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

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INVESTIGATION OF NONLINEAR COUPLED VIBRATION OF COLUMNS by JAN BRDICKO 3.Sc., U n i v e r s i t y o f I l l i n o i s ,  1970  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n the  Department 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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