EXPERIMENTAL INVESTIGATION OF NONLINEAR COUPLED VIBRATIONS OF COLUMNS by Dale P. Johnson B . A . S c , University of B r i t i s h Columbia, 1968 A'Thesis Submi tted-i-n Partial FulfiJJment Requirements for.the Degree <3f of the — Master of Applied Science In the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1970 In presenting this thesis an a d v a n c e d d e g r e e the L i b r a r y I further for agree scholarly by h i s of shall at the U n i v e r s i t y make i t that written thesis freely permission for It fulfilment of of Columbia, British available by gain shall requirements reference copying that not copying I agree and of this or Department of Mechanical The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada May 25, 1970 Engineering Columbia that study. thesis my permission. P. for or publication be a l l o w e d w i t h o u t Dale Date the t h e Head o f my D e p a r t m e n t is understood financial for for extensive p u r p o s e s may be g r a n t e d representatives. this in p a r t i a l Johnson TABLE OF CONTENTS ABSTRACT ACKNOWLEDGEMENT... LIST OF FIGURES... LIST OF TABLES LIST OF APPENDICES NOMENCLATURE CHAPTER I . INTRODUCTION P r e l i m i n a r y Remarks L i t e r a t u r e Review Limitations of Investigation,. CHAPTER II THEORY D i f f e r e n t i a l Equations of M o t i o n f o r a Column.... Theoretical Predictions.. Torsional Coupling CHAPTER III APPARATUS AND INSTRUMENTATION General Outline E l e c t r o n i c System . L o a d i n g Frame and Column Description D e t a i l s of Measuring System... CHAPTER IV TEST PROCEDURE Calibration Testing.... Photography Page CHAPTER V RESULTS AND DISCUSSION. 3 I n t e r p r e t a t i o n o f Frequency Spectra Identification of Strain Peaks Results of Flexural Strain Record Results o f Axial Strain Record CHAPTER VI SUMMARY AND CONCLUSIONS Summary Conclusions Suggestions search 8 38 39 42 48 59 59 61 f o r Further Re64 BIBLIOGRAPHY 67 APPENDICES 68 ABSTRACT Coupling vibration loading The of the flexural, modes was o f a column analytically initial crookedness inertia give tions. Further, result rise subjected and o f t h e column t h e Weber and axial investigated. longitudinal flexural-longitudmal effect between torsional to periodic and e x p e r i m e n t a l l y to coupled i n coupling longitudinal, vibra- and l o n g i t u d i n a l longitudinal and inertia torsional oscillations. To assess the validity apparatus was vibration control The s e t up t o a x i a l l y experimental theoretical exhibiting flexural ratio results results a frequency longitudinal that coupled 1:3 w a s o b s e r v e d , oscillation when was coupled a column agreement longitudinal o f 1:2 w e r e also observed, "Further, were experimental using present. with shaker. with the vibrations observed. Coupled though frequency a the experimental vibrations vibration a other than those In p a r t i c u l a r , a frequency and a c o r r e s p o n d i n g coupled a ratio of flexural present. torsional the applied sional ratio coupled expected i n good Coupled not established. suggest excite an a n d an e l e c t r o m a g n e t i c were predictions. theoretically A generator o s c i l l a t i o n s were was of the theory, frequency. mode frequency A second was was experimentally twice coupled observed the fundamental torsional mode torappeared - 11 when the excitation torsional The was to phase three times relationship between the The coupled vibrations significant \ was the fundamental frequency. observed. be frequency resonant at certain coupled frequencies. vibrations were found ACKNOWLEDGEMENT My visors, to particular gratitude Dr. C R . participate i n this out the duration I cular, also Mr. P h i l spent Sing work Ramsay, and f o r t h e i r f a c u l t y ad- f o ri n v i t i n g assistance t o thank Hurren the technical staff, and Mr. J o h n Hoar me through- Diane Johnson Leim i n assisting This project with i n typing provided by t h e D e f e n s e invaluable The care the thesis, t h e photography, was made p o s s i b l e and i n p a r t i - f o rtheir i n s e t t i n g up t h e a p p a r a t u s . by M i s s 66-9510 a n d D r . H. t o my of the project. wish co-operation No. Hazell i s extended through Research i s and time a n d b y Mr. appreciated. Research Board Grant o f Canada. - LIST i v - OF FIGURES Figure Fig.. I I - l Page Reference Axis For Displacement Measurements 9 Fig. II-2 Elastic Fig. III-l Signal Fig. III-2 Photograph o f Experimental Apparatus Fig. III-3 B a r Under Flow Schematic Axial Loading Chart View 19 o f Lower 20 Column Mount 21 Fig. III-4 Column Fig. III-5 Sectional Mounting and Loading Drawing Alignment of Loading System.. Mechanism 24 III-6 Schematic Fig. III-7 Fig. III-8 Photograph o f Specimen Loading Frame S c h e m a t i c S t r a i n Gauge Arrangement F o r F l e x u r a l S t r a i n Measurement... Fig. III-9 Fig. Fig. Fig. Fig. V - l V-2 V-3 V-4 V-5 View 22 and Fig. Fig. 15 o f Column Specimen... 25 26 28 S c h e m a t i c S t r a i n Gauge Arrangement For L o n g i t u d i n a l S t r a i n Measurement 29 F o r c e d Resonant F r e q u e n c i e s P l o t t e d V e r s u s Mode N u m b e r F o r V a r i o u s End C o n d i t i o n s 40 F l e x u r a l S t r a i n Record M i d p o i n t o f Column 42 Possible Flexural Axially Excited Obtained At Mode S h a p e s F o r Column 43 Frequencies of Experimentally Observed F l e x u r a l Resonant Strain Peaks 45 Waveforms f o r O b s e r v e d F l e x u r a l Resonances 46 Coupled - v - Figure Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. V-6 V-7 V-8 V-9 V-10 V - l l V-12 V-13 A - l Page A x i a l S t r a i n Record Obtained at M i d p o i n t o f Column. 49 A Portion o f a Column Twisting 51 Undergoing Frequencies of Experimentally Observed Resonant A x i a l Strain Peaks 51 Waveforms a t Fundamental n a l Resonance 52 Waveforms Mode a t Second Waveforms Mode at Third Longitudi- Coupled Axial 54 Coupled Axial 55 Phase R e l a t i o n s h i p Between Vibration Components a t Second C o u p l e d A x i a l Mode 56 Waveforms Modes 57 f o r Coupled E x t e n s i o n and R o t a t i o n Plane Fiber Torsional of Central 69 - LIST v i - OF TABLES Page Table V - l Permissable A c c e l e r a t i o n Levels Over V a r i o u s F r e q u e n c y Ranges F o r Column S p e c i m e n 35 - v i i - Page APPENDIX A APPENDIX B APPENDIX C S t r a i n E x p r e s s i o n f o r Column Some I n i t i a l C r o o k e d n e s s With 68 Details of Electronic Equipment Used f o r V i b r a t i o n C o n t r o l System.. 71 Linear D i f f e r e n t i a l Equations of M o t i o n f o r a Column 73 - viii - NOMENCLATURE Symbol P P = total 0 applied = constant axial applied load axial R. = amplitude of variable V" = frequency of applied 6 X = strain load applied axial of c e n t r a l plane axial load load fiber o f column SIQ = bending 6 = total u - l o n g i t u d i n a l displacement w = initial transverse displacement w = dynamic t r a n s v e r s e displacement 0 = rotation of cross-section- x,y,z = coordinate V = strain T = kinetic W = work done by e x t e r n a l v - volume A = area b = width h = heighth L = length s = fiber 1 = minimum moment o f i n e r t i a o f column section I = constant section strain strain o f a column fiber distances energy energy loading o f column of cross-section o f column o f column o f column o f column length d e p e n d i n g on d i m e n s t i o n s cross- of cross- - I o f i n e r t i-3 a b o u t moment 2 i x - Io polar moment E elastic G shear <t> density t time OC phase *tion angle longitudinal transverse freqi frequency ancies integer i w o f cross-s<. modulus resonant i nertia axis modulus fundamental p of y resonant t torsional acceleration g H, functions H (x) , 1 H (X) of f r e q u e ;cy gravity of variable x c ly , H (x) 2 3 X(x) , F ( x ) , G (y.) 1 G t w, w, x x 2 (x) u x partial respect d e r i v a t i v e o f a d splacement to the variable x Abbrevi at ions BAM Bridge amplifier CRO Cathode Hz Cycles RMS Root-mean-square a n d metex ray oscilloscope per second value of a function with CHAPT ntrocSuctiort - 1 - CHAPTER I INTRODUCTION Preliminary In Remarks t h e modern spacecraft, ment, a n d many the struggle intensified. dynamic excitation response becomes o f a column important does from often mation at frequencies investigation column can An form lar frequency of D made these vibrations which equip- i s being provide as columns. to periodic viewpoint. 10 K H z . order speed vibration sufficiently of order Simple at frequencies loading linear accurate The focus nonlinear Thus axial infor- of coupled this effects o f s e v e r a l KHz sustaining a periodic + P, c o s V t , V, may where t i s time be e x c i t e d i n t o Parametric i n the axial on s m a l l direction These free researchers have been and column. interested circu- torsional studies on t h e dynamic made periodic by a number focused vibrations i n the f i r s t straight load o f and P has a i n v e s t i g a t i o n s have lateral f o r an i n i t i a l l y under have axial transverse stability v i b r a t i o n s o f a column investigators. marily a design column of vibration. loading frequency aircraft, significant. P = P transverse o f high undesirable subjected i s on second elastic the varieties not provide vibrations, be q u i t e modes High of turbines, t o s t r u c t u r e s such theory in other against rapidly the day p r o d u c t i o n As a themselves prispatial consequence i n relatively - 2low angular frequencies Towards encountered theory the upper coupling phenomena However, a linear similar i n order Schneider [9] h a s shown discrete with i s shown that with excite of other does point causes namely, that become high reasonable to of oscillation. frequencies longitudinal i n a bar. coupling In t h i s effect work, works i n t h e i s excited i s one h a l f o r one frequency, of oscilla- i f a column longitudinal flexural t h e wave- relatively at particular nonlinear the bar into vibrations t h e two modes that a frequency resonant out that I t i s therefore frequencies also; nonlinearities. at these coupling this direction fundamental to parametric between tions tudinally inertia, frequencies. excitation reverse due t o l o n g i t u d i n a l and l o n g i t u d i n a l transverse it linear o f magnitude interaction range a simple analysis of flexural frequency investigation and a h o s t lengths mechanical excitation. end o f the a c o u s t i c i n the present i s inaccurate expect of t h i r d the i t i s o s c i l l a t i o n s with longi- possible significant amplitudes. Torsion a steel column direction the can r e s u l t Tso due of the cross i n a shortening of the square parametric movements apparent. i n rotation of the order possible tudinal resulting coupling that i n the between torsional effect of axial of the rotation. to the shortening [ 8 ] h a s shown sections Thus and longi- becomes i f a b a r i s under forced - longitudinal twice excitation a natural excite the The into equations solution. The that the order An of dic tor flexural axial the forced a and column study with those particular initial in and Chapter Y can feature allows Several methods and obtain the to Literature made to can on be the of a determine resonances. and arranged f o r more were to a closed- however, so conveniently the under perio- to moni- frequency resonant vibrations from are that realistic i s that admission longitudinal so response attempted experiments independently frequency column Observed apparatus, attempted dynamic theoretical considerations. arising experimental controlled without manipulated, vibrations crookedness I I I , was be was i n these due The are p r e d i c t e d by c o u p l i n g phenomena considered. presented nonlinear coupled linear to are and importance to viewpoint. coupled and i s close oscillations. S e v e r a l methods were responses vibrations which i t i s possible c o u p l i n g terms torsional f o r these compared torsional physical loading. frequency frequency, equations experimental spectrum Of from a of motion form second - with torsional column interpreted 3 the of inertia which parameters each spectrum strains were i s described P other. theoretical t o monitor of non- , P, This modelling. coupled f o r these 0 vibrations, oscillations. Review Previous research connected with the parametric response - Aof bars and columns has focused primarily on parametric vibrations associated with small free l a t e r a l vibrations in the f i r s t s p a t i a l mode. The investigations have been r e s t r i c t e d to r e l a t i v e l y low angular frequencies of excita- t i o n , not more than three or four times the fundamental l a t e r a l resonant Beliaev frequency. [2] was the f i r s t to analyze the parametric response of a column under time-dependent citation, longitudinal ex- and he reduced the equation of motion to the stan- dard Mathieu-Hill equation. of longitudinal i n e r t i a . He did not consider the Somerset influence [ 4 ] and Bolatin [6] experimentally v e r i f i e d that the s t a b i l i t y of the column described by Beliaev could be analyzed by investigating the s t a b i l i t y of the solutions of the Mathieu-Hill equation of motion for the column. An unstable region was found characterized by l a t e r a l column o s c i l l a t i o n s h a l f the excitation frequency. at exactly Experimentally observed oneef- fects not anticipated by this equation were attributed to longitudinal i n e r t i a , l i n e a r and nonlinear damping, nonlinear elasticity, rotary i n e r t i a , and i n t e r n a l f r i c t i o n . Somerset and Evan-Iwanowski perimentally investigated [ 7 ] theoretically and ex- the parametric i n s t a b i l i t y of straight columns sustaining a periodic axial load of low c i r c u l a r frequency. They also b r i e f l y considered the i n - fluence of damping and of nonlinearities which are amplitude dependent, such as nonlinear e l a s t i c i t y and longitudinal inertia. regions They of presence of Numerous metric render amplitude of lateral r e f e r e n c e s on found in a the survey stability of cross-section that with torsional into a an under over certain that the the column vibration topic of para- article by Evan- oscillations. the stability axial i s under which close forced i s close the to of the to the has of He longitudi- to twice excite the a the frequency fundamental longitudinal bar para- loading. F u r t h e r , when effect of of cantilever i t is possible l o a d i n g becomes torsional dynamic frequency frequency, problem elastic i f a bar frequency, torsional applied the be and to torsional longitudinal on can analytically the other space, tend companion natural of the may studied a excitation bar factors as i s unstable parametric has rectangular showed column [5 ] . [8] Tso the PI,V") unstable vibrations Iwanowski that nonlinear increases. nal (PQ, the increasingly metric found inertia to be taken thfe r e s p o n s e of a into consideration. The clamped present at equations both of longitudinal above the the been a bar ends motion considers with inertia from treated of the are transversely lower author, by of axially for i n i t i a l at strains investigated central i n the up to past. coupled i n bars plane He The and frequency. vibrations the excited. frequencies i n the [9] column crookedness longitudinal coupled Schneider and end considered resonant admission only the accounting fundamental knowledge sulting work and To re- have excited longitudinal - 6oscillations. tions the occur He The that a t 1 a n d 1/2 fundamental Limitations found of investigation form solution o f coupled i s said oscillations. limiting column about along stresses. symmetrical about in plane. stant along respect linearly The vibrations not serious Internal i s used, isotropic of developing i s assumed o f l o a d i n g so that t o be bending occurs The m a t e r i a l p r o p e r t i e s a r e assumed o f shear were elastic and capable fre- resonant of flexure to a nearly straight t h e l e n g t h o f t h e column applications, are theory the Analytically, o f coupled The c r o s s - s e c t i o n proportional transverse so only are indicated. the amplitudes the plane limitations. derived are not exact. i s not available, i t s length to the plane effects at vibration. important equations vibrations the analysis t h e same resonant The B e r n o u l l i - E u l e r uniform bending oscilla- the transverse excitation has s e v e r a l A nothing longitudinal Investigation the theoretical quencies times longitudinal Firstly, closed resonant of loading. Bending to distance deformation from limitations stiff F o r most columns plane. inertia on engineering a r e used, modes and t r a n s v e r s e with i s assumed the central at t h e lower and e x t e r n a l damping strain and o f r o t a r y are ignored. relatively and s y m m e t r i c a l con- of these oscillation, loading are not considered. Secondly, tions. the experimental I t is. not possible study to realize has numerous in practice restricthe boundary Clamped conditions column experimental mate, to amount and of signal as acceleration power' to an gauges were coupled was magnetism exact exciter the To were to used of noise the an Finally, 1 0 KHz by purely found electronic to were the level tests noise very small. measured level as high sinusoidal 'approximate the approxi- considerable signal the an means levels constant achieve spectra. frequency of A minimize strain maintain bands i s not sensitive c o n t r i b u t i o n of frequency control. the to in i s always vibrations. required since only development. approximated excitation excitation upper only column the is difficult. possible, best that caution the at analytical alignment nonlinear induced Assessing are the Further, Strain monitor ends in set-up. meaning axial. used constant were automatic limited vibration CHAPTI Theory - 8CHAPTER I I THEORY Differential Linear for plane assume no s t r a i n s motion, this means column crookedness. placement The dynamic account Lagrangian deflections deflection curve. description expression A). plane where, u x referring u = w = + assumption exhibit i n the a strain some differential expression or static of strain Mettler with w t o an + [ 3 ] developed some the strain *x x relative i s used; which dis- that i s , initial [ 1 ] d e r i v e d an a n a l a g o u s Love o f t h e column = f o r t h e un- This crookedness are measured obtained £x from definition f o r a column fiber arise i n the Considering axis a l l columns used or bar. of strain. He since any i n i t i a l o f a column commonly exist segment. The n o n l i n e a r i t i e s derived herein into the neutral line i n practice o f motion loading i s zero. that i s a straight limitations dix C) the axial equations Column (Appendix when initial for a equations plane loaded takes o f Motion differential columns neutral has Equations initial Eulerian the strain crookedness expression at the central as (V2) w 2 x ( to Fig. I I - l longitudinal displacement initial transverse displacement w - dynamic transverse displacement 2 (Appen- - i ) - 9and t h e s u b s c r i p t s denote pect t o that partial differentiation with res- variable. -INITIAL DEFLECTION CURVE •DYNAMIC DEFLECTION CURVE z,w Fig. I I - l . Reference From simple bending at a distance 6 This b expression sections mation w derived, x coupled a s was (slenderness plane axial +6 fiber axis i s assuming that during bending; The t o t a l that strain plane cross- i s , shear defor- i s therefore (2-3) b done equations by M e t t l e r greater than o f motion [3]. A slender 50) i s . c o n s i d e r e d can be n e g l e c t e d , loading of a (2-2) differential ratio inertia the neutral xx i s neglected. The rotary z Measurements the additional strain i s obtained remain 6=6 Only " f o r Displacement theory, z from = Axis i s considered. and damping i s c a n now column so that Ignored. be - 10 From usual beam bending theory, assuming that stresses occur only in the x - d i r e c t i o n , the e l a s t i c V = | £ e d s t r a i n energy i s (2-4) v where, E = modulus of elasticity v = volume {2-1) Substituting equations and (2-2) into equation (2-3), substituting the resultant expression into equation (2-4) integrating p a r t i a l l y over the cross-section and of the column, the s t r a i n energy becomes V = f J x *x x + 'J xx 0 L(u + 0 EI +(v2)w Vdx w Lw x (2-5) 2 d x 2. J where, A = cross-sectional area of column I = minimum moment of i n e r t i a of column cross-section The approximate k i n e t i c energy T = ~ J L q (u 2 t is + w ) dx 2 t (2-6) where, 0 = density of column material t = t ime Hamilton's p r i n c i p l e for a conservative system states that (V - T) dt = 0 '1 Substituting equations (2-5) and (2-6) ing into (2-7) and employ- calculus of variations provides the coupled d i f f e r e n t i a l equations of plane motion for the free vibrations of a column. EA(u Elw x + (l/2)w x x x x + ww) " EA[(u + d>Aw = 0 tt 2 x x x + x ( ) 1/2 + x Wx 2 Au + w = 0 t t ^ x ) ( ^ (2-8) + _^ . } (2-9). If the t h e n o n l i n e a r terms familiar columns equation results. provides some which f o r the longitudinal equation linearizing for lateral treatment are dropped, vibration equation vibration of the equations p r e d i c t i o n s t o b e made oscillations at (2-8) of (2-9) o f beams. Predictions qualitative allows i n equation Similarily, t h e common Theoretical A 11 - o f t h e column. coupled terms concerning In p a r t i c u l a r , are l i k e l y of motion the resonant the frequencies t o be most significant are revealed. Expanding, (2-9) xxxx EA[u form the x x level x E + A [ u u x xx x W i s also expect the axial vary x x u x xx] equation w + A4w t t = ] (2-10) excitation of the acceleration displacement + w sinusoidal column to w " of the axial constant tion and r e a r r a n g i n g gives E I w The differentiating i n the tests acceleration. with respect A double t o time time-variation at the excited sinusoidal. indicates a t any p o i n t along ting into differentiating equation to t h e column i n time, (2-11) V i s the applied frequency Appropriately that end of the u = U ( x ) c o s ( r t + OC ) where integra- I t i s therefore reasonable displacement sinusoidally i s that of (2-10) and equation yields i s some p h a s e (2-11) and angle. substitu- - E I w xxxx 12 - - EA[u x x w + u w x x x x ] + A^wtt = F(x) cos(Yt+CQ where F ( x ) i s some by the s t a t i c of equation Then (2-12) o f frequency are therefore in V . while modes, these they oscillations, for a column since time being, response can the equation (2-8) c length f> resonant flexural linear theory into equation that terms the transverse f o r the transverse response (2-8) d e s c r i b i n g mathematical the description of D i f f e r e n t i a t i n g and rearranging yields + i - utt = w u the A sinusoidal t o g e t an improved " xx where o f the coupled indicates longitudinal behavior. equation t o avoid excited. i s sinusoidal. motion t o occur resonant from since t othe However, flexural can be p r e d i c t e d (2-12) back excited, normal t o as ' l i n e a r ' the influence be s u b s t i t u t e d axial they transversly Neglecting speaking, are understood coupled i n - Strictly t othe e x c i t a t i o n . be r e f e r r e d function. oscillations i n directions with side flexural are parametrically oscillations resonances will Ther i g h t a sinusoidal transverse resonant occur forced x determined as a f o r c i n g t obe e x c i t e d . oscillations directions parallel confusing has 'Linear' oscillations excitation, o f t h e column. can be regarded expected flexural o f the coordinate effectively put parametric function displacement t h e column these (2-12) 1 3 t i l e x W x x + w w x x x + w w x x x (2-13) v e l o c i t y o f l o n g i t u d i n a l waves o f t h e column. Substituting a sinusoidal along trans- verse response equation into (2-13), of x only, 13 - the coupled t e r m s on t h e r i g h t and remembering t h a t w and w x are f u n c t i o n s x x gives = H(x)cos2tft (2-14) = H (x)cos^t 1 (2-15) ^xWxx = H ( x ) c o s t f t (2-16) w xw x x w xw X x 2 cos2 Yi Since = w w x + c o s 2 = H (x) x x t h e sum + + H (x)(H-cos2*t) 3 first ~2 tt (2-17) different into t (2-18) with longitudinal longitudinal vibration frequency. vibration of the transverse v i b r a t i o n frequency vibration a frequency vibrations, f r e q u e n c i e s are expected. as t h e e x c i t a t i o n a t an e x c i t a t i o n excitation costft X that the l o n g i t u d i n a l resonant resonant resonant frequency 2 These are t h e n o n l i n e a r c o u p l e d same f r e q u e n c y occurs (2-16) and o f two s i n u s o i d a l o s c i l l a t i o n s coupled coupled (2-17) ={H!(x) + H ( ) } u indicates r a t i o o f 1:2. two (2-15), ~ xx This equation and (2-14) becomes t (2-13) , one o b t a i n s u is y ) , equation [1 + cos2tft] 3 Substituting expressions equation side of frequency f o r the f i r s t The i s at the The second i s at twice the o f t h e column. o f j&V It where oo, i s t h e coupled longitudinal vibration. To c o n t i n u e axial response verse response. this somewhat i t e r a t i v e procedure, the c a n be u s e d t o o b t a i n a more p r e c i s e t r a n s The f i g h t side of equation (2-18) c a n be - regarded with be as two equation • from jit), + resonant questionable, effect each those iteration can i n on t h e r i g h t two..flexural side of form (2-19) 2 transverse turn coupled G (x)cos2JTt frequencies analysis a t a> a n d x due twisted close with. terms does The phenomena additional decreasingly not indicate discussed. i s a terms further second provided significant. The the significance of The e x p e r i m e n t a l investi- value. between longitudinal to the "shortening s t e e l b a r o r column shortening sections. forced the coupling the i t e r a t i o n s Coupling Coupling exists f o r continuing become i s of obvious Torsional axial since to start coupled gation For t of the functions expressions terms a consequence justification theoretical to These forcing arise. order by two . predicted x sinusoidal the coupled As G ( )cos 1 and into c a n be which The is Y (2-10) . vibrations - longitudinal frequencies substituted 14 Tso i n addition [8] s h o w s effect". subjected a natural parametrically excite simplicity, torsional dissociated from each An of the a n a l y t i c a l l y that with torsional a frequency torsional i n the present undergoes cross- which i s i t i s possible and f l e x u r a l c o u p l i n g other un- i f a b a r i s under frequency, the bar into motion initially t o torques to rotation longitudinal, vibrations to twice and t o r s i o n a l work. oscillations. are treated Following Tso's nal development displacement roidal axis can and of be referring a point written u(x,y,t) = at to F i g . II-2, the a distance y from longitudi the cent- as U(x,t) - P(y.t) Fig. Warping in as i n St. equation lation, The Elastic Venant (2-20). however. € where II-2. = The torsion i s not Loading taken axial strain i s given rj - \ y O 2 x 0 rotation of strain GAh" e displacement cross energy 2 x }dx into i n the account formu- by (2-21) 2 x longitudinal elastic Under A x i a l It i s considered later U = = Bar of cross section section becomes (2-22) - 16 where A = c r o s s - s e c t i o n a l I - area = moment o f i n e r t i a 2 about 2 - a x i s 1-^ = c o n s t a n t d e p e n d i n g 1 The 2 where I a c c o u n t s f o r S t . Venant X torsion. Q IJQ = [ A U t + - « t ]dx I e (2-23) 2 = p o l a r moment o f i n e r t i a work done by t h e a p p l i e d loads of cross-section. a t t h e end i s W = f^p^tjud^tjhdy where p ( y , t ) that represents (2-24) t h e end l o a d on t h e column. t h e mathematical form o f t h e end l o a d p(y,t) where ^ ( y ) (2-20) 1 S a n even l A (2-25) f u c t i o n o f y. Substituting i n t o e q u a t i o n (2-24) L 2 2 x 1 f b/2 R-,= - j h ^ ( y ) dy 1 A j -b/ R =r f equations yields - jR I P(t)j © 2 P(t)U(l) Assume i s separable, = P(t)*My) and ( 2 - 2 5 ) W = R where The energy i s T The 7 t e r m 3- GAh 6 kinetic on d i m e n s i o n s o f c r o s s - s e c t i o n d x { 2 _ 2 6 ) 0 2 2 hyV(y)dy Using Hamilton's P r i n c i p l e , S f ( T - V + W)dt = 0 (2-27) t2 with equations (2-22), variational motion (2-23) and ( 2 - 2 6 ) , and e m p l o y i n g a procedure, the coupled d i f f e r e n t i a l , equations o f f o r t o r s i o n a l and l o n g i t u d i n a l v i b r a t i o n s <DAUTT - EAU <H>©tt - t^GAh - |^Ii©xx©x 2 - X X + 2 * EI 0 e 0 2 x x x = 0 R i p )]e 2 2 ( t result; x x (2-28) + Ei 2 ( U x e x ) x ( " > 2 2 9 1 7 - - These e q u a t i o n s o f motion d e s c r i b e t h e l o n g i t u d i n a l and t o r s i o n a l response o f a u n i f o r m Column o f t h i n rectangular c r o s s s e c t i o n b by h and o f f i x e d l e n g t h L , l o a d e d symmetr i c a l l y about t h e OX-axis and u n i f o r m l y o v e r t h e t h i c k n e s s o f the strip. tropic. The column m a t e r i a l i s assumed l i n e a r and i s o - I f t h e n o n l i n e a r terms are n e g l e c t e d i n e q u a t i o n s (2-28) and (2-29) t h e e q u a t i o n s become u n c o u p l e d . (2-2 8) nal Equation t a k e s t h e form o f t h e f a m i l i a r e q u a t i o n f o r l o n g i t u d i - v i b r a t i o n s o f a r o d , w h i l e e q u a t i o n (2-29) Ol^tt " [GAh /3 + 2 l 2 R P ( t ) ] 0 2 xx = becomes 0 ( - > 2 30 T h i s e q u a t i o n i s d i f f e r e n t from t h e u s u a l form f o r t o r s i o n a l v i b r a t i o n o n l y i n t h e s t i f f n e s s term, which accounts f o r t h e a x i a l end l o a d . I t i s seen t h a t a compressive end l o a d such as i s used i n t h i s experiment w i l l decrease t h e t o r s i o n a l s t i f f n e s s o f t h e column. the U s i n g a p p r o p r i a t e end c o n d i t i o n s , approximate r e s o n a n t t o r s i o n a l f r e q u e n c i e s can be c a l c u - lated (Appendix C ) . Tso p o i n t s o u t t h e e x i s t e n c e o f two c r i t i c a l frequency ranges where t o r s i o n a l c o u p l i n g i s most l i k e l y t o o c c u r . The f i r s t range appears when t h e e x t e r n a l l y a p p l i e d f r e q u e n c y i s close t o twice the n a t u r a l frequency of a p a r t i c u l a r mode. torsional The second c r i t i c a l range appears when t h e a p p l i e d frequency i s c l o s e to the l o n g i t u d i n a l frequency, p a r t i c u l a r l y i f t h e dimensions o f t h e column are such t h a t the f u n damental l o n g i t u d i n a l f r e q u e n c y f r e q u e n c y f o r a t o r s i o n a l mode. i s close t o the n a t u r a l CHAPTI Apparatu CHAPTER I I I APPARATUS General A tation of AND Outline signal flow diagram o f the apparatus i s shown i n F i g . I I I - l . the actual apparatus The e x p e r i m e n t a l parts. The f i r s t includes part t h e feedback generator, rometer. to deliver ponding t o the type transmitting specimen. signal. circuit circuit, Feedback unit and making this with exciter signal the shaker since considered instruments necessary of specimen. accele- i s procorres- The s h a k e r table responds to the the vibration the excitation control i t The between i s to the output feedback accelerometer a n d the. e x c i t e r of the apparatus Strain a n d an control which signal an a c c e l e r o m e t e r unit. t o gather i n two system, of a t h e programmed e x c i t a t i o n part i n this t o the shaker wanted. using level, f o r any d i f f e r e n c e second photograph control corrections control compares i s a amplifiers 'checking' i s accomplished the excitation the t e s t of suitable to the exciter compensate through i s essential i s capable This vibration of excitation a force instrumen- and c o n s i s t s an e l e c t r o n i c monitors The can be i s the vibration The a u t o m a t i c providing apparatus electromagnetic shaker, grammed exciter F i g .III-2 and and i n s t r u m e n t a t i o n used study. by INSTRUMENTATION control so t h a t unit i tcan t h e two. consists of monitoring i n f o r m a t i o n on t h e b e h a v i o r gauges attached t o the specimen FOUR BEAM A OSCILLOSCOPE LEVEL LU ACCELEROMETER Feedback FOTONIC SENSOR Jb-o- RECORDER PREAMPLIFIER GENERATOR (Feedback and Synchronization) POWER AMPLIER Fig. IH - 1. Signal Flow Diagram For Synchronization 1> - 20 - Fig. III-2. 1. Frequency 2. Time 3. Amplifier Analyzer 4. and Spectrum of Experimental 9. Wavetek Apparatus Signal 10. Power 11. Accelerometer 12. Specimen 13. Column 14. Spring 15. Fotonic 16. Shaker Generator Amplifier Preamplifier Frame CRO Level 6. Vibration Control 8. Counter Switch 5. 7. Photograph Fotonic BAM Specimen Recorder Exciter Sensor Unit 17. Roots Sensor Blower Holder indicate flexural erometer used leration level displayed spectral on the The by automatic Fig. lower and strain circuit levels. also column mount. level the recorder, signals vibration generator) the desired signal shaker. s p e c i m e n mount actual feedback at the a CRO amplified the longitudinal were and The monitored accel- the acce- These signals were where appropriate, performed. System control m i n e d by i n the analyses of Electronic The and An III-3. at control (hereafter p r o v i d e d a pre-programmed vibration was used level to that Schematic a power t o produce accelerometer ( F i g . II1-3) vibration exciter rigidly monitored the signal of mechanical m o u n t e d on vibration the acceleration Lower Column deter- amplifier. point. View called Mount lower of the - 22 Following rometer amplification preamplifier, a n d impedance m a t c h i n g thevibration t h e programmed s e t - p o i n t control generator then correcting and of the instruments contained Loading The Fig. i nthecontrol completed any d i f f e r e n c e t h e preprogrammed level was compared generator. t h e feedback i n the vibration signal specification control loop i s column m o u n t i n g Description and l o a d i n g system i s shown i n III-4. FRAME UPPER MOUNT COLUMN SPECIMEN SPRING STRAIN GAUGES - ACCELEROMETER -LOWER MOUNT SHAKER BED BLOWER N Fig. The i n A p p e n d i x B. Frame and Column III-4. CONCRETE BASE Column M o u n t i n g a n d L o a d i n g with c i r c u i t by between t h e a c c e l e r o m e t e r s i g n a l . A more d e t a i l e d used of the accele- System The column was arranged dently. two the the ments upper shaker quite the uin) above were always complete lower column of the lower the column the much or Y could be was applied to the on P = lower tests system, and constrained arrangement the + The very specimen and by attached always of suffi- applied cyclic small displace- (of the force P The order was Q were conducted at the operating frequencies natural consisting t o move indepen- P! c o s Y t was static Hz, the shaker shown 0 were Since the than P was It i n compression. mount. of hundred and frame. varied side column variation greater shaft either mounts, springs five i n the and load the and spring-mass was P„ vertically to maintain the four mount, t mounted through so , P 0 - loaded load and negligible. cies P springs component of 200 this, that magnitude varying was static parallel cient of so The between by specimen 23 table. of The vertically. i n F i g . III-5 frequency the was of column, lower To frequen- end of the springs, the accomplish used. - 24 - Fig. Sectional Drawing 111-5 of Loading and A l i g n m e n t Mechanism 3 A — inch diameter Atlas table t o the lower lower mount bushing the directly The attached with i s shown upper mount t o t h e frame mount. using the shaft. so t h a t i n line mount specimen prevented to guide specimen column was Superior shaft The t h e moving the lower in Fig. was connected the shaker Transverse motion a Thompson shaker shaker adjustable was head specimen of the situated was ball below positioned support. The lower Ill-3. similar except that and i t d i d n o t have i t was rigidly the accelerometer - 25 attachment. The mounts were i n t e n d e d t o p r o v i d e conditions. As shown i n F i g . I I I - 3 , s p a c e r s were f o r c e d i n on each s i d e between t h e column and mount. clamped end The top i n s i d e edges o f t h e wedges were b e v e l l e d , and t h e specimen for length f l e x u r a l o s c i l l a t i o n s was t h e d i s t a n c e between t h e e x t r e - m i t i e s o f t h e b e v e l s on t h e upper and l o w e r mounts. To s e c u r e 3 the specimen i n p l a c e , a t i g h t f i t t i n g i n c h diameter m i l d s t e e l p i n was i n s e r t e d t h r o u g h t h e mount, wedges and specimen. The e x p e r i m e n t s were performed on a s t e e l specimen o f r e c t a n g u l a r c r o s s s e c t i o n f a b r i c a t e d from hot r o l l e d carbon steel, flat Fig. stock (Fig. III-6) . III-6. Schematic View o f Column Specimen The column specimen has a l e n g t h of 11.625" between and a 0.375" by 0.125" r e c t a n g u l a r c r o s s s e c t i o n . fillets, The dimen- s i o n s o f t h e specimen were chosen t o produce a measurable s t r a i n l e v e l w i t h t h e f o r c e a v a i l a b l e , and t o p r o v i d e the f i r s t c o u p l e d l o n g i t u d i n a l r e s o n a n t v i b r a t i o n i n the 8 Khz region. The f i r s t E u l e r b u c k l i n g l o a d o f t h e specimen f o r clamped end c o n d i t i o n s was 534 pounds. The s u r f a c e s were l i g h t l y ground t o s i z e on a s u r f a c e g r i n d e r , which gave t h e column a s l i g h t i n i t i a l c u r v a t u r e , t a k e n care o f i n t h e t h e o r e t i c a l model. The specimen had a w i d e r s e c t i o n on each - end t o accomodate 26 - a h o l e f o r the p i n and t o a l l o w a closer approximation to clamped ends. The column specimen was mounted i n the upper p o r t i o n o f the heavy frame, as shown i n F i g . I I I - 4 . to a s t e e l bed. the shaker was On t o p o f the bed was supported. which c i r c u l a t e d The frame was bolted a block o f wood on which Below the bed was a Roots blower a i r through the shaker t o prevent o v e r h e a t i n g . The blower and i t s d r i v i n g motor were i s o l a t e d from the t e s t bed and frame to a v o i d any unnecessary t r a n s m i s s i o n of e x t r a neous v i b r a t i o n the to the specimen. t e s t bed was mounted was minimize e x t e r n a l v i b r a t i o n isolated sources. graph of the specimen, l o a d i n g Fig. III-7. The concrete base on which from the b u i l d i n g F i g . I I I - 7 i s a photo- frame, shaker and t e s t Photograph o f Specimen to Loading Frame bed. - 27 D e t a i l s of Measuring System A p a r t from t h e a c c e l e r o m e t e r , t h e use o f two t y p e s o f transducers was attempted t o m o n i t o r the b e h a v i o r o f t h e specimen. The f i r s t of t h e s e was a non-contacting displace- ment t r a n s d u c e r , c a l l e d a F o t o n i c S e n s o r , manufactured by M e c h a n i c a l Technology, L i m i t e d . The F o t o n i c Sensor i s a s o l i d s t a t e e l e c t r o n i c i n s t r u m e n t w i t h a probe c o n s i s t i n g o f a packed bundle o f s p e c i a l l y c o n s t r u c t e d g l a s s fibres a r r a n g e d i n a random t r a n s m i t - a n d - r e c e i v e c o n f i g u r a t i o n . The t r a n s m i t t i n g f i b r e s carry l i g h t to the t a r g e t ; the r e f l e c t e d l i g h t i s returned through the r e c e i v i n g f i b r e s t o i l l u m i n a t e a l i g h t s e n s i t i v e diode. Though t h e i n s t r u m e n t o f f e r s many d e s i r a b l e f e a t u r e s , i t was not found v e r y s u c c e s s f u l i n t h i s work. The i n s t r u m e n t probe must be mounted v e r y r i g i d l y so t h a t i t undergoes no movements. With t h e apparatus used, i t was not found p o s s i b l e t o p r o v i d e a mount s u f f i c i e n t l y iso- l a t e d from a l l s o u r c e s o f v i b r a t i o n . The second type o f t r a n s d u c e r , the s t r a i n gauge, was more successful. Four BLH E l e c t r o n i c s SR-4 t y p e FAP-12-12 f o i l gauges were a t t a c h e d t o t h e m i d d l e o f t h e column specimen; two s i d e by s i d e a l l i g n e d a x i a l l y on each o f t h e w i d e r f a c e s o f t h e column specimen. the arrangement To measure bending s t r a i n levels, shown s c h e m a t i c a l l y i n F i g . I I I - 8 was where t h e specimen was used, l a t e r a l l y d i s p l a c e d t o produce d e c r e a s e d compression i n two gauges on one s i d e and i n c r e a s e d compression i n t h e two gauges on t h e o t h e r s i d e . A f o u r arm b r i d g e o f the - 28 - Fig. III-8 Schematic S t r a i n Gauge Arrangement f o r F l e x u r a l S t r a i n Measurement. type shown was used f o r s e v e r a l r e a s o n s . s e n s i t i v e bridge c o n f i g u r a t i o n , and was t u r e compensating. T h i s b r i d g e was I t was the most completely i n s e n s i t i v e t o any form l o n g i t u d i n a l s t r a i n s i n t h e specimen. Since s t r a i n l e v e l s were e n c o u n t e r e d , a f u r t h e r i n c r e a s e s e n s i t i v i t y was voltage obtained temperauni- small in by i n t r o d u c i n g as much a d d i t i o n a l as p o s s i b l e i n s e r i e s w i t h the i n t e r n a l e x c i t a t i o n of the bridge amplifier. The A m p l i f i e r and Meter used was E l l i s A s s o c i a t e s BAM c a p a b l e o f measuring dynamic s i g n a l s o v e r the range e n c o u n t e r e d . I t s frequency response i s such t h a t t h e a t t e n u a t i o n a t 10 Khz L o n g i t u d i n a l s t r a i n was 1 Bridge i s approximately measured u s i n g the b r i d g e f i g u r a t i o n shown i n F i g . I I I - 9 . 3%. con- L o n g i t u d i n a l s t r a i n cannot be measured u s i n g f o u r a c t i v e arms, so two on each s i d e o f the specimen) and two a c t i v e arms (one dummy gauges p r o v i d i n g - Schematic Strain sensitivity in a arm to torsional torsion this equal of magnitude strain sensitivity. strain oscilloscope. the at level The The of Frequency 2107, which zer. The quency gauge served level Analyzer and middle than was strain taken also spectral sensitive gauges. sensitivity longitudinal was also used cross-section, since i n both strain were i s several or from torsional t h e BAM a m p l i f i e d and analyses For were to an recorded also on recorded interest. analyzer as used a voltage recorder was the signal signal recorder, frequencies lower compensation c o n f i g u r a t i o n was strain bending orders temperature This rotations of arrangement, The and bridge. produces - Fig. III-9. Gauge A r r a n g e m e n t for Longitudinal S t r a i n Measurement increased four 29 used a Bruel was a Bruel a m p l i f i e r and i n conjunction and Kjaer Type and Kjaer spectrum with 2 305, the Type analyFre- offering a - 30 - r a n g e o f p a p e r and w r i t i n g speeds. means o f an i n k pen on 100 mm. brated paper. RMS RMS, f u n c t i o n was R e c o r d i n g s were made by logarithmic frequency DC o r peak v a l u e s c o u l d be plotted;.the chosen t o minimize t h e i n f l u e n c e extraneous s i g n a l s . F o r most t e s t s , of the recording Meter, A G e n e r a l R a d i o Company D i g i t a l Type 1151-A, excitation forms and yzer measured t h e average t o a t l h*z. a c c u r a c y . of interest were done u s i n g Time and and l e v e l Frequency frequency of the Spectral analysis o f wave- frequency c a l i b r a t e d paper anal- recorder. displayed on a T e k t r o n i x s c o p e , where t h e y c o u l d be p h o t o g r a p h e d photographs was frequency a m e c h a n i c a l f r e q u e n c y l i n k a g e between t h e f r e q u e n c y The waveforms were X sudden paper c a l i b r a t e d b y means o f an e v e n t m a r k e r and a d i g i t a l counter. cali- were t a k e n w i t h a P e n t a x directly. camera u s i n g film. ) 565 oscillo- These Kodak Plus- CHAPTER Test Piro< 4 - 31 - CHAPTER I V TEST PROCEDURE Calibration The performance instruments generator checks was checked. at the upper generator, between b y means Since scale the counter frequency of the control and t h e f r e q u e n c y was a accura- on t h e c o n t r o l used. scale which well be r e a d scale counter circuit against f r e q u e n c i e s cannot end o f t h e frequency a digital scale of a built-in on t h e f r e q u e n c y frequency. of the experimental The frequency calibrated two p o i n t s defined tely was and c a l i b r a t i o n Agreement was g o o d ata l l frequencies. A [9] factory found output, The that with favourable response BAMs noise frequency spectral was made indicated were over the frequency analysis was the accelerometer. used. within the involved. analyzer tests the accelerometer range The t h e most frequency manufacturer's of the experiments. a n a l y z e r was Frequency and t h e l e v e l was e x c e l l e n t agreed generator. and t h e one w i t h o f the frequency t h e range Schneider by t h e a c c e l e r o m e t e r the frequency response used. s e t on t h e c o n t r o l characteristics over to cool evaluated, was w e l l was d i dnot affect the acceleration specifications between level o f t h e BAM flat accelerometer cooling so no p r o v i s i o n Several tely forced acceleration exactly The calibrated comple- synchronization recorder f o r provided they were - 32 accurately of the frequency reference The signal speed records with was The it the was capable response of during an a c c u r a t e speeds providing o f measured the extent t o minimize signals sensitivity zener-diode. writing the influence on the s t r a i n record. before and d u r i n g the o f one o f t h e o s c i l l o s c o p e BAM. often necessary. A l l cables pick twisting up the course The experiment. plug-in units of the experiments, which developed from Fresh and the leads used were were gauge noisy ground signals l o o p s was were so a floating minimize always signals used i n by ground was Electro- minimized t h e gauges promptly eliminated to shielded. electromagnetic fields. i n s t r u m e n t s up w i t h through dry c e l l s t h e l e a d s t o g e t h e r and k e e p i n g out of strong arising o c c u r r e d , and e f f o r t s i n the strain leads the using internal found calibrated signals were magnetic by an The replaced. Noisy noise t o some unacceptable was high and e x t r a n e o u s performance became calibrated by was beforehand. In the present experiments, reduced oscilloscope was produced recorder levels. noise analyzer voltage level accurate of synchronized manually Any and strain replaced. hooking gauges Noise most and g r o u n d i n g of them oscilloscope. Testing The place and specimen with i n the upper the bevelled and strain lower spacers g a u g e s a t t a c h e d was (Figs. III-3 i n on e a c h side. mounts driven put i n and The III-5) - 33 specimen was replaced. by the Any output correctly and the was a were installed and of strain the as leads were on in the leakage specimen to batteries of gauge bridge least to turn advance of to adjusted were in coil rigidity to the specimen. resistance and for the and the gauge balanced. before the this point to testing. instrumentation anticipated on the A time. from wires and The excited. the a was few of the BAM By strain position switch the gauges a l l instruments automatically testing The ground. BAM, standby time excita- possibility lead circuit within At turned the in e f f e c t of of resistances BAM extended the oscilloscope and the leads the checked combined external minimized of was lead. the The shields compressive preamplifier. the connected III. which to its including Chapter specimen accelerometer according and the constant accelerometer accelerometer leads sharp of to was the the The When level side a detected acceleration connected hours the column. be be clean The connected two the their variable properly given provide on to and ground. were means a could became to loose for were achieve had amplifier. tightened flexural checked to configuration a power either gauge described gauges the gauge shaker on was the the power waveform connected appropriate, tion, any and strain springs pounds capacitance an The preamplifier The the a of from required minimum. 64 when misalignment aligned, power of only waveform attached force The removed for were at connected hours in - 34 Preliminary acceleration for tests level levels next necessary c o u l d be used the particular control were over system studied. could not provide constant over the entire choice would frequency be a c o n s t a n t which displacement range level acceleration level used would record over most of the frequency best the procedure would experimental trol. A The was that t h e power some point these the into range are used input power same t i m e t o o many segments into range bei n - The power of this that the would noise. a constant next plotj but type of con- 'approximated constant power i s the division a number level over t o t h e specimen segment. i s an o p t i m i z a t i o n delivered power Table and c o r r e s p o n d i n g stant, maximum In this power reaches plot was such a maximum a t chosen and s i z e of so t h a t t h e i s maximum w h i l e a t of the frequency IV-1 i n d i c a t e s a first of accelera- segments T h e number acceleration way each Different frequency t o t h e specimen segments. studied. o f parts over process avoiding the division column spectra i s constant. different i n the frequency segments average an constant the acceleration levels t o be so s m a l l i s not capable called approximated tion been acceleration used. the frequency which have The unsuitable since the inherent electronic system compromise, spectra', of from exciter or velocity sinusoidal however, have range investigated. This, distinguishable also what frequency The v i b r a t i o n excitation. strain was to ascertain the frequency levels used approximation achieved while range making f o r the to a con- use o f - the experimental control. system c a p a b i l i t y f o r constant The a m p l i t u d e as p o s s i b l e 35 - so t h a t o f v i b r a t i o n was as h i g h ACCELERATION LEVEL 750 Hz 7.0 750 850 4.0 850 1200 20.0 1200 2500 15.0 2500 5100 40.0 5100 8000 20.0 8000 9000 55.0 9000 10200 20 .0 to kept t h e s t r a i n s were more r e a d i l y m e a s u r e d . FREQUENCY RANGE 250 thus acceleration T a b l e IV-1 P e r m i s s i b l e A c c e l e r a t i o n L e v e l s Over V a r i o u s F r e q u e n c y Ranges f o r Column S p e c i m e n The t e s t i n g was now ready to begin. The following s t e p s were a d h e r e d t o : (1) the generator deliver (2) the desired the frequency generator c o n t r o l was programmed t o acceleration scanning speed and c o r r e s p o n d i n g level r e c o r d e r were c h o s e n speed was u s u a l l y state conditions). used level. on t h e c o n t r o l paper speed (the lowest t o approximate on t h e scanning steady- (3) 36 - the compressor stability (4) (5) i n the the frequency was speed with unit chosen to provide feedback.circuit indicated synchronized the control was was by t h e l e v e l the control put i n i t s recorder unit excitation mode (6) the proper analyzer (7) i n the frequency was s e t the attenuation recorder (8) amplification and w r i t i n g speed of the level were s e t t h e Roots blower was activated to cool the shaker (9) (10) These frequency from of interest signals the scanning steps were extended possibility At analysis were over was made performed over was each to another a few H e r t z of losing activated. segment of the of the actual due t o l a r g e so t h a t their information display change transient the frequency end points at the to avoid acceleration frequency. particular made. record level to appropriately mechanism repeated No The t e s t s changeover that the one a c c e l e r a t i o n segments were adjusted spectra. voltages. the t h e CRO w a a frequencies The p r o c e d u r e was much the frequency of interest, followed t h e same scanning as t h a t spectral i n obtaining outlined mechanism a above, was n o t analyses spectral except activated, the frequency mechanical level frequency recorder quency test a n a l y z e r was s y n c h r o n i z a t i o n was and frequency of interest was put i n the analysis was analyzer. mode, and a p r o v i d e d between t h e The e x c i t a t i o n s e t on t h e c o n t r o l generator fre- and t h e carried out. Photography Photographs were obtained particular over interest Photographs oscilloscope Spotmatic of of strain, most o f the frequency a r e shown i n Chapter o f waveforms were i n the normal Camera w i t h second, acceleration with taken triggering and power range. waveforms Those o f IV. directly mode using from the a Pentax an f / s t o p o f 2.8, an a p e r a t u r e a 55 mm. lens and a no. 3 c l o s e - u p speed lens. Oscosslooi ©f R e s u l t ; - 38 - CHAPTER RESULTS Interpretation The a chart Fig. the DISCUSSION of Frequency strain versus produced V-3). AND To by V Spectra frequency the level p l o t s were recorder correlate the record following parameters must (a) t h e gauge (b) the strain be obtained ( s e e F i g . V-2 with and the measured strain, known: f a c t o r and b r i d g e gauge on bridge arrangement excitation voltage used (c) the magnification o f t h e BAM and voltage amplifier (d) the attenuation The bridge The magnification pendent. e x c i t a t i o n voltage o f t h e BAM I t i s apparent voltage level reason, and because to study to a true was carried out at resonant cludes this peaks that not carried of these out. was was small minimized (less intent that and dependent. frequency de- o f the measured For this of the experiments i s v i b r a t i o n s , such a con- c a l c u l a t i o n s were of interest Care must records, since electromagnetic and t h e i n f l u e n c e 1 db). time i s tedious. Approximate vibrations. a c o n t r i b u t i o n from effect record frequencies of the strain time the conversion strain recorder is slightly i s both of predicted version interpretation used the primary the existence significance of the level be t o study the exercised i n the signal i n induction. on the However, resonant - 39 The records frequencies a t which peaks o c c u r r e d were s t u d i e d , and t h e s t r a i n q u e n c i e s were a n a l y z e d . sented and d i s c u s s e d Identification The several to between By to identify 'linear linear verse records obtained oscillations Resonant 1 Some o f t h e s e t h e peaks i n o r d e r flexural i n order analytically. o b j e c t i v e was n o t r e a l i z e d . where between c l a m p e d frequencies frequencies of various transverse of strain frequencies mental points resonant resonant the experimental setup this F i g . V - l i n d i c a t e s the modes theory f o r pinned accounting and c l a m p e d f o r t h e con- A l s o p l o t t e d on t h e g r a p h a r e t h e peaks b e l i e v e d t o c o r r e s p o n d o f t h e a c t u a l specimen. are obtained trans- A knowledge o f t h e e n d clamped end c o n d i t i o n s , and p i n n e d . compressive end l o a d . first. The end c o n d i t i o n s were some- ends as c a l c u l a t e d f r o m l i n e a r stant peaks. t o determine the Although was d i r e c t e d t o w a r d s o b t a i n i n g resonant to distinguish are considered c a n be c a l c u l a t e d . i s necessary correspond I t i s therefore p e a k s and n o n l i n e a r oscillations peaks a t peaks (Appendix C ) , t h e approximate frequencies frequencies have r e p r o d u c e a b l e o f t h e specimen. resonant theory conditions are pre- chapter. discrete frequencies. desirable results fre- o f S t r a i n Peaks strain resonant waveforms a t t h e s e The e x p e r i m e n t a l i n this on t h e s t r a i n from t h e f l e x u r a l quency r e c o r d . The e x p e r i m e n t a l curve the frequencies o f clamped end o s c i l l a t i o n s end vibrations. to The e x p e r i - strain versus i s seen t o f a l l The n a t u r a l f r e q u e n c i e s resonant and t h o s e of transverse fre- between of pinned vibra- - 40 - Fig. V - l . Forced Resonant Frequencies for Various Plotted Versus End C o n d i t i o n s Mode Number tion as t h e y test o f the boundary are plotted Resonant following. end loading, a t 8789 strain record Hz. the This mental Using linear theory, Hz (Appendix occurs would C). required A a t 8 700 H z . level indicate desire as a q u a l i t a t i v e and a c c o u n t i n g spike f o r the pre- at condition to oscillate increase the acceleration required increased resonant would i s t h e power of acceleration o f t h e column in on t h e l o n g i t u d i n a l Further, a probable to l i m i t are considered l o n g i t u d i n a l resonance l o n g i t u d i n a l frequency loading V - l serve conditions. a constant natural in Fig. the fundamental dicted sustain - longitudinal oscillations the to 41 at the the 8700 since funda- external to the preset level. Finally, using linear stiffness t o r s i o n a l resonances theory, considering and t h e compressive frequencies are c a l c u l a t e d are considered. the decrease end load, (Appendix C). t o r s i o n a l resonance occurs second t o r s i o n a l harmonics are calculated on No the l o n g i t u d i n a l s t r a i n 3380 H z , nance appear at respectively. so i t appears i s not excited. 10000 torsional Hz. These vibrations record that will be with strain computed Hz; the f i r s t a t 6 760 Hz strain peak fundaand and i s observed i n the neighbourhood torsional of reso- similar characteristics record discussed i n a later resonant The the fundamental Peaks on t h e e x p e r i m e n t a l a t 3380 significant in torsional the predicted mental 10140 Hz Again, a t 6920 Hz i n connection section of this and with chapter. again Results of Flexural Fig. for identified section that Strain V-2 s h o w s the midpoint 42 strain o f t h e column. versus Most flexural strain peaks i s expected • -t < < i i than even those a t t h e column i ( ( ( i if f made resonances. with frequency of the strain according t o the deductions transverse resonances This Record aflexural concerning smaller - peaks a r e i n the previous I t i s noteworthy mode n u m b e r s having record o d d mode appear as numbers. midpoint. f -c —f—r — r —r—r—r — r— r— r— r — r ed Flexural Mode -Fifth Flexural Mode E,i .,„ Second Coupled Flexural Mode= 1 —r —r—r—r—r—r —r—r—r—r—r—r—r—i' —r —r —r —t —r —i Tenth-ElexyraLModp^^ m i n l a r l Flexural F l A v u m l Mqde M o d e ^ First Coupled EXCITATION ERECIUENCY^ Flexural As shown specimen column the spike F i g . V-2 Obtained i s at o r near midpoint exhibits strain number, larger Record 9 10 kHz at Midpoint o f Column s c h e m a t i c a l l y i n F i g . V-3, t h e midpoint flexural mode a Strain will result. f o r an e v e n relatively midpoint little i s near a n d an a t t e n d a n t of the mode n u m b e r . F o r an o d d an a n t i n o d e larger The c u r v a t u r e , and i s c o r r e s p o n d i n g l y low. t h e column curvature, a node 1—r —r —r—r- —i Ninths Fl exural Mode ein amplitude having strain - 43 - Mode S h a p e at M i n i m u m Mode Shape at M a x i m u m Extension Extension Midpoint Flexural Displacement ^ M i d p o i n t Flexural ,' Displacement P Q ODD Possible not F i g . V-3. Shapes F o r A x i a l l y Mode yet accounted f o r . frequency frequency digital the EVEN NUMBER significant flexural citation tion MODE Flexural Two P Rcosrt + counter. vibration Except of at low 'fluttering' oscilloscope; of The about strain first 2900 approximately spikes 4 350 the Hz occurs second as excitation levels, feedback the v i b r a t i o n r e s p o n s e waveforms circuit i s not Column a t an a t an indicated a t e n d e n c y t o become The NUMBER appear which system has the MODE In the neighbourhood of these rapidly. PcosTt Excited of these Hz; + D two are ex- excitaby the frequencies, unstable. system flutter capable of begins on the stabili- zing the oscillation. citation the i s lowered, response the stability, on system described earlier. s t i l l same the phenomena strain waveform frequency approximately on the tion a strain record. The when fundamental excitation in the are observed bars at spikes on the and studied. of system jump to waveforms response this the is jump If flexural excitation. passes through vibration at spike appears those excita- to Hz, fle-xural while and of when the at resonances as cross-hatched strain non- peaks are coincides.with the 8700 the the to strain associated with frequency half flexural flexural frequency i s one corresponding the theory, Nonlinear excitation 4 350 of of indicates frequencies longitudinal frequency so longitudinal correspond linear peaks. frequency be oscillation. frequency F i g . V-4 strain excitation resonant unstable nonlinear strain which ordinary indicate present nal solid of ex- record. shown at a sudden fluttering that excitation small in frequency one-half can condition of the type 8700 frequencies linear be a the the becomes s t a b l e frequency, fundamental resonant bars frequency o c c u r r i n g , the Hz from by level.of 4350. Hz results excitation the and level indicated would system Hz f o r the frequencies by 2900 a snap-through histogram predicted the hand, The when flexural The as the were Finally, the as i s not snap-through at other instability acceleration however, acceleration the - I f the waveforms Increasing 44 Hz, when fundamental excitation the longitudi- frequency - 45 is one t h i r d 2900 of the fundamental longitudinal frequency Hz. EXCITATION FREQUENCY Frequencies The resonant F i g . V-4. o f Experimentally Observed Resonant S t r a i n Peaks transverse quencies are referred coupled f l e x u r a l modes histogram at at which i n F i g . V-4 coupled condition respectively. indicate f l e x u r a l modes Flexural occurring t o as t h e f i r s t , those kHz at these second, Arrows and t h i r d below t h e excitation are expected fre- frequencies from theoretical considerations. Shown i n F i g . V-5 a r e p h o t o g r a p h s corresponding The upper of several to o s c i l l a t i o n s o f the coupled waveform i n each picture waveforms flexural i s the excitation modes. voltage - 4 6 - • First Coupled Flexural Mode t (Excitation Order Frequency=8700 of Signals(top Excitation Flexural to Hz bottom) Voltage Strain Acceleration Second Coupled Excitation Order Flexural Mode Frequency=4350 of Signals(top Excitation Flexural Hz t o bottom) Voltage Strain Acceleration Third Coupled Excitation Order Flexural Frequency=2900 of Signals(top Excitation Flexural Strain F i g . V-5 Corresponding t o Coupled Flexural Modes Hz t o bottom) Voltage Acceleration Waveforms Mode supplied to the shaker, the middle waveform is the flexural s t r a i n level at the midpoint of the column, and the lower waveform i s the acceleration monitored at the lower end mount of the column. The time base i s the same for a l l the wave- forms in each picture. The s t r a i n at the column midpoint corresponding to the second and t h i r d flexural modes are comparable in magnitude to the strains associated with l i n e a r flexural resonances, while the strain level at the first coupled flexural mode i s somewhat less. The nonlinearity at an excitation frequency of approxi- mately 8700 Hz may be the result of longitudinal i n e r t i a forces, which can influence the dynamic behavior of a column when the frequency of the^external force is near the l o n g i - tudinal natural frequency of the column; that i s , when the longitudinal vibrations have a resonance character. In other- words, t h i s nonlinearity may represent the parametric influenc of resonant longitudinal vibrations which give rise to a f l e x u r a l vibration as indicated in the theory by the term (u w ) . x x x The second and t h i r d coupled flexural modes may be parametrically excited in a s i m i l a r manner. Recalling the theoretical predictions, the transverse response was seen to- be,";appT5S?mately sinusiodal on the substitution (equation 2-12) . This apparently describes first the. transverse motion quite well except when the fundamental longitudinal frequency i s excited. On the t h i r d substitution the transverse motion included a sinusoidal term with twice the excitation describes the the frequency response fundamental third from coupled the ( e q u a t i o n 2-19) when the longitudinal flexural term appearance of This excitation frequency. resonances sinusoidal . having twice frequency The exhibit second no the better equals and contribution excitation fre- quency . The indicates do occur. flexural by that the vibrations arrows lower not is compared the provided the on Axial V-6 However, a f o r the than second and at the second coupled frequencies i n the flexural strain below 1000 amplitude of these the amplitude, of the of the excitation. frequency indicated Hz. are irregularities waveform Further, No strain noted These primary response. flexural were the tests explanation is peak which is not Hz. Strain Record axial strain addition small to record the influence bridge is directly vibration the 900 In flexural V-4. remaining i s an column. contains voltage high f o r at of Fig. also frequency f o r one accounted to first anticipated nonlinearities focused Results the frequencies, particularly small were of in Fig. coupled vibrations only were discussed, since having third coupled Theoretically, Small at higher the due axial to configuration proportional to f o r the strain, the used, the midpoint the flexural the axial of record strain. measured strain and the - Axial Strain of the flexural strain a n d maximum the to order strain. axial record. mately about The 4350 H z , these third Fig. three spikes coupled V-6. frequencies when axial strain o f coupled that i s and a r e o f the contribution negligible. on t h e a x i a l strain frequency the excitation occurs when flexural the of approxi- frequency i s excitation The r e s o n a n t conditions are referred t o as t h e f i r s t , modes The e x c i t a t i o n resonant strain appear and t h e t h i r d i s 2900 H z . strain are comparable i s a t an e x c i t a t i o n 8700 Hz, t h e s e c o n d frequency and first spikes o f Column t h e maximum i t i s apparent t h e r e c o r d by t h e f l e x u r a l prominent Since strain o f 200 u i n / i n , Three to - F i g . V-6. Obtained at Midpoint Record square 49 respectively, frequencies spikes resonances second as i n d i c a t e d associated with coincide with flexural corresponding the in these excitation discussed earlier. - In general, quickly peaks than resonant V-6 these of a t two these away Hz, response the central if i t i s originally is under it c a n be strain AB 1 that i n those Fiber through AB angle metrically ©. 6920 Hz a n d 10000 The histogram shown are observed indicate torsional Hz. The shape sweeps excitation twisting of through of puts torsional the fibres Remembering t h a t t h e column will the central when to Fig. reduce fibre, t h e column torsional oscillations strain will level at excitation V-7 the such as i s twisted be are para- reduced. frequencies of Hz. frequencies bars first i n tension, t o AB' the axial at which bars The o f t h e column from when frequencies hatched have record i n system o f t h e column i n F i g . V-6 more strain l o a d i n g , and r e f e r r i n g elongate excited, i s t h e case fibre away Thus, The strain i s a t 10000 as s t e e l , twisting fibres must Such axial up oscillations parametric unloaded. compressive seen peaks. as t h e v i b r a t i o n plane build o f magnitude. the second For m a t e r i a l s such from peaks on t h e a x i a l frequencies indicates modes. strain and a x i a l order appear strain frequencies of excitation. i s a t 6920 the s t r a i n axial flexural peaks - flexural i n t h e same Inverted Fig. resonant f o r resonant amplitudes 50 strain correspond on spikes arrows strain axial frequencies The indicates those corresponding the axial t o coupled excitation resonances. i n F i g . V-8 to record. modes while associated with below the histogram excitation resonant The the crosssolid coupled indicate - 51 A Portion of A - Fig. V-7. Column U n d e r g o i n g Twisting A I 8? 8 EXCITATION FREQUENCY Frequen c i e s Fig. V-8. of Experimentally Observed Axial kHz Strain Peaks those excitation modes.were record. tions of Two axial a t which anticipated discrepancies between results i s significant the theoretical considerations. torsional mode, when t h e e x c i t a t i o n the fundamental However, frequency i t appears on t h e a x i a l though frequency The third the second t o be e x c i t e d , frequency to the third was expec- i n the neighbourhood t o be p a r a m e t r i c a l l y corresponding predic- i t i s not present Further, was excited strain the theoretical i f i t were longitudinal parametrically are apparent. mode coupled ted theoretically and e x p e r i m e n t a l coupled in frequencies a t 8700 H z . excited torsional First mode. Coupled (Fundamental Resonance) Excitation at the Axial Longitudinal Frequency=8700 Order o f Signals bottom) Excitation Axial Mode (topt o Voltage Strain Acceleration Waveforms Fig. an V-9 i s a p i c t u r e excitation excitation Fig. V-9. a t Fundamental Longitudinal frequency frequencies Resonance o f t h e waveforms o f 8700 Hz. Similar i n the neighbourhood corresponding to waveforms f o r o f 4 350 Hz a n d - 2900 Hz a r e shown A spectral 53 i n F i g . V-10 analysis, f r e q u e n c i e s was revealed: a t an e x c i t a t i o n tion frequency quency 2900 The vibration 8700 Hz Hz o f 4350 Hz exists; and terms and longitudinal having twice mode The response the e x c i t a t i o n at e x c i t a t i o n and frequency. i s therefore expected a was resonant a t an of fre- frequency 8700 t o be excita- of Hz exists. significant frequencies of development, suggested t h e sum frequency Hz, vibration (2-8) a p p e a r was these exists; of frequency theoretical the excitation o f 3700 8700 at following a t an e x c i t a t i o n i n equation Hz the frequency vibration predicted theoretically Hz. The a longitudinal as 4350 signal performed. of frequency a longitudinal coupled and V - l l r e s p e c t i v e l y . of the strain excitation longitudinal - a sinusoidal The t o be of a sinusoidal excited when that term term fundamental 8700 having longitudinal the excitation '.'i • r freq.uency i s one h a l f o f "the frequency. excitation Hz soidal tal when frequencies term vibration phase of the strain having term system moves relationship longitudinal t h e two points A phase the excitation through frequency As in Fig.V - l l change the excitation this i s illustrated sinusoidal a n d A'. <9 twice having waveforms frequencies i n the neighbourhood reveals a continuous sinusoidal longitudinal * Observation 4350 fundamental are the frequency frequency frequency frequency excited ' i n phase' range the of sinu- and t h e as t h e range. i n F i g . V-12. i s first terms between for The in Fig. This fundamenV-12(a) as i n d i c a t e d i s scanned, by t h e two - 54 - ! Excitation I , Order of Frequency Signals Excitation Axial = 4320 (top to Hz bottom) Voltage Strain A c c e l e r a t ioi. Excitation Order of Frequency Signals Excitation Axial = 4345 (top to Hz bottom) Voltage Strain Accele ration Excitation Order of Frequency Signals Excitation Axial = 4360 (top t o Voltage Strain Acceleration Waveforms Fig. V-10. Corresponding to Second Coupled Axial Mode Hz bottom) - 55- Excitation Order Frequency of Signals Excitation Axial = 2890 Hz ( t o p t o bottom) Voltage Strain Acceleration Excitation Order » Frequency of Signals Excitation Axial = 2905 Hz ( t o p t o bottom) Voltage Strain Acceleration Excitation Order Frequency of Signals Excitation Axial Fig. V - l l . Corresponding to Third Coupled Voltage Strain Axial 2910 Hz ( t o p t o bottom) Acceleration Waveforms = Mode Increasing (N Excitation Frequency Y (c) lb) F i g . V-12. I l l u s t r a t i o n Showing Changing Phase R e l a t i o n s Between T h e Two V i b r a t i o n s C o m p r i s i n g t h e S e c o n d C o u p l e d A x i a l Mode waveforms point A move i n F i g . V-12(b). two waveforms of 'out o f phase' Finally, shown by . -S7 between of a complete i n F i g . V-12(c), o f the fundamental l o n g i t u d i n a l discontinues. t h e movement t h e phase has changed by one h a l f the excitation, citation as shown and h e r e resonant of the cycle t h e excondition First Coupled Excitation Order Torsional Frequency of Signals = 6920 Hz ( t o p t o bottom) Excitation Axial Mode Voltage Strain Acceleration Second Coupled Excitation Order Torsional Frequency of Signals Excitation Axial Mode = 10000 Hz ( t o p t o bottom) Voltage Strain Acceleration Waveforms The coupled and ted, C o r r e s p o n d i n g t o F i r s t and Second T o r s i o n a l Modes waveforms torsional 10000 Hz motion modes a t e x c i t a t i o n respectively the strain coupled corresponding to the f i r s t waveform torsional mode t h e second and frequencies second o f 6920 a r e shown i n F i g . V - 1 3 . remains sinusoidal i s excited, h a s t h e same f r e q u e n c y p o n s e when Coupled coupled since when As the load. torsional i s mode expec- first the resultant as t h e a p p l i e d Hz axial The excited, res- - 58 however, able i s somewhat here, strain will due t o t h e p a r a m e t r i c strain excitation by the since the frequency be d i f f e r e n t axial nonlinear* than frequency of the variation excitation the frequency due t o t h e l o n g i t u d i n a l of the third observing A nonlinearity that a scale. torsional "linear 1 i n the axial of a torsional of the variation excitation. mode flexural i s reason- might be resonance mode, of the Parametric accounted f o r i s nearby on AjeuuuuoiQ - 59 CHAPTER V I SUMMARY, & CONCLUSIONS Summary An experimental behavior o f a column Observed resonant vibrations investigation was made o f t h e dynamic subjected to periodic forced a r e compared vibrations with those axial loading. and n o n l i n e a r p r e d i c t e d by coupled theoretical considerations. To accomplish differential to this equations study, o f motion describe the relationship oscillations particular plane o f t h e column relating from and between importance to axial inertia. interpreting longitudinal were vibrations Finally, having the second expected t o be f o r t h e column and strains motion. initial These flexural vibrations. i n the Of central terms, In a d d i t i o n , ratio and p o s s i b l y a arise and longi- the equations several predicted. with strains crookedness manipulating were a frequency axial partial i n formulating the equations vibrations anticipated. derived initial a r e assumed vibrations nonlinear and t o r s i o n a l some o f t h e c o u p l e d coupled were i s that By s u i t a b l y resonant 1:2 axial and f l e x u r a l coupled were between considerations regarding tudinal coupled nonlinear Firstly, frequency two ratio of two c o u p l e d flexural o f 1:2 w e r e expected. the third parametrically excited. and torsional modes - 60 The with experimental the theoretical was found the excitation mode the when when flexural the excitation longitudinal mental longitudinal indicated that with frequency. parametric coincided resonant These response influence resonant was with axial fundamental o f 1:2. the vibrations fundamental of the funda- A waveform t o that of response resonant at these equal when one h a l f of the and one h a l f frequency. a frequency coupled ratio exhibited response fundamental and t h e a frequency frequency the resonant sinusoidal tion frequency resonant frequency the frequency also agreement resonance to the frequency, hence response axial The second the excitation longitudinal good The l o n g i t u d i n a l coupled frequency. resonance; provided corresponded a vibration'having longitudinal was frequency fundamental contained the f i r s t resonant appeared The predictions. to exhibit longitudinal investigation two analysis frequencies of the excita- o s c i l l a t i o n s represent the of longitudinal o s c i l l a t i o n s on flexural oscillations. Parametrically ved when the longitudinal fundamental frequency third torsional was equal torsional Further and flexural quency excited excitation frequency, resonances frequency and again t o the frequency when were was twice the the corresponding obser- applied to the mode. coupled response equaled torsional o s c i l l a t i o n s were when one t h i r d observed the longitudinal of the fundamental on t h e a x i a l excitation fre- longitudinal resonant axial resonance ration nal frequency. resonance. the The that analysis of coupled having of the the•response frequency third vibration - waveform indicated exhibiting sinusoidal A 61 the the coupled contained fundamental flexural same third vib- longitudi- resonance frequency a as was the a applied loading. Both the the axial quickly coupled dynamic than Coupled those 'linear' with coupled fundamental rations modes of strain resonant record. of different phase frequency hibiting the the i s less with was passed. primary when two vibrations were discontinued tudinal higher each other ' i n phase at 1 f r e q u e n c i e s the with the component frequency 90° 'behind' having the strain two vib- coupled the the axial ex- appeared excitation the • coupled waveform became to associated at as frequency amplitude more level The the on comparable strain the out level incipient having resonant at strain the peaks dynamic frequency. longitudinal parametric and than The waveform died amplitudes The respect to The frequency, and frequencies comprising quency. the up strain flexural modes. resonant fundamental the on modes h a v e flexural modes resonant record built peaks longitudinal resonant of 'linear' flexural axial changed 'ahead' and 90° fre- maximum resonant resonant condition the fundamental primary vibration. longi- Conclusions The following theoretical and c o n c l u s i o n s were experimental drawn analysis on from the the foregoing flexural, longitudinal, to periodic 1. and t o r s i o n a l axial coupled anticipated longitudinal and f l e x u r a l longitudinal vibration applied loading resonant frequency as t h e e x t e r n a l which derations. frequency quency . ing the frequency ponse frequency though of ratio i t may a resonant frequency tains tion n o t be ratio a third o f 1:3 flexural load This o f two having a t h e same resonance also fre- exhibit- exists. indicates that vibrations the having seems the axial three excitation longitudinal The e x p e r i m e n t a l that consi- fundamental the c o n t a i n i n g two v i b r a t i o n s suggests experi- the when interpretation oscillation exhibiting frequency. frequency that i s excited development complete. condition of the longitudinal i s observed suggests coupled i s t h e sum o f 1:2. fundamental f o r by t h e t h e o r e t i c a l of the applied o f t h e column response o f the fundamental The a n a l y t i c a l between the frequency excitation vibration vibration i s one t h i r d The as loading. analysis A corresponding 3. same i s not accounted resonant to exist o f the fundamental coupled A waveform longitudinal when on t h e f l e x u r a l resonant appear oscillations. At t h i s appears mentally subjected due t o t h e i n t e r a c t i o n i s excited resonance A vibrations i s one h a l f frequency. 2. o f a column loading: Resonant theoretically response a correct, observation with response times res- the a con- excita- 4. Theoretically, the f l e x u r a l response sum o f two v i b r a t i o n s with the predicted resonances results coupled reveal no c o n t r i b u t i o n the vibration is, the f l e x u r a l response frequency having over 5. twice to the fundamental torsional ted that mode higher with the excitation range 'linear' 6. quite of f l e x u r a l resonance amplitudes. slightly peaks less. build resonances. resonance other axial phase as t h e r e s o n a n t fundamental, was also more vibrations frequency that a axial flexural are resonant are i n general resonant than comprising appear amplitudes amplitudes sharply by a t o t a l noted nearby. 'linear' resonant be This to the fact f l e x u r a l resonant as expec- corresponding t o t o r s i o n a l mode. be a t t r i b u t e d The two v i b r a t i o n s change frequency and ' l i n e a r ' up a n d d e c l i n e the was i s close t o r s i o n a l mode was resonant o f magnitude Coupled could of a l l coupled Coupled Coupled frequency, A coupled The amplitudes frequency similarily e x c i t a t i o n might t h e same o r d e r the applied resonances f o rthe third significant. torsional vibrations I t i s t o be a t an e x c i t a t i o n That investigated. parametrically. experimentally parametric from i s sinusoidal excited. frequency the experimental to t h e f l e x u r a l response torsional torsional While frequency. When i s excited o f 1:2. load parametrically the do o c c u r , The e x i s t e n c e o f c o u p l e d experimentally. ratio the external the frequency verified twice a frequency i s also the strain do f l e x u r a l a coupled axial o f 180° r e l a t i v e t o each i s passed. T h e maximum - 64 - • amplitude are resonant condition appears vibrations r First would done f o r Future Research considerations probably to date. i n suggesting aim a t overcoming To t h i s equations 'derived predictions could coupled oscillations; excitation tional could then could variables be be b e made similar t o t h e one. u s e d . and/or damping Though solution, meters through i t would might damping damping n o t be oscillations apparatus be extended and ' l i n e a r ' of viscoelastic to the and i s o l a t i o n surfaces properties i n evaluating the materials. obtained describing results. addi- theoretical then layers as f r u i t f u l solution of and temperature the application be o f i n t e r e s t equations experimental on c o u p l e d and of response an e x p e r i m e n t a l parameters would a numerical excited damping The study o f various differential plement ally probably layers the amplitudes and t h e i n f l u e n c e of c o n t r o l l i n g coupled Optimizing effectiveness the using elastic-viscoelastic the bar. for be n e c e s s a r y f o rt h e Analytical V e r i f i c a t i o n f o r these would oscillations, concerning as i n t e r n a l results research solution i s desirable. investigated; such means form t h e r e l a t i o n s h i p between considered. evaluate further t h e l i m i t a t i o n s o f t h e work end, a c l o s e d theoretical of t h e two ' i n phase . Suggestions to when as an a n a l y t i c a l on a c o m p u t e r f o r t h e column The s i g n i f i c a n c e and t h e i n f l u e n c e v i b r a t i o n s :fcould be could of parametric- of various investigated. com- para- The - 65 major not shortcoming say anything to the type is much more Other ratus the In would could tor wise be of Boundary practical the at of axial and higher of control Attention should be column behavior. BAM is essential invesigated favourable signals are a using are small. sensitive would at amplifier Fotonic the Sensor the the Fotonic to Con- pieceThe corresponding unknown. means of superior to are the the be have measured noise and to must Sensor. i s required, genera- work. amplifier electrical column useful. various since of control amplitudes The clamped control is An appro- than frequencies the appa- simultaneously. of for this on characteristics, of be be response b e t t e r than vibrations gauges. use i n the points behavior used Alternatively, e l i m i n a t e d through very be focused however, multichannel displacement i f response strain noise the vibration KHz would longitudinal the unit A oscillations monitoring very monitor A rise solution. inherent interest. acceleration coupled harmonic 10 give solution, conditions other flexural than might analytical are i t does conditions should providing constant level a several different frequencies constant an study i s desirable to displacement 1 the Such than traced simultaneously. importance to observed. on is that mechanisms which obtained closely. specimen higher stant the solution d e s i r e d boundary system capable much The way, be about response also column this numerical limitations more recording a readily used. ximated ends of of problems However, great care - 66 must the be e x e r c i s e d Sensor developed probe. in. p r o v i d i n g A t some to accurately particular frequencies measuring t h e column urements To and o p t i c a l further oscillations, axis o f t h e column sional movement conducted of oscillations subjected <7 of interest. gauges be since improved other Finally, on t h e b e h a v i o r holder be sound at of level meas- torsional a t 45°to the major the s e n s i t i v i t y Tests cross-section the effect excitation will be considered. thereby. o f more could methods excited positioned help, having Other should system f o r o f t h e column including parametrically interest. t o dynamic the length techniques would on columns practical scan might mounting expense,' a p r o b e vibration, study strain a rigid of complicated deserve t o tor- could be. geometries coupled structures attention. - 67 - BIBLIOGRAPHY 1. L o v e , A . E . H . , "A T r e a t i s e o n t h e M a t h e m a t i c a l E l a s t i c i t y " , 3rd Ed., Cambridge, U n i v e r s i t y Theory Press, of BT20 2. B e l i a e v , N.M., " S t a b i l i t y of Prismatic Rods.Subject to Variable Longitudinal Forces", Collection of Engineering C o n s t r u c t i o n and S t r u c t u r a l M e c h a n i c s (Inzhinernye s o o r z h e i i a i s t r o i t e l ' n a i a mekhanika), L e n i n g r a d , Put, 1924 3. 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 , 1 s t Ed.., F l u e g g e , W., editor, McGraw-Hill B o o k Company, I n c . 1962 4. Somerset, J.H., "Parametric Instability of E l a s t i c Columns", S y r a c u s e U n i v e r s i t y R e s e a r c h I n s t i t u t e TR SURI no. 1 0 5 3 - 7, J a n u a r y , 1963 5. Evan-Iwanoski, R.M., " P a r a m e t r i c (Dynamic) S t a b i l i t y o f E l a s t i c Systems", Proceedings of the F i r s t Southeastern C o n f e r e n c e on T h e o r e t i c a l a n d A p p l i e d Mechanics", P l e n u m P r e s s , 1 9 6 3 , p p . I l l - 130 6. B 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 ( t r a n s l a t e d from R u s s i a n ) , H o l d e n - D a y , San Calif., 1964 7. Somerset, J.H., and E v a n - I w a n o s k i , 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 o f C o l u m n s " , P r o c e e d i n g s o f the S e c o n d S o u t h e a s t e r n C o n f e r e n c e on T h e o r e t i c a l and A p p l i e d M e c h a n i c s , A t l a n t a , G a . , M a r c h , 196 4, p p . 503 525 Systems", Fransisco, S t a b i l i t y of a Bar Applied Mechanics, - 8. T s o , W.K., Parametric Torsional Axial Excitation", Journal of M a r c h 196 8, p p . 13 - 19 Under 9. S c h n e i d e r , B.C., "Experimental Investigation of Nonlinear C o u p l e d V i b r a t i o n s o f B a r s 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 o f B r i t i s h C o l u m b i a , A p r i l , 1969 Append.c< - 68 APPENDIX DEVELOPMENT OF S T R A I N ACCOUNTING The [3] results into the following column derivation exhibit, small. The Lagrangian dynamic deflections lection curve. cription tions of strain. lies dynamic provided definition that curve, while coordinate system i n which Fig. plane represent show A - l shows o f a column relative Love's plane between relative deduction the i n i t i a l takes displacement are i s used; that i s , Eulerian t h e two defdes- deriva- presented t o an assumes initial a deflection fixed o f any i s zero. undergoing plane displacement displacements which t o an i n i t i a l schematically a fiber the s t a t i c dynamic of strain i n the derivation are measured by M e t t l e r displacements [ 1 ] d e r i v e s an a n a l o g o u s i n the fact i n the neutral formulated or static these The d i f f e r e n c e deflections COLUMN f o r a column crookedness deflection point originally are measured Love FOR A CROOKEDNESS expression any i n i t i a l may EXPRESSION FOR I N I T I A L i n a strain account A motion. while relative of the central The b a r r e d the ordinary letters letters to the i n i t i a l deflection i n the central plane curve. An element o f t h e column before- deformation ds fiber has a l e n g t h (A-l) - 69 Extension The deformed ds The total length =V strain p = ds each theorem as by c + U x the u and x ) 2 A-l. of C e n t r a l element <* + Plane P'iber is 2 (A-2) 5 central fiber is - ds ds equation = the (A-3) radical binomial given of (1 of x Expanding Fig. Rotation and - + w 1 + in equations retaining (A-3) w x x + 4^ 2 x (A-l) second and order (A-2) terms, by the becomes 2 1 —w 22 xv (A-4) the strain - Assuming small i n i t i a l - 70 crookedness or s t a t i c transverse displacements, " and the 2 x « (A-5) 1 s t r a i n expression becomes 1 x x Equation f o r the static strains i n the strains central due plane fiber. plane i n two to bending f o r i n o b t a i n i n g the central (.A-bj i s t h e most g e n e r a l o r dynamic d e f l e c t i o n s Additional ted (A-6) o 2 x xx strain strain f i b e r of expression a column orthogonal or t w i s t i n g directions. must be a t some d i s t a n c e for from accouthe - 71 - APPENDIX DETAILS OF ELECTRONIC INSTRUMENTATION VIBRATION The following electronic vibration The level to peak 100 o f 0.1 i s also inches 1025 up t o peak; a maximum constant to the earth's ward or reverse over has frequency any synchronized speeds. dial A level fixed quencies frequency speeds recorder, a fixed actually dial up to change or speed angular 132 con- value of control where KHz. I t can The force. with other implies the velocity; logarithmically. range, frequency the scanning frequency the scanned with for- generator fixed that scan scan frequency control is 1 g to automatically logarithmically i s calibrated peak gravitational 10 i f used a maximum velocity value, the whole o f the range. scanning peak i s equipped spectrum scanning moves w i t h g's continuously over segments six fixed 1000 peak Constant acceleration due entire Automatic with constant the acceleration generator applied of providing t o 2 KHz and control of the the or acceleration. t o 2KHz w i t h and Kjaer capable KHz the to peak to control and 10 The or possible up work i s available inches a description i s a Bruel velocity, per second; attainable is control FOR specimen. C o n t r o l Type displacement, amplitude in this generator Exciter displacement trol used USED SYSTEM provide t o t h e column control Vibration CONTROL paragraphs apparatus B time, fre- since the The the vibration input to the control that generator. bias in The signal the level circuit table i s r e g u l a t e d back tion at regulation of the m o u n t e d on dual channel 10 mv the lower sensed slow of the constant i t deter- at the as correct shaker compressor as possible desired providing vibra- adequate a Bruel and K j a e r Type The Preamplifier Type 2622 has attenuator, which, when correctly output by was column mount. Accelerometer p r o v i d e s an g a shaker. sensitivity per since such and the time choice of speed frequencies, while s t i l l stud justed, compressor A is great, until important change the circuits integration to normal. a c c e l e r o m e t e r used built-in compressor a sudden in circuits becomes t o o i n order to avoid d i s t o r t i o n low The A The i s very with which i s available. chosen i n the accelerometer i s circuits c o r r e s p o n d i n g l y drops the is vibration appears mines speeds speed the or compressor i s obtained. compressor from operation of these table generator output vibration signal regulation i f the shaker larger the feedback the signal on the accelerometer. Bruel 43 35 and Kjaer a ad- voltage channel of - 73 APPENDIX C LINEAR EQUATIONS Equation o f Motion The bending an linear for Flexural Vibrations partial vibrations axial FOR COLUMN VIBRATIONS differential o f a uniform, elastic column s u s t a i n i n g 0 Proceeding xxxx + P oxx W + A to a solution < t t .= W 0 of equation assumed t o v i b r a t e h a r m o n i c a l l y the describing the load P i s E I in equation a normal c o n f i g u r a t i o n . { (C-l), C ~ l ) t h e column i s a t a n a t u r a l f r e q u e n c y and Thus a s o l u t i o n i s assumed having form w = X(x)(Acospt + B s i n pt) (C-2) where X(x) i s a f u n c t i o n o f x and A and B a r e c o n s t a n t s . Substituting ordinary equation (C-2) i n t o e q u a t i o n differential d X 4 EI d equation 2 Po—TT d 7 + x 2 =-Ad>p s o l u t i o n s of equation end conditions furnish The s i m p l e s t case (C-3) s a t i s f y i n g the p r e s c r i b e d t h e a p p r o p r i a t e normal f u n c t i o n s . results i - S l n i f t h e ends o f t h e b a r a r e s i m p l y "T— where i i s an i n t e g e r . resultant (C-3) T h e s e c o n d i t i o n s a r e s a t i s f i e d by x equation X dx^ x The supported. (C-l) provides a n (C-4) The f r e q u e n c y equation (C-4) i s s u b s t i t u t e d i n t o e q u a t i o n resonant flexural results i f (C-3), and t h e f r e q u e n c i e s a r e g i v e n by - 74 - (Cwhere c 2 EI = . The c o m p r e s s i v e decrease the resonant resonant transverse work V-1 f o r pinned , using transverse by t h e s p r i n g s Equation of Motion The linear vibration 2 where tes c end load used of P in Fig. = 64 0 pounds set-up. of motion f o r the axial column i s ux x ( C u c i n this Vibrations equation elastic calculated " 6 ) = as b e f o r e . Assuming modes that t h e column of v i b r a t i o n , oscilla- a solution i s o f t h e form where = X(x)(c cos procedure t h e same x = 0, level tions, pt + c s i n 3 X(x) i s a For at 2 The to EI u is = i n one o f i t s n a t u r a l taken The f o r Longitudinal differential i s seen 0 are plotted i n the experimental of a straight, utt ends compressive provided P f o r t h e column and clamped a constant load frequencies. frequencies ends axial 5) 4 function followed pt) (C-7) o f x and c^ and c^ a r e i n obtaining the frequency as t h a t outlined above t h e column studied, the deflection and at x = L t h e a p p l i e d sinusoidal acceleration. the frequency S c o s 1ft = equation - p 2 c -ss ii n z C t 5 k &- c constants. for flexural load Using vibrations. remained provided these equation zero constant boundary condi- becomes: cos pt (C-8) - where S = amplitude mode o c c u r s , resonant of applied the e x c i t a t i o n frequency u = p. frequency ^ sin x = L, t h e l o n g i t u d i n a l by static of 25 p o u n d s , uin. considerations. tion (C-9) a l o n g the experimental nal resonant The with calculated by more forces and a p p l i e d Equation The 0 a load i s about f o ru into level 200 equa- S provided i n longitudi- results. resonant frequency a f e w Hz f o r a wide of 8790 Hz d o e s n o t range of end loading levels. f o rT o r s i o n a l Vibrations equation elastic describing column the torsional sustaining an a x i a l i s 2 ~ ae Z tt = ^ - xx ^Ah^ =0 Assuming that A the b a r performs o f frequency t ^ — , w 1 conditions, (C-10) + 3 torsion Hz 9) a n d an e n d tested value '- approximated I V - 1 ) an a p p r o x i m a t e o f 8790 linear differential ® where acceleration o f a uniform, Law 1 o f t h e column acceleration o f Motion oscillations P than c a n be Hooke s normal becomes (c deflection (Table frequency then a with the ) the acceleration tests change load this c o s p t When t coincides response Using the extension Substituting acceleration. The a x i a l e^( "p2 L s When 75 - a natural , . and u s i n g mode o f v i b r a t i o n i n . ,' .. , . the following c , , boundary U = U the 0; = t t frequency 0 76 = 0 at x = 0 Xcosft, Q= equation - 0 at x = L results nrra w In l " ~ calculating a, t h e f o l l o w i n g G = 11.5 x 10 p s i A = 4.66 x IO" h = 0.125 i n . 1 I. - 5.5 P = 64 l b s . 0 equation for the f i r s t and 10140 H z . values i 2 n . 2 ( f o r uniform end ,,.-4 . 4 x 10 . i = 6 . 1 1 x 1 0 (C-ll), three a r e used: 6 1*2 = I From (C-ll) -4 i n . 4 the resonant modes loading) n torsional are calculated frequencies a s : 3380 H z , 6760 Hz
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Experimental investigation of nonlinear coupled vibrations of columns Johnson, Dale P. 1970
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Title | Experimental investigation of nonlinear coupled vibrations of columns |
Creator |
Johnson, Dale P. |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | Coupling of the flexural, longitudinal, and torsional vibration modes of a column subjected to periodic axial loading was analytically and experimentally investigated. The initial crookedness of the column and longitudinal inertia give rise to coupled flexural-longitudinal vibrations. Further, the Weber effect and longitudinal inertia result in coupling between longitudinal and torsional oscillations. To assess the validity of the theory, an experimental apparatus was set up to axially excite a column using a vibration control generator and an electromagnetic shaker. The experimental results were in good agreement with the theoretical predictions. Coupled longitudinal vibrations exhibiting a frequency ratio of 1:2 were observed. Coupled flexural oscillations were also observed, though a frequency ratio was not established. Further, the experimental results suggest that coupled vibrations other than those theoretically expected were present. In particular, a longitudinal coupled vibration with a frequency ratio of 1:3 was observed, and a corresponding coupled flexural oscillation was present. A coupled torsional mode was experimentally observed when the applied frequency was twice the fundamental torsional frequency. A second coupled torsional mode appeared when the excitation frequency was three times the fundamental torsional frequency. The phase relationship between the coupled vibrations was observed. The resonant coupled vibrations were found to be significant at certain frequencies. |
Subject |
Columns Vibration |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0102031 |
URI | http://hdl.handle.net/2429/34870 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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