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Two-dimensional structural vibration induced by fluid flow past a circular cylindrical body Chow, Yu-Min 1964-12-31

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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 University Taipei, 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Apr i l , 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y , a v a i l a b l e f o r reference and study, I f u r t h e r agree that per m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t , c o p y i n g or p u b l i  c a t i o n .of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Yu-min Chow Department of C i v i l E n g i n e e r i n g  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 , Canada D a t e May 1. 19 64 ' i i Abstract The investigation i s concerned with the vibrational response of a c i r  cular cylindrical body when subjected to oscillating l i f t and drag forces of varying frequency. The cylindrical body was mounted on a long flexible can ti l e v e r and the stiffness of the cantilever could be varied to study the effects of resonance, amplitude and induced damping. An explanation of the two-dimensional excitation due to vortex shedding i s presented. Both longitudinal and transverse vibrations were induced and the f r e  quency of the excitation i n the longitudinal direction was about twice that i n the l a t e r a l . Therefore, for equal structural natural frequencies i n the lat e r a l and longitudinal directions, resonance i n the la t e r a l direction w i l l occur at twice the velocity of the longitudinal resonance, assuming the Strouhal Number to be constant i n this range. Coefficients of the elastic response forces, i n both longitudinal and later a l directions near the l o n i - tudinal resonance, were plotted for comparison. The development of irregularity i n the la t e r a l vibrating oscillogram caused by the longitudinal vibration was demonstrated. In the c r i t i c a l Reynolds Number range the effect of a fixed eddy-starter wire was investi gated and the excitations i n both l a t e r a l and longitudinal directions were altered. The resonance of longitudinal vibration occurs at ^/v_0_ = 0.3 to 0.5. r©s Lateral amplitude i n this region i s also increased. The peak of the curve: vs Rg, i n the region of longitudinal resonance, approaches tengentially the curve: S vs R . The structural response coefficient of vibrating l i f t and drag rises at longitudinal resonance to a value of 3.0 to 8.0, but drops quickly after resonance. The damping coefficients of the structure are i n the range of 0.25 to 0.60. i i i Since the motion of the vibration may influence the oscillating l i f t and drag due to the eddy shedding, this study presents an investigation of the longitudinal and l a t e r a l resonances i n a range of Rg: 5 x 3 x 10 , and the interaction of the exciation between the oscillating l i f t and drag with the two-dimensional elastic response forces. ACKNOWLEDGEMENT The author wishes to express his gratitude to Dr. M.C. Quick for his guidance and criticism i n the experimental work and the writing of the thesis. The author i s also grateful to Prof. J.F. Muir, head of C i v i l Engineering Department, for his comments and ' the grant of the assistantship. The experimental work was carried out i n the hydraulic laboratory, Dept. of C i v i l Engineering, and appreciation i s also expressed by the author to the staffs i n the workshop. V TABLE OF CONTENTS Abstract Acknowledgement Contents Page I. Introduction 1 I I . Basic Concepts 3 III. Qualitative Explanation of the Two-dimensional Vibration Phenomena 7 IV. Apparatus and Instrumentation 14 4.1 General Consideration 4.2 Water Flume 4.3 Model Mounting 4.4 Instrumentation V. Testing Procedure and Analysis 17 5.1 Calibration 5.2 Experimental Results 5.3 Analysis of Experimental Data VI. Discussion of Results and Conclusion 30 6.1 Discussion 6.2 Conclusion VII. Appendix 38 1. Structural Vibration 2. Quasi-steady Theory Bibliography . 42 Nomenclature 44 8 Tables 36 Figures 46 3 Plates 74 v i Tables 1. Test results: 2" cylinder with steel cantilever. 2. Test results: 2" cylinder with steel cantilever and control wire. 3. Test results: 4" cylinder with steel cantilever, 4. Test results: 2" cylinder with aluminum cantilever. 5. Test results: 2" cylinder with aluminum cantilever and control wire, 6. Test results: 4" cylinder with aluminum cantilever. 7. Natural frequency of free damped vibration i n water. 8. Velocity for longitudinal and la t e r a l resonance. v i i Figures 1. C i r c u l a r cylinder i n uniform flow with von Karmants vortex street. 2. Theoretical 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. Sinusoidal v a r i a t i o n of v x and v^ .. 5. Development oflongitudinal v i b r a t i o n (1). 6. Development of longitudinal v i b r a t i o n (2). 7. Distorsion of l a t e r a l oscillogram. 8. Resonance of structure v i b r a t i o n . 9. D e f i n i t i o n sketch. 10. Water flume and te s t equipment. 11. S vs R g curve, 2" cylinder with s t e e l cantilever. 12. S vs R g curve, 4" cylinder with s t e e l cantilever. 13. S vs R g curve, 4" cylinder with aluminum cantilever. 14. S vs R curve, 2" cylinder with aluminum cantilever. 15. Control wire t e s t ( l ) . 16. Effect of lo c a t i o n of wire. 17. Control wire test ( 2 ) . 18. Relation between frequency and v e l o c i t y . 19. Variation of type of v i b r a t i o n — 2" cylinder, s t e e l cantilever. 20. V a r i a t i o n of type of v i b r a t i o n — 4" cylinder, aluminum cantilever. 21. V a r i a t i o n of type of v i b r a t i o n — 2" cylinder, aluminum cantilever. 22. Variation of type of v i b r a t i o n — 4" cylinder, s t e e l cantilever. 2 3 . Variation of type of v i b r a t i o n — 2" cylinder, s t e e l cantilever, with control wire. 2 4 . Amplitude r a t i o ( l ) . 25. Amplitude r a t i o (2). 26. Amplitude ratio (3). 27. Amplitude ratio (4). 28. Amplitude with control wire ( l ) . 29. Amplitude with control wire (2). 30. Amplitude with control wire (3). 31. Response coefficient of vibrating drag and l i f t ( l ) . 32. Response drag coefficient of vibrating drag (2). 33. Response l i f t coefficient of vibrating l i f t (3). 34. Response drag coefficient of vibrating drag (4). 35. Response l i f t coefficient of vibrating l i f t (5). 36. Water flume and test body. 37. Oscillograph and marking pen. 38. Two-dimensional vibration (photo), 39. Two-dimensional vibration (oscillogram). Two-Dimensional Structural Vibration Induced By  Fluid Flow Past A Circular Cylindrical 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 velocity. 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 longitudinal as well as l a t e r a l vibrating forces. The eddy shedding from unstable shear layers near the separation 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 rise to an osci l l a t i n g pressure distribution on the boundary of the body i n the separation zone, the resultant alternating force i n i t s vector sense w i l l give a two-dimensional excitation to the moving body. In a vibrating system with two-dimensional freedom, the vector sum of alter nating forces can give rise to longitudinal and l a t e r a l as well as torsional resultants causing vibration. This makes the vibration condition very compli cated even i n the region where the velocity i s far below the la t e r a l resonant velocity. Therefore the subject of this investigation is the relationship of lat e r a l and longitudinal vibration characteristics, and their interactions. The starting point of the shear layer intersects the boundary of the body, which stays i n the flowing f l u i d , at separation point. The location of the separation point varies with the circulation of the shedding vortex, hence the mechanics i n the shear layer may change accordingly. Oscillation of the solid boundary i n either the longitudinal or l a t e r a l direction causes the shear layer to take on a wave pattern and also increases i t s i n s t a b i l i t y . Therefore, additional distortion of the eddy shedding vortex alters the - 1 -- 2 - excitation, resulting i n an irregular vibrating path. In order to obtain evidence to explain this irregularity, oscillograms were recorded by pen recorders i n both the longitudinal and l a t e r a l direction separately and simultaneously. There are three types of excitation i n vibration: forced, self-excited and controlled. Self-excited vibration always occurs at i t s natural f r e  quency, while forced vibration follows the frequency of the alternating external force. At resonance the frequencies of forced and self-excited vibrations coincide, and large amplification of response occurs and i s only limited by the induced damping. If the vibration of the body influences the osc i l l a t i n g flow pattern, the resulting vibration i s no longer either s e l f - excited or forced, and belongs to the category of controlled vibration ( l ) . 1 The investigation of the oscillating f l u i d pressure around the cylinder has been done by several investigations, such as D.M. McGregor (5) and J.H, Gerrard (10). 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 vibration under small vibrating amplitude, represent the oscillating 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 vibrating under the f l u i d forces. The l i f t coefficient (5) i s about 0.6 while the drag coefficient i s approximately 0.06, both refer to os c i l l a t i n g parts. Sometimes the structure i s allowed to move or i s flexible i n nature, such as hydraulic gates or suspension bridges respectively. The motion of the structural vibration influences the f l u i d force, and vice versa. There fore this study investigates the structural response forces during f i n i t e «<•• 1 Numbers i n brackets refer to the bibliography. amplitude motion excited by the f l u i d force. A l l the test runs allow the structure to have f i n i t e amplitude during vibration. A structure of two- dimensional freedom with equal natural frequencies i n l a t e r a l and longitu dinal direction was used. The irregular oscillograms of controlled vibra tion are recorded and analysed for Reynolds Numbers varying from 5 x 10^ to 3 x 10 , (for velocity ratio: - 0.2 to 0.9). The longitudinal res vibration appeared at a velocity ratio as low as 0.4 which i s considerably lower than that of la t e r a l resonance. In the present investigation the maximum velocity ratio attainable was limited by the maximum rate of flow of water available, combined with limitations on depth and velocity set by c r i t i c a l flow conditions. I I . Basic concepts Introduction The wake behind a moving body i n a real f l u i d represents a discontinuous flow phenomenon which i s bounded by the thin shear layer known as a vortex sheet. The pattern of a double row of vortices originating from opposite sides of the body changes with growing R . In the range of R used i n these experiments, say 5 x 103 to 3 x 10^ the oscillating force appears to be due to eddy shedding. In considering the vibration 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 of the body, li i i c h give rise to a net o s c i l l a t i n g force acting on the structure. Secondly there is the vibra tional characteristics of the structure which may have several modes of vibration. Thirdly there i s the modification of the flow pattern by the structural vibration and the possibility of self excitation. 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 third complex aspect of interaction. - 4 - Blasius theorem From Blasius theorem the resultant f l u i d force x, y i n an ideal uniform flow with circulation can be expressed (4) by: where: w = U (z + — ) + In z dw Using the Cauchy integral theorem and i t s extension, and with resulting from flow around cylinder with circulation, 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 for ideal flow without separation but with circulation. For two-dimensional vibration, the o s c i l  lating pressure due to periodic change of circulation i s an essential factor. Oscillating pressure study: An osc i l l a t i n g pressure study by D.M. McGrefor(5) included a series of experimental and some mathematical steps. The lo c a l pressure i s obtained from the Bernoulli equation: p ,T I2 2s ?)<P Ap = | (U-v ) - p - In the complex potential, i n order to maintain the cylinder surface as a streamline, two vortices of equal strength but opposite sign were located 2 at distances a and r£ away (Fig. la) from the origin; so that one l i e s outside and one inside the cylinder: 2 . r z - b w = U (z + —) + — - I n r — z  z . a2/ b Assuming a periodic circulation of f Sin w t due to opposite m eddy shedding, the pressure coefficient w i l l be: cP - § 1 w-> tv - 5 - •A r F i g , l a 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 ideal and that i n the wake as being altered, this reveals only some aspects of local pressure variation due to the shedding of vortices. 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 relation between frequency and pressure coefficient i n a given flow, appeared i n two peaks which were called the fundamental and secondary harmonic frequencies. The fundamental frequency peak, about 165 cps, occured at 90° position 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~ tion i s excited by these oscillating pressures, the frequency of longitu dinal vibration w i l l be twice that i n the la t e r a l direction. This study i s very interesting and i t s results are supported by the finding of the present experiments. Response of elastic supported structure to eddy shedding force If a structure i s acted upon by externally applied os c i l l a t i n g f l u i d force, a time-dependent motion w i l l be set up. The forced vibration can be expressed (6) by: My + D(t) + kY = F(t) where: F(t) = F^ • q A Sin 2 n f t q = stagnation pressure D(t) = damping function A = area of body projected on f l u i d stream F R • Von Karmann vortex shedding force f » frequency of eddy shedding x Appendix I The amplitude ratio of a system with a single degree of freedom and having viscous damping, may be written: Y c l CK -^ 2/ k where: c^ i s constant for a given system Cg : coefficient. JZ : ratio of the frequency of forcing function to natural frequency 8 : damping decrement. Since there i s no complete hydrodynamic theory for separated flow, the response function of an elastic system could be treated by quasi-steady theory.* An estimate of the natural frequency of the system may be made by de flectin g i t and allowing i t to vibrate freely. The resulting damping curve can also be used to calculate the value 8. The damping w i l l change with the velocity of flow and with amplitude of vibration. It 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 vibration. III. Qualitative explanation of the two-dimensional vibration phenomenon: Shear layer In the ideal 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 1 dp V y=0 " ^ * 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 of ri s i n g pressure x Appendix II - 8 - and potential flow reversal i n the boundary layer. Two separation points and A^ are indicated (Fig. l c ) . Due to the reversal of flow, a thin shear layer i s formed with a r o l l i n g eddy train along i t s path starting from the separation point and decaying i n the down stream direction. Lateral vibrating velocity According to the Kelvin's circulation theorem there i s zero net vo r t i c i t y and therefore every time a vortex is shed there i s a circulation around the cylinder with opposite sign to that of the shedding vortex. This circulation with uniform flow generates a l i f t force and moves the separation points back and forth (Fig. l c ) . When the circu l a t i o n changes: p -» o -- - p , the separation points move to positions opposite to the previous ones, as i n Fig. l c . Hence the wake shifts repeatedly from l e f t to right. An elastic supported body acted on by this periodic force w i l l vibrate i n a direction perpendicular to the direction of the uniform flow. I f the eddy shedding frequency is lower or equal to the natural frequency of the structural vibration, the sinusoidal l a t e r a l velocity of vibration can be expressed: V • V Sin wt y ym where: = instantaneous maximum la t e r a l vibrating velocity. When the eddy shedding frequency i s higher than the natural frequency of the elastic system, the pulsating force w i l l run out of phase with the response of structure. Then the amplitude w i l l diminish quickly to a very small value. A vibration i n random nature w i l l be resulted. - 9 - Longitudinal os c i l l a t i n g force X i . With no la t e r a l vibration: When a vortex i s shedding, growing and passing downstream (Fig. lc) the tangential velocity at the circumference of rear part of the cylinder changes according to the velocity distribution of the vortex. One eddy w i l l give one complete cycle of the velocity varia tion. In the meantime, from energy theorem the oscillating pressure i n longitudinal direction also completes one periodic excitation to the vibra tion of the structure. Therefore, the oscillating drag has a frequency twice that of pulsating l i f t . i i . Reinforced by lat e r a l vibration: When the vortices are shedding alterna t i v e l y from the cylinder, the circulation around the cylinder also changes i n value from positive to negative. Therefore an assumption is made: r = r Sin cot ' ' m Considering the cylinder travelling from A to B (Fig. 2a), the sum of the velocity vectors i s : v = (ir + v y toward the cylinder with an attack angle a. From the Magnus effect, a l i f t force L exists, which gives two components i n x and y-direction: X, Y: X = p U ( T Sin tot) Sin a m Y - pU ( f Sin tot) Cos a m From the above equation, the longitudinal force i n two-dimensional vibration i s a function of l i f t and attack angle. When the cylinder, after reaching point c, vibrates back to D (Fig. 2b), the Y-component of the l i f t changes i t s direction, but the X-component s t i l l acts i n a positive x-direction. From that the angle of attack i s zero, and assuming the c i r  culation to be zero at points A, C and E, components X and Y w i l l be - 10 - F i g . 2c F i g . 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 - 11 - zero at these locations. This means that when the variation 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 excitation i n the longitudinal direc tion w i l l be twice that i n the l a t e r a l . This relation i s indicated i n Fig. 3. So far 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 distorsion (8) of the vortex along i t s axis or other small disturbance w i l l change the value of the excitation, the response of the elastic structure w i l l then change accordingly. It i s considered, although there i s at present no evidence to support i t , that once a vibration commences, a l l the vortex shedding along the cylinder w i l l be forced into phase. Resultant vibration of two-dimensional periodic phenomena In the two-dimensional vibration, the longitudinal velocity of cylinder w i l l also vary with time (Fig. 4)j this means that an additional periodic variation of V x w i l l be superposed on the uniform velocity U. Since the l i f t force i s a function of longitudinal velocity, the resulting 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 longitudinal vibration w i l l tend to excite additional lat e r a l vibrational forces, and conversely a l a t e r a l vibra tion w i l l cause additional longitudinal vibrational forces. This complex interaction i s demonstrated by some of tests i n which self-excited vibrations appear. It has not been found possible to discuss the interaction quantita t i v e l y but a qualitative understanding i s possible. The fact that the two modes di 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. It i s intersting to consider 0 a F i g . 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 V e l o c i t y Zone of V v a r i a t i o n x r , /Zone of V v a r i a t i o n -UL y F i g . 4 S i n u s o i d a l v a r i a t i o n of V and V * ' 7 V - 13 - what might happen i f the l a t e r a l vibration were i n the second mode whilst the longitudinal vibration coincided with the f i r s t mode, assuming the f i r s t and second modes to have a frequency ratio of two. Sta b i l i t y of two-dimensional vibration Since the frequency of excitation i n longitudinal direction i s twice that of the l a t e r a l , the velocity of resonance i n the longitudinal direction 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 situation of self excitation the s t a b i l i t y of the system must be considered. If the induced excitation increases more rapidly than the damping force, violent two-dimensional vibration results, hence the struc ture w i l l be overloaded. Conclusion Stationary cylinder: Starting from Von-Karman's theoretical vortex street with a stationary cylinder, due to the circulation caused by shedding vortex, the vortex street may shift l e f t and right resulting a wave-like wake path. This w i l l produce a smaller component of oscillating drag, as vortices of alternate sign grow and pass downstream. It has twice the fr e  quency of the oscillating l i f t and arises from a changing pressure distribu tion excitation i n the longitudinal direction. Vibrating cylinder: Since the later a l vibration gives a periodic V y variation, a periodic x-component arises with a frequency twice that of the la t e r a l . This strengthened oscillating drag w i l l cause a strong excitation 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 this experiment emphasized two-dimensional vibration as well as the relationship of i t s frequency, phase and amplitude, a constant flexural system with low natural frequency and recordable l i f t and drag were chosen. A hollow brass cylinder 2" i n diameter was f i r s t used as a test 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 steel bar served as a flexural structure mounted on a 5" I-beam which, supported on concrete walls, was strong enough to hold the vibrating test body. fD From the relation: S = ^- for stationary cylinder of 2" dia meter, i f the velocity of flow is i n the range of 0.50 fps to 4.35 fps, the frequency of eddy shedding w i l l be about 0.6 cps to 5.3 cps respectively. I f a flexural cantilever with a frequency about 5 cps i n both longitudinal and l a t e r a l direction i s chosen, i t w i l l give longitudinal as well as l a t e r a l resonance i n the flow range. In order to make the strain i n strain gauge recordable, strain was limited i n the range of: 50 to 2000 micro-in/in. The natural frequency was checked approximately by the equation: Four combinations, with steel and aluminum material, 2" and 4" circular cylinder, were selected and l i s t e d i n Table VII. Four strain gauges located at the upper end of the cantilever gave the vibrating response forces i n two directions, and below the resonance condition these response forces gave the frequency and amplitude of the vibration. force was required. The 30" wide water flume and steel cantilever mounting where: c: coe. including damping effect. - 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 section. The maximum discharge was about 7 cfs c i r  culated by a pumping system and stabilized by an overhead tank. The dis charge was measured by an or i f i c e flow meter and checked by a volumetric tank. A plate with holes and two steel meshes were used to keep the flow uniform. To eliminate boundary layer effect, a parabolic entrance contraction was installed i n the middle of the flume. This contraction was necessary to provide a high enough velocity. The velocity distribution i n the 15" x 15" test section was checked by a miniature propeller current meter. The devia tion of the velocity from average was + 1.7$. Water depth i n the test section was controlled by a t a i l gate. (Fig. 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 pro blem near c r i t i c a l depth. This condition limits the maximum velocity which we could use. A point gauge accurate to one hundredth of a foot was used for measurement of flow depth i n the test section. Water surface drop at the entrance of the test section was very smooth, and no standing wave appeared. Therefore f a i r uniformity of velocity and pressure distribution was predicted. 4.3 Model mounting A 3/4" square steel cantilever provided a mounting of constant f l e x i b i  l i t y and known damping coefficient for the vibrating system - a circular cylinder 2" i n diameter and one foot long. Later a 4" diameter cylinder was also used as a test body. At f i r s t the cantilever was clamped on to a 5" steel I-beam across the top of the flume. When the cylinder was subjected to even medium flows, the 5" beam vibrated i n torsion. The torsional - 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 ceiling beam and to the horizontal steel beam. To check the accuracy of measurements of strain i n the cantilever, possible movement of the support was checked by d i a l gauge. The movement appeared negligible even under cantilever resonant conditions. The total length of the cantilever system was 38 inches. The portion of the submerged square section was surrounded by a streamlined strut to eliminate the effect of form drag of the cantilever i t s e l f . The cantilever as well as the cylinder was changeable; hence the f l e x i  b i l i t y of the structure as well as the natural frequency of the system could be varied i n the tests. This gave different resonant velocities and f r e  quencies, and different types of self-excited and forced vibration. 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 strain gauge, was amplified and recorded by pen recorder (Fig. 37). In order to record l a t e r a l and longitudinal vibration simultaneously, two sets of strain gauges, amplifiers and pen recorders were used. The recorders accuracy was very high at low frequency: 0 to 20 cps. A wide variation of amplitude was permitted by changing attenuations. Three recording speeds were used: 5, 25 and 125 mm per second. A pen was provided for the purpose of marking the graph and this pen was activated simultaneously with the camera shutter, so that photo graphs could be synchronised with the vibration record (Fig. 38-39). White floats were spread on the surface of the flow so that the flow path could be photographed. - 17 - V . T e s t P r o c e d u r e a n d A n a l y s i s 5.1 C a l i b r a t i o n s T h e d i s c h a r g e o f t h e f l o w m e t e r was c a l i b r a t e d b y w e i g h i n g t a n k . The r a t i n g c u r v e shows t h e r e l a t i o n b e t w e e n d i s c h a r g e a n d f l o w m e t e r r e a d i n g s . T h e a m p l i t u d e o f t h e v i b r a t i n g c y l i n d e r was c a l i b r a t e d d i r e c t l y b y a d i a l gauge i n t h e t e s t p o s i t i o n . 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 r e c o r d e d 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 . B e c a u s e t h e b e n d i n g o f t h e s t e e l beam was w i t h i n e l a s t i c r a n g e , a n d b e c a u s e t h e c a n t i l e v e r was o s c i l l a t i n g i n i t s f u n d a m e n t a l mode , t h e a m p l i t u d e was assumed t o b e p r o p o r t i o n a l t o t h e r e a d i n g o n t h e s t r a i n g a u g e . F o r c e c a l i b r a t i o n was done b y a p p l y i n g a known p u l l i n g f o r c e o n t h e t e s t b o d y w h i l e t h e r e c o r d e r was o p e r a t i n g . The e l a s t i c r e s p o n s e f o r c e c o u l d t h e n be c a l c u l a t e d f r o m t h e c a l i b r a t i o n c u r v e . I t s h o u l d be s t r e s s e d t h a t n e a r r e s o n a n c e t h e a p p l i e d f o r c e a n d t h e c a n t i l e v e r r e s p o n s e a r e n o t e q u a l , t h e r e b e i n g a c o n s i d e r a b l e a m p l i f i c a t i o n o f r e s p o n s e a t r e s o n a n c e . The f r e q u e n c y o f f r e e damped v i b r a t i o n i n w a t e r was f o u n d b y v i b r a t i n g t h e t e s t i n g b o d y i n p o s i t i o n . T h e c y l i n d e r was d i s p l a c e d f r o m i t s e q u i l i b r i u m i n 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 i t v i b r a t e d a t i t s n a t u r a l f r e q u e n c y . 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 o f t h e d a m p i n g . T h e p r o c e d u r e was r e p e a t e d f o r t h e l a t e r a l d i r e c t i o n a n d i t was f o u n d t h a t t h e f r e q u e n c i e s o f l a t e r a l a n d l o n g i t u d i n a l v i b r a t i o n s were t h e same. F r o m t h e s e d a m p i n g c u r v e s b o t h t h e n a t u r a l f r e q u e n c y a n d t h e damping c o e f f i c i e n t , n , were d e t e r m i n e d (17) b y t h e e q u a t i o n : 6 t « • nT 6. „ e where T : p e r i o d o f v i b r a t i o n and l o g a r i t h m i c d e c r e m e n t 6: - 18 - By varying the r i g i d i t y of the elastic system, four conditions of different damping coefficients were used i n the experiment. 5.2 Experimental results Experiment was divided into the following parts, and l i s t e d i n Table I - VIII: 1. 2" circular cylinder with 3/4" square steel cantilever. II II aluminum " n n • - steel " and 1/16" diameter wire placed at four different positions, i . e . 0°, 22.5°, 45° and 67.5°. 6. 2" circular cylinder with 5/8" square aluminum cantilever and 1/8" diameter wire placed at two positions, i . e . 0° and 45°. A l l tests were run at 0.3 fps velocity increments, from the lowest flow for which the recorder was sensitive, to the highest flow for which the flow i n the test section was sufficiently sub-critical to avoid surface distur bances. The speed of the pen recorder was set to 5 or 25 mm per second for general runs and 125 mm per second for special runs. For the runs with 2" diameter cylinder and steel cantilever, a longitu dinal force was applied by a wheel mechanism to the testing body while the l a t e r a l amplide was very small. This gave additional stiffness i n the longi tudinal direction only. Frequency was recorded through a series of flows. Since this represented the l a t e r a l direction vibration only at small vibrating amplitude, the results are very close to stationary cylinder readings. These results were analysed and plotted with S vs R g i n Fig. 11. The points followed Strouhal's description and f e l l into the band zone reported by other authors (7). 2. 4" circular " " 3/4" " 3. 2" circular " " 5/8" " 4. 4" circular " " 5/8" " 5. 2" circular " " 3/4" " 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 - i O i v O h O " ^ J V O t o O O O O O i t - , l — ' O i ' P ' O l - ' o o o o o t - ' O t - o o c r o ~o o o o o o o o o o o o o o o~ •..p- - P » . p ~ t O t O h O N > N > h - » O O O O t i - ' l - i 00 -P- O ON N > 0 0 O i ' P - 0 > O i O v Q 0 V 0 O l - ' r o N J O h - o o N ) w o ^ i o o - P - t o O h o O U i L n 4 > - P ~ t j a 0 J U ) N ) O O ( - ' i — ' h - ' i — ' 4 > - o o t o o o M ^ t o h o i - ' - ^ J v o i - > h o t o - f > 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 > h O ( - i O O O O » - ' l - ' 01 01 o ~J t o a v o i h o a v o i o v o o v o o i — 1 t n h 0 h 0 0 " \ t n ' P * O N V 0 O h 0 0 0 l — ' O l — 1 I—" O O V O O N > C o O N > o o o o o o o o o o o o o o o l O l — « 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 o o o o o o o o o o o o o 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 i v O O O O O O N J h O l - ^ O O O O l - ' l - - ' 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 O O O O 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 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 U I -P- t n o o t o o i r o c y o o o o o o o o O O O O O O O • . • . « • • O O O O O O O O Ln U i to Ln -P" -P- -P- O O O O O o o L o - P - - P - t n t n O > ^ - J O O O O O O O O vO to "»l N> VO ->J O l O CT> Ol Wi O Ol I—« t— 1 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 0 f—' t—• l—' l—' o o * M JV O M U 01 00 vO to OJ VO Ol Ol r-> O 00 Cv Ol ho U *> U vO f> O H Ol O O ho to O -P- I - i .P- -P- Ol -P" -P- N> ~ J -P- 00 h-> 00 00 Ol -P* hO ON —I LO 1 -P* O O O O O O O O o \ o o o o . . . . . . . . . \ v « . « . o o o o o o o o o \ o o o o ^ • J O v - P - t O O i O i h O l - ' O \ O O O O O l h O V O O O t o O i v O t O O i \ l - ' I - I i - i N> • P - t o h O h o t o t o i - ' O O X \ O O O O . . . . . . . . . \ v • • • • o i - v i v o c o N > r < o ^ i ^ J t o \ ^ o o o i - 1 O O O O O t n ^ J L n O X -P- -P~ 00 ho h-1 I—1 »—1 l—' ho to ho O O O O • P ' - P - t o t O - P - l - ' O ' v O O O ~~" _ Ov O N O J ^ J - P - I — ' O J V O O v v C l _ i to Run No. Velocity V fps ho Relative velocity V/Vres to Reynolds No. , Re 104 Frequency f cps tn Strauhal No. fD/V 0> fn Frequency f y cps 00 Strauhal No. fyD/V fy fn Frequency fx bps Strauhal No. fxD/V ho fx fn to Amplitude .ratio Drag ^Px lb Oi Coe. B^ px Amplitude ratio <^ y/D L i f t Py lb CO Coe. o t o o 3 a. H - r t H -O 3 < cr pi n r t » a> rt n p. pj O !-• 3 to 3 a* 3* 3 1 09 j> p. r t 3 a 3 H * 3 » to 3 a w o < r t H - O 3* a - 3 (to H - pi rt r t r t c a> o o- n 3 H - to 3 f-> to 5" 3 00 H -r t C a ro f to rt n> I-J to 3 n H i 3 - a t~> o .p- o •V > CO to M O • ro o 3 . r-' o o> < t-> 3 a < H fl> CO r t 3* k to VO • o z l - h CO T3 Q -CO C to H n> < t i  CO ro (D CO O r t I I H - o •P- 3 • o> CO r t l - h ro T3 n> CO t - 1 o to 3 ti i—1 I I (0 <J n> t o X i—1 o I Ol Ml r t ho or n> o - 61 -TABLE II. 2" cylinder with 3/4" square section steel cantilever and wire in place fn : fn • 10.4 cps Vres = 9.0 fps Vres = 4.5 fps D = 0.167' (1) Run No. < 2 ) Wire placed at (3) V fps (4) V Vres (5) R E ZL 104 (6) f (7) JD V (8) %> (9) i . in (TO) fy (ID fy/ / f n (12) fvD V (13) fx (14) fx fn (15) fxD V (16) K in (17) A> (18) Sj in (19) Syy / D 37 2.18 0.24 3.21 2.80 0.214 0.272 0.008 2.8 0.272 0.214 0 0 0 0 0 0.010 0.005 35 2.53 0.28 3.74 3.56 0.234 0.343 0.018 10.4 1.0 0.682 10.4 1.0 0.755 0.116 0.058 0.170 0.085 38 4 2.89 0.32 4.27 3.94 0.227 0.382 0.021 10.4 1.0 0.597 10.4 1.0 0.670 0.233 0.117 0.184 0.092 36 3.27 0.36 4.82 4.52 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 39 3.62 0.40 5.34 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 40 3.99 0.44 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 41 4.35 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 46 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 47 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 44 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 0 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 TABLE II. Continued (1) Run No. (2) Wire placed at (3) V fps (4) V Vres (5) Re 10 4 (6) f (7) 12 V (8) (9) ^ i n (10) fy (ID f y / / f n (12) fyD V (13) fx (14) fx fn (15) fxD V (16) i x in (17) U (18) in (19) X D 51 22.5° 2.18 0.24 3.21 2.07 0.158 0.201 0.032 2.07 0.201 0.158 0 0 0 0 0 0.030 0.015 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 56 0° 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 TABLE III. 4" cylinder with 3/4" square steel cantilever f n = f n s 6.5 cps Vres = 11.8 fps Vres = 5 .9 fps E = 0.3347 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (ID (12) (13) (14) (15) (16) (17) (18) Run No. V fps y /Vres R C 4 10^ fx i s fn Sx fy f v / f n Sy ^Px X Sy Py PPy 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 fn = fn = 4.5 cps Vres - 4.0 fps Vres = 2.0 fps A 2 RdPx = ^Px 2 Run No. (2) V fps (3) V/" /Vres (4) R e 4 104 (5) fx (6) fx fn (7) Sx (8) fy (9) fy/ x f n (10) Sy (ID y in (13) Py lb (14) D p y (15) in (16) ^Px lb (17) x 101 102 103 104 105 106 107 108 109 110 111 0.752 1.090 1.450 1.810 2.180 2.530 2.890 3.270 3.620 3.990 4.350 0.186 0.271 0.361 0.450 0.486 0.632 0.722 0.814 0.902 0.992 1.080 1.11 1.61 2.14 2.68 3.21 3.74 4.27 4.82 5.34 5.89 6.43 0 4.5 4.5 4.5 4.5 4.6 4.7 4.9 0 1.0 1.0 1.0 1.0 1.02 1.04 1.09 0 0.776 0.585 0.467 0.388 0.338 0.304 0.286 0.83 1.30 4.50 2.75 2.40 2.55 3.35 3.35 3.50 3.78 3.92 0.190 0.290 1.00 0.610 0.535 0.567 0.745 0.745 0.778 0.840 0.872 0.184 0.198 0.517 0.504 0.183 0.168 0.192 0.171 0.161 0.159 0.151 0.010 0.091 0.025 0.0625 0.311 0.770 0.987 1.83 2.39 3.31 3.53 0.055 0.445 0.123 0.308 1.53 3.83 4.85 9.05 11.80 15.30 17.40 0.595 2.060 0.360 0.580 1.970 3.670 3.560 5.220 5.550 5.920 5.630 0 0.167 0.243 0.389 0.365 0.348 0.097 0.070 0.084 0.416 0.450 0 0.823 1.200 1.920 1.800 1.700 0.480 0.340 0.410 2.050 2.100 0 4.26 3.50 3.50 2.33 1.63 0.35 0.20 0.20 0.80 0.681 TABLE V. 2" cylinder with 5/8" square section aluminum cantilever and wire in position f n = fh = 4.5 cps V res = 4.0 fp s Vres r 2.0 fps (1) Run No. (2) Wire placed at (3) V fps (4) V Vres (5) Re 10* (6) fy (7) fy / f n (8) fyD, (9) fx (10) fy X f n (11) fxE> / v (12) (13) (14) ^y (15) 115 V. >° 0.752 0.187 1.11 0.91 0.202 0.202 0 0 0 0 0 0.007 0.004 116 1.090 0.272 1.61 1.36 0.302 0.208 0 0 0 0.010 0.005 0.011 0.006 117 1.450 0.361 2.14 4.50 1.00 0.218 4.5 1.00 0.515 0.232 0.116 0.118 0.059 118 1.810 0.450 2.68 2.30 0.512 0.212 4.7 1.04 0.430 0.056 0.028 0.118 0.059 119 2.180 0.542 3.21 2.35 0.523 0.180 4.6 1.02 0.350 0.196 0.098 0.360 0.180 120 2.53 0.632 3.74 3.10 0.690 0.204 4.5 1.00 0.296 0.323 0.162 0.514 0.257 121 2.89 0.722 4.72 4.00 0.890 0.229 4.5 1.00 0.259 0.120 0.060 1.010 0.505 123 3.62 0.902 5.34 4.25 0.945 0.195 4.5 1.00 0.206 0.348 0.174 2.780 1.390 125 4.35 1.080 6.43 4.85 1.080 0.186 4.5 1.00 0.172 0.243 0.122 2.090 1.045 130 131 132 C )° 0.752 1.450 2.180 0.187 0.361 0.542 1.11 2.14 3.21 0.80 1.48 2.12 0.180 0.330 0.471 0.178 0.170 0.162 0 0 4.2 0 0 0.94 0 0 0.319 0 0.010 0.091 0 0.005 0.046 0.011 0.097 0.415 0.006 0.049 0.208 133 134 / 2.890 3.62 0.722 0.902 4.72 5.34 3.92 4.78 0.651 1.060 0.225 0.220 4.2 4.2 0.94 0.94 0.241 0.192 0.007 0.208 0.004 0.104 1.670 2.920 0.835 1.460 135 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 TABLE VI. 4" cylinder with aluminum 5/8" square section cantilever fn = fh s 2.8 cps Vres =5.1 fps Vres =2.6 fps (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) Run No. V fps V Vres R« , 104 fy fy/ /fn fyD V fx 12 fn fxD V V in *2 D Py lb BP y ^x in k_ D *Px lb 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 vibration in 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 (based on the assumption: fx = 2fy) Material Cylinder diameter Vres fps Vres fps Rres 10 4 Rres 10 4 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 i 2.6 15.0 7.5 - 27 - For further study of the relationship between longitudinal and l a t e r a l vibrations, a series of runs were done. They showed how the longitudinal fibration was developed. F i r s t , as above, the lonitudinal vibration was suppressed (see Fig, -5 - 7), and the oscillogram recorded was a straight line i n the longitudinal direction, 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 direction. This curve belonged to the forced vibration category. Then, the suppressing force i n the longitudinal direction was removed and vibration i n the longitu dinal direction was b u i l t up. With the development of longitudinal s e l f - excited vibration, the l a t e r a l vibrating 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 self-excited longitudinal vibration reached i t s maximum, the l a t e r a l curve became irregu l a r with larger amplitude and indicated vibration i n a resonant condition. This also gave an explanation (Fig. 19 - the variation of vibrating types) f V of the curves with ordinates y vs ^ . In these curves longitudinal n res resonance occured at ^ = 0.30 to 0.5. Owing to the effect of resonance res i n longitudinal direction, the points of l a t e r a l frequency moved away from the line of forced vibration. Since the motion of the vibrating system changed, i t was called controlled vibration. The frequency of late r a l vibration giving an irregular oscillogram was analysed by counting the upward limbs which crossed the mean level of the oscillogram. 5.3 Analysis of experimental data Frequency: there were three different conditions under which the frequency was measured: - 28 - 1) Frequency of lat e r a l vibration alone ( f ) : i.e. with longitudinal vibra tion suppressed. The results approached those obtained.in the stationary cylinder condition, i f the amplitude was small. (Table I, column 5). 2) Lateral frequency of vibration for both x, y directions vibration (f ): this frequency varies with lateral as well as the longitudinal vibration. (Table I, column 8). 3) Longitudinal frequency of vibration for both x, y direction vibration (f ): this frequency curve was regular i n shape and belonged to the •A. category of self-excited vibration. (Table I, column 11). For the purpose of analysis, Strouhal number for variations of each of 0 3 - 3 x 10 5 constant. The following relation can be obtained the above types were calculated, as were the frequency ratios In the range of R » 5 x 1(T 10'', Strauhal Number i s almost 6 f D f D s = S = S - n V res V res V f s V f res n The above expression is a straight line inclined 45° to the horizontal (Fig. 19 - 23), representing the forced vibration due to the excitation of the eddy shedding force on a stationary cylinder. Amplitude: The lateral and longitudinal amplitudes were analysed separately. 1) Lateral Amplitude: this was the amplitude when lat e r a l and longitudinal vibration both took place. The maximum value was selected from the recording oscillograms. 2) Longitudinal Amplitude: only the maximum value was analysed because i t varied only within a small range, i.e. the shape of the vibrating curve was more regular than i n the case of lateral vibration. - 29 - Both of these were i n dimensionless form V n . With the data, amplitude ratio vs R curves (Fig. 24 - 30) were plotted. Relation between the longitudinal and l a t e r a l amplitude variation was shown i n the figures. Forces; Forces i n two directions - drag and l i f t - were calculated. There were the vibrating 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 for both response coefficients of vibrating drag and l i f t calculations. Velocity of resonance; the velocity of uniform approaching flow, when the vibration of the cylinder system was at resonance, was the velocity of resonance. Because there were two resonances which could occur, two veloci t i e s of resonance existed. Lateral 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. Similarly longitudinal resonance occurred at longitudinal natural frequency. Before longitudinal resonance the longitudinal response w i l l exist i n same phase with the os c i l l a t i n g drag. When the frequency of longitudinal excitation approaches natural frequency of the elastic system, the structure w i l l vibrate near natural frequency. Then the amplitude keeps increasing to i t s maximum. Theoretically the longitudinal resonance velocity i s one half that of l a t e r a l . But due to the distorsion of vortex, and the inter action of two-dimensional effects the experiment resulted that ^res/v v res was about 0.4. From the relation between the velocity of the uniform flow and the eddy shedding frequency under the condition of stationary cylinder (Fig. 18) the - 30 - assumed velocities of resonance for 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 of Results and Conclusion 6.1 Discussion: Frequency: The Strouhal number i s nearly constant when the Reynolds' number i s i n the range of 10^ for a stationary cylinder. But, i n two-dimensional vibration, when vibration i s excited i n the longitudinal direction, the Strouhal number of the la t e r a l direction suddenly increases. The peak of this steep curve approaches S^ vs R g curve as a tengent l i n e . After the peak, the S value drops and merges into the band region (Fig. 11) given by the stationary cylinder tests. The shape of the rising curve depends upon several variables: k, spring constant; f^, natural frequency i n longitudinal direction; and n, elastic and viscous damping coefficient. The starting point of the shear layer i s important for the generation of vortex shedding. If one of these separation points i s fixed at one side of the cylinder and the one at the other side i s l e f t free to move during vibration, the balanced periodic os c i l l a t i n g force w i l l be altered, as w i l l the periodic circulation variation. Therefore when this vortex starter wire i s i n position, the separation point i s controlled by the location of the wire. . Then the beginning of the shear layer also changes i t s location, and non-balanced shear layers give a different eddy generating condition. This unbalanced force condition affects the exciting forces, also the frequency of vibration and the Strouhal Number as well. (Fig. 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° position. 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. The lowest S occurs with the wire at 22.5°. When there i s vibration i n both directions, 2 " c y l i n d e r , aluminum c a n t i l e v e r F i g , 5 Development of l o n g i t u d i n a l v i b r a t i o n U o 1.45 f p s 4" c y l i n d e r , aluminum c a n t i l e v e r F i g . 6 Development of l o n g i t u d i n a l v i b r a t i o n i - 33 - TJ o 1.81 fps 4" c y l i n d e r , aluminum c a n t i l e v e r F i g . 7 D i s t o r s i o n of l a t e r a l o s c i l l o g r a m \ - 3U - the Strouhal Number variation is quite complex. An additional longitudinal effect makes the curve (S vs R g) distorted. But i t s t i l l bears the charac t e r i s t i c s of combined effects. The relation of frequency ratio vs velocity ratio describes the type of vibration to be expected i n the test. (Fig. 19). At low flows the ampli tude i s small and the velocity i s far from the velocity of resonance for both longitudinal and l a t e r a l directions. It i s i n forced vibration. As soon as the velocity ratio increases to near 0.3, the longitudinal vibration starts, and the l a t e r a l vibration i s gradually changed by longitudinal resonance. The points i n this region appear on the diagram as controlled vibration. When the velocity ratio approaches 0.7, the longitudinal vibration disappears, and the curve of the controlled lat e r a l vibration drops back and coincides with that of forced vibration. This depends on the value of the natural frequency and the magnitude of spring constant. If the natural frequency i n lateral direction i s small, the difference between the natural frequencies i n longi tudinal and lat e r a l direction i s also small. This means that the resonant regions are too close to each other, and the variation on the graph is not obvious. If spring constant i s small, the amplitude around longitudinal resonance w i l l be too large so that, due to the fast shifting 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 vibration then alters. In Fig. 19 to Fig. 23, the types of vibration are shown. Because the maximum attainable flow was limited by flume flow conditions, some points near l a t e r a l resonance were not attainable. Amplitude; the amplitude ratio for longitudinal direction suddenly rises to i t s peak near the point of longitudinal resonance and then f a l l s to a very - 35 - small magnitude. But the la t e r a l amplitude ratio, when the longitudinal one reaches a peak, rises slowly f i r s t , then drops and passes through a minimum value i n the region of longitudinal resonance. Then i t rises again toward the condition where la t e r a l vibration only exists. The largest amplitude ratio of longitudinal vibration i n these four sets of tests i s about 0.25, near a velocity ratio of 0.4. When the wires were i n position, a l l the maximum values were i n the region of velocity ratio about 0.4. Response coefficient of oscillating l i f t and drag of an elastic body The response coefficient of vibrating l i f t and drag due to an oscillating f l u i d force i s computed by the following equations: B A Px V x = A ?l]2/2 P B - V - P 7 A ? U 2/2 The vibrating l i f t varies with the r i g i d i t y of the structure. Its maximum values occur between velocity ratio 0.2 and 0.4 i n the region of longitudinal effect. The response coefficient of vibrating l i f t have a value above 3 for 2" cylinder and steel cantilever (Fig. 31). The maximum vibrating drag response coefficient occurs i n the region when the velocity ratio i s between 0.2 to 0.5, and i t s magnitude i s about 5 to 8 i n these test runs for the damping coefficients (n value) i n the range of 0.25 to 0.60. 6.2 Conclusion l ) The self-excited longitudinal vibration occurs between the velocity ratio of 0.2 to 0.8, when both x and y directions are allowed to vibrate. Its peak i s around 0.4. The frequency of oscillating drag, which is the - 36 - excitation force of longitudinal vibration near longitudinal resonance, i s predicted to be twice that of the la 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 late r a l direction by longitudinal vibration. The degree of distortion depends on the amplitude of the longitudinal vibration. Hence, the frequency of lateral vibration increases with the intensity of longitudinal vibration. This is referred to as a controlled vibration. The Strouhal Number of latera l vibration reaches a peak (Fig. l l ) i n the region of longitudinal resonance and approaches the curve: S vs R as a tangent. Lateral vibration i n two-dimensional vibration i s neither the self-excited nor the forced type. It is affected by longitudinal movement, and is therefore called controlled vibration. In two-dimensional vibration, the amplitude of longitudinal vibration i n the region of longitudinal resonance rises suddenly to a peak and then decreases again to a very low value. In two-dimensional vibration, the amplitude of later a l vibration i n the region of longitudinal resonance f i r s t rises and then decreases to a lower value as longitudinal amplitude approaches i t s maximum. Then the curve rises again to the original path which is the characteristic of lateral vibration alone. When the separation point i s controlled by wire location, the starting point of the shear layer i s altered. Hence the periodic excitation of the vibrating l i f t and drag i s altered. The unsymmetrical oscillating f l u i d forces produce an oscillogram with irregular shape and change the vibrating frequency, sometimes decreasing the amplitude of vibration and sometimes increasing i t . At present no general conclusion concerning the best position of the starter wire has been possible. - 37 - The response coefficient of vibrating l i f t due to oscillating 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 vibrating drag i t varies from 3 to 8 i n the region of longitudinal resonance. There would be a second maximum for amplitude of late r a l vibration at or near a velocity ratio of 1.0 but only one test was run i n this range (Fig. 21), Around this region the vibration appeared to be a controlled type. Because the longitudinal vibration appeared at a velocity ratio as low as about 0.4, the design of hydraulic structures i n this region of flow should be checked for the i n s t a b i l i t y arising from two-dimensional vibra tion, i f the structure allows to vibrate. - 38 - APPENDIX I Structural Vibration When a structure i s acted by an external disturbance the forced vibra tion w i l l take place. The forces that come into play w i l l be external force, inertia force, elastic restoring force of the structural system and damping forces. When simplified as a problem of a single concentrated mass, the equation w i l l be i n the form (17): Mx + D (t) + kx - F (t) Under the assumptions that the pulsating load: F (t) = F q Sin cot and the viscous damping: D (t) = cx by ignoring transient solution, the parti cular solution w i l l be: x = A H [ (1 - (^) 2) Sin cot - Cos cot ] p L v vp' p2 where H = — ^ - — - (called: magnification factor) p The characteristic of vibrating motion can be expressed by relation between H and w / p : where: p = circular frequency co = angular velocity n = c/m A = static deflection It says when frequency of disturbing force approaches natural frequency the amplitude w i l l be the maximum under the energy balance of disturbing and damping forces. - 39 -The force vibration of an elastic slender beam of continuous mass dis tribution i s governed by the equation: where F(t) = C„ . q . Sin 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 diferential equation, i . e . l e t the F(t) = 0, yields the solution of natural frequency of the slender, elastic beam: where: c i s a coefficient depends on the type of beam and mode of vibration . m: mass per unit length . - 41 - APPENDIX II Quasi-Steady Theory Because of no complete hydrodynamic theory for the separated flow the quasi-steady (semiempirical theory) i s employed. Using stationary hydro- dynamic forces on a vibrating body, a solution i s obtained for a non-linear differential equation. I f the f l u i d force i n the forced vibration equation i s : F y = C | A U 2 (Lateral force) where: C = f (a, C L, C D) Then with a simplified relation, using a polynomial to express the function of coefficient of l i f t and drag, the resulting equation w i l l be (15): T - e (1 - P Y 2 - Q Y 4)Y + Y = 0 where Y i s a dimensionless displacement, and e, p and Q are damping constants. A solution, involving the substitution of polynomial approxima tions to the aerodynamic force curve, has been used to predict approximately the amplitude of the vibration. - 42 - Bibliography 1. L. Rosenhead: Laminar Boundary Layers, p. 93, 102. Oxford, 1963. 2. M. Rauscher: Introduction to Aeronautical Dynamics, p. 25, 257, 366, John Wiley, 1953. 3. W. Kaufmann: Fluid Mechanics, p. 251, 257, McGraw H i l l , 1963. 4. V.L. Streeter: Fluid Dynamics, Chap. VII, p. 200. 5. D.M. McGrefor: An Experimental Investigation of the Oscillating Pressures on a Circular Cylinder i n a Fluid Stream. U. of Toronto, 1957. 6. W. Weaver: Wind Induced Vibrations i n Anttenna Members. Trans. Vol. 127, 1962, ASCE, p. 681. 7. Eduard Naudascher: On the Role of Eddies i n Flow-induced Vibrations Iowa Institute 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. Laird: Water Eddy Forces on Oscillating Cylinder, A.S.C.E. Trans. I962. 10. J.H. Gerrard: An Experimental Investigation of Oscillating L i f t and Drag, J. of Fluid Mechanics, 196l. 11. Eduard Naudashcher: Vibration of Gates During Overflow and Underflow, A.S.C.E. Trans. 1962. 12. F.B. Campbell:: Vibration Problem i n Hydraulic Structure. A.S.C.E. Trans. 1962. 13. A. Laird: Groups of Vertical Cylinder Oscillating i n Water, J. of Eng. Mechanics, Feb., 1963. 14. D. Pierce: Photographic Evidence of Formation and Growth of Vorticity, J. of Fluid Mechanics, I96I. 15. G.V. Parkinson: On the Aeroelastic Instability of Bluff Cylinders, A.S.M.E., J. of Applied Mechanics, June 1962. 16. A. Roshko: Experiments on Flow Past a Circular Cylinder at very High Reynold's Number, May 1961, J. of Fluid Mechanics. 17. Rrover: Dynamics of Frame Structure. 18. Norris: Structural Design for Dynamic Load. 19. Birhoff: Jets, Wakes and Cavities. - 43 - 20. H. Sato: Mechanism of Transition i n the Wave of a Thin Flat Plate Placed Parallel to a Uniform Flow, J. of Fluid Mechanics, Sep. 1961, P. 321. 21. Goldstien: Modern Development i n Fluid Dynamics - Wakes - Vol. II, p. 550. 22. Rouse: Engineering Hydraulics (Chapter I), 1950. - Uh - Nomenclature R e - Reynolds' number (^ p-). D, d " diameter of circular cylinder ( f t ) . h = depth of flow ( f t ) . v, u = velocity of flow (fps). = kinematic viscosity ( f t /sec). V v = velocity of eddy behind cylinder. S = Strauhal number (fD/v), vibration i n la t e r a l (y) direction only. S = Strauhal number ( f ^ / v ) , i n longitudinal direction, both x, y x vibrating. 5 = Strauhal number ( f D/v), i n la t e r a l direction, both x, y vibrating. y y f = frequency of forced vibration i n later a l direction only. f = frequency of forced vibration i n longitudinal direction, both x, y x vibrating. f = frequency of forced vibration i n l a t e r a l direction, both x, y y vibrating. f = frequency of forced vibration i n l a t e r a l direction for stationary 8 cylinder. f « natural frequency of the system, lateral direction. f ^ = natural frequency of the system, longitudinal direction. 6 X = amplitude i n l a t e r a l direction (from wave crest to trough). 6 = amplitude i n longitudinal direction (from wave crest to trough). y P = force. P x = max. hydro-elastic vibrating force i n longitudinal direction. P * max. hydro-elastic vibrating force i n lateral direction. •^ Apx ~ r e s P o n s e c o e # o f vibrating drag, */<^uz/\ P / z B » response coe. of vibrating l i f t , y ? jr ^  - 45 - V = velocity of resonance, la t e r a l vibration only. I*6S V* = velocity of resonance, longitudinal vibration only. R = Reynolds' number, when only lateral resonance occurs. 1*6 S R1 = Reynolds' number, when only longitudinal resonance occurs. X, Y = f l u i d force acting on cylinder i n x, y direction. U = velocity of uniform flow. w = complex function. z = complex variable. f = circulation. Ap = pressure difference. V = lo c a l velocity 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 iner t i a . L = length of beam. T = period. p = g , circular frequency. - 46 F l o w NXV „ F i g . 9 D e f i n i t i o n s ketch \ O O O O p M • • o • • • • o ro 4^ d\. ob o ro - 8> " 0.30 0.25 0.15 0.10 0 .05 0 P o s i t i o n of w i r e x no wire ' „ o° o 0 « o o 22.5° 45° o 67.5 • 0. 0 0 0 fS o x * x * % 0 X o ( v i b r a t i o n i n y-dire< t i o n only ) X » A A * A t i F i g . 15 Cont r o l wire t e s t ( l ) 2" j y l i n d e r with s t e e l c a n t i l e v e r and l / l 5" wire 5 I O 5 ~ VBT 5 1 0 5 5 Re - 1$ 1.0 0.8 «H 0.6 CO 0.4 0.2 0 P o s i t i o n of wire ^ 0° 22.5° * 0 ° * 67.5° ( v i b r a t i o n i n both x,y d i r e c t i o n s ) F i g . 1 6 E f f e c t * 1 of l o c a t i o n of A A n i . T > . O 111 ^..rJ-^-v, J ....... s t e e l ca i t i l e v e r o o /S ft O o 0 0 0 * A Re YD. If io-0.30 0.25 0.20 0.15 u CO 0.10 0.05 F i g . l7__C^ojatjrai^ijr_e^.e.3J;^(j2,). 2" c o c y l i n d e r with aluminum c a n t i l e v e r and 1/8" P o s i t i o n of wire -4-5- ( v i b r a t i o n i n y d i r e c t i o n only ) wire 10 • " B VD . 10' 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 V e l o c i t y r a t i o , V/ = f s . 1 res ' * n 1.0 0.8 S e l f - e x c i i e d v i b r a t ; o n / o o n t r o l l e d v i b r a t i o n / • • . / 0 '-i-b-i-a-ti-o n 0 a 0° 45° o O A o 22.5 ; . 2 5 V a r ^ . a t i o n o f ' ;ype o f v i i D r a t i o n 2" c y l i n c . e r , s t e e l c a n t i l e v e r , v ^ i t h c o n t r o l w i r e / res ' n - Z9 -- £9 " 0 .10 8 B 6.0 5.0 4.0 3.0 2.0 1.0 1 1 1 CO 0) - u ^ p X J R e s p o n s e c o e f . d r a g d u e t o l o n g i t o f a d d i t i u d i n a l v i b onaJ r a - 1 \ w VTB \ ^ p x t i o n . I f n o l o n g i t • t i o n , B ^ p x = 0. B p Y s R e s p o n s e c o e f . u d i n a l v i b o f a d d i t i r a - ona] 1 / V T) v l i f t d u e t o l a t e r a I f n o l a t e r a l v i b r B p y = 0. 1 v i b r a t i o a t i o n , i . J F i g . 31 R e s p o n s e c o e f . o f r v i b r a t i n g d r a g a n a l i f t ( l ) 2" c y l i n d e r , s t e e l c a n t i l e v e 10 4 Re VD 10" 10 / 4> ON CO - 74 - F i g . 31 O s c i l l o g r a p h a n d m a r k i n g p e n - 75 - Plow c y l i n d e r P i g . 38 C y l i n d e r v i b r a t i n g n e a r l o n g i t u d i n a l r e s o n a n c e photo-was t a k e n . ( F i g . 3 8 ) U = I . 4 4 fps 4" c y l i n d e r f x = 14.5 cps aluminum c a n t i l e v e r fy«= 4 . 6 cps Camera speed: l / 2 5 sec with white f l o a t s on s u r f a c e of water F i g . 39 Tv/o-dimensional v i b r a t i o n 

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