A Stochastic Model for Predicting Maximum Ressponse of Light Secondary Systems," thesis presented to the University of Illinois at Urbana, Illinois, in 1972, in partial fullfillment of the requirements for the degree of Doctor of Philosophy. Conference on of Light Earthquake 8) E. H . Vanmarke, " A Simplified Procedure for Predicting Amplitude Response Spectra and Equipmnent Response," Proc. Sixth World Conference on Earthquake Engineering, Vol. Ill, New Delhi, India, 1977. 9) R. H. Scanlan and 635-655, August 10) K. Sachs, "Earthquake Time Histories and Response Spectra," Journal of the Engineering Mechanics Division, ASCE, Vol. 100, No. EM4, PP 1975. S. H. Crandall and W. D. Mark, Random Academic Press, New York, N.Y., 1963. Vibration of Mechanical Systems, 11) A. Der Kiureghian and A. Asfura, " A New Floor Response Spectrum Method for Seismic Analysis of Multiply Supported Secondary Systems," Report No. UCB/EERC-84-04, Earthquake Engineering Research Center, University of California, Berkeley, California, June 1984. 12) J. Penzien and A. K. Chopra, "Earthquake Response of Appendage on a Multi-story Building," Proc. Third World Conference on Earthquake Engineering, New Zealand, 1965. 13) N. M . Newmark, "Earthquake Response Analysis of Reactor Engineering and Design, Vol. 20, No. 2, PP 303-322, 1972. 128 Structures," Nuclear 129 14) T. Nakahata, N. M . Newmark, and W. J, Hall, "Approximate Dynamic Response of Light Secondary Systems," Structural Research Series Report No. 396, Civil Engineering Studies, Unversity of Illinois, Urbana, Illinois, 1973. 15) J. L. Sackman and J. M . Kelly, "Rational Design Methods for Light Equipment in Structures Subjected to Ground Motion," Report No. UCB/EERC-78-19, Earthquake Engineering Research Center, University of California, Berkeley, California, August 1978. 16) A. R. Robinson and C. G . Ruzicka, "Dynamic Response of Tuned Secondary Systems," Report Number UILU-Eng-80-2020, Department of Civil Engineering, Unversity of Illinois, Urbana, Illinois, November 1980. 17) R. Villaverde, N. M . Newmark, "Seismic Response of Light Attachments to Buildings," Structural Research Series Report No. 469, Civil Engineering Studies, Unversity of Illinois, Urbana, Illinois, February 1980. 18) T. Igusa and A. Der . Kiureghian, "Dynamic Analysis of Multiply Tuned and Arbitary Supported Secondary Systems," Report No. UCB/EERC-83-07, Earthquake Engineering Research Center, University of California, Berkeley, California, June 1978. 19) A. D. Kiureghian, J. L. Sackman, and B. Nour Omid, "Dynamic analysis of Light Equipment in Structures: Reponse to Stochastic Input," Journal of the Engineering Mechanics Division, ASCE, Vol. 109, No. E M I , Feburary 1975. 20) W. D. Iwan, "Predicting the Earthquake Response of Resiliently Mounted Equipment with Motion Limiting Constraints," Proc. Sixth World Conference on Earthquake Engineering, New Delhi, India, 1977. 21) W. D. Iwan, "The Earthquake Systems," Journal of Earthquake Vol. 6, PP 523-534, 1978. Design and Analysis of Equipment Isolaton Engineering and Structural Dynamics, ASCE, 22) IEEE Standard 344-1975, "IEEE Recomanded Practice for Seismic Qualification of Class IE Equipment for Nuclear Power Generating Stations," New York, 1975. 23) C. W. de Silva, Dynamic Testing and Seismic Qualification Practice, Lexington Books, D. C. Heath and Company, Lexington, Massachusetts, Toronto, 1983. 24) G. L. McGavin, Earthquake Protection Wiley-Interscience, New York, 1981. of Essential Building Equipment, 25) J. C. Wilson, W. K. Tso, and A. C. Heidebrecht, "Seismic Qualification by Shake Table Testing," Proc. Third Canadian Conference on Earthquake Engineering, Montreal, June 1979. 26) J. H . Rainer, "Dynamic Testing of Civil Engineering Structures," Proc. Third Canadian Conference on Earthquake Engineering, PP 551-574, Montreal, June 1979. 27) N . N . Nielson, "Dynamic Response of Multistory Buildings," thesis presented to the California Institute of Technology, Pasadena, California, June 1964, in partial fullfillment of the requirements for the degree of Doctor of Philosophy. 130 S. Bendat and A. G . Piersol, Random Data: Analysis and Procedures, Wiley- Interscience, New York, 1971. 28) J. 29) J. 30) D. E. Newland An Introduction Longman, London, 1983. Measurement S. Bendat and A. G. Piersol, Engineering Application of Correlations and Spectral Analysis, Wiley-Interscience, New York, 1980. to Random Vibrations and Spectral Analysis, 31) G . M . Sabnis, H . G . Harris, R. N. White, and M . S. Mirzo, Structural Modelling and Experimental Techniques, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983. Dynamics of Structures, Prentice-Hall, Inc., 32) W. C. Hurty and M . F. Rubinstein, Englewood Cliffs, New Jersey, 1964. 33) R. W. Clough and York, 1975. 34) D. Gasparini and E H . Vanmarcke, "Evaluation of Seismic Safety of Buildings Report Number 2. Simulated Earthquake Motions Compatible with Prescribed Response Spectra," Research Report No. R76-4, Department of Civil Engineering, M.I.T., Cambridge, Massachusetts, January 1976. J. Penzien, Dynamics of Structures, McGraw Hill, Inc., New APPENDIX A Generation of Band Limited White Noise Record The theory of generation of band-limited white noise following discussion is quoted without reference band-limited white noise record spectral density over a specified is record presented in the from Gasparini and Vanmarcke [34]. A defined as a frequency range, cjj process having a uniform power to CJ2, as shown in Fig. A . l . By this definition, it is clear that such a process contains frequency components of equal intensity over the specified frequency range (based on squared amplitudes as a measure of intensity or power). Random band-limited records with white noise specified record, can power be spectral generated density functions, such A.sin(u> .t+0.) 1 1 a by the method of summation of sinusoids. Any periodic function can be expanded into a series of N sinusoidal N x(t)=Z i=l as waves: (A.l) 1 where Aj= the amplitude of the i ^ sinusoid. cjj= the frequency of the i ^ sinusoid. 0j= the phase angle of the i ^ 1 sinusoid. Based on the assumption of squared power, the steady state motion x(t) amplitudes as a measure of intensity or N contains a total power of Z (A2./2). When the i = l chosen number of sinusoids is large, the total under the power spectral 1 power of x(t) is equivalent to the area density function G(co) shown in Fig. A.2. A sinusoid with a 131 132 frequency OK and a frequency interval ACJ = C J . + ^- O K , contains is approximately equivalent to the area under the power spectral A 3 ./2 as power. This density function at C J . with a frequency interval of Aw so that (A J ./2) = G(o) .)Aw. Knowing a target power spectral density amplitudes A^ can be chosen to simulate content By keeping the array of generated a phase angles of amplitudes </» j. A random by randomly choosing an array computer random number generator of sinusoid a record which contains the target frequency A . fixed, different aree similar in frequency content but different arrays function G(a>), an array motion records which in form can be generated band-limited white noise from different record can be of phase angles <p.. The present study uses routine to obtain a random array of phase angles <j> ^ which has a uniform distribution between 0 and 27r. The band-limited density between the white noise interval 0 to record of interest 25 Hz. Five has a constant hundred sinusoids power with a spectral frequency- interval of 0.5 Hz were used to obtain the target band-limited white noise record. Figure A.1 Power spectral density funciion of a band-limited white noise signal 1 Figure A.2 Power spectral density function. APPENDIX B Spectral Analysis The the subject theory of spectral analysis is well documented and detailed discussions on can be found in Bendat and Piersol [28, following discussion is quoted without reference from 29] and Newland [30]. The the above mentioned authors. In working with the response characteristics of an ideal physical system, it is necessary to develop an approximate input/output relationship for the system. An ideal physical system process exhibits the following characteristics: 1. It is physically realizable which means that the system cannot respond to an input until the input has been applied to the system. 2. It is stable which means that a bounded input produces a bounded response. 3. It is linear which implies that the principle of superposition is valid such that the system is additive and homogeneous. 4. It has constant parameters which means that the physical characteristics of the system are time invariant The response of an ideal physical system which contains m degrees of freedom subjected to a single input x(t) can be described mathematically by a linear differential equation of the following form: a .(d n y./dt n ) + .„. + a n .y. = b ( d r x / d t r ) + „ . . + b n x ni l Or l r 0 where the coefficients a and b are the constants m (B.l) for a time invariant system and denotes the response of the i 1 * 1 degree of freedom of the system. 134 i=l 135 For solved a given excitation completely to yield However, simple available. Therefore, x(t) the experimental it is more and given initial response y. techniques if for convenient to conditions, equation the coefficients determining represent a these the and (B.l) can be b were known. coefficients relationship are not between the input and output signals by an alternate frequency response approach. For any arbitrary input x(t), the system output >'j(t) is given, in the time domain, by the superposition or convolution intergal yj(t)=J h.(T)x(t-r) dr where h.(r) = y.(r) i = l,....m (B.2) when x(t)=6(t) = unit impulse excitation. x(t): Figure B.1 Taking T Single input / multiple output system. a finite Fourier transform of equation (B.2) over a long record length yields the following results in the frequency domain: Y.(f,T)=H.(0X(f,T) i = l,...,m (B.3) 136 where Y The i(f-T)=J yi(i-)e_j(27rfT) d T X(f,T)=J x(r)e-j(2,rfT) dr Hi(0=J h.(r)e-j(27rfT) dr transformed functions in the frequency domain are complex such that each function contains a real part and an imaginary part It follows that Y*(f,T) = H*(f)X*(f.T) i i |Y.(f,T)| J = |H i (f)|'|X(f.T)|' (B.4) X *(f,T)Y .(f,T)=H j(f)|X(f,T)|2 where * denotes the complex conjugate. In terms of the one-sided spectral auto-spectra density functions, the cross-spectra and the are defined as G y v ( f ) = lim (2/T)X*(f,T)Y.(r.T) } i T-*-» G (f) =lim (2/T)|X(f,T)| 2 Subsututing equation (B.4) into (B.5) equation (B.5) yields the following spectral density function relations: G x y G Knowing i (f)=H(0G 1 (0=|H.(0| 2 G y. y .v ' 3 v1 the auto-spectral (f) 1 i W | recorded density (B.6> x x (f) xxv ' input and functions output can be signals, determined the input/output by the cross-spectral techniques of and fast Fourier 137 transform. And the system transfer H.(f) = i G xxy. v ( O / G m xx |H.(0|2 = G ,v I function H.(f) can be obtained as (0/G V.V. >\ ( B 7 ) (0 XX ' V " 1" 1 The and theory of discrete Fourier transform not further it will from the be sampling presented theorems and is well in this study. smoothing documented in the literature However, some important results techniques used in this study will be presented. The most important aspect of the sampling theorems lies in the choice of the sampling rate. Suppose a band-limited signal extending from t = - ° ° at a constant samples per to t = ° ° is sampled rate, one sample per interval A t At is chosen such that at least two cycle occur in the highest frequency component of the band-limited function. Defining the Nyquist frequency as the frequency which corresponds to l/2At, if the band-limited signal is sampled at a rate lower than twice the Nyquist, it will cause aliasing problems. Aliasing means fold back toward the lower frequencies, of the signal. sample at expressions an If the information at even becomes higher rate that higher therefore frequencies identically zero at the the in the signal distorting the true frequency content the Nyquist frequency because present sine Nyquist is important, one function frequency. in the has to mathematical Therefore, information cannot be transmitted unless a higher sampling rate is chosen. Smoothing techniques are needed to reduce the errors which occur in the use of the discrete Fourier transform of the recorded signal. Ideally, the recorded signal is a continuous infinite time series; however, realistically one can only obtain a discrete finite time series in an experiment. One can regard the recorded signal as the product of 0,0,0 a continuous 1,1,1 infinite 1,1,1,...,0,0,0. time series Performing a and a discrete rectangular function Fourier transform of on such the form a signal 138 will yield erroneous kernel function does function contributes resulting spectrum, techniques are results. not The real spectrum focus improperly causing properly to the leakage introduced. A cosine and may be distorted because the Dirichlet the estimation problems. taper secondary of To window is the maxima Fourier reduce such used in place of the kernal coefficient errors, in the smoothing of a rectangular window to increase the focusing power of the main lobe of the kernel and to reduce leakage. APPENDIX C Program listing 139 140 L i s t i n g of EVIS.S 1 o 4. 3 4 5 a 7 8 9 10 11 12 13 14 15 16 17 1o 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c M A I N P R O G R A M E V I S . S C C c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc \* c c c c c c c c c c c T H I S I S A TIME STEP PROGRAM WHICH CALCULATES THE RESPONSE OF A NH STORY STRUCTURE AND/OR A 3 DOF EQUIPMENT SYSTEM UNDER A BASE E X C I T A T I O N . IT ASSUMES EITHER A LINEAR VARIATION IN A C C E L E R A T I O N OR AN AVERAGE ACCELERATION BETWEEN A TIME S T E P . AND I T USES AN I T E R A T I O N AND EOUILIBRUIM CORRECTION PROCEDURE TO REDUCE THE ERROR IN RESPONSE INTRODUCED BY THE ABRUPT CHANGE IN SPRING S T I F F N E S S . THE MAXIMUM NUMBER OF STORIES IS 10. THE TOTAL DOF OF THE SYSTEM IS NB=NH+2. THE TIME S T E P . D E T A T , IS CHOSEN SUCH THAT XXX=BTI/DETAT IS A WHOLE NUMBER. BTI I S THE TIME INCREMENT OF THE BASE MOTION. ( B T I = 0 . 0 2 S E C ) INPUT BASE ACCELERATION IN MM/SEC/SEC. c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c DEFINE ALL NB NC NBB AF AFF SK SKH1R SKH1L SKV1R SKV1L SKH2 SKV2T SKV2C AM ZETA A B BL BR c = = = = » = = = = D1 D2 E DETAT = DURA = STIFF = SSK ESK DC SDC EDC AMASS = SMAS = EMAS = AMASIN= DETAF = VARIABLES THE TOTAL NUMBER OF DOF OF THE SYSTEM (MAX NB=13) NB- 1 NB-2 A C C E L . RECORD OF THE GROUND AT 0 . 0 2 SEC INCREMENT MM/SEC/SEC A C C E L . RECORD OF THE GROUND M/SEC/SEC VECTOR OF S T I F F N E S S E S OF SYSTEM (NB) N/M HORIZONTAL S T I F F N E S S OF RIGHT SPRING OF EOUIPIMENT N/M HORIZONTAL S T I F F N E S S OF LEFT SPRING OF EOUIPIMENT N/M V E R T I C A L S T I F F N E S S OF RIGHT SPRING OF EOUIPIMENT N/M V E R T I C A L S T I F F N E S S OF L E F T SPRING OF EOUIPIMENT N/M HORIZONTAL SNUBBER S T I F F N E S S OF EQUIPMENT N/M V E R T I C A L SNUBBER S T I F F N E S S OF EQUIPMENT AGAINST L I F T N/M V E R T I C A L SNUBBER S T I F F N E S S OF EQUIPMENT AGAINST DROP N/M VECTOR OF MASSES AND MOMENT OF INERTIA OF SYSTEM ( N B ) KG AND KG*M*M R E S P E C T I V E L Y VECTOR OF PERCENTAGE OF C R I T I C A L DAMPING (NB) V E R T I C A L DISTANCE BETWEEN CENTER OF MASS AND SPRINGS M HORIZONTAL DISTANCE BETWEEN SPRINGS M HORIZONTAL DISTANCE BETWEEN C . G . AND LEFT SPRING M HORIZONTAL DISTANCE BETWEEN C . G . AND RIGHT SPRING M GAP BETWEEN SPRING AND HORIZONTAL SNUBBER M GAP BETWEEN SPRING AND VERTICAL SNUBBER AGAINST L I F T M GAP BETWEEN SPRING AND VERTICAL SNUBBER AGAINST DROP M PERCENTAGE E C C E N T R I C I T Y BETWEEN C . G . AND SPRINGS LOCATION TIME STEP INCREMWNT SEC DURATION OF GROUND MOTION SEC S T I F F N E S S MATRIX OF THE SYSTEM (NB.NB ) S T I F F N E S S MATRIX OF THE STRUCTURE (NBO.NBD) S T I F F N E S S MATRIX OF THE EQUIPMENT ( 3 . 3 ) DAMPING MATRIX OF THE SYSTEM ( N B . N B ) DAMPING MATRIX OF THE STRUCTURE (NBD,NBD ) DAMPING MATRIX OF THE EQUIPMENT ( 3 . 3 ) DIAGONAL MASS MATRIX OF THE SYSTEM ( N B . N B ) DIAGONAL MASS MATRIX OF THE STRUCTURE (NBD.NBD) DIAGONAL MASS MATRIX OF THE EQUIPMENT ( 3 . 3 ) INVERSE OF THE MASS MATRIX ( N B . N B ) FORCE INCREMENT VECTOR ( N B ) 141 C C PSEUF0= PSEUD0 FORCE 60 PSEUST= PSEUDO STIFFNESS 61 C DETAX = DISPLACEMENT 62 C C DETAV = VELOCITY DETAA = ACCELERATION C FI SPFH 1 14 C C C C c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c MISCEL 1 19 1 16 c IFL.I 59 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 1O0 101 102 103 104 105 106 107 108 109 1 10 1 11 112 113 INERTIA = FD ACTUAL INCREMENT INCREMENT INCREMENT VECTOR HORIZONTAL FORCE (NB) (NB.NB) VECTOR VECTOR INCREMENT FORCE DAMPING VECTOR MATRIX (NB) (NB) VECTOR (NB) (NB) SPRING VECTOR FORCE OF FA FORCE FU UNBALANCED X RELATIVE DISPLACMENT V RELATIVE VELOCITY AC RELATIVE ACCELATION VECTOR (NB) ABC ABSOLUTE ACCELATION VECTOR (NB) VECTOR EQUIPMENT SYSTEM (NB) (NB) FORCE VECTOR (NB) VECTOR VECTOR (NB) (NB) VERTL = VERTICAL DISPLACEMENT OF LEFT VERTR = VERTICAL DISPLACEMENT OF RIGHT SPRING (COMPRESSION SPFVR = VERTICAL SPRING FORCE OF RIGHT SPRING (COMPRESSION SPFVL = VERTICAL SPRING FORCE OF LEFT XC HORIZONTAL DISPLACEMENT AT SPRING SPRING = MAX. RELATIVE DISPL. VMAX = MAX. RELATIVE VEL. ACMAX = MAX. RELATIVE ACCEL. VECTOR (NB) MAX. ABSOLUTE ACCEL. VECTOR (NB) ABCMAX= FUMAX VECTOR MAX. UNBALANCED FORCE MAX. HORIZONTAL SPRING SPMXVL= MAX. VERTICAL LEFT SPMXVR= MAX. VERTICAL RIGHT XCMAX = MAX. HORIZONTAL VLMAX = MAX. VERTICAL DISPLACMENT OF LEFT VRMAX = MAX. VERTICAL DISPLACMENT OF RIGHT MAX. ERROR SPRING IN VECTOR (NB) FORCE SPRING OF EQUIPMENT FORCE SPRING FORCE DISPLACMENT FORCE OF OF OF TO SYSTEM EQUIPMENT EQUIPMENT DUE SYSTEM EQUIPMENT AT SYSTEM SPRING SPRING ABRUPT CHANGE STIFFNESS. ACF ABSOLUTE VECTOR WHICH CONTAINS THE LOWER HALF BAND OF STIFF DB VECTOR WHICH CONTAINS THE LOWER HALF BAND OF AMASS DE VECTOR WHICH CONTAINS THE TAC VECTOR OF VECTOR VECTOR = LEVEL SPRING DA TABC TX +VE) +VE) (NB) = IN +VE) (NB) SPMAXH= DFSMAX= (COMPRESSION +VE) SPRING XMAX VECTOR (COMPRESSION ACCELATION RECORD DF THE SYSTEM EIGENVALUES OF TIME TO MAXIMUM RELATIVE OF TIME TO MAXIMUM ABSOLUTE ACCEL. OF TIME TO MAXIMUM RELATIVE DISPL. VEL. TV VECTOR OF TIME TO MAXIMUM RELATIVE TFU VECTOR OF TIME TO MAXIMUM UNBALANCED TSP TIME TO MAXIMUM HORIZONTAL SPRING TSPVR = TIME TO MAXIMUM VERTICAL RIGHT TSPVL TDF = TIME TO MAXIMUM VERTICAL LEFT TIME TO MAX. TIME TO MAXIMUM ERROR HORIZONTAL IN TVL TIME TO MAXIMUM VERTICAL LEFT TVR TIME TO MAXIMUM VERTICAL RIGHT DV MATRIX WHICH OF CONTAINS THE SPRING FORCE FORCE FORCE SPRING DISPLACMENT SPRING DISPLACMENT SPRING EIGENVECTOR SYSTEM = VECTOR ALPHA = MULTIPLICATION FACTOR DEPENDING ON THE THE TIME STEP « METHOD CHOSEN. MULTIPLICATION FACTOR DEPENDING ON THE THE TIME STEP METHOD FLAGS PARTICIPATION DISPLACMENT OF GAMMA BETA MODAL SPRING FORCE. FORCE SPRING TXC SYSTEM ACCEL. FACTORS CHOSEN. AND CHECKS FA, I A , I Q , J Q . K Q . I E R , I : FLAG.IFLAA.IFLAAA,CHECK,ICHECK.11CHECK 142 1 17 1 18 1 19 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 C IMPLICIT REAL*8(A-H.0-Z) DIMENSION STIFF(13.13),AMASS(13.13).AMASIN(13.13),X(13).V(13) DIMENSION ESK(3.3).EMAS(3.3),SMAS(10.10),SDA(100),EDA(9) DIMENSION S D B ( 5 5 ) . E D B ( 6 ) . S D E ( 1 0 ) . E D E ( 3 ) . S D V ( 10. 1 0 ) . E D V I 3 . 3 ) OIMENSION D E T A F ( 1 3 ) ,PSEUFO(13) , P S E U S T ( 1 3 , 1 3 ) , D E T A X ( 1 3 ) DIMENSION F I ( 1 3 ) . F S ( 1 3 ) , F A ( 1 3 ) . A C ( 1 3 ) ,DETAVl 1 3 ) , A C F ( 9 G 0 1 2 ) DIMENSION D B ( 9 1 ) , D A ( 1 6 9 ) . D E ( 1 3 ) , A F F ( 6 0 0 0 0 ) , A F ( 7 6 0 0 ) DIMENSION D C ( 1 3 . 13),FD< 13),DV< 13. 13) .IQ( 18 ) .IA< 3) ,dQ< 18 ) ,KQ(18) DIMENSION XMAX(13).VMAX(13),ACMAX(13),ABC(13),ABCMAX( 13) DIMENSION T A C ( 1 3 ) , T A B C ( 1 3 ) . T X ( 13 ) . T V ( 1 3 ) , D E T X ( 1 3 ) . S D C < 1 0 , 1 0 ) DIMENSION A M ( 1 3 ) , S K ( 1 3 ) , Z E T A ( 13).IPERM(24 ) , T ( 1 3 , 1 3 ) ,EDC(3,3) DIMENSION E Z E T A ( 3 , 3 ) , S Z E T A ( 1 0 . 1 0 ) . Y A C L ( 1 0 0 O 0 ) . Y A C R ( 1 0 0 0 0 ) DIMENSION DETAA(13).FU(13),FUMAX(13),TFU(13) INTEGER S T A R ( 1 ) / ' * ' / C C READ IN PROGRAM OPTIONS C WRITE(5,199) READ(5,STAR)MET WRITE(5,200) READ(5,STAR ) NB WRITE(5,201) READ(5,STAR ) I F L I F ( I F L . N E . O ) IFA = 7 I F ( I F L . N E . O ) GOTO 40 WRITE(5,202) READ(5.STAR)IFA 40 W R I T E ( 5 , 2 0 3 ) READ(5,STAR ) ICHECK I F ( I C H E C K . E O . O ) GOTO 25 WRITE(5,204) READ(5,STAR ) I I C H E C C C C C c c c C A L L SUBROUTINE EQUIPMENT DATA TO READ THE INPUT STRUCTURAL AND 25 C A L L READ(SK.ZETA,AM,AFF,DURA.N.SKH2,A,BL,BR,C,NB,DETAT,IFL.IT, 1T0L,IFA,SKH1L.SKH1R,SKV1L.SKV1R.SKV2T.SKV2C.D 1 ,D2,EMAS.SMAS 1,EZETA,SZETA,B,E,FMU) SET DO INITIAL CONDINTIONS 100 1=1,NB X ( I ) =0.DO V(I)=0.D0 AC(I)=0.D0 A B C ( I )=0.D0 ACMAX(I)=O.DO ABCMAX(I)=0.D0 XMAX(I)=0.D0 DETAX(I)=0.DO DETAA(I)=0.D0 DETAV(I)=0.DO FU(I)=0.DO FUMAX(I)=0.00 TAC(I)=0.00 TABC(I)=0.DO 143 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 • 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 TX(I)=0.D0 TFU(I)=O.DO TV(I)=O.DO VMAX(I)=O.DO 100 CONTINUE DO 101 1=1.18 IO(I)=0 J0(I)=O K0(I)=O 101 CONTINUE I FLAA=0 XCMAX=0.D0 SPMAXH'O.DO DFSMAX=O.DO SPMXVL=O.DO SPMXVR=O.DO VRMAX=O.DO VLMAX=0.D0 VBL=O.DO VBR=0.D0 TSPVL=0.D0 TSPVR=O.DO TVL=O.DO TVR=0,. DO TDF=o\DO TSP=0.D0 TXC=0.D0 ICOUNT=0 DT1=0.D0 NC=NB-1 NBB=NB-2 NBD=NB-3 NBV=NB BT=DETAT DETVL=0.D0 DETVR=0.D0 I F ( I F A . E Q . I ) NBV=NB-3 I F ( I F A . E O . O ) NBV = 3 I F ( M E T . E O . O ) BETA=1.DO/6.DO I F ( M E T . E O . O ) ALPHA=0.5D0 I F ( M E T . E Q . I ) BETA=0.25DO I F ( M E T . E O . I ) ALPHA=0.5D0 ALBE *ALPHA/BETA ALBR=1.DO-(0.5D0*ALBE) C C C SET C C C SET UP MASS MATRIX C C TIME STEP P INTERVAL < (SMALLEST XX=DURA/DETAT+1.5DO L=IDINT(XX) WRITE(6.205)L WRITE(6,206) TIME=-DETAT CALL M A S S ( A M , A M A S S . A M A S I N , N B V , I F L ) BEGIN TIME STEP NATURAL PERIOD /10) 144 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 C DO 30 I - 1 . L IERR=0 IER=0 NUM=0 11=1+1 JE=1 IFLAG=0 IFLAAA=0 IDIREC=0 IXX=0 AS=(AFF(II)-AFF(I))/DETAT CALL D R I V E ( X . S K . A M A S S . N B . I F L . A , B L , B R , C , S K H 2 . D E T A T . AM , A F F , I . N . N B B . N C . I T . V . A C , I ERR,ABC.ZETA,DETAX.TIME,NBV . JE,IFA.ICOUNT.XCC,IER,AMASIN,AS.IFLAG.XC.DX1.DX2.T0L,BT .DETAFS.IFLAA,IXX,ID IREC.ALPHA.BETA.ALBE,ALBR.SDC.EDC ,VERTL.VERTR,SKH1L.SKH1R,SKV1L.SKV1R.SKV2T.SKV2C,D1,D2 ,VBL,VBR.SPFH.SPFVR.SPFVL.IA.10.JO.KQ.IFLAAA,EMAS.SMAS .EZETA.SZETA,NBD,E,B,YAL,YAR,FU.DETVL,DETVR,FMU) C C C PERFORM ITERATION PROCEDURE TO MINIMIZE SPRING FORCE 10 12 C c c c c c c I F ( I E R . N E . 1 ) GOTO 310 DT1=DETAT IFLAG= 1 DT1=DT1/DX2*DX1 IFLAA= 1 CALL D R I V E ( X , S K , A M A S S , N B . I F L . A , B L , B R . C , S K H 2 . D T 1 , A M , A F F , I . N . N B B . N C , I T , V . A C . I ERR.ABC,ZETA.DETAX.TIME.NBV ,JE,IFA,ICOUNT.XCC,IER.AMASIN.AS.I FLAG,XC.DX1.DX2.T0L.BT .DETAFS.IFLAA.IXX,IDIREC.ALPHA.BETA.ALBE.ALBR.SDC,EDC .VERTL.VERTR.SKH1L,SKH1R.SKV1L,SKV1R,SKV2T,SKV2C.D1.D2 .VBL,VBR.SPFH.SPFVR.SPFVL.IA.10.JQ.KO.IFLAAA,EMAS.SMAS .EZETA.SZETA,NBD,E.B.YAL.YAR.FU.DETVL.DETVR.FMU ) I F ( I F L A G . E 0 . 2 ) GOTO 177 NUM=NUM+1 I F ( N U M . G E . 1 0 0 ) GOTO 12 GOTO 10 WRITE(6.207)TIME GOTO 4 0 0 COMPLETE THE TIME STEP 177 AFTER THE SUCCESSFUL ITERATION. DT2=DETAT-DT1 IFLAA= 1 C A L L D R I V E ( X , S K , A M A S S , N B , I F L . A . B L . B R , C , S K H 2 . D E T A T . AM , A F F , I . N . N B B . N C , I T , V . A C , I ERR.ABC.ZETA,DETAX.TIME,NBV , J E , I FA.ICOUNT.XCC.IER,AMASIN,AS.I FLAG,XC,DX1.DX2.T0L.BT .DETAFS,IFLAA,IXX,IDIREC.ALPHA.BETA.ALBE.ALBR.SDC,EDC ,VERTL,VERTR.SKH1L,SKH1R,SKV1L.SKV1R.SKV2T,SKV2C,D1,D2 ,VBL,VBR.SPFH.SPFVR.SPFVL.IA.10,JO.KQ.IFLAAA,EMAS.SMAS . E Z E T A . S Z E T A , N B D , E . B . Y A L , Y A R . F U . D E T V L . D E T V R , FMU) FIND MAX. X, V. A C , AND DETAFS AT THEIR 310 ERROR. TIME=TIME+DETAT DO 37 K=1,NBV ABC, X C . SPFH SPFVR. S P F V L , VERTR, R E S P E C T I V E TIME OF OCCURENCE VERTL 145 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 37 C C C C C C C STORE ABSOLUTE ACCEL. ID=0 I F ( I C H E C K . E O . O ) GOTO DO 78 IP=1,NBV ACF(ID+I)=ABC(IP) ID=ID+8001 78 CONTINUE Y A C L ( I ) = Y A L * 1000.DO YACR(I)=YAR*1000.DO 30 CONTINUE RECORD OF EACH FLOOR 169 173 170 IN ACF 30 OUTPUT THE A C C E L . RECORD OF EACH DOF. AND THE MAXIMUM VALUES IN UNIT 6 . 172 C C I F ( D A B S ( A C ( K ) ) . G T . D A B S ( A C M A X ( K ) ) ) TAC(K)=TIME IF(DABS(ABC(K) ) .GT.DABS(ABCMAX(K))) TABC(K)=TIME I F ( D A B S ( V ( K ) ) . G T . D A B S ( V M A X ( K ) ) ) TV(K)=TIME I F ( D A B S ( X ( K ) ) . G T . D A B S ( X M A X ( K ) ) ) T X ( K ) = TIME I F ( D A B S ( F U ( K ) ) . G T . D A B S ( F U M A X ( K ) ) ) TFU(K)=TIME I F ( D A B S ( X ( K ) ) . G T . D A B S ( X M A X ( K ) ) ) XMAX(K)=X(K) I F ( D A B S ( V ( K ) ) . G T . D A B S ( V M A X ( K ) ) ) VMAX(K) = V ( K ) I F ( D A B S ( A C ( K ) ) . G T . D A B S ( A C M A X ( K ) ) ) ACMAX(K )=AC(K ) IF(DABS(ABC(K)).GT.DABS(ABCMAX(K))) ABCMAX(K)=ABC(K) I F ( D A B S ( F U ( K ) ) . G T . D A B S ( F U M A X ( K ) ) ) FUMAX(K)=FU(K ) CONTINUE I F ( D A B S C X C ) . G T . D A B S ( X C M A X ) ) TXC = TIME I F ( D A B S ( V E R T R ) . G T . D A B S ( V R M A X ) ) TVR = TIME I F ( D A B S ( V E R T L ) . G T . D A B S ( V L M A X ) ) TVL = TIME I F ( D A B S ( S P F H ) . G T . D A B S ( S P M A X H ) ) TSP = TIME IF(DABS(SPFVL).GT.DABS(SPMXVL)) TSPVL=TIME IF(DABS(SPFVR).GT.DABS(SPMXVR)) TSPVR=TIME IF(DABS(DETAFS).GT.DABS(DFSMAX)) TDF=TIME I F ( D A B S ( X C ) . G T . D A B S ( X C M A X ) ) XCMAX = XC I F ( D A B S ( S P F V L ) . G T . D A B S ( S P M X V L ) ) SPMXVL = SPFVL I F ( D A B S ( S P F V R ) . G T . D A B S ( S P M X V R ) ) SPMXVR = SPFVR I F ( D A B S ( S P F H ) . G T . D A B S ( S P M A X H ) ) SPMAXH=SPFH I F ( D A B S ( V E R T R ) . G T . D A B S ( V R M A X ) ) VRMAX = VERTR I F ( D A B S ( V E R T L ) . G T . D A B S ( V L M A X ) ) VLMAX = VERTL DFSMAX=DMAX1(DFSMAX.DABS(DETAFS)) I F ( I C H E C K . E O . O ) GOTO 197 ID=0 DO 170 I Z M . N B V I F ( I I C H E C . E O . O ) GOTO 172 I F ( I I C H E C . N E . I Z ) GOTO 173 IDD=ID+1 IDL=ID+L WRITE(8,208)L,DETAT DO 169 I ' I D D . I D L ACF(I)=ACF(I)*1000.D0 CONTINUE WRITE(8,209)(ACF(I).I=IDD.IDL) ID=ID+8001 CONTINUE I F ( I I C H E C . N E . O ) GOTO 197 WRITE(10.208)L.DETAT WRITE(10.209)(YACR(I).I=1.L) (MM/SEC/SEC) IN UNIT 146 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 C C C WRITE(11,208)L,DETAT WRITE(11,209)(YACL(I),I=1.L) 197 W R I T E ( 6 . 2 1 0 ) WRITE(6.21 1 ) WRITE(6.212) WRITE(6.213)(XMAX(K),K=1,NBV ) WRITE(6.214)(TX(K).K=1.NBV) WRITE!6.215)(VMAX(K).K=1.NBV) WRITE(6.214)(TV(K),K=1,NBV ) W R I T E ( 6 , 2 1 6 ) ( A C M A X ( K ) .K=1 . N B V ) WRITE(6.214)(TAC(K),K=1.NBV) WRITE(6,217)(ABCMAX(K),K=1,NBV) WRITE(6.214)(TABC(K),K=1,NBV) WRITE(6.218)(FUMAX(K),K=1,NBV) WRITE(6.214)(TFU(K),K=1,NBV) WRITE(6.219)SPMAXH,TSP WRITE(6.220)XCMAX,TXC WRITE(6.221)IC0UNT WRITE(6.222)DFSMAX,TDF WRITE(6.223)SPMXVL,TSPVL WRITE(6.224)VLMAX.TVL WRITE(6,225)SPMXVR,TSPVR WRITE(6,226)VRMAX,TVR GOTO 4 0 0 26 W R I T E ( 5 . 2 2 7 ) WRITE(6,227) C 199 FORMAT('WHICH TIME INTEGRATION METHOD DO YOU W A N T ? ' , 1 ' ( 0 = L I N E A R ACCEL . , 1=AVERAGE A C C E L . ) ' ) 2 0 0 FORMAT('HOW MANY DOF ARE THERE IN THE SYSTEM? (MAX N B = 1 3 ) ' ) 201 F O R M A T ( ' T O WHICH FLOOR IS THE EOUIPMENT A T T A C H E D ? ' , 1 '(0=NOT A T T A C H E D . 1 = F I R S T , 2 = SEC0ND )') 202 FORMAT('WHICH RESPONSE DO YOU WANT?(0=EOUIPMENT.1^STRUCTURE)') 203 FORMAT('DO YOU WANT THE A C C E L . RECORD?(YES=1,N0 = 0 ) ' ) 204 FORMAT('WHICH DOF RESPONSE RECORD DO YOU W A N T ? ' , 1 ' (0=ALL . 1 = F I R S T . 2 = SEC0ND ...)') 205 FORMAT(I 10) 206 F O R M A T ( / , '******** RESULTS * * • * * * * • ' ) 207 F O R M A T ( ' I T E R A T I O N F A I L E D @ TIME = ' . F 1 5 . 7 ) 208 F O R M A T ( I 1 0 . F 1 5 . 6 ) 209 F O R M A T ( 8 F 1 0 . 0 ) 210 F O R M A T ( / / , ' U N I T S OF MAXIMUM DISPLACEMENTS = M OR R A D ' ) 211 F O R M A T ( ' U N I T S OF MAXIMUM V E L O C I T I E S = M/SEC OR R A D / S E C ) 212 F O R M A T ( ' U N I T S OF MAXIMUM ACCELERATIONS = M/SEC/SEC OR ' . 1 'RAD/SEC/SEC') 213 F O R M A T ( / / , 'MAXIMUM R E L . D I S P L . ' . 5 F 1 6 . 5 ) 214 FORMAT( ' «» R E S P . TIME = '.5F16.4) 215 FORMAT(/,'MAXIMUM R E L . V E L . '.5F16.5) 216 FORMAT(/,'MAXIMUM R E L . ACCEL.',5F16.5) 217 F O R M A T ( / , 'MAXIMUM A B S . ACCEL . ' . 5 F 1 6 . 5 ) 2 18 FORMAT( / , 'MAX. UNBALANCED FORCE' , 5 F 1 6 . 5 ) 219 F O R M A T ( / , ' M A X HORIZONTAL SPRING FORCE = ' . F 1 5 . 5 . 1 ' P TIME = ' . F 1 0 . 5 ) 220 F O R M A T ( / , 'MAX HORIZONTAL SPRING D I S P L = ' . F 1 5 . 5 . 1 ' (S TIME = ' . F 1 0 . 5 ) 221 F O R M A T ( / , ' T H E H OF TIMES THAT THE E Q U I P . H I T S HORIZONTAL SNUBBER 1 ,110) 147 407 408 409 410 411 412 413 414 415 416 417 4 18 4 19 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 222 F O R M A T ( / , ' M A X SPRING FORCE ERROR = ' . F 1 5 . 5 , ' © TIME = ' . F 1 0 . 5 ) 223 F O R M A T ( / , ' M A X V E R T I C A L LEFT SPRING FORCE = ' . F 1 5 . 5 . 1 ' & TIME = ' . F 1 0 . 5 ) 224 F O R M A T ( / , ' M A X V E R T I C A L L E F T SPRING D I S P L = ' . F 1 5 . 5 , 1 * TIME = ' . F 1 0 . 5 ) 225 F O R M A T ( / , ' M A X V E R T I C A L RIGHT SPRING FORCE = ' . F 1 5 . 5 , 1 9 TIME = ' . F 1 0 . 5 ) 226 F O R M A T ( / , ' M A X V E R T I C A L RIGHT SPRING D I S P L = ' . F 1 5 . 5 . 1 0 TIME = ' , F 1 0 . 5 ) 227 F O R M A T ( ' S O L U T I O N F A I L E D ) 4 0 0 STOP END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C C S U B R O U T I N E R E A D C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C T H I S SUBROUTINE READS FROM UNIT 4 THE STRUCTURAL DATA FLOOR C BY FLOOR AND THEN THE EOUIPMENT DATA. IT ALSO READS THE C GROUND MOTION DATA FROM UNIT 7. THE GROUND ACCELERATION TIME C INCREMENT IS B T I . C SUBROUTINE R E A D ( S K , Z E T A , A M , A F F , D U R A , N , S K H 2 , A . B L , B R , C . N B . D E T A T , 1IFL,IT.TOL,IFA.SKH1L,SKH1R,SKV1L,SKV1R,SKV2T.SKV2C,01,D2,EMAS, 1SMAS,EZETA,SZETA,B,E,FMU) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION T I T L E ( 8 0 ) DIMENSION A F ( 7 6 0 0 ) , A F F ( 6 0 0 0 0 ) , A M ( 13 ) . S K ( 1 3 ) , Z E T A ( 1 3 ) DIMENSION EMAS ( 3 , 3 ) , SMAS ( 10, 10 ) . E ZET A ( 3 . 3 ) ,-SZET A ( 10, 10) SI=1000.DO NBB=NB-2 NBD=NB-3 NC=NB-1 DO 8 1=1.NBD READ(4,20)SK(I),ZETA(I),AM(I) 8 CONTINUE R E A D ( 4 , 2 0 ) S K H 1 L , S K H 1 R , Z E T A ( N B B ) , A M ( NBB) READ(4,20)SKV1L,SKV1R,ZETA(NC).AM(NC) READ!4,20)SKH2,SKV2T,SKV2C,ZETA(NB),AM(NB) READ(4,20)A,B,C.D1,D2.DETAT.E READ(4,21)T0L.FMU READ(7,22)TITLE READ(7,23)N,BTI READ(7,24)(AF(L+1),L=1,N) C C CALCULATE LOCATION OF C . G . KNOWING THE ECCENTRICITY C BL=B/2.D0-E*B/100.D0 BR=B/2.D0+E*B/100.D0 C C ECHO INPUT STRUCTURAL AND EOUIPMENT DATA C 10=5 7 WRITE(I0.22)TITLE I F ( I F L . N E . O ) WRITE(10,25)IFL IF(IFA.EQ.O) WRITEU0.26) IF(IFA.EO.I) WRITEU0.27) 148 465 466 467 468 469 470 47 1 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 51 1 512 513 514 515 516 517 518 519 520 521 522 WRITE(10.28) DO 10 1 = 1 , N B D W R I T E ( 1 0 , 2 9 ) 1 , 5 K ( I ) , Z E T A ( I ) ,AMI I ) 10 CONTINUE IF(I0.E0.5)WRITE(5,51) IF(I0.EQ.5)READ!5.52)NXX I F ( ( I 0 . E 0 . 5 ) . A N D . ( N X X . E Q . 1) ) STOP WRITE(10,30) WRITE(I0,31 ) WRITE(10.32)AM(NB).AM(NC) WRITE(I0.33)ZETA(NBB).ZETA(NO.ZETA(NB) W R I T E ( 1 0 . 3 4 )SKH1L WRITE(I0.35)SKH1R WRITE(I0.36)SKV1L WRITE(I0,37)SKV1R WRITE(I0.38)SKH2 WRITE(10.39)SKV2T WRITE(I0.40)SKV2C W R I T E ( 1 0 , 4 1 )A W R I T E ( 1 0 . 4 2 )BL WRITE(IO,43)BR WRITE(I0.44)C WRITE(I0.45)D1 WRITE(I0.46)D2 WRITE(I0.47)E IF(I0.EQ.5)WRITE(5,51) IF(I0.EQ.5)READ(5.52)NXX I F ( ( 1 0 . E Q . 5 ) . A N D . ( N X X . E O . 1 ) ) STOP WRITE(I0,48)DETAT WRITE(I0.49)T0L WRITE(I0.50IN.BTI 10=10+1 IF(I0.E0.6) G0T07 C 20 F 0 R M A T ( 9 F 2 0 . 4 ) 2 1 FORMAT(2F19.15) 22 F O R M A T ( 8 0 A 1 ) 23 F O R M A T ( I 1 0 . F 1 5 . 6 ) 24 F 0 R M A T ( 8 F 1 0 . 0 ) 25 F O R M A T ( / . ' C O U P L E D A N A L Y S I S - - E Q U I P M E N T ATTACHED TO FLOOR /»'.I3) 26 F O R M A T ( / , 'UNCOUPLED A N A L Y S I S - - E Q U I P M E N T RESPONSE ONLY' I 27 F O R M A T ! / , ' E Q U I P M E N T NOT ATTACHED - STRUCTURAL RESPONSE O N L Y ' ) 28 F O R M A T ( / , ' F L O O R ' , 3 X , ' I N T E R S T O R Y S T I F F N E S S ( N / M ) ' . 3 X . 1 'DAMPING R A T I O ' , 6 X . ' M A S S ( K G ) ' ) 29 F O R M A T ! 1 5 , 3 F 2 0 . 4 ) 30 F O R M A T ! / / . ' E Q U I P M E N T D A T A ' ) 31 FORMAT( '*****»•»***»*•• ) 32 F O R M A T ! ' M A S S ! K G ) = ' , F 1 1 . 4 . 4 X . 1 'MOMENT OF INERTIA !KG*M*M) = ' . F 1 1 . 4 ) 33 FORMAT!'EQUIPMENT DAMPING RATIOS = ' . 3 F 1 0 . 4 ) 34 F O R M A T ! ' H O R I Z O N T A L S T I F F N E S S OF L E F T SPRING (N/M) = ' , F 1 3 . 4 ) 35 F O R M A T ! ' H O R I Z O N T A L S T I F F N E S S OF RIGHT SPRING (N/M) = ' . F 1 3 . 4 ) 36 F O R M A T ! ' V E R T I C A L S T I F F N E S S OF L E F T SPRING (N/M) = ' . F 1 3 . 4 ) 37 F O R M A T ! ' V E R T I C A L S T I F F N E S S OF RIGHT SPRING (N/M) = ' , F 1 3 . 4 ) 38 F O R M A T ! ' H O R I Z O N T A L SNUBBER S T I F F N E S S (N/M) = ' . F 1 6 . 4 ) 39 F O R M A T ! ' V E R T I C A L SNUBBER S T I F F N E S S AGAINST L I F T (N/M) = ' . F 1 G 4 ) 40 F O R M A T * ' V E R T I C A L SNUBBER S T I F F N E S S AGAINST DROP !N/M) = ' , F 1 6 . 4 ) 4 1 F O R M A T ! ' V E R T I C A L DISTANCE BETWEEN CENTRE OF MASS AND S P R I N G S ' 149 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 54B 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 1 , ' (M) = ' , F 7 . 4 ) 42 F0RMAT('HORIZONTAL DISTANCE BETWEEN C . G . AND L E F T SPRINGS (M) =' 1,F7.4) 43 FORMAT!'HORIZONTAL DISTANCE BETWEEN C . G . AND RIGHT SPRINGS (M) =' 1 .F7.4) 44 FORMAT('GAP BETWEEN SPRING AND HORIZONTAL SNUBBER (M) = ' , F 7 . 5 ) 45 F O R M A T ( ' G A P BETWEEN SPRING AND V E R T I C A L SNUBBER AGAINST L I F T ' 1 . ' (M) = ' . F 7 . 5 ) 46 FORMAT('GAP BETWEEN SPRING AND VERTICAL SNUBBER AGAINST DROP' 1 . ' (M) = ' . F 7 . 5 ) 47 FORMAT('PERCENTAGE E C C E N T R I C I T Y BETWEEN C . G . AND SPRINGS = ' . F 7 . 4 ) 48 F O R M A T ( / / . ' T I M E INCREMENT ( S E C ) = ' . F 7 . 4 ) 49 F O R M A T ( ' T O L E R E N C E (M) = ' , F 1 5 . 1 3 ) 50 F O R M A T ( 1 5 . ' INPUT A C C E L . P T . * ' . F 8 . 4 . ' S E C S . ' ) 51 F O R M A T ( / / , ' D O YOU WISH TO CONTINUE ? 0 => YES : 1 = > N O ' ) 52 FORMAT(11 ) C C C C ASSUMING A LINEAR VARIATION BETWEEN TIME THE INPUT ACELERATION ACCORDINGLY. STEP SUBDIVIDE AF(1)=0.DO DURA=BTI*N XXX=BTI/DETAT+0.5D0 IST=IDINT(XXX) I F ( I S T . L E . 1) GOTO 11 ISS=IST-1 M=N+1 DO 1 d=1.M dd=J*IST-ISS AFF(dd)=AF(d)/SI 1 CONTINUE DO 2 K = 1 , N DO 3 d = 1 , I S S d«Jd=K*IST+1 dd=IST*K+d-ISS dddd=K*IST-ISS AFF(Jd) = (AFF(JdJ)-AFF(dddd))/IST*J+AFF(ddJJ) 3 CONTINUE 2 CONTINUE AFF(ddd+1)=O.DO GOTO 12 1 1 00 9 1=1,N AFF(I)=AF(I)/SI 9 CONTINUE AFF(N+1)=0.D0 C C C ZERO EOUIPMENT 12 CALL CALL CALL CALL C C C AND STRUCTURE AND DAMPING VECTOR DGSET(EMAS,3,3.3.0.D0) DGSET(SMAS,10.10.10,0.DO) DGSET(EZETA,3.3,3.0.D0) DGSET(SZETA,10,10,10,0.DO) SET UP EOUIPMENT DO MASS MATRIX 13 1=1,NBD SMAS(I.I)=AM(I) 13 CONTINUE AND STRUCTURE MASS MATRIX 150 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 DO 14 1 = 1 , 3 EMAS(I,I)=AM(I+NBD) 14 CONTINUE 0 0 15 1=1.NBD SZETA(I,I)=ZETA(I) 15 CONTINUE 00 16 1 = 1 , 3 EZETA(I,I)=ZETA(I+NBD) 16 CONTINUE I F ( I F L . E 0 . N B - 3 ) GOTO 5 I F ( I F L . E O . O ) GOTO 6 C C C C REARRANGE LOCATION. THE MASSES AND DAMPING VECTORS TO THE EOUIPMENT EMASSX=AM(NBB) EMASSY=AM(NC) E A<J = AM( NB) EOZETA=ZETA(NBB) EOZETB=ZETA(NC) EOZETC=ZETA(NB) IFJ=IFL+4 IFK=IFL+3 IFD=IFL+2 IFE=IFL+1 DO 4 I = I F d , N B AM(I)=AM(1-3) ZETA(I ) = ZETA(1-3) 4 CONTINUE AM(IFE)=EMASSX AM(IFD)=EAd A M ( I F K ) = EMASSY ZETA(IFE)=EOZETA ZETA(IFD)=EOZETB ZETA(IFK)=EQZETC GOTO 5 6 I F ( I F A . E O . 1 ) GOTO 5 AM(1)=AM(NBB) AM(2)=AM(NC) AM(3)=AM(NB) ZETA(1 ) = ZETA(NBB) ZETA(2)=ZETA(NC) ZETA(3)=ZETA(NB) 5 CONTINUE RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C C S U B R O U T I N E M A S S C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C T H I S SUBROUTINE S E T S UP THE MASS MATRIX AND ITS INVERSE C SUBROUTINE M A S S ( A M , A M A S S , A M A S I N , N B V , I F L ) IMPLICIT R E A L * 8 ( A - H . O - Z ) DIMENSION A M A S S ! 1 3 , 1 3 ) , A M A S I N ! 13, 13),AMI 13) CALL D G S E T ( A M A S S . 1 3 . 1 3 . 1 3 . 0 . D O ) 151 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 CALL D G S E T ( A M A S I N , 1 3 , 1 3 , 1 3 , 0 . D O ) DO 1 1=1,NBV AMASSfI,I)=AM(I) AMASIN(I,I)=1.DO/AMASS(1,1) 1 CONTINUE WRITE(6,74) 74 FORMAT (// , ' »»»»* MASS MATRIX ***<•*<) CALL D P R M A T ( A M A S S , 1 3 . 1 3 . N B V . N B V , 1 , 1 , 1 3 . 1 ) RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C C S U B R O U T I N E S T I F F S C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C T H I S SUBROUTINE SETS UP THE S T I F F N E S S MATRIX C SUBROUTINE S T I F F S ( S K . S T I F F . X C , A . B L . B R . C . S K H 2 . N B . I F L . I F A . I A . 1 0 1,VERTL,VERTR,SKH1L.SKH1R.SKV1L.SKV1R,SKV2T,SKV2C,D1,D2 1.SSK.ESK) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION S T I F F ( 1 3 , 1 3 ) .SK< 1 3 ) , S S K ( 1 0 , 1 0 ) , I A ( 3 ) , I 0 ( 1 8 ) , E S K ( 3 . 3 ) CALL D G S E T ( S T I F F , 1 3 , 1 3 , 1 3 , 0 . D O ) CALL D G S E T ( S S K , 1 0 , 1 0 , 1 0 , 0 . D O ) CALL D G S E T ( E S K . 3 . 3 , 3 . 0 . D 0 ) NC=NB-1 NBB=NB-2 NBD=NB-3 NBS=NB-4 IFJ=IFL+4 IFK=IFL+3 IFD=IFL+2 IFE=IFL+1 U1=D1*SKV2T*A U2=D2*SKV2C*A SK(NBB)=SKH1L+SKH1R SKDUMY=SKV1L*BL*BL+SKV1R*BR*BR SK(NC)=SKV1L+SKV1R SK(NB)=SKV1R*BR-SKV1L*BL I F ( I F A . N E . O ) GOTO 77 C C SET UP A S T I F F N E S S MATRIX FOR A UNCOUPLED EOUIPMENT ANALYSIS C I F ( D A B S ( X C ) . G E . C ) SK(NBB ) = SKH1L + SKH2+SKH1R + SKH2 IF ( V E R T R . L E . ( - D 1 ) ) SKDUMY = (SKV1R+SKV2T)»BR*BR+SKV1L*BL*BL-U1 IF ( V E R T R . L E . ( - D 1 ) ) SK(NC)=SKV1R+SKV2T+SKV1L IF ( V E R T R . L E . ( - D 1 ) ) SK(NB) = (SKV1R+SKV2T)*BR-SKV1L*BL IF ( V E R T R . G E . ( D 2 ) ) SKDUMY«(SKV 1R+SKV2C)*BR*BR+5KV1L*BL*BL + U2 IF ( V E R T R . G E . ( D 2 ) ) SK(NC ) = SKV1R+SKV2C+SKV1L IF ( V E R T R . G E . ( D 2 ) ) SK(NB ) = (SKV1R+SKV2C)»BR-SKV1L*BL IF ( V E R T L . L E . ( - D 1 ) ) SKDUMY = (SKV1L + S K V 2 T ) * B L *BL + SKV1R*BR*BR-U1 IF ( V E R T L . L E . ( - D 1 ) ) SK(NC)=SKV1L+SKV2T+SKV1R IF ( V E R T L . L E . ( - D 1 ) ) SK(NB ) = S K V 1 R * B R - ( S K V 1 L + S K V 2 T ) * B L I F ( V E R T L . G E . ( D 2 ) ) SKDUMY = (SKV1L + S K V 2 C ) * B L * B L + SKV1R*BR*BR+U2 I F ( V E R T L . G E . ( D 2 ) ) SK(NC ) = SKV1L + SKV2C+SKV1R IF ( V E R T L . G E . ( D 2 ) ) SK(NB)=SKV1R*BR-(SKV1L+SKV2C)*BL I F ( ( V E R T R . L E . ( - D 1 ) ) . A N D . ( V E R T L . G E . ( D 2 ) ) ) SKDUMY = ( S K V 1 L + S K V 2 C ) * 152 697 1BL*BL+(SKV1R+SKV2T)*BR*BR-U1+U2 698 I F ( ( V E R T R . L E . ( - D 1 ) ) . A N D . ( V E R T L . G E . ( D 2 ) ) ) IF ( ( V E R T R . L E . ( - D 1 ) ) . A N D . ( V E R T L . G E . 700 I F ( ( V E R T R . G E . ( D 2 702 703 SK(NB)=(SKV1R+SKV2T)* ) ) . A N D . ( V E R T L . L E . ( - D 1 ) ) ) SKDUMY=(SKV1L+SKV2T)* 1BL*BL+(SKV1R+SKV2C)*BR*BR-U1+U2 704 I F ( ( V E R T R . G E . ( D 2 ) ) . A N D . ( V E R T L . L E . ( - D 1 ) ) ) 705 S K ( N C ) =S K V 1 L + S K V 2 T + 1SKV1R+SKV2C I F ( ( V E R T R . L E . ( D 2 ) ) . A N D . ( V E R T L . L E . ( - D 1 ) ) ) 706 707 1BR-(SKV1L+SKV2T E S K ( 1 , 1 709 E S K ( 1 , 2 ) = - S K ( N B B ) * A 7 1 0 E S K ( 2 , 1 ) = - S K ( N B B ) * A 711 E S K ( 2 . 2 712 E S K ( 2 . 3 ) = S K ( N B ) 713 E S K ( 3 . 2 ) = S K ( N B ) 714 SK(NB )= (SKV1R+SKV2C)* )*BL 708 )= SK(NBB) ) =S K ( N B B ) * A * A + SKDUMY E S K ( 3 . 3 ) = S K ( N C ) 715 C 716 C 7 C PUT 718 DO 719 EQUIPMENT 9 S T I F F N E S S MATRIX INTO SYSTEM STIFFNESS MATRIX 1=1.3 DO 10 J = 1 , 3 S T I F F ( I , J ) = E S K ( 1 , 0 ) 720 721 10 722 9 723 CONTINUE CONTINUE GOTO 724 C 725 C 726 C 727 PUT 77 DO 79 STRUCTURE 11 S T I F F N E S S E S INTO A DUMMY MATRIX I=1,NBS 728 11=1+1 729 S S K ( I . I ) = S K ( I ) + S K ( I I ) 730 S S K ( I . I I ) = - S K ( 1 1 ) S S K ( I I , I ) = - S K ( 1 1 ) 731 1 1 732 CONTINUE 733 SSK (NBD , NBD ) = SK (NBD ) 734 I F Z = I F L 735 I F ( I F L . E Q . O ) IFZ=NBD 736 C 737 C IF THE EQUIPMENT IS 738 C IN THE S T I F F N E S S MATRIX 739 C DO 740 741 12 DO 1 = 13 NOT ATTACHED TO WITH STRUCTURAL THE THE STRUCTURE. 13 744 1,IFZ J = 1 , I F Z 12 CONTINUE CONTINUE 745 I F ( I F L . E Q . O ) 746 I F ( I F L . E Q . N B D ) 747 C 748 C IF 749 C REARRANGE 750 c 751 DO THE 14 DO 752 GOTO EQUIPMENT THE 79 GOTO IS 78 NOT S T I F F N E S S CONNECT I = I F L , N B S 15 K=1,NBD S T I F F ( K . I + 4 ) = S S K ( K , 1 + 1 ) 753 15 CONTINUE TO THE ACCORDINGLY. TOP F I L L S T I F F N E S S . S T I F F ! I . J ) = S S K ( I , J ) 742 743 754 (D2 ) ) ) 1 B R - ( S K V 1 L + S K V 2 C ) * B L 701 17 SK(NC)=SKV1L+SKV2C+ 1SKV1R+SKV2T 699 FLOOR. 153 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 C C C 14 CONTINUE DO 16 I = I F L . N B S DO 17 KK=NBB,NB STIFF(I+4.KK )=STIFF(1+1,KK) STIFF(1+1,KK)=O.DO 17 CONTINUE 16 CONTINUE DO 18 1=1,NB DO 19 J = 1 , N B STIFF(J,I)=STIFF(I.J) 19 CONTINUE 18 CONTINUE PUT 78 EOUIPMENT STIFFNESSES INTO THE S T I F F N E S S MATRIX. IF ( O A B S ( X C ) , G E . C ) SK(NBB ) =SKH1L + SKH2 + SKH1R + SKH2 IF ( V E R T R . L E . ( - D 1 ) ) SKDUMY = (SKV1R+SKV2T)*BR*BR+SKV1L*BL*BL-U1 IF (VERTR . LE . ( - D 1 ) ) SK(NC ) = SKV1R+SKV2T + SKV1L IF ( V E R T R . L E . ( - D 1 )) SK(NB ) = <SKV 1R+SKV2T)*BR-SKV1L>BL IF ( V E R T R . G E . (D2 ) ) SKDUMY=(SKV1R+SKV2C)*BR*BR+SKV1L*BL*BL+U2 IF ( V E R T R . G E . ( D 2 ) ) SK(NC ) = SKV1R + SKV2C + SKV1L IF ( V E R T R . G E . ( D 2 ) ) SK(NB ) = (SKV1R + S K V 2 C ) * B R - S K V 1 L * B L IF ( V E R T L . L E . ( - D 1 ) ) SKDUMY = (SKV1L + SKV2T ) *BL*BL+SKV1R*BR*BR-U1 IF ( V E R T L . L E . ( - D D ) SK( NC ) = SKV 1 L + SKV2T + SKV1R IF ( V E R T L . LE . (-D1 ) ) SK(NB ) =SKV 1R*BR-(SKV1L+SKV2T)*BL IF ( V E R T L . G E . ( D 2 ) ) SKDUMY = (SKV1L + SKV2C)*BL*BL+SKV1R*BR*BR + U2 IF ( V E R T L . G E . ( D 2 ) ) SK(NC)=SKV1L+SKV2C+SKV1R IF ( V E R T L . G E . ( D 2 ) ) SK(NB ) =SKV1R*BR-(SKV1L + SKV2C)*BL I F ( ( V E R T R . L E . ( - D 1 ) ) - AND . ( V E R T L . G E . ( D 2 ) ) ) SKDUMY = (SKV1L + SKV2C) 1BL*BL+(SKV1R+SKV2T)*BR*BR-U1+U2 IF((VERTR.LE.(-D1)).AND.(VERTL.GE.(D2))) SK(NC)=SKV1L+SKV2C+ 1SKV1R+SKV2T I F ( ( V E R T R . L E . ( - D 1 ) ) . A N D . ( V E R T L . G E . ( D 2 ) ) ) SK(NB ) = (SKV1R+SKV2T) 1BR-(SKV1L+SKV2C )*BL IF((VERTR.GE.(D2 ) ) .AND.(VERTL.LE.(-D1))) SKDUMY=(SKV1L+SKV2T) 1BL»BL+(SKV1R+SKV2C)*BR*BR-U1+U2 I F ( ( V E R T R . G E . ( D 2 ) ) . A N D . ( V E R T L . L E . ( - D 1 ) ) ) SK(NC) = SKV1L + SKV2T + 1SKV1R+SKV2C I F ( ( V E R T R . L E . ( D 2 ) ) . A N D . ( V E R T L . L E . ( - D 1 )) ) SK(NB ) = (SKV1R + SKV2C) 1BR»BR-(SKV1L+SKV2T)*BL *BL STIFF(IFE,IFE)=SK(NBB) STIFF(IFE,IFD)=-SK(NBB)«A STIFF(IFD,IFE)=-SK(NBB)*A S T I F F ( I F D , I F D ) = SK(NBB ) *A*A+SKDUMY STIFF(IFD,IFK)=SK(NB) STIFF(IFK.IFD)=SK(NB) STIFF(IFK.IFK)=SK(NC) STIFF(IFL.IFL)=STIFF(IFL.IFL)+SK(NBB) STIFF(IFL.IFE)=-SK(NBB) STIFF(IFE,IFL)=-SK(NBB) S T I F F ( I F L , I F D ) r SK(NBB)* A S T I F F ( I F D , I F L) = SK(NBB)* A ESK( 1 , 1 ) = SK(NBB) ESK( 1 , 2 ) = - S K ( N B B ) * A ESK(2,1)=-SK(NBB)*A ESK(2,2)=SK(NBB)*A*A+SKDUMY ESK(2,3)=SK(NB) ESK(3,2)=SK(NB) 154 813 814 815 816 817 618 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 84 1 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 C C C E5K(3.3)=SK(NC) OUTPUT STIFFNESS MATRIX IF NEEDED 79 IF(DABS(XC) LT.C) IA(1 )=0 IF(DABS(XC).GE.C) IA(1)=1 IF (( VERTR. GT.(-DD). AND. (VERTR.LT.(D2))) IA<2)=0 IF (VERTR.LE.(-DD) IA(2)=1 IF (VERTR.GE.(D2) ) IA(2)=2 IF((VERTL GT.(-D1) ) .AND.(VERTL.LT.(D2))) IA(3 )=0 IF (VERTL.LE.(-DO) IA(3) = 1 IF (VERTL GE.(D2)) IA(3)=2 I JK= 1 IF(IA(1 ) .EO.1) IJK=10 IF(IA(2).EO.O IJK=IJK+3 IF(IA(2).E0.2) IJK=IJK+6 IF(IA(3).EO.O) IQ(IJK)=IQ(IJK)+1 IF((IA(3).E0.0).AND.(I0(IJK).E0.O) G O T O 81 IF(IA(3) .EO.1) I0(IJK+1) = I0(IJK+1) + 1 IF((IA(3).E0.O.AND.(10(1 JK+O.E0.1)) G O T O 81 IF(IA(3) EQ.2) I0(IJK+2)=IQ(IJK+2)+1 IF((IA(3).EO.2).AND.(I0(IdK+2).EO.1)) G O T O 81 G O T O 82 81 WRITE(6.84) 84 FORMAT(//, ' STIFFNESS MATRIX ***•*') N B V = N B IF(IFA.EO. 1 ) NBV=NB-3 IF(IFA.EO.O) N B V=3 CALL DPRMAT(STIFF, 13, 13,NBV.NBV. 1,1.13,1) 82 CONTINUE RETURN E N D cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c C c SUBROUTINE FORCE C c C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c THIS SUBROUTINE SETS UP T H E INCREMENTAL FORCE VECTOR c SUBROUTINE FORCE(AFF,DETAT,DETAF.AMASS.I,N,FA,XC.A,SKH2,C, 1NBV,IFL,I FLAG,JE.IT,A 1,AS.IFA,SKV2T,SKV2C,D1,D2,VERTL.VERTR, 1,DETVL,DETVR,ROR,DETAX,FMU,DETAXC) IMPLICIT REAL*8(A-H.O-Z) DIMENSION AFF(60000).DETAF( 13) ,FA( 13) .AMASS( 13, 13 ),DETAX( 13 ) 11=1+1 A2 = AFF(11 ) A 1 =AFF( I ) IF(IFLAG.E0.2) G O T O 8 IF(IFLAG.EO.O) G O T O 5 c c IF ITERATION IS NEEDED. USE T H E APPROPRIATE PORTION O F c THE INPUT ACCELERATON BASED O N T H E ITERATION TIME STEP c A N DT H E ASSUMPTION O F A LINEAR VARIATION IN INPUT c ACCELERATION. c A2=AS*DETAT+A1 G O T O 5 155 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 C C C C C AFTER THE ITERATION PROCEDURE IS COMPLETED USE UP THE REMAINING INPUT ACCELERATION IN ORDER TO COMPLETE THE TIME S T E P . 8 A1=A2 A2=AFF(II) 5 DO 3 J = 1 . N B V DETAF(d)=-(A2-A1)*AMASS)J, J) FA(J) = -AMASS(J,J)*A 1 3 CONTINUE I F ( I F A . E O • 1 ) GOTO 4 DUMMY=0.D0 DUMMY 1=0.DO DUMMY2=0.D0 DUMMY3=0.DO DUMMY4=0.D0 I F ( D A B S ( X C ) . G E . C ) DUMMY = D S I G N ( 2 . D 0 * S K H 2 * C , X C ) I F ( D A B S ( X C ) . G E . C ) FFR=DSIGN(FMU'DETAXC*SKH2,DETVR) I F ( D A B S ( X C ) . G E . C ) F F L = DSIGN(FMU*DETAXC*SKH2,DETVL) I F ( D A B S ( X C ) . G E . C ) DUMMY3=FFL*BL-FFR*BR+(FFL+FFR)»A*ROR I F ( D A B S ( X C ) . G E . C ) DUMMY4=FFL+FFR IF ( V E R T R . L E . ( - D 1 ) ) DUMMY 1=-BR*D1*SKV2T IF ( V E R T R . G E . ( D 2 ) ) DUMMY1=BR*D2*SKV2C IF ( V E R T L . L E . ( - D I ) ) DUMMY2 = BL*D1*SKV2T IF ( V E R T L G E . ( D 2 ) ) DUMMY2=-BL*D2*SKV2C I F ( I F A . E O . O ) GOTO 6 F A ( I F L ) = - A M A S S ( I F L , I F L ) * A 1-DUMMY 6 F A ( I F L + 1 ) = - A M A S S ( I F L + 1 . I F L + 1 ) * A 1+DUMMY F A ( I F L + 2) = - DUMMY * A+DUMMY1+DUMMY 2+DUMMY 3 FA(IFL+3)=DUMMY1/BR-DUMMY2/BL-DUMMY4 DETAF(IFL+2)=0.D0 DETAF(IFL+3)=0.DO 4 CONTINUE RETURN END cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc C c C S U B R O U T I N E P F O R C E c c c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c T H I S SUBROUTINE SETS UP THE PSEUDO FORCE VECTOR c c SUBROUTINE P F O R C E ( D E T A F , D E T A T , A M A S S , A C , V , P S E U F O . D C , N B V , BET A 1,ALBE,ALBR ) IMPLICIT REAL*8(A-H.O-Z ) DIMENSION D E T A F ( 1 3 ) . A M A S S ( 1 3 , 1 3 ) , A C ( 1 3 ) . V ( 1 3 ) . P S E U F O ( 1 3 ) DIMENSION DUM1(13),DUM( 13) .DC( 13, 13) ,DUM2( 13 ) ,DUM3( 13) DO 1 1=1,NBV D U M ( I ) = 0 . 5 D 0 * A C ( I ) / B E T A + V ( I )/DETAT/BETA 1 CONTINUE CALL DGMATV(AMASS,DUM.DUM1,NBV.NBV.13) DO 2 1=1,NBV D U M 2 ( I ) " A L B E * V ( I ) - A C ( I ) *DETAT * ALBR 2 CONTINUE CALL D G M A T V ( D C , D U M 2 , D U M 3 , N B V , N B V , 1 3 ) DO 3 1=1.NBV 156 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 P S E U F O U )=DETAF(I)+DUM1(I)+DUM3<I) 3 CONTINUE RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C C S U B R O U T I N E P S T I F F C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C T H I S SUBROUTINE SETS UP THE PSEUDO S T I F F N E S S MATRIX C SUBROUTINE P S T I F F ( S T I F F , D E T A T , A M A S S , P S E U S T , D C . N B V , B E T A . A L B E ) IMPLICIT R E A L * 8 ( A - H . O - Z ) DIMENSION S T I F F ( 1 3 . 1 3 ) , A M A S S ( 1 3 . 1 3 ) . P S E U S T ( 1 3 . 1 3 ) . D C ( 1 3 . 1 3 ) DO 1 1=1.NBV DO 2 d =1,NBV PSEUST(I.J) = AMASS(I.J)/(DETAT*DETAT*BETA)+STIFF(I.J) + 1 A L B E * D C ( I , d l/DETAT 2 CONTINUE 1 CONTINUE RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C C S U B R O U T I N E S T E P C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C T H I S SUBROUTINE C A L C U L A T E S THE EIGENVECTORS. EIGENVALUES AND C MODAL P A T I C I P A T I O N FACTORS OF THE SYSTEM OF EQUATIONS TO BE C SOLVED SO THAT A CHECK ON THE TIME STEP CAN BE MADE. IT USES C A UBC LIBARY SUBROUTINE TO SOLVE THE EIGENVALUE PROBLEM. C SUBROUTINE S T E P ( S T I F F , A M A S S . C H E C K . D E . D V . X C . C . N B V , I F L . I A . J Q 1.VERTR,VERTL,D1.D2.SDE,SDV.EDE.EDV.SMAS.EMAS.SSK.ESK.NBD) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION S T I F F ( 1 3 , 1 3 ) , A M A S S ( 13, 1 3 ) , D B ( 9 1 ),DA( 1 6 9 ) , D E ( 13) DIMENSION DV( 13. 1 3 ) , G A M M A ( 1 3 ) , V I ( 1 3 ) , D U M ( 13).DUM1( 1 3 ) . V V ( 13) DIMENSION E S K ( 3 . 3 ) , S D A ( 1 0 0 ) . E D A ( 9 ) , S D B ( 5 5 ) . E D B ( 6 ) . S D E ( 1 0 ) DIMENSION E D E ( 3 ) ,SDV( 10. 10 ) , E D V ( 3 , 3 ) . S S K ( 1 0 , 1 0 ) DIMENSION SMAS( 1 0 . 1 0 ) . E M A S ( 3 , 3 ) . I A ( 3 ) . J Q ( 18) CALL D G S E T ( D V . 1 3 , 1 3 , 1 3 . O . D 0 ) CALL D G S E T ( S D V , 1 0 , 1 0 . 1 0 . 0 . D O ) CALL D G S E T ( E D V , 3 . 3 . 3 , O . D O ) KK=0 DO 1 1=1.NBV DO 2 K= 1 , I KK=KK+1 DA(KK ) = S T I F F ( I , K ) DB(KK ) = A M A S S ( I , K ) 2 CONTINUE 1 CONTINUE CALL D R S G A L ( D A . D B . N B V , D E . I ERROR.1) DO 3 1 = 1 ,NBV IOI = ( 1 - 1 )*NBV+1 DO 4 J= 1,NBV D V ( J . I )=DA(IOI ) 157 937 101=101+1 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 4 CONTINUE 3 CONTINUE KK=0 DO 11 1=1,NBD DO 12 K = 1 , I KK=KK+1 SDA(KK)=SSK(I,K) SDB(KK)=SMAS(I,K) 12 CONTINUE 11 CONTINUE CALL D R S G A L ( S D A , S D B . N B D . S D E . I E R R O R , 1 ) DO 13 1=1,NBD IOI=(1-1)*NBD+1 DO 14 J=1,NBD SDV(J,I)=SDA(101) 1003 101=101+1 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 14 CONTINUE 13 CONTINUE KK=0 DO 21 1 = 1 , 3 ' DO 22 K = 1 , I KK=KK+1 EDA(KK)=ESK(I,K) EDB(KK ) = E M A S ( I , K ) 22 CONTINUE 21 CONTINUE CALL D R S G A L ( E D A , E D B , 3 , E D E , I E R R O R , 1 ) DO 23 1=1,3 I0I=(I-1)*3+1 DO 24 J=1 , 3 EDV(J,I)=EDA(101) 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 24 CONTINUE 23 CONTINUE 1019 101=101+1 C C C C OUTPUT NATURAL PERIODS, EIGENVALUES AND EIGENVECTOR I F ( D A B S ( X C ) . L T . C ) IA(1)=0 I F ( D A B S ( X C ) . G E . C ) IA(1)=1 I F ( ( V E R T R . G T . ( - D 1 ) ) . A N D . ( V E R T R . L T . ( D 2 ) ) ) IA(2)=0 IF ( V E R T R . L E . ( - D 1 ) ) I A ( 2 ) = 1 IF ( V E R T R . G E . ( D 2 ) ) I A ( 2 ) = 2 I F ( ( V E R T L . G T . ( - D 1 ) ) . A N D . ( V E R T L . L T . ( D 2 ) ) ) IA(3)=0 IF ( V E R T L . L E . ( - D 1 ) ) I A ( 3 ) = 1 IF ( V E R T L . G E . ( D 2 ) ) IA(3)=2 IJK=1 I F ( I A ( 1 ) . E O . 1 ) IJK=10 I F ( I A ( 2 ) E O . 1 ) IJK=IJK+3 I F ( I A ( 2 ) . E 0 . 2 ) IJK=IJK+6 I F ( I A ( 3 ) . E O . O ) JO(IJK) = JO(IJK)+1 I F ( ( I A ( 3 ) . E O . O ) . A N D . ( J 0 ( I J K ) . E O . 1 ) ) GOTO 81 I F ( I A ( 3 ) . E O . 1) J 0 ( I J K + 1 ) = J 0 ( I J K + 1 ) + 1 I F ( ( I A ( 3 ) . E O . 1 ) . A N D . ( J 0 ( I J K + 1 ) . E O . 1 ) ) GOTO 81 I F ( I A ( 3 ) . E O . 2 ) J0(IJK+2) = J0(IJK+2)+1 I F ( ( I A ( 3 ) . E Q . 2 ) . A N D . ( J O ( I J K + 2 ) . E O . O ) GOTO 81 GOTO 82 IF NEEDED 158 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 105G 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 108 1 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1 100 1 101 1 102 C C C O M P U T E THE M O D A L PARTICIPATION FACTORS 81 D O 5 1=1,13 GAMMA(I)=0.D0 VI(I)=O.DO VV(I)=0.D0. 5 CONTINUE D O 6 1=1,NBV VI(I ) = AMASS(I,I) 6 CONTINUE D O 7 1=1,NBV D O 8 J=1,NBV VV(J)=DV(J.I ) 8 CONTINUE VB=DGVV(VV,VI.NBV) CALL DGMATV(AMASS,VV,DUM1,NBV.NBV, 13) VD=DGVV(VV.DUM1.NBV) GAMMA(I)=VB/VD 7 CONTINUE WRITE(6.77) 77 F0RMAT(//,5X, 'EIGENVALUES' ,7X, 'NATURAL PERIODS' ,6X. 1 M ' ODAL PARTICIPATION FACTORS) WRITE(6,78) 78 FORMAT(/ ) D O 79 1=1,NBV PERI0D=(2.D0*3.141592654D0)/DSORT(DE(I)) WRITE(6,80)DE(I),PERIOD,GAMMA(I ) 80 F0RMAT(3F20.5) 79 CONTINUE 82 CHECK=(0.2D0*3.14 15926540O)/DSORT(DE(NBV)) RETURN E N D cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c C c SUBROUTINE DAMP C c C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c THIS SUBROUTINE SETS UP T H E DAMPING MATRIX ASSUMING c CLASSICAL DAMPING. c SUBROUTINE DAMP(ZETA,AMASS.DE.DC.DV,XC.C.NBV.IFL.ST IFF,IA.KQ 1.VERTR,VERTL.D1.D2,SDC,EDC,SDV.EDV,NB.EZETA,SZETA.IFA,SDE,EDE 1,EMAS,SMAS ) IMPLICIT REAL*8(A-H.O-Z) DIMENSION DC( 13. 13).AMASS(13.13 ) ,DE( 13 ) ,DV(13. 13 ) ,SZ(10. 101 DIMENSION VDUM(13.13),VDUM1(13,13),VDUM2(13,13),EZ(3.3) DIMENSION EVDUM(3.3).EVDUM1(3,3) ,EVDUM2(3.3).ETDV(3,3) DIMENSION SVDUM(10.10),SVDUM1(10,10),SVDUM2(10,10),EZETA(3.3) DIMENSION STDV(10.10),TDV(13,13),ZETA(13).STIFF(13.13).IA(3) DIMENSION K0(18),SDC( 10. 10).EDC(3.3),SDV( 10, 10) .EDV(3.3) DIMENSION SDE(10),EDE(3),EMAS(3.3).SMASf10.10).SZETA(10.10) CALL DGSET(DC. 13. 13. 13.0.DO) CALL DGSET(SDC.10,10.10,0.DO) CALL DGSET(EDC.3.3,3.0.DO) CALL DGSETCSZ.10.10.10.0.D0) CALL DGSET(EZ.3,3.3.0.DO) NC=NB-1 159 1103 1104 1105 1106 1107 1108 1109 11 10 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1 123 1 124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1 134 1135 1136 1137 1138 1139 1140 1141 1 142 1143 1144 1 145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1 156 1157 1158 1 159 1160 NBB=NB-2 NBD=NB-3 NB5=NB-4 IFd=IFL+4 IFK=IFL+3 IFD=IFL+2 IFE=IFL+1 C C C FIND STRUCTURE S T I F F N E S S MATRIX ASSUMING C L A S S I C A L DAMPING DO 1 1=1,NBD SZ(I,I)=2.D0*SZETA(I.I)*(DSQRT(SDE(I))) 1 CONTINUE CALL DGTRAN(SDV,STDV,NBD.NBD,10,10) CALL DGMULT(STDV,SMAS,SVDUM,NBD,NBD,NBD,10,10,10) CALL DGMULT(SZ,SVDUM,SVDUM1.NBD,NBD,NBD,10,10,10) CALL DGMULT(SDV,SVDUM1.SVDUM2.NBD,NBD,NBD,10,10,10) CALL D G M U L T ( S M A S , S V D U M 2 . S D C , N B D , N B D , N B D , 1 0 , 1 0 , 1 0 ) C C C FIND EOUIPMENT S T I F F N E S S MATRIX ASSUMING C L A S S I C A L DAMPING DO 2 1 = 1 . 3 EZ(I,I)=2.D0*EZETA(I,I)*(DSORT(EDE(I))) 2 CONTINUE CALL D G T R A N ( E D V , E T D V , 3 , 3 , 3 . 3 ) CALL D G M U L T ( E T D V . E M A S , E V D U M , 3 , 3 , 3 , 3 , 3 , 3 ) CALL D G M U L T ( E Z , E V D U M , E V D U M 1 , 3 , 3 . 3 , 3 , 3 , 3 ) CALL D G M U L T ( E D V , E V D U M 1 , E V D U M 2 , 3 . 3 , 3 , 3 , 3 , 3 ) CALL D G M U L T ( E M A S , E V D U M 2 , E D C , 3 . 3 . 3 , 3 , 3 , 3 ) I F ( I F A . E O . O ) GOTO 3 GOTO 4 C C C BUILT DAMPING MATRIX 3 DO 6 1 = 1 , 3 DO 7 d=1 , 3 OC(I.d)=EDC(I,d) 7 CONTINUE 6 CONTINUE GOTO 8 0 4 IFZ=IFL I F ( I F A . E O . I ) IFZ=NBD C C C C IF THE EOUIPMENT IS NOT ATTACHED TO THE STRUCTURE. IN THE DAMPING MATRIX WITH THE STRUCTURAL DAMPING. DO 8 1 = 1 , I F Z DO 9 d = 1 , I F Z DC(I,d ) = SDC(I,d) 9* CONTINUE 8 CONTINUE I F ( I F A . E O . I ) GOTO 8 0 IF ( I F L . E O . N B D ) GOTO 79 C C C C IF THE EOUIPMENT IS NOT CONNECT TO THE TOP REARRANGE THE DAMPING ACCORDINGLY. DO 10 I=IFL,NBS FLOOR, FILL 160 1 161 1 162 1 163 1 164 1 165 1 166 1 167 1 168 1169 1170 1 171 1 172 1 173 1 174 1 175 1 176 1 177 1 178 1 179 1 180 1 181 1 182 1 183 1 184 1 185 1 186 1 187 1 188 1 189 1 190 1191 1 192 1 193 1 194 1 195 1 196 1 197 1 198 1 199 120O 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 121 1 1212 1213 1214 12 15 1216 1217 1218 C C c D O 11 K=1,NBD DC(K,I+4)=SDC(K.1+1) 1 1 CONTINUE 10 CONTINUE D O 12 I=IFL.NBS D O 13 KK=NBB,NB DC(I+4,KK)=DC(1+1.KK) DC(I+1,KK)=0.D0 13 CONTINUE 12 CONTINUE D O 1 4 I=1.NB D O 15 d = 1 ,NB DC(d,I)=DC(I.J) 15 CONTINUE 14 CONTINUE 79 DC(IFE.IFE)=EDC(1.1) DC(IFE,IFD)=EDC(1,2) DC(IFE,IFK)=EDC(1,3) DC(IFD,IFE)=EDC(2, 1 ) DC(IFD,IFD) = EDC(2.2 ) DC(IFD,IFK)=EDC(2,3) DC(IFK,IFE)=EDC(3. 1 ) DC(IFK,IFD)=EDC(3.2) DC(IFK,IFK)=EDC(3,3) DC(IFL,IFL)=DC(IFL,IFL)+EDC(1,1) DC(IFL,IFE)=-EDC(1.1) DC(IFL,IFD)=-EDC(1,2) DC(IFL,IFK)=-EDC(1,3) DC(IFE,IFL)*-EDC(1.1) DC(IFD,IFL)=-EDC(2.1) DC(IFK,IFL)=-EDC(3.1) O U T P U T DAMPING MATRIX IF NEEDED 80 IF(DABS(XC) LT.C) IA(1)=0 IF(DABS(XC).GE.C) IA(1)=1 IF((VERTR.GT.(-D1)).AND.(VERTR.LT.(D2))) IA(2)= IF (VERTR.LE.(-D1)) IA(2)=1 IF (VERTR.GE.(D2)) IA(2)=2 IF((VERTL.GT.(-D1)).AND.(VERTL.LT.(02))) IA(3)= IF (VERTL.LE.(-DD) IA(3) = 1 IF (VERTL.GE.(D2)) IA(3)=2 IJK=1 IF(IA(1).EO.1) IJK=10 IF(IA(2).EO.1) IJK=IdK+3 IF(IA(2) .E0.2) IJK = IdK+G IF(IA(3).EO.O) K0(IdK)=KQ(IdK)+1 IF((IA(3).EO.O).AND.(K0(IdK).EO.1)) G O T O 81 IF(IA(3) .EO.1) KQ(IJK+1)=K0(IdK+1)+1 IF((IA(3 ) .EO. 1 ) .AND.(K0(IdK+1).EO. 1)) G O T O 81 IF(IA(3) .EO.2) K0(IdK+2)=K0(IdK+2)+1 IF((IA(3).EO.2).AND.(K0(IJK+2).EO.1)) G O T O 81 G O T O 82 81 WRITE(6,83) 83 FORMAT(//,' * * * * * EIGEN V E C T O R •»***' ) CALL DPRMAT(DV. 13. 13,NBV,NBV. 1.1.13,1) WRITE(6.84) 84 FORMAT(//.' * * * * * DAMPING MATRIX * * * * + 0 0 161 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 82 CALL DPRMAT(DC . 13, 1 3 , N B V , N B V . 1 . 1 , 1 3 . 1 ) CONTINUE RETURN END cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c c S U B R O U T I N E D R I V E C C C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c c c c c c c c c c c T H I S SUBROUTINE IS THE DRIVER OF THE PROGRAM. IT CALCULATES THE DISPLACEMENTS, V E L O C I T I E S , ACCELERATIONS. AND FORCES. IT USES A UBC LIBARY SUBROUTINE ( S L E ) TO SOLVE THE SYSTEM OF EQUATIONS. SUBROUTINE DRIVE(X,SK,AMASS,NB.IFL,A,BL,BR,C.SKH2,DETAT, 1AM,AFF,I,N.NBB.NC,IT,V.AC,I ERR,ABC,ZETA,DETAX,TIME,NBV.UE. 11 F A , I C O U N T . X C C , I E R . A M A S I N , A S , I F L A G , X C , D X 1 , D X 2 , T 0 L , B T , 1DETAFS,IFLAA,IXX,IDIREC,ALPHA,BETA,ALBE.ALBR.SDC,EDC 1,VERTL,VERTR,SKH1L,SKH1R,SKV1L,SKV1R,SKV2T,SKV2C,D1,D2 1,VBL,VBR.SPFH.SPFVR.SPFVL,IA,10.JQ.KQ,IFLAAA,EMAS,SMAS 1,EZETA,SZETA,NBD,E,B,YAL,YAR,FU,DETVL,DETVR.FMU) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION S T I F F ( 13, 13 ) .AMASS( 13, 13),AMASIN( 1 3 , 1 3 ) . X ( 1 3 ) . V ( 1 3 ) DIMENSION D E T A F ( 1 3 ) . P S E U F 0 ( 1 3 ) , P S E U S T ( 1 3 , 1 3 ) , D E T A X ( 1 3 ) DIMENSION F I ( 1 3 ) . F S ( 1 3 ) . F A ( 1 3 ) , A C ( 13) , D E T A V ( 1 3 ) . A C F ( 9 6 0 1 3 ) DIMENSION D B ( 9 1 ) , D A ( 1 6 9 ) , D E ( 1 3 ) , A F F ( 6 0 0 0 0 ) , A F ( 7 6 0 0 ) , E D C ( 3 . 3 ) DIMENSION D C ( 1 3 . 13 ) ,FD( 13) , D V ( 1 3 , 13 ) .DETX( 13 ) , A B C ( 1 3 ) DIMENSION A M ( 1 3 ) , S K ( 1 3 ) ,ZETAI 13) . I P E R M ( 2 4 ) , T ( 1 3 , 1 3 ) , S D C ( 1 0 . 1 0 ) DIMENSION I A ( 3 ) . I Q ( 1 8 ) . J Q ( 1 8 ) , K Q ( 1 8 ) , S M A S ( 1 0 . 1 0 ) , E M A S ( 3 . 3 ) DIMENSION E S K ( 3 , 3 ) , S D A ( 1 0 0 ) , E D A ( 9 ) . S D B l 5 5 ) , E D B ( 6 ) . S D E ( 1 0 ) DIMENSION E D E ( 3 ) , S D V ( 10, 1 0 ) , E D V ( 3 . 3 ) , S S K ( 1 0 , 1 0 ) . S Z E T A ( 10. 10) DIMENSION E Z E T A ( 3 , 3 ) , D E T A A ( 1 3 ) , F U ( 1 3 ) FIND SPRING DISPLACEMENT IC=IFL+1 ID=IFL+2 IF(IFA.EQ.O) IF(IFA.NE.1) IF(IFA.NE. 1 ) I F ( I F A . N E . 1) IF(IFA.NE.1) IF(IFA.EO.1) IF(IFA.EQ.1) IF(IFA.EQ.1) IF(IFL.NE.O) CHECK XC=X(IC)-A*(X(ID)) VERTL=-BL*X(ID)+X(ID+1) VERTR=BR*X(ID)+X(ID+1) YAL = - ( . 5 8 4 2 D 0 - E * B / 1 0 0 . D O ) * A B C ( I D ) + ABC ( ID+1) YAR=(.5842D0+E*B/100.DO)*ABC(ID)+ABC(ID+1) XC=O.DO VERTL=O.DO VERTR=O.DO XC=X(IC)-X(IFL)-A*(X(ID>) I F NEW S T I F F N E S S AND DAMPING MATRIX ARE NEEDED I F ( ( V B L . L T . D 2 ) . A N D . ( V E R T L . G E . D 2 ) ) IF LAAA=1 I F ( ( V B L . G E . D 2 ) . A N D . ( V E R T L . L T . D 2 ) ) IF LAAA-1 I F ( ( V B L . G T . ( - D 1 ) ) . A N D . ( V E R T L . L T . ( - D 1 ) ) ) IFLAAA= 1 I F ( ( V B L . L E . ( - D 1 ) ) . A N D . ( V E R T L . G T . ( - D 1 ) ) ) IF LAAA= 1 I F ( ( V B R . L T . D 2 ) . A N D . ( V E R T R . G E D2) ) I FLAAA= 1 I F ( ( V B R . G E , D 2 ) . A N D . ( V E R T R . L T .D2 ) ) IF LAAA= 1 I F ( ( V B R . G T . ( - D 1 ) ) . A N D . ( V E R T R . L T . ( - D I ) ) ) IFLAAA = 1 I F ( ( V B R . L E . ( - D 1 ) ) . A N D . ( V E R T R . G T . ( - D 1 ) ) ) I FLAAA= 1 162 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 IF((IFLAA.NE.2).OR.(IFLAAA.EQ.1)) GOTO 76 75 I F ( I F L A A . N E . 2 ) I FLAA = IFLAA+1 C C C SET UP S T I F F N E S S GOTO 75 MATRIX CALL STIFFS(SK.STIFF.XC.A.BL,BR.C.SKH2,NB,IFL.IFA.IA.10 1.VERTL.VERTR.SKH1L.SKH1R.SKV1L.SKV1R.SKV2T,SKV2C.D1.02 1.SSK.ESK) C C C CHECK IF THE TIME INCREMENT IS TOO LARGE CALL S T E P ( S T I F F . A M A S S . C H E C K . D E . D V . X C . C . N B V , I F L . I A. JO 1.VERTR.VERTL.D1.D2.SDE.SDV.EDE.EDV.SMAS.EMAS.SSK.ESK.NBD) I F ( C H E C K . G E . B T ) GOTO 77 WRITE(5.61) WRITE(5,62)CHECK,BT WRITE(6,62)CHECK,BT WRITE(6,61) 62 FORMAT('MAX TIME INCREMENT = ' . F 1 5 . 8 , ' DETAT = ' , F 1 5 . 8 ) 61 FORMAT('***»*• TIME INCREMENT IS TOO LARGE ****»**') STOP C C C SET UP DAMPING MATRIX 77 C A L L DAMP(ZETA,AMASS,DE,DC,DV.XC.C,NBV.IFL.STIFF,IA.KQ 1.VERTR,VERTL.D1,D2,SDC,EDC,SDV.EDV,NB.EZETA,SZETA,IFA,SDE,EDE 1.EMAS,SMAS) 76 CONTINUE C C C C C C SET CALL SET UP THE PSEUDO STIFFNESS MATRIX PSTIFF(STIFF.DETAT,AMASS.PSEUST,DC.NBV,BETA,ALBE) UP FORCE INCREMENT VECTOR CALL FORCE(AFF.DETAT.DETAF.AMASS.I.N,FA.XC,A,SKH2,C,NBV,IFL 1,IFLAG.JE,IT.A1,AS,IFA,SKV2T,SKV2C.D 1 .D2.VERTL.VERTR,BL.BR 1.DETVL,DETVR,ROR,DETAX,FMU,DETAXC) C C C c c COMPUTE THE ERROR IN THE SPRING FORCE ABRUPT CHANGE IN SPRING S T I F F N E S S INTRODUCED BY THE I F ( I F A . E O . 1 ) GOTO 799 VBL = - B L * ( X ( I D ) - D E T A X ( I D ) ) + ( X ( I D + 1 ) - D E T A X ( I D + 1 ) ) VBR=BR*(X(ID)-DETAX(ID))+(X(ID+1)-DETAX(ID+1)) I F ( IFA . E0..0) XB = X ( I C ) - D E T A X ( I C ) - A * ( X ( I D ) - D E T A X ( I D ) ) I F ( I F L . N E . 0 ) XB = X ( I C ) - D E T A X ( I C ) - X ( I F L ) - D E T A X ( I F L ) - A * ( X ( I D 1 )-DETAX(ID)) I F ( ( D A B S ( X B ) . L T . C ) . A N D . ( D A B S ( X C ) G E . C ) ) ICOUNT = ICOUNT+1 I F ( I F L A G . N E . 2 ) GOTO 799 GAP=C I F ( ( X B . L T . O . D O ) . A N D . ( X C . L T . 0 . D O ) ) GAP=-C I F ( ( D A B S ( X B ) .GT C ) . A N D . ( D A B S ( X C I . L E C ) ) GOTO 797 DETAFS=SKH2*(XC-GAP) GOTO 799 797 D E T A F S = - S K H 2 * ( X C - G A P ) 163 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 C C APPLY 799 CALL CALL CALL CALL CALL CALL CALL C C C FIND EQUILBRUIM CORRECTION PROCEDURE DGMATV(STIFF,X,FS,NBV,NBV,13) DGMATV(DC,V,FD,NBV,NBV,13) DGMATV(AMASS,AC,FI,NBV,NBV,13) D G A D D ( F I , F D , F I . N B V , 1 . 1 3 . 1 3 . 13) D G A D D ( F I . F S . F I . N B V . 1 . 13, 1 3 . 1 3 ) DGSUB(FA.FI.FU.NBV,1,13.13.13) DGADD(DETAF,FU,DETAF,NBV.1,13,13.13) SPRING FORCE DUMX=O.DO IF(DABS(XC) .GE.C) DUMX=DSIGN(SKH2*C,XC) SPFH=FS(IC)-DUMX SPFVR=VERTR*SKV1R SPFVL=VERTL*SKV1L IF ( V E R T R . L E . ( - D 1 ) ) SPFVR = (SKV2T + SKV1R)*VERTR+SKV2T*D1 IF ( V E R T R . G E . ( D 2 ) ) SPFVR=(SKV2C+SKV1R)*VERTR-SKV2C*D2 IF ( V E R T L . L E . ( - D 1 ) ) SPFVL=(SKV2T+SKV1L)*VERTL+SKV2T*D1 IF ( V E R T L . G E . ( D 2 ) ) S P F V L = ( S K V 2 C + S K V 1 L ) * V E R T L - S K V 2 C * D 2 I F ( I F A . E Q . I ) SPFH=O.DO C C C SET UP THE PSEUDO FORCE VECTOR CALL PFORCE(DETAF,DETAT,AMASS,AC,V,PSEUFO,DC,NBV,BETA 1,ALBE,ALBR) C C C SOLVE FOR DISPLACMENT INCREMENT DETAX BY SUBROUTINE (SLE) CALL S L E ( N B V , 1 3 . P S E U S T , 1, 1 3 , P S E U F O , D E T X , I P E R M , 1 3 , T , D E T , J E X P ) IF(DET)215,26,215 215 I F ( I F A . E O . 1 ) GOTO 231 I F ( I F A . E Q . O ) XCC = X ( I C )+DETX(IC ) - A * ( X ( I D ) + D E T X ( I D ) ) I F ( I F L . N E . O ) XCC = X ( I C ) + D E T X ( I C ) - X ( I F L ) - D E T X ( I F L ) - A * 1 (X(ID )+DETX(ID)) C C C C CHECK FOR ABRUPT CHANGE IN SPRING S T I F F N E S S , D I R E C T I O N AND PREPARE FOR I T E R A T I O N . THE LOADING I F ( ( D A B S ( X C ) . L T . C ) . A N D . ( D A B S ( X C C ) . G T . C ) ) IER=1 IF((DABS(XC).GT.C).AND.(DABS(XCC).LT.C)) IERM I F ( I X X . E O . I ) GOTO 269 I F ( ( O A B S ( X C ) . L T . C ) . A N D . ( D A B S ( X C C ) . G T . C ) ) IDIREC=1 I F ( ( D A B S ( X C ) . G T . C ) . A N D . ( D A B S ( X C C ) . L T . C ) ) IDIREC = 2 IXX=1 269 CONTINUE GAP=C I F ( X C C . L T . O . D O ) GAP*-C DX1=GAP-XC DX2=XCC-XC BT0L=DABS(DX1)-DABS(DX2) I F ( I F L A G . N E . 1 ) GOTO 271 I F U D I R E C . E Q . 2 ) GOTO 273 I F ( ( D A B S ( B T O L ) . L E . T O L ) . A N D . ( D A B S ( X C C ) . G T . C ) ) IFLAG = 2 GOTO 271 273 I F ( ( D A B S ( B T O L ) . L E . T O L ) . A N D . ( D A B S ( X C C ) . L T . C ) ) I F L A G = 2 271 I F ( I F L A G . E Q . 2 ) GOTO 231 164 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 IF ( I F L A G . E O . 1 ) GOTO 4 0 0 I F ( I E R . E Q . I ) GOTO 4 0 0 C C C FIND D E T A X . DETAV. X. AND V 231 DO 2 0 0 K=1,NBV DETAX(K)=DETX(K) DETAV(K)=ALBE*(DETAX(K)/DETAT-V(K))+DETAT•AC(K)*ALBR DETAA(K ) = ( D E T A X ( K ) / D E T A T / D E T A T - V ( K ) / D E T A T - 0 . 5 D 0 * A C ( K ) ) / B E T A X(K)=X(K)+DETAX(K) V(K)=V(K)+DETAV(K) AC(K)=AC(K)+DETAA(K) 200 CONTINUE DETVL=-BL*DETAX(ID)+DETAX(ID+1) DETVR=BR*DETAX(ID)+DETAX(ID+1) ROR=DETAX(ID) I F ( I F A . E O . O ) DETAXC=DETAX(IC)-A*DETAX(ID) I F ( I F L . N E . O ) DETAXC = DETAX<IC ) - D E T A X ( I F L ) - A * D E T A X ( I D ) C C C FIND ABSOLUTE 87 26 60 400 ACCEL. OF EACH FLOOR DO 87 J J = 1 , N B V ABC(Jd)=AC<JJ)+A1 CONTINUE I F O F A . N E . 1 ) ABC( ID ) = AC( ID ) I F ( I F A . N E . I ) ABC( ID+1 ) =AC(ID+1) GOTO 4 0 0 WRITE(5.60) WRITE(6,60) FORMAT('SOLUTION F A I L E D ) RETURN END 165 Listing 1 of cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 2 3 4 C C C 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 C C C C C C C C C C C C C C 5 FOUR.S M A I N P R O G R A M F O U R . S C C C cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc T H I S PROGRAM READS IN A D I G I T I Z E D TIME S E R I E S RECORD AND COMPUTES THE POWER SPECTRAL DENSITY OF THE RECORD. INPUT RECORD X ( I ) IN I/O UNIT 7 . OUTPUT RECORD Y d ) IN I/O UNIT 4 . RESULTS IN I/O UNITS 1,2.3.6.8 DEFINE VARIABLES N DT N*DT - NUMBER OF POINTS IN THE RECORD - TIME INCREMENT OF RECORD - LENGTH OF RECORD IN SECONDS IMPLICIT R E A L * 8 ( A - H . O - Z ) INTEGER STAR( 1 ) / ' * ' / C 0 M P L E X M 6 DATAX ( 4096 ) , DATA Y ( 4096 ) . CXY ( 4096 ) DIMENSION X ( 4 0 9 6 ) , X S I 4 0 9 6 ) , Y < 4 0 9 6 ) , Y S ( 4 0 9 6 ) . X Y ( 4 0 9 6 ) . X Y S ( 4 0 9 6 ) DIMENSION C H ( 4 0 9 6 ) . A H ( 4 0 9 6 ) . P H I ( 4 0 9 6 ) . C 0 ( 4 0 9 6 ) . P H I S ( 4 0 9 6 ) C C C SET VALUES PI=3. 1415926535897933 C C C READ IN DATA WRITE(5.87) 87 FORMAT('SPECTRUM TYPE . . . O => AUTOSPECTRUM 1 => CROSSSPECTRUM') READ(5,STAR)ISPEC READ(4,88)INY,DTY I F ( I S P E C . N E . 1 ) GOTO 9 0 R E A D ( 7 , 8 8 ) INX.DTX 88 F O R M A T ( I 1 0 . F 1 5 . 6 ) IF(DTX.NE.DTY) WRITE(5.89) 8 9 FORMAT('ERROR : - INPUT/OUTPUT RECORDING INTERVAL ARE D I F F E R E N T ' ) I F ( ( I N X . N E . I N Y ) . O R . ( D T X . N E . D T Y ) ) STOP 9 0 CONTINUE WRITE(5,91) 91 F O R M A T ( ' I N P U T F I L E FORMAT . . . O => 8 1 1 0 1 => 8 F 1 0 . 5 ' ) READ(5,STAR)IFLAG IN=INY DT =DTY WRITE(5,93)IN.DT 9 3 F O R M A T ( ' # OF POINTS IN THE RECORD = ' . I 5 , 2 X , 1 ' • A RECORDING INTERVAL ( S E C ) = ' . F 1 0 . 5 ) WRITE(5,94) 94 F O R M A T ( ' N OF POINTS SAMPLED = 2 * * I P ' ) WRITE(5,95) 95 F O R M A T ( ' I P . G E . O AND I P . L E . 1 2 IP=?') READ(5,STAR)IP N=DINT(2.D0**IP) WRITE(5,96)N,DT 9 6 F O R M A T ( ' M OF POINTS SAMPLED = ' . I 5 . 2 X . 166 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 1 1 1 112 113 1 14 115 1 16 97 98 99 100 101 102 103 104 105 106 107 108 C C C 1 '@ A SAMPLING INTERVAL ( S E C ) = ' , F 1 0 . 5 ) DURA=DT*(2.D0**IP) WRITE(5,97)DURA F0RMAT( 'DURATION OF RECORD S A M P L E D ' , F 1 2 . 5 ) I F ( I F L A G . E O . O ) GOTO 99 READ(4,98)(Y(I),I=1,N) I F ( I S P E C . E O . 1 ) R E A D ( 7 , 9 8 ) ( X ( I ) ,1 = 1,N) FORMAT(8F10.5) GOTO 101 READ(4,100)(Y(I).1=1,N) I F ( I S P E C . E O . 1) R E A D ( 7 , 1 0 0 ) ( X ( I ) , 1 = 1,N) F0RMAT(8F10.0) WRITE(5.102) F O R M A T C N F S = 0 => NO FREQUENCY SMOOTHING' ) WRITE(5.103) F O R M A T C N F S = NFS => SMOOTH 2*NFS+1 ADJACENT SPECTRAL ORDINATES' READ(5,STAR)NFS WRITE(5,104) FORMAT('NW = 0 => NO DATA TAPERING PRIOR TO FIND P S D ' ) WRITE(5,105) FORMAT('NW = 1 => DATA TAPERING PRIOR TO FIND P S D ' ) READ(5,STAR)NW WRITE(5.106) F O R M A T ( ' S C A L E FACTOR = ? ' ) READ(5,107)SCALE1 F0RMAT(F12.5) AX=O.DO AY=O.DO DO 108 I = 1 , N Y ( I ) = Y ( I )*SCALE 1 AY = AY + Y( I ) IF(ISPEC.EQ.O) X(I)=0.D0 I F ( I S P E C . E O . 1) X ( I ) =X(I ) *SCALE1 • I F ( I S P E C . E Q . 1 ) AX = AX + X ( I ) CONTINUE 1 TAKE OUT THE MEAN AX=AX/N AY=AY/N DO 109 I = 1 . N Y(I)=Y(I)-AY X(I)=X(I)-AX 109 CONTINUE CALL P S D ( X , D A T A X , Y , D A T A Y , D T , N , N F S , N W , P I , I S P E C . A H , C H , C O , P H I . X Y ) C C C C DT N - SAMPLING INTERVAL - NUMBER OF SAMPLE POINTS DELF= 1 .DO/(N*DT) N2=N/2 WRITE(6,STAR)N2 IF ( I S P E C . N E . 1 ) GOTO WRITE(1,STAR)N2 WRITE(2,STAR)N2 WRITE(3.STAR)N2 WRITE(8,STAR)N2 WRITE(10,STAR)N2 111 167 1 17 1 18 1 19 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 C 111 CONTINUE DO 112 I = 1 , N 2 F=(I-1)« DELF W R I T E ( 6 . 110) F . Y ( I ) I F ( I S P E C . N E . 1 ) GOTO 112 W R I T E ( 1 . 110) F . C H ( I ) WRITE(2,110) F . P H I ( I ) W R I T E O . 1 10) F , AH( I ) W R I T E ( 8 . 110) F . C O d ) WRITE(10.110) F , XY(I ) 112 CONTINUE 1 10 FORMAT(F 12 . 5 , 1 X . F 3 0 . 2 0 ) STOP END C C C C C C SUBROUTINE PSD COMPUTES THE POWER SPECTAL DENSITY OF A DISCRETE OUTPUT TIME S E R I E S Y ( I ) OR COMPUTES THE CROSS SPECTRAL DESNSITY BETWEEN TWO DICRETE TIME SERIES INPUT X ( I ) AND OUTPUT TIME S E R I E S Y ( I ) USING FFT TECHNIQUES. C C - D I S C R E T E OUTPUT TIME S E R I E S - THE OUTPUT PSD IS RETURNED IN Y d ) X(I ) - D I S C R E T E INPUT TIME S E R I E S - THE INPUT PSD IS RETURNED IN X ( I ) X Y ( I ) - THE CROSS PSD IS RETURNED IN X Y ( I ) AH( I ) - THE AUTO-TRANSFER FUNCTION IS RETURNED IN A H ( I ) CH( I ) - THE CROSS-TRANSFER FUNCTION IS RETURNED IN C H d ) P H I ( I ) - THE PHASE RELATIONSHIP BETWEEN INPUT AND OUTPUT - IS RETURNED IN P H I ( I ) C0( I ) - THE COHERENCE BETWEEN INPUT AND OUTPUT IS RETURNED - IN C0( I ) Y(I ) C C C c c c c c c c c c c c c c c c c c c c c c c - THE ABOVE RESULTS ARE RETURNED AT - O . D F , 2 * D F . . . .WHERE DF = 1.0/(N*DT ) DT N NFS NW DATAX DATAY FREQUENCIES - SAMPLING INTERVAL OF THE TIME SERIES - 2 * * P , P INTEGER NUMBER OF SAMPLES OF Y d ) - 0 NO FREQUENCY SMOOTHING - NFS SMOOTH 2*NFS+1 ADJACENT SPECTRAL ORDI NATES SEE NEWLANDS P . 1 4 5 - 0 NO DATA TAPERING PRIOR TO FINDING PSD - 1 COSINE DATA TAPER A P P L I E D TO THE TIME SERIES PRIOR TO FINDING THE PSD SEE NEWLANDS P. 146-147 - COMPLEX ARRAY OF X ( I ) - COMPLEX ARRAY OF Y ( I ) PSD(X.DATAX,Y,DATAY.DT.N,NFS.NW.PI ,I SPEC.AH.CH.CO,PHI ,XY) IMPLICIT R E A L * 8 ( A - H . O - Z ) DIMENSION X ( 4 0 9 6 ) , X S ( 4 0 9 6 ) , Y ( 4 0 9 6 ) . Y S ( 4 0 9 6 ) . X Y ( 4 0 9 6 ) , X Y S ( 4 0 9 6 ) DIMENSION CH( 4096 ) ,AH( 4 0 9 6 ) .PHI (4096 ) . CO ( 4096 )'. PHI S( 4096 ) COMPLEX* 16 D A T A X ( 1 ) , O A T A Y ( 1 ) . C X Y I 4 0 9 6 ) PER=DT*N DF=1.DO/PER DO 1 I = 1 . N DATAY(I ) = D C M P L X ( Y ( I ) . 0 . D O ) SUBROUTINE 1 168 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 1 C C C I F ( I S P E C . E O . 1 )DATAX(I ) = D C M P L X ( X ( I ) , 0 . D O ) IF(ISPEC.E0.O)DATAX(I)=DCMPLX{Y(I),0.DO) CONTINUE I F ( N W . E Q . O ) SCALEX=1.DO I F ( N W . E O . O ) SCALEY=1.DO I F ( N W . E Q . O ) GO TO 8 APPLY COSINE DATA TAPER XJ=0.9D0*N SUMY=0.DO SUMX=0.D0 DO 2 1 = 1,N J=I-1 T=J*DT D=1.D0 IF(J.LT.N/10.D0) D=0.5D0*(1.DO-DCOS(10.DO*PI*T/PER)) I F ( J . G T . X J ) D=0.5D0*(1.DO+DC0S(10.DO*PI* ( T-0.9D0*PER)/PER)) D A T A Y ( I ) = D A T A Y ( I )*D SUMY=SUMY+D**2 IF(ISPEC.NE.1)G0T0 2 DATAX(I)=DATAX(I)*D SUMX=SUMX+D**2 2 CONTINUE SCALEY=SUMY/N SCALEX=SUMX/N C C C CALCULATE FOURIER AMPLITUDES 8 CALL DF0UR2(DATAY.N. 1 , - 1 , 1 ) I F ( I S P E C . E Q . 1 ICALL D F 0 U R 2 ( D A T A X . N , 1 , -1 , 1 ) DO 4 1=1,N DATAY(I)=DATAY(I)/SCALEY I F ( I S P E C . E Q . 1 ) DATAX(I)=DATAX(I)/SCALEX 4 CONTINUE C DO 5 1=1,N REY=DREAL(DATAY(I)) AIY = D A I M A G ( D A T A Y ( I ) ) Y ( I ) = (REY**2+AIY**2)*2.DO*DT*DT/PER I F ( I S P E C . N E . 1 ) GOTO 5 REX=DREAL(DATAX(I)) AIX=DAIMAG(DATAX(I)) X ( I ) = (REX**2+AIX**2)*2.D0*DT*DT/PER CXY(I)=(DCONJG(DATAX(I))*DATAY(I))*2.DO*DT*DT/PER REXY=DREAL(CXY(I)) AIXY = D A I M A G ( C X Y ( I ) ) XY(I)*DSQRT(REXY*»2+AIXY**2) IF(REXY.EQ.O.O) PHI(I)=O.DO I F ( R E X Y . E O . O . O ) GOTO 5 P H I ( I ) = D A T A N ( - A I X Y / R E X Y ) * 180.DO/PI 5 CONTINUE C I F ( N F S . E Q . O ) GO TO 12 DO 6 1 = 1 , N YSU)-Y(I) I F U S P E C . N E . 1 ) GOTO 6 XS(I)=X(I) 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 XYS(I)=XY(I) PHIS(I)=PHI(I ) 6 CONTINUE C C C APPLY FREQUENCY SMOOTHING NF2=2*NFS+1 DO 7 1 = 1 ,N SUMY=0.DO SUMX=O.DO SUMXY=0.D0 SUMPHI=O.DO MSUM=NF2 DO 9 J M . N F 2 K=I+J-NFS-2 I F 1 K . L T . 0 ) K=-K K=K+1 I F ( K . G T . N ) MSUM=MSUM-1 I F ( K . G T . N ) GO TO 9 SUMY=SUMY+YS(K) I F ( I S P E C . N E . 1 ) GOTO 9 SUMX=SUMX+XS(K) SUMXY=SUMXY+XYS(K) SUMPHI=SUMPHI+PHIS(K) 9 CONTINUE Y(I)=SUMY/MSUM I F ( I S P E C . N E . 1 ) GOTO 7 X(I)=SUMX/MSUM XY(I)=SUMXY/MSUM PHI(I)=SUMPHI/MSUM 7 CONTINUE C C C C COMPUTE AUTO GAIN FACTOR SQUARED CH AND COHERENCE SQUARED AH . CROSS GAIN FACTOR FUNCTION 12 CONTINUE I F ( I S P E C , N E . 1 ) GOTO 10 DO 11 1 = 1.N C H ( I ) = X Y ( I )*XY( I ) / X ( I )/X( I I AH(I )=Y(I )/X(I ) C0( I ) = XY( I )«XY< I ) / Y ( I )/X( I ) 11 CONTINUE 10 CONTINUE C RETURN END
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Analytical and experimental studies of the behaviour of equipment vibration isolators under seismic conditions Lam, Frank C. F. 1985
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Title | Analytical and experimental studies of the behaviour of equipment vibration isolators under seismic conditions |
Creator |
Lam, Frank C. F. |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | Analytical and experimental studies of the behaviour of equipment vibration isolators under seismic conditions are presented. A preliminary parametric study of the effect of equipment-structure interaction on the ultimate equipment response of general equipment-structure systems is considered first. The results of this study indicate the conditions under which a non-interactive approach can yield adequate ultimate equipment response estimates. A model of a prototype air handling unit mounted on vibration isolators was constructed for use in the experimental studies. Two types of vibration isolators -elastomeric isolators and open spring isolators with uni-directional restraint - were tested under static and dynamic conditions. The frequency and seismic response characteristics of these vibration isolated systems were obtained. The experimental results indicate that the vibration isolators have nonlinear stiffness characteristics and high damping values. The results also show that the elastomeric isolators can survive a substantially higher level of base excitation than the open spring isolators with uni-directional restraint Analytical models of the vibration isolated systems, based on the model identification test results, have been formulated. A numerical procedure, utilizing time series analysis, was used to solve the equations of motion of the systems. Good agreement between the experimental results and the analytical results was observed. This study indicates that the analytical procedure can be used to accurately predict the response characteristics of vibration isolated equipment systems subjected to known base excitation inputs. |
Subject |
Vibration - Measurement |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062953 |
URI | http://hdl.handle.net/2429/25110 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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