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Investigation on nonlinear coupled vibration of columns. Bridicko, Jan 1972

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INVESTIGATION OF NONLINEAR COUPLED VIBRATION OF COLUMNS by JAN BRDICKO 3.Sc., U n i v e r s i t y o f I l l i n o i s , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department o f M e c h a n i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May, 1972 In present ing th i s thes is i n . . - p a r t i a l ' f u l f i lmen t .o£ the .„ requ i rements f o r an advanced degree at the Un ivers i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying o f th i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department of ^Q.ck<x\nic<xl Eingiyie.^ri The Un ivers i t y of B r i t i s h Columbia Vancouver 8, Canada Date M<x~, 2 S (972 i ABSTRACT The o s c i l l a t i o n of a column subjected to periodic axial end excitation was analytically and experimentally investigated. The i n i t i a l crookedness of the column and the longitudinal inertia of a column element give rise to coupled, longitudinal-flexural oscillations. A snap-thru phenomenon and complex subharmonics of natural flexural modes of osc i l l a t i o n also occur at certain axial end excitation frequencies. Furthermore; at certain excitation frequencies, a coupling between longitudinal and torsional oscillations i s found to exist. A theory providing qualitative and quantitative information about coupled longitudinal-flexural oscillations and complex subharmonics was developed for a column with hinged ends. In order to test the valid i t y of the theory an experimental apparatus was set up to excite the column axially, with transducers monitoring the response of the column. The experimental results were in very good agreement with the theoretical predictions. A column with b u i l t - i n ends was also tested and i t s response was similar to the column with hinged ends. Thus, the results of the experimental investigation suggest that the results of the theory developed for a column with hinged ends are also applicable to a column with b u i l t - i n ends. Coupled longitudinal-flexural oscillations were observed when the axial end excitation frequency was equal to the natural longitudinal frequency of the column. Large amplitude oscillations in both longitudinal i i and flexural vibration modes occurred at this frequency. When the frequency of axial end excitation was equal to the natural flexural frequencies of the column, large amplitude flexural oscillations resulted, Flexural oscillations were also observed when the frequency of the axia l end excitation was one half, one third up to one eighth of the natural flexural frequencies of the column, A spectrum analysis of the strain signal showed that the flexural response then comprised two fundamental motions, one with the frequency of the axial excitation and one with frequency equal to the associated natural frequency. The resulting amplitudes of flexural oscillations at these frequencies were smaller than those observed when the frequency of the axial end excitation was equal to the natural flexural frequencies of the column. The flexural oscillations occurring at these axial end excitation frequencies were identified as the complex subharmonics of natural flexural frequencies, A snap-thru phenomenon occurred when the axial end excitation frequency was twice the frequency of natural flexural frequencies. Under certain circumstances the column then oscillated flexurally with one half of the excitation frequency. The amplitudes of flexural oscillations were comparable to those occurring when the frequency of the axial end excitation was equal to natural flexural frequencies of the column. Large amplitude flexural oscillations occurring at natural flexural frequencies, complex subharmonics and snap-thru phenomena, though excited by the axi a l end excitation, did not cause appreciable increase in amplitudes of longitudinal oscillations. i i i Finally, large amplitude torsional oscillations occurred when the axial end excitation was of the same frequency as the predicted torsional natural frequencies. Again no appreciable increase in amplitudes of longitudinal oscillations was observed. Flexural o s c i l l a t i o n phenomena here described, also occurred during flexural in-plane o s c i l l a t i o n of a column. TABLE OF CONTENTS Page ABSTRACT i ACKNOWLEDGEMENT xiv LIST OF FIGURES v i i i LIST OF TABLES ' ix LIST OF APPENDICES ' x NOMENCLATURE x i CHAPTER 1 INTRODUCTION 1 Preliminary Remarks 1 Literature Review 3 Limitation of Investigation..... 6 CHAPTER 2 NONLINEAR THEORY 7 Theoretical Considerations 8 Governing Differential Equations 8 Solution of the governing d i f f e r e n t i a l equation by the perturbation method 22 Material damping, 31 Theoretical predictions.* 36 CHAPTER 3 LINEAR THEORY... ^0 Longitudinal o s c i l l a t i o n , 41 Free flexural o s c i l l a t i o n of a prismatic column with hinged ends, , 4-3 Free flexural o s c i l l a t i o n of a prismatic column with hinged ends - effect of rotary inertia and shear terms , 44 Free flexural o s c i l l a t i o n of a prismatic column with b u i l t - i n ends 45 Free flexural o s c i l l a t i o n of a prismatic column with b u i l t - i n ends - effect of rotary inertia and shear terms 46 Torsional o s c i l l a t i o n of a prismatic column with b u i l t - i n ends.... 49 V Page CHAPTER 4 APPARATUS AND INSTRUMENTATION 51 Design of the column with hinged ends.... 52 Design of the column with b u i l t - i n ends.. 56 Design of the test bench 58 Vibration control apparatus.... 58 Transducers and associated electronics... 61 CHAPTER 5 TEST PROCEDURE 65 Calibration 65 Testing preliminaries... 65 Mounting of a column. 66 Testing 6? Additional testing 68 Total damping measurements 69 Undertesting and .overtesting 70 Photography 70 Loading of a column by a constant axial force 71 CHAPTER 6 RESULTS AND DISCUSSION 72 Identification and analysis of strain vs. frequency records 72 Analysis of flexural strain vs. frequency records - natural flexural frequencies.... 72 Analysis of flexural strain vs. frequency records - complex subharmonics 74 Phase angle shift 77 Analysis of flexural strain vs. frequency records - snap-thru phenomenons 80 Discontinuities of strain vs. frequency curves 84 Analysis of longitudinal strain vs. frequency records 84 Study of os c i l l a t i o n modes obtained by sprinkling of the column with salt 85 Intensity of coupling between vibration modes 89 Agreement between experimental results and theoretical predictions...... 90 v i Page Influence of a constant axial load.... 92 Strain magnitudes., 93 CHAPTER 7 SUMMARY AND CONCLUSIONS 96 Summary of theoretical investigation.. 96 • Summary of experimental investigation. 98 Suggestions for future research 101 BIBLIOGRAPHY 103 APPENDICES 105 v i i LIST OF FIGURES Figure Page Fig. 1. Coordinate system of the column 7 Fig. 2. Coordinate system of the column element 9 Fig. 3. Approximation (23) i = 1 17 Fig. 4. Approximation (23) i = 6 18 Fig. 5. Magnitude of parameter q 23 Fig. 6. Design #1 53 Fig. 7. Design #2 53 Fig. 8. Design #3 54 Fig. 9. Column with hinged ends 55 Fig. 10. Column with b u i l t - i n ends 57 Fig. 11. Test bench 59 Fig. 12. Signal flow diagram 60 Fig. 13. Placement of strain gages on a column 63 Fig. 14. Arrangement of strain gages i n the 64 Wheatstone bridges Fig. 15. Oscillation of a column in natural o s c i l l a t i o n 73 modes Fig. 16. Oscillation of a column at second order 74 subharmonics Fig. 17. Oscillation of a column at third order 75 subharmonics Fig. 18. Oscillation of a column at fourth order 75 subharmonics Fig. 19. Oscillation of a column at f i f t h order 75 subharmonics Fig. 20. Oscillation of a column at sixth order 76 subharmonics v i i i Figure Page Fig. 21. Oscillation of a column at seventh order 76 subharmonics Fig. 22. Oscillation of a column at third order 76 subharmonics (weak) Fig. 23. Oscillation of a column ar F7/4 subharmonics - 77 phase angle shift (912 Hz) Fig. 24. Oscillation of a column at F7/4 subharmonics - 78 phase angle shift (920 Hz) Fig. 25. Oscillation of a column at F7/4 subharmonics - 78 phase angle shift (923 Hz) Fig. 26. Oscillation of a column at F7/4 subharmonics - 78 phase angle shift (928 Hz) Fig. 27. Oscillation of a column in snap-thru o s c i l l a t i o n 80 modes Fig. 28. Strain vs. frequency record for a column with 82 hinged ends Fig. 29. Strain vs. frequency record for a.column with 83 b u i l t - i n ends Fig. 3 0 . Nodal line pattern occurring when a column 86 oscillates i n the f i r s t natural torsional o s c i l l a t i o n mode Fig, 3 1 . Nodal line pattern occurring when a column 86 oscillates in the second natural torsional o s c i l l a t i o n mode Fig. 32. Nodal line pattern occurring when a column 87 oscillates in the third natural torsional o s c i l l a t i o n mode Fig. 33. Nodal line pattern occurring when a column 87 oscillates i n the f i f t h natural flexural o s c i l l a t i o n mode Fig. 34. Nodal line pattern occurring when a column 87 oscillates in the tenth natural flexural o s c i l l a t i o n mode Fig. 35. Theoretical nodal line patterns 88 ix LIST OF TABLES Natural frequencies of the column with hinged ends Natural frequencies of the column with b u i l t - i n ends LIST OF APPENDICES Appendix A List of equipment Appendix B Examples of application of the nonlinear theory of Chapter 2 Appendix C Magnitudes of strains NOMENCLATURE parameter parameter constant Fourier Sine series coefficients describing i n i t i a l geometry imperfection of a column Fourier Sine series coefficients amplitudes of polynomials associated with cosine terms amplitudes of polynomials associated with sine terms width of column heighth of column length of column weight density of column material mass density of column material area of cross section of column normalized damping coefficient of internal (material) damping c r i t i c a l damping coefficient damping factor specific damping energy factor logarithmic decrement amplitude of forcing function amplitude of axial end excitation acceleration of gravity polar moment of inertia of column cross section modulus of el a s t i c i t y shear modulus x i i 3? = shear c o e f f i c i e n t r = radius c 0 = v e l o c i t y o f l o n g i t u d i n a l wave propagation i n column ^ = t o r s i o n a l s t i f f n e s s o f column cross section k, = constant ( r e l a t e d to t o r s i o n a l s t i f f n e s s ) t = r e a l time z = normalized time x,y,z = coordinate distances u = a x i a l displacement v = in-plane displacement w = f l e x u r a l displacement 9- = t o r s i o n a l displacement p(x,t) = d i s t r i b u t e d load P = t o t a l applied a x i a l load P Q = constant applied a x i a l load ^ = frequency o f a x i a l end e x c i t a t i o n ° ° n = n-th natural f l e x u r a l frequency i n rad/sec f„ = n-th natural f l e x u r a l frequency i n Hz = n-th natural l o n g i t u d i n a l frequency ^ L = fundamental l o n g i t u d i n a l frequency u 0 ( x ) = s p a t i a l form of l i n e a r l o n g i t u d i n a l o s c i l l a t i o n # o f column w D(x) = s p a t i a l form of l i n e a r f l e x u r a l o s c i l l a t i o n of column 0"o(x) = s p a t i a l form of l i n e a r t o r s i o n a l o s c i l l a t i o n of column w,(t), w ( t ) , . . = functions d e s c r i b i n g time v a r i a t i o n of amplitudes of i n d i v i d u a l components of the Fourier Sine s e r i e s d e s c r i b i n g the f l e x u r a l shape of column F = "F l e x u r a l mode" P = l"in-Plane mode" x i i i L = "Longitudinal mode" T = "Torsional mode" i , j = integers n = mode number f . = phase angle r.i C, , C2 ... = constants D = di f f e r e n t i a l operator R = resistance of column material E = applied voltage V = measured voltage y*- = mass per unit length 0 f column " • • = rOOtS w(x) = i n i t i a l geometry imperfection ( i n i t i a l crookedness) of column w n i(z) = series expansion terms of w n(z) Abbreviations BAM = Bridge amplifier and meter Hz = cycles per second RMS = Root-mean-square value of a function xiv ACKNOWLEDGEMENT I wish to express my gratitude to my faculty advisors, Dr. H. Vaughan and Dr. H. Ramsey, for giving me an opportunity to work on this project. I found their assistance and their advice very helpful in overcoming the d i f f i c u l t i e s arising during the investigation. Further, I wish to thank a l l the technicians and secretaries i n the Department for their contribution to this research. This study was made possible through Research Grant No. 67-5563 provided by the National Research Council of Canada. CHAPTER 1 1 Preliminary remarksi In todays engineering practice, a design of components exposed, or subjected to high frequency excitation must often be tackled. In particular, problems associated with a design of turbine blades, sonar equipment, etc. sometimes lead to the study of axially excited columns. The linear theory provides some information about the behaviour of a column, but this may be insufficient as, at very high frequencies the nonlinear effects can be quite significant. For this reason the os c i l l a t i o n of a column subjected to periodic axial end excitation was analytically and experimentally investigated at relatively high frequency. The i n i t i a l crookedness of a column and the longitudinal inertia of a column element give rise to coupled longitudinal-flexural oscillations. Furthermore; at certain excitation frequencies, coupling between longitudinal and torsional oscillations i s found to exist, A study of coupled longitudinal-flexural oscillations of a column with hinged ends was done. The study eventually yielded a solution giving a complete description of the combined longitudinal-flexural motion of the column. The examination of this solution allows us to make predictions about the response of the column. Large amplitude flexural oscillations w i l l occur when the excitation frequency equals natural-flexural frequencies. The column then oscillates flexurally with the same frequency as the excitation axial frequency. The theory also predicts large amplitude flexural oscillations when the excitation frequencies equal to 1/2, l/3, l/4, l/n,...of the natural flexural frequencies. The column then oscillates flexurally with the same 2 frequency as the axial excitation frequency and also with the natural flexural frequency (that i s the frequency equal to 2, 3, 4, n ....times the excitation frequency). These types of os c i l l a t i o n are identified as subharmonics of natural flexural frequencies. Resonant, large amplitude longitudinal oscillations w i l l occur only when the excitation frequency equals natural longitudinal frequencies. The column then oscillates longitudinally with the same frequency as the axial excitation frequency. These predictions were then checked experimentally, and the results of the experimental investigation agreed very closely with theoretical predictions. A Literature Reviewj 3 Most of the research connected with the parametric response of bars and columns has been limited to small free lat e r a l oscillations in the f i r s t natural vibration mode , Some researchers studied the parametric response of columns subjected to small axial or lateral periodic excitation. The excitation was usually of the form of sinusoidally variable force or acceleration with time, and sometimes a constant axial force was superimposed on the variable excitation. Nearly a l l of the theoretical and experimental analysis has been limited to cases where the frequencies of column o s c i l l a t i o n (and external excitation) were well below the frequency of fundamental longitudinal vibration mode. Variety of boundary conditions, damping, excitation and i n i t i a l crookedness were considered in theoretical and experimental analyses of this problem. Coupled longitudinal-flexural and longitudinal-torsional oscillations of a column were also studied by several researchers. A short literature review of the work done on this subject i s presented here. Other a r t i c l e s and references can be found in Journal of Applied Mechanics, numerous vibration handbooks etc. One of the f i r s t researchers to analyze the response of a column subjected to a periodic axial end excitation was Beliaev (3). He reduced the equation of motion of the column to the standard form of Mathieu equation by neglecting the longitudinal inertia of a column element. The s t a b i l i t y analysis of Mathieu equation predicted i n s t a b i l i t i e s to occur when the excitation frequency equals 2, 1, 2/3. 1/2, 2/5,....multiplies of natural flexural frequencies. In absence of damping i n s t a b i l i t i e s represent an unbounded growth of amplitudes of o s c i l l a t i o n with time. The experimental verification of Beliaev's theory was done by Bolotin (4), Somerset (5) and others. Analytical and experimental investigations have beer, performed in a study of longitudinal inertia effects upon the rarar.etrio response of a colnr.r under sr axial load P (t) = P0 + P, cos ft. The analysis by Evan-Iwap.owski and EVensen (6) has led to s t a b i l i t y c r i t e r i a , which have beer, plotted i r the form of two bounding surfaces of an in s t a b i l i t y region in a three-d 1 r.ens 1 on? 1 taraneter sre.ee (P0 P, * ) , Use of this region permits dejscrir-tie" of the conditions and responses associated with paranetrie ins t a b i l i t y , and allows evaluation of the effects of disturbances. A straight beam with fixed ends, excited by the periodic motion of i t s supporting base in a direction normal to the beam span, was investigated analytically and experimentally by Tseng, W, Y., and Dugundji, J, (7 )• By using C-alerkin's method (one mode approximation) the governing partial d i f f e r e n t i a l equation (not coupled to longitudinal motion) reduces to the well-known Duffing equation. The harmonic balance method is applied to solve the Duffing equation. Besides the solution of simple harmonic motion, many other solutions, involving superharmonic motion and subharmonic motion are found experimentally and analytically. The s t a b i l i t y problem i s analysed by solving a corresponding variational Hill-type equation.( The column tested was (steel) 18 in long 0.021*0,5 in with b u i l t - i n ends. The fundamental flexural frequency of the li g h t l y (^= 0.0006) damped column was approximate! 20 Hz. The testing was limited to 7. - 50 Hz frequency range) . Parametric torsional s t a b i l i t y of a bar under axial excitation was analytically and experimentally treated by Tso ( 8 ). His theory predicts th when a. straight column is excited axially at frequency twice the natural torsional frequency an unstable ter clonal c s c i l l ^ t i c n r.?v take -Is cr. Ar.c*hr unstable tcrsirr.al oscillation ray occur when th° excitation, frecuency is eoual or c l r s ^ to the natural longitudinal frecverey of the cclvr.r. 5 Johnson (2) experimentally analyzed the osci l l a t i o n of a prlsmatical column with b u i l t - i n ends. The excitation was that of constant level sinusoidal acceleration in time, Imposed at one end of the column, 64 lbs. constant force was superimposed on this excitation. He claimed to have detected the large amplitude coupled longitudinal-flexural o s c i l l a t i o n when the excitation frequency was equal to the fundamental longitudinal frequency, to one-half and also one third of i t . Schneider ( l ) experimentally analysed the parametric response of a prismatical column subjected to small l a t e r a l excitation. The later a l excitation of the form of constant level sinusoidal acceleration was imposed at the center of the column. Schneider detected large amplitude coupled flexural-longitudinal oscillations at la t e r a l excitation frequencies equal to the fundamental longitudinal frequency. Similar i n s t a b i l i t i e s were also detected when excitation frequency was equal to l / 2 and l / 3 of fundamental longitudinal frequency. Limitations of Investigation: In crrior to develop a theory predicting the behaviour of a column subjected tc sinusoidal axial end excitation seme assumptions had to be made. Nearly straight elastic isotropic prismatical column vrith rerfectly hinged ends is assumed. The material properties are assumed constant alcrs.g the entire length of the column and symmetrical with respect to the plane of leading. Bending strain is assumed tc be linearly proportional to tho distance from the central nlane. Tho effects cf shear and rotary inertia, are neglected. Also neglected is surmort, surface and air damning. The excitation is assu-ed to he truly axia.1. Other limitations pertaining to the theory of nonlinear coupled longitudinal-flexural oscillations are discussed in chanter ( 2 ). Limitations--related tc linear theory calculations are presented in chapter ( 3 ) where the individual vibration nodes vrith various boundary conditions are analysed. The experimental analysis was also subject to numerous restrictions. The boundary conditions are not exact but only approximated. The degree of approximation was much better in case of a column with b u i l t - i n ends than in case of a column with hinged ends, especially at low excitation frequencies. A small error could arise from, possible misalignment of the shaker vrith the rest of the setup. The amplitude of excitation was rather small 6 - 60 g's over a frequency range 100 - l6000 Hz. Only two prismatical, mild steel, cold rolled columns were used. Their dimensions were 1/8 in x3/8 in x12 in approx. Strains were detected by strain gages and spatial shapes found by sprinkling the column with salt. CHAPTER 2 NONLINEAR THEORY 7 F i g . 1. Coordinate system o f the column 8 NONLINEAR THEORY Theoretical Considerations: In this section, the transverse and axial displacements which occur in an i n i t i a l l y imperfect prismatic column, when one end of the column Is subjected to a-periodic axial excitation, are considered. The theory consists of two main parts; the reduction of the coupled partial d i f f e r e n t i a l equations to ordinary d i f f e r e n t i a l equations, and the solution of the governing ordinary d i f f e r e n t i a l equation by the perturbation method. We start by considering the general equations. Differential Equations Governing the Coupled Transverse and Axial Motion  of a Prismatic Column; The equations governing the motion of a rod are given (up to third order terms) by Mettler (9) and are given i n his a r t i c l e in "Handbook of Engineering Mechanics" by Fluegge (9). These equations are presented heret (1) Elw XXXX where; u = u(x,t) axial displacement w = w(x,t) transverse displacement w = w(x) i n i t i a l geometry imperfection ( i n i t i a l crookedness) mass per unit length of the column A cross section area of the column 9 Z,w (F) Y , v (P) Fig. 2. Coordinate system of the column element 10 E Young's modulus of the column material I . minimum moment of inertia of the column cross section /S ...... coefficient of internal (material) damping p(x,t)...distributed load The column i s excited axially, and this excitation then i s a source of steady state oscillations of the column. Therefore, the steady state longitudinal o s c i l l a t i o n may be expected to the negligibly or very weakly affected by an accompanying transverse os c i l l a t i o n . (This deduction w i l l be eventually supported by the results of experimental investigation which show that the only and verv small influence of transverse oscillations on the longitudinal o s c i l l a t i o n occurs when the column oscillates at any of i t s natural flexural frequencies). Thus, especially at lower frequencies, that i s for excitation frequencies lower than one half of the fundamental longitudinal frequency, the two terms: -§wx and w^ w^  can be neglected from equation ( l ) . The same two terms w i l l be neglected from equation (2) since in equation (2) they are of third order. It i s reasonable to assume that the influence of third order terms on the osc i l l a t i o n of the column w i l l be much smaller than that of f i r s t and second order terms. In addition,second order terras of equation (2) are the terms coupling the longitudinal and transverse motion and should therefore be retained. The internal damping force /8w^ w i l l not be neglected as i t w i l l ultimately limit the amplitudes of coupled o s c i l l a t i o n . The effect of a distributed load upon a coupled longitudinal-flexural o s c i l l a t i o n w i l l not be examined either theoretically or experimentally, so that p(x,t) may be taken as zero. 11 After making these assumptions and restrictions the two equations ( l ) and (2) reduce to: -£AMxk + y*M^ - 0 (3) In the particular case considered here, one end of the column i s clamped, and the other end is subjected to a sinusoidal acceleration of magnitude S . Thus the boundary conditions on the axial displacement are: u(o,t) = 0 (5) u ( l , t ) = S cos e t (6) The exact solution to equation ( 3 ) , satisfying boundary conditions (5) and ( 6 ) , was found in chapter 3« section l ) equation l ) equation (77) as: f S SI* ( C * X ) <HJ fr,\ sin ( ±4) 9- * 0 (8) sLn(fl) 4 0 (9) 6 frequency of axial excitation 1 c 0 velocity of longitudinal wave propagation in the column 12 Restriction (8) means that the static case w i l l not be considered. Restriction (9) implies i n f i n i t e amplitudes for sin rr £ - 0 which li k e the results of any linear theory results needs to be interpreted correctly} i t defines the resonant (large amplitude) frequencies for the column. Substitution of equation (7) in (4) yields: which after differentiation, division by j*. , and rearranging becomes: /* EI tt /*• * y<- **** y<-w/ ^ - EAS f 9- . /+ x - cos -cos(£x)-w„JcosM ( l l ) We now consider the transverse displacements. The boundary conditions on the transverse displacements are: w(0,t) = 0 , w(l,t) = 0 (12) The specification of transverse displacements at the ends of the column are not sufficient for the solution. We need to stipulate additional physical conditions. In this case we consider the column to be freely hinged at the ends, so that the remaining boundary conditions are: tf « x x ( 0 , t ) = w x x ( l , t ) = 0 (13) The physical significance and practical imposition of these end conditions are discussed later in chapter 4 . The terms containing i n i t i a l crookedness were put on the right side of the equation ( l l ) and the right side then essentially represents the transverse excitation forcing function. The i n i t i a l crookedness in a certain sense converts axial excitation into l a t e r a l excitation. The transverse displacement w(x,t) i s to be found by solving the partial d i f f e r e n t i a l equation ( l l ) . There is no standard procedure to obtain a solution to such an equation. Usually, a solution based on the knowledge about the behaviour of the system the equation represents^, i s assumed. This w i l l be done here. Imagine a column subjected to the axial excitation of constant frequency 9 , Now look at the column at a given instant of time t * . The spatial shape of a column in this instant of time t* could be described by a Fourier series of period 1 . The series must satisfy the boundary conditions ( 1 2 ) and ( 1 3 ) . Fourier sine series satisfies the boundary conditions identically, and therefore1 w(x,t*) = Cf s i n y + G 2 s i n — + ... + s i n ^ y + (14) If we now l e t the time change ,the magnitude of C s w i l l l i k e l y change as well. It i s l i k e l y that C( w i l l change differently than C £ etc. Thus constant C s are independent of each other and no longer constant but time dependent. -It i s also reasonable to assume that C^s w i l l be affected by the frequency and amplitude of excitation. Thus ( 1 4 ) should be rewritten 14 for the real time ast w(x,t) = w (t) sin • • » • + H n ( t ) sin>2^< + (15) w,(t), w 2(t) , • • » . i • • • are then functions describing the time variation of amplitudes of individual components of the Fourier sine series representing the flexural shape of the column at a l l time. Theoretically, series (15) has an i n f i n i t e number of terms. However, we may anticipate convergence and generally expect that the early terms in the series describe the motion adequately. It w i l l be shown that for certain exciting frequencies, one term w.(t) w i l l be of sufficiently large amplitude as to completely dominate the motion of the column. The substitution of equation (15) into equation ( l l ) yields the following equation1 2 Ac&sin (*£) EAS cos cos 9-i (16) 15 Eq. (16) can be rearranged asj •+ - ^ c . » « . (**) L ( 5 s i n ? c " X + T M T 7 / T ^ + d~t ( Before proceeding further l e t us consider the i n i t i a l crookedness of the column which i s represented here by w(x). Since both ends of the column are clamped we can say: w(0) = w(£) = 0 (18) w(x) can be also represented by a Fourier series. The Fourier sine series of period 1 w i l l be chosen as i t identically satisfies the condition (18). w(x) = a* s i n y - + a* s i n ~ - + .... + a* sin ~ - + ... (19) 16 •+ where a* , a* ....a* ...are constants depending on the form of the i n i t i a l crookedness w(x). The i n i t i a l crookedness w(x) as expressed by equation (19) transforms the right side of equation (17) into: \c0 c e Z A q, z / £ z + (co^ToC0S-T Ycos-stn-j-J-j-a* + .- Jcos*t (20) Comparison of equations (17) and (20) shows that the both sides of equation (17) contain the terms: \Vin^.c*s~ + 7 C W ^ 7 / I - c - i ^ . - . (21) Many workers replace this term by simpler term (i1T/l) sin (iffx/l) with the restriction that 9 i s much smaller than £AL , the fundamental longitudinal frequency of the column. We w i l l now show that this simplification can be made without limiting 6 to be so small. In particular, i t w i l l be shown that expression (21) can be adequately represented by (iTT/l) sin (iT x/l) for 9 as large as ^1/2 . From chapter 3. (78) i t is seen that: — s (22) After substituting eq. (22) in expression (21), we need to show that: Fig. 4. Approximation (23) , i - 6 19 The greatest error in the approximation (23) w i l l occur for i = 1 as i t makes the f i r s t term on the l e f t hand side of the (23) largest. To judge the accuracy of the approximation (23) the approximation i s shown graphically in Figures 3 and 4 , In producing these charts the length of the column 1 was assumed 12 in. and i t s fundamental longitudinal frequency jtL was assumed to be 8.1kHz . These values are very close to the values for the column actually experimentally analyzed. Both charts show the function to be approximated i.e. the l e f t side of eq.(23) and the approximating function i.e. the right side of (23). The approximation (23) is shown for 1500 Hz, 2500 Hz and 3500 Hz values of forcing frequencies «• , Fig. 3. applies to the case i = 1 and therefore shows the worst case that can arise in approximation (23). Fig. 4. applies to the case i = 6. A conclusion based on the two charts can be drawn. The approximation (23) is better the lower the frequency 0" , the smaller the coordinate x , and the higher the mode number i Thus i t seems that the approximation (23) is of good quality, at least for low values of forcing frequency & . With this restriction imposed, and with use of equations (l9)» and (20)-and (23) the governing d i f f e r e n t i a l equation (l?) becomesj [ wt(i) si* jr- + wt(t) sen ^ ~ + + *n(*)sC» ^ -h J u » X +  SC"  + + ^ f t ) « > x + J £AS f / T\ Z . ffx /ZTT) 2 ,^ / 2fa x + (ff w^ t)siy> IT  + - • ] cos ** ' (24) 20 The equation (24) can be uncoupled by equating the coefficients of like sine terms. A set of d i f f e r e n t i a l equations is thus obtained: w V * & $ ~ "•M"'pl * From linear theory (see chapter 3)formulas giving natural flexural and longitudinal frequencies can be used to rewrite equation (25) 2 £I/hir\ The natural flexural frequencies are: COn - — [ ~ £ ~ J (26) The natural longitudinal frequencies are: - = y>f<L Substitution of (26) in (25) after slight rearrangement gives the d i f f e r e n t i a l equations: 2 2 « n ( t ) + £w..(t) + («- - ' S \ cos 9t) w n(t) = n 2 f * a > s «t (27) n = 1 ,2 ,3, 21 Changing the dependent variable from t to z where St the relationships: 8t = 2z With these substitutions eq.(27) becomes: Introducing c, a , q , and A by: c = 4S#lna'a* Enables eq.(3l) to be written in the form: » = .... Where prime denotes differentiation with respect to z . 22 Solution of the governing d i f f e r e n t i a l equation by the perturbation  method: Equation (33) is an inhomogenous, second order, ordinary d i f f e r e n t i a l equation with a variable coefficient associated with the zeroth order term. The solution of this equation w i l l consist of the complementary function plus a particular integral. The complementary function i s the solution of the d i f f e r e n t i a l equation with the right-hand side set equal to zero. However; since damping i s present in the system^the complementary function w i l l decay exponentially with time and rapidly approach zero. It w i l l be present in the i n i t i a l stages of motion or as a decaying free vibration which follows after the cessation of the forcing term A^cos 2z . Thus for the general description of the motion we need to consider only the particular solution of eq,(33). The particular solution, which i s a solution satisfying the complete d i f f e r e n t i a l equation, represents a part of the motion which w i l l occur continuously while the forcing condition is present. An exact particular solution to eq.(33) cannot be readily found; however, there are numerous procedures for finding approximate solutions analytically. The perturbation method (12) w i l l be used as i t s application to such an equation as (33) presents l i t t l e d i f f i c u l t y . The method w i l l be applied successively in order to obtain additional terms in the series solution, thereby achieving better accuracy. Unfortunately, as w i l l become obvious later, each successive application becomes significantly more tedious. When a perturbation method i s applied to an equation such as (33) and the forced solution i s sought, a solution i s assumed of the form; 23 0 2 4 6 8 10 12 14 16 18 Vibration mode number (n) Fig. 5« Magnitude of parameter q n 24 To assure that the solution converges we w i l l consider | fn | -4. I Since q^ depends on S, 9, and physical properties and dimensions of a column, the restriction j^n|< I clearly limits the choice of these variables. Thus, when the results predicted hereafter are applied to any specific test, one must ensure that the values of q defined by experimental input are less than unity. In Fig. 5« a plot of values of the parameter q^ , where q = q (n,9,S), is given for several values of excitation level amplitude S , Also shown is the region where the experimental investigation was performed. The region i s bounded by curve E . Substitution of the assumed solution (34) in the governing d i f f e r e n t i a l equation (33) yields the following equation* I f terms with lik e powers of q^ are collected the result to the i - th order i s : W % : <"»a(z) + cw„0cz) + ahw„9(z) = A^cesZ*. : ; • : : . : (36) These equations w i l l be solved successively. The equations are linear, of second order, inhomogenous and with constant coefficients. Exact particular solutions can be obtained i n a straightforward manner. 25 The solution of eq.(36 i ) i s : = ^-IT 1 Ahcos2z + ^ A^sinZz. (37) Substituting eq.(37) into eq.(36 ii).the solution for w (z) is found to be: , x a » - 4 . (at,-4)(at,-<t) - (2c)(4c) L fa«-4;4c + faH-/<;2c . . . ,,oX + - o ~z TAhsi»4z (38) fa- 4)\ 4czj[(an'l<) + 16C*J When eq.(38) i s substituted in e q . ( i i i ) of eq.(36), the solution to " ^ ( z ) is found as: , . 2(aH-4f . o 4c(a»-4') A _ wm <*) = —r 5 T  A» cos2z- + —ir 3 7 /!„ 2 z q h / jvo + a„ / r«*-4; + 4c 2j (an-4)(a„-i6)- 4c2(ah-4) -(2cf(a»-i6) . , [(a„-4J L+4c z][(a„'/6f+ He*] , 4c^H-4^Q,-/6)-r2c)r4c; + Ca--4/f4c) . „ + 1 r = A»Sih Zz [(a„-4) +4c 1][(an-/0 + I6c*] _{ (an-4Xa»-KXaH-U)- (ZcXtcX** X) - (€cX4cXa»-4) - (2c X6cXa„-/6 ) ^ • [Can-4f+ 4cl][(a^f+ l6c 2][(a„-36) Z+36c*] + r 1 Tr 2 Tr a n Ahs,*6z (39) [(am-4) +4cx][(a„-/t) + l6c 2][(aH-36) * 3Cc'J Since this extremely tedious proces and each successive solution i s much more complex than the preceeding one, only the f i r s t three solutions are shown here. However; already a pattern i s emerging. The parameter c i s 26 related to internal (material) damping and therefore i s very small in magnitude. For certain values of parameter a^ as a matter of fact for 2. a M= 0, k, 16, 36, . . . . (2m). •... m = 1,2,...,some terms in the solution ^ o r w h o ( z ) ' W W | (z)« •••••• • w * i ( z ) ^ y become extremely large, since for these values of the parameter a h .division by the small parameter c (or a power of) occurs. Let us c a l l such a term w „(z) (l= large). The division by. c may make w^(z) so large as to completely dominate the solution of a particular w n(z) . Rewriting eq.(34): we see that for the case discussed here w j z ) can be closely approximated ast w o^ , the second largest term of eq.(l5), i s included here as well, to improve the accuracy of approximation. Remembering that the parameter a^ was defined by eq.(32) as: from which the c r i t i c a l forcing frequencies can be calculated: a « = ( | < o 0 = ° » k > 1 6 » 3 6 » 6 4 or: £co„ = 0,. 2, 4, 6, 8 (41) Since only dynamic cases are considered here such that 0 <c9"<oo , and 2 such physical systems for which co h> 0 , case — cc - 0 need not be $ " examined. 27 2 e q . ( 4 l ) gives ©• = _ _ co t t (kz) on 6 = i c o n n = 1, 2,3, . . . (^3) i = 1, 2, 3i ... Thus f o r the values o f f o r c i n g frequency 9- as def i n e d by eq.(43) the s o l u t i o n o f a p a r t i c u l a r w^(z) may be approximated by the dominant terms o f the s o l u t i o n , - as given by eq.(40). F u r t h e r a n a l y s i s o f eq!s(37)»(38),(39) shows t h a t when the f o r c i n g frequency i s any o f those d e f i n e d by eq.(43) the frequency o f w„£( z) approximating w h ( z ) i s the same as th a t o f the a s s o c i a t e d n a t u r a l frequency. For example when a n = 16 the l a r g e s t term w i l l be the l a s t term o f eq.(38) having frequency |/a^Z , o r 4z, which i s e q u i v a l e n t to 26t us i n g eq . (30). Thus, the case a„ = 16 o c c u r r i n g when the e x c i t a t i o n frequency ©• = -~— oj - -r co r t , see eq.(42), w i l l make the column to o s c i l l a t e w i t h frequency 29- . But 28- = co h thus c o n f i r m i n g our e a r l i e r c l a i m . And f o r t h i s case: wh(z)* " * 0 ( z ) + l ' * " * : ( z ) = c n o A h c o s 6 t +  s » o K s i n e t + V = c » A " c o s m + + q' s„, A„ s i n 2Gt (44a) c h Q , c n | the amplitudes o f polynomials a s s o c i a t e d w i t h cosine terms s«e> » s * i ^'ie amplitudes o f polynomials a s s o c i a t e d w i t h s i n e terms Terms o f eq.(44a) can be combined by means o f phase angles {fho } f , and by increased amplitudes s n o , o f the s i n e terms, r e s u l t i n g i n approximation: Ci) * S* Ah sin (*-t + ?„) * jns* Ah si» (29-t + ?J - (44b) o r : Wh(t) =r S*Q Ah si» (9* + %J + f's* A„ s/» (coj + %,) (44c) 28 and i n general when equation (43) applies: *n (*) *  s * o K (& t + Y»o) * K  s i h (<** * f»* J (^) ™» Ct) « S* Ah sen (9i + fM) + j?S* AH sm (coH* + <f„jt) (44e) . (44f) We now wish to examine the response of the column when the fo r c i n g frequency 9- i s not any of those defined by eq.(43). By far the largest term i s solution f o r a l l w h ^ ( z ) ( n = 1» 2, 3» ) i s then the f i r s t term of eq.(37) having frequency 2z, or equivalent 9t . That i s , for these forcing frequencies the column o s c i l l a t e s with the same frequency as the e x c i t a t i o n frequency. That t h i s i s indeed so, can be e a s i l y shown. Remembering eq.(l5) shich shows that the t o t a l response of the column is» w(x,t) = w, (t) s i n ^ L + w 2 ( t ) s i n £ p +\ + w „ ( t ) s i n ^  + ... (15). and since each term i n the series o s c i l l a t e s mainly with frequency 9t , each w • ( t ) can be clo s e l y approximated as» w . ( t ) « c, A . cos 6t (45a) where C £ 0 1 S the amplitude of polynomial associated with the f i r s t cosine term of eq.(37) . 29 E q . ( l 5 ) can therefore be written as: JL Factoring of cos 9t yields; Eq.(46) describes small late r a l oscillations of nearly perfect column with hinged ends which i s subjected to a periodic axial excitation of the form u(l,t) = S cos 6t (or equivalent u(l,t) = :cos 0t) . It shows that the response of the column to the excitation i s of the same frequency as the excitation frequency. This i s nearly true as long as the excitation frequency 9 i s not any of those defined by eq.(43) and other restrictions, imposed earlier in this chapter also apply. At this point we are in a better position to examine the total flexural response of a column to the axial excitation given by eq.(5), when the frequencies of the excitation are those defined by eq.(43). We have established that for these excitation frequencies, one term of expansion eq.(l5) can be approximated as given by eq.(44e) or eq.(44g) respectively. ,With this exception a l l the other terms can be approximated, as given by eq. (45a), yielding; 4 8 So \ c o s 9 i 5,h T + c " A*cos** si" IT + -+U?<~"sns'"C^* 30 (*,*) * ca A,  c ° s ®t S'» — + czo A 2 Cos frism =Y + + c, 4 -4 cos &t s i n C* 'f* + s *A^s/h (Vt + * )si» ^ £ + + C A cos *t sin —" lf2ftd»«f i-2,3,.....' (48) Equations (47) and (48) can also be written as» + C,_ A, _ Sin 7, + J COS&t + i - 1 (49) A sin — Cn-oo <*—) J& 7Tx ) ~ , . , J f „ -x I < - i o »• y + ^ I ^ ^ f w ^ + S ^ ; ^ ^ n - l f 2 f . . . i-2,3,... (50) Eq.(43) applies to eq's (47), (48), (49) and (50) . If the excitation frequencies satisfy eq.(43), then equations (47), (48), (49) and (50) describe small l a t e r a l oscillations of nearly perfect column with hinged ends which i s subjected to a periodic a x i a l excitation of the form u(l,t) = - (s/e*)cos 9t . 31 Material damping; Effect of internal (material) damping on column os c i l l a t i o n needs to be examined in somewhat greater detail, so that some reasonable theoretical predictions can be made. Only linear d i f f e r e n t i a l equations w i l l be considered in the derivation of a formula relating material damping properties, (as given by most researchers), to the damping parameter c introduced in this chapter. It is l i k e l y that, were the nonlinear terms included in the derivation, hereafter presented, the f i n a l relationship would not change appreciably. Equation (4) of this chapter, with nonlinear terms, neglected becomes an equation governing free flexural vibration of a damped column; A column with hinged ends o s c i l l a t i n g in i t s natural modes of o s c i l l a t i o n can be well described by; El w 0 (51) W„(*,t) ~ w„cV; sin nTx. (52) The unknown time variation function w M(t) can be determined by solving the eq . ( 5 l ) with assumed solution (52) for w^(x,t) substituted in i t . The substitution yields; 0 (53) which implies: 32 Recalling identity (86) defining the natural flexural frequencies of a column as: The eq.(5^) then becomes: + ~0 (55) A solution w i l l be sought of the form: wnct) = C e . s i C £0 (5°)' Substitution of eq.(56) in eq.(55) yields: C ls< **-s +»>. / - o from which: *A " / ( W ~ W " (58) The c r i t i c a l damping /5c producing non-oscillating solution of eq.(56) can be found from condition: and: ^ = ^ (60) A damping factor ^ expressing the ratio of actual material damping present in the column to the c r i t i c a l damping can be introduced as: (61) 33 I f the damping f a c t o r i s g i v e n , o r otherwise determined, the damping o f the column i s c a l c u l a t e d from eq.(6l) as: /8 - f/% (62) Most m a t e r i a l damping data i s given e i t h e r by a s p e c i f i c damping energy f a c t o r ^ , o r b y the l o g a r i t h m i c decrement R e l a t i o n s h i p s between f , c T . ls > and sr) are given f o r example i n "Damping o f M a t e r i a l s i n S t r u c t u r a l Mechanics" ls [ll] as: = -u = (63) f - ^ / AT (64) * f * X (65) R e c a l l i n g eq.(32) which d e f i n e s the damping parameter c : C = i - ^  (32) 9-Using eq.(50), (52), (55), equation (32) becomes: A - ^ ^ c o ^ (66) C " = Z % ~ ¥ ( 6 7 ) Since values o f ^ are given by Lazan /^"J f o r d i f f e r e n t m a t e r i a l s o s c i l l a t i n g under v a r i o u s c o n d i t i o n s , the equation (67) i s a convenient one f o r c a l c u l a t i o n o f the parameter c . For the column a c t u a l l y e x p e r i m e n t a l l y i n v e s t i g a t e d the m a t e r i a l was a c o l d - r o l l e d m i l d s t e e l , t e s t i n g took place a t room temperature, the maximum s t r a i n s were l e s s than 1 * 10 4 % , and the frequency range up to 16000 Hz . 34 Under these conditions values f o r ^  , as given by Lazan In] , were: -7 6 * io < 7 < 2 * id4 The l a r g e s t amplitudes ( l a r g e s t s t r a i n s ) of column o s c i l l a t i o n occur when a column o s c i l l a t e s i n i t s natural v i b r a t i o n modes. The amplitudes of column o s c i l l a t i o n are smaller when a column o s c i l l a t e s subharmonically, and the amplitudes ( s t r a i n s ) decrease with i n c r e a s i n g order o f a subharmonics. Taking into consideration that greater s t r a i n s involve greater damping, a b e t t e r guess o f values of ^ associated with a p a r t i c u l a r v i b r a t i o n mode o f a column can be made. In the examples o f t h e o r e t i c a l p r e d i c t i o n s presented i n Appendix B the values o f ^ w i l l be taken as follows: "? = 43.8 x to'7 i = 1,2. "2s = 1.67 * ld? i = 3,4 where i i s the order of a subharmonics i . e . the r a t i o of a na t u r a l frequency to the f o r c i n g frequency. C - ^ (68) Using eq.(66) and eq.(68) a following r e l a t i o n s h i p i s obtained: ch = 2ooi (69) I t i s of course assumed here (and was shown before, see eq.(49) ) that when a column o s c i l l a t e s a t i t s subharmonics, i t s s p a t i a l shape e s s e n t i a l l y resembles the shape of the column o s c i l l a t i n g a t the natural v i b r a t i o n mode from which the subharmonics i s derived. In other words, the la r g e s t term comprising the subharmonic o s c i l l a t i o n of a column occurring when the 35 f o r c i n g f r e q u e n c y i s n-times s m a l l e r than a n a t u r a l f r e q u e n c y o f the column, i s a s s o c i a t e d w i t h the c o e f f i c i e n t s i n • From t h e examples p r e s e n t e d i n t h e Appendix B , we see t h a t f o r : i = 1 1 = 0 i = 2 1 = 1 i = 3 1 = 2 i = 4 1 = 4 Thus a conclusion may be made: l i i - 1 (70) E x a m i n a t i o n o f t h e g e n e r a l s o l u t i o n a s g i v e n be eq's (34), (38) and (39) r e v e a l s t h a t eq.(70) i s t r u e i n d e e d . 36 Theoretical predictions j F i r s t we wish to examine the response of the column, when the excitation frequencies are at, or are very close to values which are equal to natural flexural frequencies of the column. That i s when: 9 = <o„ n = 1 ,2 ,3 The theoretical predictions based on equations ( 3 2 ) , ( 3 4 ) , ( 3 ? ) ,(48), ( 5 0 ) ares 1) The column w i l l oscillate l a t e r a l l y with the following frequencies: a) the same frequency as the excitation frequency 9 (this frequency i s now equal to the natural flexural frequency of the column) b) frequencies J's which are integral multiples of the excitation frequency. - J e , j = 1 , 2 , . . . . ) 2) The amplitude of la t e r a l o s c i l l a t i o n of the column w i l l be largest for n = 1 and w i l l decrease with increasing mode number n . 3) Relative amplitudes of the terms of the same frequency as the excitation frequency are very much larger than the amplitude of terms of other frequencies than 9 , 4) An apparent phase angle shift in time variation of the dominant term of frequency © w i l l occur. The dominant term i s the last term of eq . (50) associated with a phase angle ^ 0 . This angle w i l l change drastically as the forcing frequency 8 passes thru the value 9 = OL> H 37 5) The dominant s p a t i a l shape w i l l be assoc ia ted with s i n ^ and therefore (n - l ) nodal l i n e s could be de tec ted . We w i l l look a t the response o f the column, when the e x c i t a t i o n f requencies are a t , or are very c lose to values which are equal to f r a c t i o n s o f na tu ra l f l e x u r a l f requencies o f the column. That i s , wheni S = 4- O J „ n = 1,2,3 i = 2,3,4 The t h e o r e t i c a l p red i c t i ons based on ana l y s i s o f equations (34),(47),(49), (37),(38),(39) a r e , 1) The column w i l l o s c i l l a t e l a t e r a l l y with the fo l l ow ing f requencies 1 a) the same frequency as the e x c i t a t i o n frequency 9 , b) the na tu ra l f l e x u r a l frequency o f the column G J „ , which i s an i n t e g r a l mu l t i p l e o f the e x c i t a t i o n frequency. (co n =i^') f.i S which are other i n t e g r a l m u l t i p l i e s o f the e x c i t a t i o n frequency. (^ = j 9 ; j £ i , l ) 2) The l a t e r a l o s c i l l a t i o n o f the column w i l l be o f r e l a t i v e l y l a rge ampl i tudes, and the magnitude o f these amplitudes depends on the parameters i and n as fo l l ows : a) f o r a given value o f n , the l a rge r the parameter i , the smal ler w i l l be the amplitude o f o s c i l l a t i o n b) f o r a given value o f i , the l a rge the parameter n , the smal ler w i l l be the amplitude o f o s c i l l a t i o n . 3) Eq's ( 4 0 ) , ( 4 4 e ) and ( 4 9 ) imply that the relative amplitudes of the terms of the same frequency as the excitation frequency 8 , and the term of the natural frequency of the -column O J N , are much larger than the amplitudes of terms of other frequencies than 6 and C J M . 4 ) There may occur an apparent phase angle shift between the time variation of the two dominant terms, one of frequency 6 and the other of frequencyco„. This is apparent from e q . ( 4 9 ) where the last term includes a phase angle VJ,jg • 1 S l i k e l y that as the forcing frequency 9 passes thru neighbourhood of the value 6 = 4- cOn c the value of ^ . . w i l l change differently than the phase angle . The apparent phase angle shift in this case would be related to the change in ^  and as well. I f the second term of e q , ( 4 9 ) the one including the phase angle i s very small as compared to the last term of e q . ( 4 9 ) , than the apparent phase angle shift w i l l be equal to the change in the phase angle • 5) The dominant spatial shape (not as strong as in case i = l ) w i l l be associated with sin , and therefore (n - l ) nodal lines could be detected. Finally we w i l l consider the case when the forcing frequencies are not fractions of, or equal to the natural flexural frequencies of the column. That i s when 6 f \°°*\ > n = 1.2,3.... } i = 1,2,3 As predicted by e q . ( 4 6 ) the column w i l l oscillate l a t e r a l l y with the same frequency as the excitation frequency. Terms o s c i l l a t i n g at multiplies of 39 e x c i t a t i o n frequency may al s o be present, but t h e i r amplitudes w i l l be much smaller than those of eq.(46). The amplitude o f o s c i l l a t i o n w i l l depend on various parameters as given i n eq . ( 3 2 ) . Thus f o r a l l parameters except 6 held constant, the amplitude o f l a t e r a l o s c i l l a t i o n s should decrease r a p i d l y with i n c r e a s i n g frequency 9 . However; the amplitude o f l a t e r a l o s c i l l a t i o n should 'peak' when the f o r c i n g frequencies equal to nat u r a l l o n g i t u d i n a l frequencies of the column. This can be seen from examination o f eq's (3) and (4) as follows: Neglecting the nonlinear terra EA(u xw x ) x from eq.(4) but keeping the term r e l a t e d to i n i t i a l crookedness o f the column -EA(u "W ) F a f t e r some rearrangement r e s u l t s i n eq,(7l); The term o f the r i g h t side of.(71) can be interpreted as a f o r c i n g function. This term 'peaks' when the f o r c i n g frequency equals to natural l o n g i t u d i n a l frequencies o f the column; see eq.(7) and (9). The steady state l a t e r a l o s c i l l a t i o n o f the column a r i s i n g from the f o r c i n g function should therefore a l s o 'peak' a t these frequencies. tt •+ (71) CHAPTER 3 LINEAR THEORY 40 LINEAR THEORY Linear d i f f e r e n t i a l equations describing the motion of the two columns when o s c i l l a t i n g in longitudinal, flexural, or torsional vibration modes are considered in this chapter. Resonant frequencies and spatial forms of the columns o s c i l l a t i n g in these natural vibration modes w i l l be determined. The influence of rotary inertia and shear terms affecting the frequencies at which natural flexural o s c i l l a t i o n modes occur w i l l be also studied. The information about the behaviour of the two columns derived from the linear d i f f e r e n t i a l equations was used to check and to complement the theory developed in chapter 2 . Oscillations of the two columns w i l l be analysed in this chapter in the following order: l ) Longitudinal o s c i l l a t i o n of a prismatic column with one end fixed and the other end subjected to excitation of the form u(l,t) = (- S/0 1)cos et 2a) Free flexural o s c i l l a t i o n of a prismatic column with hinged ends. 2b) Free flexural o s c i l l a t i o n of a prismatic column with hinged ends -effect of rotary inertia and shear terms. 3a) Free flexural o s c i l l a t i o n of a prismatic column with b u i l t - i n ends. 3b) Free flexural o s c i l l a t i o n of a prismatic column with b u i l t - i n ends -effect of rotary inertia and shear terms. 4) Torsional os c i l l a t i o n of a prismatic column with b u i l t - i n ends. 41 l ) L o n g i t u d i n a l o s c i l l a t i o n s In the f o l l o w i n g c o n s i d e r a t i o n i t i s assumed t h a t d u r i n g the l o n g i t u d i n a l v i b r a t i o n o f a p r i s m a t i c bar the cross s e c t i o n s o f the bar . remain i n plane and the p a r t i c l e s i n these cross s e c t i o n s perform only motion i n an a x i a l d i r e c t i o n o f the bar. Under these c o n d i t i o n s the d i f f e r e n t i a l equation o f motion o f an element o f the bar i s given by Timoshenko (13) : (72) 2 2 3^c ° ° 9 x * cz _ (73) The s o l u t i o n o f t h i s d i f f e r e n t i a l equation o f the form (74), f o r a steady s t a t e o s c i l l a t i o n i s soughtj u(x,t) = u o ( x ) cos 6t (74) The boundary c o n d i t i o n s f o r l o n g i t u d i n a l motion considered i n t h i s experiment are s u(0,t) = 0 (75) =— =3 6 COS TTt Where S i s the amplitude o f imposed end a c c e l e r a t i o n . S u b s t i t u t i n g eq.(74) i n eq.(72) y i e l d s : -eu (x) cos et = e ^ ' ^ c o s et .2 2 From which: aM0(x) <&• u^(x) = A cos — x + B s i n — x 0 c c / v / , v u (x, t ) = { A cos — + B s i n — ) cos 8t Cg (76) 42 Using the boundary conditions (75) and (76) we get: u (0,t) = 0 = (A)' cos 8t .*• . A = 0 =~'= S cos et = - e (B sin — ) cos et .c B = -^ sin The solution to eq.(72) satisfying the boundary conditions (75) and (76) i s given by: u (x tt) = - Sl" — S c o s 9-6 fr*0 (77) 9- lsih & From equation (77) the resonant (large amplitude) frequencies of longitudinal o s c i l l a t i o n can be found by noting that when: Sin—- — • 0 M.C*,t) »*• 00 (*>o) The condition: s i n — = 0 implies: — •= *7T ^ C0 «• - ( 7 8 ) and using equation (73) we get: =• — - y - §T„ (79) From equation (69) the forcing frequencies 9 at which large amplitude longitudinal o s c i l l a t i o n occurs can be calculated. F i r s t longitudinal natural frequency w i l l be denoted by . Value of f£L for the two columns are given in tables VI-1 and VI-2 in column ( l ) . In the formula (79) "the effect of a i r , material, and support damping and a pos s i b i l i t y of flexural o s c i l l a t i o n are not included. These effects may 43 change the v a l u e o f resonant f r e q u e n c i e s somewhat. Indeed^the e x p e r i m e n t a l l y observed resonant f r e q u e n c i e s d i f f e r s l i g h t l y from the p r e d i c t e d v a l u e s . 2a) Free f l e x u r a l o s c i l l a t i o n o f a p r l s m a t i c a l column w i t h h inged endst By making' the u s u a l assumptions o f l i n e a r t h e o r y , n e g l e c t i n g f r i c t i o n , and assuming t h a t the n e u t r a l a x i s o f the column undergoes no s t r e t c h i n g o r compress ion a t any p o i n t a l i n e a r d i f f e r e n t i a l e q u a t i o n can be o b t a i n e d . The l i n e a r d i f f e r e n t i a l e q u a t i o n o f mot ion i s g i v e n by Timoshenko (13) as» tfw + . a 2 t f v m Q ( 8 Q ) 9t2 dx4 Genera l s o l u t i o n to (80) 1st (xtt) = (CtStyiw£ + CZCOS ojijfCj COSJI/3K + C4 si»k/3x + Cs Cos/Zx. + C( si»/2x) 4 ^Aco ( 8 3 ) ' €1 The boundary c o n d i t i o n s f o r h inged ends end c o n d i t i o n s a r e j w ( 0 , t ) = 0 no end d i s p l a c e m e n t s w ( l , t ) = 0 Z (82) 9 X 2 - no end moments ax2-44 Upon substitution of boundary conditions (84) in eq.(82), a frequency equation i s obtained: -2 sinh/31 s i n / 5 l = 0 ( 8 5 ) Using e q . ( 8 3 ) and ( 8 5 ) the formula giving the values of resonant frequencies for a column with hinged ends i s obtained: This expression also appears in the theory of chapter 2 which shares the same limitations and assumptions that were made in derivation of e q . ( 8 6 ) . Using e q . ( 8 6 ) resonant frequencies were evaluated and are presented in tables VI - 1 and VI-2 in column ( l ) . 2b) Free flexural o s c i l l a t i o n of a prismatic column with hinged ends - effect of rotary inertia and shear terms: In order to obtain the values of resonant flexural frequencies more accurately the effect of rotary inertia and shear should be considered. The d i f f e r e n t i a l equation including these two terms i s : aj? + w ~ r (^xGjzfi?  +r r ^ G = ° (87) 2 I 2 f r °f where: r = — and a = — (88) A for a rectangular cross section = . 8 3 3 45 Equation ( 8 ? ) and the hinged end conditions w i l l be s a t i s f i e d by assuming the solution for w(x,t) to bet w = C s i n — cos c o t ( 8 9 ) JL Upon substitution of ( 8 9 ) i n eq . ( 8 7 J and using ( 8 8 ) and assuming that the formula ( 9 0) giving values for natural f l e x u r a l frequencies of the column i s obtained: The values of co for f i r s t 13 natural modes are given i n tables VI-1 and VI-2 i n column (2) . 3a) Free f l e x u r a l o s c i l l a t i o n of a prismatic column with b u i l t - i n ends: The equations (80), (81), (82) and (83) also apply for t h i s case. The boundary conditions for b u i l t - i n end conditions are: w ( 0 f t ) =0 no ends displacement w ( l , t ) =0 lw<0,4) 0  (91) ^ x no end slope = 0 3x Upon sub s t i t u t i o n of these boundary conditions i n equation (82) a frequency equation i s obtained: 1 - cosh/3/cos^= 0 (92) 46 wherei . / ? * = A co El Equation (92) i s solved numerically for /2Zs and by using (94) resonant frequencies for this column were evaluated and are presented in tables VI-1 and VI-2 in column (3) . I s. 3b) Free flexural o s c i l l a t i o n of a prismatic column with b u i l t - i n ends - effect of rotary inertia and shear termsi The same linear d i f f e r e n t i a l equation as that in section 2c) applies in this case as wellt a ^? 9? r U *G J 9x z9S r q*G 9t4 ( 9 5 ) where i ** » — a. = A A f„ For a rectangular cross section 92. = . 833 Also define, A„ = rz(I + Sr) B* = ~ ( 9 6 ) Assuming thatt w(xtt) - %(x) cos cot ( 9 7 ) A solution of the form ( 9 7 ) w i l l be sought. The unknown function w c(x) must satisfy the b u i l t - i n ends end conditions ( 9 8 ) . w (0,t) = 0 .'. wo(0) = 0 w ( l , t ) = 0 .-. w 0(l) = 0 47 = o • • —-— » ° Substitution of assumed solution (97) ineq . ( 9 5 ) yields: Or: (98) aV^ + w" + -.a**) w o = 0 (99) (100) If we l e t , fc,-^T . h - * K £ $ r ! ) (101) CL And using operator D : (D + 2 bj> * k>3 ) W0 » 0 (102) (D 2 -<).(D* -^) w0 = 0 (103) Where, the roots <*T( and e£j are found as: -(Z){*>(£)^i+ -[(itf- *.]7 - a*) Or by making use of (96) Thus eq.(l03) can be written as, (106) 43 The d i f f e r e n t i a l equation (106) has characteristic roots: -/c(? - /TcC P and the primitive i s : > > -fi?X ^ i T ^ I 7 %(*)  9 C,  e •+ Cze + C3 cos/ia(j x + C+sm/i*Ki x or: -ya(x) = C, coskfTfx + C^sink fZ?X + C3cos JlJj'x + C* sin /UJ X (107) Application of the boundary conditions (98) to eq . ( l 0 7 ) after some calculation results in a condition: (cosh - cos /iz?*Xfi?cosh fc1* ~ ^cos/uj 1^ ) -/of 1 Sinh/Z?JL + /iZJ'sin /U^pjeXsmh rW^. -.r—^p Sir, fiJ^A) - 0 (108) Bq,(l08) can be written i n an alternate form more suitable for a computer use: Z - Z cosh fZtZ cos/JZ?^ + ^i'1**3-1 Sj„^ fZ?£ sim/iTj 7* = 0 (109) From equation (105) i t is apparent that both roots </( and <J2 are functions of physical constants related the column dimensions and material properties, and on the frequency O J . Such cos which satisfy (109) are the resonant frequencies of the column. The smallest w being the fundamental resonant frequency of free lat e r a l oscillations. F i r s t 13 resonant frequencies were found with an aid of computer and appear in tables VI-1 and VI-2 in column ( 4 ) . 49 4) Torsional os c i l l a t i o n of a prismatic column with b u i l t - i n endst Free small oscillations of a prismatic column with rectangular cross section w i l l be considered in this section. Damping and high order terms are neglected. It is assumed that the cross sections of the column during torsional vibration remain plane and no other vibration mode i s occurring at the same time. The linear d i f f e r e n t i a l equation of motion, under these restrictions, i s given by Flugge (9) « b > h are cross section dimensions In our case: — = 3 and from "Advanced Mechanics of Materials" by Hugh (no) Where $ is the torsional stiffness of the particular cross section, and i s found from: § = k,i>h 3G ( i l l ) k = ,263 in this case. Equation ( 1 1 0 ) can be transformed intot 2 19 (112) where 1 (113) 50 From eq.(ll2) a frequency equation can be readily obtained. General solution of eq.(ll2) is of the formj and the boundary conditions for fixed ends are: 6(0,t) = 0 6(1,t) =0 <n5) Substitution of (114) and (115) in eq.(ll2) eventually yields a formula giving natural torsional frequencies: = —r- Ca (116) By means of this formula f i r s t three natural frequencies were calculated and are presented in tables VT-1 and VI-2 in column ( l ) . CHAPTER 4 APPARATUS AND INSTRUMENTATION 51 APPARATUS AND INSTRUMENTATION In order to t e s t the theory developed i n chapter II a s u i t a b l e experimental setup had to be conceived. The setup had to ensure t h a t the boundary c o n d i t i o n s assumed f o r the column i n the theory would be a c c u r a t e l y reproduced under experimental c o n d i t i o n s . The boundary c o n d i t i o n s r e q u i r e d f o r the f l e x u r a l o s c i l l a t i o n mode of a column w i t h hinged ends were: ( l a ) zero displacements i n f l e x u r a l d i r e c t i o n a t both ends ( l b ) no moments a r n l i e d a t both ends To examine the i n f l u e n c e of v a r i o u s boundary c o n d i t i o n s on the behaviour of a column subjected to a x i a l end e x c i t a t i o n a column w i t h b u i l t - i n ends was chosen to be t e s t e d as w e l l . This choice was made because the b u i l t - i n boundary c o n d i t i o n s are easy- to achieve e x p e r i m e n t a l l y and the experimental r e s u l t s f o r t h i s column could a l s o be compared to those of D. Johnson (2) who had t e s t e d a column of the same dimensions w i t h b u i l t - i n boundary c o n d i t i o n s before. The boundary c o n d i t i o n s f o r the f l e x u r a l o s c i l l a t i o n mode of a column w i t h b u i l t - i n ends were: (2a) zero displacements a t both ends (2b) zero slone a t both ends Two more boundary c o n d i t i o n s r e q u i r e d f o r the longitudinal o s c i l l a t i o n mode of the column were: •(3) (4) zero displacement i n longitudinal d i r e c t i o n a t one end a c c e l e r a t i o n of the form Scos9t at the other end 52 The design of the experimental setup then entailedj a) design of two columns, one with hinged ends and one with b u i l t - i n ends b) design of test bench that would accommodate the two columns, one at a time, and which would conform to before-mentioned boundary conditions c) choice of proper apparatus which would ensure constant acceleration level imposed at one end of the column (boundary condition No. 4) d) choice of transducers and associated electronics by means of which the column to axial end excitation could be observed. The solution to the above mentioned problems w i l l be discussed in this chapter Design of the column with hinged ends: The design of a column with hinged ends proved quite d i f f i c u l t as the boundary condition la, lb, 3 and 4 had to be satisfied simultaneously. In addition to satisfying these boundary conditions the column had to exhibit a well defined and unique fundamental longitudinal frequency in order to make a check for subharmonics of this vibration mode possible. Several columns were designed and tested. Each column was clamped by means of i t s threaded ends. This way of clamping of a column was chosen to achieve perfect axial symetry of a l l moving parts of setup. In addition, i t also minimized a total weight of a l l moving parts and completely eliminated any possible chatter between them. Successive designs of a column with hinged ends w i l l be shown and their inadequacies w i l l be pointed out. It w i l l be shown that the elimination of these inadequacies eventually lead to the design of a column having a l l desirable characteristics. 53 Fig. 6. Design #1 3f The design of the f i r s t column i s similar to the one commonly used by other researchers to test the flexural response of longer columns (up to 3 f t ) at very low axial excitation frequencies as compared to the fundamental longitudinal frequency of the column. Due to stress concentration at i t s ends where therefore much deformation took place, the column did not exhibit a unique fundamental longitudinal frequency but rather several ill-defined ones. Because of "two-piece" construction of the column a good axial alignment was not possible. However, the column satisfied quite well the boundary conditions of hinged ends due to extremely small end moments opposing rotation which resulted from a negligible f r i c t i o n at i t s ends. Fig. ?. Design #2 54 In the second design, a uniform cross section was retained even at the ends of the column, which therefore resulted in a unique fundamental longitudinal frequency. However, the column s t i l l suffered from poor alignment inherent in "two-piece" design, and because of strong end moments opposing rotation which resulted from high f r i c t i o n at i t s ends, the column oscillated flexurally as though i t had b u i l t - i n ends. Fig. 8. Design #3 ( ^ The third design had a good axial alignment " b u i l t - i n " due to "one-piece" construction, and the end conditions were quite close to the hinged-ends condition because of small end moment opposing rotation which was accomplished by the reduction of height of cross-section at i t s ends. While the reduced height of the cross-section at the ends of the column helped in an approximation of hinged-ends end conditions, i t also resulted in stress concentration in the reduced cross-section which again caused a column not to exhibit a unique fundamental longitudinal frequency. The fourth and the f i n a l design incorporated positive features of previous designs. It i s of "one-piece" construction and therefore a good axial alignment of the column with the rest of moving parts was assured. Ey 9.75 i n 12.250 i n 12.375 i n F i g . 9. Column with hinged ends 56 reducing the height and i n c r e a s i n g the width of the column c r o s s - s e c t i o n a t i t s ends approximately uniform area of c r o s s - s e c t i o n was r e t a i n e d . The constant area of c r o s s - s e c t i o n along the whole column r e s u l t e d i n a unique and w e l l . d e f i n e d fundamental longitudinal frequency of the column. F i n a l l y , the dimensions of modified c r o s s - s e c t i o n of the column a t i t s ends caused the column to be f l e x i b l e a t i t s ends thus approximating .hinged-ends end c o n d i t i o n s . The closeness of t h i s approximation may by judged by comparing the numerical values of a x i a l e x c i t a t i o n f r e q uencies at which n a t u r a l f l e x u r a l o s c i l l a t i o n modes occur t o the t h e o r e t i c a l values f o r a column w i t h b u i l t - i n . e n d s and a column with hinged ends. A l l these values are given i n t a b l e VI-1. As can be seen, e s p e c i a l l y a t high frequencies of a x i a l e xcita.tion a column behaves as though i t has hinged ends r a t h e r than b u i l t - i n ends. This column i s shown i n Fig. 9. Design of the column w i t h b u i l t - i n ends; The design of the column w i t h b u i l t - i n ends was not d i f f i c u l t as most of the design problems were a.lready solved d u r i n g the design of the column with hinged ends. The same means of clamping the column was used, and a l l boundary c o n d i t i o n s were s a t i s f i e d -by keeping the c r o s s - s e c t i o n of the column 1/8 i n by 3/8 i n constant along the e n t i r e l e n g t h of the column. The column e x h i b i t e d a unique fundamental longitudinal frequency and the n a t u r a l f l e x u r a l o s c i l l a t i o n r^.cdes occurred very c l o s e to t h e o r e t i c a l l y p r e d i c t e d v a l u e s . This can be seen from t a b l e VI-2. The column i s shown i n Fig. 10. Accelerometer Moving end 11.9 in 1 3 Cross section -Q i n * g in Fixed end 12.0 in Fig. 10. Column K i t h b u i l t - i n ends 58 Design of the t e s t bench: The f i n a l design o f the t e s t bench i s shown i n F i g . 11. . The bench accommodates one column at a time and a column i s placed i n a h o r i z o n t a l plane to allow t e s t i n g of the column by s a l t s p r i n k l i n g . To eliminate creation and transmission of any mechanical noise an adjustable t e f l o n bearing rather than b a l l bearing i s used, there are no loose connections anywhere, and rubber padding i n s u l a t e s the bench from most o f the base motion. The boundary condition (3) i s accomplished by threading one end of the column i n a heavy s t e e l block, large i n e r t i a of which almost completely eliminates any motion o f the end o f the column i n the l o n g i t u d i n a l d i r e c t i o n . The t e s t bench a l s o had to house the shaker and allow easy mounting o f the two columns. S o l u t i o n of these and other minor problems presented no d i f f i c u l t y . V i b r a t i o n c o n t r o l apparatus: The purpose of the v i b r a t i o n c o n t r o l apparatus i s to provide a s i n u s o i d a l a x i a l e x c i t a t i o n of constant l e v e l a t the moving end of the column, or i n other words to s a t i s f y the boundary condition (4). A s i g n a l flow diagram showing the arrangement o f e l e c t r o n i c and mechanical components by means o f which t h i s i s accomplished i s shown i n F i g , 12. Desired amplitude and frequency o f a x i a l e x c i t a t i o n i s set. by means of the v i b r a t i o n e x c i t e r c o n t r o l . The output e l e c t r o n i c s i g n a l corresponding to the desired form of a x i a l e x c i t a t i o n i s then amplified i n a power a m p l i f i e r and the amplified e l e c t r o n i c s i g n a l i s converted to the a c t u a l a x i a l e x c i t a t i o n by means of electromagnetic shaker. The a x i a l e x c i t a t i o n as d e l i v e r e d by the shaker i s monitored by an accelerometer which i s placed a t the moving end Fig. 11. Test bench FREQUENCY COUNTER POWER AMPLIFIER VIBRATION EXCITER CONTROL FOUR BEAM OSCILLOSCOPE ACCELEROMETER PREAMPLIFIER SHAKER LEVEL RECORDER VOLTAGE AMPLIFIER SPECTRUM ANALYSER BAM BAND PASS FILTER AND AMPLIFIER ACCELEROMETER - ELECTRIC STRAIN GAGES SPECIMEN 7777777777777777777777777. 4J-Fig. 12. Signal flow diagram 61 o f the column. The accelerdmeter output s i g n a l i s a m p l i f i e d and fed back i n t o the v i b r a t i o n e x c i t e r c o n t r o l u n i t where the amplitude and frequency of t h i s s i g n a l i s compared to the d e s i r e d amplitude and frequency. I f c o r r e c t i o n s are necessary the v i b r a t i o n e x c i t e r c o n t r o l e f f e c t s a p p r o p r i a t e changes i n i t s output e l e c t r o n i c s i g n a l u n t i l the a c t u a l a x i a l e x c i t a t i o n has d e s i r e d amplitude and frequency. The accelerometer output s i g n a l i s a l s o d i s p l a y e d on the o s c i l l o s c o p e screen and fed i n the frequency counter. Thus a t a glance the amplitude, time v a r i a t i o n and frequency o f the a x i a l e x c i t a t i o n can be checked. Transducers and a s s o c i a t e d e l e c t r o n i c s ; S i x s t r a i n gages (BLH SR-4 type FAP-12-12) were attached to the surface o f each o f the two columns as shown i n F i g , 13. . Depending on the choice o f s t r a i n gages and t h e i r arrangement i n the Wheatstone b r i d g e , which i s an i n t e g r a l p a r t o f BAM, the output s i g n a l o f BAM represented e i t h e r a m p l i f i e d l o n g i t u d i n a l f l e x u r a l (normal) o r f l e x u r a l ( i n - p l a n e ) s t r a i n . The choice o f s t r a i n gages and t h e i r arrangement i n the Wheatstone bridge to o b t a i n these three p a r t i c u l a r s t r a i n s i s shown i n F i g . 14, The output s i g n a l o f BAM passed t h r u band pass f i l t e r and a m p l i f i e r where the s i g n a l was f u r t h e r a m p l i f i e d and low and high frequency s i g n a l content was suppressed. Low and high frequency s i g n a l content contained some e l e c t r o n i c noise and no i n f o r m a t i o n was l o s t by i t s suppression. The use o f a band pass f i l t e r a l s o r e s u l t e d i n sharper and w e l l d e f i n e d s t r a i n waveform more s u i t a b l e f o r a n a l y s i s and photographing. In a d d i t i o n to being d i s p l a y e d on the o s c i l l o s c o p e screen a f t e r p a s s ing t h r u the band pass f i l t e r the s t r a i n gage s i g n a l was 62 also further amplified in the voltage amplifier and i t s RKS level recorded by a level recorder. The level recorder was used to record the variation in amplified RKS values of particular strain as a function of axial end excitation frequency. Whenever i t was desired, a spectrum analysis of the strain gage signal was performed by making use of the spectrum analyser. The voltage amplifier and the spectrum analyser were contained in the single unit B&K 2107 , and a switch was used to choose between the two operational modes of this unit. 63 Z,w (F) Y,v (P) For bending i n X-Z plane i ARI = *R3 <=- ^ R2 =-AR4 = ziR For bending i n X-Y plane t - ^ R l = -«*R2 = AR3 = A R4 =» <aR* -AR6 = ^ R5 =AR For tension along X axis t <iRl = AR2 = AR3 =AR4 =/iR5 = 4R6 = ^ R Fig. 13. Position of strain gages on a column 64 C In general: C For bending in X-Z plane: D G For bending in X-Y plane: D Fig. 14. Arrangement of strain gages In the Wheatstone bridges CHAPTER 5 TEST PROCEDURE 65 TEST PROCEDURE Calibration; Before the actual testing of a column was commenced i t was necessary to ascertain that'all electronic and mechanical components that would be used during the testing were functioning properly. Improper or poor ' performance of any of the components, i f not detected immediately, might give rise to erroneous experimental results. To avoid this, a l l components comprising the experimental setup were calibrated and tested according to manufacturers specifications. A l l electronic components were properly connected and turned to the standby position for at least an hour before testing. Testing preliminaries: The amplitudes of flexural vibration of the column depended on the frequency and amplitude of the imposed sinusoidal axial end excitation. When the amplitude of axial end excitation was held constant over the entire frequency range considered here, the following happened: for some frequency range segments the response of the column was strong and the excitation power consumption high, while for some other frequency range segments the response of the column was weak - almost undetectable, and the excitation power consumption low. To make the response of the column of comparable and detectable strength at a l l axial end excitation frequencies, different amplitudes of the excitation were used over different frequency range segments. The number and size of the segments was chosen so that the excitation power consumption remained approximately constant over the entire range of axial end excitation frequencies considered in this experiment. This compromise 66 is called an "approximated constant power spectra" and is discussed in somewhat greater detail in Johnson's thesis (2) .To determine "approximate constant power spectra" each of the two columns had to be pre-tested and desired acceleration levels established. In this experiment the actual levels of imposed end excitation varied from 6g to 40g and no record of these is presented here as. i t is of secondary concern. Mounting of a column: The column with strain gages and two accelerometers attached was mounted in the test setup in the following way: ( 1 ) The shaker shaft and the bearing shaft were screwed very tightly together. The shaker shaft was then attached to the shaker by three alien screws. (2) The teflon bearing was adjusted u n t i l a tight s l i p f i t between the bearing and the bearing shaft was achieved. The adjustment had to be done with great care to avoid bending of the shaft so that when the adjustment was completed the shaft remained aligned with the axis of the shaker. (3) Thin brass washers were used to separate the fixed end of the column and the steel block so that when these two parts were screwed tightly together the f l a t side of the column was horizontal to make salt-sprinkling test possible. (4) The moving end of the column was screwed into a hole in the bearing shaft. Then a nut on the moving end of the column was screwed tightly against the face of the hearing shaft to eliminate possible chatter between the end of the column and the bearing shaft. While this was done the f l a t , wide side of the column was held in a 67 horizontal plane, (5) The remaining three heavy steel blocks together with the block already attached to the fixed end of the column were then attached to the test bench frame by means of nuts and two long bolts. At this point, the f l a t , wide side of the column lay in a horizontal plane and the column was aligned and firmly connected with the other parts of the setup. Testing; The column with strain gages and in test setup and the testing was now steps were: (1) two accelerometers attached was mounted ready to begin. The test procedure The strain gage leads were connected to the BAM in a configuration depending on the strain to be meassured, and the BAM was switched on. After approximately fifteen minutes of warming up the BAM stabilised and could be balanced and calibrated. To minimize a pick-up of electronic noise the strain gage leads were twisted together and wraped in an aluminium f o i l . Leads of the two accelerometers were connected to the accelerometer preamplifier. The preamplifier inputs were then- adjusted according to the combined voltage gain of each accelerometer and i t s lead. The vibration exciter control unit was set to deliver a sinusoidal signal of desired level. Frequency scanning speed on the vibration exciter control unit was set and was manually changed during the experiment to remain at approximately 3-4 cps. Proper paper speed on the level recorder was chosen. (2) (3) CO (5) 6a (6) Compressor speed was chosen to assure s t a b i l i t y of the feedback ci r c u i t , (7) The power amplifier was switched from stand-by to on position. ( 8 ) The control unit was put in i t s excitation mode. ( 9 ) - Proper amplification in the frequency analyser was set. (10) Attenuation and writing speed on the level recorder was set. (11) The oscilloscope was adjusted to display the signals of interest, (12) The frequency counter was switched on. (13) The scanning mechanism was activated. This procedure was repeated for each frequency range segment and for different vibration modes of the two columns. Boundaries of frequency range segments can be recognized from the strain vs. frequency records as discontinuities in the strain vs. frequency curve. The tests were performed so that the frequency segments extended a few Hertz over their end points to avoid possible undertesting of the column . At particular frequencies of interest a spectrum analysis of the strain signal was performed to identify i t s components. While this was done the B&K 2107 was switched from i t s usual amplifier mode to analyser mode. The scanning of the analysed signal was done manually. Additional testing; • To obtain more information about the response of the column to axial end excitation frequency additional tests were carried out. Salt sprinkling was used to gather information about the spatial form of the column at certain frequencies of axial end excitation. At an axial end excitation frequency of interest fine crystal salt was sprinkled on the column and the axial end excitation frequency was varied 69 slowly u n t i l the nodal pattern created by salt crystal was sharpest. The axial end excitation frequency was recorded and a nodal pattern classified. This information supplemented the information about the response of the column to axial end excitation at a given point of a column obtained by means of strain gages. Vibration exciter control and the accelerometer preamplifier have usable frequency range from 50 Hz to 10000 Hz . The rest of electronic components had even greater usable frequency range, thus a l l components of the setup were well suited for testing of the column in range of frequencies between 300 Hz and 10000 Hz . For testing of the column up to 16000 Hz , which is the upper usable range of the shaker, an alternate vibration control was set up. The Wavetek signal generator and a special B&K 4336 accelerometer were used. This setup, however, had not a feedback c i r c u i t and consequently a constant level of sinusoidal axial end excitation could not be maintained. S t i l l , this setup was satisfactory for detection of natural frequencies of os c i l l a t i o n of a column in 10000 Hz to 16000 Hz frequency region. Total damping measurements: In order to gain an understanding of the effect of damping i t was desirable to find the actual value of damping which affected the motion of a column. Damping arises from several sources but most of i t i s derived from the column supports. That support damping i s the single largest contributor to total damping was also noted by Lazan ( l l ) . Surface damping due to attached strain gages, a i r damping and material damping also contributed to the total damping of the system. By plucking the centre of the column and observing the decay of amplitudes of free o s c i l l a t i o n i t was possible to determine the total damping affecting the ?0 o s c i l l a t i o n of the column vibrating in i t s fundamental flexural vibration mode. Determination of a material damping from the total damping was not possible and an approximate handbook value had to be used. Undertesting and overtesting: To eliminate a possibility of undertesting or overtesting several steps were taken; 1) Each strain was measured independently at 2 randomly chosen points of the column to eliminate the influence of transducer location. 2) Two accelerometers were detecting the imposed excitation level at the moving end of the column. The two accelerometers were B&K 4335 and B&K 4336 . Their outputs were monitored by oscilloscope and any disagreement between the two would be easily noted, 3) To eliminate high frequency electronic noise, influence of house current and any magnetic f i e l d influence a l l cables were shielded; a l l electronic equipment was grounded to the oscilloscope and the band pass f i l t e r f i l t e r e d out unwanted high and low frequency signals 4) A l l tests were repeated several times to assure that a l l phenomenons detected were consistent and none overlooked. Photography: Photographs of strain and acceleration waveforms typical for a particular vibration mode were obtained. Photographs of waveforms were taken directly from the oscilloscope in normal triggering mode (waveforms were stable enough) using a Pentax Spotmatic Camera with an f/stop of 2 .8 , an aperture speed of 71 1/250 second, with normal 55 nun lens and a No. 2 close-up lens. KODAK 24-75 recording film (developed in KODAK D-19 contrast developer) was used. S t i l l photographs of several nodal salt patterns were also taken. KODAK TRI-X 35mm film (developed in KODAK D-19 contrast developer)- was used. Loading of a column by a constant axial force: In order to study the effect of axial force on vibration, a column was loaded axiall y by a constant force and tested. The load was imposed by means of two variable tension springs. The ends of each spring were attached to fixed and to moving (excited) ends of the column respectively. The load could be varied from zero up to 200 lbs. 64 lbs constant axial load was chosen but other loads were tried also. CHAPTER 6 RESULTS AND DISCUSSION 72 RESULTS AND DISCUSSION A theory predicting the behaviour of a column subjected to sinusoidal axial end excitation was developed in chapter II. In order to test the validity of theoretical predictions that were made, the response of a column was observed experimentally. Strain vs. frequency records, and photographs of nodal line patterns and strain waveforms were obtained for a hinged-end column and for a b u i l t - i n column. These experimental data and their analysis is presented in this chapter. Identification and analysis of strain vs. frequency records: Because the variation of amplitude of column os c i l l a t i o n as a function of the axial end excitation frequency is of particular interest, the RMS-values of strain vs. the axial end excitation frequency records were produced for the two columns, and for longitudinal flexural, and in-plane modes. Numerous peaks appear on the records and i t is of paramount importance to interpret their significance correctly. The peaks are characterised by their height and by the axial end excitation frequency at which they occur. Analysis of flexural strain vs. frequency records - natural flexural frequencies: Highest strain peaks occur when the axial end excitation frequency is equal to the natural frequency of flexural o s c i l l a t i o n of the column. A column then oscillates flexurally with large amplitude and with the same frequency as the axial end excitation frequency. The spectrum analysis of 73 the flexural strain was carried out and no strain components of other frequencies comprising flexural strain waveform were observed. In general, the waveform representing time variation of axial end excitation was always a pure sinusoid while the waveform of the resultant strain was either a pure sinusoid or a complex curve resulting from addition of two or more sinusoids. The waveform showing the flexural strain when the column with hinged ends oscillates at i t s eighth natural frequency of flexural o s c i l l a t i o n i s shown here. In this particular example (F8) the frequencies of both waveforms are 4760 Hz. Axial end excitation Flexural strain for a column os c i l l a t i n g at natural frequencies of flexural o s c i l l a t i o n modes Fig. 15. Oscillation of a column in natural o s c i l l a t i o n modes The strain peaks corresponding to osc i l l a t i o n of a column at natural frequencies of any of i t s vibration modes are identified by a letter representing the particular vibration mode as follows: F Flexural (normal) os c i l l a t i o n mode P flexural (in-Plane) os c i l l a t i o n mode L Longitudinal (axial) o s c i l l a t i o n mode T.....Torsional o s c i l l a t i o n mode The letter representing a particular vibration mode i s followed by a number 74 which corresponds to n-th natural frequency. Thus F3 is associated with the third natural frequency of flexural o s c i l l a t i o n mode etc. Analysis of flexural strain vs. frequency records - complex subharmonics: When the axial end excitation frequency i s equal to one half, one third,.. ..up to one eighth of any of natural frequencies of either flexural (normal) or flexural (in-plane) o s c i l l a t i o n mode, smaller strain peaks may appear on the record. Flexural oscillations at these axial end excitation frequencies were identified as complex subharmonics. The spectrum analysis performed at these frequencies revealed that the flexural strain waveform consists of components having: a) the same frequency as the natural frequency with which the complex subharmonics i s associated b) the same frequency as the axial end excitation frequency c) frequencies which are integral multiples of axial end excitation frequency Only two strain components a) and b) had comparable and sufficiently large amplitudes to be detected visually from the oscilloscope display. (The pictures of these strain waveforms are presented here) / V \ / V V \ / Axial end excitation Flexural strain for a column o s c i l l a t i n g at second order subharmonics of flexural oscillation modes. Fig. 16. Oscillation of a column at second order subharmonics 75 Axial end excitation Flexural strain for a column o s c i l l a t i n g at fourth order subharmonics of flexural o s c i l l a t i o n modes Fig. 18. O s c i l l a t i o n of a column at fourth order subharmonics Axial end excitation Flexural strain for a column o s c i l l a t i n g at f i f t h order subharmonics of flexural o s c i l l a t i o n modes Fig. 19. O s c i l l a t i o n of a column at f i f t h order subharmonics 76 Axial end excitation Flexural strain for a column os c i l l a t i n g at seventh order subharmonics of flexural o s c i l l a t i o n modes Fig. 21. Oscillation of a column at seventh order subharmonics Axial end excitation Flexural strain for a column os c i l l a t i n g at -third order {weak) subharmonics of flexural o s c i l l a t i o n modes Fig. 22. Oscillation of a column at i h i r d order subharmonics ( w e a k ) 77 Phase angle shifts Tne two strain components a) and b) exhibit an apparent phase angle shift. As the axial end excitation frequency i s increased continuously through the narrow frequency band where the complex subharmonics occurs the phase angle between these two components changes by as much as 180°, At a l l frequencies within the narrow frequency band under consideration, the ratio of axial end excitation frequency to the natural frequency (which i s an integral multiple of axial end excitation frequency) remains constant. The amplitude of the complex subharmonics reaches maximum at the center of the frequency band and declines to the steady state o s c i l l a t i o n amplitude at the upper and lower limits of the band. An example showing the phase angle shift i s given here for a complex subharmonics F?/k . Axial end excitation (912 Hz) Flexural strain for a column os c i l l a t i n g at fourth order subharmonics of flexural o s c i l l a t i o n mode F7 Fig. 23. Oscillation of a column at F7/4 subharmonics - phase angle s h i f t 78 Axial end excitation (920 Hz) Flexural strain for a column os c i l l a t i n g at fourth order subharmonics of flexural o s c i l l a t i o n mode F? Fig. 2k. Axial end excitation (923 Hz) Flexural strain for a column os c i l l a t i n g at fourth order subharmonics of flexural o s c i l l a t i o n mode F7 Axial end excitation (928 Hz) Flexural strain for a column os c i l l a t i n g at fourth order subharmonics of flexural o s c i l l a t i o n mode F7 Fig. 26. 79 The complex subharmonics are identified by the lett e r of a natural frequency with which the subharmonics is associated and by the ratio (order) of the axial end excitation frequency to the associated natural frequency. Thus P5/2 is associated with the second order subharmonics of f i f t h natural frequency of flexural (in-plane) o s c i l l a t i o n mode. A particular ordering of flexural strain peaks was noticed. I f for a certain axial end excitation frequency (or a narrow band width of) a natural frequency osc i l l a t i o n as well as a complex subharmonics os c i l l a t i o n of another natural frequency were predicted to occur, i t would be the natural frequency os c i l l a t i o n that was observed. Similar ordering of complex subharmonics oscillations was observed. The lower order complex subharmonics o s c i l l a t i o n would be preferred to the higher order ones. Because of the ordering, then, not a l l of predicted subharmonics were observed. In like manner, heights of strain peaks associated with complex subharmonics of any given natural flexural frequency would decrease rapidly with increasing order of the subharmonics. Often no notable strain peak was associated with a complex subharmonics at a l l and i t s presence could only be inferred from the particular strain waveform corresponding to the complex subharmonics. The highest order of complex subharmonics detected was eight. Undoubtedly even higher order complex subharmonics existed, however; because of their extremely small amplitudes their detection was too d i f f i c u l t . Experimental and theoretical values of axial end excitation frequencies at which natural modes of oscillation }complex subharmonics,and snap-thru phenomenons were observed are also given in the tables V l - i and VI-2 . 80 To prevent cluttering of strain vs. frequency records only some complex subharmonics were identified. The complete survey of complex subharmonics detected visually fron the oscilloscope screen i s presented in the tables VI-1 and VI-2 . Analysis of flexural strain vs. frequency records - snap-thru phenomenons: When the axial end excitation frequency i s twice the natural flexural frequency a high strain peak identified as a snap-thru phenomenon may appear on the record. A column then oscillates flexurally with large amplitudes and at i t s natural frequency which i s equal to one half of frequency of axial end excitation. Axial end excitation Flexural strain for a column os c i l l a t i n g in a snap-thru os c i l l a t i o n mode Fig. 27. Oscillation of a column in snap-thru o s c i l l a t i o n modes The identification of snap-thru phenomenons is consistent with identification of complex subharmonics. For example 2F2 would be associated with the snap-thru phenomenon derived from second natural frequency of flexural oscillation mode. Only two snap-thru phenomenons appear on the flexural strain vs. 8 1 frequency record for the column with hinged end. None appears on the record for the column with b u i l t - i n ends. Their existence seemed to depend mainly upon the amplitudes of axial end excitation and to a lesser degree on the number of natural frequency from which they are derived. For small amplitudes of axial end excitation the column oscillated with small amplitudes (no flexural strain peak) with the same frequency as the axial end excitation frequency. As the amplitude of axial end excitation was increased a transition zone was encountered in which two frequencies of flexural o s c i l l a t i o n were present: The same frequency as frequency of axial end excitation and the frequency equal to one half of axial end excitation frequency. The latter i s the natural frequency. As the amplitude of axial end excitation increased more, the transition zone was passed, and the column then oscillated flexurally with natural frequency only. High strain peak corresponds to this o s c i l l a t i o n . Smaller strain peak would correspond to os c i l l a t i o n in transition zone. Since relatively large amplitudes of axial end excitation were imposed at low axial end excitation frequencies a snap-thru phenomenon 2F2 occurred. The other snap-thru phenomenon 2F8 occurred at very high frequency and a strain peak is not nearly as high as the f i r s t one. This is because the amplitudes of axial end excitation at high frequencies are much smaller. It occurred, probably, because F8 seemed to be preferred mode of flexural o s c i l l a t i o n . 82 F R K X J E N C T ( H z l FLEXURAL STRAIN (F) VS. FREQUENCY RECORD 750 KX» ^ 1500 3000 4000 5000 6000 8000 CJOO FREQUENCY (Hz) FLEXURAL STRAIN (P) VS. FREQUENCY RECORD Fig. 28. STRAIN vs. FREQUENCY - COLUMN WiTH HINGED ENDS 83 1500 2000 3000 1000 sooo 6000 7000 FLEXURAL STRAIN (F) VS FREQUENCY RECORD 8000 9000 FREQUENCY (Hz) 1000 2 000 3500 5000 8000 BOX 1000 2000 3000 CP noa 6000 8000 9500 FREQUENCY (Hz) LONGITUDINAL STRAIN (L) VS. FREQUENCY RECORD FLEXURAL STRAIN (P) AND LONGITUDINAL STRAIN (L) VS. FREQUENCY RECORD •o SCO 1000 1500 2000 300C 3b00 45X 5000 6500 8000 9500 OP noa FREQUENCY (Hz) FLEXURAL STRAIN (P) VS. FREQUENCY RECORD Fig. 29. STRAIN vs. FREQUENCY - COLUMN WITH BUILT-IN ENDS 84 Discontinuities of strain vs. frequency curves; Because approximately constant power was to be supplied to the osc i l l a t i n g system, i t was necessary to change the amplitude of imposed end acceleration S at several points of the frequency range considered here. This resulted in discontinuities in strain at these points, and these discontinuities appear on the records. Analysis of longitudinal strain vs. frequency records; Because the axial end excitation was of the form u(l,t) = - (S/&- )cos Gt the amplitudes of the excitation are then|u(l,t)| = (s/62 ) where S is a constant (the magnitude of imposed end acceleration) and 8 is the axial end excitation frequency. Thus the amplitude of axial end excitation decreases with square of i t s frequency. Amplitudes of longitudinal strains which are directly related to the amplitudes of longitudinal o s c i l l a t i o n should then decrease with square of the axial end excitation frequency. This i s nicely shown on the record of longitudinal strain vs. frequency for the column with b u i l t - i n ends. In this case amplitude S was held at constant 20 g's for the whole frequency range and monotonically decreasing curve of longitudinal strain vs. frequency results. Because the amplitude S was changed several times in the frequency range considered a different distorted curve representing longitudinal strain vs. frequency resulted for the column with hinged ends. There i s only one high strain peak present on each of these records. It occurs when the axial end excitation frequency is equal to the f i r s t longitudinal natural frequency of the column. A column oscillates longitudinally with large amplitudes and with the same frequency as the axial 85 end excitation frequency. The spectrum analysis confirmed that no other frequencies comprising longitudinal o s c i l l a t i o n at these frequencies were observed. Several very small peaks which show effect of large amplitude flexural o s c i l l a t i o n at natural flexural frequencies upon the amplitudes of longitudinal strain are present as well. No subharmonics of the f i r s t longitudinal natural frequency were observed. For neither of the two columns at excitation frequency equal to one half, and one third of the f i r s t natural longitudinal frequency a strain peak occurs and the spectrum analysis showed only one osci l l a t i o n frequency present - the same one as that of axial end excitation. "Dummy" subharmonics of the fundamental longitudinal frequency might appear on the strain record in flexural (in-plane) strain and longitudinal strain signals were added. This would be l i k e l y to happen i f one neglected the p o s s i b i l i t y of existence of in-plane flexural osc i l l a t i o n . Such a strain record was created on purpose to show this, and i s presented here. "Dummy" subharmonics of the fundamental longitudinal frequency could also possibly arise from not truly sinusoidal axial end excitation. Such an excitation would also contain components with periods being integral multiples of the desired period. These components would then excite the fundamental longitudinal mode when the "imposed" frequency i s one half, one third,.... of the fundamental longitudinal frequency. Study of osc i l l a t i o n modes obtained by sprinkling of the column with salt: When fine table salt i s sprinkled on the surface of os c i l l a t i n g column, i t i s shaken off the surface except at the points where the amplitudes of osc i l l a t i o n are zero or very small. This requirement is satisfied by the 8€ n o d a l l i n e s o f a column o s c i l l a t i n g a t any o f i t s n a t u r a l f r e q u e n c i e s o f the f o u r p o s s i b l e v i b r a t i o n nodes. T h e o r e t i c a l n o d a l l i n e p a t t e r n s f o r s e v e r a l f l e x u r a l and t o r s i o n a l o s c i l l a t i o n modes a r e shown i n t h e F i g . 35. p r e s e n t e d h e r e . The n o d a l l i n e p a t t e r n s o b t a i n e d e x p e r i m e n t a l l y were found t o be i n good agreement w i t h the p r e d i c t e d ones and a r e a l s o shown h e r e . F i g . 31. Nodal l i n e p a t t e r n o c c u r r i n g when a column o s c i l l a t e s i n the second n a t u r a l t o r s i o n a l o s c i l l a t i o n mode Fig. 32. Nodal line pattern occurring when a column oscillates in the third natural torsional o s c i l l a t i o n mode fi g . 33. Nodal line pattern occurring when a column oscillates i n the f i f t h natural flexural o s c i l l a t i o n mode Fig. 34 . Nodal line pattern occurring when a column oscillates In the tenth natural flexural o s c i l l a t i o n mode 88 Torsional mode #1 1 Torsional mode #2 1 t A Torsional mode #3 _ _ — — — 1 i — -i A ^ ^ S a l ^ here Magnified view A - A of the column Flexural mode #2 Fig. 35. Theoretical nodal line patterns 89 Intensity of coupling between vibration modest A high strain peak on a l l three strain vs. frequency records appears when the axial end excitation frequency is the same as the natural longitudinal frequency of the column. This experimental result implies strong coupling between the vibration modes at this excitation frequency. A very weak coupling also occurs between the two flexural vibration modes and the longitudinal vibration mode when the column oscillates at natural frequencies of the flexural (normal) and flexural (in-plane) vibration modes. Very small strain peaks may appear on the longitudinal strain vs. frequency record at these axial end excitation frequencies, 0 Weak coupling between the two flexural vibration modes occurs. This i s shown by the presence of small strain peaks on flexural (in-plane) strain vs. frequency record when the column oscillates at any natural frequency of flexural (normal) vibration mode and vice-versa. When the axial end excitation frequency i s equal to the natural torsional frequency of the column, the column oscillates torsionally with large amplitudes. Virtually no strain peaks appear on any of the strain vs. frequency records. This indicates that the intensity of coupling between torsional and other vibration modes is very small indeed. 90 Agreement between experimental results and theoretical predictionsi A theory was developed for a coupled longitudinal-flexural (normal) osc i l l a t i o n in a column with' hinged ends. The theory predictedj Occurrence ofj - coupled longitudinal-flexural (normal) os c i l l a t i o n - natural flexural (normal) vibration modes - natural longitudinal vibration modes - complex subharmonics - strong coupling between longitudinal and flexural (normal) modes of os c i l l a t i o n Frequencies: - of axial end excitation at which the above described phenomena occur - which are present in column o s c i l l a t i o n as a response to the axial end excitation Relative amplitudes: - of coupled longitudinal-flexural (normal) o s c i l l a t i o n in stable regions - of natural flexural (normal) os c i l l a t i o n - of natural longitudinal o s c i l l a t i o n - of individual components comprising osc i l l a t i o n associated with complex subharmonics - which are associated with strong coupled longitudinal-flexural (normal) osc i l l a t i o n 91 Experimental results as described in this chapter confirm validity of most of these predictions with following exceptions: - some complex subharmonics of low order were not observed - complex subharmonics of order higher than eight were not observed - very high order components comprising low order complex subharmonics, natural modes of flexural o s c i l l a t i o n and stable region oscillation were not observed - the numerical values of predicted and observed frequencies might d i f f e r slightly in their magnitudes - two snap-thru phenomenons not predicted by theory occurred - weak coupling between flexural (normal) and longitudinal modes of osc i l l a t i o n which occurred when axial end excitation frequency was equal to the natural longitudinal frequency of the column was not predicted by the theory. No theory was developed for a coupled longitudinal-flexural (normal) os c i l l a t i o n or longitudinal-flexural (in-plane) o s c i l l a t i o n for a column with b u i l t - i n ends. B u i l t - i n end conditions also apply to the flexural (in-plane) o s c i l l a t i o n modes of a column with hinged ends. Yet the experimental investigation shows that the same osci l l a t i o n phenomenons as those observed in a column with hinged ends occur in these cases as well. One exception being that no snap-thru phenomenons were observed for flexural o s c i l l a t i o n where b u i l t - i n boundary conditions apply. Increased r i g i d i t y of the column which limits the amplitudes of flexural o s c i l l a t i o n i s a probable cause. Shear and rotary inertia were not included in the theory for coupled 92 longitudinal flexural (normal) oscillations of a column. The influence of the two terms is more pronounced at high frequencies where high flexural modes of osc i l l a t i o n occur. The natural frequencies of o s c i l l a t i o n as calculated from linear theory with and without shear and rotay inertia for the two columns are given in tables V l - i and VI-2 . As can be seen, when these two terms are included in a linear d i f f e r e n t i a l equation for flexural o s c i l l a t i o n of a column, the numerical values of natural frequencies are closer to the values observed experimentally, It may be concluded then that shear and rotary inertia affect s l i g h t l y the numerical values at which the o s c i l l a t i o n phenomenons occur but do not preclude the existence of these phenomenons. Natural frequencies of longitudinal and torsional o s c i l l a t i o n were calculated from formulas derived from linear d i f f e r e n t i a l equations. Shear and rotary inertia terms were not included in the d i f f e r e n t i a l equation for torsional osci l l a t i o n . S t i l l the agreement between calculated and observed values of natural frequencies for these two osc i l l a t i o n modes was quite good, Influence of a constant axial loads A constant axial load of up to 64 lbs, was imposed by means of springs in addition to the periodic axial end excitation and the response of the column was observed. Except for a very slight change in frequencies at which os c i l l a t i o n phenomenon occurred no other effect was observed. Therefore, in a l l other experimental investigation, with this exception, no constant axial load was imposed. An effect of greater axial loads on the column behaviour was not investigated. 93 Strain magnitudes! In the range of frequencies considered, magnitudes of strain for a l l three vibration modes were usually greater than . ly*.-in/in and smaller than 200/<-in/in . Strain vs. frequency records give only the relative amplitudes of osc i l l a t i o n . No attempt was made to obtain numerical values of strains at a l l frequencies as this was not needed and would be extremely tedious to do. Actual amplitudes of strains for several axial end excitation frequencies are given in Appendix G . Tables VI-1 and VI-2 The two tables, VI-1 and VI-2, contain numerical values of various natural frequencies of the two columns considered in this experiment. Table VI-1 applies to the column with hinged ends as shown in Fig. 9. on page 55» Table VI-2 applies to the column with b u i l t - i n ends as shown in Fig. 10. on page 57. The values of natural frequencies of these two columns calculated by using eq's. (86), (90), (94), and (109) are given in columns ( l ) , (2), (3), and (4) respectively. Only one of these four columns i s not shaded, It gives the most precise calculated values of natural frequencies (shear and rotary inertia terms are included) of a particular column for i t s actual end conditions. The other three columns are shaded and give the values of natural frequencies calculated either by less precise formula (shear and rotary inertia terms are not included), or by considering the other (not actual) boundary conditions. By comparing particular numerical values presented in these tables the accuracy of approximation of actual end conditions of the two columns can be examined. The effect of rotary inertia and shear terms can be studied as well. 94 Natural frequencies of the column with hinged ends ( a l l frequencies in Hz ) vibration mode expt'ly observed natural freq's calculated values of natural frequencies experimentally observed harmonics of natural frequencies hinged ends b u i l t in ends (1) (2) (3) (4) Fl - n 75 170 170 -F2 365 300 300 : 46B 2 F3 806 675 920 216- : -F4 1291 1201 1198 1513 h F5 1952 1869 2686 2271 " 3172" i 1/3 F6 2756 '<} vj 3146 i 1/3 i F? 3693 3679 3648 4223 Mao i 1/3 i 1/5 F8 4760 4753 5424 I sm- 2 i 1/3 i 1/5 1/6 1/7 F9 6006 €?S1 5998 C??5 * (1/5) F10 7360 75*J7 7380 5123 1/3 1/7 F l l 8902 9034 8898 9923 1/3 F12 10410 iO C l l 10547 **(. 1/3 F13 12460 1263? 12325 -Pl 484 2?0 6ll 608 -P2 1674 10?! 1667 i 1/3 P3 3213 3301 3237 ( i ) i/3 P4 4806 4I&5 5284 (i) (i/3) 1/5 P5 7750 6735 8150 7773 (1/3) P6 IOO53 : 9&9f i 9063 ! a m N 10663 1/3 Tl 3005 2904 -T2 5995 5808 -T3 8707 8712 -LI 8215 8103 -Table VI-1 (The best calculated values are in the unshaded columns). 95 Natural frequencies of the rfcolamn? with built in ends ( a l l frequencies i n Hz ) vibration mode expt'ly observed natural freq's calculated values of natural frequencies experimentally , observed harmonics of natural frequencies hinged ends bu i l t in ends (1) (2) (3) (4) FI 170 80 80 172 -F2 488 31B }lt 497 491 -F3 984 716 i 715 975 976 -F4 1615 1269 l & i 1603 1 2 F5 2392 19&0 240? 2391 i 1/3 i F6 3330 • m $ 3361 3338 i 1/3 1/5 F7 4410 3S64 ¥*?5 442? 11/3 i F8 5630 5092 T033 5?*& 5675 i 1/3 F9 6950 6Ui4 6350 ?I6Q 7062 1 2 F10 8620 7956 7313 b??l 8595 -F l l 10269 9 ^ 7 10521 - 10272 • * 1/3 F12 12000 t t i 6 o l<'r3& 12020 1 1/3 1/6 F13 14212 -PI 502 234 530 -P2 1458 959 1453 i 1/3 P3 2845 Zlik 2106 sa?5 2826 1/3 1/5 P4 4500 37>6 3?4? 4753 4620 1 2 P5 6440 5006 5-^ 62 6809 ± 1/3 P6 9310 6^42 9360 1/3 1/5 Tl 3023 2989 -T2 6035 5979 -T3 9039 8968 -L l 8286 8272 IPillllll -Table VI-2 (The best calculated values are in the unshaded columns). CHAPTER 7 SUMMARY AND CONCLUSIONS SUMMARY 96 Summary of theoretical investigation: A theoretical investigation of the behaviour of an i n i t i a l l y imperfect column with hinged ends subjected to periodic, axial, end excitation was made. Two coupled, partial, nonlinear d i f f e r e n t i a l equations governing the motion of such a column are given by Mettler [ 9 ] • The equations are extremely complex, and therefore the f i r s t step of the theoretical analysis was to simplify these equations by neglecting the unimportant, uncoupled, terms. The f i r s t p a r t i a l d i f f e r e n t i a l equation was reduced to an ordinary d i f f e r e n t i a l equation by assuming the solution of longitudinal motion to be of variable separable type. Then an exact solution of the f i r s t d i f f e r e n t i a l equation governing the longitudinal motion of the column, was found. Unknown integration constants of this solution were determined from consideration of the end conditions imposed in this problem, The second partial d i f f e r e n t i a l equation govex-ning the flexural motion was inhomogenous, of second order, and with variable coefficients in x and t . At this point i t became necessary to guess a solution describing the flexural motion of the column. A Fourier sine series in x, with period 1, with unknown time variable coefficients, was chosen as a possible form of a variable separable type of the solution, as i t identically satisfies the hinged end conditions. With this assumed solution, the second partial d i f f e r e n t i a l equation became an ordinary, inhomogenous linear, second order d i f f e r e n t i a l equation with variable coefficients in x and t . By rest r i c t i n g the excitation frequencies to values smaller than one half of fundamental longitudinal frequency of the column, a l l the coefficients in x could be assumed alike, and were factored out. Equating of the terms associated 97 with like coefficients in x, yielded an i n f i n i t e set of governing d i f f e r e n t i a l equations. Each of the governing d i f f e r e n t i a l equations was an ordinary inhomogenous linear second order d i f f e r e n t i a l equation with one variable coefficient in t associated with zeroth order term. Each equation governed the time variation of one coefficient of the Fourier sine series. An approximate particular (forced) solution of the n-th governing d i f f e r e n t i a l equation was obtained by perturbation method, accuracy of which depends on magnitudes of excitation, physical properties and dimensions of the column. Now the approximately solved time variable coefficients were substituted in the assumed Fourier s ine series giving the approximate solution governing the flexural motion of the column. This solution together with the solution of the longitudinal motion give a complete description of the combined longitudinal-flexural motion of the column. Within the restrictions applicable to the theory the theory provides quantitative as well as qualitative information about the forced coupled longitudinal-flexural oscillations of a column. Some theoretical predictions are b r i e f l y discussed here. Large amplitude flexural oscillations w i l l occur when the excitation frequency equals the natural flexural frequencies. The column oscillates flexurally with the same frequency as the excitation axial frequency. The theory also predicts large amplitude flexural oscillations when the excitation frequency equals to l/2, l/3, 1/4,.,,.l/n... of the natural flexural frequencies. The column then oscillates flexurally with the same frequency as the axial excitation frequency and also with the natural flexural frequency (that i s the frequency equal to 2, 3( 4 ....n...times 98 the excitation frequency). These types of os c i l l a t i o n are identified as subharmonics of natural flexural frequencies. Resonant, large amplitude longitudinal oscillations w i l l occur only when the excitation frequency equals to natural longitudinal frequencies. The column then oscillates longitudinally with the same frequency as the axial excitation frequency. Summary of experimental investigation: An experimental investigation of the behaviour of an i n i t i a l l y imperfect column with hinged ends subjected to periodic, axial, end excitation was done to verify the theoretical predictions, and to complement the theory in the excitation frequency region where the theory i s not applicable. A possibility of existence of subharmonics of the fundamental longitudinal vibration mode was checked. The experimental investigation of the behaviour of an i n i t i a l l y imperfect column with b u i l t - i n ends subjected to periodic, axial end excitation was also done. Its purpose was to examine the effect of different boundary conditions on the response of a column. The results of the experimental investigation of the two columns were compared. The experimental investigation was accomplished in essentially two steps; the design of the two columns and testing setup, and the actual testing of the two columns. The design of the two columns, one with hinged ends and one with b u i l t - i n ends, involved a solution of a close approximation of desired end conditions, and a satisfactory means of clamping the column. A testing bench was designed to accommodate one column at a time, a shaker, and the springs. Its design assures that with a specimen mounted, the desired 99 end conditions of the column are accurately approximated, the axial alignment is excellent, and a generated or transmitted mechanical noise i s minimum. Strain gages and salt sprinkling were chosen as a means of monitoring the behaviour of the two columns. Strain gages were found quite satisfactory in a l l aspects. Their output, after processing, yielded various strain vs. frequency records at one point of a column. At some values of excitation frequencies, a spectrum analysis of the strain gage signal was done to determine the individual components comprising the strain at a point of a column. From the oscilloscope display of the signal a continuous variation of strain with excitation frequency was observed. Pictures of typical strain waveforms were taken directly of the oscilloscope screen. Salt sprinkling was Used to check the spatial form of a column os c i l l a t i n g flexurally with large amplitudes, which occurred for particular values of excitation frequencies. Salt sprinkling also complemented the information supplied by the strain gages. The results of the experimental investigation of the column with hinged ends agreed very closely with theoretical predictions. Natural flexural o s c i l l a t i o n modes, and their subharmonics occurred at, or close to values of predicted excitation frequencies. Components of different frequencies comprised the flexural subharmonics. A particular ordering of magnitudes of subharmonics was noted. No subharmonics were observed when the forcing frequencies were greater than one half of the fundamental longitudinal frequency. Large amplitude coupled longitudinal-flexural o s c i l l a t i o n occurred only when the excitation frequency was equal to the fundamental longitudinal frequency of the column. 100 No other large amplitude coupled longitudinal-flexural o s c i l l a t i o n was detected. No subharmonics of fundamental longitudinal o s c i l l a t i o n mode were observed. Some other o s c i l l a t i o n phenomenons, in addition to those theoretically predicted, were observed. Two snap-thru phenomenons were observed when the excitation frequency was equal twice the frequency of the second and the eighth natural flexural o s c i l l a t i o n mode respectively. The column then oscillated flexurally with frequency equal to one half of the excitation frequency. Snap-thru phenomenons may also be identified as superharmonics of natural flexural o s c i l l a t i o n modes. In-plane flexural o s c i l l a t i o n also occurred. In-plane natural flexural o s c i l l a t i o n modes and their subharmonics were observed. The subharmonics consisted of components of different frequencies. Torsional natural o s c i l l a t i o n modes were excited when the excitation frequencies were equal to natural torsional frequencies of the column.-Very weak coupled flexural-longitudinal o s c i l l a t i o n occurred when the excitation frequencies were equal to natural flexural frequencies. The experimental investigation of the column with b u i l t - i n ends produced basically the same results as the investigation of the column with hinged ends. The only differences were that no snap-thru phenomenons were observed, and the subharmonics of natural flexural frequencies occurred also at frequencies higher than one half of the fundamental longitudinal frequency (but lower than the fundamental longitudinal frequency). Thus i t would seem, that the behaviour of the column i s not much different whether the column has hinged or b u i l t - i n ends. The superharmonics are more d i f f i c u l t to excite in a column with b u i l t - i n ends probably due to i t s increased flexural stiffness. 101 Suggestions for future research: An exact solution of the partial d i f f e r e n t i a l equations governing the coupled longitudinal-flexural motion of a column is desirable. If an exact solution cannot be obtained, then perhaps a solution predicting the existence of superharmonics and weak coupled flexura]-longitudinal oscillations in addition to coupled longitudinal-flexural oscillations already considered here, would be desirable. Again, some restrictions on values of parameters of the system would probably have to be imposed. An approximate solution of the parti a l d i f f e r e n t i a l equations governing coupled longitudinal-torsional motion of a column could be found. The procedure for obtaining the solution might be similar to the one presented here for a case of coupled longitudinal-flexural motion of a column. Solutions meant to describe the motion of a column' at very high frequencies should include effects of rotary inertia and shear to make the solutions sufficiently accurate. Boundary conditions other than hinged ends or b u i l t - i n ends might be of interest. In particular a fixed-free ends end conditions should be investigated as results of such an investigation could be of practical value. A turbine blade mounted on a slightly bent or an imperfect shaft turning at very high speed, could be thought of as an axially excited column with one end free and the other b u i l t - i n . As i t i s often desirable in the engineering practice to limit amplitudes of oscillations in order to reduce energy transmission, or just to lower a noise generation, various means of damping ofparametrically induced oscillations of a column should be investigated. Damping could be induced, for example, by coating the surface of a column with a viscoelastic or elastic-viscoelastic material, 102 Some other means of monitoring the response of a specimen than strain gages could be tried, such as a fotonic sensor, liquid crystal coatings, or the .use of holography could be considered. Conclusion; The nonlinear theory of chapter 2 has been developed for a column with hinged ends,and the column was assumed to have some i n i t i a l crookedness. The theoretical considerations included a small material (internal) damping as well. The theory was eventually limited to the excitation frequencies as large as one half of the fundamental longitudinal frequency, which i s where this theory i s unlike the theories developed by some other researchers. These researchers have usually restricted their theories to values of forcing frequencies much smaller than the fundamental longitudinal frequency of the column. Furthermore, most of the researchers have considered oscillations of undamped, i n i t i a l l y straight columns. Their theories usually lead to Mathieu equation predicting i n s t a b i l i t i e s of column o s c i l l a t i o n to occur at certain excitation frequencies. o The theory of chapter 2 also gives more detailed information about the behaviour of the column than many of other theories. For a known set of systems parameters, the theory predicts approximate amplitudes of components of various frequencies comprising the flexural o s c i l l a t i o n of the column. The phase angle shift between these components i s predicted too. 103 BIBLIOGRAPHY 1 . Schneider., B.C., "Exoori~er-t.a.l I n v e s t i g a t i o n cf Nonlinear Coupled V i b r a t i o n s of Bars and P l a t e s " , M.A.Sc. Thesis, The U n i v e r s i t y of B r i t i s h Columbia, A p r i l , 1969 2. Johnson, D.F., "Experimental I n v e s t i g a t i o n of Nonlinear Coupled V i b r a t i o n s of Columns", M.A.So. Thesis, The U n i v e r s i t y of B r i t i s h Columbia, May, 1970 3. B e l i a e v , N.M., " S t a b i l i t y of P r i s m a t i c Reds Subject to V a r i a b l e Longitudinal Forces", C o l l e c t i o n of Engineering C o n s t r u c t i o n and S t r u c t u r a l Mechanics (i n z h i n e r n y e s c o r z h e i n i a i s t r o i t e l ' n a i a mekha.nika), Leningrad, Put, 1924 4. E o l o t i n , T.V., "Dynamic S t a b i l i t y of E l a s t i c Systems", ( t r a n s l a t e d from R u s s i a n ) , Holden-Pay, San F r a n c i s c o , C a l i f . , 1964 5. Somerset, J.H., and Evan-Iwanc-vski, R.M., "Experiments on Parametric I n s t a b i l i t y of Columns", Proceedings of the Second Southeastern, Conference cn T h e o r e t i c a l and A p p l i e d Mechanics, Atlanta., Ga,, March, 1964, pp. 503 - 525 6. Evan-Iwanow.ski, R.M., and Bvensen, K.A,, " E f f e c t s of I n e r t i a Upon the Parametric Response of P l a s t i c Columns", J o u r n a l of A p p l i e d Mechanics, March, 1966, pp. l 4 l - 148 ?. Tseng, W.Y., and Dugundji, J . , "Nonlinear V i b r a t i o n s of a Beam Under Harmonic E x c i t a t i o n " , J o u r n a l of Apr-lied. Mechanics, June, 1970, PP. 292 - 29.7 8. Tso, W.K., "Parametric T o r s i o n a l S t a b i l i t y of a Bar Under A x i a l E x c i t a t i o n " , J o u r n a l of A p p l i e d Mechanics, March, 1968, pp. 13 - 19 9, M e t t l e r , E,, Dynamic B u c k l i n g , "Handbook of Engineering Mechanics", 1st Ed., Fluegge, W,, e d i t o r , McGraw-Hill Book Company, Inc., 1962 10. Schmidt, G., "Coupling c f F l e x u r a l and Longitudinal Resonances of Columns", (translated, from German), Archiwum Mechaniki Stosowanej, B e r l i n , 1 965, pp. 233 - 24? 11. Lazan, B„J., "Damping of M a t e r i a l s and Members i n S t r u c t u r a l Mechanics", Pergamon Press, London, 1968, Fig. 2.6 and pp. 2 l 4 - 216 and p. 203 12. Cunningham, W.J,, " I n t r o d u c t i o n to Nonlinear A n a l y s i s " , McGraw-Hill Bock Company, Inc., 1958, p. 1?2 13. Timoshenko, S,, " V i b r a t i o n Problems i n Engineering", D. Van Nostrand Company, Inc., May, 1 ° ^ , pp. 2Q? - 37? 104 1.4, H a r r i s , CM., and Creole, C.E,, "Shock an'i Vibration Handbook", v o l . l . , e d i t o r , McGraw-Hill Bock Company, Inc.,. 1961 , p. 7 - 16 15• Ford, H,, "Advanced Mechanics of M a t e r i a l s " , Longmans, Green and Co. L t d . , London, England, 1969, p. 380 ' ' APPENDICES 105 APPENDIX A List of equipment: B&K Automatic Vibration Exciter Control Type 1025 capable of providing desired peak to peak displacement, velocity or acceleration. In this case B&K 1025 was used to provide peak to peak acceleration of 6 to 60 g's from approximately 100 Hz to 10 kHz. The frequency range i s scanned logarithmically with time. This unit was used together with B&K Accelerometer Preamplifier Type 2622 and with 2250 MB Power Amplifier. B&K Accelerometer Type 4335 having constant voltage sensitivity of 17.8 mV/g up to 10 kHz. This accelerometer was stud mounted on the moving end of the column and was used with B&K Accelerometer Preamplifier Type 2622. B&K Accelerometer Type 4336 having constant voltage sensitivity of 4.08 raV/g up to 45 kHz. This accelerometer was stud mounted on the moving end of the column as B&K 4335 "but on ax i a l l y opposed side of i t . This accelerometer was used in conjunction with B&K Accelerometer Preamplifier Type 2616. B&K Accelerometer Preamplifier Type 26l6 is a battery driven unit designed to be used with different types of B&K accelerometers. In this experiment i t was used with B&K Accelerometer Type 4336 and in frequency range 100 Hz to 16 kHz. B&K Accelerometer Preamplifier Type 2622 which has a b u i l t - i n sensitivity attenuator, which, when correctly adjusted for a given accelerometer, provides an'output voltage signal of 10 mV/g as sensed by the accelerometer. This unit was used together with B&K Automatic Vibration 106 Exciter Control Type 1025 and with B&K Accelerometer Type 4335. 2250 MB Power Amplifier made by MB Electronics, has frequency range 5 Hz to 20 kHz. It was used together with EA 1500 Exciter and with B&K Automatic Vibration Exciter Control Type 1025. EA 1500 Exciter (shaker) made by MB Electronics. It has 50 lbs.force rating, frequency range 5 Hz to 20 kHz and possible acceleration level over 100 g's. EA 1500 Exciter was used together with 2250 M3 Power Amplifier. B&K Frequency Analyser Type 2107 consists of an input amplifier, a number of weighting networks, a selective amplifier section, and an output amplifier. Usable frequency range for this unit was 5 Hz to 10 kHz. BAM-1 Bridge Amplifier and Meter, and B&K Level Recorder Type 2305 were used together with this unit. B&K Level Recorder Type 2305 having a wide range of paper and writing speeds and f a c i l i t i e s enabling plotting of RMS, DC or peak to peak values. Frequency response was well in excess of 10 kHz range. RMS of output of B&K Frequency Analyser Type 2107 was recorded by this unit. BAM-1 Bridge Amplifier and Meter measures and amplifies dynamic signals over a frequency range of 0-20 kHz, SR-4 strain gages were inputs to this unit and the output was delivered to KH 335 Variable F i l t e r . KH 335 Variable F i l t e r made by Krohn-Hite has low pass, high pass, and band pass f i l t e r settings. Usable frequency range i s .02 Hz to 20 kHz. BAM-1 Bridge Amplifier and Meter and B&K Frequency Analyser Type 2107 were input and output connections respectively. 107 Digital Time and Frequency Meter Type 1151-A made by General Radio Company was used to measure frequencies with ± 1 Hz accuracy. Inputs to this unit were either from B&K Automatic Vibration Exciter Control Type 1025 or B&K Frequency Analyser Type 2107 or others. Function Generator Model 110 made by Wavetek was used to check B&K Automatic Vibration Exciter Control Type 1025 in 0 to 10 kHz range and to substitute i t in 10 kHz to 16 kHz range of testing. It can provide triangle, square or sinusoidal (actually used) wave signal of up to 1 MHz frequency. 2250 MB Power Amplifier was used to amplify i t s output signals Type 565 Dual-Beam Oscilloscope made by Tektronix was used to display up to four signals simultaneously. Its inputs were any of the components mentioned here, EA-06-125BT-120 Electric Strain Gages made by Micro-Measurements were used to detect strains in the surface of the column. These gages had 120 ohms resistance and 2,11 gage factor. BAM-1 Bridge Amplifier and Meter was used to process strain gage output. Other equipment was also used to substitute, check or complement above mentioned equipment but i t was not used consistently throughout the experimental testing and i s not of sufficient importance to be li s t e d here. 108 APPENDIX B Examples of application of the nonlinear theory of Chapter 2 i-Solution of eq.(33) was obtained for i = 1,2.3, and 4 by solving eq.(36) up to sixth power of a parameter q n for several typical values of the mode number n . Eq.(36) was solved exactly, but in the end only several dominant terms of each frequency were retained, as by inspection many other terms were of negligible magnitude. A l l of the parameters used here are identical to those actually experimentally imposed. Thus, the examples presented here, may be also used to check the validity of theoretical predictions experimentally. Analysis of these solutions shows that the amplitude of second term of eq.(50) and of the third term of eq.(49) are very much affected by the magnitude of internal (material) damping and by other parameters of the system. It also suggests a possi b i l i t y of a phase angle change as the forcing frequency passes thru values given by eq.(43). Phase angle change arises from the presence of sine and cosine terms, the magnitudes of which change drastically and differently as the forcing frequency i s varied. Addition of sine and cosine terms of varying magnitudes results in varying phase angles. 109 Example # 1t 1=1 n = 5 (a = 4) Eq.(36) H a s solved to sixth power of the parameter q^for a h= 4 . This solution i s presented here with only dominant terms of each frequency retained: W (z) ( ¥A °r A 5 V /52 c 1 7640 c* f A • o f A 0 JA . 0 fA ' • \ + I ~~ Sin 2z •+ cos 2-z. -i -1 sm 2z. + — —- cos 2TL -t . -. - / V 2C 4fcz 230 . 5SZO c* J Y fA • * f3A , J tSA • j, \ + I ~ Sin Cz + — cos 6 2 + si" 6-z +•-••) V 76&C IS4ooc^ 8s*00c2 / / f $ A <f- sA \ + / -« J. sir, Sz. ~ 1 cos <f z - .... ) v 46 ISO c I24OOOOC2- 7 46 ISO c /24oeaOcz 4 4Z0 coo c •+ 1 sin io z + H.O.T.'S The case n = 5 » i = 1 represents the f i f t h natural flexural o s c i l l a t i o n mode for the column with hinged ends. Parameters experimentally imposed were: S = 20 g calculated 0- = 1877 Hz (experimentally observed 0- = 1952 Hz) n = 5 a = h 2 2S C$ = 2y i = 2(63.8 *id7)l * /. J * /0 -.T 110 With these parameters the solution for w$ (-z") becomes» =c 2.03 A - (-S'/S *io'7)A + ---+ 33 0C0As!r>2z •+ 2oZOAcos2z, + S30 A sin 2 z. -h ^ 7.4 A cos - / 3 A sin 4z - 0.677 A cos 4z. - O.I43 A sin 4z - ... •*• (i.63 x idZ) A Sin 6z + ( S,S£ < /Q~s) Acos. 62. •+ (2,2 */a~sJ A sin € - (/-OS * io'7) A sin fz ~ ( S-.04- * /d9)AcoS 8z - ... •+ (4.S x io"'2) A £t'*> /Oz + .... And this solution can be well approximated ast W5(z) ^ 33 530 Assi'n 2* 20S7 A£cos 2z In order to obtain the other terms of eq.(l5)» eq.(45a) w i l l be used to solve Wj-Ct), J = 1,2,.. for <Sr = 1877 Hz : A' w. =s _ —J.— cos 2, z J 4-a.. °-1L = °" W/. « - cos 2 Z Z 3. S3 - 0 . 5 3 W, « - cas Zz. 2. It A* a, = 8.25 « — — cos 2*. 6 4 4,2S _ „ /I7 Where • A. - ; ~ — / a / = A 1 J C0Sr3s,n(t?j J 4 J J I l l Substitution of w-(z), j = 1,2,...7 in eq.(l5) yields: - [ - — T T ^ — - -JL.S;„2-*L . JL. S/T, 21* fr= [ X S.-fS A 3.4! J? ' A 4 . 4Tx i S7TX A 6 , 6fo A; . 7Tx . , . Sin -f ZOS7 Ar S,r> -*• Sin — — •+• S/>i — — i- - - • / cos pfc 2.36 A A 4.2S il. 3 A j + 33530 Ac Sl'm sin frt i A or: W(x.t) ~ / - ~ — sin 1 sun ~ + ^ ' ' I 3.334 A 2.S3 A 3.48 £ 2.36 A -f (2oS7Xa.S)a<. s i n - 7 - •+ — — •+ sty, — z - + ' " A<=°s.&-£ + •> *• 4-2S £ u. 3 <• J A (39 S30X2 s) v s'n S'"P* If a , j = 1,2,,.. are within three orders of magnitude this equation can be approximated as: 9 " ~ A * . ST* , \ ~ A 39/aoo 4.J Sin — — cos (<&•£ - 87y STTx JL" A The above equation shows that when the axial excitation frequency i s equal to natural flexural frequency of the column, the amplitude of one of the components comprising the spatial form of the column is greatly amplified. It also shows that as the forcing frequency passes thru natural flexural frequency, large phase angle s h i f t occurs. 112 Example § 2: 1=2 n = 5 (a = l6) Eq.(36) was solved to f i f t h power of the parameter q for a = 16 . This solution is presented here with only dominant terms of each frequency retained: Wc (z) = ( + -J- + - + - - • • J / A • , r xA . , r 4A • \ + ( - r cos 2z H- - r — ; — sin 2z. •+ cos- -Zz. - - - ) f Iri- s/Wz - -2^— cos4z. + — ^ 5 , ^ 4 2 •+ - ) V 4SC SSOOc2: 174 ooo c 3 y -I- / - -1 s/n£z. + • c o s CT. + - - • ) ^ 36o c ll£ ooo c2- ' + f -I s i w *z - — T 1 " f ^ + " - J V 46 too c ^ 5 7 0 0 0 0 c ' S/>> 10 Z •+ •+ U. O.T'S 3 «?oooo c The case n = 5 » i = 2 represents the second subharmonics of the f i f t h natural flexural o s c i l l a t i o n mode for the column with hinged ends. Parameters experimentally imposed werej S = 10 g calculated 0- = 938 Hz (experimentally observed & = 976 Hz) n = 5 a = 16 c$ = 2<yt = Z(63.8*I07)2 » 2.6*/o' S 113 With these parameters the solution for w- (z) becomes: W5 (z) ~ + ^ cos ^z +' 4r s/'»i ^ 2 •+ ^~ cos 2z. -f '2. K.fc 42,2 4 2£..2 <4 s/*7 4z - 3.77 A cos 4z + It.3 A sin 4 z + A • A sm 6z -f cos 6z + Z2.6. 70.4- 354 no 00a 392 coo Sm IO -z_ •+.-.. And this solution can be well approximated as: ("z ) =s - c o s ^ t + -—£/«-?z -f J7. i"/4s/«4z - S.77 Acos4z. 5 5 14.2 4 • • 4 ^ - ~ Sin 6z. + —! cos 6 Z In order to obtain the other terras of eq.(l5)i eq,(45a) w i l l be used to solve w ( t ) , j = 1,2,.. for 9- = 9?6 Hz : W. - " J 4 - */ * 3 « * 33.3 a7 - C/.7 vv. = -IV, 56 Cos 2: CO s 2 z. 1.3 A+ 2-56 Ae 23. 3 Ay 57.7 Co s 2z cos •Sz. cos 2z. COS 2z 114 Substitution of j • IV (x,t) r t' 3.98 II z A, 3.S6 •+ *s . S7l~x T , A' + As S'n J4-Z sin 9-t „ • 2Fx Ai • 3Tx ,  A< • ir* sm — r — - sm ——- •+ sm + * 1.6 * S.S6 A. - sm —— — 1 .... / cos Pr JL S7.7 £ />•? —-— sm 2 V-L - 9.77As Sin S**-cos29>t - diL_ SLH £ 3 L s/„ 2 9 t + A^_ ^ ££* ^ or: / 3.9? * 3.S6 Z 1,3 S m Z 3.SC X * -A — — — sv>i — -f 1 <r/>7 -— + — — i s m —± + . . . . I A cos&t + 3 Z z<3.3 Z S7.7 s€ J 1 — STx + UL- cos I t t <tcA sin . 70A / X or: , . N. * • €T% *. . 7/7* 7 r a u + 1.23 a, sm - j - -i- . 8S a7 sin ~ •+ • • j A C o S ft + + 3.Z3 asA sin — Co* (*t ~3ZaJ + a j ^ <r/W ^ c c r f w * -The above equation shows that when the axial excitation frequency i s equal to one half of natural flexural frequency of the column the column responds with at least two different frequencies. Namely, i t responds with the same frequency as the excitation frequency and also with the natural frequency of the column. The latter frequency i s now twice the excitation frequency. 115 The amplitude of the s i n . — — component comprising the spatial form of the column i s greatly amplified. This term also undergoes a large phase s h i f t as the forcing frequency passes thru natural flexural frequency. Very small term having frequency three times the axial excitation frequency is also present. 116 Example #3: i = 3 n = 10 (a = 36) Eq.(36) was solved to fourth power of the parameter q for a = 36 . This solution i s presented here with only dominant terms of each frequency retained: W Cz) ~ ( - ^ i + ) -+ ( A 2-2. T • • •+ — — — s m 2 z + • •• ) -+ ( -— cos 4z -+ . .. + — sfto 4z. •+••• y / rzA . , <?4A \ •+ I sim 6z - ... - — cos 6z + - - - j •+ ( ' , 1,A~ s/" #Z~ + • • -) * ( —^ sin lOz. •+ • • • ) V 107 600c J \£es>c,oooc. Z ' The case n = 10 , i = 3 represents the t h i r d subhamonics of the tenth natural f l e x u r a l o s c i l l a t i o n mode for the column with hinged ends. Parameters experimentally imposed were: S = 40 g calculated 0- = 2502 Hz (experimentally observed 9- = 2453 Hz) n = 10 a = 36 <1 = (3 x 10s)S —r- = O * ios)(iS44o) r =s /. 3 *io c = Z^i = 2(Mf*idT)3 - / * id 6 With these parameters the solution for w (z) becomes: A A A <o{ ' 90 000 32, 9.6 *io? A A A A A cos 4z. + - - • •+ ••S/t-i 4z + - - + si'n 6z -h - • •— cos 6z. So 000 JSJoo Z4 63 117 And this solution can be well auuroximated as: -4 A A IV,»CZ) ~ TT cos 2z + — sin ^ z - — Cos 6z 10 ' 31 24 63 or: w 10 (z) = j^cos2z - 0.0447 cos (£z _ 75") In order to obtain the other terms of eq.(l5), eq.(45a) w i l l be used to solve w-(t), j = 1,2,.. for 0"= 2502 Hz : 0 O036 a 2 = 0-OS7 d, - 0-293 =r 0.32. cxs r- 2.26 A cos- 2 r 3.7 _ A 2. OS — cor 2z. co r Zz 1.74 cos 2z at = 4.77 w, cos: 2z 6 0.77 £7 = S.6S as = /4.7 ^ 7 io.7 cos 2.z cos -2 2. H4 = — cos2z 3 13.7 a// = S3 w it 43 cos 2z <*fi = 74. £ IV, - — c o s -2z 118 Substitution of w-(z), j = 1,2, 12 in eq.(l5) yields: 3. . 2 7 x Sm — — — JL A* 3.7 . 3Tx Sm — — 4. 3.OS Sm —— -+ Z A7 -4.6% 10.7 As , sir* A6 . 6r* y . 7jrK Ae „ . swx Sm —r- ~- -+ sm —-~ -+ — Sm 1.74 -t 0.77 6%. £ 10.7 - 0. 044 7 A S/'n cos (3$-t - 7S°) or: ~A~[- o.ZSf. sin — - i.a a*sir, ^ _ 2.43 a* si» — - S.'3e£si„ ~ - ,4.4 cts st^--f + 46.8% si„ <2L -f /O.Sa*sin?f + S.3Sa*sm + + 4.IZ cT3sin -j- •+ 3.a a.;0 S,h + 2.47a~sin — * 2.03 a*Si» -~ + - - - j c o s ^ t - 4. 47A~c%0 sir, cos (3<H ~7S°) MJ cysln(U) J J J J The above equation for w(x,t) shows that when the axial excitation frequency is equal to one third of natural flexural frequency of the column, the column responds with at least two different frequencies. Namely, i t responds with the same frequency as the excitation frequency and also with the natural frequency of the column. 119 Example # 4: 1=4 n=7 (a = 64) Eq.(36) was solved to f i f t h power of the parameter q for a = 64 . This solution i s presented here with only dominant terms of each frequency retained: 3840 fA 880 ( cos 4-z + f — cos 6z ( ( So 700 fA 644ooo c Sin 8z •+ f A — Cos 2z -f -60 (iUO)(£44oeo)c (2tX6*4o°o)C ... ) Sin 4z + sin 6z •+ (loo,)(644ooo) C 2 " sin /Oz + - • • • ^  •+ ( cos &Z + ) ) ) (Z $SO)(644oOo)C Si'r> 12 2 + ...) The case n = 7 , i = 4 represents the fourth subharmonics of the seventh natural flexural o s c i l l a t i o n mode for the column with hinged ends. Parameters experimentally imposed were: calculated 0 n S = 20 g 919 Hz (experimentally observed 0- = 923 Hz) 7 = 64 2. 43 = (jx/oS)S ~ = (3 * iOS)(772o) - = h S 6 9134 = - 2 (/.6r*/o~ r)4 - 1.34 * id6 120 With these parameters the solution for w ?(z) becomes! A A . - - - - . COS 2 2 + •• + • - • , 24 soo 60 l&*oo I.Z7 '/o7 Wr(z) — — •+ -— coS 2z •*• • • •*• ;—-— cos 4z •+ —+ — — son 4z_ A A A Cos 6z * - + sin 6z - - - • -+ sm $z. + 3.1 * io6 4IZSO " ZZ8 A A A cos Sz +....— S/'n 10z. •+ • - + — sin i2z 12-5 53 coo 2-7' 10  7 • And this solution can be well approximated ass A A A W (z) =s cos 2z + sin 8z — cos Sz 7 £0 229 12-5 A or: WL/Z) «s — c o s 2z - 0. Soi A cos(8z.-3a ) In order to obtain the other terms of eq.(l5), eq.(45a) w i l l be used to solve w.(t), j = 1,2,... for9 -=919Hzj «•/ - = ( h 7S)= °-0265 * - - 7 3 7 c o s Z * K a2 = 0.4Z wz = — cos2z 3.S8 A3 a, - 2.Z6 w, = - cos 2z J /• 84 4-A, 2.8 &s = 16-6 ssr - —— cos 2z. s 12.6 <X. = J4.S iv, - cos 2z. 6 6 30.5 A3 ag =io$ ys„ - cos 2z_ Ac A9 = 174- W3 3 -> Wa - cos 2z 170 121 Substitution of wj( z)« j c 1.2,....9 in eq.(l5) yields: / - sin —— — 1 # t / l 3.37 * 3.S8 4 I. 84 * 2- 8 ~h sm — Sin — ~ + Sin —r- + — - Sm + 12.£ Z so.S £ 60 £ icS * A3 . ST*. 7, . a ^ „ „„. < • '7> • j COS &t - 0. SOI A7 sm ~ ~ Cos (4frt — 3 or: i<- a.~ sm — -r- /-'^a, —J- +• 0.8/a7 Sin — + 0.6/ oTg Sm —j- + -h 0.476 a* sin ^ j- + - - - J1 Acos&t - 39.3 A'a* sin — - 3') . 2 where: ^ • - -3 — — / a- = A a. The above equation for w(x,t) shows that when the axial excitation frequency is equal to one fourth of natural flexural frequency of the column, the column responds with at least two different frequencies. Namely, i t responds with the same frequency as the excitation frequency and also with the natural frequency of the column. 122 APPENDIX G Magnitudes of strains: During a steady state o s c i l l a t i o n of a column in i t s possible vibration modes, strains of various amplitudes exist. As is well shown on the strain vs. frequency charts obtained for the two columns investigated here, the magnitudes of strains depend very much on the frequency and level of the axial end excitation. The strain vs. frequency charts serve well for a qualitative analysis of a column osc i l l a t i o n , however; due to extensive processing of the strain signal, the actual magnitudes of strains are d i f f i c u l t to determine from the strain vs, frequency charts. This inadequacy is not of great consequence since i t is the qualitative analysis that i s more important in this investigation. Yet, to gain more insight, and to be in a good position to evaluate the internal (material) damping, typical values of strain in the surface of a column with hinged ends were meassured. The axial end excitation level was held constant at 20 g's during this test. The procedure for meassuring of strains suggested by the manufacturer of BAM-1 was used. The excitation frequency and the kind, and the amplitude of meassured strains are given belowt 9 * - w„ L " e = l^oo Hz e- = 1291 Hz F' - strain = 0.3 /<in/in F - strain = 26.5 /«in/in 0 = GO, 9- = 1400 Hz e = 1674 Hz P - strain = O.38 /<.in/in P - strain = 3.5 / * i n / i n s * -f. 9 * 4-e- = & = 1400 Hz 0 = 6500 Hz 0 = 8215 Hz L - strain = 2.5 / * i n / i n L - strain = 0.125 /<• in/in L - strain = 7.5 yx.in/in 

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