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Two-dimensional structural vibration induced by fluid flow past a circular cylindrical body Chow, Yu-Min 1964

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TWO-DIMENSIONAL STRUCTURAL VIBRATION INDUCED BY FLUID FLOW PAST A CIRCULAR CYLINDRICAL BODY  by YU-MIN CHOW B.S., National Taiwan U n i v e r s i t y T a i p e i , Taiwan, Republic of China. 1954  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Applied Science i n the Department of CIVIL ENGINEERING  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1964  In the  presenting  this thesis i n partial  r e q u i r e m e n t s f o r an a d v a n c e d  British  Columbia, I agree  available mission  f o r reference  f o r extensive  representatives.  the L i b r a r y  copying of t h i s thesis  agree  of  by the Head o f my Department  a  t  e  May  o r by  shall  n o t be a l l o w e d  permission*  C i v i l Engineering  1. 19 64  Columbia,  '  per-  that,copying or p u b l i -  f o r f i n a n c i a l gain  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 D  that  f o r scholarly  Yu-min Department  of •  s h a l l make i t f r e e l y ,  I further  I t i s understood  c a t i o n .of t h i s t h e s i s w i t h o u t my w r i t t e n  degree a t the U n i v e r s i t y  and s t u d y ,  p u r p o s e s may be g r a n t e d his  that  f u l f i l m e n t of  Chow  ii Abstract The i n v e s t i g a t i o n i s concerned with the v i b r a t i o n a l response of a c i r cular c y l i n d r i c a l body when subjected t o o s c i l l a t i n g l i f t and drag forces of varying frequency.  The c y l i n d r i c a l body was mounted on a long f l e x i b l e can-  t i l e v e r and the s t i f f n e s s of the cantilever could be varied to study the e f f e c t s of resonance, amplitude and induced damping.  An explanation of the  two-dimensional e x c i t a t i o n due to vortex shedding i s presented. Both longitudinal and transverse vibrations were induced and the f r e quency of the e x c i t a t i o n i n the longitudinal d i r e c t i o n was about twice that i n the l a t e r a l .  Therefore, f o r equal s t r u c t u r a l natural frequencies i n the  l a t e r a l and longitudinal d i r e c t i o n s , resonance i n the l a t e r a l d i r e c t i o n w i l l occur at twice the v e l o c i t y of the longitudinal resonance, assuming the Strouhal Number to be constant i n t h i s range.  C o e f f i c i e n t s of the e l a s t i c  response forces, i n both longitudinal and l a t e r a l directions near the l o n i tudinal resonance, were plotted f o r comparison. The development of i r r e g u l a r i t y i n the l a t e r a l v i b r a t i n g oscillogram caused by the longitudinal v i b r a t i o n was demonstrated.  I n the c r i t i c a l  Reynolds Number range the e f f e c t of a f i x e d eddy-starter wire was i n v e s t i gated and the excitations i n both l a t e r a l and longitudinal directions were altered. The resonance of l o n g i t u d i n a l v i b r a t i o n occurs at ^/v_ _ r©s 0  L a t e r a l amplitude i n t h i s region i s also increased.  =  0.3 to 0.5.  The peak of the curve:  vs R , i n the region o f longitudinal resonance, approaches tengentially g  the curve:  S  vs R .  The s t r u c t u r a l response c o e f f i c i e n t of v i b r a t i n g  l i f t and drag r i s e s at l o n g i t u d i n a l resonance to a value o f 3.0 t o 8.0, but drops quickly a f t e r resonance.  The damping c o e f f i c i e n t s of the structure  are i n the range of 0.25 to 0.60.  iii Since the motion of the v i b r a t i o n may influence the o s c i l l a t i n g  lift  and drag due to the eddy shedding, t h i s study presents an i n v e s t i g a t i o n of the longitudinal and l a t e r a l resonances i n a range of  R: g  5 x  3 x 10 ,  and the i n t e r a c t i o n of the exciation between the o s c i l l a t i n g l i f t and drag with the two-dimensional e l a s t i c response forces.  ACKNOWLEDGEMENT  The author wishes to express h i s gratitude to Dr. M.C. Quick for h i s guidance and c r i t i c i s m i n the experimental work and the w r i t i n g of the t h e s i s .  The author i s also g r a t e f u l to Prof. J.F.  Muir, head of C i v i l Engineering Department, f o r h i s comments and ' the grant of the assistantship. The experimental work was c a r r i e d out i n the hydraulic laboratory, Dept. of C i v i l Engineering, and appreciation i s also expressed by the author t o the s t a f f s i n the workshop.  V  TABLE OF CONTENTS  Abstract Acknowledgement Contents Page I.  Introduction  1  II.  Basic Concepts  3  III.  Qualitative Explanation of the Two-dimensional V i b r a t i o n Phenomena  7  IV.  Apparatus and Instrumentation 4.1 4.2 4.3 4.4  V.  General Consideration Water Flume Model Mounting Instrumentation  Testing Procedure and Analysis 5.1 5.2 5.3  VI.  Discussion of Results and Conclusion  30  Discussion Conclusion  Appendix 1. 2.  17  Calibration Experimental Results Analysis of Experimental Data  6.1 6.2 VII.  14  38  Structural Vibration Quasi-steady Theory  Bibliography .  42  Nomenclature  44  8 Tables 36 Figures  46  3 Plates  74  vi  Tables  1.  Test r e s u l t s :  2" cylinder with s t e e l c a n t i l e v e r .  2.  Test r e s u l t s :  2" cylinder with s t e e l cantilever and control wire.  3.  Test r e s u l t s :  4" cylinder with s t e e l c a n t i l e v e r ,  4.  Test r e s u l t s :  2" cylinder with aluminum cantilever.  5.  Test r e s u l t s :  2" cylinder with aluminum cantilever and control wire,  6.  Test r e s u l t s :  4" cylinder with aluminum c a n t i l e v e r .  7.  Natural frequency of free damped v i b r a t i o n i n water.  8.  V e l o c i t y f o r longitudinal and l a t e r a l resonance.  vii Figures 1.  C i r c u l a r c y l i n d e r i n uniform flow w i t h von Karman s v o r t e x s t r e e t .  2.  T h e o r e t i c a l o s c i l l a t i n g l i f t and drag.  3.  Frequency of i d e a l o s c i l l a t i n g l i f t and drag.  4.  S i n u s o i d a l v a r i a t i o n of v  5.  Development o f l o n g i t u d i n a l v i b r a t i o n ( 1 ) .  6.  Development of l o n g i t u d i n a l v i b r a t i o n ( 2 ) .  7.  D i s t o r s i o n of l a t e r a l oscillogram.  8.  Resonance of s t r u c t u r e v i b r a t i o n .  9.  D e f i n i t i o n sketch.  t  x  and v^..  10.  Water flume and t e s t equipment.  11.  S vs R  g  curve, 2" c y l i n d e r w i t h s t e e l c a n t i l e v e r .  12.  S vs R  g  curve, 4" c y l i n d e r w i t h s t e e l c a n t i l e v e r .  13.  S vs R  g  curve, 4" c y l i n d e r w i t h aluminum c a n t i l e v e r .  14.  S vs R  15.  C o n t r o l wire t e s t ( l ) .  16.  E f f e c t of l o c a t i o n of w i r e .  17.  C o n t r o l wire t e s t ( 2 ) .  18.  R e l a t i o n between frequency and v e l o c i t y .  19.  V a r i a t i o n of type of v i b r a t i o n — 2" c y l i n d e r , s t e e l c a n t i l e v e r .  20.  V a r i a t i o n of type of v i b r a t i o n — 4" c y l i n d e r , aluminum c a n t i l e v e r .  21.  V a r i a t i o n of type of v i b r a t i o n — 2" c y l i n d e r , aluminum c a n t i l e v e r .  22.  V a r i a t i o n of type of v i b r a t i o n — 4" c y l i n d e r , s t e e l c a n t i l e v e r .  23.  V a r i a t i o n of type of v i b r a t i o n — 2" c y l i n d e r , s t e e l c a n t i l e v e r , w i t h control wire.  24.  Amplitude r a t i o ( l ) .  25.  Amplitude r a t i o ( 2 ) .  curve, 2" c y l i n d e r w i t h aluminum c a n t i l e v e r .  26.  Amplitude r a t i o (3).  27.  Amplitude r a t i o (4).  28.  Amplitude with control wire ( l ) .  29.  Amplitude with control wire (2).  30.  Amplitude with control wire (3).  31.  Response c o e f f i c i e n t of v i b r a t i n g drag and l i f t ( l ) .  32.  Response drag c o e f f i c i e n t of v i b r a t i n g drag (2).  33.  Response l i f t c o e f f i c i e n t of v i b r a t i n g l i f t (3).  34.  Response drag c o e f f i c i e n t of v i b r a t i n g drag (4).  35.  Response l i f t c o e f f i c i e n t of v i b r a t i n g l i f t (5).  36.  Water flume and test body.  37.  Oscillograph and marking pen.  38.  Two-dimensional  v i b r a t i o n (photo),  39.  Two-dimensional  vibration  (oscillogram).  Two-Dimensional S t r u c t u r a l V i b r a t i o n Induced By F l u i d Flow Past A C i r c u l a r C y l i n d r i c a l Body  I.  Introduction: Violent vibrations of various types of structures submerged i n moving  water are sometimes observed even i n flows of moderately low v e l o c i t y . Slender t a l l bodies such as a stack, an antenna or a sounding-rod, which are subject to f l u i d flow, may vibrate i n response to l o n g i t u d i n a l as w e l l as l a t e r a l v i b r a t i n g forces. near the separation  The eddy shedding from unstable shear layers  points behind a body i n a moving f l u i d or a body moving  i n a f l u i d w i l l give r i s e t o an o s c i l l a t i n g pressure d i s t r i b u t i o n on the boundary of the body i n the separation  zone, the resultant a l t e r n a t i n g force  i n i t s vector sense w i l l give a two-dimensional e x c i t a t i o n to the moving body. In a v i b r a t i n g system with two-dimensional freedom, the vector sum of a l t e r nating forces can give r i s e t o l o n g i t u d i n a l and l a t e r a l as w e l l as t o r s i o n a l resultants causing v i b r a t i o n .  This makes the v i b r a t i o n condition very compli-  cated even i n the region where the v e l o c i t y i s f a r below the l a t e r a l resonant velocity.  Therefore the subject of t h i s i n v e s t i g a t i o n i s the r e l a t i o n s h i p of  l a t e r a l and l o n g i t u d i n a l v i b r a t i o n c h a r a c t e r i s t i c s , and t h e i r i n t e r a c t i o n s . The  s t a r t i n g point of the shear layer intersects the boundary of the  body, which stays i n the flowing f l u i d , at separation of the separation  point.  The l o c a t i o n  point varies with the c i r c u l a t i o n of the shedding vortex,  hence the mechanics i n the shear layer may change accordingly.  Oscillation  of the s o l i d boundary i n e i t h e r the l o n g i t u d i n a l or l a t e r a l d i r e c t i o n causes the shear l a y e r t o take on a wave pattern and also increases i t s i n s t a b i l i t y . Therefore, a d d i t i o n a l d i s t o r t i o n of the eddy shedding vortex a l t e r s the  - 1 -  - 2 e x c i t a t i o n , r e s u l t i n g i n an i r r e g u l a r v i b r a t i n g path. evidence to explain t h i s i r r e g u l a r i t y , oscillograms  In order to  obtain  were recorded by  recorders i n both the l o n g i t u d i n a l and l a t e r a l d i r e c t i o n separately  pen and  simultaneously. There are three types of e x c i t a t i o n i n v i b r a t i o n : and controlled.  forced, s e l f - e x c i t e d  S e l f - e x c i t e d v i b r a t i o n always occurs at i t s natural f r e -  quency, while forced v i b r a t i o n follows the frequency of the external force.  alternating  At resonance the frequencies of forced and s e l f - e x c i t e d  vibrations coincide, and large amplification of response occurs and i s only l i m i t e d by the induced damping.  I f the v i b r a t i o n of the body influences  the  o s c i l l a t i n g flow pattern, the r e s u l t i n g v i b r a t i o n i s no longer either s e l f excited or forced, and belongs to the category of c o n t r o l l e d v i b r a t i o n ( l ) . The i n v e s t i g a t i o n of the o s c i l l a t i n g f l u i d pressure around the has been done by several investigations, such as D.M. J.H,  Gerrard (10).  McGregor (5)  1  cylinder and  The measurements, obtained either from the study of  stationary cylinders or from cylinders held r i g i d l y enough to produce forced v i b r a t i o n under small v i b r a t i n g amplitude, represent the o s c i l l a t i n g f l u i d force i n various flow conditions.  This gives design data, presuming the  structure to be s t i f f enough to prevent i t s e l f from v i b r a t i n g under the forces.  The l i f t c o e f f i c i e n t (5) i s about 0.6  i s approximately 0.06,  fluid  while the drag c o e f f i c i e n t  both r e f e r to o s c i l l a t i n g parts.  Sometimes the structure i s allowed to move or i s f l e x i b l e i n nature, such as hydraulic gates or suspension bridges respectively.  The motion of  the s t r u c t u r a l v i b r a t i o n influences the f l u i d force, and v i c e versa.  There-  fore t h i s study investigates the s t r u c t u r a l response forces during f i n i t e «<••  1  Numbers i n brackets r e f e r to the  bibliography.  amplitude motion excited by the f l u i d force.  A l l the t e s t runs allow the  structure t o have f i n i t e amplitude during v i b r a t i o n .  A structure of two-  dimensional freedom with equal natural frequencies i n l a t e r a l and l o n g i t u d i n a l d i r e c t i o n was used.  The i r r e g u l a r oscillograms of controlled v i b r a -  t i o n are recorded and analysed f o r Reynolds Numbers varying from 5 x 10^ to 3 x 10 , ( f o r v e l o c i t y r a t i o :  - 0.2 t o 0.9). The l o n g i t u d i n a l res  v i b r a t i o n appeared at a v e l o c i t y r a t i o as low as 0.4 which i s considerably lower than that of l a t e r a l resonance.  I n the present i n v e s t i g a t i o n the  maximum v e l o c i t y r a t i o attainable was l i m i t e d by the maximum rate of flow of water available, combined with l i m i t a t i o n s on depth and v e l o c i t y set by c r i t i c a l flow conditions. II.  Basic concepts  Introduction  The wake behind a moving body i n a r e a l f l u i d represents a  discontinuous flow phenomenon which i s bounded by the t h i n shear layer known as a vortex sheet.  The pattern of a double row of v o r t i c e s o r i g i n a t i n g  from opposite sides o f the body changes with growing R  used i n these experiments, say 5 x 10  3  R .  In the range of  to 3 x 10^ the o s c i l l a t i n g force  appears to be due to eddy shedding. In considering the v i b r a t i o n of a body i n a moving f l u i d there are three major aspects to be considered.  F i r s t l y these are the f l u i d pressures  and shears transmitted to the surface o f the body, l i i i c h give r i s e to a net o s c i l l a t i n g force acting on the structure.  Secondly there i s the v i b r a -  t i o n a l c h a r a c t e r i s t i c s of the structure which may have several modes of vibration.  T h i r d l y there i s the modification of the flow pattern by the  s t r u c t u r a l v i b r a t i o n and the p o s s i b i l i t y of s e l f e x c i t a t i o n .  The f i r s t two  aspects w i l l now be considered independently and then some attempts w i l l be made to discuss the t h i r d complex aspect o f i n t e r a c t i o n .  - 4Blasius theorem  From Blasius theorem the resultant f l u i d force  x, y  i n an i d e a l uniform flow with c i r c u l a t i o n can be expressed (4) by:  where:  w  =  U (z + — ) +  In z  dw Using the Cauchy i n t e g r a l theorem and i t s extension, and with r e s u l t i n g from flow around cylinder with c i r c u l a t i o n , i t i s found that: X = 0 i . e . Drag i s zero Y  = f p  U  i . e . l i f t depends on  U  and P .  This gives the force acting on the cylinder f o r i d e a l flow without separation but with c i r c u l a t i o n .  For two-dimensional  v i b r a t i o n , the o s c i l -  l a t i n g pressure due to periodic change of c i r c u l a t i o n i s an e s s e n t i a l f a c t o r . O s c i l l a t i n g pressure study:  An o s c i l l a t i n g pressure study by D.M.  included a series of experimental and some mathematical  steps.  McGrefor(5)  The l o c a l  pressure i s obtained from the B e r n o u l l i equation: Ap  p , T I 2 2s ?)<P = | (U-v ) - p -  In the complex p o t e n t i a l , i n order to maintain the cylinder surface as a streamline, two v o r t i c e s o f equal strength but opposite sign were located  2  at distances  a  and r£  away ( F i g . l a ) from the o r i g i n ;  so that one l i e s  outside and one inside the cylinder:  2 . U (z + —) + — - I n  z - b r —  r  w  =  z  z  . a2/  b  Sin w t m eddy shedding, the pressure c o e f f i c i e n t w i l l be: Assuming a periodic c i r c u l a t i o n of f  c  P  -§  1  w-> t  v  due t o opposite  - 5 -  •A  r  Fig,  la  where  are the arguments i n polar ordinates. Since, at any  instant the flow i n front of the point of separation can be treated as i d e a l and that i n the wake as being altered, t h i s reveals only some aspects of l o c a l pressure v a r i a t i o n due t o the shedding of v o r t i c e s . The o s c i l l a t i n g pressure change around the various positions on the cylinder measured by McGregor, i n a plotted r e l a t i o n between frequency and pressure c o e f f i c i e n t i n a given flow, appeared i n two peaks which were c a l l e d the fundamental and secondary harmonic frequencies. The fundamental frequency peak, about 165 cps, occured at 90° p o s i t i o n from the upstream stagnation point of the cylinder, while the secondary harmonic frequency, about 330 cps, appeared at 180°.  Therefore, i f the two-dimensional  vibra~  t i o n i s excited by these o s c i l l a t i n g pressures, the frequency of l o n g i t u d i n a l v i b r a t i o n w i l l be twice that i n the l a t e r a l d i r e c t i o n . This study i s very i n t e r e s t i n g and i t s r e s u l t s are supported by the f i n d i n g of the present experiments. Response of e l a s t i c supported structure t o eddy shedding force I f a structure i s acted upon by e x t e r n a l l y applied o s c i l l a t i n g f l u i d force, a time-dependent motion  w i l l be s e t up. The forced v i b r a t i o n can  be expressed (6) by: My + D(t) + kY where:  F(t) =  F^ •  q  =  stagnation pressure  D(t)  =  damping function  A  =  area of body projected on f l u i d stream  •  Von Karmann vortex shedding force  »  frequency of eddy shedding  F  R  f  x  Appendix I  q A  = F(t) Sin 2 n f t  The amplitude r a t i o of a system with a single degree of freedom and having viscous damping, may be written:  Y  where:  c  lK -^ / C  2  k  c^  i s constant f o r a given system  Cg  :  coefficient.  JZ :  r a t i o of the frequency of forcing function to natural frequency  8  damping decrement.  :  Since there i s no complete hydrodynamic theory f o r separated flow, the response function of an e l a s t i c system could be treated by quasi-steady theory.* An estimate of the natural frequency of the system may be made by def l e c t i n g i t and allowing i t to v i b r a t e f r e e l y . can also be used to calculate the value  8.  The r e s u l t i n g damping curve  The damping w i l l change with  the v e l o c i t y of flow and with amplitude of v i b r a t i o n .  I t should also be  remembered that the eddy shedding frequency and the width of the vortex street w i l l be altered by the amplitude of v i b r a t i o n .  III.  Qualitative explanation of the two-dimensional  Shear layer  v i b r a t i o n phenomenon:  I n the i d e a l flow, a uniform flow and a doublet gives the  symmetrical flow pattern with two stagnation points  S^  and S,,  (Fig. l b ) .  From the boundary layer theory and the energy theorem, an expression: (b_us  V  y=0  "  1  dp  ^  *  can be obtained which reveals that the separation phenomenon w i l l take place on the rear surface of the cylinder, which i s a region o f r i s i n g pressure x  Appendix I I  - 8 -  and p o t e n t i a l flow reversal i n the boundary l a y e r . Two separation points  and A^  are indicated ( F i g . l c ) .  Due to  the r e v e r s a l o f flow, a t h i n shear layer i s formed with a r o l l i n g eddy t r a i n along i t s path s t a r t i n g from the separation point and decaying i n the downstream  direction.  Lateral vibrating velocity  According t o the Kelvin's c i r c u l a t i o n theorem  there i s zero net v o r t i c i t y and therefore every time a vortex i s shed there i s a c i r c u l a t i o n around the cylinder with opposite sign to that of the shedding vortex.  This c i r c u l a t i o n with uniform flow generates a l i f t force  and moves the separation points back and f o r t h ( F i g . l c ) . l a t i o n changes:  When the c i r c u -  p -» o -- - p , the separation points move to positions  opposite to the previous ones, as i n F i g . l c . Hence the wake s h i f t s repeatedly from l e f t to r i g h t . An e l a s t i c supported body acted on by t h i s periodic force w i l l vibrate i n a d i r e c t i o n perpendicular to the d i r e c t i o n of the uniform flow.  I f the  eddy shedding frequency i s lower or equal to the natural frequency of the s t r u c t u r a l v i b r a t i o n , the sinusoidal l a t e r a l v e l o c i t y of v i b r a t i o n can be expressed: V  • y  where:  =  V S i n wt ym  instantaneous maximum l a t e r a l v i b r a t i n g v e l o c i t y .  When the eddy shedding frequency i s higher than the natural frequency of the e l a s t i c system, the pulsating force w i l l run out of phase with the response of structure. small value.  Then the amplitude w i l l diminish quickly to a very  A v i b r a t i o n i n random nature w i l l be resulted.  - 9 Longitudinal i.  o s c i l l a t i n g force X  With no l a t e r a l v i b r a t i o n :  When a vortex i s shedding, growing and passing  downstream ( F i g . l c ) the tangential v e l o c i t y at the circumference of rear part of the cylinder changes according to the v e l o c i t y d i s t r i b u t i o n o f the vortex. tion.  One eddy w i l l give one complete cycle of the v e l o c i t y v a r i a -  I n the meantime, from energy theorem the o s c i l l a t i n g pressure i n  longitudinal d i r e c t i o n also completes one periodic excitation t o the v i b r a t i o n of the structure.  Therefore, the o s c i l l a t i n g drag has a frequency  twice that of pulsating  lift.  i i . Reinforced by l a t e r a l v i b r a t i o n :  When the vortices are shedding alterna-  t i v e l y from the cylinder, the c i r c u l a t i o n around the cylinder also changes i n value from positive to negative.  r =r Sin ' m  '  Therefore an assumption i s made:  cot  Considering the cylinder t r a v e l l i n g from  A  to B  ( F i g . 2a), the sum  of the v e l o c i t y vectors i s :  v =  (ir + v y  toward the cylinder with an attack angle force  L  a. From the Magnus e f f e c t , a l i f t  e x i s t s , which gives two components i n x X  =  p U ( T S i n tot) S i n a m  Y  -  p U ( f S i n tot) Cos a m  and y-direction:  X, Y:  From the above equation, the longitudinal force i n two-dimensional v i b r a t i o n i s a function o f l i f t and attack angle. reaching point lift  When the cylinder, a f t e r  c, vibrates back to D ( F i g . 2b), the Y-component of the  changes i t s d i r e c t i o n , but the X-component s t i l l  x-direction.  acts i n a p o s i t i v e  From that the angle of attack i s zero, and assuming the  culation to be zero at points  A, C  and E, components  X  cir-  and Y w i l l be  -  10  -  Fig. Fig.  2  Theoretical  oscillating  2c lift  and  drag  - 11 zero at these locations.  This means that when the v a r i a t i o n of l i f t  force  completes one cycle, the longitudinal force w i l l complete two cycles i n the same time.  Therefore, the frequency of e x c i t a t i o n i n the l o n g i t u d i n a l d i r e c -  t i o n w i l l be twice that i n the l a t e r a l .  This r e l a t i o n i s indicated i n F i g . 3.  So f a r i t has been assumed that vortex shedding i s two dimensional and i s therefore  i n phase along the axis of the cylinder.  In a real f l u i d this  may not be true and the phase of eddy shedding may change along the axis of the cylinder.  Any d i s t o r s i o n (8) of the vortex along i t s axis or other small  disturbance w i l l change the value of the e x c i t a t i o n , the response of the e l a s t i c structure w i l l then change accordingly.  I t i s considered, although  there i s at present no evidence to support i t , that once a v i b r a t i o n commences, a l l the vortex shedding along the cylinder w i l l be forced i n t o phase. Resultant v i b r a t i o n of two-dimensional periodic phenomena In the two-dimensional v i b r a t i o n , the longitudinal v e l o c i t y of cylinder w i l l also vary with time ( F i g . 4)j t h i s means that an additional periodic v a r i a t i o n of V  x  w i l l be superposed on the uniform v e l o c i t y  U.  Since the  l i f t force i s a function of longitudinal v e l o c i t y , the r e s u l t i n g l i f t  forces  and displacements w i l l present themselves as a resultant of two-dimensional periodic combinations. Therefore i t can be seen that a l o n g i t u d i n a l v i b r a t i o n w i l l tend to excite additional l a t e r a l v i b r a t i o n a l forces, and conversely a l a t e r a l v i b r a t i o n w i l l cause additional longitudinal v i b r a t i o n a l forces.  This complex  i n t e r a c t i o n i s demonstrated by some of t e s t s i n which s e l f - e x c i t e d vibrations appear.  I t has not been found possible t o discuss the i n t e r a c t i o n  t i v e l y but a q u a l i t a t i v e understanding i s possible.  quantita-  The f a c t that the two  modes d i f f e r i n frequency by a factor of two i s a l l important, because i t presents a double resonance at the fundamental.  I t i s i n t e r s t i n g to consider  0 a  Fig.  3  Frequency  of i d e a l  oscillating  lift  and  drag  Velocity Zone o f V  x  variation  r , /Zone o f V y  -UL  Fig.  4  Sinusoidal  variation  V  of V  *  and V  '7  variation  - 13 what might happen i f the l a t e r a l v i b r a t i o n were i n the second mode whilst the longitudinal v i b r a t i o n coincided with the f i r s t mode, assuming the f i r s t and second modes to have a frequency r a t i o of two. S t a b i l i t y of two-dimensional  vibration  Since the frequency of e x c i t a t i o n i n longitudinal d i r e c t i o n i s twice that of the l a t e r a l , the v e l o c i t y of resonance i n the longitudinal d i r e c t i o n w i l l be half that of the l a t e r a l , i f natural frequencies are the same i n both directions.  In a s i t u a t i o n of s e l f e x c i t a t i o n the s t a b i l i t y of the system  must be considered.  I f the induced e x c i t a t i o n increases more r a p i d l y than  the damping force, v i o l e n t two-dimensional v i b r a t i o n r e s u l t s , hence the structure w i l l be overloaded. Conclusion Stationary  cylinder:  S t a r t i n g from Von-Karman's t h e o r e t i c a l vortex  street with a stationary cylinder, due to the c i r c u l a t i o n caused by shedding vortex, the vortex street may s h i f t l e f t and right r e s u l t i n g a wave-like wake path.  This w i l l produce a smaller component of o s c i l l a t i n g drag, as  vortices of alternate sign grow and pass downstream.  I t has twice the f r e -  quency of the o s c i l l a t i n g l i f t and arises from a changing pressure d i s t r i b u t i o n e x c i t a t i o n i n the longitudinal d i r e c t i o n . Vibrating cylinder:  Since the l a t e r a l v i b r a t i o n gives a periodic  V y  v a r i a t i o n , a periodic x-component arises with a frequency twice that of the lateral.  This strengthened o s c i l l a t i n g drag w i l l cause a strong e x c i t a t i o n  i f i t s frequency i s near the natural frequency of the structure.  - 14 IV.  Apparatus and Instrumentation 4.1  General consideration:  Since t h i s experiment emphasized two-dimensional v i b r a t i o n as w e l l as the r e l a t i o n s h i p of i t s frequency, phase and amplitude, a constant f l e x u r a l system with low natural frequency and recordable l i f t and drag force was required.  The 30" wide water flume and s t e e l c a n t i l e v e r mounting  were chosen. A hollow brass cylinder 2" i n diameter was f i r s t used as a t e s t body. At both ends, 6" diameter discs were provided as s p l i t walls to prevent secondary v o r t i c i t y .  A 3/4" square s t e e l bar served as a f l e x u r a l structure  mounted on a 5" I-beam which, supported on concrete walls, was strong enough to hold the v i b r a t i n g t e s t body. fD From the r e l a t i o n :  S  =  ^-  f o r stationary cylinder of 2" d i a -  meter, i f the v e l o c i t y of flow i s i n the range of 0.50 fps t o 4.35 f p s , the frequency of eddy shedding w i l l be about 0.6 cps to 5.3 cps respectively. I f a f l e x u r a l cantilever with a frequency about 5 cps i n both l o n g i t u d i n a l and l a t e r a l d i r e c t i o n i s chosen, i t w i l l give longitudinal as w e l l as l a t e r a l resonance i n the flow range.  In order to make the s t r a i n i n s t r a i n gauge  recordable, s t r a i n was l i m i t e d i n the range of:  50 to 2000 micro-in/in.  The natural frequency was checked approximately by the equation: where:  c: coe. including damping e f f e c t .  Four combinations, with s t e e l and aluminum material, 2" and 4" c i r c u l a r cylinder, were selected and l i s t e d i n Table V I I .  Four s t r a i n gauges located  at the upper end of the cantilever gave the v i b r a t i n g response forces i n two d i r e c t i o n s , and below the resonance condition these response forces gave the frequency and amplitude of the v i b r a t i o n .  - 15 4.2  Water flume:  The 40-ft. long water flume 30" wide and 3| f t . high provided a calm flow ahead of the test s e c t i o n . The maximum discharge was about 7 cfs c i r culated by a pumping system and s t a b i l i z e d by an overhead tank.  The d i s -  charge was measured by an o r i f i c e flow meter and checked by a volumetric tank.  A plate with holes and two s t e e l meshes were used to keep the flow  uniform. To eliminate boundary layer e f f e c t , a parabolic entrance contraction was i n s t a l l e d i n the middle of the flume. provide a high enough v e l o c i t y .  This contraction was necessary t o  The v e l o c i t y d i s t r i b u t i o n i n the 15" x 15"  t e s t s e c t i o n was checked by a miniature propeller current meter.  The devia-  t i o n of the v e l o c i t y from average was + 1.7$. Water depth i n the test section was  controlled by a t a i l gate. ( F i g . 10). C r i t i c a l flow was avoided by con-  t r o l l i n g the height of the t a i l water gate, because surface waves are a problem near c r i t i c a l depth. we could use.  This condition l i m i t s the maximum v e l o c i t y which  A point gauge accurate to one hundredth of a foot was used  for measurement o f flow depth i n the t e s t section. Water surface drop at the entrance of the t e s t section was very smooth, and no standing wave appeared.  Therefore f a i r uniformity o f v e l o c i t y and  pressure d i s t r i b u t i o n was predicted. 4.3  Model mounting  A 3/4" square s t e e l cantilever provided a mounting of constant  flexibi-  l i t y and known damping c o e f f i c i e n t f o r the vibrating system - a c i r c u l a r cylinder 2" i n diameter and one foot long. also used as a test body.  Later a 4" diameter cylinder was  At f i r s t the c a n t i l e v e r was clamped on t o a 5"  s t e e l I-beam across the top of the flume.  When the cylinder was subjected  to even medium flows, the 5" beam vibrated i n t o r s i o n .  The t o r s i o n a l  - 16 r i g i d i t y of the I-beam being found to be too small, another 5" I-beam was clamped v e r t i c a l l y to the concrete c e i l i n g beam and to the h o r i z o n t a l s t e e l beam.  To check the accuracy of measurements of s t r a i n i n the cantilever,  possible movement of the support was  checked by d i a l gauge.  The movement  appeared n e g l i g i b l e even under cantilever resonant conditions. The t o t a l length of the cantilever system was 38 inches. of the submerged square section was  The portion  surrounded by a streamlined strut to  eliminate the e f f e c t of form drag of the cantilever i t s e l f . The cantilever as w e l l as the cylinder was changeable;  hence the f l e x i -  b i l i t y of the structure as w e l l as the natural frequency of the system could be v a r i e d i n the t e s t s .  This gave d i f f e r e n t resonant v e l o c i t i e s and f r e -  quencies, and d i f f e r e n t types of s e l f - e x c i t e d and forced v i b r a t i o n . 4.4  Instrumentation:  Since measurements of the frequency and the displacement of the test body were required, the oscillograph was used.  The e l e c t r i c a l current, which  varied according to the mechanical changes i n the s t r a i n gauge, was amplified and recorded by pen recorder ( F i g . 37).  In order to record l a t e r a l and  l o n g i t u d i n a l v i b r a t i o n simultaneously, two sets of s t r a i n gauges, amplifiers and pen recorders were used. frequency:  0 to 20 cps.  changing attenuations. mm  per second.  t h i s pen was  The recorders accuracy was very high at low  A wide v a r i a t i o n of amplitude was permitted by Three recording speeds were used:  5, 25 and  125  A pen was provided f o r the purpose of marking the graph and  activated simultaneously with the camera shutter, so that photo-  graphs could be synchronised with the v i b r a t i o n record ( F i g . 38-39). White f l o a t s were spread on the surface of the flow so that the flow path could be photographed.  V.  Test  Procedure  5.1  Calibrations  The d i s c h a r g e rating  curve  gauge  of the  shows t h e  i n the  f l o w meter  elastic  test  range,  on the  strain  Force  position.  recorder from the  the  Because the  the  bending of the  and the  free  It  beam was  proportional to  response  s h o u l d be  cantilever  response  response  at  the  T h e c y l i n d e r was  force  T h e p r o c e d u r e was the  frequencies  From t h e s e  coefficient,  n, 6  6. logarithmic  reading  force  stressed are  on the  not  t  repeated  for  of l a t e r a l  damping curves  both the  d e t e r m i n e d (17)  «•  nT  was f o u n d b y v i b r a t i n g  v i b r a t e d at  decrement  6:  lateral  natural  by the  where  e  there  its  equilibrium  natural of  the  d i r e c t i o n and i t  was  and l o n g i t u d i n a l v i b r a t i o n s were  were  „  the  near  equal,  displaced from i t s  it  T:  test  could then  that  The a m p l i t u d e o f v i b r a t i o n s t e a d i l y d i m i n i s h e d b e c a u s e  found that  its  resonance.  damped v i b r a t i o n i n w a t e r  a l o n g i t u d i n a l d i r e c t i o n a n d when r e l e a s e d  same.  be  The e l a s t i c  c a l i b r a t i o n curve.  the t e s t i n g body i n p o s i t i o n .  damping.  steel  recorded  was o s c i l l a t i n g i n  done b y a p p l y i n g a known p u l l i n g  amplification of  The f r e q u e n c y o f  frequency.  c a l i b r a t e d d i r e c t l y by a  cantilever  was o p e r a t i n g .  applied force  being a considerable  in  readings.  The c o r r e s p o n d i n g d e f l e c t i o n s were  a m p l i t u d e was a s s u m e d t o  c a l i b r a t i o n was  calculated  resonance  and flow meter  The  gauge.  body w h i l e the be  c a l i b r a t e d by weighing tank.  v i b r a t i n g c y l i n d e r was  and because  fundamental mode, t h e  was  r e l a t i o n between d i s c h a r g e  s i m u l t a n e o u s l y b y pen r e c o r d e r . within  -  and A n a l y s i s  The a m p l i t u d e o f t h e dial  17  f r e q u e n c y and the  the  damping  equation:  p e r i o d o f v i b r a t i o n and  - 18 By varying the r i g i d i t y of the e l a s t i c system, four conditions  of  d i f f e r e n t damping c o e f f i c i e n t s were used i n the experiment. 5.2  Experimental r e s u l t s  Experiment was  divided i n t o the following parts, and l i s t e d i n Table I -  VIII: 1.  2" c i r c u l a r cylinder with 3/4"  square s t e e l cantilever.  2.  4" c i r c u l a r  "  "  3/4"  "  3.  2" c i r c u l a r  "  "  5/8"  "  4.  4" c i r c u l a r  "  "  5/8"  "  5.  2" c i r c u l a r  "  "  3/4"  "  II  II  aluminum n  " n • -  steel wire placed at four d i f f e r e n t positions, 6.  "  and 1/16"  diameter  i . e . 0°, 22.5°, 45° and 67.5°.  2" c i r c u l a r cylinder with 5/8"  square aluminum cantilever and 1/8"  wire placed at two  i.e.  positions,  A l l t e s t s were run at 0.3 f o r which the recorder was i n the test section was bances.  The  0° and  mm  45°.  fps v e l o c i t y increments, from the lowest flow  s e n s i t i v e , to the highest flow f o r which the flow  s u f f i c i e n t l y s u b - c r i t i c a l to avoid surface  speed of the pen recorder was  general runs and 125  diameter  set to 5 or 25 mm  distur-  per second f o r  per second f o r s p e c i a l runs.  For the runs with 2" diameter cylinder and s t e e l cantilever, a l o n g i t u d i n a l force was  applied by a wheel mechanism to the t e s t i n g body while the  l a t e r a l amplide was  very small.  tudinal d i r e c t i o n only.  This gave additional s t i f f n e s s i n the l o n g i -  Frequency was  recorded through a series of flows.  Since t h i s represented the l a t e r a l d i r e c t i o n v i b r a t i o n only at small v i b r a t i n g amplitude, the r e s u l t s are very close to stationary cylinder readings. r e s u l t s were analysed and plotted with  S  vs  R  g  i n F i g . 11.  The  points  followed Strouhal's description and f e l l into the band zone reported by other authors (7).  These  Run No.  C T ^ U i - P " L o N > i — ' O 0 0 ^ J O I _ n 4 > U > N i l — ' • P » u > o j u > r o N J N > t o » - i o o o o o i O i v O h O " ^ J V O t o O O O O O i t -  Velocity V fps  l — ' O i ' P ' O l - '  ,  o o o o o t - ' O t - o o c r o  ~o  o  o  •..p00 r o O U 4 >  -PN i L - o  o  o  o  o  P » . p ~ t O t O O ON N > 0 0 J O h - o o N n 4 > - P ~ t j a 0 o t o o o M ^ t o  o  o  o  N > N > h - » ' P - 0 > O i w o ^ i o ) N ) O O ( - ' i - ' - ^ J v o  h O O i ) J U h o  o  o  o  o  o~  O O O O t - ' l O v Q 0 V 0 O l - ' o - P - t o O h o i — ' h - ' i — ' i - > h o t o - f > i  i  t o v O - P ' N > > J * * » ' J l - ' . p - O v O i - ' O i - P - v o O l - P - - P ~ t o t o N > N > 01 01 o ~J t o a v o i h o t n h 0 h 0 0 " \ t n ' P * O N V O  o  o  o  o  o  o  o  o  h O ( - i O O O O » - ' l - ' a v o i o v o o v o o i — 0 O h 0 0 0 l — ' O l — I—" O V O O N > C o O N > 1  o  o  o  o  o  o  o  Relative velocity V/Vres Reynolds No. , Re 10  o  o  o  o  o  o  o  o  o  o  o  o  o  0>  Strauhal No. fyD/V  O  O  O  O  O  O  O  O  O  K > O J - P ~ U i O i l w n l - ' l - ' l - ' l — ' l - ' l - ' l - ' t - ' l - ' N) M M N 00 U OS ~-J O p — I 00 00 00 00 00 < y > O i O o a v O o O O i O i ' P h o O i O O O i t o ,  O  O  l  —  '  l  —  '  I  —  '  O  O  O  O  O  O  O  O  O  fy  O • Ln  Frequency fx bps  O  O  O . Ln  O  O «  O •  -P" -P-  O  O  o  O •  O  -P-  O  O  O  O  O  O  O  o  t—  I—•  I—'  I—•  »—* t—*  O  O  O  O  O  O  o  o  o  o  o  ~o  O  o o o  O  O  O  O  O  O  fx  O  r-> O  00  U Ol  *> O  vO f > ho t o  U O  O O  -P.P- -P- O l -P" -P- N> ~J -P- 00 h-> 00 00 O l -P* hO ON —I LO -P* O  O  O  O  O  O  O  O  .  .  .  .  .  .  .  .  o  o  o  o  o  .  o  \v  o  \  ^ • J O v - P - t O O i O i h O l - ' O O l h O V O O O t o O i v O t O O i • P - t o h O h o t o t o i - ' O .  .  .  .  .  .  .  .  O  .  o i - v i v o c o N > r < o ^ i ^ J t o O h-  O 1  O  I—  1  O  O  t  n  ^  »— l—' ho t o 1  J  M  O •  .  L ho  n O  O  X  \  \  o  o  o  «  .  «  .  o  o  o  o  O  O  O  O  o  o  o  i-  -P-  -P~ 00 ho  O O O O \ l - ' I - I i - i N>  \v  •  \ ^ X  •  O  • P ' - P - t o t O - P - l - ' O ' v O O O O N O J ^ J - P - I — ' O J V O O v v C  o  3  to a*  a  <  H fl>  3* 3 1 09 j> p. 3 3  rt  CO  3*  k  rt  to  VO  a  •  o  z  3 »  l-h  CO  H*  T3  Q-  C  CO  <  to 3 a w o rt  O 3* a- 3 ho (to H - pi H-  rt r t r t  c a> o o- n to  3 H - to 3 f->  to  to  H  n>  <  ti-  CO  CO  O  ro  •  O ~~"  _  •  1  O  Ov l_i to  - 61 -  Coe.  (D  rt  H-  II  •P• o>  o 3 CO  l-h  rt  CO  t-  ro n>  T3  1  o to 3 i— (0  II  <J  3 00 Oi  Hrt  C  a ro  to  X i—  1  I  Ol  f to rt CO  n> I-J to  1  n>  o  Amplitude ratio <^y/D Lift Py l b  ro  3 o r-' o < o>  3  00  B^px o \  >  to  t->  Coe.  1  o  fn  Drag ^Px lb  H  I-i  o  p. pj O !-• 3  5"  O l ho  Cv  cr pi  Amplitude .ratio  f—' t—• l—' l—' o M JV O M U 00 vO t o OJ VO O l  Ol  o •V CO  ti  0 * 01  1  a  n rt » a> rt n  fn  Strauhal No. fxD/V  L o - P - - P - t n t n O > ^ - J O O O O O O O O vO t o "»l N> VO ->J O l O CT> O l Wi O O l I—«  3 -  <  O  O i 0 0 O O O 0 0 N 3 N > l - ' O O O O O t - - ' ^ J ^ J O O O > - ' 4 > K ) U i U l O > ~ - J O O V O O UI -Pt n o o t o o i r o c y o o o o o o o o O O . • Ui to  Hi  t~> o .p-  Strauhal No. fD/V  O  O  n  H-  rt  O 3  tn  Frequency f y cps  3  H-  fn  ,  O  to  Frequency f cps  v £ > O - P - U > N > - P - U i M 0 - N U i 0 - * 0 0 V O O > - ' u i O O u i L f i o o N ^ o o r o o o i — ' O i — • O VO O ho 00 O hO O  o a. 3  o  O i v O O O O O O N J h O l - ^ O O O O l - ' l - - '  O  o  4  L n . f > L O t O L o h O h O h O | — ' O O O O O h ^ L O U ) ' ~ O O N t - ' U l 4 ^ t o o i U i a > ~ ^ J O O ' - 0 0 v O v O O O i O v O ^ O O t O O i h O O v O O O O O O O O  O  ho  1  lOl—«t—«|—.1—i|—•»—1|—1|—•!—»|—•!—«|-«|—»|—i I - ^ O O O O V O O O - ^ I O O ^ J O O ^ - J O O O O O O O O O O t-» O Cn i-» qs • P ' O i O i . p - h O t n O O o i t o  o  o t  i  Ml rt ho  or n>  o  TABLE II. 2" cylinder with 3/4" square section steel cantilever and wire i n place fn : fn • 10.4 cps  (7)  (8)  (9)  (TO)  JD V  %>  i .  fy  (12) (13)  (3)  (4)  V fps  V Vres  37  2.18  0.24  3.21  2.80  0.214 0.272 0.008  35  2.53  0.28  3.74  3.56  0.234 0.343 0.018 10.4 1.0  0.682 10.4  38  2.89  0.32  4.27  3.94  0.227 0.382 0.021 10.4 1.0  3.27  0.36  4.82  4.52  39  3.62  0.40  5.34  40  3.99  0.44  41  4.35  46 47  2)  R  E  (6)  Vres = 4.5 fps  < Wire Run placed at No.  (1)  (5)  Vres = 9.0 fps  ZL  10  f  4  in  (ID fy/ /fn  fvD V  fx  D = 0.167'  (14)  (15)  (16)  fx fn  fxD V  K  0  0  in  (17)  A>  (18)  (19)  Sj  Syy in  /D  0.010  0.005  1.0  0.755 0.116 0.058 0.170  0.085  0.597 10.4  1.0  0.670 0.233 0.117 0.184  0.092  0.230 0.439 0.026 10.4 1.0  0.528 10.4  1.0  0.595 0.405 0.203 0.110  0.055  4.90  0.227 0.472 0.034 10.4 1.0  0.478 10.4  1.0  0.525 0.452 0.226 0.084  0.043  5.89  6.31  0.264 0.612 0.036  7.8 0.75  0.326 10.4  1.0  0.476 0.452 0.226 0.170  0.083  0.48  6.43  7.30  0.279 0.708 0.031  6.3 0.61  0.241 10.4  1.0  0.430 0.135 0.068 0.170  0.085  2.18  0.24  3.21  2.44  0.186 0.237 0.021  2.5 0.24  0.191 10.4  1.0  0.798 0.028 0.014 0.025  0.013  2.53  0.28  3.74  2.96  0.195 0.287 0.038  9.2 0.885 0.602 10.4  1.0  0.755 0.165 0.083 0.037  0.019  2.89  0.32  4.27  3.52  0.202 0.342 0.032 10.4 1.0  0.663 10.4  1.0  0.670 0.172 0.086 0.116  0.058  48  3.27  0.36  4.82  3.92  0.199 0.381 0.045 10.4 1.0  0.531 10.4  1.0  0.595 0.295 0.148 0.061  0.031  45  3.62  0.40  5.34  4.10  0.188 0.398 0.037 10.4 1.0  0.478 10.4  1.0  0.525 0.306 0.153 0.110  0.055  49  3.99  0.44  5.89  4.26  0.178 0.413 0.030  6.5 0.625 0.270 10.4  1.0  0.476 0.485 0.243 0.177  0.088  50  4.35  0.48  6.43  5.00  0.195 0.485 0.025  5.5 0.530 0.210 10.4  1.0  0.430 0.300 0.150 0.178  0.089  36  4  44 0  2.8 0.272 0.214  0  0  0  TABLE II.  Continued  (3)  (4)  (5)  (6)  V fps  V Vres  Re 10  2.18  0.24  3.21  2.07  0.158 0.201 0.032  2.07 0.201 0.158  52  2.53  0.28  3.74  2.68  0.176 0.260 0.048  5.60 0.540 0.367 10.4  1.0  0.755 0.092 0.046  0.080 0.040  42  2.89  0.32  4.27  2.92  0.168 0.283 0.075  8.50 0.816 0.488 10.4  1.0  0.670 0.165 0.083  0.122 0.061  53  3.27  0.36  4.82  3.34  0.170 0.324 0.115 10.4  1.0  0.528 10.4  1.0  0.595 0.160 0.080  0.160 0.080  43  3.62  0.40  5.34  3.80  0.175 0.369 0.110 10.4  1.0  0.478 10.4  1.0  0.525 0.257 0.128  0.227 0.114  54  3.99  0.44  5.89  4.06  0.169 0.394 0.080 10.4  1.0  0.345 10.4  1.0  0.476 0.307 0.154  0.184 0.092  55  4.35  0.48  6.43  4.20  0.160 0.408 0.114 10.4  1.0  0.398 10.4  1.0  0.430 0.355 0.178  0.215 0.108  2.18  0.24  3.21  2.36  0.180 0.227 0.031 10.4  1.0  0.796 10.4  1.0  0.798 0.068 0.034  0.086 0.043  57  2.53  0.28  3.74  2.86  0.187 0.275 0.055 10.4  1.0  0.683 10.4  1.0  0.755 0.116 0.058  0.120 0.060  58  2.89  0.32  4.27  3.42  0.196 0.329 0.043 10.4  1.0  0.597 10.4  1.0  0.670 0.165 0.083  0.141 0.071  59  3.27  0.36  4.82  3.76  0.191 0.362 0.040  3.60 0.346 0.183 10.4  1.0  0.595 0.153 0.087  0.120 0.060  60  3.62  0.40  5.34  4.06  0.186 0.390 0.061  4.06 0.390 0.186  0  0  0  0.037 0.019  0.074 0.037  61  3.99  0.44  5.89  4.3  0.179 0.413 0.055  4.30 0.413 0.179  0  0  0  0.025 0.013  0.055 0.028  62  4.35  0.48  6.43  5.1  0.195 0.490 0.123  5.10 0.490 0.194  0  0  0  0.037 0.019  0.120 0.060  (2) Wire Run placed at No. (1)  51  56  22.5°  0°  f 4  (7) 12 V  (8)  (9) ^in  (10)  (ID  (12)  fy  fy/ /fn  fyD V  (13) (14) fx 0  fx fn 0  (15) fxD V 0  (16) i  (17) U  x  in  in 0  (18)  0  (19) X  D  0.030 0.015  TABLE I I I . 4" cylinder with 3/4" square steel cantilever fn = (2)  (1) Run No.  V fps  (3)  y  /Vres  (4) R C  4 10^  fn  6.5 cps  s  (5) (6) fx  is  Vres = 11.8 fps  (7)  (8)  Sx  fy  fn  (9) f  v  (10)  Vres = 5 .9 fps  (ID  (12)  Sy  (13)  E = 0.334  7  (14)  ^Px  X  (15)  (16)  Sy  (17) Py  (18) P  Py  /fn  70  1.78  0.153  5.26  0  0  0  1.07  0.164  0.200  0  0  0  0  0.019  0.005  0.582  1.13  71  2.14  0.183  6.34  0  0  0-  1.28  0.195  0.239  , 0  0  0  0  0.049  0.012  1.45  1.94  73  2.48  0.213  7.32 6.5 1.0 0.871 1.66  0.254  0.317  0.025  0.006  0.60  0.601  0.055  0.014  1.53  1.52  74  2.85  0.244  8.45 6.5 1.0 0.763 2.60  0.397  0.464  0.147  0.038  4.30  3.22  0.116  0.029  3.37  2.54  75  3.22  0.275  9.53 6.5 1.0 0.679 3.00  0.458  0.475  0.171  0.043  5.00  2.96  0.153  0.038  4.48  2.66  76  3.56  0.305  10.50 6.5 1.0 0.610 3.40  0.520  0.486  0.074  0.019  2.15  1.05  0.110  0.028  3.20  1.54  77  3.91  0.336  11.60 6.5 1.0 0.552 4.00  0.610  0.518  0.074  0.019  2.15  0.865  0.153  0.038  4.48  1.80  78  4.28  0.367  12.70 6.5 1.0 0.506 5.20  0.795  0.615  0.098  0.027  2.85  0.955  0.110  0.028  3.20  1.70  TABLE IV.  2" cylinder with 5/8" square aluminum cantilever  Vres - 4.0 fps  fn = fn =4.5 cps  Vres = 2.0 fps  RdP = ^Px x  A  (2)  (3) V/" /Vres  (5)  (4)  fx  (7)  (8)  fx fn  Sx  fy  (6)  (9)  (10)  (ID  Sy  y  (13) Py  2  2  (14)  (15)  D  (16)  (17)  ^Px  Run No.  V  101  0.752  0.186  1.11  0  0  0  0.83  0.190  0.184  0.010  0.055  0.595  0  0  0  102  1.090  0.271  1.61  4.5  1.0  0.776  1.30  0.290  0.198  0.091  0.445  2.060  0.167  0.823  4.26  103  1.450  0.361  2.14  4.5  1.0  0.585  4.50  1.00  0.517  0.025  0.123  0.360  0.243  1.200  3.50  104  1.810  0.450  2.68  4.5  1.0  0.467  2.75  0.610  0.504  0.0625  0.308  0.580  0.389  1.920  3.50  105  2.180  0.486  3.21  4.5  1.0  0.388  2.40  0.535  0.183  0.311  1.53  1.970  0.365  1.800  2.33  106  2.530  0.632  3.74  4.6  1.02  0.338  2.55  0.567  0.168  0.770  3.83  3.670  0.348  1.700  1.63  107  2.890  0.722  4.27  4.7  1.04  0.304  3.35  0.745  0.192  0.987  4.85  3.560  0.097  0.480  0.35  108  3.270  0.814  4.82  4.9  1.09  0.286  3.35  0.745  0.171  1.83  9.05  5.220  0.070  0.340  0.20  109  3.620  0.902  5.34  3.50  0.778  0.161  2.39  11.80  5.550  0.084  0.410  0.20  110  3.990  0.992  5.89  3.78  0.840  0.159  3.31  15.30  5.920  0.416  2.050  0.80  111  4.350  1.080  6.43  3.92  0.872  0.151  3.53  17.40  5.630  0.450  2.100  0.681  fps  R e  4 10 4  fy/ fn  in  x  lb  p  in  y  x  lb  TABLE V.  2" cylinder with 5/8" square section aluminum cantilever and wire i n p o s i t i o n Vres = 4.0 fp s  f n = fh = 4.5 cps  (1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  fyD,  fx  Vres r 2.0 fps  (10)  (11)  fy  fxE>  (12)  Wire Run placed No. at  V fps  V Vres  115  0.752  0.187  1.11  0.91  0.202  0.202  0  0  0  116  1.090  0.272  1.61  1.36  0.302  0.208  0  0  0  0.010  117  1.450  0.361  2.14  4.50  1.00  0.218  4.5  1.00  0.515  118  1.810  0.450  2.68  2.30  0.512  0.212  4.7  1.04  119  2.180  0.542  3.21  2.35  0.523  0.180  4.6  120  2.53  0.632  3.74  3.10  0.690  0.204  121  2.89  0.722  4.72  4.00  0.890  123  3.62  0.902  5.34  4.25  125  4.35  1.080  6.43  0.752  0.187  131  1.450  132 133  130  >° V.  C)°  134 135  /  Re  10*  fy  fy  X  /fn  fn  (13)  (14)  (15)  ^y  /v 0  0  0.007  0.004  0.005  0.011  0.006  0.232  0.116  0.118  0.059  0.430  0.056  0.028  0.118  0.059  1.02  0.350  0.196  0.098  0.360  0.180  4.5  1.00  0.296  0.323  0.162  0.514  0.257  0.229  4.5  1.00  0.259  0.120  0.060  1.010  0.505  0.945  0.195  4.5  1.00  0.206  0.348  0.174  2.780  1.390  4.85  1.080  0.186  4.5  1.00  0.172  0.243  0.122  2.090  1.045  1.11  0.80  0.180  0.178  0  0  0  0.011  0.006  0.361  2.14  1.48  0.330  0.170  0  0  0  0.010  0.005  0.097  0.049  2.180  0.542  3.21  2.12  0.471  0.162  4.2  0.94  0.319  0.091  0.046  0.415  0.208  2.890  0.722  4.72  3.92  0.651  0.225 4.2  0.94  0.241  0.007  0.004  1.670  0.835  3.62  0.902  5.34  4.78  1.060  0.220  4.2  0.94  0.192  0.208  0.104  2.920  1.460  4.35  1.080  6.43  4.67  1.040  0.178  4.2  0.94  0.160  0.208  0.104  2.780  1.390  0  0  TABLE VI.  4" cylinder with aluminum 5/8" square section cantilever  fn = fh s 2.8 cps  (1) Run No.  (4)  (2)  (3)  fps  V Vres  R« ,  V  10  Vres = 5 . 1  (5)  (6)  (7)  (8)  (9)  fy  fy/ /fn  fyD V  fx  12 fn  4  (10) fxD V  Vres = 2 . 6 fps  fps (11)  V  in  (12)  (13)  *2  Py  D  (14) B  lb  (15) ^x  P  in  y  (16)  k_  (17)  (18)  *Px lb  D  79  0.74  0.144  2.18  0.43  0.154  0.20  0  0  0  0.014  0.004  0.069 0.78  0  0  0  0  80  1.07  0.210  3.15  0.58  0.207  0.18  0  0  0  0.036  0.009  0.178 0.95  0  0  0  0  81  1.43  0.279  4.20  0.69  0.246  0.16  2.8  1.0  0.672  0.470  0.118  1.16  3.49  0.390  0.098  1.89  6.11  82  1.24  0.242  3.69  0.63  0.225  0.17  2.8  1.0  0.794  0.256  0.064  1.26  5.00  0.250  0.062  1.21  5.15  83  1.17  0.229  3.50  0.62  0.222  0.18  2.8  1.0  0.815  0.110  0.028  0.548 2.44  0.153  0.038  0.74  3.55  84  1.78  0.348  5.26  0.96  0.343  0.18  2.8  1.0  0.520  0.612  0.153  1.50  2.91  0.700  0.175  3.37  7.02  85  2.14  0.419  6.34  1.15  0.411  0.17  2.9  1.03 0.436  0.808  0.202  3.97  4.01  1.150  0.288  5.56  8.05  86  2.48  0.487  7.32  1.46  0.522  0.19  2.9  1.03 0.372  1.26  0.315  6.23  6.24  0.975  0.244  4.71  5.07  87  2.85  0.557  8.45  2.15  0.768  0.25  2.8  1.0  0.325  1.57  0.392  8.55  6.42  0.236  0.059  1.15  0.94  88  3.22  0.629  9.53  2.47  0.882  0.26  2.8  1.0  0.290  2.37  0.592 11.60  6.88  0.220  0.055  1.10  0.71  89  3.56  0.695  10.50  2.66  0.950  0.25  0  0  0  3.56  0.890 17.50  8.45  0.139  0.035  0.69  0.35  90  3.91  0.768  11.60  2.72  0.970  0.23  0  0  0  3.73  0.932 18.40  7.35  0.167  0.042  0.82  0.35  91  4.28  0.836  12.70  2.77  0.990  0.22  0  0  0  3.95  0.990 19.50  6.54  0.110  0.023  0.55  0.20  - 26 -  TABLE VII Natural frequency of free damped v i b r a t i o n i n water:  Material  Cylinder  diameter  ?  N  Natural frequency fn cps  fn, ^2  Steel  2"  0.0567  0.590  10.4  5.2  Steel  4"  0.0414  0.269  6.5  3.3  Aluminum  2"  0.0961  0.432  4.5  2.3  Aluminum  4"  0.111  0.312  2.8  1.4  TABLE VIII Velocity of resonance  Material  Cylinder  diameter  (based on the assumption: fx = 2fy)  Vres  Vres fps  fps  Rres 10  Rres 4  10  Steel  2"  9.0  4.5  13.2  6.6  Aluminum  2"  4.0  2.0  5.9  2.9  Steel  4"  11.8  5.9  34.5  17.3  Aluminum  4"  5.1  2.6  15.0  7.5  i  4  - 27 For further study of the r e l a t i o n s h i p between longitudinal and l a t e r a l v i b r a t i o n s , a series of runs were done. f i b r a t i o n was developed.  They showed how the l o n g i t u d i n a l  F i r s t , as above, the l o n i t u d i n a l v i b r a t i o n was  suppressed (see F i g , -5 - 7), and the oscillogram recorded was a straight l i n e i n the longitudinal d i r e c t i o n , i . e . there was stationary drag alone.  Simul-  taneously a sinusoidal o s c i l l a t i n g curve was recorded i n the l a t e r a l d i r e c t i o n . This curve belonged to the forced v i b r a t i o n category.  Then, the suppressing  force i n the l o n g i t u d i n a l d i r e c t i o n was removed and v i b r a t i o n i n the l o n g i t u d i n a l d i r e c t i o n was b u i l t up.  With the development of longitudinal s e l f -  excited v i b r a t i o n , the l a t e r a l v i b r a t i n g curves changed i n shape.  At the  beginning, the l a t e r a l curve was distorted, i n that i t was sharper i n shape and wavy along the r i s i n g or descending limbs.  Later, when the s e l f - e x c i t e d  l o n g i t u d i n a l v i b r a t i o n reached i t s maximum, the l a t e r a l curve became i r r e g u l a r with l a r g e r amplitude and indicated v i b r a t i o n i n a resonant  condition.  This also gave an explanation ( F i g . 19 - the v a r i a t i o n of v i b r a t i n g types) f V of the curves with ordinates y vs ^ . I n these curves l o n g i t u d i n a l n res resonance occured at ^ = 0.30 to 0.5. Owing to the e f f e c t of resonance res i n l o n g i t u d i n a l d i r e c t i o n , the points of l a t e r a l frequency moved away from the l i n e of forced v i b r a t i o n . changed, i t was  Since the motion of the v i b r a t i n g system  c a l l e d controlled v i b r a t i o n .  The frequency of l a t e r a l v i b r a t i o n giving an i r r e g u l a r oscillogram  was  analysed by counting the upward limbs which crossed the mean l e v e l of the oscillogram. 5.3 Frequency:  Analysis of experimental data there were three d i f f e r e n t conditions under which the frequency  was measured:  - 28 1)  Frequency of l a t e r a l v i b r a t i o n alone ( f ) : i . e . with longitudinal v i b r a t i o n suppressed.  2)  The r e s u l t s approached those obtained.in the  cylinder condition, i f the amplitude was  small.  L a t e r a l frequency of v i b r a t i o n f o r both  x, y  stationary  (Table I, column 5). directions v i b r a t i o n ( f ):  t h i s frequency varies with l a t e r a l as well as the longitudinal v i b r a t i o n . (Table I, column 8). 3)  Longitudinal ( f ):  frequency of v i b r a t i o n f o r both  t h i s frequency curve was  x, y  direction vibration  regular i n shape and belonged to the  •A.  category of s e l f - e x c i t e d v i b r a t i o n .  (Table I, column 11).  For the purpose of analysis, Strouhal number f o r variations of each of the above types were calculated, as were the frequency r a t i o s In the range of  0 » 5 x 1(T  R  3  -  10 Strauhal Number i s almost 3 x 10'', 5  6  constant.  The following r e l a t i o n can be obtained f D s  V  =  S  =  S  res  -  f D n  V res  V V  res  s f n  f  The above expression i s a straight l i n e i n c l i n e d 45° t o the ( F i g . 19 - 23), representing  the forced v i b r a t i o n due to the e x c i t a t i o n of  the eddy shedding force on a stationary Amplitude: 1)  cylinder.  The l a t e r a l and longitudinal amplitudes were analysed  L a t e r a l Amplitude:  t h i s was  v i b r a t i o n both took place. recording 2)  horizontal  the amplitude when l a t e r a l and The maximum value was  separately.  longitudinal  selected from the  oscillograms.  Longitudinal Amplitude:  only the maximum value was  analysed because i t  varied only within a small range, i . e . the shape of the v i b r a t i n g curve was  more regular than i n the case of l a t e r a l v i b r a t i o n .  - 29 Both of these were i n dimensionless form With the data, amplitude r a t i o vs  R  Vn. curves ( F i g . 24 - 30) were plotted.  R e l a t i o n between the l o n g i t u d i n a l and l a t e r a l amplitude v a r i a t i o n was shown i n the f i g u r e s . Forces;  Forces i n two directions - drag and l i f t - were calculated.  There  were the v i b r a t i n g drag and l i f t which were additional to the steady drag and l i f t .  For the sake of comparison, the dynamic energy of the uniform flow  was used as a standard base f o r both response c o e f f i c i e n t s of v i b r a t i n g drag and l i f t  calculations.  V e l o c i t y of resonance;  the v e l o c i t y of uniform approaching flow, when the  v i b r a t i o n of the cylinder system was at resonance, was the v e l o c i t y of resonance.  Because there were two resonances which could occur, two v e l o c i -  t i e s of resonance existed.  L a t e r a l resonance happened at the frequency of  eddy shedding, which coincided with the natural frequency i n the l a t e r a l direction.  S i m i l a r l y l o n g i t u d i n a l resonance occurred at longitudinal natural  frequency. Before l o n g i t u d i n a l resonance the l o n g i t u d i n a l response w i l l exist i n same phase with the o s c i l l a t i n g drag.  When the frequency of longitudinal  e x c i t a t i o n approaches natural frequency of the e l a s t i c system, the w i l l vibrate near natural frequency.  structure  Then the amplitude keeps increasing  to i t s maximum. T h e o r e t i c a l l y the longitudinal resonance v e l o c i t y i s one h a l f that of l a t e r a l .  But due to the d i s t o r s i o n of vortex, and the i n t e r -  action of two-dimensional e f f e c t s the experiment resulted that  ^res/v v  was about  res  0.4.  From the r e l a t i o n between the v e l o c i t y of the uniform flow and the eddy shedding frequency under the condition of stationary cylinder ( F i g . 18) the  - 30 assumed v e l o c i t i e s of resonance f o r both l a t e r a l and longitudinal directions were selected and l i s t e d i n Table VIII.  VI.  Discussion 6.1  of Results and Conclusion  Discussion:  Frequency:  The Strouhal number i s nearly constant when the Reynolds' number  i s i n the range of  10^ f o r a stationary cylinder.  But, i n two-dimensional  v i b r a t i o n , when v i b r a t i o n i s excited i n the l o n g i t u d i n a l d i r e c t i o n , the Strouhal number of the l a t e r a l d i r e c t i o n suddenly increases. t h i s steep curve approaches peak, the S  S^ vs R  curve as a tengent l i n e .  g  A f t e r the  value drops and merges into the band region ( F i g . 11) given by  the stationary cylinder t e s t s . several variables: direction;  The peak of  and  The shape of the r i s i n g curve depends upon  k, spring constant;  f ^ , natural frequency i n l o n g i t u d i n a l  n, e l a s t i c and viscous damping c o e f f i c i e n t .  The s t a r t i n g point of the shear layer i s important f o r the generation of vortex shedding.  I f one of these separation  points i s f i x e d at one side  of the cylinder and the one at the other side i s l e f t free to move during v i b r a t i o n , the balanced periodic o s c i l l a t i n g force w i l l be altered, as w i l l the periodic c i r c u l a t i o n v a r i a t i o n . i s i n position, the separation  Therefore when t h i s vortex s t a r t e r wire  point i s controlled by the l o c a t i o n of the  wire. . Then the beginning of the shear layer also changes i t s l o c a t i o n , and non-balanced shear layers give a d i f f e r e n t eddy generating condition.  This  unbalanced force condition a f f e c t s the e x c i t i n g forces, also the frequency of v i b r a t i o n and the Strouhal Number as w e l l .  ( F i g . 15 and 16).  The mag-  nitude of the Strouhal Number w i l l be the highest when the wire i s placed at a 45° p o s i t i o n .  Then the next ones are that when the wires are placed at  positions 67.5°, 0°, 22.5°, and f i n a l l y no wire condition. occurs with the wire at 22.5°.  The lowest  S  When there i s v i b r a t i o n i n both d i r e c t i o n s ,  2"cylinder,  Fig,  5  aluminum  Development  cantilever  of l o n g i t u d i n a l  vibration  U o 1.45 f p s 4" c y l i n d e r , a l u m i n u m  Fig.  6  Development  cantilever  of l o n g i t u d i n a l  vibration  i  - 33  -  TJ o 1.81 fps 4" c y l i n d e r , aluminum  Fig.  7  Distorsion  of  lateral  \  cantilever  oscillogram  -  3U  -  the Strouhal Number v a r i a t i o n i s quite complex. e f f e c t makes the curve (S vs R )  distorted.  g  An additional longitudinal  But i t s t i l l bears the charac-  t e r i s t i c s of combined e f f e c t s . The r e l a t i o n of frequency r a t i o vs v e l o c i t y r a t i o describes the type of v i b r a t i o n to be expected i n the t e s t .  ( F i g . 19).  At low flows the ampli-  tude i s small and the v e l o c i t y i s f a r from the v e l o c i t y of resonance f o r both l o n g i t u d i n a l and l a t e r a l d i r e c t i o n s .  I t i s i n forced v i b r a t i o n .  the v e l o c i t y r a t i o increases to near 0.3,  As soon as  the longitudinal v i b r a t i o n s t a r t s ,  and the l a t e r a l v i b r a t i o n i s gradually changed by longitudinal resonance. The points i n t h i s region appear on the diagram as controlled v i b r a t i o n . the v e l o c i t y r a t i o approaches 0.7,  When  the l o n g i t u d i n a l v i b r a t i o n disappears, and  the curve of the controlled l a t e r a l v i b r a t i o n drops back and coincides with that of forced v i b r a t i o n .  This depends on the value of the natural frequency  and the magnitude of spring constant.  I f the natural frequency i n l a t e r a l  d i r e c t i o n i s small, the difference between the natural frequencies i n l o n g i tudinal and l a t e r a l d i r e c t i o n i s also small.  This means that the resonant  regions are too close to each other, and the v a r i a t i o n on the graph i s not obvious.  I f spring constant i s small, the amplitude around longitudinal  resonance w i l l be too large so that, due to the f a s t s h i f t i n g of  separation  points and the increasing rate of eddy generation, the flow pattern at rear part of the cylinder becomes almost random.  The  character of the v i b r a t i o n  then a l t e r s . In F i g . 19 to F i g . 23, the types of v i b r a t i o n are shown. maximum attainable flow was  l i m i t e d by flume flow conditions, some points  near l a t e r a l resonance were not Amplitude;  Because the  attainable.  the amplitude r a t i o f o r longitudinal d i r e c t i o n suddenly r i s e s to  i t s peak near the point of l o n g i t u d i n a l resonance and then f a l l s to a very  - 35 small magnitude. But the l a t e r a l amplitude r a t i o , when the longitudinal one reaches a peak, r i s e s slowly f i r s t , then drops and passes through a minimum value i n the region of longitudinal resonance.  Then i t r i s e s again toward  the condition where l a t e r a l v i b r a t i o n only e x i s t s . The largest amplitude r a t i o of longitudinal v i b r a t i o n i n these four sets of tests i s about 0.25, near a v e l o c i t y r a t i o of 0.4.  When the wires were i n  position, a l l the maximum values were i n the region of v e l o c i t y r a t i o about  0.4. Response c o e f f i c i e n t of o s c i l l a t i n g l i f t and drag of an e l a s t i c body The response c o e f f i c i e n t of v i b r a t i n g l i f t and drag due to an o s c i l l a t i n g f l u i d force i s computed by the following equations: AP  B  Vx  x  2  = A  ?l]  P  B  P  7  /2  V-  A ? U 2 /2  The v i b r a t i n g l i f t varies with the r i g i d i t y of the structure. I t s maximum values occur between v e l o c i t y r a t i o 0.2 and 0.4 i n the region of longitudinal e f f e c t .  The response c o e f f i c i e n t of v i b r a t i n g l i f t have a  value above 3 f o r 2" cylinder and s t e e l cantilever ( F i g . 31). The maximum v i b r a t i n g drag response c o e f f i c i e n t occurs i n the region when the v e l o c i t y r a t i o i s between 0.2 to 0.5, and i t s magnitude i s about 5 t o 8 i n these t e s t runs f o r the damping c o e f f i c i e n t s (n value) i n the range of 0.25 to 0.60. 6.2 Conclusion l)  The s e l f - e x c i t e d l o n g i t u d i n a l v i b r a t i o n occurs between the v e l o c i t y r a t i o of 0.2 to 0 . 8 , when both Its  peak i s around 0.4.  x  and y  directions are allowed to vibrate.  The frequency of o s c i l l a t i n g drag, which i s the  - 36 e x c i t a t i o n force of longitudinal v i b r a t i o n near l o n g i t u d i n a l resonance, i s predicted to be twice that of the l a t e r a l o s c i l l a t i n g l i f t . In the region of longitudinal resonance, the oscillogram  i s affected and  distorted i n the l a t e r a l d i r e c t i o n by l o n g i t u d i n a l v i b r a t i o n .  The degree  of d i s t o r t i o n depends on the amplitude of the l o n g i t u d i n a l v i b r a t i o n . Hence, the frequency o f l a t e r a l v i b r a t i o n increases with the i n t e n s i t y of longitudinal v i b r a t i o n .  This i s referred to as a controlled v i b r a t i o n .  The Strouhal Number of l a t e r a l v i b r a t i o n reaches a peak ( F i g . l l ) i n the region of longitudinal resonance and approaches the curve:  S  vs R  as  a tangent. L a t e r a l v i b r a t i o n i n two-dimensional v i b r a t i o n i s neither the s e l f - e x c i t e d nor the forced type.  I t i s affected by longitudinal movement, and i s  therefore c a l l e d controlled v i b r a t i o n . In two-dimensional v i b r a t i o n , the amplitude of longitudinal v i b r a t i o n i n the region of l o n g i t u d i n a l resonance r i s e s suddenly to a peak and then decreases again to a very low value. In two-dimensional v i b r a t i o n , the amplitude of l a t e r a l v i b r a t i o n i n the region of l o n g i t u d i n a l resonance f i r s t r i s e s and then decreases t o a lower value as longitudinal amplitude approaches i t s maximum. Then the curve r i s e s again to the o r i g i n a l path which i s the c h a r a c t e r i s t i c of l a t e r a l v i b r a t i o n alone. When the separation  point i s controlled by wire l o c a t i o n , the s t a r t i n g  point of the shear layer i s altered.  Hence the periodic e x c i t a t i o n of  the v i b r a t i n g l i f t and drag i s altered.  The unsymmetrical  oscillating  f l u i d forces produce an oscillogram with i r r e g u l a r shape and change the v i b r a t i n g frequency, sometimes decreasing the amplitude of v i b r a t i o n and sometimes increasing i t .  At present no general conclusion  the best position of the starter wire has been possible.  concerning  - 37 The response c o e f f i c i e n t of v i b r a t i n g l i f t due to o s c i l l a t i n g f l u i d force i s about 5 i n these four conditions of different spring constant, damping y constant and natural frequency values.  The peaks appear at  ^  = res  0.2 to 0.5.  For v i b r a t i n g drag i t varies from 3 to 8 i n the region of  longitudinal resonance. There would be a second maximum f o r amplitude of l a t e r a l v i b r a t i o n at or near a v e l o c i t y r a t i o of 1.0 but only one t e s t was run i n t h i s range ( F i g . 21),  Around t h i s region the v i b r a t i o n appeared to be a controlled  type. Because the longitudinal v i b r a t i o n appeared at a v e l o c i t y r a t i o as low as about 0.4, the design of hydraulic structures i n t h i s region of flow should be checked f o r the i n s t a b i l i t y a r i s i n g from two-dimensional v i b r a t i o n , i f the structure allows to v i b r a t e .  - 38 APPENDIX I  Structural Vibration When a structure i s acted by an external disturbance the forced v i b r a t i o n w i l l take place.  The forces that come into play w i l l be external force,  i n e r t i a force, e l a s t i c restoring force of the s t r u c t u r a l system and damping forces.  When s i m p l i f i e d as a problem of a single concentrated mass, the  equation w i l l be i n the form (17): Mx + D (t) + kx  -  (t)  F  Under the assumptions that the pulsating load: the viscous  damping:  cular s o l u t i o n w i l l x  where  H  D (t) = cx  F (t) = F  q  S i n cot  and  by ignoring transient s o l u t i o n , the p a r t i -  be: =  p  AH  [ (1 - (^) ) p' 2  L  v  =  v  —^  S i n cot -  -—-  p  2  (called:  Cos cot ]  magnification  factor)  p The  c h a r a c t e r i s t i c of v i b r a t i n g motion can be expressed by r e l a t i o n  between where:  H  and  w  /  p  :  p  =  c i r c u l a r frequency  co  =  angular v e l o c i t y  n  =  A  =  c  /m  static deflection  I t says when frequency of disturbing force approaches natural frequency the amplitude w i l l be the maximum under the energy balance of disturbing damping forces.  and  - 39 -  The force v i b r a t i o n of an e l a s t i c slender beam of continuous mass d i s t r i b u t i o n i s governed by the equation:  where  F ( t ) = C„ . q  . S i n tot  i f external sinusoidal force i s the  l i f t caused by a f l u i d flow. The solving of the homogenious boundary problem d i f e r e n t i a l equation, i . e . l e t the F ( t ) = 0, y i e l d s the s o l u t i o n of natural frequency o f the slender, e l a s t i c beam:  where:  c  i s a c o e f f i c i e n t depends on the type of beam and mode of  vibration . m:  mass per u n i t length .  - 41 APPENDIX I I Quasi-Steady Theory Because of no complete hydrodynamic theory f o r the separated flow the quasi-steady (semiempirical theory) i s employed.  Using stationary  hydro-  dynamic forces on a v i b r a t i n g body, a s o l u t i o n i s obtained f o r a non-linear d i f f e r e n t i a l equation.  I f the f l u i d force i n the forced v i b r a t i o n equation  is: F where:  C  =  y  =  C | A U  ( L a t e r a l force)  2  f (a, C , L  C) D  Then with a s i m p l i f i e d r e l a t i o n , using a polynomial to express the function of c o e f f i c i e n t of l i f t and drag, the r e s u l t i n g equation w i l l be (15): T - e (1 - P Y where  Y  constants.  2  - Q Y )Y 4  + Y  i s a dimensionless displacement, and  =  0 e, p  and  Q  are damping  A solution, involving the s u b s t i t u t i o n of polynomial approxima-  tions to the aerodynamic force curve, has been used to predict the amplitude of the v i b r a t i o n .  approximately  - 42 Bibliography 1.  L. Rosenhead:  Laminar Boundary Layers, p. 93, 102.  Oxford, 1963.  2.  M. Rauscher: Introduction to Aeronautical Dynamics, p. 25, 257, John Wiley, 1953.  3.  W. Kaufmann:  4.  V.L. Streeter:  5.  D.M. McGrefor: An Experimental Investigation of the O s c i l l a t i n g Pressures on a C i r c u l a r Cylinder i n a F l u i d Stream. U. of Toronto, 1957.  6.  W. Weaver: Wind Induced Vibrations i n Anttenna Members. 1962, ASCE, p. 681.  7.  Eduard Naudascher: On the Role of Eddies i n Flow-induced Vibrations Iowa I n s t i t u t e of Hydraulic Research; I.A.H.R. Congress, London, (1963 sep.).  8.  G.H. Toebes: Hydroelastic Forces on Hydraulic Structures due to Turbulent Wake Flows, Purdue University. I.A.H.R. Congress, Dubrovnik, 1961.  9.  A. L a i r d :  F l u i d Mechanics, p. 251, 257, McGraw H i l l ,  366,  1963.  F l u i d Dynamics, Chap. VII, p. 200.  Trans. V o l . 127,  Water Eddy Forces on O s c i l l a t i n g Cylinder, A.S.C.E. Trans. I962.  10.  J.H. Gerrard: An Experimental Investigation of O s c i l l a t i n g L i f t and Drag, J . of F l u i d Mechanics, 196l.  11.  Eduard Naudashcher: V i b r a t i o n of Gates During Overflow and Underflow, A.S.C.E. Trans. 1962.  12.  F.B. Campbell:: Trans. 1962.  13.  A. L a i r d : Groups of V e r t i c a l Cylinder O s c i l l a t i n g i n Water, Mechanics, Feb., 1963.  14.  D. Pierce: Photographic Evidence of Formation and Growth of V o r t i c i t y , J . of F l u i d Mechanics, I96I.  15.  G.V. Parkinson: On the Aeroelastic I n s t a b i l i t y of B l u f f Cylinders, A.S.M.E., J . of Applied Mechanics, June 1962.  16.  A. Roshko: Experiments on Flow Past a C i r c u l a r Cylinder at very High Reynold's Number, May 1961, J . of F l u i d Mechanics.  17.  Rrover:  Dynamics of Frame Structure.  18.  Norris:  S t r u c t u r a l Design f o r Dynamic Load.  19.  Birhoff:  V i b r a t i o n Problem i n Hydraulic Structure.  Jets, Wakes and C a v i t i e s .  A.S.C.E.  J . of Eng.  - 43 20.  H. Sato: Mechanism of T r a n s i t i o n i n the Wave of a Thin F l a t Plate Placed P a r a l l e l to a Uniform Flow, J . of F l u i d Mechanics, Sep. 1961, P. 321.  21.  Goldstien: p. 550.  22.  Rouse:  Modern Development i n F l u i d Dynamics - Wakes -  Engineering Hydraulics (Chapter I ) ,  1950.  Vol. II,  - Uh Nomenclature  R  (^p-).  -  Reynolds' number  D, d  "  diameter of c i r c u l a r cylinder ( f t ) .  h  =  depth of flow ( f t ) .  v, u  =  v e l o c i t y of flow ( f p s ) .  =  kinematic v i s c o s i t y ( f t / s e c ) .  =  v e l o c i t y of eddy behind c y l i n d e r .  S  =  Strauhal number (fD/v), v i b r a t i o n i n l a t e r a l (y) d i r e c t i o n only.  S  =  Strauhal number ( f ^ / v ) , i n longitudinal d i r e c t i o n , both vibrating.  =  Strauhal number ( f D/v), i n l a t e r a l d i r e c t i o n , both x, y y frequency of forced v i b r a t i o n i n l a t e r a l d i r e c t i o n only.  e  V  v  x  5 y f  =  f  x, y vibrating.  =  frequency of forced v i b r a t i o n i n l o n g i t u d i n a l d i r e c t i o n , both vibrating.  =  frequency of forced v i b r a t i o n i n l a t e r a l d i r e c t i o n , both vibrating.  =  frequency of forced v i b r a t i o n i n l a t e r a l d i r e c t i o n f o r stationary cylinder.  f  «  natural frequency of the system, l a t e r a l d i r e c t i o n .  f^  =  natural frequency of the system, longitudinal d i r e c t i o n .  6  =  amplitude i n l a t e r a l d i r e c t i o n (from wave crest to trough).  x  f y  f 8  X  x, y  x, y  6 y P  =  amplitude i n l o n g i t u d i n a l d i r e c t i o n (from wave crest to trough).  =  force.  P  =  max. hydro-elastic v i b r a t i n g force i n l o n g i t u d i n a l d i r e c t i o n .  P  *  max. hydro-elastic v i b r a t i n g force i n l a t e r a l d i r e c t i o n .  •^Apx  ~  B  »  x  r e s  P  o n s e  c o e #  o  f  v i b r a t i n g drag,  */<^u /\ z  P / z response coe. o f v i b r a t i n g l i f t ,  y ? jr ^  - 45 V  =  v e l o c i t y of resonance, l a t e r a l v i b r a t i o n only.  V*  =  v e l o c i t y of resonance, l o n g i t u d i n a l v i b r a t i o n only.  R  =  Reynolds' number, when only l a t e r a l resonance occurs.  =  Reynolds' number, when only longitudinal resonance occurs.  X, Y  =  f l u i d force acting on cylinder i n x, y d i r e c t i o n .  U  =  v e l o c i t y of uniform flow.  w  =  complex function.  z  =  complex v a r i a b l e .  f  = circulation.  Ap  =  pressure difference.  V  =  l o c a l v e l o c i t y i n the flow.  f  =  mass density.  t  =  time.  M  =  mass of the system.  k  =  spring constant.  E  =  modulus of e l a s t i c i t y .  I  =  moment of i n e r t i a .  L  =  length of beam.  T  =  p  = g ,  I*6S  1*6 S  R  1  period. c i r c u l a r frequency.  -  46  Flow  NXV  Fig.  „  9  Definition  sketch  \  O  o  O  O  O  •  •  o  ro  4^  d\.  - 8> "  p  ••  ob  M  •  •  o  ro  Position x  0  0.30  '  o  „  no  of  wire  wire  o° 22.5° 45°  o «  0.25  0  67.5  o  •  ( v i b r a t i o n i n y-dire< tion only )  0.  0 0 o x * X  fS  x  0 *  »  0 % A  A  X o t  *  i  A  0.15  F i g . 15 2"  Cont r o l w i r e  j y l i n d e r with s t e e l  test  o  (l)  cantilever  and l / l 5" w i r e  0.10  0.05 5  I  O  5  ~ Re  VBT  -  1$  5  10  5  5  Position  1.0  ^  of wire 0°  22.5° 0.8  * °  67.5°  0  *  ( v i b r a t i o n i n both x,y d i r e c t i o n s )  0.6  «H  * 1 A  CO  Fig.  0.4  16  E f f e c t of  location  ni.T>.  O 111 ^..rJ-^-v, J  steel  ca i t i l e v e r  of  A  ....... O  o 0  o o  0.2  /S  0 ft  0 *  A  0 Re  YD.  If  io-  0.30  Position  0.25  c  0.20  o  (vibration only  0.15 u CO  0.10  Fig. 2"  l7__C^ojatjrai^ijr_e^.e.3J;^(j2,). c y l i n d e r w i t h aluminum  cantilever  and  1/8" w i r e  0.05  10  • "  B  VD .  10'  of wire  -4-5in y direction )  1.2  0  0.2  0.4 Velocity  0.6 ratio,  1.0  0.8 V/ 1  = f res  s  . '*  n  1.2  1.4  S e l f - e x c i i ed  /  v i b r a t ; on  1.0  o ontrolled  0.8  vibration  / 0°  ••./ 0 0  a  '-i-b-i-a-ti-o n  o  O  A  ;.  2"  V a r ^. a t i o n  25  cylinc.er,  /  res  steel  '  n  of  45°  ';ype  cantilever,  of  v^ith  o  22.5  v i iD r a t i o n control  wire  -  Z9  -  - £9 "  0 .10  8  6.0  1 1 1  ^ p  CO 0)  5.0  1  \  drag  - u  tion.  w  ^px  Bp  4.0  l i f t If  B  no  1/V  a d d i t i onaJ  to  l o n g i tu d i n a l  v i br a -  no  l o n g i tu d i n a l  v i br a -  B^px  due  =  to  p  0.  coef.  of  l a t e r a1  lateral B  3.0  of  If  Response  s  Y  coef.  due  •tion,  VTB \  Response  J  X  =  y  a d d i t i ona] vibratioi .  v i b ration,  ON CO  0.  T)v J  2.0  F i g .  31  Response  v i b r a t i n g 2"  coef .  drag  cylinder,  ana  steel  of l i f t  (l)  cantileve r  1.0  10  4  Re  VD  10"  /  4>  10  - 74 -  F i g .  31  Oscillograph  and  marking  pen  - 75 -  cylinder Plow Pig.  38  Cylinder  vibrating  near  longitudinal  resonance  photo-was U = I . 4 4 fps f = 14.5 cps f «= 4 . 6 cps x  y  Fig.  39  taken.(Fig.  38)  4" c y l i n d e r aluminum c a n t i l e v e r Camera s p e e d : l / 2 5 s e c w i t h w h i t e f l o a t s on s u r f a c e  Tv/o-dimensional  vibration  of  water  

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