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Response of a flexible marine column to base excitation Vernon, Thomas A. 1984

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RESPONSE OF A FLEXIBLE MARINE COLUMN TO BASE EXCITATION  by  Thomas A. Vernon B.A.Sc, The University of B r i t i s h Columbia, 1982  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering  We accept this thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA February 1984 ©  Thomas A. Vernon, 1984  In presenting  this thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an  advanced degree at the University of B r i t i s h Columbia, I agree that Library s h a l l make i t freely available for reference  and study.  the  I further  agree that permission for extensive copying of this thesis for scholarly purposes may  be granted by the Head of my Department or by his or her  representatives.  It i s understood that copying or publication of this thesis  for f i n a n c i a l gain s h a l l not be allowed without my  Department of C i v i l Engineering  The University of B r i t i s h Columbia 2324 Main Mall Vancouver, B.C. V6T 1W5  written permission.  ii ABSTRACT  The displacement response of a f l e x i b l e , surface-piercing cylinder subjected to a u n i d i r e c t i o n a l base motion i s considered i n this study. Laboratory experiments have been performed  with a c i r c u l a r , fixed-base  model using sinusoidal and scaled seismic input motion.  Sinusoidal tests  were designed to investigate the dependence of cylinder t i p response on the r a t i o of base motion frequency to cylinder natural frequency and base displacement amplitude, for a fixed water depth, i n e r t i a r a t i o and damping r a t i o .  Further tests with a base motion corresponding to past  earthquake records were then used to determine the cylinder's response to seismic e x c i t a t i o n . The sinusoidal test results are compared with predictions derived from analyses of the motion i n terms of the f i r s t and second undamped mode shapes of a cantilever beam.  The Morison equation i s used to  estimate hydrodynamic loads i n this formulation, and three treatments of the drag term i n the equation of motion are considered:  neglect of drag,  drag l i n e a r i z a t i o n and retention of the complete nonlinear form.  A  closed-form solution for the former two approximations i s developed, and a numerical approach i s adopted for the complete nonlinear formulation. The numerical method i s used to predict the response of the column to seismic input. The dependence of cylinder t i p displacement on frequency r a t i o and base motion amplitude follows predictable patterns of dynamic response. Peak amplitudes occur at resonance and increase with base motion amplitudes.  However, this relationship i s not linear because of the  damping contribution of nonlinear drag forces near the free surface.  iii The numerical and l i n e a r i z e d drag predictions agree well with the experimental response i f a suitable choice of drag c o e f f i c i e n t i s made. The neglect of drag r e s u l t s i n very conservative resonant response predictions for large excitation amplitudes i n which the free surface Keulegan-Carpenter number exceeds about 4. Drag forces can generally be neglected, however, i n the estimation of response to earthquake motion because displacement amplitudes are small. In extreme cases, small l i f t  forces can result from flow separation about  the cylinder near the free surface.  iv  TABLE OF CONTENTS PAGE ABSTRACT  i i  LIST OF FIGURES  vi  LIST OF TABLES  viii  ACKNOWLEDGEMENTS 1.  ix  INTRODUCTION  1  1.1  Introduction  1  1.2  Background  2  1.3  Literature Review  6  1.3.1  6  Earthquake Response of Structures i n Fluids ..  1.3.2. Response i n O s c i l l a t i n g Flows  7  1.3.3. Hydrodynamic Drag and Damping  9  1.3.4. Related Topics 2.  3.  10  THEORETICAL DEVELOPMENT  12  2.1.  Problem D e f i n i t i o n  12  2.2.  Dimensional Analysis  13  2.3.  Hydrodynamic Force Formulation  16  2.4.  Modal Analysis  17  2.4.1. Neglect of Drag Forces  18  2.4.2. Linearized Drag  24  2.4.3. Numerical Method with Nonlinear Drag  28  TESTING FACILITIES AND MODEL PARAMETERS  32  3.1.  Testing F a c i l i t i e s  32  3.2.  Model Parameters  33  3.3.  Data Acquisition  35  V PAGE  4.  5.  6.  DESCRIPTION OF EXPERIMENTS  36  4.1.  Damping Tests and System Characteristics  36  4.2.  Sinusoidal Tests  37  4.3.  Seismic Motion Tests  39  DISCUSSION OF RESULTS  41  5.1.  Response Functions from Sinusoidal Tests  41  5.2.  Response Predictions for Sinusoidal Motion  42  5.3.  Transverse Response  45  5.4.  Response to Seismic Input  46  5.5.  Numerical Prediction of Seismic Response  48  CONCLUSIONS AND RECOMMENDATIONS  49  6.1.  Conclusions  49  6.2.  Recommendations for Further Study  51  BIBLIOGRAPHY  72  APPENDIX A  -  MODAL ANALYSIS  76  APPENDIX B  -  DERIVATION OF RESPONSE AMPLITUDES  82  APPENDIX C  -  DESCRIPTION OF COMPUTER PROGRAM  87  APPENDIX D  -  INSTRUMENT CALIBRATION AND TYPICAL DATA  89  vi LIST OF FIGURES FIGURE  PAGE  1  Coordinate System D e f i n i t i o n  52  2  Mode Shapes for a Cantilever Beam  52  3  Schematic of Test F a c i l i t y  53  4  Shake Table  54  5  Testing Tank Over Shake Table  54  6  Instrumented Rod for Strain Measurement  55  7  Photo of Instrumented Rod  55  8  Free Vibration Damping i n A i r  56  9  Free Vibration Damping i n Water  56  10  Photo of Sinusoidal Test of 4.5 Hz  57  11  Photo of Sinusoidal Test at 28 Hz  57  12  Fourier Spectra for E l Centro 1940  58  13  Fourier Spectra for San Fernando 1971  58  14  Response Function for Y /D = .020  59  15  Response Function for Y /D = .036  60  16  61  18  Response Function f o r Y /D = .054 g Response Function f o r Y /D = .072 g Drag Coefficient versus Reynolds Number f o r Various Values of K  19  Ratio of Y  17  g  , ,Y at Resonance versus Drag Coefficient at pred/ exp Various Base Displacement Amplitudes  62 65  65  20  Numerical Simulation of the Formation of Asymmetric Vortices ... 66  21  In-line and Transverse Response Near Resonance f o r Y /D = .054 . 66  vii PAGE  FIGURE  22  Transverse Response Functions for Various Base Displacement Amplitudes  23  67  Tip Locus at Resonance for Y /D = .054  68  8 24  Tip Locus o f f Resonance for Y /D = .054 g  68  25  Response comparison for E l Centro  69  26  Response comparison for San Fernando  69  27  Transverse Time History f o r San Fernando  70  28  Response Prediction Comparison for E l Centro  70  29  Response Prediction Comparison f o r San Fernando  71  30  Calibration Curve for Instrumented Rod  90  31  Typical Time History at 3rd Subharmonic  91  32  Fourier Spectra for Record of Figure 31  91  33  In-line Resonance Time History at Y /d = .036 g Transverse Resonance Time History at Y /d = .054  92  34  92  viii  LIST OF TABLES  TABLE 1  PAGE Resonant Response Ratios Y ,/Y pred exp Experimental Data  63  r  2  >  64  ix  ACKNOWLEDGEMENTS  I wish to express many thanks to Dr. M. Isaacson and Dr. S. Cherry for their guidance and support i n the preparation of t h i s t h e s i s , and to Chris Dumont for his help i n the Earthquake Laboratory.  1 1.  1.1  INTRODUCTION  Introduction The p r e d i c t i o n of s t r u c t u r a l response to hydrodynamic loads i s of  major importance i n the design of many marine f a c i l i t i e s .  As a  contribution to t h i s general subject, t h i s thesis endeavours to Investigate methods of response prediction applicable to compliant marine columns undergoing base motion. Hydrodynamic forces are generated when r e l a t i v e motion e x i s t s between the f l u i d and  structure such as occurs as a r e s u l t of wave and/or  current a c t i o n or seismic e x c i t a t i o n of the structure.  In the case of a  compliant marine structure, the f l u i d - s t r u c t u r e system becomes i n t e r a c t i v e since the response and applied forces are then coupled. the available damping i s low, dynamic amplification at resonance become important.  If  may  An investigation of a quasi-resonance phenomena i s of  merit i n the case of seismic excitation because system damping i s often low (<10%  c r i t i c a l ) and natural frequencies of larger structures often  tend to coincide with earthquake spectral density peaks (1-5 Hz).  This  proximity does not generally occur i n the case of wave loading because of the lower frequencies associated with large ocean waves. The hydrodynamic forces generated by the motion of a body i n a s t i l l f l u i d result from i n e r t i a forces associated with the acceleration of a volume of f l u i d , and from drag forces associated with flow  separation.  For small r e l a t i v e motions t y p i c a l of earthquake ground motions, these forces w i l l be mainly i n e r t i a l .  For f l e x i b l e structures, dynamic  amplification enhances the p o s s i b i l i t y of larger displacements leading to flow separation.  Where separation occurs, form drag, vortex  formation  2  and possibly l i f t forces can r e s u l t .  In the case of base motion, form  drag i s energy d i s s i p a t i v e , and i s treated as an added damping.  However,  the  associated l i f t forces may synchronize with the natural frequency of  the  structure and i n extreme cases, lead to large transverse  oscillations.  Such a mechanism has been the cause of f a i l u r e of numerous  free-standing p i l e s i n waves. For  purposes of t h i s t h e s i s , a r e l a t i v e l y simple system i s  considered: a surface-piercing c i r c u l a r c y l i n d r i c a l structure i s excited with u n i d i r e c t i o n a l , horizontal base motion.  The i n - l i n e and transverse  responses to a range of input motion amplitudes and frequencies have been recorded and the i n - l i n e responses compared to predictions based on a modal analysis assuming both linear and nonlinear hydrodynamic Although several powerful methods, such as lumped-parameter  loading.  modelling and  f i n i t e element methods, exist for the treatment of this problem, the use of modal techniques for the continuous structure provides a convenient and straightforward approach without the need to resort to the extensive computing or modelling demanded by other methods.  (See K i r k l e y and  Murtha, 1975, for the former; Liaw and Chopra, 1973, for the l a t t e r ) . Transverse responses are not compared to any numerical predictions because a simple predictive mathematical model of l i f t generally a v a i l a b l e at t h i s time.  force i s not  It must be noted, however, that  transverse o s c i l l a t i o n s can fundamentally affect the near resonant response of a marine p i l e or tower. 1.2  Background Fluid-structure i n t e r a c t i o n i n the marine environment i s generally  assumed to occur i n either a separated or unseparated (potential) flow regime.  Viscous e f f e c t s can be included i n the former, but must be  3 neglected In p o t e n t i a l flow solutions. assumption  The determining factor i n the  of an appropriate flow regime i s the magnitude of r e l a t i v e  displacement between the structure and f l u i d , often expressed i n terms of the Keulegan-Carpenter  number, K.  This i s defined, for a  two-dimensional  sinusoidal o s c i l l a t i o n of a c i r c u l a r cylinder, as (Keulegan, Carpenter (1958)). K where  A  amplitude of o s c i l l a t i o n displacement  D  cylinder diameter  The Keulegan-Carpenter separation i n the problem.  number indicates the significance of flow For K values of roughly 10 or more, the  o s c i l l a t i o n amplitude to cylinder diameter r a t i o w i l l be r e l a t i v e l y large, and the hydrodynamic loads are primarily form drag associated with the flow separation. small body regime.  The problem i s then defined as lying within the  On the other hand, i f K i s small, flow separation  does not generally occur, i n e r t i a forces are dominant and potential flow theory can provide a good flow f i e l d aproximation.  Overlaps of these  c l a s s i f i c a t i o n s c e r t a i n l y exist (for a f u l l e r account see Sarpkaya and Isaacson, 1981), and a p a r t i c u l a r problem may encompass several flow classes. The s o l u t i o n approach d i f f e r s i n the case of large and small body problems.  The former i s generally formulated using potential flow theory  for an i n v i s c i d f l u i d and an i r r o t a t i o n a l flow f i e l d .  The v e l o c i t y of  any point i s specified as the gradient of a scalar potential function which s a t i s f i e s the Laplace equation and appropriate boundary conditions.  4 The pressure may be obtained  from the p o t e n t i a l using the unsteady  Bernoulli equation and thus once the potential function has been determined, the hydrodynamic loads may be calculated from an integration of the pressure around the body. the determination  This type of approach i s applicable to  of wave or seismic loading of large structures such as  o i l storage tanks or gravity platform bases where A/D  i n (1.1) i s small.  Local flow separation due to sharp geometries can occur on large structures and leads to inaccuracies i n solutions obtained flow assumptions.  from potential  The s i g n i f i c a n c e of these errors should be  given  consideration i n such cases. In the small body regime, viscous e f f e c t s are included i n the hydrodynamic force derivation.  Experimental evidence suggests that  viscous forces r e s u l t i n g from flow separation can become important at Keulegan-Carpenter numbers greater than about 2 (A/D o s c i l l a t i n g flows about b l u f f bodies. can occur for K > 5 (A/D  > .7).  - 0.3 i n (1.1)) for  Vortex shedding and l i f t  forces  Although seismic base motion amplitudes  are u s u a l l y small such that K < 5 over most portions of a structure, the viscous forces may become s i g n i f i c a n t near the free surface. viscous e f f e c t s , the hydrodynamic loads are u s u a l l y formulated  To include using the  well known Morison equation, deriving from the work of Morison, et a l (1950).  This equation i s based on the simultaneous addition of an  i n e r t i a force associated with i n v i s c i d f l u i d acceleration and a drag force analogous to that i n a steady flow, and i s given i n simplest form for a stationary c i r c u l a r cylinder as:  F' = C m  DO  (1.2)  5 Here, F' i s the f l u i d force per unit length, p i s the f l u i d density, D i s the body diameter, U and U are the f l u i d v e l o c i t y and acceleration respectively (using the dot notation to imply time d i f f e r e n t i a t i o n ) , and C  ffl  and Cj are i n e r t i a and drag c o e f f i c i e n t s r e s p e c t i v e l y .  The  n o n l i n e a r i t y of the forcing function i s apparent i n the drag term.  The  force c o e f f i c i e n t s C,, C depend on the dimensionless flow and structure d m parameters such as l o c a l Reynolds number, Keulegan-Carpenter number and surface roughness.  These c o e f f i c i e n t s are not fundamental constants and  must be determined experimentally. The problem considered i n t h i s study encompasses both flow regimes. Near the structure base, the displacement i s e s s e n t i a l l y the base motion input.  In the case of seismic input these displacements are generally  less than 20 cm and for an average size p i l e or platform l e g , the maximum Keulegan-Carpenter number would then be roughly 0.5 or l e s s .  However,  many marine structures are highly dynamic, with fundamental frequencies of o s c i l l a t i o n t y p i c a l l y i n the range 0.2 to 1.0 Hz, a range of frequency also t y p i c a l of earthquake spectral i n t e n s i t y maxima (See Figures 12 and 13).  The proximity of these frequencies leads to the possible quasi-  resonance condition commonly associated with seismic excitation of large structures.  Because dynamic amplification at resonance can be of order  10 or more, a Keulegan-Carpenter number of 5 can be attained near the free surface. lift  This value indicates that flow separation and possibly  forces may become important i n the problem. A bottom fixed marine structure can consequently span both flow  regimes:  a potential flow near the base (and nodes If we consider the  response to be modal) and a separated flow i n regions of large s t r u c t u r a l displacements.  The present study uses the Morison equation to estimate  6 hydro-dynamic loads, since that formulation w i l l e s s e n t i a l l y be v a l i d i n the case of small structures, for the entire span, aside from effects of wave generation near the free surface.  1.3  L i t e r a t u r e Review A number of more recent investigations of structure response to  hydrodynamic loads are relevant to the present study. divided here into several broad categories:  These have been  response to seismic loading,  response i n o s c i l l a t i n g flows, damping i n f l u i d s and a d d i t i o n a l miscellaneous t o p i c s .  1.3.1.  Earthquake Response of Structures i n Fluids  Clough (1960) performed v i b r a t i o n tests of f l e x i b l y mounted c y l i n d r i c a l shapes i n a i r and water.  The study determined added mass and  damping values f o r f i r s t and second mode natural frequencies using the different frequencies of o s c i l l a t i o n i n the two mediums. Clough found agreement between the experimental values of added mass and those derived from potential flow theory. continuous  The response of a pseudo-  structure subjected to base loading was a l s o investigated.  However, the applied displacement  i n these tests was generated by a  pendulum s t r i k i n g the edge of a table and d i d not simulate r e a l i s t i c seismic input.  Clough concluded  that structural o s c i l l a t i o n s r e s u l t i n g  from seismic e x c i t a t i o n would not be large enough to cause flow separation. If this conclusion i s accepted, potential flow theory can be used to predict marine structure response during earthquakes. A comprehensive a n a l y t i c a l study of the response of e l a s t i c structures surrounded by a f l u i d using p o t e n t i a l flow theory was  7  presented by Liaw and Chopra (1973).  The authors u t i l i z e d a modal  analysis technique i n combination with a f i n i t e element method to predict s t r u c t u r a l response. No experimental v e r i f i c a t i o n s were given i n t h i s study. Much of the work on the seismic response of marine structures has been concerned with large, non-elastic structures such as submerged o i l tanks.  In these cases, most authors use a p o t e n t i a l flow formulation  similar to that used for load prediction i n the wave d i f f r a c t i o n regime. Such studies include those by Byrd (1978), Tung (1979), Mei (1979), Westermo (1980), and Isaacson (1983).  A review of many of the methods  used by the above authors, as well as others, i s presented by Eatock Taylor (1981), for the case of dams and offshore structures. Kirkley and Murtha (1975) have studied the response of offshore structures i n earthquakes, comparing several techniques of response prediction using lumped-parameter models.  A Morison forcing function was used and a  direct integration procedure compared with an uncoupled approximation. Their r e s u l t s indicate that the l i n e a r i z a t i o n technique i s accurate i n higher frequency ranges, but unconservative for low frequencies.  The  same authors have presented response spectra for offshore structure design based on the numerical solution of the linearized equations ( K i r k l e y and Murtha,  1.3.2.  1975).  Response of F l e x i b l e Structures i n O s c i l l a t i n g Flows  There exists a very s i g n i f i c a n t volume of l i t e r a t u r e concerning the hydrodynamic  loads imposed on structures i n o s c i l l a t i n g flows. However,  very l i t t l e of t h i s work a c t u a l l y concerns e l a s t i c structure response to these loads.  8 An e a r l y work by Selna and Cho (1972) presented a numerical integration and time step technique for d i s c r e t i z e d models of marine structures.  This approach can incorporate the nonlinear coupled drag  c h a r a c t e r i s t i c s , although such direct integration methods are very c o s t l y for more than a few degrees of freedom.  Blevins (1977) has considered  the i n - l i n e response of continuous structures i n o s c i l l a t o r y flows with zero and nonzero means. He provides a solution based on a Fourier expansion of the linearized equation of motion.  An assumption of small  structure response i n comparison to the f l u i d motion i s inherent i n this l i n e a r i z a t i o n and solution approach.  Such an assumption cannot be  made i n an analysis of seismic response and other l i n e a r i z a t i o n techniques must be used. A number of authors have presented numerical methods for the response solution using integration of the motion equations derived for d i s c r e t i z e d models. Anagnostopoulos  These include Sawaragi et a l (1977), and  (1982), who uses an uncoupled Morison forcing function.  These numerical techniques are expensive, but very powerful and the limitations on their accuracy are imposed mainly by uncertainty i n the input data. The transverse response of e l a s t i c a l l y restrained cylinders i n o s c i l l a t o r y flows has been investigated by several authors.  Sarpkaya  (1979, 1981), has performed extensive experimentation relevant to this subject and presented correlations of transverse response with system parameters such as reduced v e l o c i t y and Keulegan-Carpenter  number.  Similar correlations were presented by Isaacson and Maull (198l) for linear mode models i n progressive waves.  Zedan and Yeung (1980)  investigated p i l e dynamics under lock-on conditions i n waves at moderate  9  Keulegan-Carpenter  numbers.  Lock-on was found to occur over only a  narrow bandwidth and to be highly interactive with the i n - l i n e response. McConnel and Park (1982) o s c i l l a t e d e l a s t i c a l l y restrained cylinders i n a s t i l l f l u i d and concluded that the frequency r a t i o parameter i s more of a c o n t r o l l i n g factor i n transverse response than K or reduced v e l o c i t y as suggested by other authors.  Skop and G r i f f i n (1975) presented an  a n a l y t i c a l model of the fluctuating l i f t force c o e f f i c i e n t and used this model i n conjunction with modal superposition to obtain transverse response p r e d i c t i o n s .  The model predictions were compared to experiments  and the results follow at least similar trends.  1.3.3. Hydrodynamic Drag and Damping The drag forces acting on marine structures are nonlinear and, i n a dynamic system, coupled to the structure response.  Because of this  coupling the drag force can be expressed i n terms of an exciting force, associated with the incident flow and a damping force associated with the s t r u c t u r a l motion.  The estimation of such damping forces can be very  important when resonant response i s a p o s s i b i l i t y .  Several authors have  considered the treatment of these exciting forces and damping terms.  Dao  and Penzien (1980) have presented a comprehensive comparison of methods of  handling nonlinear drag forces using d i s c r e t i z e d models.  methods again involve assumptions  These  on the r e l a t i v e magnitudes of the  displacements (uncoupling) or i n some cases assume harmonic response. The l i n e a r i z a t i o n techniques are modified to give an approximation of the response to non-harmonic motion. Moe and Verley (1980) have considered the hydrodynamic damping of offshore structures i n various flow situations.  They conclude that the  10 use of the Morison equation may give unconservative response predictions because the normally assumed values of drag c o e f f i c i e n t s are too high. This i s a r e s u l t of an assumption of separated flow inherent i n the Morison approach.  Separated flow does not occur where motion amplitude  to diameter r a t i o s are small and i n such cases drag damping i s overestimated by a choice of drag c o e f f i c i e n t based on that  assumption.  Sugiyama and Ito (1981) have considered the treatment of nonlinear drag damping as a function of vibration amplitude for a single degree of freedom system. A polynomial function of Reynolds number was found appropriate for the drag c o e f f i c i e n t .  The function was  determined  empirically from experiments.  1.2.4.  Miscellaneous Related Topics  Two related areas of investigation are the prediction of marine r i s e r response, and the non-deterministic approach to offshore structure response.  Kirk, et a l (1979) have developed a normal mode solution i n the  frequency domain for a marine r i s e r subjected to wave loading and top e x c i t a t i o n from horizontal motions of the support platform. nonlinear drag term was conditions.  The  found to be very important at resonance  A dynamic analysis of a multi-tube r i s e r has been given by  Grecco and Utt (1982). Chakrabarti and Frampton (1982) have presented a  comprehensive  review of the many techniques used i n the analysis of marine r i s e r s . These solutions are i n v a r i a b l y based on numerical methods because of the complexity of the governing equations of motion. Non-deterministic methods of response analysis have been presented by Malhotra and Penzien (1970), Penzien and Dao (1980).  Other references  and a review of the p r i n c i p l e s of non-deterministic analyses can be found in Sarpkaya and Isaacson (1981).  12 2.  THEORETICAL DEVELOPMENT  The purpose of this chapter i s to formalize the problem under consideration and to develop the methods of response p r e d i c t i o n used i n this study.  A dimensional analysis i s presented which defines the  relevant parameters i n the problem.  The modal analysis method of  response prediction i s developed for three forms of the hydrodynamic drag force i n the equation of motion; forces.  zero, l i n e a r i z e d and nonlinear drag  A closed-form solution i s obtained for the zero and l i n e a r i z e d  drag cases;  a numerical approach i s given for the nonlinear form.  2.1. Problem D e f i n i t i o n This study i s concerned with the general subject of response prediction for submerged f l e x i b l e structures undergoing base motion.  As  a fundamental case characterizing t h i s problem, a s i n g l e , f l e x i b l e , surface-piercing c i r c u l a r cylinder i s considered.  For a constant mass,  length and diameter of cylinder, and constant f l u i d properties, the displacement response w i l l be a function of base motion amplitude and frequency. This dependence i s investigated using sinusoidal input.  The  response of f l e x i b l e columns to earthquakes i s then investigated using scaled seismic inputs. The s o l u t i o n approaches used i n t h i s study necessitate a number of assumptions about the flow and structure c h a r a c t e r i s t i c s i n the problem. These assumptions include the following: 1)  The problem exists i n the small body flow regime where the Morison equation can be used to predict hydrodynamic loads.  2)  The structure can be considered as a uniform cantilever beam and  appropriate beam theory used to develop the equation of motion. The response i s assumed to be primarily i n the f i r s t mode and the undamped mode shapes are assumed to adequately describe the response. 3)  The column i s free standing, with no t i p mass and i s v i r t u a l l y submerged.  The c r i t i c a l response i s taken as that at the free  surface. 4)  Seismic base motion amplitude i s t y p i c a l of that recorded on firm ground at moderate distances from the focus of a strong motion earthquake but i s limited to a u n i d i r e c t i o n a l and horizontal component.  5)  Base motion i s prescribed, circumventing the problem of s o i l structure Interaction.  With these assumptions, response prediction methods are developd using the Morison equation to estimate hydrodynamic loads and a modal analysis to solve the equation of motion.  Response predictions are  compared to experimental results from model t e s t s .  2.2. Dimensional Analysis Dimensional analysis can be used to define the dimensionless parameters important i n a particular problem.  The present study intends  to model the displacement response of a f l e x i b l e marine column undergoing base motion.  The relevant independent variables are then c h a r a c t e r i s t i c s  of the structure, f l u i d and motion.  The variables are:  D  diameter of cylinder  f  fundamental frequency of structure i n water  14 m  structure mass per unit length damping r a t i o i n a i r  v  kinematic v i s c o s i t y of f l u i d  P  f l u i d density  H  depth of f l u i d  U  v e l o c i t y amplitude at base ( f o r sinusoidal motion)  t  time  f  forcing frequency ( f o r sinusoidal motion)  Dimensional analysis w i l l provide seven independent dimensionless groups from these ten v a r i a b l e s .  Using conventional groups, time-  invariant representative values of cylinder t i p response can be written as:  Y/D.Z/D = f l ( f / f , C, D/H, m/pD , K, Re) 2  n  Here, Y and Z are i n - l i n e and transverse displacement amplitudes respectively, of the column t i p . are  The dynamic properties of the structure  represented by the non-dimensional groups f/f » the frequency r a t i o , n  £, the damping r a t i o , and m/pD , the i n e r t i a r a t i o . 2  The natural  frequency and damping r a t i o may be taken as values either i n a i r or water. UD/v,  The Keulegan-Carpenter  number K = U/fD, and Reynolds number, Re =  characterize the motion at the structure base. In general, the a p p l i c a b i l i t y of model test r e s u l t s depends upon the  similtude observed between prototype and model values of these dimensionless groups.  As i s common i n many f l u i d - s t r u c t u r e modelling  15 problems, Reynolds number similtude Is d i f f i c u l t to maintain and i t s significance i s not investigated i n this study.  A constant damping r a t i o  i s also d i f f i c u l t to maintain because s t r u c t u r a l damping i s not e a s i l y controlled.  However, damping ratios of both model and prototype are  expected to be low (< 10% c r i t i c a l ) . For a s p e c i f i c choice of cylinder and water depth, the damping r a t i o , diameter to depth r a t i o D/H,  and i n e r t i a r a t i o are constants.  The  non-dimensional i n - l i n e response investigated here can then be described as:  Y/D = f ( f / f , 2  n  U/fD)  The dependence of the response on the frequency r a t i o r e s u l t s i n a dimensionless transfer type function.  The Keulegan-Carpenter number  characterizes the response dependence on the amplitude of base motion. Although K i s defined at the structure base, the response dependence on K i s important only at higher values of this parameter such as occurs at the free surface at resonance. The choice of dimensionless groups i s not unique, and i n - l i n e and transverse response can be expressed i n terms of alternate parameters such as the reduced v e l o c i t y U = U/f D = K/(f / f ) , for which ' r n n correlations exist i n the case of r i g i d cylinder response (Sarpkaya, 1979). of K.  However, the reduced v e l o c i t y i s more suitable at larger values In his investigations of the hydroelastic o s c i l l a t i o n s of a r i g i d  cylinder, Sarpkaya has also included a roughness factor, K /D r  and  combined the Reynolds and Keulegan-Carpenter numbers as Re/K = D /vT. 2  16 Isaacson & Maull (1981) and Sarpkaya (1979) have expressed transverse displacements of cylinders i n harmonically o s c i l l a t i n g flow as a function of the parameter  A = (U/f D) /(m£/pD ). 2  2  n  for the case of f l e x i b l e column motion;  Similar correlations may exist however, these correlations are  much more d i f f i c u l t to obtain experimentally and are not addressed i n this thesis.  2.3  Hydrodynamic Force Formulation The Morison equation i s used i n t h i s study to estimate the  hydrodynamic loads acting on the structure as i t o s c i l l a t e s i n water. With reference to Figure 1, the most general form of the equation, f o r a body moving i n a uniform f l u i d flow f i e l d i s given as (Sarpkaya & Isaacson, 1981).  F» = PV(1 + C )U - V C Y a  P  a  + j  t  p^ApCU-Y^IU-Yj  (2.1)  Here, F' i s the force per unit length, p the f l u i d density, U(x,t) **  •  represents the flow f i e l d , Y , Y  and Y  are the t o t a l body motions as  defined i n Figure 1, the dot notation implying time derivative, V i s the displaced volume of the body per unit length, A normal to the plane of motion and C mass c o e f f i c i e n t s .  n  and C  p  i s the projected width  are empirical drag and added  The added-mass and drag components are seen to depend  on the r e l a t i v e motion between structure and f l u i d . the undisturbed f l u i d i s stationary.  In the present case,  Equation (2.1) therefore reduces to  17  (2.2)  where the projected width Ap i n (2.1) has been replaced by the diameter D of the cylinder.  2.4  Modal Analysis The c y l i n d r i c a l marine tower considered i n t h i s study can be  considered as a uniform cantilever beam and the governing d i f f e r e n t i a l equation of motion developed from appropriate beam theory (see Appendix A).  For the support motion problem, defined i n Figure 1, the  d i f f e r e n t i a l equation i n terms of t o t a l displacement can be separated into two components;  6 ( t ) , the known input at the structure base, and  Y(x,t) the displacement r e l a t i v e to the base.  The d i f f e r e n t i a l equation  including the s t r u c t u r a l damping term, can be written as (Clough and Penzien,  1975).  m (x)Y(x,t) + EIY  (x,t) + CIY ( x , t ) Y ( x , t ) = V  - m (x) 6(t) + p(x,t)  (2.3)  Q  where the superscript denotes d i f f e r e n t i a t i o n with respect to x, ^ ( x ) i s the mass per unit length, C i s the c o e f f i c i e n t of i n t e r n a l damping and p(x,t) i s the forcing function here defined as i n Equation (2.2). the e l a s t i c modulus and I the moment of i n e r t i a of the section.  E is  The same  separation technique applied to t o t a l displacement can be applied to the forcing function p(x,t) since both drag and added mass depend on t o t a l r e l a t i v e motion between f l u i d and structure.  Equation (2.3) can then be  18 rewritten as  mY + CIY Y + EIY lv  iv  =  -  m5  - ^. C D|Y4 P  D  (2.4)  |(Y-^)  i n which m now includes the added mass,  m = m  o  + p—-.— C  (2.5)  a  and the dependence of Y on x and t i s now implied. Equation (2.4) i s a coupled nonlinear d i f f e r e n t i a l equation and must be solved numerically or simplified by making certain assumptions about the r e l a t i v e importance of the various terms.  Two  s i m p l i f i c a t i o n methods  and a numerical approach are considered. 2.4.1  Neglect of Drag Forces For small amplitudes of response and forcing, the nonlinear term i n  Equation (2.4) i s small i n comparison to the i n e r t i a l terms.  If this  term i s neglected, Equation (2.4) becomes l i n e a r and modal analysis can be used to obtain a solution for the response.  In the case of harmonic  base motion forcing 6 ( t ) , a closed-form solution i s obtainable.  In the  present study, a solution including only the two lowest modes (See Figure 2) i s developed, because the large frequency increments between modes precludes any s i g n i f i c a n t contribution from higher modes.  In p r i n c i p l e ,  a solution involving any number of modes can be developed.  The f a m i l i a r  techniques of modal analysis are applied i n the linear aproximation Clough and Penzien, 1975).  Y(x,t) = I <j>(x) 5 ( t ) r  r  (see  A separable series solution i s assumed:  (2.6)  19 where £ (t) r  *  sa  time dependent amplitude function and ^ ( x ) i s the r - t h  mode shape of a cantilever beam given by  Ax <j) (x) = cosh — c o s r  Ax —£  Ax o (sinh -j-  Here, £ i s the length of the c a n t i l e v e r , A  cosAcoshA  Ax sin -y- )  r  (2.7)  i s a solution of  +1=0  (2.8)  and sinhA -sinA o  = r  (2.9) coshA +cosA r r  Substituting Equation Equation  (2.6) into the linear ( C = 0) form of Q  (2.4) and using orthogonality of mode shapes to decompose the  p a r t i a l d i f f e r e n t i a l equation (see Appendix A) we obtain the system of single degree of freedom equations  i ( t ) + 25 w I (t) + w 5 ( t ) = P  (t)  2  r  r  (2.10)  r  Here P^ (t) i s the generalized i n e r t i a l load. r  An assumption of  equivalent viscous damping has been assumed for the s t r u c t u r a l damping, with the damping r a t i o given as  (2.11)  20 and the generalized i n e r t i a l load given as  P  (t) - o 6 ( t )  x  (2.12)  r  where t \* t(x)dx \a J< m(x)(}> o  (2.13)  l  r  = —n  «  /Vx)^(x)dx o  For  harmonic excitation given by  6(t) = Y s i n ut  (2.14)  the generalized i n e r t i a l forcing function i s given as P (t) = ct u>Y s i n uit I r g r 2  T  x  (2.15) '  In this case, a steady state solution can be found d i r e c t l y by substitution of the harmonic form i n (2.10).  a coY 2  5 (t) r  = - — u  S  sin(oot - 6 ) -  Ld-("/  r  w  = - a A sin(oot-9 ) r  r  r  (2.16)  ) ) +(2? ^ ) ] 2  r  This yields  2  2  r  r  where the phase angle 6^ i s given by  2£ u)/u 6  r  =  t a n _ 1  ^  l4^T)r  2)  ( 2  *  1 7 )  and wY 2  A  r  = w i(l-(u)/u) ) ) + (2C u/u ) J r r r r 2  2  2  2  1/2  (2.18)  21  The response of the column r e l a t i v e to the base i s then  Y(x,t) = - I • (x) oc A sin(wt - 0 ) r=l r  (2.19)  r  and the t o t a l response i s  Y  t  = Y(x,t) + 6(t)  (2.20)  For the uniform cantilever beam used i n the present study  I  *r< > * X  (2.21)  d  i <t>(x)dx o  r  2  where X and f  1977).  are defined as i n Equation (2.8) and (2.9) (See Blevins,  The t i p response can be obtained by evaluating <J>(JO and r  substituting this r e s u l t , with those of (2.16) and (2.19), into (2.20). Using a f i r s t mode approximation, t h i s yields  = 2 [ I + 0.25 Y + Y S i c o s e J / i g g 2  tip  L  2  1  1  (2.22)  2  J  where the bar indicates amplitude, and Q\ i s as i n (2.17).  The addition  of higher mode contributions i s straightforward but a l g e b r a i c a l l y inconvenient because of the increasing number of phase angles which must  22  be combined (See Appendix B).  The two mode approximation i s given as  Y.,  tip  =  2[I  2  1  +  I  2  2  +  0.25  Y  2  g  +  Y  (I  cos  Bj-  g 1  + 25^2 cos (0! - e ) ]  l  2  c o s  6 ) 2  (2.23)  1 / 2  2  and 6 i s as i n (2.17). f  As discussed e a r l i e r , the large modal frequency separation makes the i n c l u s i o n of more than the two lowest modes unnecessary i n t h i s analysis. The functional dependence of t i p displacement  on the frequency r a t i o  parameter can be generated from Equation (2.23) by simple calculation at discrete frequencies and specified forcing amplitudes and compared to experimental  results.  For a random input function such as seismic base motion the formulation cannot be reduced to give a closed-form  solution i n the time  domain and must be integrated numerically to obtain the time s e r i e s . this case, i t i s usually more convenient  In  to work i n the frequency domain,  using Fast Fourier Transform methods commonly a v a i l a b l e .  Because time  series were desired, this study uses numerical integration to obtain the response.  Here again, the addition of higher mode contributions i s  straightforward.  However, i n the present case, seismic input frequencies  23 would not r e s u l t i n a s i g n i f i c a n t second mode v i b r a t i o n contribution because of the high second fundamental frequency of the model, and a f i r s t mode approximation i s used. For  design purposes, a base shear prediction F^(t) can r e a d i l y be  obtained once the structure response i s known, from an integration over the  structure span given by  F (t) b  = /£m(x)Y (x,t)dx t  and s i m i l a r l y for base moment.  (2.24)  Response spectra can be obtained i n the  usual manner f o r these and other functions for s p e c i f i e d input motions. There are several assumptions inherent i n the above approach which must be noted.  F i r s t i s the assumption, used i n a l l parts of t h i s study,  of a constant added mass.  Other authors have shown that added mass i s i n  fact a function of mode shape and frequency (Liaw and Chopra 1973; Byrd 1978;  Pegg 1983).  However, i t i s generally accepted that the error  introduced by neglecting these dependencies i s minimal for slender structures vibrating i n the lower mode shapes, for which the present analysis i s concerned.  The second assumption involves the neglect of two  forms of energy dissipation;  viscous drag and surface wave generation.  Near resonance conditions where t i p o s c i l l a t i o n s can be large, these terms w i l l not be i n s i g n i f i c a n t .  For this reason, the above approach  w i l l overestimate the response of the structure, the largest discrepancies occurring at resonance. i s thus overconservative.  This method of response prediction  24 The q u a n t i f i c a t i o n of damping due to surface wave generation i s difficult.  That term i s r e a d i l y accounted for as a free surface boundary  condition i n the p o t e n t i a l flow solution, but has no d i r e c t interpretation i n the small body regime.  An approximation as viscous  damping ignores the frequency dependence of t h i s term although resonance is perhaps the only c r i t i c a l may have merit.  frequency and hence such an approximation  Because, i n general, t h i s damping cannot be separated  from the viscous drag damping i n the experiments, the t o t a l external damping i n the analysis i s controlled by the choice of drag c o e f f i c i e n t and wave generation damping values are not s p e c i f i e d . A l o g i c a l approach to the problem of including drag forces i s to uncouple or l i n e a r i z e the nonlinear drag term i n Equation (2.4). However, the commonly used practise of uncoupling the drag terms (See Blevins, 1977) i s not an alternative i n an analysis of base motion problems because the r e l a t i v e motion which i s neglected i n such an approach i s exactly the structure response.  Linearization techniques are  tractable i n c e r t a i n problems and one such method i s considered i n the next section. 2.4.2.  Linearized Drag Forces  The nonlinear drag term can be l i n e a r i z e d to obtain a l i n e a r added damping i n the equation of motion.  This method, based on a Fourier  expansion or error minimization procedure, r e s u l t s i n amplitude dependent damping and forcing terms, which necessitates an i t e r a t i v e solution technique.  For continuous structures, the amplitude and hence the  damping i s p o s i t i o n a l l y dependent, and this dependence prevents a complete modal decomposition as i n the purely i n e r t i a l formulation. However, a f i r s t mode approximation can be made as follows.  25 The drag term i n Equation (2.4) can be written as:  P (x,t) = - V j Y j  (2.25)  Kp = j PC D  (2.26)  D  where  D  To l i n e a r i z e (2.25), l e t  |Y |Y t  t  = A Y (x,t) t  (2.27)  where A includes the p o s i t i o n a l dependence, A = A(x). For a zero mean Gaussian process i t can be shown that minimizing the mean square error of Equation (2.27) for x constant yields  |Y |Y t  t  = 1.2 a-  (2.28)  where 0£ i s the root-mean-square of the t o t a l r e l a t i v e v e l o c i t y and i s a function of p o s i t i o n because of the dynamic responses.  For a harmonic  process, (2.28) becomes  1.2 0£ = a Y  and  a  = 8/3n  t  (2.29)  26 where the bar represents amplitude.  Using a f i r s t mode approximation, we  write the p o s i t i o n a l dependence as  Y (x) - 5 + 4 . ! (x) ?!  (2.30)  t  Combining the above, we have an expression for the l i n e a r i z e d drag force,  -aides'  P (x,t) = D  + <h (x)5i ) ( 6 ( t ) + 4,!  (x)li ( t ) )  (2.31)  The generalized drag load becomes, after manipulation  P (t) = - ^ D  {6 6 (t) Bi + 3  2  (6 ki (t) + 6  (t)4i )  +5i£i(t)6 } 3  where  3i  =  i  = J* 4,3 (x) dx  3  32  Combining  (2.10),  (2.32)  4)1 (x) dx  (2.33)  = /* <t> (x) dx  (2.32)  2  with the generalized i n e r t i a force P j ( t ) i n Equation r  and reorganizing, the equation of motion becomes  27  + C—  C  (2.34)  •Pl(t)  Taking due account of phases, (See Appendix B), P i ( t ) Is defined as  (aK a Y (e a)Y +g Q) ] sin(a)t-6 )  [(ma^S!)^ ?  1  2  D  1  ^  g  i  g  1  2  LD  2  mp  where  9  LD  =  t  a  ?  n  o>Y  m  I, g  <' > 2 36  i s the phase angle inherent i n the forcing function due to drag damping and Cg = s t r u c t u r a l damping c o e f f i c i e n t C  Q  = added damping from drag forces  Assuming viscous damping, the linearized drag formulation gives  s  5  +  a  V  [6B2+SiB j 3  2w m3? n *  (2.37)  From equations (2.35) and (2.37) i t can be seen that both force and damping are amplitude dependent. The s o l u t i o n amplitude of (2.34) i s given i n the usual form as  28  2  5i -  2 11/2  [(m3iO) Y ) +(aK^toY (a)Y ^ + 5 i t S 2 ) ) ] 2  2  m3 wi[(l-(a)/io ) ) 2  2  2  1  + (25i0)/a) ) ] 2  (2.38)  1 / 2  1  A solution of (2.38) can be obtained by choosing an i n i t i a l value o f 5l,  t y p i c a l l y zero, and i t e r a t i n g to a prescribed s o l u t i o n accuracy. The  t i p response i s then given, as before, as  Y(t) = 4 > i ( £ K i ( t ) + 6(t) tip  (2.39)  and the amplitude of cylinder t i p motion i s  Y„. = 2 [ I + 0.25 Y tip l g 2  2  + Y 5 cos(Q + %n)] 1/2 g l i LD  / J  (2.40)  As before, Q\ i s the phase angle a r i s i n g from s t r u c t u r a l damping:  0! =  2.4.3.  tan"  1  250/n>i) { } l-Cw/G)!)  (2.41)  2  Numerical Method  The nonlinear form drag term can be included i n a numerical solution of the equation of motion.  Beginning again with the separated form of  the equation, we have, f o r the motion r e l a t i v e to the base  29  m I 5<|>(x) + CI I l < t . r=l r=l r  r  r  - - m6(t) -  I r=l  iV r  l + ( x ) + 6(t) r  I 5 ^ (x) r=l  ( x ) + EI  r  where again m includes the added mass.  V  r  | I 54> (x) + 6 r=l r  (2.42)  Making a f i r s t mode  approximation  (2.43)  Y(x,t) = <h(x) € i ( t )  and assuming that S(t) «  Y(x,t) i n regions where drag forces are  s i g n i f i c a n t , we have  il<n<(>i(x) + £iCI<h (x) + 5iEI()) (x) lv  (2.44)  lv  1  = - {m6(t) + K |li<(>i(x)|c: <t. (x)} D  1  1  Multiplying by <i>i(x), integrating over the length and rearranging, we obtain,  l\  /* CI <j) (x)<t) (x)dx + K | 5 | 5 i / ^ i ( x ) d x  /* m^(x)dx +  lv  1  + ?! /* EI <h(x)<h (x) dx - - 6(t) ±V  D  1  1  m<p (x)dx 1  (2.45)  Orthogonality concepts can be used to reduce the fourth order terms i n Equation (2.44);  however, the nonlinear term cannot be s i m p l i f i e d  30  further.  Performing the integrations, and assuming viscous damping, a  single degree of freedom equation i s obtained:  33  ?l(t) + 2Ci<o ? (t) 4 — K j l i C t O l ^ C t ) + 0 ) ^(t) =  (2.46)  2  1  1  -a ^ t )  where a i s given by Equation (2.21) and 3 and 3 are defined as i n 2  3  Equation (2.33)  Equation (2.46) can be solved numerically using one of a number of solution algorithms for nonlinear d i f f e r e n t i a l equations.  In the present  study a fourth order Runge-Kutta integration method i s used. This method reduces Equation (2.46) to two f i r s t order equations by appropriate substitution (See Appendix C).  The time h i s t o r y response i s  generated by time stepping the integration process.  This method i s  p e r f e c t l y general and can be used for random or sinusoidal forcing functions.  For numerical s t a b i l i t y and accuracy, the time step i s chosen  as approximately a tenth of the smallest forcing function period. The numerical analysis can be extended to include higher mode contributions and the base v e l o c i t y i n the drag term which was o r i g i n a l l y assumed small.  For a two mode formulation, Equation (2.43) i s replaced  by  Y(x,t) = 4>l(*Kl(t) + 4> (xH (t) 2  2  (2.47)  31 The nonlinear term i n t h i s case cannot be separated into time and p o s i t i o n dependent functions and integrals involving E,^ must be evaluated at each time step.  The same problem r e s u l t s from the i n c l u s i o n  of the base v e l o c i t y i n the drag term i n a f i r s t mode approximation.  In  that case, a solution i s sought for the equation  ll + j^  {  m  S ( t ) K  +  m  g  D  2*510)! +  KJJ  J* <h (x)|4>i(x)li + 6(t)|dx  (2.48)  rX, , • • , J* +i(x)|4>i(x)e + 6(t)|dx + 0)i5i = - ai6(t) 2  1  An exact solution to Equation (2.48) can evidently be found only by iteration.  The combination of a Runge-Rutta method, i t e r a t i o n and time  stepping can be time consuming and hence expensive. applicable  However, i n the  frequency range, a two mode solution approximation, which  included the base v e l o c i t y , was found to d i f f e r very l i t t l e from a f i r s t mode approximation, which neglected base v e l o c i t y . approach i s to use the previously the i n t e g r a l evaluation.  An a l t e r n a t i v e  calculated value of the 5 variable i n  If the time step i s small, this method  provides very close agreement with an iterated solution and decreases the cost s i g n i f i c a n t l y . by Anagnostopoulos  A similar approach for a d i s c r e t i z e d system was used  (1981).  second mode contribution  Because the i n c l u s i o n of base v e l o c i t y and a  does not s i g n i f i c a n t l y a l t e r the predicted  solution, the response predictions  for the nonlinear drag case are  generated from the solution of Equation (2.46).  32 3.  TESTING FACILITIES AND MODEL PARAMETERS  The present study attempts to model and p r e d i c t  the response of a  s l e n d e r c a n t i l e v e r marine s t r u c t u r e undergoing h o r i z o n t a l base motion. The e x i s t i n g  facilities  at the U n i v e r s i t y of B r i t i s h Columbia  Earthquake  L a b o r a t o r y used f o r t h i s purpose i n c l u d e a shaking t a b l e , water tank and data a c q u i s i t i o n  system.  A s u i t a b l e model, i n s t r u m e n t a t i o n system, and  t e s t procedure must be chosen t o be c o m p a t i b l e with these f a c i l i t i e s .  3.1  Testing All  Facilities  t e s t s were performed i n the Earthquake L a b o r a t o r y o f the  Department of C i v i l  E n g i n e e r i n g at U.B.C.  The l a b o r a t o r y  facilities  i n c l u d e a 3.3m x 3.3m s i n g l e degree o f freedom shaking t a b l e ( F i g u r e 4) supported by an MTS d r i v e system and a PDP-11 mini-computer.  The shaking  t a b l e o p e r a t e s i n a s i n g l e h o r i z o n t a l d i r e c t i o n i n the frequency range 030 hz, with very l i m i t e d motion c a p a b i l i t y at the h i g h e r f r e q u e n c i e s . The MTS system can e x c i t e the t a b l e with a v a r i e t y of waveforms, o f variable or  frequency and amplitude, as w e l l as random motions from  other d i s c r e t i z e d  records.  were used i n these t e s t s .  S i n u s o i d a l and s e i s m i c base motion  seismic input  The PDP-11 p r o v i d e s the data a c q u i s i t i o n  system with a p r o c e s s i n g c a p a b i l i t y o f 17 channels i n any one t e s t . channels  were used i n these t e s t s .  Four  The output s i g n a l s are processed  through an analogue t o d i g i t a l c o n v e r t e r and s t o r e d d i r e c t l y on f l o p p y discs. The water tank s t r a d d l e s the shake t a b l e so as not to be i n f l u e n c e d by i t s v i b r a t i o n is lined  ( F i g u r e 5 ) . The tank can be f i l l e d  to a depth of lm and  with h o r s e h a i r - t y p e mats to damp waves r a d i a t e d by the c y l i n d e r  33 and thereby minimize the e f f e c t s of r e f l e c t i o n on cylinder response.  The  model to be tested was attached to the table through a hole i n the tank bottom sealed with a natural rubber diaphragm.  This rubber seal  e f f e c t i v e l y allows the f u l l table displacements of 15 cm peak to peak at low frequencies.  A schematic of the test f a c i l i t y i s presented i n Figure  3. 3.2  Model Parameters Model c h a r a t e r i s t i c s must be chosen c a r e f u l l y i n order to obtain  experimental results which are applicable to prototype s i t u a t i o n s .  As  discussed i n Chapter 2, dimensional analysis provides the relevant nondimensional parameters i n the problem.  In the present study, Reynolds  number s i m i l a r i t y was not maintained and no attempt was made to maintain consistent damping r a t i o s .  An attempt  remaining dimensionless groups: r a t i o D / H , i n e r t i a r a t i o m/pD  2  was made to e f f e c t i v e l y model the  the frequency r a t i o f / f > geometric n  and Keulegan-Carpenter  number K.  The  choice of model i s controlled by the s t r u c t u r a l parameters, f / f , D / H , n  and to a lesser extent by the damping r a t i o C« Seismic e x c i t a t i o n i s severest i n the frequency range 0-20  hz,  whereas t y p i c a l structure fundamental frequencies are 0.5 - 2.0 hz. a r e a l i s t i c r a t i o of f / f n i s 0-10. the shaking table, a model f  Thus  In order to maintain t h i s r a t i o using  of 1-3 hz i s required.  Prototype values of D/H vary but are usually 1/20 -1/30.  This r a t i o  is limited i n the model tests by the lm depth to which the water tank can be f i l l e d . 1/20.  For a model diameter of 5cm, maximum submergence gives D/H =  With the diameter known from the geometric scaling, the sectional  mass, m, can be found for a s p e c i f i c i n e r t i a r a t i o m / p D . 2  type values are 0.8-2.0, giving m of 20-50 g/cm  Typical proto-  for a diameter of 5 cm.  34 F u l l scale damping r a t i o s are usually 3 - 10%. The parameter cannot e a s i l y be controlled i n the model and measured model values of 3.8% for f i r s t mode and 6.6% for the second mode were considered  acceptable.  The modulus-mass c h a r a c t e r i s t i c s of dynamic prototypes are d i f f i c u l t to model. Convenient geometric scaling often requires e l a s t i c moduli which are not available, p a r t i c u l a r l y when continuous models are desired. Sectional d i s t o r t i o n can be used i n some cases where the dynamic behaviour i s primarily f l e x u r a l v i b r a t i o n . l i n e a r mode models could be considered.  A l t e r n a t i v e l y , sectional or  Neither of these methods w i l l  give accurate modal response which i s desired i n this investigation. Therefore, a continuous model of high mass and low s t i f f n e s s was sought. After several t r i a l s were made with hollow p l a s t i c pipe f i l l e d with heavy materials, a s o l i d Teflon cylinder with the following c h a r a c t e r i s t i c s was f i n a l l y chosen as the test specimen:  Material  :  Teflon  (polytetrafluoroethylene)  Modulus  :  9.9 x 10 kg/cm.sec  Diameter  : 5 cm  Length free standing  : 99 cm  Sectional Mass  :  6  43 g/cm 4.6 hz  f (air)  28.7 hz  Water depth H  97 cm  Free height  2 cm  35 m/pD  1.72  Modal damping r a t i o s 1  3.8%  2  2  6.6  %  1/20  D/H  Although T e f l o n i s s u s c e p t i b l e to creep a t low s t r e s s e s , exhibits hysteritic  response  dynamic t e s t s would be  3.3  Data The  behaviour,  i t was  felt  and  that i t s use i n  feasible.  Acquisition t e s t instrumentation c o n s i s t e d of a v e r t i c a l  aluminum bar connected i n F i g u r e s 6 and 7.  to the c y l i n d e r  H o r i z o n t a l motion  strain-gauged  t i p with a l i g h t of the c y l i n d e r  s p r i n g as shown t i p induces  f l e x u r e i n the bar which can be measured with the s t r a i n gauges. system measures displacement d i r e c t i o n of base motion. each d i r e c t i o n . signal  calibrated tained.  The  with s t a t i c  t r a n s v e r s e to the  s t r a i n gauges and a b r i d g e are used f o r  the system  performed  instrumented bar and  would have i n c r e a s e d the  adequately at a l l but  the  s p r i n g were p e r i o d i c a l l y  d i s p l a c e m e n t s and a l i n e a r  calibration  was  main-  Table d i s p l a c e m e n t s and a c c e l e r a t i o n s were measured with the  l i n e a r voltage d i f f e r e n t i a l to the shaking t a b l e . processed discs.  Two  A l t h o u g h a f o u r gauge system  to n o i s e r a t i o ,  smallest outputs.  response both i n - l i n e and  This  through  The  t r a n s d u c e r (LVDT) and f o u r channels monitored  i n each  an a n a l o g u e - t o - d i g i t a l c o n v e r t e r and  The output sampling r a t e was  g i v e minimum sampling frequency.  accelerometer attached test  were  stored d i r e c t l y  on  s e t v i a the a c q u i s i t i o n software to  f r e q u e n c i e s of approximately 20 times  the  testing  36 4.  EXPERIMENTS  The o b j e c t i v e of the t e s t s performed  i n t h i s study was  i n v e s t i g a t e the behaviour of a compliant marine d i r e c t i o n a l h o r i z o n t a l base motion.  s t r u c t u r e e x c i t e d by  response p r e d i c t i o n s d i s c u s s e d i n Chapter 2.  response u s i n g one of the t h r e e approaches would be adequate  uni-  The e x p e r i m e n t a l l y determined  response can then be used as the b a s i s f o r comparison the  to  and e v a l u a t i o n of  I f the p r e d i c t e d  i s a c c u r a t e , then t h a t method  f o r p r e d i c t i o n of earthquake  loads and d i s p l a c e m e n t s .  (As w e l l , the t e s t c y l i n d e r i s instrumented to o b t a i n t r a n s v e r s e d i s p l a c e m e n t s and a l t h o u g h no comparisons  w i l l be made to any  particular  t h e o r y , i t i s i n f o r m a t i v e to note the c h a r a c t e r i s t i c s of t h i s m o t i o n ) . The  t e s t s and  comparisons  were of two  types:  sinusoidal  tests,  u s i n g f r e q u e n c i e s i n the range 2 to 28 hz, and random motion u s i n g a c t u a l scaled  seismic records.  97 cm and f r e e b o a r d of 2  4.1  A l l t e s t s were performed  of  with a water depth  cm.  Damping T e s t s and System C h a r a c t e r i s t i c s The damping c h a r a c t e r i s t i c s of the model and m o d e l - i n s t r u m e n t a t i o n  system are extremely important i n terms of r e a l i s t i c a l l y dynamic behaviour of a p r o t o t y p e s t r u c t u r e . initial  t e s t s conducted  determine the  were f r e e v i b r a t i o n  the system damping.  I t was  m o d e l l i n g the  With t h i s f a c t i n mind, the t e s t s i n a i r and  water  to  hoped that the n a t u r a l damping  T e f l o n dowel p l u s the damping i n t r o d u c e d by the v e r t i c a l  spring  of would  be low enough so as to reproduce the range of t y p i c a l p r o t o t y p e damping r a t i o s , u s u a l l y about of  5% of c r i t i c a l .  F i g u r e 8 shows the r e c o r d o b t a i n e d  a f r e e v i b r a t i o n t e s t i n a i r , F i g u r e 9 the r e s u l t of a s i m i l a r t e s t i n  37  water.  Second mode free v i b r a t i o n frequency tests were also performed by  o s c i l l a t i n g the model at 28 hz and suddenly stopping the excitation, to determine a second modal damping r a t i o . determined from this method.  An approximate value could be  The log decrement was used to find the  damping r a t i o , C, i n each mode and an averaged r e s u l t gave c,\= 3.8% c r i t i c a l i n the f i r s t mode and £2 = 6.6% i n the second mode. The tests i n water were to determine the e f f e c t of added damping on the natural period of the model.  A small decrease i n natural frequency was noted and  t h i s reduced frequency was used for the response predictions. The instrumentation system used i n these tests, as described i n Chapter 3, has c e r t a i n c h a r a c t e r i s t i c s which should be noted.  Whereas i t  would be b e n e f i c i a l i n terms of response accuracy to have very l i g h t spring forces acting on the model t i p , the strain-gauge system used required a moderate spring tension to give recoverable readings at low o s c i l l a t i o n amplitudes. This fact i n turn necessitated the use of a spring system with at least a nominal mass which resulted i n a system resonance at approximately 22 hz, close to the second mode frequency of the  model.  Although this problem may have been correctable to some  extent, observations Indicated that second mode e f f e c t s did not contribute s i g n i f i c a n t l y to the magnitude of cylinder t i p response, even though their presence was noted i n the output signals (See Appendix D) at certain subharmonic  4.2  frequencies.  Sinusoidal Tests The sinusoidal tests were performed at four amplitudes of base  displacements:  1.0, 1.9, 2.8 and 3.6 mm.  The choice of amplitude was  38  l i m i t e d by the p o s s i b l e d e s t r u c t i o n o f the model from l a r g e resonances.  At each amplitude  of base motion,  amplitude  a p p r o x i m a t e l y 13 f r e q u e n -  c i e s were used i n the range 2 t o 28 h z . The h i g h e r f r e q u e n c i e s c o u l d not be combined with the l a r g e r amplitudes because table  of l i m i t s on the shaking  system. The h i g h e r frequency t e s t s were an attempt  response of the model.  t o induce second mode  U n f o r t u n a t e l y the second mode frequency was v e r y  c l o s e to a resonant frequency i n the i n s t r u m e n t a t i o n system, as p r e v i o u s l y d i s c u s s e d , and the output was then o b v i o u s l y i n e r r o r . problem  This  was compounded by the i n t e n s e water spray generated by the  c y l i n d e r motion frequencies. existing  which caused  e x c e s s i v e n o i s e i n the s i g n a l output a t h i g h  The a c c u r a t e measurement o f second mode response u s i n g  facilities  appears  f e a s i b l e only i f the model  fundamental  frequency i s lower than the v a l u e o f 4.5 hz o b t a i n e d i n the p r e s e n t tests. Second mode response c o u l d be important f o r s t r u c t u r e s with low fundamental  f r e q u e n c i e s (<1.5 h z ) .  However, the a d d i t i o n of h i g h e r mode  c o n t r i b u t i o n s i n g e n e r a l d e c r e a s e s the s t r u c t u r e t i p d i s p l a c e m e n t s , but increases remains  those at more deeply submerged p o i n t s .  The combined  response  everywhere l e s s than the f i r s t mode t i p response and hence i s not  critical. I n - l i n e and t r a n s v e r s e d i s p l a c e m e n t s were r e c o r d e d i n most The  t r a n s v e r s e d i s p l a c e m e n t s showed a very peaked frequency  c e n t e r i n g on the fundamental Chapter 5 ) .  resonant frequency ( d i s c u s s e d  tests.  response further i n  V i r t u a l l y no output was a t t a i n a b l e o u t s i d e a 1.5 hz band-  width c e n t e r e d a t resonance and the t r a n s v e r s e s i g n a l was not recorded at f r e q u e n c i e s o u t s i d e t h i s range.  The output sampling  r a t e was s e t at  39  frequencies of 20- 50 times the d r i v i n g frequency and 5 seconds of steady state motion were recorded i n each test. filters  At low amplitudes of response,  were required to reduce noise i n the s i g n a l .  A 10 hz low-pass  f i l t e r was used at test frequencies below 4.5 hz, a 40 hz low-pass  filter  on a l l other t e s t s . Figure 10 shows a test at the fundamental resonance (4.5 hz); Figure 11 a test at the second mode resonance (28 hz).  Note the intense  water spray which affected response measurements near the second mode frequency.  4.3  Seismic Motion Tests The acceleration records of two actual earthquakes, E l Centro N-S  (1940) and San Fernando N21E (1971) were used as random motion input. These records were chosen because they have reasonably representative frequency spectra of earthquakes i n the P a c i f i c region (See Figures 12 and 13) and they are r e l a t i v e l y severe. The San Fernando record i n particular contains s i g n i f i c a n t energy i n the region of the fundamental frequency of the test cylinder (4.5 hz), as might occur from s o i l filtering. The d i g i t i z e d acceleration input data can be scaled to an appropriate l e v e l on the MTS system.  In this investigation i t was  desired to keep peak-to-peak base displacements of the model geometrically similar to those which can occur i n actual strong motion seismic a c t i v i t y (40 - 60 cm). This resulted i n 0.15 f u l l scale displacements i n the E l Centro test and 1.5 f u l l scale displacements i n the  San Fernando t e s t .  A peak and time-averaged Keulagan-Carpenter  number i s then maintained between prototype and model since that non-  AO  dimensional group i s proportional to the r a t i o of displacement to cylinder diameter.  Because the actual earthquake time scale i s  maintained the test accelerations are scaled by the same factors. tests were performed at peak-to-peak displacements of 3.5 2.0) and 2.8 cm  ( ase ~ K  i n  t h e  E 1  C  e  n  t  r  o  a n d  S a n  F  e  r  n  cm a  n  d  o  D  The  ^'Sase tests  respect i v e l y . The random motion test data were sampled at a frequency which would allow direct use of table displacement and acceleration records in the numerical program described i n Chapter 2. approximation this frequency was 50 hz.  For the f i r s t mode This gives a time step of  seconds, or approximately 1/11 of the fundamental period, a value s u f f i c i e n t l y small to assure numerical accuracy and  stability.  0.02  41 5.  5.1  DISCUSSION OF RESULTS  Response Functions from Sinusoidal Tests The sinusoidal test data as recorded are presented In Table 2.  The frequency dependence of cylinder t i p response can be presented i n non-dimensional form f o r the sinusoidal tests and predictions.  These  comparison p l o t s , sometimes termed receptance functions, are presented i n the Figures 14, 15, 16 and 17 for the tested r a t i o s of base amplitude to diameter r a t i o Y /D. g  The frequency r a t i o f / f n  i s defined i n terms of  the fundamental frequency of the cylinder i n water. r e s u l t s are indicated on these f i g u r e s .  Several important  As anticipated, the response  near resonance i s very peaked and dynamic amplifications on the order of 15 - 20 occurred at the fundamental frequency.  Unfortunately, as  discussed i n Chapter 4, a true second mode response was not obtainable i n part because of l i m i t a t i o n s on the shake table and p a r t l y because large system resonances led to erroneous data above a frequency r a t i o of about 4.0.  The general trend i n the data towards a second response peak i s  evident i n the figures. For a l i n e a r system, the dynamic a m p l i f i c a t i o n i s a system constant dependent  on the s t i f f n e s s and damping parameters and independent of the  amplitude of applied motion or force.  The nonlinear behaviour of the  system considered here i s apparent i n the amplitude dependence of the dynamic a m p l i f i c a t i o n factor Y/Y . This factor ranges from a high of 19 at Y /D = 0.02, decreasing s t e a d i l y with amplitude increases to a low of 13 at Y /D = 0.072 (See Figures 14 to 17).  This decrease i s a r e s u l t of  the increased damping associated with the occurrence of flow separation  near the cylinder t i p , and to increased surface wave generation at the larger forcing amplitudes.  An increase i n the viscous damping r a t i o of  approximately 45% i s required to achieve the t o t a l a m p l i f i c a t i o n reduction.  Although this i s a large increase, the absolute damping  values are quite small and the change i s of the order of 1.5%  critical.  However, as we note from the peaked response c h a r a c t e r i s t i c , the response i s very s e n s i t i v e to small changes i n damping.  5.2  Response Predictions for Sinusoidal Motion The methods of response prediction used i n t h i s study have been  developed i n Chapter 2.  The simplest approach neglects the nonlinear  drag force term i n the Morison equation and a closed-form solution i s obtained for the resulting linear equation of motion.  The neglect of  drag forces i n the formulation e f f e c t i v e l y lowers the damping by eliminating the drag damping present i n the actual system.  This method  consequently overestimates the response of the cylinder t i p . The discrepancy w i l l increase with forcing amplitude due to the increasing importance of drag forces near the free surface. The response function for this solution method i s compared with other methods and the experimental data i n Figures 14 to 17. of predicted response to measured response at resonance Y  The r a t i o s  ,/Y for pred exp  the four amplitudes used i n the tests' are presented i n Table 1.  The  r e s u l t s of t h i s method exhibit the c h a r a c t e r i s t i c s anticipated.  Only for  the lowest amplitude case, Figure 14, does this method provide an accurate estimation of structure response. forces in the experiment  were n e g l i g i b l e .  In that case, nonlinear drag Because the experimental data  for second mode response i s not considered v a l i d , for reasons discussed,  43 i t i s i m p o s s i b l e t o make c o n c l u s i o n s r e g a r d i n g t h e a c c u r a c y o f the second mode response p r e d i c t i o n based on t h i s method. The second p r e d i c t i o n method i n c l u d e s a l i n e a r i z e d form o f t h e M o r i s o n drag term i n the e q u a t i o n of m o t i o n .  A c l o s e d - f o r m s o l u t i o n can  be o b t a i n e d f o r a f i r s t mode response assumption.  The response  function  generated by t h i s method i s i n c l u d e d i n F i g u r e s 14 t o 17 and the resonant response r a t i o s a r e t a b u l a t e d i n T a b l e 1.  I t i s e v i d e n t from Table 1  t h a t the a c c u r a c y of a l i n e a r i z e d drag p r e d i c t i o n i s dependent on the c h o i c e o f d r a g c o e f f i c i e n t , (Lj, as c o u l d be a n t i c i p a t e d .  The i n i t i a l  c h o i c e o f d r a g c o e f f i c i e n t was based on an e x t r a p o l a t i o n o f e x i s t i n g c o r r e l a t i o n s of t h i s c o e f f i c i e n t w i t h Reynolds number and K e u l e g a n C a r p e n t e r number as i n d i c a t e d i n F i g u r e 18 ( a f t e r Sarpkaya and I s a a c s o n ) . A c y l i n d e r t i p Reynolds number of o r d e r l&  was assumed.  Accordingly a  drag c o e f f i c i e n t o f 0.7 was s e l e c t e d as a r e p r e s e n t a t i v e v a l u e f o r t h e range of K and Re expected near the c y l i n d e r t i p and was used i n both t h e l i n e a r i z e d d r a g and n u m e r i c a l s o l u t i o n s . For Cp = 0.7, the l i n e a r i z e d drag p r e d i c t i o n s a r e v e r y u n c o n s e r v a t i v e a t resonance and d i v e r g e from t h e e x p e r i m e n t a l r e s u l t s w i t h i n c r e a s i n g base motion a m p l i t u d e ( T a b l e 1 ) . I n t h i s c a s e , drag f o r c e l i n e a r i z a t i o n r e s u l t s i n an o v e r e s t i m a t e o f drag damping.  A more  a c c u r a t e and n o n - d i v e r g e n t response e s t i m a t i o n i s o b t a i n e d w i t h C^ = 0.25 and an o p t i m a l s o l u t i o n i s reached w i t h  - 0.20.  The l i n e a r i z e d  method appears t o g i v e a c c u r a t e response p r e d i c t i o n s f o r t h i s when an a p p r o p r i a t e drag c o e f f i c i e n t i s used.  Of c o u r s e , t h i s  i n f o r m a t i o n would not g e n e r a l l y be known a p r i o r i .  drag  problem  44 As a t h i r d approach, the nonlinear drag term was included i n a numerical solution of the equation of motion derived from a f i r s t mode response assumption.  A drag c o e f f i c i e n t of  = 0.7 was again used  i n i t i a l l y and the results are presented i n Figures 14 to 17 and Table 1. The response predictions for  = 0.7 are once more unconservative, but  consistent over the range of base displacement amplitudes.  This  indicates that the numerical method can provide good resonant response estimates i f the drag c o e f f i c i e n t i s chosen c o r r e c t l y .  Figure 19 shows  the relationship between resonant response amplitude ratios and drag c o e f f i c i e n t used i n the numerical approach.  Evidently a very low drag  c o e f f i c i e n t i s required to obtain an accurate prediction.  This i s a  result of the low Reynolds number associated with the motion away from the free surface. The amplitude dependence of the drag terra i s apparent from the v a r i a t i o n of response ratios at fixed C^. would be amplitude invariant.  In a linear system, this r a t i o  In the present case a general requirement  for agreement of experimental and numerical response i s an increasing drag c o e f f i c i e n t for increasing base motion amplitude.  However, the  interaction of transverse and i n - l i n e o s c i l l a t i o n s at the largest test amplitude caused a reversal i n this trend.  Based on Figure 19 a drag  c o e f f i c i e n t of about 0.3 would give reasonable r e s u l t s except i n the case of the lowest input amplitude.  45 5.3  Transverse Response The cylinder response near resonance at the larger base displacement  r a t i o s , Y /D, was influenced by l i f t S  forces.  The occurrence and  magnitude of l i f t forces i s a function of the degree of flow separation, hence the Keulegan-Carpenter number.  For the two largest base motion  amplitudes, (2.8 and 3.6 mm), the free surface Keulegan-Carpenter numbers based on peak response values, were 5.1 and 6.2 respectively.  The  l i t e r a t u r e suggests that vortex shedding (Figure 20) and l i f t forcing can occur for K > 5.0 (Sarpkaya, 1979).  Thus, i t could be expected that  forces would be present near the free surface. experiments these l i f t  lift  In the present  forces trigger ovalling of the cylinder t i p . The  interaction of i n - l i n e and transverse o s c i l l a t i o n s decreases the i n - l i n e response over a narrow band-width centered at resonance, from the response expected with no l i f t forces present.  The i n t e r a c t i o n i s  evident i n the results presented i n Table 2 for Y /D = .054 and .07 2. 8  As  resonance (4.5 hz) i s approached, transverse and i n - l i n e amplitudes increase. of  Between 4.3 and 4.7 hz, the i n - l i n e response drops as a result  the energy d i s s i p a t i o n i n transverse o s c i l l a t i o n s .  Maximum i n - l i n e  response occurs, i n t h i s case, at a frequency below resonance.  Figure 21  indicates such response characteristics near resonance for Y^/D = 0.054. The very narrow bandwidth of this phenomenon i s evident i n the transverse response function presented as Figure 22.  The response at the  lowest Y /D r a t i o (0.20) was negligible and i s not shown i n the figure. The displacement responses of the cylinder t i p are combined as t i p l o c i i n Figures 23 and 24.  Figure 23, a resonant response at Y /D =  0.54, shows a c l e a r l y defined orientation of the o s c i l l a t i o n axes. c h a r a c t e r i s t i c was evident i n a l l  the responses which included large  This  46 transverse components, and the axes appeared stable only i n such orientations.  This s t a b i l i t y may indicate either a hydrodynamic  interaction favouring that orientation, such as a s p e c i f i c l i f t  force  phase, or simply some inherent structural c h a r a c t e r i s t i c or both. However, i f the cylinder exhibited a tendency to o s c i l l a t e diagonally because of a material defect, this behaviour would presumably be evident in the response i n a i r .  Also, the response would be altered i f the  cylinder were rotated i n i t s base clamp.  Because only i n - l i n e response  occurred i n a i r tests and cylinder rotation had no e f f e c t on the preferred axes orientations i t can be concluded that the directions of the e l l i p t i c axes are controlled by a hydrodynamic mechanism.  The exact  details of this interactive mechanism are no doubt complex and beyond the scope of the present study.  5.4  Response to Seismic Input The displacement response of the cylinder during earthquakes was  tested using r e a l earthquake accelerograms for base motion input.  These  seismic records were scaled to give maximum displacements consistent with geometric scaling while the time frame was maintained f u l l scale.  As a  r e s u l t , accelerations i n the test are lower than recorded values.  Two  earthquake records were used i n the tests;  the E l Centro 1940, N-S  component and the San Fernando, 1971, N21E component.  The test input was  scaled to give a maximum base displacement r a t i o , Y /D, of about 0.4, consistent with strong motion earthquake displacements. The E l Centro earthquake energy i s distributed f a i r l y evenly over a range of frequencies (see Figure 12) and this particular record i s often treated as white noise input.  Such input would not be expected to  produce large dynamic responses.  Figure 25 shows the t i p response for  the E l Centro input, and indicates that dynamic response of the cylinder i s minimal during t h i s earthquake. negligible during this test.  Transverse o s c i l l a t i o n s were also  The hydrodynamic loads i n this case could  be estimated quite adequately by a pseudo-static a n a l y s i s . whereas most of the energy i n the El Centro earthquake i s equipartitioned at frequencies below 3 hz, the San Fernando  1971  earthquake has a large spectral peak near 4.5 Hz, the fundamental frequency of the cylinder (see Figure 13).  Such spectral peaks r e s u l t  from s o i l f i l t e r i n g or r e f l e c t i o n interaction of white noise earthquakes such as E l Centro.  The magnitude of t h i s peak i s a l s o s i g n i f i c a n t l y  greater than any peaks in the El Centro record (Figure 12).  This  earthquake provides a better i l l u s t r a t i o n of the e f f e c t s of coincidence between structure natural frequency and earthquake spectral density peaks. The base input and i n - l i n e t i p response for the San Fernando earthquake are compared i n Figure 26.  It i s apparent that dynamic  amplification i s s i g n i f i c a n t i n this case because of the effects.  The resulting t i p displacements  resonant  and v e l o c i t i e s are s u f f i c i e n t  to cause l i f t forces which r e s u l t i n transverse o s c i l l a t i o n s during the peak excitation period.  A transverse displacement  presented  Although the transverse o s c i l l a t i o n s are small,  i n Figure 27.  time h i s t o r y i s  they can contribute to the stresses on the structure and should be included i n a seismic design load estimation. develop from a longer exposure to resonant  Larger responses could  frequencies.  This could be a  p o s s i b i l i t y under c e r t a i n conditions, although the scaling of the  San  Fernando record used here, i n conjunction with the proximity of cylinder  48  natural frequency and spectral density peak probably represents a reasonably extreme event.  5.5  Numerical Prediction of Seismic Response With several small modifications, the numerical time step program  discussed i n Chapter 2 can be used to predict the response of the cylinder to seismic input.  The actual base motion i s used as input and  the integration performed at each time step.  A linear approximation with  no drag loading can be obtained by setting the drag c o e f f i c i e n t to zero. The response time h i s t o r i e s presented i n Figures 28 and 29 were generated using a drag c o e f f i c i e n t of 0.2, i . e . assumming that non-linear effects would be minimal.  A l t e r n a t i v e l y , the drag c o e f f i c i e n t could be set to  zero and the s t r u c t u r a l damping increased s l i g h t l y to account for the hydrodynamic  damping i f the l a t t e r i s low.  Figures 28 and 29 indicate that the response prediction using the numerical solution approach i s quite accurate.  The peak responses are i n  close agreement, and there are only minor differences i n phase.  The  accuracy obtained at this low value of drag c o e f f i c i e n t indicates that viscous e f f e c t s are almost n e g l i g i b l e i n seismic response, a conclusion previously reached by other researchers (Clough 1960; 1973;  Byrd 1978).  Liaw and  Chopra  49 6.  6.1  CONCLUSIONS AND RECOMMENDATIONS  Conclusions The displacement response of a f l e x i b l e , surface-piercing c i r c u l a r  cylinder to sinusoidal and random base motion has been investigated.  The  variables considered i n the study have been the frequency r a t i o and nondimensional base motion amplitude.  Experimental response has been  compared to predictions based on modal analyses using a Morison equation representation of the hydrodynamic forces.  Three treatments of the  nonlinear drag term i n the r e s u l t i n g equation of motion have been considered:  neglect of the drag terra, l i n e a r i z a t i o n , and inclusion i n  i t s nonlinear form.  Based on results discussed i n Chapter 5, conclusions  are made as follows: Cylinder response to small base displacements can be accurately predicted neglecting f l u i d drag forces.  However, the nonlinear drag term  becomes increasingly important at higher displacement amplitudes.  In the  present case, this term i s equivalent to a nonlinear added damping which reduces the dynamic amplification as input motion amplitude increases. Neglect of this term leads to overly conservative response predictions. The method of drag l i n e a r i z a t i o n developed herein can give accurate response estimates i f a suitable drag c o e f f i c i e n t i s used i n the analysis.  A suitable c o e f f i c i e n t w i l l be lower than that chosen on the  basis of anticipated cylinder t i p motion, a value which gives consistently unconservative r e s u l t s . Numerical solution of the equation of motion including the nonlinear drag term can give accurate response predictions for a suitable choice of drag c o e f f i c i e n t .  This method i s numerically much more time consuming  50 than the linearized drag approach, and suffers from the same limitations on accuracy.  The drag l i n e a r i z a t i o n technique appears, then, to be a  more r e a l i s t i c approach for response prediction i n the case of sinusoidal motion. The ground displacements of strong motion earthquakes with spectral density peaks near structure resonance frequencies can be amplified to l e v e l s which cause flow separation near the free surface.  Small  transverse o s c i l l a t i o n s are evident i n such cases and this additional response would contribute to the stresses i n the structure.  However, i t  is concluded that the i n - l i n e response can be adequately predicted neglecting drag forces.  The more common l i n e a r analysis methods i n the  frequency domain can thus be used i n the solution of the seismic response problem, as an alternative to the time step integration method used here. Inherent i n the above conclusions i s the f e a s i b i l i t y of using modal analysis with a Morison type forcing function i n the problem formulation. In fact, this approach appears to be quite tractable and can give good results with the inclusion of only the f i r s t mode. It should be noted that the present experiments have generated data, such as modal damping ratios which are then used as input for numerical predictions.  In the design process this data i s not known a p r i o r i and  estimations of such quantities as added mass and structural damping must be made.  These estimations w i l l undoubtedly affect the accuracy of  response predictions based on any of the techniques presented i n this study.  51 6.2  Recommendations for Further Study The work of this study could be extended into several areas i n terras  of the base motion problem.  «»  These areas might include:  Investigation of possible c o r r e l a t i o n of nonlinear drag and added viscous damping over s p e c i f i c frequency and amplitude ranges. correlations would allow the accurate  Such  prediction of response using  only linear terras i n the equation of motion. «>  Similar studies of response for multi-leg structure with various deck-leg  o  linkage arrangements.  A similar dynamic response analysis which includes a base system for modelling  s o i l structure i n t e r a c t i o n .  Y . (t) tip Y(x,t) Y (x,t) t  P(x,t)  6(t)  Figure 1.  Coordinate System D e f i n i t i o n  Figure 2.  Mode Shapes For a Cantilever Beam  WATER TANK SPRING  SHAKING TABLE  Figure 3. Schematic of Test F a c i l i t y  Figure 5.  Testing Tank Over Shaking Table  55  TO M I K E  MWLiritu  S T U m SAUGES ! ' . - SIDE  » l U M i m m HOC (25>m SIU*«E:  TO C T l l U D f K T l »  Figure 6. Instrumented Rod For S t r a i n Measurement  Figure 7.  Photo of Instrumented Rod  56  57  Figure 10.  Photo of Sinusoidal Test at 4 . 5 H z .  Figure 1 1 . Photo of Sinusoidal Test at 28Hz.  53  Figure  13.  Fourier  S p e c t r a f o r San  Fernando  1971  59  a  o EXPERIMENTAL NON-LINERR DRAG  LINEARIZED DRAG NO DRAG  Dynamic Amplification  fl.O  T  1  0.8  1  1  16  1  1  2.4  Figure 14.  1  1  1  1  1  (Y /Y ) = 19 t g max  1  9.2 4RATIO .D 4.8 FREQUENCY F/F  1  In  1  5.6  1  1  6.4  Response Function For Y /D =.020  1  1  1.2  r— 9.3  60  o  s  EXPERIMENTAL NON-LINERR  DRAG  LINEARIZED DRAG NO ORflC  . "  -  a  o EXPERIMENTAL NON-LINERR DRAG LINEARIZED DRAG  NO DRAG  Figure 16.  Response Function For Y /D =.054  62  a  o EXPERIMENTAL NON-LINERR DRRG  LINEARIZED DRAG NO DRAG  Figure 17. Response Function For Y /D =.072  CD = .7 Y /D g  CD = .25  CD = .20  NO DRAG LINEARIZED LINEARIZED LINEARIZED  CD = .7 NONLINEAR  .02  1.07  .65  .85  .91  .74  .036  1.30  .60  .90  .97  .76  .054  1.53  .5 7  .92  1.00  .79  .072  1.55  '.49  .84  .93  .75  TABLE 1 :  Resonant Response Ratio Y v  ,/Y pred exp  64  IN-LINE RESPONSE Y /D = .02 g  Y /D = .036 g  RESPONSE  Y /D = .054 g  Y /D = .072 g  FREQ.  RESPONSE  FREQ.  2.0 3.0  1.5 2.3  2.0 3.0  2.8 4.5  2.0  3.9  2.0  5.2  3.0  6.5  3.0  8.5  3.5 4.0  3.4 6.0  3.5 4.0  6.7  3.5  9.3  13.8  4.0  4.5 4.7  19.3 10.4  4.3  17.9  4.3  19.5 38.0  3.5 4.0 4.3  30.5 48.0  4.5  30.1  4.5  26.0  4.5  25.0  5.0 6.0  8.5 3.9  4.7 5.0  4.7 5.0  4.7 5.0  22.0  9.0  6.0  20.0 20.4 11.0  6.0  9.0  5.6  9.0  2.9  6.0  18.1 15.8 6.8  14.0  2.2  9.0  4.8  18.0 22.0  5.0 6.5  14.0 18.0 28.0  4.4 7.6 23.0  FREQ.  RESPONSE  14.0 18.0 28.0  7.1 14.0 23.8  FREQ.  14.0 18.0  RESPONSE  12.3  24.0 12.0 7.0 10.0 12.0  TRANSVERSE RESPONSE Y /D = .02 g 4.5 4.7  2.3 1.4  Y /D - .036 g  Y /D = .054 g  :  g  4.0  2.1  4.0  1.7  4.0  2.7  4.3 4.5 4.7 5.0  3.5 15.6  4.3 4.5 4.7 5.0  6.0 28.0 25.0 2.4  4.3 4.5 4.7 5.0 5.1  8.0 33.0  1.6 1.0  FREQUENCIES ARE IN Hz, RESPONSE VALUES IN mm. TABLE 2  Y /D = .072  Sinusoidal Test Data  f  = 4.5 Hz  29.0 20.0 2.5  •••10  _i 0.1  Figure 18.  I  I  I  I 0.3  I  I  I  I  I  1  J. J  I  3  , I 4  I 5  '  '  '  '  ' 10  15  Drag Coefficient versus Reynolds Number For Various Values of K (after Sarpkaya and Isaacson)  66  (2P Figure 20. Numerical Simulation of the Formation of Asymmetric Vortices (after Sarpkaya and Isaacson)  J  »  e TRANSVERSE  i  + IN-LINE  a. m l  ST U  1  1  IM  1  1  IBS  1  1  (L92  1 p 1 1 1 1 IX ID 104 FREQUENCY R f l T I O F/F1N 1  1  1M  1  1  1J2  1  1  1JI  r12  Figure 21. In-line and Transverse Response Near Resonance For Y /D =.054 g  67  Figure 22.  Transverse Response Functions For Various Base Displacement Amplitudes  68  V I  Figure 24.  Tip Locus off Resonance For Y /D =.054  69  Figure  25.  Figure  Response  26.  Comparison  Response  For  Comparison  El  For  Centro  San  Fernando  70  f i g u r e 28.  Response P r e d i c t i o n Comparison For El Centro  71  Figure 29.  Response P r e d i c t i o n Comparison For San Fernando  72 BIBLIOGRAPHY  Anagnostopoulos, S., 1982, "Dynamic Response of Offshore Platforms to Extreme Waves Including F l u i d Structure Interaction", Engineering Structures, V o l . 4, pp 179-185. Blevins, R.D., 1977,  "Flow Induced Vibrations", Van Nostrand Reinhold  Co., New York. Blevins, R.D., 1979, "Formulas f o r Natural Mode Shape and Frequency", Van Nostrand Reinhold, New York. Byrd, R., 1978, "A study of the F l u i d Sructure Interaction of Submerged Tanks and Caissons i n Earthquakes", Earthquake Engineering Research Report 78/08, May 1978. Chakrabarti, S.K. and Frampton, R., 1982, "Review of Riser Analysis Techniques", Applied Ocean Research, V o l . 4, No. 2, pp. 73-90. Clough, R.W.,  1960, "Effects of Earthquakes on Underwater Structures",  Proceedings, Second World Conference on Earthquake Engineering, Tokyo 1960, V o l . I l l , pp. 815-831. Clough, R. and Penzien, J . , 1975, "Dynamics of Structures", McGraw-Hill. Eatock-Taylor, R., 1981, "A Review of Hydrodynamic Load Analysis f o r Submerged Structures Excited by Earthquakes", Engineering Structures, V o l . 3, pp. 131-139. Fish, P.R., et a l . , 1980, "Fluid-structure Interaction i n Morison's Equation f o r the Design of Offshore Structures", Engineering Structures, V o l . 2, pp. 15-26. Grecco, M. and Utt, M., 1982, "Dynamic Analysis of a Multi-Tube Production Riser", Proceedings, Ocean Structural Dynamics Symposium, Oregon State University, pp. 289-305.  73 Isaacson, M.,  1983,  "Earthquake Loading on Axisymmetric Offshore  Structures", Proceedings, Fourth Canadian Conference  on Earthquake  Engineering, Vancouver, Canada. Isaacson, M. and Maull, D.,  1981,  "Dynamic Response of V e r t i c a l P i l e s i n  Waves", Proceedings, Hydrodynamics i n Ocean Engineering, Trondheim, Norway. Keulegan, G.H.  and Carpenter, L.H.,  1958,  "Forces on Cylinders and Plates  in an O s c i l l a t i n g F l u i d " , Journal of Research of the National Bureau of Standards, V o l . 30, No. 5, pp. 423-440. Kirk, C.L.,  et a l . , 1979,  "Dynamic and Static Analysis of a Marine  Riser", Applied Ocean Research, V o l . 1, No. 3, pp. 125-135. Kirkley, O.M., Ph.D.  1973,  "Earthquake Response of Fixed Offshore Structures",  Thesis, U n i v e r s i t y of I l l i n o i s , Urbana, U.S.A.  Kirkley, O.M.  and Murtha, J . , 1975,  "Earthquake Response of Offshore  Structures", Proceedings, C i v i l Engineering i n the Oceans I I I , pp. 865-879. Liau, C.Y.  and Chopra, A., 1973,  "Dynamics of Towers Surrounded by  Water", Earthquake Engineering Research Centre Report, 73/25. Malhotra, A.K.  and Penzien, J . , 1969,  "Stochastic Analysis of Offshore  Tower Structures", Earthquake Engineering Research Center  Report,  69/6. Mei, C.C.,  et a l . , 1979,  "Exact and Hybrid-Element Solutions for the  Vibration of a Thin E l a s t i c Structure Seated on the Seafloor", Applied Ocean Research, V o l . 1, No. 2, pp. 79-88. Moe,  G. and Verley, R.L.P., 1980,  "Hydrodynamic Damping of Offshore  Structures i n Waves and Currents", Proceedings, 12th Offshore Technology Conference,  Houston, pp. 37-44.  74 Morison, J.R.,  et a l . , 1950,  "The Force Exerted by Surface Waves on  P i l e s " , Transactions, American Institute of Mining and Metalurgical Engineers, V o l . 189, pp. 149-154. Murtha, J.P. and Kirkley, O.M.,  1975,  "Response Spectra for Ocean  Structures", Proceedings, 7th Offshore Technology Conference,  pp.  985-990. Newnark, N.M.  and Rosenbleuth, E., 1971,  "Fundamentals of Earthquake  Engineering, Prentice-Hall Inc., Englewood C l i f f s , Pegg, N.G.,  1983,  N.J.  "An Experimental Study of the Seismic Forces on  Submerged Structures", MASc Thesis, University of B r i t i s h Columbia, Vancouver, Canada. Penzien, J . and Kaul, M.K., Motion Earthquakes",  1972,  "Response of Offshore Towers to Strong  Earthquake Engineering and Structural Dynamics,  Vol. 1, pp. 55-68. Sarpkaya,T., 1979,  "Lateral O s c i l l a t i o n s of Smooth and Sand-Roughened  Cylinders i n Harmonic Flow", Mechanics of Wave-Induced Forces on Cylinders (ed. T.L. Shaw), Pitman, London, pp. 421-435. Sarpkaya, T. and Isaacson, M.,  1981,  "Mechanics of Wave Forces on  Offshore Structures", Van Nostrand Reinhold, New York. Sawaragi, T. et a l . , 1977,  "Dynamic Behaviour  of a Circular P i l e due to  Eddy Shedding i n Waves", Coastal Engineering i n Japan, V o l . 20,  pp.  109-120. Selna, L. and Cho, D.,  1972,  "Resonant Response of Offshore Structures",  Journal of Waterways, Harbours and Coastal Engineering Division ASCE, V o l . 98, WW1,  pp. 15-24.  Skop, R.A.  and G r i f f i n , O.M.,  1975,  "Vortex-Excited O s c i l l a t i o n s of  E l a s t i c Cylinders", Proceedings, C i v i l Engineering i n the Oceans I I I , Delaware, pp. 535-545. Stark, P., 1970,  "Introduction to Numerical Methods", Macmillan Company,  New York. Stelson, T.E. and Mavis, F.T.,  1955,  " V i r t u a l Mass and Acceleration i n  Fluids", Transactions ASCE 2870, pp. 518-530. Sugiyama, T. and Ito, M.,  1981,  "Dynamic Characteristics of Structures i n  Water", Theoretical and Applied Mechanics, V o l . 30, University of Tokyo Press, pp. 373-379. Tung, C.C.,  1979,  "Hydrodynamic Forces on Submerged V e r t i c a l C i r c u l a r  C y l i n d r i c a l Tanks Under Ground Excitation", Applied Ocean Research, Vol. 1, No. 2, pp. 75-78. Van Dao,  B. and Penzien, J . , "Treatment of Nonlinear Drag Forces Acting  on Offshore Platforms", Earthquake Engineering Research Center Report, 80/13, University of C a l i f o r n i a , Berkeley, C a l i f o r n i a , Warburton, G.B.  and Hutton, S.G.,  1978,  USA.  "Dynamic Interaction for  Idealized Offshore Structures", Earthquake Engineering and Structural Dynamics, V o l . 6, pp. 557-567. Westermo, B.D.,  1980,  "Hydrodynamic Interaction of E l a s t i c Structures",  Proceedings, 7th World Conference  on Earthquake Engineering,  Istanbul, Turkey, V o l . 6, pp. 129-132. Zedan, M.F.  and Yeung, J.Y.,  1980,  "Dynamic Response of a Cantilever P i l e  to Vortex Shedding i n Regular Waves", Proceedings, 12th Offshore Technology Conference,  pp. 45-59.  76 APPENDIX A  MODAL ANALYSIS  The methods of response prediction used i n this study are based on the technique of modal a n a l y s i s . obtained  This technique uses the mode shapes  from the solution of the governing equations of motion for an  Euler beam to describe the dynamic response of a continuous structure undergoing external loading.  Modal analysis e f f e c t i v e l y reduces the  governing p a r t i a l d i f f e r e n t i a l equation to a system of ordinary d i f f e r e n t i a l equations for which solutions e x i s t .  The response functions  are l i n e a r l y superimposed to describe the o v e r a l l response.  This method  i s very useful f o r linear systems and can be applied successfully to nonlinear problems provided  that the degree of nonlinearity i s small.  The p a r t i a l d i f f e r e n t i a l equation describing the f l e x u r a l motion of a slender beam, neglecting the e f f e c t s of damping, shear deformation and rotatory i n e r t i a i s  m (x)Y(x,t) + (EIY"(x,t))" = p(x,t) o  where m  Q  (Al)  i s the mass of the beam per unit length, y i s the normal  displacement from the l o n g i t u d i n a l a x i s , x i s the coordinate along the beam axis and p the externally applied load per unit length (See Clough and Penzien, 1975).  In general, E and I can be functions of position,  however, i n the case of a prismatic beam, the equation becomes, for free vibrations  m Y(x,t) + E I Y ( x , t ) = 0 iV  (A2)  77  subject to the appropriate geometric  and kinematic boundary conditions,  Variable separation i s used to solve Equation (A2); l e t  Y(x,t) - 4>(x)5(t)  (A3)  Substituting (A3) into (A2) and re-arranging, we have  iv, „ -m (x) _ o S(t) _ u a • (x) EI ? ( t ) 4>  (A4)  4  The solution to the fourth order d i f f e r e n t i a l equation defines the p o s i t i o n a l dependence given i n general as  <j> (x) = Asinax + Bcosax + Csinhax + Dcoshax  (A5)  where the c o e f f i c i e n t s are determined from the boundary conditions.  The  v i b r a t i o n frequency can be related to the constant 'a' by l e t t i n g  2 0)  m  * - TT* in the solution for the time dependent amplitude  ( A 6 )  £(t)  For the cantilever beam, the appropriate boundary conditions are expressed  as  78 . , Geometric r o n m o (  H /  4> (0) = 0 ^'(o) = 0 (A7)  Natural  • .,* ^  ' °  Q  These conditions are used to obtain a system of equations from (A5).  The  c o e f f i c i e n t matrix of this system i s set equal to zero for n o n t r i v i a l solutions resulting i n the transcendental equation  1 + cos a l cosh a l = 0  For a particular solution  (A8)  = a^i  of (A8), the ratio of c o - e f f i c i e n t s  i n the system of equations derived from (A3) and (A7) becomes  cosX X, + coshX SL sinX % + sinhX i. n n  n  K  *  J  and the nth mode shape function i s then given by  Xx <P (x) = cosh -£ n  Xx cos -£  O  r  Xx (sinh —  sin  Xx —)  (A10)  It can readily be shown that the natural modes of beams with c l a s s i c a l boundary conditions are orthogonal over the span of the beam, where orthogonality, as defined herein, implies  0 m * n <j> (x>j> (x)dx = { o £/2m=n  I  /  n  m  (All)  The free v i b r a t i o n frequency associated determined from (A6),  with a p a r t i c u l a r mode shape i s  as  (A12)  -  "  The  *2  "o  time dependent amplitude function becomes  K (t) = A n n  1  sinw t + B n n  1  cosu t n  where the c o e f f i c i e n t s A^ , B^ conditions.  The  (A13) v  i n (A13)  are determined by the  complete solution to (A2) i s the sum  /  initial  of the modal  solutions,  Y(x,t) = ) (A n=l  The  sinw t + B  1  1  COSUJ  t) q> (x)  (A14)  formulation of the more general problem of beam vibration w i l l  include damping and external f o r c i n g . formed from the substitution of (A14) including damping.  ao  I m<|> n=l  In that case, the solution i s into the general equation of motion  For the present problem, this y i e l d s  oo  oo  .  (x) 5 (t) + 1 CI<t> (x)£ (t) + I EI^ (x)S (t) n=l n=l v  n  n  n  = (x,t) P  (A15)  80 Multiplying by <J> (x), integrating over the length of the beam, and using m  ( A l l ) , we obtain  M  J n  (  t  )  +  C  n^n  ( t )  +  K  n n ?  ( t )  ^  =  * (  x  )  p  n  (  x  »  t  )  d  (A16)  x  o  where M C  n  i s the equivalent mass, given here simply as the t o t a l mass, m£,  n  i s the generalized modal damping c o e f f i c i e n t and K  stiffness.  K = n  r  i s the generalized  For orthogonal mode shapes, i t can be shown that  2 0)  M n  n  v  (A17)  If the system damping i s approximated by viscous damping, as i s usually done, equation  (A16) can be written as  2  5 (t) + 25 w £ (t) + u> 5 (t) =P (t) n ' n n n n n n v  v  v  v  (A18) '  where the equivalent viscous damping ratio for each mode has been defined as , g  m  n  fn  (A19)  2m  n and  t h i s term w i l l c o n t a i n any damping c o n t r i b u t i o n present  generalized  i n the  f o r c i n g f u n c t i o n P ( t ) , g i v e n as n  / %(x,t)<j> (x)dx P (t) n  -2  (A20)  Solutions to (A18) can be found using either convolution integrals, Laplace transforms or numerical methods.  Although i n p r i n c i p a l , a  solution based on modal analysis can include contributions from many modes, generally only the lowest modes are Important i n the analysis slender beams at low frequencies.  82 APPENDIX B : DERIVATION OF RESPONSE AMPLITUDES  For the nth damped single degree of freedom system defined as  E  2 + 2c u) £ + u> E - P ( t ) n n n n n  n  (Bl)  n  a steady state amplitude i s given as  f . I* n  ? or  (>  1  [ (l-(w/w ) ) +(£ w/w ^ l n n n 2  n  L  B2  2  1  J  '  2  In the case of base e x c i t a t i o n given by  (5 (t) = Y sinwt  (B3)  the generalized i n e r t i a l forcing function becomes  P I  = ct ID Y n g 2  T  n  (B4)  and a solution to (Bl) including only i n e r t i a forces i s given then as  £  where  n  = a A sin(u)t-© ) n n n  (B5)  83 u> Y = * u [(l-(w/w ) ) n n' '  1  2  A n  2  2  1v  (B6) 2  v  + ( 2 ; u/d n n  v  / J  )]  1 / 2  and 0  2? w/u) = tan" ! — M l-(u>/ f n  (B7)  1  n  U  For  two modes, we have the modal and base motions as  Si (t)  = ? i sin(o)t-<pi )  (B8)  £ (t) = S sin(wt^> ) 2  6 (t)  2  2  = Y sinw t 8  The cylinder t i p response i s given by combining and expanding (B8) to yield  Y(x,t) = <Pi (£ )Ci (sinwtSi - cosw t s i n ^ ) 2  + <t>2 (*•  (siao tcosS2 ~ cosw tsin32 )  + Y sinwt g  For  the present formulation  • l (A ) - 2  (B9)  84  4>2(*> = "2  (BIO)  Using (BIO) i n (B9) and c o l l e c t i n g terms,  Y  = [(25icosOi - 25 cos© 2  2  + Y )  + (2S sinG - 2 ^ i s i n 0 i ) ] 2  2  2  (Bll)  1 / 2  2  Simplifying  Y . = 2k + \ tip 1 2 2  2  + 0.25Y g  + Y (£icos0i - £ c o s © ) g  2  - 2Ci"i2Cos(0i-O )]  2  2  (B12)  1 / 2  2  In the case of l i n e a r i z e d drag, the generalized forcing function complex.  i s more  The sum of the generalized i n e r t i a l and drag components i s  given as  aK  P(t) =  2  g  + 3 (li^Y 2  with  D  sinwt  aiU) Y  l g 1  m3  2  oa Y cosu)t 2  2  8  + coliY coswt) +  (B13)  85  01 " J * <t>i(x)dx  3  - /* <t>(x)dx  (B14)  2  2  33 = /* <t> (x)dx 3  i  The components o f (B13) involving I i are included as viscous damping, leaving the forcing function as  P(t) = aiu Y 2  8  sincot + —-—& m&  (8].u>Y + e li)cosu>t 2  (B15)  8  2  which can be rewritten i n amplitude form.  The response i s then given, as before, as  il  OJ  The constituent  2  [(l-(u)/u>i) ) 2  2  (B16)  + (2tiu/ui) \ 2  l  1  2  responses of the cylinder t i p are  Y(A,t) = 4> (A)5 sin(u>t-e 1  1  LD  - ©x) (B17)  6(t) = Y sinut g  86 where  aK^uY (BiUY  •  = tan'l  + B E) 2  ^ — §  (B18)  moj^Y B i g  ©1 as i n (B7)  Combining and simplifying, we obtain  Y  tip  =  2  +  0 , 2 5 Y  g  2 +  Y  g  ^l  c o s  ( l 0  +  e  LD  )J  1/2  (B19)  87 APPENDIX C  DESCRIPTION OF COMPUTER PROGRAM  The numerical method used to solve the linearized and  nonlinear  d i f f e r e n t i a l equation derived i n Chapter 2 involves a small Fortran IV program.  The program requires the following input to be contained  in a  data f i l e attached to the main program:  1)  I n i t i a l values of displacement and v e l o c i t y .  Zero i n i t i a l  values  are u s u a l l y assumed since t h i s allows a stable s o l u t i o n to be reached more quickly.  2)  Length of structure, diameter, e l a s t i c modulus, moment of i n e r t i a , sectional mass and damping r a t i o .  3)  Frequency of o s c i l l a t i o n , i n hertz, and amplitude of base motion, for  sinusoidal motion.  For random forcing, these variables are  replaced by the appropriately d i s c r e t i z e d input.  4)  Drag c o e f f i c i e n t  5)  Number of time steps, and the time increment.  The program calculates the natural frequencies and c o e f f i c i e n t s of the generalized equation of motion using cantilever beam theory. second order d i f f e r e n t i a l equation i s replaced by two f i r s t  The  order  equations which are solved using a fourth order Runge-Kutta integration  88 scheme.  A time h i s t o r y response i s generated f o r the t i p displacement by  time stepping.  Time increments of approximately a tenth of the minimum  forcing function period are recommended  f o r numerical accuracy.  At high  frequencies, modulated responses are evident for many cycles N > 600 for low damping values.  Near the resonant frequency, the output s t a b i l i z e s  reasonably quickly (N < 300).  89  APPENDIX D  INSTRUMENT CALIBRATION AND TYPICAL DATA  The strain-gauged rod and spring system was calibrated p e r i o d i c a l l y by s t a t i c displacement t e s t s .  Figure 30 shows a t y p i c a l c a l i b r a t i o n  curve indicating a linear relationship between bridge voltage and i n - l i n e and transverse cylinder t i p displacement. A time history and Fourier spectra for a test at 9 Hz and g / ° Y  .036 are presented i n Figure 31 and 32 respectively.  =  These records  i l l u s t r a t e the presence of the second mode response at c e r t a i n forcing frequencies.  Such response was only noticeable i n the 9 and 14 Hz  records, frequencies near subharmonics of the second modal frequency, 28 Hz.  It should be noted that the single mode approximations developed  i n t h i s study w i l l not reproduce t h i s type of response. The average half-peak-to-peak response of the actual record was used as representative response i n the receptance functions rather than the Fourier amplitudes of the output at the forcing frequency.  Typical i n -  l i n e and transverse records a t resonance (4.5 Hz) are presented i n Figures 33 and 34 respectively.  90  F i g u r e 30.  C a l i b r a t i o n Curve f o r the Instrumented  Rod  91  O  03 -1  Figure 31.  Typical Time History at 3rd Subharmonic  ID  0.0  4.0  B.O  12.0  16.0  FREQUENCY  Figure 32.  20..0  24.0  28.0  32.0  (HZ)  Fourier Spectra For Record of Figure 31  36.0  Figure  34.  Transverse  Resonance  Time  History  at  Y /D =.054  

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