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 of Mechanical Engineering We a c c e p t t h i s t h e s i s as required standard conforming to THE UNIVERSITY OF BRITISH COLUMBIA May, 1972 the In p r e s e n t i n g t h i s thesis i n . . - p a r t i a l ' f u l f i l m e n t .o£ the .„ requ i rements f o r an advanced degree at the U n i v e r s i t y the L i b r a r y s h a l l make i t f r e e l y of B r i t i s h available Columbia, I agree f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s for that thesis s c h o l a r l y purposes may be granted by the Head o f my Department o r by his representatives. of this written It thesis f o r financial gain s h a l l not be allowed without my permission. Department o f ^Q.ck<x\nic<xl The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada Date i s understood that copying o r p u b l i c a t i o n M<x~, 2 S Eingiyie.^ri Columbia (972 i ABSTRACT The o s c i l l a t i o n o f a column subjected to periodic a x i a l end e x c i t a t i o n was a n a l y t i c a l l y and experimentally The i n i t i a l crookedness investigated. o f the column and the l o n g i t u d i n a l i n e r t i a of a column element give r i s e to coupled, l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n s . A snap-thru phenomenon and complex subharmonics o f natural f l e x u r a l modes o f oscillation also occur a t certain a x i a l end e x c i t a t i o n frequencies. Furthermore; at c e r t a i n e x c i t a t i o n frequencies, a coupling between l o n g i t u d i n a l and t o r s i o n a l o s c i l l a t i o n s i s found to e x i s t . A theory providing q u a l i t a t i v e and quantitative information about coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n s and complex subharmonics was developed f o r a column with hinged ends. In order to t e s t the v a l i d i t y o f the theory an experimental apparatus was set up to excite the column a x i a l l y , with transducers monitoring the response o f the column. The experimental r e s u l t s were i n very good agreement with the t h e o r e t i c a l predictions. A column with b u i l t - i n ends was also tested and i t s response was s i m i l a r to the column with hinged ends. Thus, the r e s u l t s o f the experimental i n v e s t i g a t i o n suggest that the r e s u l t s o f the theory developed column with hinged ends are also applicable for a to a column with b u i l t - i n ends. Coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n s were observed when the a x i a l end e x c i t a t i o n frequency was equal to the natural frequency o f the column. longitudinal Large amplitude o s c i l l a t i o n s i n both longitudinal ii and f l e x u r a l v i b r a t i o n modes occurred a t t h i s frequency. When the frequency o f a x i a l end excitation was equal to the natural f l e x u r a l frequencies o f the column, large amplitude f l e x u r a l o s c i l l a t i o n s resulted, Flexural o s c i l l a t i o n s were also observed when the frequency o f the a x i a l end e x c i t a t i o n was one half, one t h i r d natural f l e x u r a l frequencies o f the column, up to one eighth o f the A spectrum a n a l y s i s o f the s t r a i n s i g n a l showed that the f l e x u r a l response then comprised two fundamental motions, one with the frequency o f the a x i a l e x c i t a t i o n and one with frequency equal to the associated natural frequency. The r e s u l t i n g amplitudes o f f l e x u r a l o s c i l l a t i o n s a t these frequencies were smaller than those observed when the frequency o f the a x i a l end excitation was equal to the natural f l e x u r a l frequencies o f the column. occurring a t these a x i a l end e x c i t a t i o n The f l e x u r a l o s c i l l a t i o n s frequencies were i d e n t i f i e d as the complex subharmonics o f natural f l e x u r a l frequencies, A snap-thru phenomenon occurred when the a x i a l end e x c i t a t i o n was twice the frequency o f natural f l e x u r a l frequencies. frequency Under certain circumstances the column then o s c i l l a t e d f l e x u r a l l y with one h a l f o f the excitation frequency. The amplitudes o f f l e x u r a l o s c i l l a t i o n s were comparable to those occurring when the frequency o f the a x i a l end e x c i t a t i o n was equal to natural f l e x u r a l frequencies o f the column. Large amplitude f l e x u r a l o s c i l l a t i o n s occurring a t natural f l e x u r a l frequencies, complex subharmonics and snap-thru phenomena, though excited by the a x i a l end excitation, amplitudes o f l o n g i t u d i n a l d i d not cause appreciable increase i n oscillations. iii F i n a l l y , large amplitude t o r s i o n a l o s c i l l a t i o n s occurred when the a x i a l end e x c i t a t i o n natural frequencies. longitudinal was o f the same frequency as the predicted torsional Again no appreciable increase i n amplitudes o f o s c i l l a t i o n s was observed. F l e x u r a l o s c i l l a t i o n phenomena here described, a l s o occurred during in-plane o s c i l l a t i o n o f a column. flexural TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGEMENT viii LIST OF TABLES ' ix xi INTRODUCTION 1 NONLINEAR THEORY Theoretical Considerations Governing D i f f e r e n t i a l Equations Solution o f the governing d i f f e r e n t i a l equation by the perturbation method Material damping, Theoretical predictions.* CHAPTER 3 x NOMENCLATURE Preliminary Remarks L i t e r a t u r e Review Limitation of Investigation..... CHAPTER 2 xiv LIST OF FIGURES LIST OF APPENDICES ' CHAPTER 1 i LINEAR THEORY... Longitudinal o s c i l l a t i o n , Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with hinged ends, , Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with hinged ends - e f f e c t of rotary i n e r t i a and shear terms , Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with b u i l t - i n ends 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 - e f f e c t of rotary i n e r t i a and shear terms Torsional o s c i l l a t i o n o f a prismatic column with b u i l t - i n ends.... 1 3 6 7 8 8 22 31 36 ^0 41 4-3 44 45 46 49 V Page CHAPTER 4 APPARATUS AND INSTRUMENTATION Design o f the column with hinged ends.... Design o f the column with b u i l t - i n ends.. Design o f the t e s t bench Vibration control apparatus.... Transducers and associated e l e c t r o n i c s . . . CHAPTER 5 TEST PROCEDURE Calibration Testing preliminaries... Mounting o f a column. Testing Additional testing T o t a l damping measurements Undertesting and .overtesting Photography Loading o f a column by a constant a x i a l force CHAPTER 6 RESULTS AND DISCUSSION I d e n t i f i c a t i o n and a n a l y s i s o f s t r a i n vs. frequency records Analysis o f f l e x u r a l s t r a i n vs. frequency records - natural f l e x u r a l frequencies.... Analysis o f f l e x u r a l s t r a i n vs. frequency records - complex subharmonics Phase angle s h i f t Analysis o f f l e x u r a l s t r a i n vs. frequency records - snap-thru phenomenons D i s c o n t i n u i t i e s o f s t r a i n vs. frequency curves Analysis o f l o n g i t u d i n a l s t r a i n vs. frequency records Study o f o s c i l l a t i o n modes obtained by s p r i n k l i n g o f the column with s a l t Intensity o f coupling between v i b r a t i o n modes Agreement between experimental r e s u l t s and t h e o r e t i c a l predictions...... 51 52 56 58 58 61 65 65 65 66 6? 68 69 70 70 71 72 72 72 74 77 80 84 84 85 89 90 vi Page Influence o f a constant a x i a l load.... S t r a i n magnitudes., CHAPTER 7 SUMMARY AND CONCLUSIONS Summary o f t h e o r e t i c a l i n v e s t i g a t i o n . . • Summary o f experimental i n v e s t i g a t i o n . Suggestions f o r future research 92 93 96 96 98 101 BIBLIOGRAPHY 103 APPENDICES 105 vii LIST OF FIGURES Figure Page F i g . 1. Coordinate system o f the column 7 F i g . 2. Coordinate system o f the column element 9 F i g . 3. Approximation (23) i = 1 17 F i g . 4. Approximation (23) i = 6 18 F i g . 5. Magnitude o f parameter q 23 F i g . 6. Design #1 53 F i g . 7. Design #2 53 F i g . 8. Design #3 54 F i g . 9. Column with hinged ends 55 F i g . 10. Column with b u i l t - i n ends 57 F i g . 11. Test bench 59 F i g . 12. Signal flow diagram 60 F i g . 13. Placement o f s t r a i n gages on a column 63 F i g . 14. Arrangement o f s t r a i n gages i n the Wheatstone bridges 64 F i g . 15. O s c i l l a t i o n o f a column i n natural o s c i l l a t i o n modes 73 F i g . 16. O s c i l l a t i o n of a column at second order subharmonics 74 Fig. 17. O s c i l l a t i o n of a column a t t h i r d order subharmonics 75 F i g . 18. O s c i l l a t i o n o f a column a t fourth order subharmonics 75 Fig. 19. O s c i l l a t i o n o f a column at f i f t h order subharmonics 75 Fig. 20. O s c i l l a t i o n o f a column a t s i x t h order subharmonics 76 viii Figure Page Fig. 21. O s c i l l a t i o n o f a column a t seventh order subharmonics 76 Fig. 22. O s c i l l a t i o n o f a column at t h i r d order subharmonics (weak) 76 Fig. 23. O s c i l l a t i o n o f a column a r F7/4 subharmonics phase angle s h i f t (912 Hz) 77 Fig. 24. O s c i l l a t i o n o f a column a t F7/4 subharmonics phase angle s h i f t (920 Hz) 78 Fig. 25. O s c i l l a t i o n o f a column a t F7/4 subharmonics phase angle s h i f t (923 Hz) 78 Fig. 26. O s c i l l a t i o n o f a column a t F7/4 subharmonics phase angle s h i f t (928 Hz) 78 Fig. 27. Oscillation modes 80 Fig. 28. S t r a i n vs. frequency record f o r a column with hinged ends 82 Fig. 29. S t r a i n vs. frequency record f o r a.column with b u i l t - i n ends 83 Fig. 3 0 . Nodal l i n e pattern occurring when a column o s c i l l a t e s i n the f i r s t natural t o r s i o n a l o s c i l l a t i o n mode 86 Fig, 3 1 . Nodal l i n e pattern occurring when a column o s c i l l a t e s i n the second natural t o r s i o n a l o s c i l l a t i o n mode 86 Fig. 32. Nodal l i n e pattern occurring when a column o s c i l l a t e s i n the t h i r d natural t o r s i o n a l o s c i l l a t i o n mode 87 Fig. 33. Nodal l i n e pattern occurring when a column o s c i l l a t e s i n the f i f t h natural f l e x u r a l o s c i l l a t i o n mode 87 Fig. 34. Nodal l i n e pattern occurring when a column o s c i l l a t e s i n the tenth natural f l e x u r a l o s c i l l a t i o n mode 87 Fig. 35. Theoretical nodal l i n e patterns 88 o f a column i n snap-thru o s c i l l a t i o n ix LIST OF TABLES Natural frequencies o f the column with hinged ends Natural frequencies o f the column with b u i l t - i n ends LIST OF APPENDICES Appendix A L i s t of equipment Appendix B Examples o f a p p l i c a t i o n o f the nonlinear theory of Chapter 2 Appendix C Magnitudes o f s t r a i n s NOMENCLATURE parameter parameter constant Fourier Sine s e r i e s c o e f f i c i e n t s describing i n i t i a l geometry imperfection o f a column Fourier Sine s e r i e s c o e f f i c i e n t s amplitudes o f polynomials associated with cosine terms amplitudes o f polynomials associated with sine terms width o f column heighth o f column length o f column weight density o f column material mass density o f column material area o f cross section o f column normalized damping c o e f f i c i e n t o f i n t e r n a l (material) damping c r i t i c a l damping c o e f f i c i e n t damping factor s p e c i f i c damping energy f a c t o r logarithmic decrement amplitude o f f o r c i n g function amplitude o f a x i a l end e x c i t a t i o n acceleration o f gravity polar moment o f i n e r t i a o f column cross section modulus o f e l a s t i c i t y shear modulus xii 3? = shear r = radius c coefficient = velocity of longitudinal 0 ^ = torsional k, = constant ( r e l a t e d t = real z = normalized 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- = torsional p(x,t) = distributed P = total applied a x i a l P s t i f f n e s s o f column c r o s s to torsional section stiffness) time time displacement load load = constant a p p l i e d a x i a l Q wave p r o p a g a t i o n i n column load ^ = f r e q u e n c y o f a x i a l end e x c i t a t i o n °°n = n-th n a t u r a l f l e x u r a l frequency i n rad/sec f„ = n - t h n a t u r a l f l e x u r a l f r e q u e n c y i n Hz = n-th n a t u r a l l o n g i t u d i n a l ^ L = fundamental l o n g i t u d i n a l frequency frequency u (x) = s p a t i a l form o f l i n e a r l o n g i t u d i n a l column w (x) = s p a t i a l form o f 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 o f column 0" (x) = s p a t i a l form o f l i n e a r t o r s i o n a l 0 D o w,(t), w ( t ) , . . oscillation of # o s c i l l a t i o n o f column = f u n c t i o n s d e s c r i b i n g time v a r i a t i o n o f a m p l i t u d e s o f i n d i v i d u a l components o f the F o u r i e r S i n e 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 o f column F = "Flexural P = " i n - P l a n e mode" l mode" xiii L = "Longitudinal mode" T = "Torsional mode" i,j = integers n = mode number f . = phase angle r.i C, , C ... 2 = constants D = differential R = resistance o f column material E = applied V = measured voltage y*- operator voltage = mass per unit length " •• 0 f column = rOOtS w(x) = i n i t i a l geometry imperfection of column w (z) = s e r i e s expansion terms o f w (z) ni (initial n Abbreviations BAM = Bridge a m p l i f i e r and meter Hz = cycles per second RMS = Root-mean-square value o f a function crookedness) xiv ACKNOWLEDGEMENT I wish to express my gratitude to my f a c u l t y advisors, Dr. H. Vaughan and Dr. H. Ramsey, f o r g i v i n g me an opportunity to work on t h i s project. I found t h e i r assistance and t h e i r advice very h e l p f u l i n overcoming the d i f f i c u l t i e s a r i s i n g during the i n v e s t i g a t i o n . Further, I wish to thank a l l the technicians and s e c r e t a r i e s i n the Department f o r t h e i r contribution to t h i s research. This study was made possible through Research Grant No. provided by the National Research Council o f Canada. 67-5563 CHAPTER 1 1 Preliminary remarksi In todays engineering practice, a design o f components exposed, or subjected to high frequency e x c i t a t i o n must often be tackled. In p a r t i c u l a r , problems associated with a design o f turbine blades, sonar equipment, e t c . sometimes lead to the study o f a x i a l l y excited columns. The l i n e a r theory provides some information about the behaviour o f a column, but t h i s may be i n s u f f i c i e n t as, a t very high frequencies the nonlinear e f f e c t s can be quite s i g n i f i c a n t . For t h i s reason the o s c i l l a t i o n o f a column subjected to p e r i o d i c a x i a l end e x c i t a t i o n was a n a l y t i c a l l y and experimentally investigated a t r e l a t i v e l y high frequency. The i n i t i a l crookedness o f a column and the l o n g i t u d i n a l i n e r t i a o f a column element give r i s e to coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n s . Furthermore; a t c e r t a i n e x c i t a t i o n frequencies, coupling between l o n g i t u d i n a l and t o r s i o n a l o s c i l l a t i o n s i s found to e x i s t , A study o f coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n s o f a column with hinged ends was done. The study eventually yielded a s o l u t i o n g i v i n g a complete d e s c r i p t i o n o f the combined l o n g i t u d i n a l - f l e x u r a l motion o f the column. The examination o f t h i s s o l u t i o n allows us to make predictions about the response o f the column. Large amplitude f l e x u r a l o s c i l l a t i o n s w i l l occur when the e x c i t a t i o n frequency equals n a t u r a l - f l e x u r a l frequencies. The column then o s c i l l a t e s f l e x u r a l l y with the same frequency as the e x c i t a t i o n a x i a l frequency. The theory also predicts large amplitude f l e x u r a l o s c i l l a t i o n s when the e x c i t a t i o n frequencies equal to 1/2, f l e x u r a l frequencies. l/3, l/4, l/n,...of the natural The column then o s c i l l a t e s f l e x u r a l l y with the same 2 frequency as the a x i a l e x c i t a t i o n frequency and also with the natural f l e x u r a l frequency (that i s the frequency equal to 2, 3, 4, e x c i t a t i o n frequency). n ....times the These types o f o s c i l l a t i o n are i d e n t i f i e d as subharmonics o f natural f l e x u r a l frequencies. Resonant, large amplitude l o n g i t u d i n a l o s c i l l a t i o n s w i l l occur only when the e x c i t a t i o n frequency equals natural l o n g i t u d i n a l frequencies. The column then o s c i l l a t e s l o n g i t u d i n a l l y with the same frequency as the a x i a l e x c i t a t i o n frequency. These predictions were then checked experimentally, and the r e s u l t s o f the experimental investigation agreed very c l o s e l y with t h e o r e t i c a l predictions. A 3 L i t e r a t u r e Reviewj Most of the research connected with the parametric response o f bars and columns has been limited to small free l a t e r a l o s c i l l a t i o n s i n the f i r s t natural v i b r a t i o n mode , of Some researchers studied the parametric response columns subjected to small a x i a l or l a t e r a l periodic e x c i t a t i o n . The e x c i t a t i o n was usually o f the form o f s i n u s o i d a l l y variable force or a c c e l e r a t i o n with time, and sometimes a constant a x i a l force was superimposed on the variable e x c i t a t i o n . Nearly a l l o f the t h e o r e t i c a l and experimental a n a l y s i s has been limited to cases where the frequencies o f column o s c i l l a t i o n (and external e x c i t a t i o n ) were w e l l below the frequency o f fundamental l o n g i t u d i n a l v i b r a t i o n mode. Variety o f boundary conditions, damping, e x c i t a t i o n and i n i t i a l crookedness were considered i n t h e o r e t i c a l and experimental analyses o f t h i s problem. Coupled l o n g i t u d i n a l - f l e x u r a l and l o n g i t u d i n a l - t o r s i o n a l o s c i l l a t i o n s of a column were a l s o studied by several researchers. A short l i t e r a t u r e review o f the work done on t h i s subject i s presented here. Other a r t i c l e s and references can be found i n Journal o f Applied Mechanics, numerous v i b r a t i o n handbooks etc. One o f the f i r s t researchers to analyze the response o f a column subjected to a periodic a x i a l end e x c i t a t i o n was Beliaev (3). He reduced the equation of motion o f the column to the standard form o f Mathieu equation by neglecting the l o n g i t u d i n a l i n e r t i a o f a column element. The s t a b i l i t y a n a l y s i s o f Mathieu equation predicted i n s t a b i l i t i e s to occur when the e x c i t a t i o n frequency equals 2, 1, 2/3. 1/2, 2/5,....multiplies o f natural f l e x u r a l frequencies. In absence of damping i n s t a b i l i t i e s represent an unbounded growth o f amplitudes of o s c i l l a t i o n with time. The experimental v e r i f i c a t i o n o f Beliaev's theory was done by B o l o t i n (4), Somerset (5) and others. A n a l y t i c a l and experimental investigations have beer, performed study of l o n g i t u d i n a l in a i n e r t i a e f f e c t s upon the rarar.etrio response of a colnr.r under sr a x i a l load P (t) = P Iwap.owski and EVensen (6) + 0 P, cos ft. The analysis by Evan- 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 i n s t a b i l i t y region i n a three-d 1 r.ens 1 on? 1 taraneter sre.ee (P P, * ) , 0 Use of this region permits dejscrir-tie" of the conditions and responses associated with paranetrie i n s t a b i l i t y , and allows evaluation of the e f f e c t s of disturbances. A s t r a i g h t beam with f i x e d ends, excited by the periodic motion of i t s supporting base i n a d i r e c t i o n normal to the beam span, was investigated a n a l y t i c a l l y and experimentally by Tseng, W, Y., and Dugundji, J , (7 )• By using C-alerkin's method (one mode approximation) the governing p a r t i a l d i f f e r e n t i a l equation (not coupled to longitudinal well-known Duffing equation. the Duffing equation. motion) reduces to the The harmonic balance method i s applied to solve Besides the solution of simple harmonic motion, many other solutions, involving superharmonic motion and subharmonic motion are found experimentally and a n a l y t i c a l l y . The s t a b i l i t y problem i s analysed by solving a corresponding v a r i a t i o n a l H i l l - t y p e equation.( The column tested was (steel) 18 i n long 0.021*0,5 i n with b u i l t - i n ends. The fundamental f l e x u r a l frequency of the l i g h t l y (^= 0.0006) damped column was approximate! 20 Hz. The t e s t i n g was limited to 7. - 50 Hz frequency range) . 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 was a n a l y t i c a l l y and experimentally treated by Tso ( 8 ). His theory predicts th when a. s t r a i g h t column i s excited a x i a l l y at frequency twice the natural torsional frequency an unstable ter c l o n a l c s c i l l ^ t i c n r.?v take -Is cr. Ar.c*hr unstable tcrsirr.al o s c i l l a t i o n ray occur when th° excitation, frecuency i s eoual or c l r s ^ to the natural longitudinal frecverey of the cclvr.r. 5 Johnson (2) experimentally analyzed the o s c i l l a t i o n o f a prlsmatical column with b u i l t - i n ends. The e x c i t a t i o n was that o f constant l e v e l sinusoidal acceleration i n time, Imposed at one end o f the column, constant force was superimposed on t h i s e x c i t a t i o n . 64 l b s . He claimed to have detected the large amplitude coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n when the e x c i t a t i o n frequency was equal to the fundamental l o n g i t u d i n a l frequency, to one-half and also one t h i r d o f i t . Schneider ( l ) experimentally analysed the parametric response o f a prismatical column subjected to small l a t e r a l e x c i t a t i o n . The l a t e r a l e x c i t a t i o n of the form o f constant l e v e l sinusoidal a c c e l e r a t i o n was imposed at the center o f the column. Schneider detected large amplitude coupled f l e x u r a 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 a t l a t e r a l e x c i t a t i o n frequencies equal to the fundamental l o n g i t u d i n a l frequency. Similar i n s t a b i l i t i e s were also detected when e x c i t a t i o n frequency was equal to l / 2 and l / 3 o f fundamental l o n g i t u d i n a l frequency. Limitations of Investigation: In crrior to develop a theory predicting the behaviour of a column subjected tc sinusoidal a x i a l end excitation seme assumptions had to be made. Nearly straight e l a s t i c i s o t r o p i c prismatical column vrith r e r f e c t l y hinged ends i s 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 i s assumed tc be linearly proportional to tho distance from the central nlane. Tho effects cf shear and rotary inertia, are neglected. Also neglected i s surmort, surface and a i r damning. The excitation is assu-ed to he truly axia.1. Other limitations pertaining to the theory of nonlinear coupled l o n g i t u d i n a l 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 r e s t r i c t i o n s . 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. r o l l e d columns were used. Their Only two prismatical, mild steel, cold dimensions were 1/8 i n x3/8 i n x12 in approx. Strains were detected by strain gages and spatial shapes found by s p r i n k l i n g the column with s a l t . CHAPTER 2 NONLINEAR THEORY 7 F i g . 1. C o o r d i n a t e system o f t h e column 8 NONLINEAR THEORY Theoretical Considerations: In t h i s section, the transverse and a x i a l displacements which occur i n an i n i t i a l l y imperfect prismatic column, when one end o f the column Is subjected to a-periodic a x i a l e x c i t a t i o n , are considered. The theory consists o f two main parts; the reduction of the coupled p a r t i a l 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 s o l u t i o n o f the governing ordinary d i f f e r e n t i a l equation by the perturbation method. We s t a r t by considering the general equations. D i f f e r e n t i a l Equations Governing the Coupled Transverse and A x i a l Motion of a Prismatic Column; The equations governing the motion o f a rod are given (up to t h i r d order terms) by Mettler (9) and are given i n h i s a r t i c l e i n "Handbook o f Engineering Mechanics" by Fluegge (9). These equations are presented heret (1) Elw XXXX where; u = u(x,t) a x i a l 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 o f the column A cross section area o f the column 9 Z,w (F) Y,v F i g . 2. Coordinate system o f the column element (P) 10 E Young's modulus o f the column material I . minimum moment o f i n e r t i a o f the column cross section /S ...... c o e f f i c i e n t o f i n t e r n a l (material) damping p ( x , t ) . . . d i s t r i b u t e d load The column i s excited a x i a l l y , and t h i s e x c i t a t i o n then i s a source of steady state o s c i l l a t i o n s o f the column. Therefore, the steady state l o n g i t u d i n a l o s c i l l a t i o n may be expected to the n e g l i g i b l y or very weakly affected by an accompanying transverse o s c i l l a t i o n . (This deduction w i l l be eventually supported by the r e s u l t s o f experimental i n v e s t i g a t i o n which show that the only and verv small influence o f transverse o s c i l l a t i o n s on the l o n g i t u d i n a l o s c i l l a t i o n occurs when the column o s c i l l a t e s at any o f i t s natural f l e x u r a l frequencies). Thus, e s p e c i a l l y a t lower frequencies, that i s f o r e x c i t a t i o n frequencies lower than one h a l f o f the fundamental longitudinal frequency, the two terms: from equation ( l ) . -§w x and w^w^ can be neglected The same two terms w i l l be neglected from equation (2) since i n equation (2) they are o f t h i r d order. I t i s reasonable to assume that the influence o f t h i r d order terms on the o s c i l l a t i o n o f the column w i l l be much smaller than that o f f i r s t and second order terms. In addition,second order terras o f equation (2) are the terms coupling the longitudinal and transverse motion and should therefore be retained. The i n t e r n a l damping force /8w^ w i l l not be neglected as i t w i l l ultimately l i m i t the amplitudes o f coupled o s c i l l a t i o n . The e f f e c t o f a d i s t r i b u t e d load upon a coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n w i l l not be examined e i t h e r t h e o r e t i c a l l y or experimentally, so that p(x,t) may be taken as zero. 11 A f t e r making these assumptions and r e s t r i c t i o n s the two equations ( l ) and ( 2 ) reduce t o : -£AM xk + y*M^ (3) - 0 In the p a r t i c u l a r case considered here, one end of the column i s clamped, and the other end i s subjected to a s i n u s o i d a l acceleration o f magnitude S . Thus the boundary conditions on the a x i a l displacement are: u(o,t) = 0 (5) u ( l , t ) = S cos e t (6) The exact solution to equation ( 3 ) , s a t i s f y i n g boundary conditions ( 5 ) and ( 6 ) , was found i n chapter 3« section f l ) equation l ) equation ( 7 7 ) a s : S SI* ( C * ) sin ( ±4) X <HJ fr,\ 9- * 0 sLn(fl) 6 c frequency of a x i a l e x c i t a t i o n 0 velocity of longitudinal (8) 40 (9) 1 wave propagation i n the column 12 R e s t r i c t i o n (8) means that the s t a t i c case w i l l not be considered. R e s t r i c t i o n (9) implies i n f i n i t e amplitudes f o r s i n rr £ - 0 which l i k e the r e s u l t s of any l i n e a r theory r e s u l t s needs to be interpreted correctly} i t defines the resonant (large amplitude) frequencies f o r the column. S u b s t i t u t i o n o f equation ( 7 ) i n ( 4 ) y i e l d s : which a f t e r d i f f e r e n t i a t i o n , d i v i s i o n byj*. , and rearranging becomes: w/ tt ^ /* - /*• * EI y<- **** y<- EAS f 9- . /+ x - cos -cos(£x)-w„JcosM (ll) We now consider the transverse displacements. The boundary conditions on the transverse displacements are: w(0,t) = 0 , w(l,t) = 0 The s p e c i f i c a t i o n of transverse displacements column are not s u f f i c i e n t f o r the s o l u t i o n . physical conditions. (12) at the ends of the We need to s t i p u l a t e a d d i t i o n a l In t h i s case we consider the column to be f r e e l y hinged at the ends, so that the remaining boundary conditions are: tf « (0,t) = w (l,t) = 0 x x (13) x x The physical s i g n i f i c a n c e and p r a c t i c a l imposition of these end conditions are discussed l a t e r i n chapter 4 . The terms containing i n i t i a l crookedness were put on the r i g h t side of the equation ( l l ) and the r i g h t side then e s s e n t i a l l y represents the transverse e x c i t a t i o n f o r c i n g function. The i n i t i a l crookedness i n a c e r t a i n sense converts a x i a l e x c i t a t i o n into l a t e r a l e x c i t a t i o n . The transverse displacement w(x,t) i s to be found by s o l v i n g the p a r t i a l d i f f e r e n t i a l equation ( l l ) . There i s no standard procedure obtain a s o l u t i o n to such an equation. to Usually, a solution based on the knowledge about the behaviour o f the system the equation represents^, i s assumed. This w i l l be done here. Imagine a column subjected to the a x i a l e x c i t a t i o n of constant frequency 9 , Now look a t the column at a given instant of time t * . The s p a t i a l shape of a column i n t h i s instant o f time t * could be described by a Fourier series of period 1 . and (13) . and therefore1 Fourier sine s e r i e s s a t i s f i e s the boundary conditions i d e n t i c a l l y , w(x,t*) = Cf I f we now well. The series must s a t i s f y the boundary conditions ( 1 2 ) siny + G s i n — + ... + l e t the time change ,the magnitude o f C s I t i s l i k e l y that C constant C s dependent. 2 ( sin^y + (14) w i l l l i k e l y change as w i l l change d i f f e r e n t l y than C £ etc. Thus are independent of each other and no longer constant but time -It i s also reasonable to assume that C^s w i l l be affected by the frequency and amplitude of e x c i t a t i o n . Thus ( 1 4 ) should be rewritten 14 for the r e a l time ast w(x,t) = w ( t ) s i n w,(t), w ( t ), • • » . 2 • •» • i •• • + H n ( t )s i n >2^< + (15) are then functions describing the time v a r i a t i o n of amplitudes o f i n d i v i d u a l components o f the Fourier sine s e r i e s the f l e x u r a l shape o f the column at a l l time. has an i n f i n i t e number o f terms. representing T h e o r e t i c a l l y , s e r i e s (15) However, we may a n t i c i p a t e convergence and generally expect that the early terms i n the s e r i e s describe the motion adequately. I t w i l l be shown that f o r c e r t a i n e x c i t i n g frequencies, one term w.(t) w i l l be o f s u f f i c i e n t l y large amplitude as to completely dominate the motion o f the column. The s u b s t i t u t i o n o f equation (15) into equation ( l l ) y i e l d s the following equation1 2 EAS A &sin c (*£) 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). are clamped we can say: w(0) w(x) Since both ends of the column = w(£) = 0 (18) can be also represented by a Fourier s e r i e s . The Fourier sine s e r i e s of period 1 w i l l be chosen as i t i d e n t i c a l l y s a t i s f i e s the condition w(x) = a* s i n y - + a* s i n ~ - + .... + a* s i n ~ - + ... (18). (19) 16 where a* , a* ....a* ...are constants depending on the form o f 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 r i g h t side o f equation (17) into: •+ \c c 0 + (c ^T -T Z A Ycos-stn-j-J-j-a* C0S o e o q, z / £ + .- Jcos*t z (20) Comparison o f equations (17) and (20) shows that the both sides o f equation (17) contain the terms: + \V in^. c* s~ 7 C W ^ 7 / I - c - i ^ . - . Many workers replace t h i s term by simpler term ( i 1 T / l ) r e s t r i c t i o n that 9 i s much smaller than frequency o f the column. (21) L (iffx/l) with the , the fundamental l o n g i t u d i n a l We w i l l now show that t h i s s i m p l i f i c a t i o n can be made without l i m i t i n g 6 to be so small. expression £A sin (21) In p a r t i c u l a r , i t w i l l be shown that can be adequately represented by (iTT/l) s i n (iT x / l ) f o r 9 as large as ^1/2 . From chapter 3. (78) i t i s seen that: — s A f t e r s u b s t i t u t i n g eq. (22) i n expression (21), we need to show that: (22) Fig. 4. Approximation (23) , i - 6 19 The greatest error i n the approximation (23) w i l l occur f o r i = 1 as i t makes the f i r s t term on the l e f t hand side o f the (23) largest. To judge the accuracy o f the approximation (23) the approximation i s shown graphically i n Figures 3 and 4 , In producing these charts the length of the column 1 was assumed 12 i n . 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 a c t u a l l y experimentally analyzed. show the function to be approximated i . e . the l e f t side o f approximating function i . e . the r i g h t side o f (23). i s shown f o r 1500 Hz, 3. Fig. 2500 Hz and 3500 Hz values i = 1 applies to the case the (23) The approximation o f f o r c i n g frequencies «• , Fig. 4. i = 6. applies to the case A conclusion based on the two charts can be drawn. and the higher the mode number eq.(23) and and therefore shows the worst case that can a r i s e i n approximation (23). i s better the lower the frequency Both charts The approximation 0" , the smaller the coordinate (23) x , i Thus i t seems that the approximation (23) i s o f good q u a l i t y , a t l e a s t f o r low values o f f o r c i n g frequency with use o f equations & . With t h i s r e s t r i c t i o n imposed, and (l9)» and (20)-and (23) the governing differential equation ( l ? ) becomesj [ wt(i) si* jr- u £AS + wt(t) sen ^~ » X + f / T\ Z SC " . + + ffx + *n(*)sC» + /ZTT) 2 + (ff w^ t)siy> ^ -h ^ f t ) « > x ,^ / 2fa J + J x IT + - • ] cos ** ' (24) 20 The equation (24) can be uncoupled by equating the c o e f f i c i e n t s of l i k e sine terms. w V A set of d i f f e r e n t i a l equations i s thus obtained: *& $ ~ "• "' * M pl From l i n e a r theory (see chapter 3)formulas giving natural f l e x u r a l longitudinal (25) frequencies can be used to rewrite equation 2 The natural f l e x u r a l frequencies are: CO n and £I/hir\ —[~£~J - (26) The natural l o n g i t u d i n a l Substitution of (26) - frequencies are: i n (25) = y>f< L a f t e r s l i g h t rearrangement gives the d i f f e r e n t i a l equations: 2 « n ( t ) + £w..(t) + («- - ' S \ 2 cos 9t) w ( t ) = n n 2 f * a>s «t (27) n = 1,2,3, 21 Changing the dependent variable from t to z where St the r e l a t i o n s h i p s : 8t = 2z With these substitutions eq.(27) becomes: Introducing c, a , q , and A by: c = 4S#ln 'a* a Enables eq.(3l) to be written i n the form: » = .... Where prime denotes d i f f e r e n t i a t i o n 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) i s an inhomogenous, second order, ordinary d i f f e r e n t i a l equation with a variable c o e f f i c i e n t associated with the zeroth order term. The solution of t h i s equation w i l l consist of the complementary function plus a p a r t i c u l a r i n t e g r a l . The complementary function i s the s o l u t i o n 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 i n the system^the complementary function w i l l decay exponentially with time and r a p i d l y approach zero. I t w i l l be present i n the i n i t i a l stages of motion or as a decaying free v i b r a t i o n which follows a f t e r the cessation of the f o r c i n g term A ^ c o s 2z . Thus for the general d e s c r i p t i o n of the motion we need to consider only the p a r t i c u l a r solution of eq,(33). The p a r t i c u l a r solution, which i s a s o l u t i o n s a t i s f y i n g 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 f o r c i n g condition i s present. An exact p a r t i c u l a r s o l u t i o n to eq.(33) cannot be r e a d i l y found; however, there are numerous procedures for f i n d i n g approximate solutions a n a l y t i c a l l y . The perturbation method (12) equation as (33) w i l l be used as i t s a p p l i c a t i o n to such an 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 i n order to obtain a d d i t i o n a l terms i n the s e r i e s s o l u t i o n , thereby achieving b e t t e r accuracy. Unfortunately, as w i l l become obvious l a t e r , each successive a p p l i c a t i o n becomes s i g n i f i c a n t l y 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 F i g . 5« 10 12 14 16 18 Vibration mode number Magnitude o f parameter q n (n) 24 To assure that the solution converges we w i l l consider Since q^ depends on S, 9, and physical properties and dimensions of a column, the r e s t r i c t i o n variables. j^n|< I c l e a r l y l i m i t s the choice of these Thus, when the r e s u l t s predicted hereafter are applied to any s p e c i f i c test, one must ensure that the values of input are less than unity. where q | fn | -4. I q defined by experimental In F i g . 5« a plot of values of the parameter q^ , = q (n,9,S), i s given for several values of e x c i t a t i o n l e v e l amplitude S , was performed. Also shown i s the region where the experimental The region i s bounded by curve investigation E . Substitution of the assumed solution (34) i n the governing d i f f e r e n t i a l equation (33) y i e l d s the following equation* I f terms with l i k e powers of q^ are collected the r e s u l t to the i - th order i s : W % : <"» ( ) z + a : ; c w„ cz) + a w„ (z) : : 0 • h = A^cesZ*. 9 . : (36) These equations w i l l be solved successively. The equations are l i n e a r , of second order, inhomogenous and with constant c o e f f i c i e n t s . Exact p a r t i c u l a r solutions can be obtained i n a straightforward manner. 25 The s o l u t i o n o f eq.(36 i ) i s : ^-IT 1 = Ahcos2z + A^sinZz. ^ (37) S u b s t i t u t i n g eq.(37) into eq.(36 i i ) . t h e s o l u t i o n f o r w (z) i s found to be: , x a»-4 L . (at,-4)(at,-<t) fa«-4;4c + fa -/<;2c + - fa- 4 )\ o (2c)(4c) . H ~z 4c zj[(an'l<) - T . . ,, Ahsi»4z oX (38) + 16C*J When eq.(38) i s substituted i n e q . ( i i i ) o f eq.(36), the s o l u t i o n to " ^ ( z ) i s found as: w m . o T A» cos2z- , . 2(aH-4f <*) = —r 5 q 4c(a»-4') + —ir 3 h /jvo + /!„ 2 -(2cf(a»-i6) [(a„-4J L+4c z][(a„'/6f+ . , He*] , 4c^H-4^Q,-/6)-r2c)r4c; + Ca--4/f4c) + 1 r = [(a„-4) +4c 1][(an-/0 + I6c*] . „ A»Sih Zz _{ (an-4Xa»-KXaH-U)- (ZcXtcX** X) - (€cX4cXa»-4) - (2c X6cXa„-/6 ) [Can-4f+ 4c l][(a^f+ + r [(am-4) 1 Tr +4c x][(a„-/t) _ 2z a„ / r«*-4; + 4c j 4c 2(ah-4) (an-4)(a„-i6)- A 7 ^ • l6c 2][(a„-36) Z+36c*] 2 Tr + l6c 2][(aH-36) a n Ahs,*6z * 3Cc'J (39) Since t h i s extremely tedious proces and each successive s o l u t i o n 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 is 26 related to i n t e r n a l (material) damping and therefore i s very small i n magnitude. For certain values o f parameter a ^ as a matter o f fact f o r 2. 0, k, 16, 3 6 , . . . . (2m). •... a= M ^ o r w ( )' z h o W | ( )« •••••• • * i ( ) z W m = 1,2,...,some terms i n the s o l u t i o n w ^ y become extremely large, since z for these values o f the parameter (or a power o f ) occurs. The d i v i s i o n by. c the a .division by the small parameter h Let us c a l l such a term w „(z) may make solution o f a p a r t i c u l a r c ( l = large). w ^ ( z ) so large as to completely dominate w (z) . Rewriting eq.(34): n we see that f o r the case discussed here w j z ) can be c l o s e l y approximated ast w^ o , the second largest term o f eq.(l5), i s included here as well, to improve the accuracy o f approximation. Remembering that the parameter a^ was defined by eq.(32) as: from which the c r i t i c a l f o r c i n g frequencies can be calculated: a or: « (| = < o 0 = °» > k 1 6 » 3 6 » 6 4 £co„ = 0,. 2, 4, 6, 8 (41) Since only dynamic cases are considered here such that such physical systems f o r which examined. 2 co > 0 , case — cc - 0 $ " h 0 <c9"<oo , and need not be 27 2 eq.(4l) gives ©• = _ _ c o on 6 = ico (kz) tt n = 1, 2,3, . . . n i Thus f o r t h e v a l u e s o f f o r c i n g f r e q u e n c y solution of a particular of (^3) 1, 2, 3i ... = 9- a s d e f i n e d b y eq.(43) t h e w ^ ( z ) may be a p p r o x i m a t e d by t h e dominant terms 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 t h e f o r c i n g f r e q u e n c y i s any o f t h o s e d e f i n e d b y eq.(43) t h e f r e q u e n c y o f approximating w „£( ) z w ( z ) i s t h e same a s t h a t o f t h e a s s o c i a t e d n a t u r a l f r e q u e n c y . h F o r example when a = 16 n t h e l a r g e s t term w i l l be t h e l a s t term o f eq.(38) h a v i n g f r e q u e n c y |/a^Z , o r 4 z , w h i c h i s e q u i v a l e n t t o 2 6 t u s i n g eq.(30). ©• = -~— oj - -r c o f r e q u e n c y 29- . And for this w (z)* h a „ = 16 Thus, t h e case "* ( ) 0 z rt o c c u r r i n g when t h e e x c i t a t i o n frequency , s e e e q . ( 4 2 ) , w i l l make t h e column t o o s c i l l a t e w i t h B u t 28- = c o t h u s c o n f i r m i n g o u r e a r l i e r claim. h case: + l'*"*: ( ) z = c no h A 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 c h Q , c n | (44a) the amplitudes o f polynomials associated with cosine terms «e> » s s * i ^' ie a m p l i t u d e s o f p o l y n o m i a l s 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 b y means o f phase a n g l e s f { ho increased amplitudes s n o , h Wh(t) =r S*Q A si» (9* + %J h f , and by o f t h e s i n e terms, r e s u l t i n g i n a p p r o x i m a t i o n : Ci) * S* A sin (*-t + ?„) or: } * j s* A si» (29-t + ?J + f's* A„ s/» (coj n h + %,) - (44b) (44c) 28 and i n general when equation (43) a p p l i e s : *n (*) *o K S* A sen (9i s * ™» Ct) « ( & t + * Y»o) + fM) h K + j?S* A s i h H (<** * f»* sm (co * + H J (^) <f„jt) (44e) . (44f) We now wish to examine the response o f the column when the f o r c i n g frequency 9- i s not any o f those defined by eq.(43). term i s s o l u t i o n f o r a l l w ^( ) z h term o f eq.(37) having frequency ( n = 1» 2, 3» 2z, By f a r the l a r g e s t ) i s then the f i r s t or equivalent 9t . That i s , f o r these f o r c i n g 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 o f the column is» w(x,t) = w, ( t ) s i n ^ L + w ( t ) s i n £ p +\ + w „ (t) s i n ^ 2 (15). + ... and since each term i n the s e r i e s o s c i l l a t e s mainly with frequency 9t , each w • (t) can be c l o s e l y approximated w . ( t ) « c, where C £ 1 S 0 as» A . cos 6t (45a) the amplitude o f polynomial a s s o c i a t e d 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 y i e l d s ; Eq.(46) describes small l a t e r a l o s c i l l a t i o n s of nearly perfect column with hinged ends which i s subjected to a periodic a x i a l e x c i t a t i o n of the form u ( l , t ) = S cos 6t (or equivalent u ( l , t ) = :cos 0t) . I t shows that the response of the column to the e x c i t a t i o n i s of the same frequency as the e x c i t a t i o n frequency. frequency 9 This i s nearly true as long as the e x c i t a t i o n i s not any of those defined by eq.(43) and other r e s t r i c t i o n s , imposed e a r l i e r i n t h i s chapter also apply. At t h i s point we are i n a better p o s i t i o n to examine the t o t a l f l e x u r a l response of a column to the a x i a l e x c i t a t i o n given by eq.(5), when the frequencies of the e x c i t a t i o n are those defined by eq.(43). We have established that for these e x c i t a t i o n frequencies, one term of expansion eq.(l5) can be approximated as given by eq.(44e) or eq.(44g) r e s p e c t i v e l y . ,With t h i s exception a l l the other terms can be approximated, as given by eq. (45a), y i e l d i n g ; 48 So \ c o s 9 i 5,h T + " * ** c A cos " IT si + - U?<~" '" ^* + sns C 30 (*,*) * c , a A c ° s ®t S'» — + c, + C cos A zo A 2 + Cos frism =Y cos &t s i n * 'f* + s *A^s/h (Vt + * -4 4 + c )si» ^£ C + sin *t —" l f 2 f t d » « f i-2,3,.....' (48) i - 1 (49) Equations (47) and (48) can also be written as» A sin — Cn-oo <*—) + C,_ + A, ., „ _ Sin J f 7Tx 7, + -x - i o »• y JI <COS&t ) ^ I ^ ^ f w ^ + S ^ ; ^ ^ ~ , J& + n-l 2 ... f f i-2,3,... Eq.(43) (50) applies to eq's (47), (48), (49) and (50) . I f the excitation frequencies s a t i s f y eq.(43), then equations (47), (48), (49) and (50) describe small l a t e r a l o s c i l l a t i o n s 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; E f f e c t of i n t e r n a l (material) damping on column o s c i l l a t i o n needs to be examined i n somewhat greater d e t a i l , so that some reasonable t h e o r e t i c a l predictions can be made. Only l i n e a r d i f f e r e n t i a l equations w i l l be considered i n the d e r i v a t i o n of a formula r e l a t i n g material damping properties, (as given by most to the damping parameter c introduced i n t h i s chapter. researchers), I t i s l i k e l y that, were the nonlinear terms included i n the derivation, hereafter presented, the f i n a l r e l a t i o n s h i p would not change appreciably. Equation (4) of t h i s chapter, with nonlinear terms, neglected becomes an equation governing free f l e x u r a l v i b r a t i o n of a damped column; El w (51) 0 A column with hinged ends o s c i l l a t i n g i n i t s natural modes o f o s c i l l a t i o n can be well described by; W„(*,t) ~ w„cV; sin The unknown time v a r i a t i o n function nTx. w (t) M (52) can be determined by s o l v i n g the e q . ( 5 l ) with assumed s o l u t i o n (52) f o r w^(x,t) substituted i n i t . The substitution yields; 0 which implies: (53) 32 Recalling i d e n t i t y (86) d e f i n i n g the natural f l e x u r a l frequencies o f a column as: The eq.(5^) then becomes: + ~0 (55) A s o l u t i o n w i l l be sought of the form: w ct) = n C e . s C i £0 (5°)' Substitution of eq.(56) i n eq.(55) y i e l d s : C ls< +»>. / - o **-s from which: *A " /( W ~ W " (58) The c r i t i c a l damping /5 producing n o n - o s c i l l a t i n g s o l u t i o n o f eq.(56) can be c found from condition: and: ^ = ^ (60) A damping factor ^ expressing the r a t i o of actual material damping present i n the column to the c r i t i c a l damping can be introduced as: (61) 33 I f t h e damping f a c t o r i s g i v e n , o r o t h e r w i s e d e t e r m i n e d , t h e damping o f t h e column i s c a l c u l a t e d from eq.(6l) a s : /8 - f/% (62) Most m a t e r i a l damping d a t a i s g i v e n 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 t h e 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 . s > and sr) a r e g i v e n 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" l s l [ll] as: = -u (63) = f - ^ / AT (64) *f *X (65) R e c a l l i n g eq.(32) w h i c h d e f9i- n e s t h e damping parameter c : C = i- ^ U s i n g eq.(50), (52), (55), e q u a t i o n (32) becomes: (66) A - ^ ^ c o ^ C Since values o f ^ " = Z % ~ ¥ ( 6 7 ) a r e g i v e n by L a z a n /^"J under v a r i o u s c o n d i t i o n s , t h e e q u a t i o n o f t h e parameter c (32) f o r different materials o s c i l l a t i n g (67) i s a c o n v e n i e n t one f o r c a l c u l a t i o n . F o r t h e 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 t h e 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 p l a c e a t room t e m p e r a t u r e , t h e maximum s t r a i n s were l e s s t h a n 1 * 10 4 % , and t h e f r e q u e n c y range up t o 16000 Hz . 34 Under t h e s e c o n d i t i o n s v a l u e s f o r ^ * -7io < 7 6 The l a r g e s t amplitudes a column o s c i l l a t e s , a s g i v e n by Lazan < 2 * id the a m p l i t u d e s ( l a r g e s t s t r a i n s ) o f column o s c i l l a t i o n o c c u r when i n i t s n a t u r a l v i b r a t i o n modes. The a m p l i t u d e s o f t h a t g r e a t e r s t r a i n s i n v o l v e g r e a t e r damping, a b e t t e r guess o f v a l u e s o f ^ a s s o c i a t e d with a p a r t i c u l a r mode o f a column can be made. in Appendix B "? = "2 = s where i frequency subharmonically, ( s t r a i n s ) d e c r e a s e w i t h i n c r e a s i n g o r d e r o f a subharmonics. Taking into consideration presented , were: 4 column o s c i l l a t i o n a r e s m a l l e r when a column o s c i l l a t e s and In] I n t h e 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 the values o f ^ 43.8 x vibration to' 7 1.67 * ld i = 1,2. i = ? w i l l be t a k e n a s f o l l o w s : 3,4 i s t h e o r d e r o f a subharmonics i . e . t h e r a t i o o f a n a t u r a l to the f o r c i n g frequency. C - ^ Using (68) eq.(66) and eq.(68) a f o l l o w i n g r e l a t i o n s h i p i s o b t a i n e d : c h = I t i s o f course assumed here (69) 2ooi (and was shown b e f o r e , see eq.(49) ) t h a t 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 resembles essentially t h e shape o f the column o s c i l l a t i n g a t the n a t u r a l v i b r a t i o n mode from which the subharmonics i s d e r i v e d . c o m p r i s i n g the subharmonic I n o t h e r words, t h e l a r g e s t term o s c i l l a t i o n o f a column o c c u r r i n g when t h e 35 forcing is frequency associated with From t h e examples i s n-times s m a l l e r than a n a t u r a l the c o e f f i c i e n t presented sin = 1 1=0 i = 2 1 = 1 i = 3 1=2 i = 4 1=4 o f t h e column, • i n the Appendix B i frequency , we see t h a t f o r : Thus a conclusion may be made: l i i - 1 Examination o f the general s o l u t i o n reveals that eq.(70) i s t r u e indeed. (70) a s g i v e n b e e q ' s (34), (38) a n d (39) 36 Theoretical predictionsj F i r s t we wish to examine the response o f the column, when the e x c i t a t i o n frequencies are at, or are very close to values which are equal to natural f l e x u r a l frequencies o f the column. That i s when: n = 1,2,3 9 = <o„ The t h e o r e t i c a l predictions based on equations ( 3 2 ) , ( 3 4 ) , ( 3 ? ) , ( 4 8 ) , ( 5 0 ) 1) ares 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 following frequencies: a) the same frequency as the e x c i t a t i o n frequency (this frequency i s now equal to the natural 9 flexural frequency o f the column) b) frequencies J's which are i n t e g r a l multiples o f the e x c i t a t i o n frequency. 2) e j = 1,2,....) The amplitude o f 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 largest for 3) - J , n = 1 and w i l l decrease with increasing mode number n . Relative amplitudes o f the terms o f the same frequency as the e x c i t a t i o n frequency are very much larger than the amplitude o f terms o f other frequencies than 4) 9 , An apparent phase angle s h i f t i n time v a r i a t i o n o f the dominant term o f frequency © w i l l occur. The dominant term i s the l a s t term o f e q . ( 5 0 ) associated with a phase angle ^ w i l l change d r a s t i c a l l y as the f o r c i n g frequency the value 9 = OL> H 0 . This angle 8 passes thru 37 5) The dominant s p a t i a l shape w i l l be a s s o c i a t e d w i t h s i n and t h e r e f o r e (n - l ) n o d a l l i n e s c o u l d be d e t e c t e d . We w i l l look a t the r e s p o n s e o f the column, when the frequencies are a t , or are very 4- excitation c l o s e t o v a l u e s which a r e e q u a l to o f n a t u r a l f l e x u r a l f r e q u e n c i e s o f the column. S = ^ That i s , fractions wheni n = 1,2,3 O J „ i = 2,3,4 The t h e o r e t i c a l p r e d i c t i o n s based on a n a l y s i s o f e q u a t i o n s (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 laterally w i t h the following frequencies 1 a) t h e same f r e q u e n c y as the e x c i t a t i o n f r e q u e n c y 9 , b) the n a t u r a l f l e x u r a l f r e q u e n c y o f the column G J „ , which an i n t e g r a l m u l t i p l e o f the e x c i t a t i o n f r e q u e n c y . f .i S 2) The l a t e r a l (co =i^') n which a r e o t h e r i n t e g r a l m u l t i p l i e s o f e x c i t a t i o n frequency. (^ = j 9 ; j £ is the i,l) o s c i l l a t i o n o f the column w i l l be o f relatively l a r g e a m p l i t u d e s , and the magnitude o f these a m p l i t u d e s depends on the parameters a) i and n as follows: f o r a given value o f n , the l a r g e r the parameter the s m a l l e r w i l l be the a m p l i t u d e o f b) f o r a given value o f i i oscillation , the l a r g e the parameter the s m a l l e r w i l l be the a m p l i t u d e o f o s c i l l a t i o n . n , , 3) Eq's ( 4 0 ) , ( 4 4 e ) and ( 4 9 ) imply that the r e l a t i v e amplitudes o f the terms of the same frequency as the e x c i t a t i o n frequency and the term of the natural frequency of the -column O J N 8, , are much larger than the amplitudes o f terms o f other frequencies than 6 and C J . M 4) There may occur an apparent phase angle s h i f t between the time v a r i a t i o n o f the two dominant terms, one o f frequency other o f frequencyco„. 9 and the This i s apparent from e q . ( 4 9 ) where the l a s t term includes a phase angle frequency 6 VJ,jg • 1 passes thru neighbourhood S l i k e l y that as the f o r c i n g o f the value 6 = 4- cO n c the value o f ^ . . w i l l change d i f f e r e n t l y than the phase angle . The apparent phase angle s h i f t i n t h i s case would be r e l a t e d to the change i n ^ and as well. I f the second term o f e q , ( 4 9 ) the one including the phase angle i s very small as compared to the l a s t term o f e q . ( 4 9 ) , than the apparent phase angle s h i f t w i l l be equal to the change i n the phase angle 5) • The dominant s p a t i a l shape (not as strong as i n case be associated with s i n i = l) will , and therefore (n - l ) nodal l i n e s could be detected. F i n a l l y we w i l l consider the case when the f o r c i n g frequencies are not fractions of, or equal to the natural f l e x u r a l frequencies o f 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 o s c i l l a t e l a t e r a l l y with the same frequency as the e x c i t a t i o n frequency. Terms o s c i l l a t i n g a t m u l t i p l i e s o f 39 e x c i t a t i o n f r e q u e n c y may a l s o be p r e s e n t , b u t t h e i r a m p l i t u d e s w i l l be much s m a l l e r than those o f eq.(46). The a m p l i t u d e o f o s c i l l a t i o n w i l l depend on v a r i o u s parameters eq.(32). Thus f o r a l l parameters 6 except as given i n 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; t h e a m p l i t u d e o f l a t e r a l o s c i l l a t i o n s h o u l d 'peak' when t h e f o r c i n g frequencies equal to n a t u r a l l o n g i t u d i n a l column. T h i s c a n be seen from e x a m i n a t i o n o f eq's ( 3 ) and (4) a s f o l l o w s : N e g l e c t i n g t h e n o n l i n e a r terra the term r e l a t e d EA(u w ) t o i n i t i a l crookedness a f t e r some rearrangement tt frequencies o f the x x x from eq.(4) b u t k e e p i n g o f t h e column -EA(u "W ) F r e s u l t s i n eq,(7l); (71) •+ The term o f t h e r i g h t s i d e of.(71) can be i n t e r p r e t e d a s a f o r c i n g function. T h i s term longitudinal 'peaks' when t h e f o r c i n g f r e q u e n c y e q u a l s t o n a t u r a l f r e q u e n c i e s o f t h e column; see eq.(7) and (9). The s t e a d y s t a t e l a t e r a l o s c i l l a t i o n o f t h e column a r i s i n g from the f o r c i n g should t h e r e f o r e a l s o 'peak' a t t h e s e f r e q u e n c i e s . function CHAPTER 3 LINEAR THEORY 40 LINEAR THEORY Linear d i f f e r e n t i a l equations d e s c r i b i n g the motion o f the two columns when o s c i l l a t i n g i n l o n g i t u d i n a l , are considered i n t h i s chapter. flexural, or t o r s i o n a l v i b r a t i o n modes Resonant frequencies and s p a t i a l forms o f the columns o s c i l l a t i n g i n these natural v i b r a t i o n modes w i l l be determined. The influence o f rotary i n e r t i a and shear terms a f f e c t i n g the frequencies at which natural f l e x u r a l 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 o f the two columns derived from the l i n e a r d i f f e r e n t i a l equations was used to check and to complement the theory developed i n chapter 2 . Oscillations o f the two columns w i l l be analysed i n t h i s chapter i n the following order: l) Longitudinal o s c i l l a t i o n o f a prismatic column with one end fixed and the other end subjected to e x c i t a t i o n o f the form u ( l , t ) = (- S/0 )cos et 1 2a) Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with hinged ends. 2b) Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with hinged ends e f f e c t o f rotary i n e r t i a and shear terms. 3a) Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with b u i l t - i n ends. 3b) Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with b u i l t - i n ends e f f e c t o f rotary i n e r t i a and shear terms. 4) Torsional o s c i l l a t i o n o f a prismatic column with b u i l t - i n ends. 41 l) Longitudinal oscillations I n t h e 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 t h e 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 sections o f the bar . r e m a i n i n p l a n e and t h e p a r t i c l e s i n t h e s e c r o s s s e c t i o n s p e r f o r m motion i n an a x i a l d i r e c t i o n o f the bar. only Under t h e s e c o n d i t i o n s t h e d i f f e r e n t i a l e q u a t i o n o f m o t i o n o f a n element o f t h e b a r i s g i v e n by Timoshenko (13) : 2 2 3^c °° c z (72) 9x* _ (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 e q u a t i o n o f t h e form (74), f o r a s t e a d y s t a t e o s c i l l a t i o n i s soughtj u ( x , t ) = u ( x ) cos 6t (74) o 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 c o n s i d e r e d i n t h i s experiment are s u(0,t) = 0 =— (75) 6 =3 COS TTt (76) Where S i s t h e a m p l i t u d e 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 .2 From w h i c h : aM0(x) <&• 2 u ^0 ( x ) = A cos — c / v u (x, t ) = / ^ ' ^ c o s et { x + B s i n — x c , A cos — + B s i n — Cg v ) cos 8t 42 Using the boundary conditions (75) and (76) we u (0,t) = 0 = (A)' cos 8t =~'= S cos get: .*• .A = 0 et = - e (B s i n — B = - ) cos et .c ^ sin The s o l u t i o n to eq.(72) s a t i s f y i n g the boundary conditions (75) and (76) i s given by: u (x t) = - Sl t — S c o s 9-6 " 9- sih l From equation (77) the resonant fr*0 (77) & (large amplitude) frequencies of l o n g i t u d i n a l o s c i l l a t i o n can be found by noting that when: Sin—- The condition: —• sin— ^ M.C*,t) »*• 00 0 = 0 implies: — C •= (*>o) *7T 0 «• and using equation (73) we get: (78) =• — - y - §T„ From equation (69) the forcing frequencies 9 at which large amplitude l o n g i t u d i n a l o s c i l l a t i o n occurs can be calculated. natural frequency w i l l be denoted by . (79) Value of First longitudinal f£ L for the two columns are given i n tables VI-1 and VI-2 i n column ( l ) . In the formula (79) "the e f f e c t of a i r , material, and support damping and a p o s s i b i l i t y of f l e x u r a l o s c i l l a t i o n are not included. These e f f e c t s may 43 change the v a l u e o f r e s o n a n t f r e q u e n c i e s observed resonant frequencies 2a) somewhat. Indeed^the experimentally 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 values. F r e e 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 hinged endst By making' t h e u s u a l a s s u m p t i o n s o f l i n e a r t h e o r y , neglecting and a s s u m i n g t h a t t h e n e u t r a l a x i s o f the column undergoes no friction, stretching o r c o m p r e s s i o n 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 obtained. 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 m o t i o n i s g i v e n by Timoshenko (13) as» tfw 9t t + a 2 t f v m Q ( Z 4 = 0 w (l,t) = 0 2- ax2 ) 4 ojijfCj COSJI/3K ^Aco + C 4 si»k/3x + C s Cos/Zx. + C ( (82) Z ( 8 3 ) €1 no end 9X Q dx The boundary c o n d i t i o n s f o r h i n g e d ends end c o n d i t i o n s a r e j w (0,t) 8 1st C COS ' . 2 G e n e r a l s o l u t i o n t o (80) (xtt) = (C Styiw£ + displacements no end moments si»/2x) 44 Upon s u b s t i t u t i o n o f boundary conditions (84) i n eq.(82), a frequency equation i s obtained: -2 sinh/31 s i n / 5 l = 0 (85) 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 i n the theory o f chapter 2 which shares the same l i m i t a t i o n s and assumptions that were made i n d e r i v a t i o n o f e q . ( 8 6 ) . Using e q . ( 8 6 ) resonant frequencies were evaluated and are presented i n tables V I - 1 and VI-2 i n column ( l ) . 2b) Free f l e x u r a l o s c i l l a t i o n o f a prismatic column with hinged ends - e f f e c t o f r o t a r y i n e r t i a and shear terms: In order to obtain the values o f resonant f l e x u r a l frequencies more accurately the e f f e c t o f r o t a r y i n e r t i a 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 : a j? + w ~ 2 where: r r (^xGjzfi? I = — a r r^G = ° (87) f r °f 2 and + = — (88) A for a rectangular cross section = .833 45 Equation ( 8 ? ) and the hinged end c o n d i t i o n s w i l l be s a t i s f i e d by assuming the s o l u t i o n f o r w(x,t) to bet w = C s i n — cos c o t JL (89) Upon s u b s t i t u t i o n o f ( 8 9 ) i n e q . ( 8 7 J and u s i n g ( 8 8 ) and assuming t h a t the formula ( 9 0 ) g i v i n g values f o r n a t u r a l f l e x u r a l frequencies o f the column i s obtained: The values o f co f o r f i r s t 13 n a t u r a l modes are given i n t a b l e s 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 o f a p r i s m a t i c column with b u i l t - i n ends: The equations (80), (81), (82) and (83) a l s o apply f o r t h i s case. The boundary c o n d i t i o n s f o r b u i l t - i n end c o n d i t i o n s a r e : w (0 t) =0 f w ( l , t ) =0 lw<0,4) ^ (91) 0 no end slope x 3x no ends displacement = 0 Upon s u b s t i t u t i o n o f these boundary c o n d i t i o n s 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 frequencies for t h i s column were evaluated VI-1 and VI-2 and by using (94) resonant and are presented i n tables i n column (3) . I s. 3b) 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 - e f f e c t of rotary i n e r t i a and shear termsi The same l i n e a r d i f f e r e n t i a l equation as that i n section 2c) a p p l i e s i n t h i s case as wellt a ^? 9? where i ** r » *G U — J 9x z9S Assuming thatt (95) 4 A f„ = r (I + Sr) z w(x t) t 833 92. = . For a rectangular cross section A„ 9t a. = A Also define, q*G r - %(x) A s o l u t i o n of the form ( 9 7 ) * B = (97) cos cot w i l l be sought. The unknown function must s a t i s f y the b u i l t - i n ends end conditions ( 9 8 ) w (0,t) = 0 w (l,t)= 0 .'. .-. (96) ~ w (0) o = 0 w (l) = 0 0 . w (x) c 47 = o •• »° —-— (98) Substitution of assumed s o l u t i o n (97) i n e q . ( 9 5 ) y i e l d s : aV^ + w" + -.a**) w o = 0 (99) Or: (100) I f we l e t , . fc,-^T h - * K £ $ r ! ) (101) CL And using operator (D 2 D : (D -<).(D* - ^ ) w 0 + 2 bj> * k>3 ) W 0 » 0 = 0 (102) (103) Where, the roots <*T and e£j are found as: ( -(Z){*>(£)^i+ -[(itf- *.]7 - a*) Or by making use o f (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 c h a r a c t e r i s t i c roots: and -/c(? - /TcC P the primitive i s : > %(*) or: 9 C, e > -fi?X ^ -y (x) = C, coskfTfx + C^sink fZ?X a ^ T i + C3 cos/ia(j •+ Cze x I + C+sm/i*Ki + C cos JlJj'x 3 7 x + C* sin /UJ X (107) Application o f the boundary conditions (98) to e q . ( l 0 7 ) a f t e r some c a l c u l a t i o n r e s u l t s i n a condition: (cosh - cos /iz?*Xfi? cosh ~ ^cos/uj ^) fc * 1 - 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 s u i t a b l e f o r a computer use: Z - Z cosh fZ Z t cos/JZ?^ + ^i' ** - j„^ 1 3 1 S fZ?£ sim/iTj 7* = 0 (109) From equation ( 1 0 5 ) i t i s apparent that both roots </ and <J are functions ( 2 of physical constants related the column dimensions and material properties, and on the frequency O J . Such cos which s a t i s f y ( 1 0 9 ) are the resonant frequencies o f the column. The smallest w oscillations. being the fundamental resonant frequency o f free l a t e r a l F i r s t 13 resonant frequencies were found with an a i d o f computer and appear i n tables VI-1 and VI-2 i n column ( 4 ) . 49 4) Torsional o s c i l l a t i o n of a prismatic column with b u i l t - i n endst Free small o s c i l l a t i o n s of a prismatic column with rectangular cross section w i l l be considered i n t h i s section. are neglected. Damping and high order terms I t i s assumed that the cross sections of the column during t o r s i o n a l v i b r a t i o n remain plane and no other v i b r a t i o n mode i s occurring at the same time. The l i n e a r d i f f e r e n t i a l equation of motion, under these r e s t r i c t i o n s , i s given by Flugge (9) « (no) Where $ i s the t o r s i o n a l s t i f f n e s s of the p a r t i c u l a r cross section, and i s found from: § = k,i>h 3G b > h are cross section dimensions In our case: — k (ill) = ,263 = 3 and from "Advanced Mechanics o f Materials" by Hugh i n t h i s case. Equation ( 1 1 0 ) can be transformed intot 2 19 (112) where 1 (113) 50 From eq.(ll2) s o l u t i o n of and a frequency equation can be r e a d i l y obtained. eq.(ll2) General i s o f the formj the boundary conditions f o r fixed ends are: 6(0,t) = 0 <n5) 6(1,t) = 0 Substitution o f (114) and (115) in eq.(ll2) eventually y i e l d s a formula g i v i n g natural t o r s i o n a l frequencies: = —r- Ca (116) By means of t h i s formula f i r s t three natural frequencies were calculated and are presented i n tables VT-1 and VI-2 i n column ( l ) . CHAPTER 4 APPARATUS AND INSTRUMENTATION 51 APPARATUS AND INSTRUMENTATION In order t o t e s t the theory developed e x p e r i m e n t a l s e t u p had t o be c o n c e i v e d . i n chapter I I a s u i t a b l e The s e t u p had t o e n s u r e t h a t t h e boundary c o n d i t i o n s assumed f o r t h e column i n t h e t h e o r y would be a c c u r a t e l y reproduced under e x p e r i m e n t a l 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 t h e f l e x u r a l o s c i l l a t i o n mode o f a column w i t h hinged ends were: (la) zero displacements i n f l e x u r a l d i r e c t i o n a t both (lb) no moments a r n l i e d a t both ends ends To examine t h e i n f l u e n c e o f v a r i o u s boundary c o n d i t i o n s on t h e b e h a v i o u r o f a column s u b j e c t e d t o 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 was chosen t o be t e s t e d as w e l l . ends T h i s c h o i c e was made because t h e b u i l t - i n boundary c o n d i t i o n s a r e easy- t o a c h i e v e e x p e r i m e n t a l l y and t h e e x p e r i m e n t a l r e s u l t s f o r t h i s column c o u l d a l s o be compared t o t h o s e o f D. Johnson ( 2 ) who had t e s t e d a column o f t h e same d i m e n s i o n s 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 t h e f l e x u r a l o s c i l l a t i o n mode o f a column w i t h b u i l t - i n ends were: (2a) z e r o d i s p l a c e m e n t s a t b o t h ends (2b) z e r o s l o n e a t b o t h 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 t h e longitudinal oscillation mode o f t h e 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 o f t h e form Scos9t a t t h e o t h e r 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 c) boundary conditions choice of proper apparatus which would ensure constant a c c e l e r a t i o n l e v e l imposed at one end of the column (boundary condition No. d) 4) choice of transducers and associated e l e c t r o n i c s by means of which the column to a x i a l end e x c i t a t i o n could be observed. The s o l u t i o n to the above mentioned problems w i l l be discussed i n t h i s 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 l a , l b , 3 and 4 had to be s a t i s f i e d simultaneously. In a d d i t i o n to s a t i s f y i n g these boundary conditions the column had to exhibit a w e l l defined and unique fundamental longitudinal frequency i n order to make a check f o r subharmonics of t h i s v i b r a t i o n mode possible. Several columns were designed and tested. i t s threaded ends. Each column was clamped by means of This way of clamping of a column was chosen to achieve perfect a x i a l symetry of a l l moving parts of setup. In addition, i t also minimized a t o t a l 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 t h e i r inadequacies w i l l be pointed out. I t 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 c h a r a c t e r i s t i c s . 53 3f Fig. 6. Design #1 The design o f the f i r s t column i s s i m i l a r to the one commonly used by other researchers to test the f l e x u r a l response o f longer columns (up to 3 f t ) at very low a x i a l e x c i t a t i o n frequencies as compared to the fundamental longitudinal frequency o f the column. Due to stress concentration a t i t s ends where therefore much deformation took place, the column d i d not exhibit a unique fundamental frequency but rather several i l l - d e f i n e d ones. longitudinal Because o f "two-piece" construction o f the column a good a x i a l alignment was not possible. However, the column s a t i s f i e d quite well the boundary conditions o f hinged ends due to extremely small end moments opposing r o t a t i o n which resulted from a n e g l i g i b l e f r i c t i o n a t i t s ends. Fig. ?. Design #2 54 In the second design, a uniform cross section was retained even a t the ends of the column, which therefore resulted i n a unique fundamental l o n g i t u d i n a l frequency. However, the column s t i l l suffered from poor alignment inherent i n "two-piece" design, and because o f strong end moments opposing r o t a t i o n which resulted from high f r i c t i o n a t i t s ends, the column o s c i l l a t e d f l e x u r a l l y as though i t had b u i l t - i n ends. Fig. 8. ( ^ Design #3 The t h i r d design had a good a x i a l 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 o f small end moment opposing r o t a t i o n which was accomplished by the reduction of height o f cross-section at i t s ends. While the reduced height o f the cross-section a t the ends of the column helped i n an approximation o f hinged-ends end conditions, i t also resulted i n stress concentration i n the reduced cross-section which again caused a column not to exhibit a unique fundamental l o n g i t u d i n a l frequency. The fourth and the f i n a l design incorporated p o s i t i v e features of previous designs. I t i s o f "one-piece" construction and therefore a good a x i a l alignment o f the column with the rest of moving parts was assured. Ey 9.75 in 12.250 i n 12.375 i n F i g . 9. Column with hinged ends 56 r e d u c i n g t h e h e i g h t and i n c r e a s i n g t h e w i d t h o f t h e column c r o s s - s e c t i o n a t i t s ends a p p r o x i m a t e l y u n i f o r m a r e a o f c r o s s - s e c t i o n was r e t a i n e d . The c o n s t a n t a r e a o f c r o s s - s e c t i o n a l o n g t h e 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 F i n a l l y , the dimensions ends caused longitudinal f r e q u e n c y o f t h e column. o f m o d i f i e d c r o s s - s e c t i o n o f t h e column a t i t s t h e column t o be f l e x i b l e a t i t s ends t h u s .hinged-ends end c o n d i t i o n s . approximating The c l o s e n e s s o f t h i s a p p r o x i m a t i o n may by judged by comparing t h e n u m e r i c a l v a l u e s o f a x i a l e x c i t a t i o n f r e q u e n c i e s 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 o c c u r t o t h e t h e o r e t i c a l v a l u e s f o r a column w i t h b u i l t - i n . e n d s and a column w i t h h i n g e d ends. v a l u e s a r e g i v e n i n t a b l e VI-1. A l l these As can be seen, e s p e c i a l l y a t h i g h f r e q u e n c i e s o f a x i a l e x c i t a . t i o n a column behaves as though i t has h i n g e d ends r a t h e r than b u i l t - i n ends. T h i s column i s shown i n F i g . 9. D e s i g n o f t h e column w i t h b u i l t - i n ends; The d e s i g n o f t h e column w i t h b u i l t - i n ends was n o t d i f f i c u l t as most o f the d e s i g n problems were a.lready s o l v e d d u r i n g t h e d e s i g n o f t h e column w i t h h i n g e d ends. The same means o f c l a m p i n g t h e 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 k e e p i n g t h e c r o s s - s e c t i o n o f t h e column 1/8 i n by 3 / 8 i n c o n s t a n t a l o n g t h e e n t i r e l e n g t h o f t h e column. e x h i b i t e d a unique fundamental longitudinal The column f r e q u e n c y and t h e 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 o c c u r r e d v e r y c l o s e t o 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 . can be seen from t a b l e VI-2. The column i s shown i n F i g . 10. This Accelerometer 11.9 i n Moving end Cross section Fixed end 1 3 -Q i n * g i n 12.0 i n Fig. 10. Column Kith b u i l t - i n ends 58 Design o f the t e s t The bench: f i n a l d e s i g n o f the t e s t bench i s shown i n F i g . 11. accommodates one column a t a time and a column i s p l a c e d i n a p l a n e t o a l l o w t e s t i n g o f the column by s a l t s p r i n k l i n g . c r e a t i o n and . To The bench horizontal eliminate t r a n s m i s s i o n o f any m e c h a n i c a l n o i s e an a d j u s t a b l e t e f l o n b e a r i n g r a t h e r than b a l l b e a r i n g i s used, t h e r e a r e no l o o s e c o n n e c t i o n s anywhere, and r u b b e r p a d d i n g i n s u l a t e s motion. The boundary condition the bench from most o f the base (3) i s a c c o m p l i s h e d by t h r e a d i n g one o f the column i n a heavy s t e e l b l o c k , l a r g e i n e r t i a o f which end almost c o m p l e t e l y e l i m i n a t e s 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 direction. The t e s t bench a l s o had t o house the shaker and a l l o w easy mounting o f the two p r e s e n t e d no Vibration columns. Solution o f t h e s e and o t h e r minor problems difficulty. c o n t r o l apparatus: The purpose o f the v i b r a t i o n c o n t r o l a p p a r a t u s i s t o p r o v i d e 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 o f c o n s t a n t l e v e l a t the moving end o f the column, o r i n o t h e r words t o s a t i s f y the boundary flow diagram showing by means o f which the arrangement components i s shown i n F i g , 12. The o u t p u t e l e c t r o n i c s i g n a l c o r r e s p o n d i n g t o the form o f a x i a l e x c i t a t i o n i s t h e n a m p l i f i e d the a m p l i f i e d A signal f r e q u e n c y o f a x i a l e x c i t a t i o n i s set. by means o f the vibration exciter control. desired (4). o f e l e c t r o n i c and m e c h a n i c a l t h i s i s accomplished D e s i r e d a m p l i t u d e and condition i n a power a m p l i f i e r and e l e c t r o n i c s i g n a l i s c o n v e r t e d 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 o f e l e c t r o m a g n e t i c 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 s h a k e r i s monitored by an a c c e l e r o m e t e r which by the i s p l a c e d a t the moving end Fig. 11. Test bench FOUR BEAM OSCILLOSCOPE FREQUENCY COUNTER VOLTAGE AMPLIFIER LEVEL RECORDER POWER AMPLIFIER SPECTRUM ANALYSER VIBRATION EXCITER CONTROL ACCELEROMETER PREAMPLIFIER BAM ACCELEROMETER 4J- - ELECTRIC STRAIN GAGES SHAKER SPECIMEN 7777777777777777777777777. F i g . 12. Signal flow diagram BAND PASS FILTER AND AMPLIFIER 61 of the column. The a c c e l e r d m e t e r o u t p u t s i g n a l i s a m p l i f i e d and f e d 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 a m p l i t u d e and of frequency t h i s s i g n a l i s compared t o the d e s i r e d a m p l i t u d e and f r e q u e n c y . If 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 o u t p u t e l e c t r o n i c s i g n a l u n t i l t h e a c t u a l a x i a l has d e s i r e d a m p l i t u d e and f r e q u e n c y . excitation The a c c e l e r o m e t e r o u t p u t 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 s c r e e n and f e d i n the f r e q u e n c y c o u n t e r . Thus a t a g l a n c e t h e a m p l i t u d e , time v a r i a t i o n and f r e q u e n c y o f t h e a x i a l e x c i t a t i o n can be checked. T r a n s d u c e r s and a s s o c i a t e d Six electronics; s t r a i n gages (BLH SR-4 t y p e FAP-12-12) were a t t a c h e d t o t h e s u r f a c e o f each o f t h e two columns as shown i n F i g , 13. . Depending on t h e c h o i c e o f s t r a i n gages and t h e i r arrangement i n t h e Wheatstone b r i d g e , w h i c h i s a n i n t e g r a l p a r t o f BAM, either amplified longitudinal the o u t p u t s i g n a l o f BAM represented f l e x u r a l (normal) o r f l e x u r a l (in-plane) strain. The to c h o i c e o f s t r a i n gages and t h e i r arrangement i n t h e Wheatstone b r i d g e o b t a i n t h e s e t h r e e 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 o u t p u t s i g n a l o f BAM t h e s i g n a l was was suppressed. passed t h r u band p a s s f i l t e r and a m p l i f i e r where f u r t h e r a m p l i f i e d and low and h i g h f r e q u e n c y s i g n a l c o n t e n t Low and h i g h f r e q u e n c y s i g n a l c o n t e n t c o n t a i n e d some e l e c t r o n i c n o i s e and no i n f o r m a t i o n was of l o s t by i t s s u p p r e s s i o n . The use a band pass f i l t e r a l s o r e s u l t e d i n s h a r p e r 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 t o b e i n g d i s p l a y e d on t h e o s c i l l o s c o p e s c r e e n a f t e r p a s s i n g t h r u the band pass f i l t e r t h e s t r a i n gage s i g n a l was 62 also further amplified i n the voltage a m p l i f i e r and i t s RKS l e v e l recorded by a l e v e l recorder. The l e v e l recorder was used to record the v a r i a t i o n in amplified RKS values of p a r t i c u l a r s t r a i n as a function of a x i a l end e x c i t a t i o n frequency. Whenever i t was desired, a spectrum a n a l y s i s o f the s t r a i n gage s i g n a l was performed by making use of the spectrum analyser. The voltage a m p l i f i e r and the spectrum analyser were contained i n the single unit B&K 2107 , and a switch was used to choose between the two operational modes o f t h i s unit. 63 Z,w ( F ) Y,v For bending i n X-Z plane i ARI = *R3 (P) <=- ^R2 =-AR4 = ziR For bending i n X-Y plane t - ^ R l = -«*R2 = AR3 = A R4 =» <aR* -AR6 = For tension along X a x i s t F i g . 13. ^ R5 =AR <iRl = AR2 = AR3 =AR4 =/iR5 = 4R6 = ^R P o s i t i o n o f s t r a i n gages on a column 64 C C In general: For bending i n X-Z plane: D G For bending i n X-Y plane: D F i g . 14. Arrangement o f s t r a i n gages In the Wheatstone bridges CHAPTER 5 TEST PROCEDURE 65 TEST PROCEDURE Calibration; Before the actual t e s t i n g of a column was commenced i t was necessary to ascertain t h a t ' a l l e l e c t r o n i c and mechanical components that would be used during the t e s t i n g were functioning properly. Improper or poor ' performance of any o f the components, i f not detected immediately, might give r i s e to erroneous experimental r e s u l t s . To avoid t h i s , a l l components comprising the experimental setup were calibrated and tested according to manufacturers s p e c i f i c a t i o n s . A l l e l e c t r o n i c components were properly connected and turned to the standby p o s i t i o n f o r at least an hour before testing. Testing preliminaries: The amplitudes o f f l e x u r a l v i b r a t i o n o f the column depended on the frequency and amplitude o f the imposed sinusoidal a x i a l end e x c i t a t i o n . When the amplitude of a x i a l end e x c i t a t i o n was held constant over the entire frequency range considered here, the following happened: f o r some frequency range segments the response o f the column was strong and the e x c i t a t i o n power consumption high, while f o r some other frequency range segments the response o f the column was weak - almost undetectable, and the e x c i t a t i o n power consumption low. To make the response o f the column o f comparable and detectable strength at a l l a x i a l end e x c i t a t i o n frequencies, d i f f e r e n t amplitudes o f the e x c i t a t i o n were used over d i f f e r e n t frequency range segments. The number and size of the segments was chosen so that the e x c i t a t i o n power consumption remained approximately constant over the entire range o f a x i a l end e x c i t a t i o n frequencies considered i n t h i s experiment. This compromise 66 i s called an "approximated constant power spectra" and i s discussed i n somewhat greater d e t a i l i n Johnson's thesis ( 2 ) .To determine "approximate constant power spectra" each of the two columns had to be pre-tested and desired a c c e l e r a t i o n l e v e l s established. imposed In t h i s experiment the a c t u a l l e v e l s of end e x c i t a t i o n varied from 6g to 40g and no record of these i s presented here as. i t i s of secondary concern. Mounting of a column: The column with s t r a i n gages and two accelerometers attached was mounted i n the test setup i n the following way: (1) The shaker shaft and the bearing shaft were screwed very t i g h t l y together. The shaker shaft was then attached to the shaker by three a l i e n screws. (2) The t e f l o n bearing was adjusted u n t i l a t i g h t 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 s t e e l block so that when these two parts were screwed t i g h t l y together the f l a t side of the column was h o r i z o n t a l to make s a l t - s p r i n k l i n g test possible. (4) The moving end of the column was screwed into a hole i n the bearing shaft. Then a nut on the moving end of the column was screwed t i g h t l y against the face o f the hearing shaft to eliminate possible chatter between the end of the column and the bearing shaft. While t h i s was done the f l a t , wide side o f the column was held i n a 67 horizontal plane, (5) The remaining three heavy s t e e l 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 b o l t s . At t h i s point, the f l a t , wide side of the column lay i n a h o r i z o n t a l plane and the column was aligned and f i r m l y connected with the other parts of the setup. Testing; The column with s t r a i n gages and two accelerometers attached was mounted i n t e s t setup and the t e s t i n g was now ready to begin. The test procedure steps were: (1) The s t r a i n gage leads were connected to the BAM i n a configuration depending on the s t r a i n to be meassured, and the BAM on. After approximately was switched f i f t e e n minutes of warming up the s t a b i l i s e d and could be balanced and c a l i b r a t e d . BAM To minimize a pick-up of e l e c t r o n i c noise the s t r a i n gage leads were twisted together and wraped i n an aluminium f o i l . (2) Leads of the two accelerometers were connected preamplifier. to the accelerometer The preamplifier inputs were then- adjusted according to the combined voltage gain of each accelerometer and i t s lead. (3) The v i b r a t i o n exciter control unit was set to d e l i v e r a s i n u s o i d a l s i g n a l of desired l e v e l . CO Frequency scanning speed on the v i b r a t i o n e x c i t e r control unit was set and was manually changed during the experiment to remain at approximately 3-4 (5) cps. Proper paper speed on the l e v e l recorder was chosen. 6a (6) Compressor speed was chosen to assure s t a b i l i t y o f the feedback circuit, (7) The power a m p l i f i e r was switched from stand-by to on position. (8) The control unit was put i n i t s e x c i t a t i o n mode. (9) (10) - Proper a m p l i f i c a t i o n i n the frequency analyser was set. Attenuation and w r i t i n g speed on the l e v e l 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 f o r each frequency range segment and f o r d i f f e r e n t v i b r a t i o n modes o f the two columns. Boundaries of frequency range segments can be recognized from the s t r a i n vs. frequency records as d i s c o n t i n u i t i e s i n the s t r a i n vs. frequency curve. The tests were performed so that the frequency segments extended a few Hertz over t h e i r end points to avoid possible undertesting of the column . At p a r t i c u l a r frequencies o f i n t e r e s t a spectrum analysis of the s t r a i n s i g n a l was performed to i d e n t i f y i t s components. While t h i s was done the B&K was switched from i t s usual amplifier mode to analyser mode. 2107 The scanning o f the analysed s i g n a l was done manually. Additional testing; • To obtain more information about the response o f the column to a x i a l end e x c i t a t i o n frequency a d d i t i o n a l tests were carried out. S a l t s p r i n k l i n g was used to gather information about the s p a t i a l form of the column at certain frequencies of a x i a l end e x c i t a t i o n . At an a x i a l end e x c i t a t i o n frequency of i n t e r e s t fine c r y s t a l s a l t was sprinkled on the column and the a x i a l end e x c i t a t i o n frequency was varied 69 slowly u n t i l the nodal pattern created by s a l t c r y s t a l was sharpest. The a x i a l end e x c i t a t i o n frequency was recorded and a nodal pattern c l a s s i f i e d . This information supplemented the information about the response of the column to a x i a l end e x c i t a t i o n at a given point of a column obtained by means of s t r a i n gages. Vibration e x c i t e r control and the accelerometer preamplifier have 50 Hz to 10000 Hz . The r e s t of e l e c t r o n i c usable frequency range from components had even greater usable frequency range, thus a l l components of the setup were well suited for t e s t i n g of the column i n range of frequencies between 300 Hz and 10000 Hz . For t e s t i n g of the column up to 16000 Hz , which i s the upper usable range o f the shaker, an alternate v i b r a t i o n control was set up. were used. The Wavetek s i g n a l generator and a s p e c i a l B&K 4336 accelerometer This setup, however, had not a feedback c i r c u i t and consequently a constant l e v e l of s i n u s o i d a l a x i a l end e x c i t a t i o n could not be maintained. S t i l l , t h i s setup was s a t i s f a c t o r y f o r detection o f natural frequencies of 10000 Hz to 16000 Hz frequency region. o s c i l l a t i o n o f a column i n Total damping measurements: In order to gain an understanding of the e f f e c t of damping i t was desirable to find the a c t u a l value of damping which affected the motion of a column. Damping a r i s e s 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 t o t a l damping was also noted by Lazan ( l l ) . Surface damping due to attached s t r a i n gages, a i r damping and material damping also contributed to the t o t a l 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 t o t a l damping a f f e c t i n g the ?0 o s c i l l a t i o n o f the column v i b r a t i n g i n i t s fundamental f l e x u r a l v i b r a t i o n mode. Determination of a material damping from the t o t a l damping was not possible and an approximate handbook value had to be used. Undertesting and overtesting: To eliminate a p o s s i b i l i t y of undertesting or overtesting several steps were taken; 1) Each s t r a i n was measured independently a t 2 randomly chosen points of the column to eliminate the influence o f transducer l o c a t i o n . 2) Two accelerometers were detecting the imposed e x c i t a t i o n l e v e l 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 e a s i l y noted, 3) To eliminate high frequency e l e c t r o n i c noise, influence o f house current and any magnetic f i e l d influence a l l cables were shielded; a l l e l e c t r o n i c equipment was grounded to the o s c i l l o s c o p e 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 t e s t s were repeated several times to assure that a l l phenomenons detected were consistent and none overlooked. Photography: Photographs o f s t r a i n and a c c e l e r a t i o n waveforms t y p i c a l for a p a r t i c u l a r v i b r a t i o n mode were obtained. Photographs o f waveforms were taken d i r e c t l y from the oscilloscope i n normal t r i g g e r i n g mode (waveforms were stable enough) using a Pentax Spotmatic Camera with an f/stop o f 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 f i l m (developed i n KODAK D-19 contrast developer) was used. S t i l l photographs o f several nodal s a l t patterns were also taken. KODAK TRI-X 35mm f i l m (developed i n KODAK D-19 contrast developer)- was used. Loading o f a column by a constant a x i a l force: In order to study the e f f e c t o f a x i a l force on v i b r a t i o n , a column was loaded a x i a l l y by a constant force and tested. of two variable tension springs. The load was imposed by means The ends o f each spring were attached to fixed and to moving (excited) ends o f the column r e s p e c t i v e l y . could be varied from zero up to 200 l b s . chosen but other loads were t r i e d a l s o . The load 6 4 lbs constant a x i a l load was CHAPTER 6 RESULTS AND DISCUSSION 72 RESULTS AND DISCUSSION A theory predicting a x i a l end excitation the behaviour of a column subjected to was developed i n chapter I I . v a l i d i t y of t h e o r e t i c a l sinusoidal In order to test the predictions that were made, the response of a column was observed experimentally. S t r a i n vs. frequency records, and photographs of nodal l i n e patterns and s t r a i n waveforms were obtained for a hinged-end column. column and for a b u i l t - i n These experimental data and t h e i r analysis i s presented i n t h i s chapter. I d e n t i f i c a t i o n and analysis of s t r a i n vs. frequency records: Because the v a r i a t i o n of amplitude of column o s c i l l a t i o n as a function of the a x i a l end e x c i t a t i o n frequency i s of p a r t i c u l a r RMS-values of s t r a i n vs. the a x i a l end excitation produced f o r the two columns, and for interest, the frequency records were longitudinal f l e x u r a l , and in-plane modes. Numerous peaks appear on the records and i t i s of paramount importance to interpret t h e i r significance correctly. height and by the a x i a l end e x c i t a t i o n The peaks are characterised by t h e i r frequency at which they occur. Analysis of f l e x u r a l s t r a i n vs. frequency records - natural f l e x u r a l frequencies: Highest s t r a i n peaks occur when the a x i a l end excitation frequency i s equal to the natural frequency of f l e x u r a l o s c i l l a t i o n of the column. A column then o s c i l l a t e s f l e x u r a l l y with large amplitude and with the same frequency as the a x i a l end excitation frequency. The spectrum analysis of 73 the f l e x u r a l s t r a i n was carried out and no s t r a i n components o f other frequencies comprising f l e x u r a l s t r a i n waveform were observed. In general, the waveform representing time v a r i a t i o n was o f a x i a l end excitation always a pure sinusoid while the waveform o f the resultant s t r a i n was either a pure sinusoid or a complex curve r e s u l t i n g from addition o f two or more sinusoids. The waveform showing the f l e x u r a l s t r a i n when the column with hinged ends o s c i l l a t e s at i t s eighth natural frequency o f f l e x u r a l o s c i l l a t i o n i s shown here. In t h i s p a r t i c u l a r example (F8) the frequencies o f both waveforms are 4760 Hz. A x i a l end e x c i t a t i o n Flexural s t r a i n for a column o s c i l l a t i n g at natural frequencies of f l e x u r a l o s c i l l a t i o n modes Fig. 15. The O s c i l l a t i o n o f a column i n natural o s c i l l a t i o n modes s t r a i n peaks corresponding to o s c i l l a t i o n o f a column a t natural frequencies o f any o f i t s v i b r a t i o n modes are i d e n t i f i e d by a l e t t e r representing the p a r t i c u l a r v i b r a t i o n mode as follows: F Flexural (normal) o s c i l l a t i o n mode P flexural L (in-Plane) o s c i l l a t i o n mode Longitudinal ( a x i a l ) 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 l e t t e r representing a p a r t i c u l a r v i b r a t i o n mode i s followed by a number 74 which corresponds to n-th natural frequency. Thus F3 i s associated with the t h i r d natural frequency o f f l e x u r a l o s c i l l a t i o n mode etc. Analysis o f f l e x u r a l s t r a i n vs. frequency records - complex subharmonics: When the a x i a l end e x c i t a t i o n frequency i s equal to one half, one t h i r d , . . ..up to one eighth o f any o f natural frequencies o f e i t h e r f l e x u r a l (normal) or f l e x u r a l (in-plane) o s c i l l a t i o n mode, smaller s t r a i n peaks may appear on the record. Flexural o s c i l l a t i o n s a t these a x i a l end e x c i t a t i o n frequencies were i d e n t i f i e d as complex subharmonics. The spectrum analysis performed at these frequencies revealed that the f l e x u r a l s t r a i n 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 a x i a l end e x c i t a t i o n frequency c) frequencies which are i n t e g r a l multiples o f a x i a l end e x c i t a t i o n frequency Only two s t r a i n components a) and b) had comparable and s u f f i c i e n t l y large amplitudes to be detected v i s u a l l y from the oscilloscope display. (The pictures o f these s t r a i n waveforms are presented here) / V\/VV\/ A x i a l end e x c i t a t i o n Flexural s t r a i n for a column o s c i l l a t i n g a t second order subharmonics o f f l e x u r a l o s c i l l a t i o n modes. Fig. 16. O s c i l l a t i o n o f a column a t second order subharmonics 75 A x i a l end e x c i t a t i o n Flexural s t r a i n f o r a column o s c i l l a t i n g a t fourth order subharmonics o f f l e x u r a l oscillation modes F i g . 18. O s c i l l a t i o n o f a column a t fourth order subharmonics A x i a l end e x c i t a t i o n Flexural s t r a i n f o r a column o s c i l l a t i n g a t f i f t h order subharmonics of f l e x u r a l modes F i g . 19. O s c i l l a t i o n o f a column a t f i f t h order subharmonics oscillation 76 A x i a l end e x c i t a t i o n Flexural s t r a i n f o r a column o s c i l l a t i n g at seventh order subharmonics o f f l e x u r a l o s c i l l a t i o n modes Fig. 21. O s c i l l a t i o n o f a column a t seventh order subharmonics A x i a l end e x c i t a t i o n Flexural s t r a i n f o r a column o s c i l l a t i n g at -third order { w e a k ) subharmonics of f l e x u r a l o s c i l l a t i o n modes Fig. 22. O s c i l l a t i o n o f a column at i h i r d order subharmonics ( w e a k ) 77 Phase angle s h i f t s Tne two s t r a i n components a) and b) exhibit an apparent phase angle shift. As the a x i a l end e x c i t a t i o n 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 r a t i o o f a x i a l end excitation frequency to the natural frequency (which i s an i n t e g r a l multiple o f a x i a l end e x c i t a t i o n frequency) remains constant. The amplitude of the complex subharmonics reaches maximum a t 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 l i m i t s o f the band. An example showing the phase angle s h i f t i s given here f o r a complex subharmonics F?/k . A x i a l end e x c i t a t i o n (912 Hz) Flexural s t r a i n f o r a column o s c i l l a t i n g a t fourth order subharmonics of f l e x u r a l o s c i l l a t i o n mode F7 Fig. 23. O s c i l l a t i o n o f a column at F7/4 subharmonics - phase angle s h i f t 78 A x i a l end e x c i t a t i o n (920 Hz) Flexural s t r a i n f o r a column o s c i l l a t i n g a t fourth order subharmonics of f l e x u r a l oscillation mode F? Fig. 2k. A x i a l end e x c i t a t i o n (923 Hz) Flexural strain f o r a column o s c i l l a t i n g a t fourth order subharmonics o f f l e x u r a l oscillation mode F7 A x i a l end e x c i t a t i o n (928 Hz) Flexural s t r a i n f o r a column o s c i l l a t i n g at fourth order subharmonics o f f l e x u r a l mode F7 Fig. 26. oscillation 79 The complex subharmonics are i d e n t i f i e d by the l e t t e r o f a natural frequency with which the subharmonics i s associated and by the r a t i o (order) of the a x i a l end e x c i t a t i o n frequency to the associated natural frequency. Thus P5/2 i s associated with the second order subharmonics o f f i f t h natural frequency of f l e x u r a l (in-plane) o s c i l l a t i o n mode. A p a r t i c u l a r ordering o f f l e x u r a l s t r a i n peaks was noticed. I f for a c e r t a i n a x i a l end e x c i t a t i o n frequency (or a narrow band width o f ) a natural frequency o s c i l l a t i o n as well as a complex subharmonics o s c i l l a t i o n of another natural frequency were predicted to occur, i t would be the natural frequency o s c i l l a t i o n that was observed. S i m i l a r ordering o f complex subharmonics o s c i l l a t i o n s 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 l i k e manner, heights o f s t r a i n peaks associated with complex subharmonics of any given natural f l e x u r a l frequency would decrease r a p i d l y with increasing order o f the subharmonics. Often no notable s t r a i n peak was associated with a complex subharmonics a t a l l and i t s presence could only be inferred from the p a r t i c u l a r s t r a i n waveform corresponding to the complex subharmonics. The highest order o f complex subharmonics detected was eight. Undoubtedly even higher order complex subharmonics existed, however; because o f t h e i r extremely small amplitudes t h e i r detection was too d i f f i c u l t . Experimental and t h e o r e t i c a l values o f a x i a l end e x c i t a t i o n frequencies at which natural modes o f oscillation complex subharmonics,and snap-thru } phenomenons were observed are also given i n the tables V l - i and VI-2 . 80 To prevent c l u t t e r i n g o f s t r a i n vs. frequency records only some complex subharmonics were i d e n t i f i e d . The complete survey o f complex subharmonics detected v i s u a l l y fron the oscilloscope screen i s presented i n the tables VI-1 and VI-2 . Analysis of f l e x u r a l s t r a i n vs. frequency records - snap-thru phenomenons: When the a x i a l end e x c i t a t i o n frequency i s twice the natural flexural frequency a high s t r a i n peak i d e n t i f i e d as a snap-thru phenomenon may appear on the record. A column then o s c i l l a t e s f l e x u r a l l y with large amplitudes and at i t s natural frequency which i s equal to one h a l f o f frequency o f a x i a l end e x c i t a t i o n . A x i a l end e x c i t a t i o n Flexural s t r a i n f o r a column o s c i l l a t i n g i n a snap-thru o s c i l l a t i o n mode Fig. 27. O s c i l l a t i o n o f a column i n snap-thru o s c i l l a t i o n modes The i d e n t i f i c a t i o n o f snap-thru phenomenons i s consistent with i d e n t i f i c a t i o n of complex subharmonics. For example 2F2 would be associated with the snap-thru phenomenon derived from second natural frequency o f f l e x u r a l o s c i l l a t i o n mode. Only two snap-thru phenomenons appear on the f l e x u r a l s t r a i n vs. 81 frequency record f o r the column with hinged end. for the column with b u i l t - i n ends. None appears on the record Their existence seemed to depend mainly upon the amplitudes of a x i a l end e x c i t a t i o n and to a lesser degree on the number of natural frequency from which they are derived. For small amplitudes of a x i a l end e x c i t a t i o n the column o s c i l l a t e d with small amplitudes (no f l e x u r a l s t r a i n peak) with the same frequency as the a x i a l end excitation frequency. As the amplitude of a x i a l end excitation was increased a t r a n s i t i o n zone was encountered i n which two frequencies of f l e x u r a l o s c i l l a t i o n were present: end excitation frequency. The same frequency as frequency of a x i a l and the frequency equal to one h a l f of a x i a l end The l a t t e r i s the natural frequency. end excitation excitation As the amplitude of a x i a l increased more, the t r a n s i t i o n zone was passed, and the column then o s c i l l a t e d f l e x u r a l l y with natural frequency only. corresponds to this o s c i l l a t i o n . High s t r a i n peak Smaller s t r a i n peak would correspond to o s c i l l a t i o n i n t r a n s i t i o n zone. Since r e l a t i v e l y large amplitudes of a x i a l end e x c i t a t i o n low a x i a l end e x c i t a t i o n were imposed at frequencies a snap-thru phenomenon 2F2 occurred. The other snap-thru phenomenon 2F8 occurred at very high frequency and a s t r a i n peak i s not nearly as high as the f i r s t one. This i s because the amplitudes of a x i a l end excitation a t high frequencies are much smaller. It occurred, probably, because F8 seemed to be preferred mode of f l e x u r a l oscillation. 82 FRKXJENCT (Hzl FLEXURAL STRAIN (F) VS. FREQUENCY RECORD 750 KX» ^ 1500 FLEXURAL 3000 STRAIN (P) 4000 5000 6000 8000 VS. FREQUENCY RECORD CJOO FREQUENCY Fig. 28. STRAIN vs. F R E Q U E N C Y - COLUMN WiTH HINGED E N D S (Hz) 83 1500 2000 3000 sooo 1000 6000 7000 8000 9000 FREQUENCY (Hz) FLEXURAL STRAIN (F) VS FREQUENCY RECORD 1000 2 000 3500 5000 8000 BOX 1000 2000 3000 6000 CP noa 8000 9500 FREQUENCY (Hz) LONGITUDINAL STRAIN (L) FLEXURAL STRAIN (P) AND LONGITUDINAL VS. FREQUENCY RECORD 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 D i s c o n t i n u i t i e s o f s t r a i n vs. frequency curves; Because approximately constant power was to be supplied to the o s c 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 o f the frequency range considered here. This resulted i n d i s c o n t i n u i t i e s i n s t r a i n at these points, and these d i s c o n t i n u i t i e s appear on the records. Analysis of l o n g i t u d i n a l s t r a i n vs. frequency records; Because the a x i a l end excitation was o f the form u ( l , t ) = - (S/&- )cos Gt the amplitudes o f the e x c i t a t i o n are then|u(l,t)| = (s/6 ) where S i s a 2 constant (the magnitude o f imposed end acceleration) and 8 i s the a x i a l end e x c i t a t i o n frequency. Thus the amplitude o f a x i a l end e x c i t a t i o n decreases with square o f i t s frequency. Amplitudes o f l o n g i t u d i n a l d i r e c t l y r e l a t e d to the amplitudes of l o n g i t u d i n a l o s c i l l a t i o n should then decrease with square o f the a x i a l end excitation frequency. shown on the record of l o n g i t u d i n a l with b u i l t - i n ends. s t r a i n s which are This i s n i c e l y s t r a i n vs. frequency f o r the column In t h i s case amplitude S was held at constant 20 g's for the whole frequency range and monotonically decreasing curve o f longitudinal s t r a i n vs. frequency r e s u l t s . Because the amplitude S was changed several times i n the frequency range considered a d i f f e r e n t d i s t o r t e d curve representing l o n g i t u d i n a l s t r a i n vs. frequency resulted f o r the column with hinged ends. There i s only one high s t r a i n peak present on each o f these records. It occurs when the a x i a l end excitation frequency i s equal to the f i r s t longitudinal longitudinally natural frequency o f the column. A column o s c i l l a t e s with large amplitudes and with the same frequency as the a x i a l 85 end e x c i t a t i o n frequency. frequencies comprising observed. The spectrum analysis confirmed that no other l o n g i t u d i n a l o s c i l l a t i o n at these frequencies were Several very small peaks which show e f f e c t o f large amplitude f l e x u r a l o s c i l l a t i o n at natural f l e x u r a l frequencies upon the amplitudes of l o n g i t u d i n a l s t r a i n are present as well. No subharmonics o f the f i r s t l o n g i t u d i n a l natural frequency were observed. For neither o f the two columns a t e x c i t a t i o n frequency and one t h i r d o f the f i r s t natural equal to one half, l o n g i t u d i n a l frequency a s t r a i n peak occurs and the spectrum a n a l y s i s showed only one o s c i l l a t i o n frequency present - the same one as that o f a x i a l end e x c i t a t i o n . "Dummy" subharmonics o f the fundamental l o n g i t u d i n a l frequency might appear on the s t r a i n record i n f l e x u r a l (in-plane) s t r a i n and l o n g i t u d i n a l s t r a i n 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 o f existence o f in-plane f l e x u r a l o s c i l l a t i o n . record was created on purpose to show t h i s , and i s presented "Dummy" subharmonics o f the fundamental Such a s t r a i n here. l o n g i t u d i n a l frequency could also possibly a r i s e from not t r u l y s i n u s o i d a l a x i a l end e x c i t a t i o n . Such an e x c i t a t i o n would also contain components with periods being i n t e g r a l multiples o f the desired period. fundamental These components would then excite the l o n g i t u d i n a l mode when the "imposed" frequency one t h i r d , . . . . o f the fundamental i s one h a l f , l o n g i t u d i n a l frequency. Study o f o s c i l l a t i o n modes obtained by s p r i n k l i n g o f the column with salt: When fine table s a l t i s sprinkled on the surface o f o s c i l l a t i n g column, i t i s shaken o f f the surface except a t the points where the amplitudes o f o s c i l l a t i o n are zero or very small. This requirement i s s a t i s f i e d by the 8€ nodal the lines four several possible flexural presented to o f a column o s c i l l a t i n g here. v i b r a t i o n nodes. and torsional The n o d a l b e i n good a g r e e m e n t Fig. 31. Nodal second line pattern natural Theoretical oscillation line with a t any o f i t s n a t u r a l patterns torsional line modes a r e shown obtained the predicted occurring nodal frequencies patterns f o r i n t h e F i g . 35. experimentally ones and a r e a l s o when a c o l u m n oscillation mode of were shown oscillates found here. i n the Fig. 32. Nodal l i n e pattern occurring when a column o s c i l l a t e s i n the third natural t o r s i o n a l o s c i l l a t i o n mode fig. 33. Nodal l i n e pattern occurring when a column o s c i l l a t e s i n the f i f t h natural f l e x u r a l o s c i l l a t i o n mode Fig. 34 . Nodal l i n e pattern occurring when a column o s c i l l a t e s tenth natural f l e x u r a l o s c i l l a t i o n mode In the 88 Torsional mode #1 1 Torsional mode #2 1 t Torsional mode A 1 __— —— A #3 — ^^Sal^ Magnified view A - here A o f the column Flexural mode F i g . 35. i i #2 T h e o r e t i c a l nodal l i n e patterns 89 Intensity o f coupling between v i b r a t i o n modest A high s t r a i n peak on a l l three s t r a i n vs. frequency records appears when the a x i a l end e x c i t a t i o n frequency l o n g i t u d i n a l frequency o f the column. i s the same as the natural This experimental r e s u l t implies strong coupling between the v i b r a t i o n modes a t t h i s e x c i t a t i o n frequency. A very weak coupling also occurs between the two f l e x u r a l v i b r a t i o n modes and the l o n g i t u d i n a l v i b r a t i o n mode when the column o s c i l l a t e s at natural frequencies o f the f l e x u r a l (normal) and f l e x u r a l v i b r a t i o n modes. (in-plane) Very small s t r a i n peaks may appear on the l o n g i t u d i n a l s t r a i n vs. frequency record a t these a x i a l end e x c i t a t i o n frequencies, 0 Weak coupling between the two f l e x u r a l v i b r a t i o n modes occurs. This i s shown by the presence o f small s t r a i n peaks on f l e x u r a l (in-plane) s t r a i n vs. frequency record when the column o s c i l l a t e s a t any natural frequency o f f l e x u r a l (normal) v i b r a t i o n mode and vice-versa. When the a x i a l end e x c i t a t i o n frequency i s equal to the natural t o r s i o n a l frequency o f the column, the column o s c i l l a t e s t o r s i o n a l l y with large amplitudes. frequency records. V i r t u a l l y no s t r a i n peaks appear on any of the s t r a i n vs. This indicates that the i n t e n s i t y o f coupling between t o r s i o n a l and other v i b r a t i o n modes i s very small indeed. 90 Agreement between experimental r e s u l t s and t h e o r e t i c a l p r e d i c t i o n s i A theory was developed f o r a coupled l o n g i t u d i n a l - f l e x u r a l (normal) o s c i l l a t i o n i n a column with' hinged ends. The theory predictedj Occurrence o f j - coupled l o n g i t u d i n a l - f l e x u r a l (normal) o s c i l l a t i o n - natural f l e x u r a l (normal) v i b r a t i o n modes - natural l o n g i t u d i n a l v i b r a t i o n modes - complex subharmonics - strong coupling between l o n g i t u d i n a l and f l e x u r a l (normal) modes o f o s c i l l a t i o n Frequencies: - o f a x i a l end e x c i t a t i o n a t which the above described phenomena occur - which are present i n column o s c i l l a t i o n as a response to the a x i a l end e x c i t a t i o n Relative amplitudes: - o f coupled l o n g i t u d i n a l - f l e x u r a l (normal) o s c i l l a t i o n i n stable regions - o f natural f l e x u r a l (normal) o s c i l l a t i o n - o f natural l o n g i t u d i n a l oscillation - o f i n d i v i d u a l components comprising o s c i l l a t i o n associated with complex subharmonics - which are associated with strong coupled f l e x u r a l (normal) o s c i l l a t i o n longitudinal- 91 Experimental r e s u l t s as described i n t h i s chapter confirm v a l i d i t y 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 f l e x u r a l o s c i l l a t i o n and stable region o s c i l l a t i o n were not observed - the numerical values of predicted and observed frequencies might d i f f e r s l i g h t l y i n t h e i r magnitudes - two snap-thru phenomenons not predicted by theory occurred - weak coupling between f l e x u r a l (normal) and l o n g i t u d i n a l modes of o s c i l l a t i o n which occurred when a x i a l end e x c i t a t i o n frequency was equal to the natural l o n g i t u d i n a l frequency of the column was not predicted by the theory. No theory was developed o s c i l l a t i o n or f o r a coupled l o n g i t u d i n a l - f l e x u r a l (normal) l o n g i t u d i n a l - f l e x u r a l (in-plane) o s c i l l a t i o n f o r a column with b u i l t - i n ends. B u i l t - i n end conditions also apply to the f l e x u r a l (in-plane) o s c i l l a t i o n modes o f a column with hinged ends. Yet the experimental i n v e s t i g a t i o n shows that the same o s c i l l a t i o n phenomenons as those observed i n a column with hinged ends occur i n these cases as well. One exception being that no snap-thru phenomenons were observed f o r f l e x u r a l 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 l i m i t s the amplitudes of f l e x u r a l o s c i l l a t i o n i s a probable cause. Shear and rotary i n e r t i a were not included i n the theory f o r coupled 92 longitudinal flexural (normal) o s c i l l a t i o n s of a column. The influence of the two terms i s more pronounced at high frequencies where high modes of o s c i l l a t i o n occur. calculated from l i n e a r flexural The natural frequencies of o s c i l l a t i o n as theory with and without shear and rotay i n e r t i a f o r the two columns are given i n tables V l - i and VI-2 . As can be seen, when these two terms are included i n a l i n e a r d i f f e r e n t i a l equation for f l e x u r a l 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, and r o t a r y i n e r t i a a f f e c t I t may be concluded then that shear 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 l o n g i t u d i n a l and t o r s i o n a l o s c i l l a t i o n were calculated from formulas derived from l i n e a r d i f f e r e n t i a l equations. Shear and r o t a r y i n e r t i a terms were not included i n the d i f f e r e n t i a l equation f o r torsional oscillation. S t i l l the agreement between calculated and observed values of natural frequencies f o r these two o s c i l l a t i o n modes was quite good, Influence of a constant a x i a l loads A constant a x i a l load of up to 64 l b s , was imposed by means of springs i n addition to the periodic a x i a l end e x c i t a t i o n and the response of the column was observed. Except f o r a very s l i g h t change i n frequencies at which o s c i l l a t i o n phenomenon occurred no other e f f e c t was observed. Therefore, i n a l l other experimental investigation, with t h i s exception, no constant axial load was imposed. An e f f e c t behaviour was not investigated. of greater a x i a l loads on the column 93 S t r a i n magnitudes! In the range o f frequencies considered, magnitudes o f s t r a i n f o r a l l three v i b r a t i o n modes were usually greater than than 200/<-in/in . . ly*.-in/in and smaller S t r a i n vs. frequency records give only the r e l a t i v e amplitudes o f o s c i l l a t i o n . No attempt was made to obtain numerical values of s t r a i n s at a l l frequencies as t h i s was not needed and would be extremely tedious to do. Actual amplitudes o f strains for several a x i a l end e x c i t a t i o n frequencies are given i n Appendix G . Tables VI-1 and VI-2 The two tables, VI-1 and VI-2, contain numerical values o f various natural frequencies o f the two columns considered i n t h i s experiment. applies to the column with hinged Table VI-1 ends as shown i n F i g . 9. on page 55» Table VI-2 applies to the column with b u i l t - i n ends as shown i n F i g . 10. on page 57. The values o f natural frequencies o f these two columns calculated by using eq's. (86), (90), (94), and (109) are given i n columns ( l ) , (4) r e s p e c t i v e l y . Only one o f these four columns i s not shaded, (2), (3), and I t gives the most precise calculated values o f natural frequencies (shear and rotary i n e r t i a terms are included) o f a p a r t i c u l a r column f o r i t s a c t u a l end conditions. The other three columns are shaded and give the values o f natural frequencies calculated e i t h e r by less precise formula (shear and rotary i n e r t i a terms are not included), or by considering the other (not actual) boundary conditions. By comparing p a r t i c u l a r numerical values presented i n these tables the accuracy of approximation of actual end conditions o f the two columns can be examined. The e f f e c t of rotary i n e r t i a and shear terms can be studied as well. 94 Natural frequencies o f the column with hinged ends ( a l l frequencies i n Hz ) vibration mode calculated values of natural frequencies expt'ly observed natural freq's Fl - F2 365 F3 806 F4 1291 F5 1952 F6 2756 F? 3693 F8 4760 F9 6006 F10 hinged ends (1) b u i l t i n ends (2) n 75 300 300 (3) (4) 170 170 - 46B 2 : 920 675 1201 experimentally observed harmonics of natural frequencies 216- - 1513 h : 1198 1869 2271 '<} vj 2686 " 3172" 3146 3679 3648 4223 Mao 4753 5424 I sm- €?S1 5998 C??5 7360 75*J7 7380 Fll 8902 9034 8898 F12 10410 iOCll 10547 F13 12460 1263? 12325 i 1/3 i 1/3 i i 1/3 i 1/5 2 i 1/3 i 1/5 1/6 1/7 * (1/5) 5123 1/3 1/7 9923 1/3 **(. 1/3 - 6ll - Pl 484 2?0 P2 1674 10?! P3 3213 P4 4806 P5 7750 P6 IOO53 Tl 3005 2904 T2 5995 5808 T3 8707 8712 - LI 8215 8103 - 608 3237 3301 5284 4I&5 6735 : i 1667 7773 8150 9&9f i 9063 ! a m N 10663 1/3 ( i ) i/3 (i) (i/3) 1/5 (1/3) 1/3 Table VI-1 (The best calculated values are i n the unshaded columns). 95 Natural frequencies o f the rfcolamn? with b u i l t i n ends ( a l l frequencies i n Hz ) vibration mode expt'ly observed natural freq's calculated values of natural frequencies hinged ends b u i l t i n ends (3) (2) (1) experimentally , observed harmonics of natural frequencies (4) - FI 170 80 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 F6 3330 m $ 3361 3338 i 1/3 1/5 F7 4410 3S64 ¥*?5 442? 11/3 i F8 5630 T033 5?*& 5675 i 1 2 • 5092 172 80 F9 6950 6Ui4 6350 ?I6Q 7062 F10 8620 7956 7313 b??l 8595 Fll 10269 9^7 F12 12000 tti6o F13 14212 10521 - 10272 l<'r3& 12020 •* 1/3 i 1/3 1/3 1 1/3 1/6 - PI 502 234 530 P2 1458 959 1453 i P3 2845 Zlik 2106 sa?5 2826 1/3 1/5 P4 4500 37>6 3?4? 4753 4620 P5 6440 5006 5-^62 1 2 6809 P6 9310 6^42 9360 ± 1/3 1/3 1/5 Tl 3023 2989 T2 6035 5979 T3 9039 8968 - Ll 8286 8272 - 1/3 - - IPillllll Table VI-2 (The best calculated values are i n the unshaded columns). CHAPTER 7 SUMMARY AND CONCLUSIONS 96 SUMMARY Summary of t h e o r e t i c a l i n v e s t i g a t i o n : A t h e o r e t i c a l investigation of the behaviour of an i n i t i a l l y imperfect column with hinged ends subjected to periodic, a x i a l , end e x c i t a t i o n was made. Two coupled, p a r t i a l , 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 t h e o r e t i c a l analysis was to s i m p l i f y 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 l o n g i t u d i n a l of v a r i a b l e separable type. motion to be Then an exact s o l u t i o n of the f i r s t d i f f e r e n t i a l equation governing the l o n g i t u d i n a l motion of the column, was found. integration constants of t h i s solution were determined Unknown from consideration of the end conditions imposed i n t h i s problem, The second p a r t i a l d i f f e r e n t i a l equation govex-ning the f l e x u r a l motion was inhomogenous, of second order, and with variable c o e f f i c i e n t s i n x and t . At t h i s point i t became necessary to guess a s o l u t i o n d e s c r i b i n g the f l e x u r a l motion of the column. A Fourier sine series i n unknown time variable c o e f f i c i e n t s , was x, with period 1, with chosen as a possible form of a variable separable type of the solution, as i t i d e n t i c a l l y s a t i s f i e s the hinged end conditions. With t h i s assumed solution, the second p a r t i a l d i f f e r e n t i a l equation became an ordinary, inhomogenous l i n e a r , second order d i f f e r e n t i a l equation with variable c o e f f i c i e n t s i n x and t . By r e s t r i c t i n g the e x c i t a t i o n frequencies to values smaller than one h a l f of fundamental l o n g i t u d i n a l frequency of the column, a l l the c o e f f i c i e n t s i n assumed a l i k e , and were factored out. x could be Equating of the terms associated 97 with l i k e c o e f f i c i e n t s i n 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 o f the governing d i f f e r e n t i a l equations was an ordinary inhomogenous l i n e a r second order d i f f e r e n t i a l equation with one variable c o e f f i c i e n t i n t associated with zeroth order term. Each equation governed of one c o e f f i c i e n t sine s e r i e s . o f the Fourier the time v a r i a t i o n An approximate particular (forced) s o l u t i o n o f the n-th governing d i f f e r e n t i a l equation was obtained by perturbation method, accuracy o f which depends on magnitudes of excitation, physical properties and dimensions of the column. Now the approximately solved time variable c o e f f i c i e n t s were substituted i n the assumed Fourier s ine series giving the approximate the f l e x u r a l motion o f the column. of the l o n g i t u d i n a l solution governing This s o l u t i o n together with the solution motion give a complete d e s c r i p t i o n o f the combined l o n g i t u d i n a l - f l e x u r a l motion o f the column. Within the r e s t r i c t i o n s applicable to the theory the theory provides quantitative as well as q u a l i t a t i v e information about the forced coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n s o f a column. Some t h e o r e t i c a l predictions are b r i e f l y discussed here. Large amplitude f l e x u r a l o s c i l l a t i o n s w i l l occur when the e x c i t a t i o n frequency equals the natural f l e x u r a l frequencies. The column o s c i l l a t e s f l e x u r a l l y with the same frequency as the e x c i t a t i o n a x i a l frequency. The theory also predicts large amplitude f l e x u r a l o s c i l l a t i o n s when the e x c i t a t i o n frequency equals to l/2, l/3, 1/4,.,,.l/n... natural f l e x u r a l frequencies. of the The column then o s c i l l a t e s f l e x u r a l l y with the same frequency as the a x i a l e x c i t a t i o n frequency and also with the natural f l e x u r a l frequency (that i s the frequency equal to 2, 3 ( 4 ....n...times 98 the e x c i t a t i o n frequency). These types o f o s c i l l a t i o n are i d e n t i f i e d as subharmonics o f natural f l e x u r a l frequencies. Resonant, large amplitude l o n g i t u d i n a l o s c i l l a t i o n s w i l l occur only when the e x c i t a t i o n frequency equals to natural l o n g i t u d i n a l frequencies. The column then o s c i l l a t e s l o n g i t u d i n a l l y with the same frequency as the a x i a l e x c i t a t i o n frequency. Summary o f experimental i n v e s t i g a t i o n : An experimental i n v e s t i g a t i o n o f the behaviour o f an i n i t i a l l y imperfect column with hinged ends subjected to periodic, a x i a l , end e x c i t a t i o n was done to v e r i f y the t h e o r e t i c a l predictions, and to complement the theory i n the e x c i t a t i o n frequency region where the theory i s not applicable. of existence o f subharmonics o f the fundamental l o n g i t u d i n a l A possibility v i b r a t i o n mode was checked. The experimental i n v e s t i g a t i o n o f 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, a x i a l end e x c i t a t i o n was also done. I t s purpose was to examine the e f f e c t o f d i f f e r e n t boundary conditions on the response o f a column. The r e s u l t s o f the experimental i n v e s t i g a t i o n o f the two columns were compared. The experimental i n v e s t i g a t i o n was accomplished i n e s s e n t i a l l y two steps; the design of the two columns and t e s t i n g setup, and the actual t e s t i n g of the two columns. The design o f the two columns, one with hinged ends and one with b u i l t - i n ends, involved a s o l u t i o n o f a close approximation o f desired end conditions, and a s a t i s f a c t o r y means o f clamping the column. A t e s t i n g 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 o f the column are accurately approximated, the a x i a l alignment i s excellent, and a generated or transmitted mechanical noise i s minimum. S t r a i n gages and s a l t s p r i n k l i n g were chosen as a means o f monitoring the behaviour o f the two columns. S t r a i n gages were found quite s a t i s f a c t o r y i n a l l aspects. Their output, a f t e r processing, yielded various s t r a i n vs. frequency records at one point of a column. At some values of e x c i t a t i o n frequencies, a spectrum analysis o f the s t r a i n gage s i g n a l was done to determine the i n d i v i d u a l components comprising the s t r a i n a t a point o f a column. From the oscilloscope display of the s i g n a l a continuous v a r i a t i o n o f s t r a i n with e x c i t a t i o n frequency was observed. Pictures o f t y p i c a l s t r a i n waveforms were taken d i r e c t l y o f the oscilloscope screen. S a l t s p r i n k l i n g was Used to check the s p a t i a l form o f a column o s c i l l a t i n g f l e x u r a l l y with large amplitudes, which occurred for p a r t i c u l a r values of e x c i t a t i o n frequencies. supplied The Salt s p r i n k l i n g also complemented the information by the s t r a i n gages. r e s u l t s o f the experimental investigation o f the column with hinged ends agreed very c l o s e l y with t h e o r e t i c a l predictions. Natural f l e x u r a l o s c i l l a t i o n modes, and t h e i r subharmonics occurred at, or close to values o f predicted e x c i t a t i o n frequencies. frequencies comprised the f l e x u r a l subharmonics. magnitudes o f subharmonics was noted. Components of d i f f e r e n t A p a r t i c u l a r ordering of No subharmonics were observed when the f o r c i n g frequencies were greater than one h a l f o f the fundamental longitudinal frequency. Large amplitude coupled l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n occurred only when the e x c i t a t i o n frequency was equal to the fundamental l o n g i t u d i n a l of the column. frequency 100 No other large amplitude coupled detected. l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n was No subharmonics o f fundamental l o n g i t u d i n a l 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, i n addition to those t h e o r e t i c a l l y predicted, were observed. Two snap-thru phenomenons were observed when the e x c i t a t i o n frequency was equal twice the frequency o f the second and the eighth natural f l e x u r a l o s c i l l a t i o n mode respectively. The column then o s c i l l a t e d f l e x u r a l l y with frequency equal to one h a l f o f the e x c i t a t i o n frequency. may also be i d e n t i f i e d as superharmonics Snap-thru phenomenons o f natural f l e x u r a l o s c i l l a t i o n modes. In-plane f l e x u r a l o s c i l l a t i o n also occurred. In-plane natural f l e x u r a l o s c i l l a t i o n modes and t h e i r subharmonics were observed. The subharmonics consisted o f components o f d i f f e r e n t frequencies. T o r s i o n a l natural o s c i l l a t i o n modes were excited when the e x c i t a t i o n frequencies were equal to natural t o r s i o n a l frequencies o f the column.Very weak coupled f l e x u r a l - l o n g i t u d i n a l o s c i l l a t i o n occurred when the e x c i t a t i o n frequencies were equal to natural f l e x u r a l frequencies. The experimental i n v e s t i g a t i o n o f the column with b u i l t - i n ends produced b a s i c a l l y the same r e s u l t s as the i n v e s t i g a t i o n o f the column with hinged ends. The only differences were that no snap-thru phenomenons were observed, and the subharmonics o f natural f l e x u r a l frequencies occurred also a t frequencies higher than one h a l f o f the fundamental l o n g i t u d i n a l the fundamental l o n g i t u d i n a l frequency (but lower than frequency). Thus i t would seem, that the behaviour of the column i s not much d i f f e r e n t 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 i n a column with b u i l t - i n ends probably due to i t s increased flexural stiffness. 101 Suggestions f o r future research: An exact solution o f the p a r t i a l d i f f e r e n t i a l equations governing the coupled l o n g i t u d i n a l - f l e x u r a l motion o f a column i s desirable. I f an exact solution cannot be obtained, then perhaps a solution predicting the existence o f superharmonics i n addition to coupled and weak coupled f l e x u r a ] - l o n g i t u d i n a l oscillations l o n g i t u d i n a l - f l e x u r a l o s c i l l a t i o n s already considered here, would be desirable. Again, some r e s t r i c t i o n s on values o f parameters of the system would probably have to be imposed. An approximate solution o f the p a r t i a l d i f f e r e n t i a l equations governing coupled l o n g i t u d i n a l - t o r s i o n a l motion o f a column could be found. The procedure f o r obtaining the s o l u t i o n might be s i m i l a r to the one presented here f o r a case of coupled l o n g i t u d i n a l - f l e x u r a l motion of a column. Solutions meant to describe the motion o f a column' at very high frequencies should include e f f e c t s of rotary i n e r t i a and shear to make the solutions s u f f i c i e n t l y accurate. Boundary conditions other than hinged ends or b u i l t - i n ends might be o f i n t e r e s t . In p a r t i c u l a r a fixed-free ends end conditions should be investigated as r e s u l t s o f such an investigation could be of p r a c t i c a l value. A turbine blade mounted on a s l i g h t l y bent or an imperfect shaft turning a t very high speed, could be thought o f as an a x i a l l y excited column with one end free and the other b u i l t - i n . As i t i s often desirable i n the engineering practice to l i m i t amplitudes of o s c i l l a t i o n s i n order to reduce energy transmission, or just to lower a noise generation, various means of damping o f p a r a m e t r i c a l l y induced o s c i l l a t i o n s of a column should be investigated. Damping could be induced, f o r example, by coating the surface o f a column with a v i s c o e l a s t i c or e l a s t i c - v i s c o e l a s t i c material, 102 Some other means o f monitoring the response o f a specimen than s t r a i n gages could be t r i e d , such as a f o t o n i c sensor, l i q u i d c r y s t a l coatings, or the .use o f holography could be considered. Conclusion; The nonlinear theory o f chapter 2 has been developed f o r a column with hinged ends,and the column was assumed to have some i n i t i a l crookedness. The t h e o r e t i c a l considerations included a small material ( i n t e r n a l ) damping as well. The theory was eventually l i m i t e d to the e x c i t a t i o n frequencies as large as one h a l f o f the fundamental l o n g i t u d i n a l frequency, which i s where t h i s theory i s unlike the theories developed by some other researchers. These researchers have usually r e s t r i c t e d t h e i r theories to values o f f o r c i n g frequencies much smaller than the fundamental l o n g i t u d i n a l frequency o f the column. Furthermore, most o f the researchers have considered o s c i l l a t i o n s of undamped, i n i t i a l l y s t r a i g h t columns. Their theories usually lead to Mathieu equation p r e d i c t i n g i n s t a b i l i t i e s o f column o s c i l l a t i o n to occur a t c e r t a i n e x c i t a t i o n frequencies. o The theory o f chapter 2 a l s o gives more d e t a i l e d information about the behaviour o f the column than many o f other theories. For a known set o f systems parameters, the theory p r e d i c t s approximate amplitudes of components of various frequencies comprising the f l e x u r a l o s c i l l a t i o n o f the column. phase angle s h i f t between these components i s predicted too. The 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 N o n l i n e a r Coupled V i b r a t i o n s o f Bars and P l a t e s " , M.A.Sc. T h e s i s , 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., " E x p e r i m e n t a l I n v e s t i g a t i o n of N o n l i n e a r Coupled V i b r a t i o n s of Columns", M.A.So. T h e s i s , 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 S u b j e c t t o V a r i a b l e Longitudinal F o r c e s " , C o l l e c t i o n o f E n g i n e e r i n g 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), L e n i n g r a d , P u t , 1924 4. E o l o t i n , T.V., "Dynamic S t a b i l i t y o f 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 P a r a m e t r i c I n s t a b i l i t y of Columns", P r o c e e d i n g s of the Second S o u t h e a s t e r n , 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 o f I n e r t i a Upon the P a r a m e t r i c Response o f P l a s t i c Columns", J o u r n a l of A p p l i e d M e c h a n i c s , March, 1966, pp. l 4 l - 148 ?. Tseng, W.Y., and D u g u n d j i , J . , " N o n l i n e a r V i b r a t i o n s o f a Beam Under Harmonic E x c i t a t i o n " , J o u r n a l of Apr-lied. M e c h a n i c s , June, 1970, PP. 292 - 29.7 8. Tso, W.K., " P a r a m e t r i c 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 J o u r n a l o f A p p l i e d M e c h a n i c s , March, 1968, pp. 13 - 19 9, M e t t l e r , E,, Dynamic B u c k l i n g , "Handbook o f E n g i n e e r i n g M e c h a n i c s " , 1st Ed., F l u e g g e , W,, e d i t o r , M c G r a w - H i l l Book Company, I n c . , 1962 Excitation", 10. Schmidt, G., " C o u p l i n g c f F l e x u r a l and Longitudinal Resonances of Columns", ( t r a n s l a t e d , from German), Archiwum M e c h a n i k i S t o s o w a n e j , B e r l i n , 1 965, pp. 233 - 24? 11. Lazan, B„J., "Damping o f 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 P r e s s , London, 1968, F i g . 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 N o n l i n e a r A n a l y s i s " , M c G r a w - H i l l Bock Company, I n c . , 1958, p. 1?2 13. Timoshenko, S,, " V i b r a t i o n Problems i n E n g i n e e r i n g " , D. Van N o s t r a n d Company, I n c . , 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 , M c G r a w - H i l l Bock Company, Inc.,. 1961 , p. 7 - 16 15• F o r d , H,, "Advanced Mechanics of M a t e r i a l s " , Longmans, Green and Co. L t d . , London, E n g l a n d , 1969, p. 380 ' ' APPENDICES 105 APPENDIX A L i s t o f equipment: B&K Automatic Vibration E x c i t e r Control Type 1025 capable o f providing desired peak to peak displacement, v e l o c i t y or a c c e l e r a t i o n . In t h i s case B&K 1025 was used to provide peak to peak a c c e l e r a t i o n o f 6 to 60 g's from approximately 100 Hz to 10 kHz. with time. The frequency range i s scanned logarithmically 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 s e n s i t i v i t y o f 17.8 mV/g up to 10 kHz. This accelerometer was stud mounted on the moving end o f the column and was used with B&K Accelerometer Preamplifier Type 2622. B&K Accelerometer Type 4336 having constant voltage s e n s i t i v i t y o f 4.08 raV/g up to 45 kHz. This accelerometer was stud mounted on the moving end o f the column as B&K 4335 "but on a x i a l l y opposed side o f i t . This accelerometer was used i n conjunction with B&K Accelerometer Preamplifier Type 2616. B&K Accelerometer Preamplifier Type 26l6 i s a battery driven unit designed to be used with d i f f e r e n t types o f B&K accelerometers. In t h i s experiment i t was used with B&K Accelerometer Type 4336 and i n frequency range 100 Hz to 16 kHz. B&K Accelerometer Preamplifier Type 2622 which has a b u i l t - i n s e n s i t i v i t y attenuator, which, when c o r r e c t l y adjusted f o r a given accelerometer, provides an'output voltage s i g n a l o f 10 mV/g as sensed by the accelerometer. This unit was used together with B&K Automatic Vibration 106 E x c i t e r Control Type 1025 and with B&K Accelerometer Type 4335. 2250 MB Power Amplifier made by MB E l e c t r o n i c s , has frequency range 5 Hz to 20 kHz. I t was used together with EA 1500 E x c i t e r and with B&K Automatic Vibration E x c i t e r Control Type 1025. EA 1500 E x c i t e r (shaker) made by MB E l e c t r o n i c s . I t has 50 lbs.force r a t i n g , frequency range 5 Hz to 20 kHz and possible a c c e l e r a t i o n l e v e l over 100 g's. EA 1500 E x c i t e r was used together with 2250 M3 Power Amplifier. B&K Frequency Analyser Type 2107 consists o f an input amplifier, a number o f weighting networks, a s e l e c t i v e a m p l i f i e r section, and an output amplifier. Usable frequency range for t h i s unit was 5 Hz to 10 kHz. BAM-1 Bridge Amplifier and Meter, and B&K Level Recorder Type 2305 were used together with t h i s u n i t . B&K Level Recorder Type 2305 having a wide range o f paper and w r i t i n g speeds and f a c i l i t i e s enabling p l o t t i n g o f RMS, DC or peak to peak values. Frequency response was well i n excess o f 10 kHz range. RMS o f output o f B&K Frequency Analyser Type 2107 was recorded by t h i s u n i t . BAM-1 Bridge Amplifier and Meter measures and amplifies dynamic signals over a frequency range o f 0-20 kHz, SR-4 s t r a i n gages were inputs to t h i s unit and the output was delivered to KH 335 Variable Filter. 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 D i g i t a l Time and Frequency Meter Type 1151-A made by General Radio Company was used to measure frequencies with ± 1 Hz accuracy. Inputs to t h i s unit were either from B&K Automatic Vibration E x c i t e r 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 i n 0 to 10 kHz range and to substitute i t i n 10 kHz to 16 kHz range o f t e s t i n g . I t can provide t r i a n g l e , square or sinusoidal (actually used) wave s i g n a l o f 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 d i s p l a y up to four signals simultaneously. I t s inputs were any o f the components mentioned here, EA-06-125BT-120 E l e c t r i c S t r a i n Gages made by Micro-Measurements used to detect s t r a i n s i n the surface o f the column. ohms resistance and 2,11 gage factor. were These gages had 120 BAM-1 Bridge Amplifier and Meter was used to process s t r a i n 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 t e s t i n g and i s not o f s u f f i c i e n t importance to be l i s t e d here. 108 APPENDIX B Examples of a p p l i c a t i o n o f the nonlinear theory o f Chapter 2 i- Solution o f eq.(33) was obtained f o r i = 1,2.3, and 4 by s o l v i n g eq.(36) up to s i x t h power o f a parameter q n for several t y p i c a l values o f the mode number n . Eq.(36) was solved exactly, but i n the end only several dominant terms of each frequency were retained, as by inspection many other terms were o f n e g l i g i b l e magnitude. a c t u a l l y experimentally A l l o f the parameters used here are i d e n t i c a l to those imposed. Thus, the examples presented here, may be also used to check the v a l i d i t y o f t h e o r e t i c a l predictions experimentally. Analysis o f these solutions shows that the amplitude o f second term o f eq.(50) and of the t h i r d term o f eq.(49) are very much affected by the magnitude of i n t e r n a l (material) damping and by other parameters of the system. I t also suggests a p o s s i b i l i t y o f a phase angle change as the f o r c i n g frequency passes thru values given by eq.(43). Phase angle change a r i s e s from the presence o f sine and cosine terms, the magnitudes o f which change d r a s t i c a l l y and d i f f e r e n t l y as the f o r c i n g frequency i s varied. Addition o f sine and cosine terms o f varying magnitudes r e s u l t s i n varying phase angles. 109 Example # 1t 1=1 n = 5 (a = 4) Eq.(36) H a s solved to s i x t h power of the parameter q^for a = 4 . This h s o l u t i o n i s presented here with only dominant terms of each frequency retained: W (z) 5 f f • o 2z A + I ~~ V 2 •+ Sin Y + f I ~ Sin /52 c 46 ISO c v •+ 1 4 4Z0 coo c The case A 7640 c* 1 .0 sm 2z. 1 , fA 3 + — fA tA J • S cos ' 62 \ si" 6-z io z n = 5 »i = 8s*00c <f- A cos <f z 1 /24oeaOcz I24OOOOC2- / - .... \ ) 7 H.O.T.'S + 1 +•-••) 2 s sin J j, + IS4ooc^ sir, Sz. ~ • \ + — —- cos 2TL -t . -. - / . 5SZO c* 230 z $ + °r A JA -i - 0 Cz V 76&C / f A / -« J. 46 ISO c V cos 2-z. • * A ¥ A 4fc C ( represents the f i f t h natural f l e x u r a l o s c i l l a t i o n mode f o r 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 C = $ 2S -.T 2y i = 2(63.8 *id )l * /. J * /0 7 110 With these parameters the s o l u t i o n f o r w (-z") becomes» $ =c 2.03 A - (-S'/S *io' )A 7 + 33 0C0As!r>2z - / 3 A sin 4z •+ 2oZOAcos2z, + S30 A sin 2 z. -h ^7.4 A cos - 0.677 A cos 4z. - O.I43 A sin 4z •*• (i.63 x id ) A Sin 6z (/-OS * io' ) A sin fz ... s s ~ ( S-.04- * /d )AcoS 7 - + ( S,S£ < /Q~ ) Acos. 62. •+ (2,2 */a~ J A sin € Z - + --- 8z 9 •+ (4.S x io"' ) A £t'*> /Oz - ... .... 2 + And t h i s s o l u t i o n can be well approximated ast W (z) 5 ^ 33 530 A si'n 2* s In order to obtain the other terms 20S7 A cos 2z £ of eq.(l5)» eq.(45a) w i l l be used to solve Wj-Ct), J = 1,2,.. f o r <Sr = 1877 Hz : A' w. J °-1L = °" W/. « - Z -0.53 W, « =s _ —— .J cos 2, z 4-a.. cas - 2. a, 6 = 8.25 A* « 4,2S —— 4 _„ Where cos 2 Z 3. S3 It cos 2*. /I7 • A. J - ; ~ — C Sr s,n(t?j 3 0 / / a J 4 = Zz. A 1 J J Ill Substitution of w-(z), j = 1,2,...7 fr [ - — T T ^ — = - -JL. ;„ -*L S.-fS . 4Tx S7TX A i Sin -f ZOS7 Ar S,r> -*• 4 2.36 A A + 33530 Ac Sl'm S/T A 6 4.2S , 6fo A; Sin — — •+• il. 3 . 7Tx . , . S/>i — — i- - - • / cos pfc A j sin frt A i . JL. , 21* 3.4! J? 2 S X [ 'A in eq.(l5) yields: or: W(x.t) ^ ~ /- ' ' ~ 3.334 I A -f (2oS7Xa.S)a<. •> A If a (39 S30X , j = 1,2,,.. 2 — A 2.S3 s i n - 7 - •+ — *• 4-2S s) v ' s n S 3.48 — £ sin •+ 1 £ sun ~ + 2.36 sty, — z - + ' " u. 3 <• A A<=°s.&-£ J '"P* are within three orders of magnitude t h i s equation can be approximated as: STTx JL" 9 " ~ ~ A * . ST* , A 39/aoo 4.J Sin — — cos (<&•£ A - \ 87y The above equation shows that when the a x i a l e x c i t a t i o n frequency i s equal t o natural f l e x u r a l frequency of the column, the amplitude of one of the components comprising the s p a t i a l form of the column i s greatly amplified. I t also shows that as the f o r c i n g frequency passes thru natural f l e x u r a l 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 f o r a = 16 . This s o l u t i o n i s presented here with only dominant terms of each frequency retained: W + / A ( - r ( = (z) c + -J- •, r xA cos 2z H- - r — ; — f I r i - s/Wz V 4SC s i w + r 4A . , sin 2z. • c o s CT. + ll£ ooo c - V 46 too c 1" —T ^ f ^ + 10 Z ) ) y -- • ) ' " - J ' 5 7 00 0 0 c S/>> - - - 5 , ^ 4 2 •+ 3 2 *z - \ cos- -Zz. 2 + --• • J • •+ - - 2 ^ — cos4z. + — ^ SSOOc : 174 ooo c -I- / - -1 s/n£z. ^ 36o c + f -I + - •+ 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 f l e x u r a l o s c i l l a t i o n mode f o r the column with hinged ends. experimentally imposed werej S = 10 g calculated 0- = 938 Hz (experimentally observed n =5 a = 16 c$ = 2<yt = Z(63.8*I0 )2 7 » 2.6*/o' S & = 976 Hz) Parameters 113 With these parameters the s o l u t i o n f o r w- (z) becomes: W (z) 5 + ^'2. ^ +' K.fc 4r cos ~ 4 2£..2 <4 s/*7 4z A • Z2.6. sm 6z z - 3.77 A -f A cos 4z cos 6z 70.4- s/'»i ^ + It.3 + •+ 2 A ^~ cos 2z. 42,2 4 sin z + 354 no 00a Sm IO -z_ •+.-.. coo 392 And t h i s s o l u t i o n can be well approximated as: ("z ) =s 5 - c o s ^ t 5 4 - ~ + -—£/«-?z 14.2 • 6•z . Sin + —4! - S.77 Acos4z. -f J7. i " / 4 s / « 4 z 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,.. f o r 9- = 9?6 Hz : - W. J " 4 - */ Cos 2: CO s 2 z. vv. = - 56 * « 3 1.3 A+ A a 7 - C/.7 Co 2z cos 2z. COS 2z e 23. 3 IV, s cos •Sz. 2-56 * 33.3 -f Ay 57.7 114 Substitution of j • r IV (x,t) II z 3.98 t' •+ *s A, „ • 2Fx , ' S7l~x T . As + - S s —— - sm — S7.7 JL £ diL_ £ 3 L /„ SLH s 2 9 + t + 1 .... / cos Pr />•? —-— sm 2 sin 9-t S'n J4-Z Sin **-cos29>t - 9.77A A A • 3Tx , < • ir* sm ——- •+ sm 1.6 * S.S6 A. i A sm — r — 3.S6 * A^_ ^ V-L ££* ^ or: / * 3.9? -A — — — sv>i — -f Z 3 + UL- 70A 3.S6 Z S />7 1 / Z — <t A sin c Z m + — — i 1 <r - — <3.3 z cos I t t 1,3 S7.7 s m 3.SC X * —± + . . . . I A cos&t s€ J + STx . X or: , . . * + + *. • €T% . 7/7* N 1.23 a, sm - j - -i- . 8S a 3.Z3 aA s sin — Co* (*t 7 ~3Z J a sin ~ + 7 r• • j aA u •+ C o S aj^ <r/W ^ ft + ccrfw * - The above equation shows that when the a x i a l e x c i t a t i o n frequency i s equal to one h a l f of natural f l e x u r a l frequency of the column the column responds with at l e a s t two d i f f e r e n t frequencies. Namely, i t responds with the same frequency as the e x c i t a t i o n frequency and also with the natural frequency of the column. The l a t t e r frequency i s now twice the e x c i t a t i o n frequency. 115 The amplitude of the sin.—— column i s greatly amplified. component comprising the s p a t i a l form of the This term also undergoes a large phase s h i f t as the f o r c i n g frequency passes thru natural f l e x u r a l frequency. Very small term having frequency three times the a x i a l e x c i t a t i o n frequency i s also present. 116 #3: Example i = 3 n = 10 (a = 36) Eq.(36) was solved t o fourth power o f the parameter f o r a = 36 . q This s o l u t i o n i s presented here with only dominant terms o f each frequency retained: W ( - ^ i + Cz) ~ -+ ( -— / rA ' ., z s/ A + " #Z~ 4z. •+••• y \ 4 sim 6z , 1, ~ — sfto <? A - ... - — • • -) cos 6z + - —^ * ( J 107 600c V The case -+ . .. + cos 4z •+ I •+ ( 2-2. T • • •+ — — — s m 2 z + • •• ) ) -+ ( A -- j sin lOz. •+ • • • ) \ >c,oooc. ' Z £es n = 10 , i = 3 represents the t h i r d subhamonics o f the tenth 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 mode f o r the column with hinged ends. Parameters experimentally imposed were: S = 40 g 0- = 2502 Hz calculated (experimentally observed 9- = 2453 Hz) n = 10 a = 36 <1 = (3 x 10 )S —r- = O * io )(iS44o) s - / * id 6 c = Z^i = 2(Mf*id )3 T With these parameters the s o l u t i o n f o r w A <o { ' A So 000 (z) becomes: A A 32, 90 000 r =s /. 3 *io s A cos 4z. + - - • •+ 9.6 A JSJoo *io? A Z4 ••S/t-i 4z + - - + A 63 si'n 6z -h - • •— cos 6z. 117 And this solution can be well auuroximated as: IV ,»C ) 10 Z ' ~ -4 31 T T cos 2z w (z) 10 or: = + A 24 — sin ^z j^cos2z - A — 63 - 0.0447 Cos 6z cos (£ _ 75") z In order to obtain the other terms o f eq.(l5), eq.(45a) w i l l be used to solve w-(t), j = 1,2,.. f o r 0"= 2502 Hz : 0 A O036 a = 2 d, 0-OS7 cos- 2 r — c o r 2z. - 0-293 3.7 =r _A 0.32. 2. OS cxs r- co r Zz 2.26 1.74 a = 4.77 £ = S.6S t 7 a cos: 2z w, 0.77 6 ^ 7 = /4.7 s H4 = // = S3 a <* = 74. £ fi cos 2.z io.7 3 w it IV, cos 2z — cos -2 2. cos2z 13.7 43 cos 2z - — c o s -2z 118 Substitution o f w-(z), j = 1,2, 12 Sm 3. A s 1.74 Sm , sir* - A 6 —r- -t 0.77 0. 044 7A Sm i n eq.(l5) y i e l d s : . 27x — — JL . 6r* —— ~- A* — 3.7 A A 7y -+ -+ Z Sm — — 3.OS 4. sm 4.6%. 4.6% cos (3$-t S/'n . 3Tx . jr 7 - K —-~ -+ £ A e — 10.7 10.7 „ . swx Sm - 7S°) or: ~A~[- - i.a a*sir, o.ZSf. sin — - ,4.4 ct s ^--f s + 4.IZ cT3sin -j- •+ 3.a a.; 0 - 4. 47A~c% sir, M J _ 2.43 a* si» 46.8% si„ <2L + t ^ 0 cy (U) + S,h J - S.'3e£si„ -f /O.Sa*sin?f 2.47a~sin cos (3<H J — — ~ + S.3Sa*sm * 2.03a*Si» -~ + + - --jcos ~7S°) J J sln The above equation f o r w(x,t) shows that when the a x i a l e x c i t a t i o n frequency i s equal to one t h i r d of natural f l e x u r a l frequency of the column, the column responds with at least two d i f f e r e n t frequencies. Namely, i t responds with the same frequency as the e x c i t a t i o n frequency and also with the natural frequency of the column. 119 # 4: Example 1=4 (a = 64) n=7 Eq.(36) was solved to f i f t h power o f the parameter f o r a = 64 . q This solution i s presented here with only dominant terms o f each frequency retained: — Cos 2z 3840 ( 60 fA So 700 fA fA ( The case (loo,)(644ooo) C " 2 sin /Oz + - • • • ^ n =7 ,i =4 ) 4z + ) sin 6z •+ (2tX6*4o°o)C Sin 8z •+ 644ooo c Sin (iUO)(£44oeo)c cos 6z f — ( + cos 4-z 880 ...) -f - •+ ( (Z cos &Z ) + $SO)(644oOo)C Si'r> 12 2 + represents the fourth subharmonics o f the seventh natural f l e x u r a l o s c i l l a t i o n mode f o r the column with hinged ends. experimentally imposed were: S = 20 g calculated 0 919 Hz n 7 (experimentally observed 0- = 923 Hz) = 64 2. = (j /o )S S x ~ 43 = (3 * iO )(772o) - S 913 = ...) - 2 (/.6r*/o~ r)4 - 1.34 * id 4 6 = h S 6 Parameters 120 With these parameters the solution for W (z) — r — •+. - - 24 soo - w (z) becomes! ? A A COS 2 2 + •• + -— ;—-— 60coS 2z •*• • • •*• l&*oo . A A 3.1 * io 6 A 12-5 Cos 6z * - + 4IZSO A cos Sz +....— 53 coo , •+ — +— I.Z7 '/o7— • - 4z • cos A sin " 6z - - - • -+ZZ8 sm $z. + A S/'n 10z. •+ •-7' - 10 +• — sin i2z 7 2 And t h i s solution can be well approximated ass W (z) =s 7 A £0 cos 2z + A 229 sin 8z A WL/Z) «s — c o s 2z - or: — 0. Soi A 12-5 cos Sz A cos(8z.-3 ) a In order to obtain the other terms o f eq.(l5), eq.(45a) w i l l be used to solve w.(t), «•/ - j = 1,2,... =( h ) 7S = for9-=919Hzj °- * - -737 0265 a2 = 0.4Z K wz = — 3.S8 A a, - 2.Z6 A = 174- 3 —— r cos 2z. iv, 6 30.5 3 cos 2z_ ys„ - A3 c 9 * cos 2z A =io$ Z A, 2.8 s 6 g s ss cos 2z. 12.6 <X. = J4.S a o /• 84 4- = 16-6 c cos2z w, = J &s son 4z_ 3 Wa W - 170 -> cos 2z 121 Substitution o f j ( ) « w 1 # t / / l j z 1.2,....9 c sin —— 3.37 — 3.S8 * ~h 12.£ — sm A 4 Sin Z . 3 i n eq.(l5) y i e l d s : I. — ~ so.S * + Sin 60 £ 7, . a ^ • j COS &t ST*. 84 - 2 —r£ + 8 — icS Sm + * 77 „ „„. < • '> 0. SOI A sm ~ ~ - 7 Cos (4frt — 3 or: i<- -h — a.~ sm 0.476 a* sin ^j- + . where: ^ • - -r- /-'^a, - 3 -- - —J- +• 0.8/a7 J Acos&t 1 - Sin 39.3 — + 0.6/ A'a* sin oTg Sm —j- — + - 3') 2 — — The above equation f o r w(x,t) / a- = A a. shows that when the a x i a l e x c i t a t i o n frequency i s equal to one fourth o f natural f l e x u r a l frequency o f the column, the column responds with a t least two d i f f e r e n t frequencies. Namely, i t responds with the same frequency as the e x c i t a t i o n frequency and also with the natural frequency o f the column. 122 APPENDIX G Magnitudes o f s t r a i n s : During a steady state o s c i l l a t i o n o f a column i n i t s possible modes, strains o f various amplitudes e x i s t . vibration As i s well shown on the s t r a i n vs. frequency charts obtained f o r the two columns investigated here, the magnitudes o f s t r a i n s depend very much on the frequency and l e v e l o f the a x i a l end excitation. The s t r a i n vs. frequency charts serve well for a q u a l i t a t i v e analysis o f a column o s c i l l a t i o n , however; due to extensive processing o f the s t r a i n s i g n a l , the actual magnitudes o f strains are d i f f i c u l t to determine from the s t r a i n vs, frequency charts. This inadequacy i s not o f great consequence since i t i s the q u a l i t a t i v e analysis that i s more important i n t h i s investigation. in a good position Yet, to gain more insight, and to be to evaluate the i n t e r n a l (material) damping, t y p i c a l values o f s t r a i n i n the surface o f a column with hinged ends were meassured. The test. a x i a l end e x c i t a t i o n l e v e l was held constant a t 20 g's during t h i s The procedure f o r meassuring o f strains suggested by the manufacturer of BAM-1 was used. The e x c i t a t i o n frequency and the kind, and the amplitude of meassured strains are given belowt 9 * - w„ L 0 " = GO, -f. s* 9 * e- = 4- e = l^oo Hz F' - s t r a i n = 0.3 /<in/in e- = 1291 Hz F - s t r a i n = 26.5 /«in/in 9- = 1400 Hz P - s t r a i n = O.38 /<.in/in e = 1674 Hz P - s t r a i n = 3.5 / * i n / i n & = 1400 Hz L - s t r a i n = 2.5 / * i n / i n 0 = 6500 Hz L - s t r a i n = 0.125 /<• i n / i n 0 = 8215 Hz L - s t r a i n = 7.5 yx.in/in
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Investigation on nonlinear coupled vibration of columns. Bridicko, Jan 1972
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Title | Investigation on nonlinear coupled vibration of columns. |
Creator |
Bridicko, Jan |
Publisher | University of British Columbia |
Date Issued | 1972 |
Description | The oscillation of a column subjected to periodic axial end excitation was analytically and experimentally investigated. The initial 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 oscillation also occur at certain axial end excitation frequencies. Furthermore; at certain excitation frequencies, a coupling between longitudinal and torsional oscillations is 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 validity 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 built-in ends was also tested and its 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 built-in 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 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 axial 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 axial end excitation, did not cause appreciable increase in amplitudes of longitudinal oscillations. 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 oscillation phenomena here described, also occurred during flexural in-plane oscillation of a column. |
Subject |
Columns Vibration |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-04-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0101538 |
URI | http://hdl.handle.net/2429/33290 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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