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Coupled-oscillator models for vortex-induced oscillation of a circular cylinder Wood, Kelvin Norman 1976

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COUPLED-OSCILLATOR MODELS FOR VORTEX-INDUCED OSCILLATION OF A CIRCULAR CYLINDER BY KELVIN NORMAN WOOD B.A.Sc, University of Br i t i s h Columbia, 1973 A.THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1976 (cT) Kelvin Norman Wood, 1976 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date August 17, 1976 (i) ABSTRACT The vortex-induced os c i l l a t i o n of a circular cylinder is modelled by a non-linear system with two degrees of freedom. The periodic l i f t acting on the cylinder due to the vortex-street wake is represented by a self-excited oscillator, which i s coupled to the cylinder motion. Approximate solutions and s t a b i l i t y c r i t e r i a are presented which are valid over restricted intervals. Changes to the form of the coupled-oscillator model and i t s approximate solution are examined i n order to improve the comparison between predicted model and experimental results. The changes are motivated by the study of experimental evidence, and by comparison with the known properties of similar systems of non-linear equations. ~- Significant improvement in the coupled-oscillator model performance i s obtained through the inclusion of an effective structural damping term which i s dependent on wind speed and cylinder displacement. TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. PRELIMINARY 3 3. MODEL FORMULATION . 7 . 3.1 Higher Order Non-linearity 7 3.2 Combination-Oscillation Solution 8 3.3 Variable Damping 11 (i) Harmonic Solution .13 ( i i ) Combination-Oscillation Solution 17 4. DISCUSSION 22 REFERENCES 23 APPENDIX A ; Hartlen and Currie's Original System of ELifferential Equations 24 APPENDIX B Extension to Seventh Order Non-linearity i n C 1 ........ 30 J-i APPENDIX C Combination-Oscillation Solution to Hartlen and Currie's Original System 36 APPENDIX D Variable Structural Damping 39 ( i i i ) LIST OF FIGURES Page Figure I Experimental Results for Vortex-Induced Oscillation of a Circular Cylinder (Feng) 42 Figure II Schematic Diagram of Experimental Configuration 43 Figure III Characteristic Domains of Vortex-Induced Oscillation 44 Figure IV Theoretical Predictions for Hartlen and Currie's Original Model 45 Figure V Theoretical Prediction for a Higher Order Non-linearity i n C' 46 Figure VI Theoretical Predictions for Combination-Oscillation Solution Applied to Hartlen and Currie's Original Model 47 Figure VII Theoretical Predictions of the Effect of Variable Structural Damping-Harmonic ~" Solution 48 Figure VIII Theoretical Predictions of the Effect of Variable Structural Damping-Combination-Oscillation Solution 49 (iv) LIST OF TABLES . ... Page Table I Effective Structural Damping During Vortex-Induced Oscillation 11 Table II Damping Parameter Determination 12 (v) LIST OF SYMBOLS Non-dimensional transverse cylinder displacement amplitude. Component of A at ft) (free component) F Component of A at U)^  (harmonic component) Instantaneous l i f t coefficient Amplitude of l i f t coefficient Amplitude of l i f t coefficient for stationary cylinder Amplitude of the component of C at w (harmonic component) J_i c Amplitude of the component of C at oo (free component) V F h <ov g Sizrouhal number 2 IT V Free stream velocity Instantaneous transverse cylinder displacement X c Non-Dimensional transverse cylinder displacement = — — 2 Mass parameter = ^ ^ Coupling parameter Damping parameter Cylinder diameter Cylinder mass per unit length Detuned frequency of cylinder o s c i l l a t i o n (wind-on) Natural frequency of spring-cylinder system ( s t i l l - a i r ) Vortex formation frequency for the e l a s t i c a l l y mounted cylinder (vi.) Vortex formation frequency for stationary cylinder Vortex formation frequency approximately at to^  ( e lastically mounted cylinder) s to • - c CO n to V s CO n to to n C r i t i c a l damping ratio (wind-on) C r i t i c a l damping ratio (wind-off) Coefficients of non-linear damping terms Phase angle by which C leads X J_i Detuning parameter for cylinder o s c i l l a t i o n frequency Fluid density Non-dimensional time = to t n ACKNOWLEDGEMENT The author would l i k e to thank Dr. G.V. Parkinson for his advice and guidance i n the course of this research. Financial support was received from the National Research Council of Canada, Grant A586. 1. 1. INTRODUCTION Dating from the early 1960's, there has been an active program in this department to study the effects on fixed or e l a s t i c a l l y supported bluff bodies of the wakes produced by them. In the Reynolds Number range 4 which i s of interest [0(10 ) ] , the wake i s characterized by periodically shed vortices, the frequency of which i s governed by the Strouhal relation-ship. This work i s concerned with the interaction of an e l a s t i c a l l y mounted circular cylinder with i t s wake, for the case i n which the Strouhal frequency i s close to the resonance frequency, of the cylinder-mounting system. Detailed experimental studies have been carried out by Ferguson (1) and Feng (2) to document the vortex-induced o s c i l l a t i o n of just such a system. - As direct solution of the governing dynamic equations for the cylinder and i t s wake i s not feasible at present, a variety of simplified mathematical models have been suggested to describe the interaction [a summary of the more promising suggestions i s given by Parkinson (3)]. A proposal by Hartlen and Currie (4) seems to have particular merit. They consider the l i f t acting on the cylinder (due to i t s periodic wake) to be governed by a second order non-linear d i f f e r e n t i a l equation (of the type studied by van der Pol) which i s coupled to the cylinder motion. Over a restricted interval, the results predicted by their model bear good re-semblance to certain of the experimentally observed features. They f a i l to produce some important characteristics however. Using the coupled-oscillator concept i t is the intention of this work to suggest changes in the form of non-linear terms and examine the effects on the solution. The stimulus for this comes from the need to obtain better correlation between model predictions and experimental results. 3. 2. PRELIMINARY Figure I provides a summary of Feng's results for the vortex-induced o s c i l l a t i o n of a circular cylinder (for given input conditions). As Feng determined only three values of l i f t coefficient amplitude, transient behaviour was used i n establishing the location of the jumps in C [Parkinson (5)]. The results demonstrate that over a discrete range of flow speeds (the lock-in range), cylinder displacement and fluctuating l i f t are periodic i n time, with the same frequency, which i s close to that of the natural frequency of the spring-cylinder system. The amount by which the phase of the exciting force leads the cylinder displacement i s measured as well. Important features to note are the hysteresis loops which exist for both amplitude (of displacement and l i f t ) and phase. Also significant i s the response for u)„ > 1.4 (outside of lock-in), where c y l -inder oscillations persist at frequency close to while the frequency of the predominant excitation i s considerably higher ( w ) . r Figure II describes the configuration and the important elements of the spring-cylinder system. With the effect of the vortex-street wake on the cylinder included as a forcing function, the di f f e r e n t i a l equation for transverse displacement X c i s : mX + 2g0) mX + mu) 2X = CT (§V2h) c n c n c L I . - • To nondimensionalize the equation, introduce T = to t n 4. h CO • - S V = (Strouhal Relationship) and obtain r . X" + 23X' + X = aco 0 2C L (2.1) For modelling purposes, the problem now reduces to determining an ex-pression for C . ' Ju Hartlen and Currie originally suggested that the l i f t co-efficient be governed by the following d i f f e r e n t i a l equation C" - atOoC; + ^  C'3 + w 0 2 C = bX' (2.2) i-i Jj C 0 o Li Jj This form was chosen because of i t s simplicity, and because away from resonance of the spring-cylinder system (bX' •> 0 ) , self-excited os-d i l a t i o n of amplitude and frequency approximately equal toy— — and co0 w 3 y respectively i s predicted for C L (provided a, y are small). This behaviour i s consistent with experimental observation i f — ' i s set equal to the am-Y plitude of the l i f t coefficient for a stationary cylinder (C ). L 0 The coupling term fliX ') was included to provide a dependence of on cylinder motion. Its presence leads to the prediction of interesting C behaviour for C0o close to co . Drawing a comparison between this system Li n and the well-studied forced o s c i l l a t i o n of the van der Pol equation [Stoker (6) ] , one would expect a range of co0 for which C and X have the same os-Li d i l a t i o n frequency (lock-in), bounded by a range of co0 for which C has components close to co0 and to (combination-oscillation). Figure III dem-n onstrates that the postulated regions of characteristic response are consistent with experimental evidence - region A being associated with the 5. typical forced response of an elastic system, region B with the trans-i t i o n a l range in which frequency components close to co0 and 0) are present, n and region C with the lock-in range. It i s not possible to make further assumptions concerning the detailed nature of the response, as the forcing function is i t s e l f dependent on C through Equation (2.1). Hartlen and Currie obtained an approximate solution to the system of coupled d i f f e r e n t i a l equations [Equations (2.1) and (2.2)] valid within the lock-in region, by assuming X and to be given as follows (method of van der Pol) X = sin fix CT = C„ sin (fix + <{>„) (2.3) The~actual analysis and a summary of results i s included i n Appendix A. Figure IV summarizes model predictions for the indicated input values. The results demonstrate the model's a b i l i t y to generate certain of the features of vortex-induced oscil l a t i o n . The s t a b i l i t y of the approximate solution i s not given directly by the method of van der Pol. An alternate method^ which does provide such i n -formation i s the K-B method [Minorsky (7)]. This analysis i s introduced and developed i n Appendix A. The results obtained allow one to confirm that the solutions summarized by Figure IV are stable, and that the two approximate methods of solution yield identical results provided that fi, ft2 = 1. The results obtained are encouraging. The model f a i l s to pro-duce a double-amplitude response, however, and since the approximate sol-ution i s valid only within the lock-in region, the system behaviour for 0)o > 1.4 cannot be produced. The following work i s concerned with an investigation of the form of model and solution used, with a view to improving the comparison between predicted and experimental results. 7. 3. MODEL FORMULATION 3.1 HIGHER_ORDER_NON=L^ f " It was decided to investigate the effect of increasing the order of non-linearity i n the governing equation for C T. Following a suggestion by Landl (8), odd power terms to seventh order in C' were Li included. The equation for C then takes the form J-i C" ~ O0)oC' + X_ ( c ' ) 3 - (CM 5 +'7^f (CM 7 + 0) o 2 C = bX* L L 0Jo L C0o3 L l0o~> L L (3.1) where a, y, n, <5 > 0 The j u s t i f i c a t i o n for including f i f t h and seventh powers of C' Ju comes from examining the homogeneous form of Equation (3.1) (bX1 - y 0). For a, y, r\, & small, then CT = C_, sin OJ 0T and C„ may have one or three positive real roots. In the latter case F the middle root would be unstable, and the t r i v i a l solution C^ , = 0 i s unstable i n either case. Considering the inhomogeneous form, i t was hoped that the increase i n non-linearity would result in the exist-ence of two stable C amplitudes for a given co0 within the lock-in region; L i a hysteresis effect possibly resulting from the manner of the dependence on 0)o. Approximate solutions (by the methods of van der Pol and K-B) to the system of Equations (2.1) and (3.1) are included i n Appendix B. Values for the non-linear coefficients a, y, n, 6 are determined by requiring that three positive real roots C exist within lock-in (two of 8. which are known from experiment), and that one real root C exist away from ° lock-in (bX' 0). r In order to match predicted with experimental values of l i f t -coefficient amplitude within lock-in, the non-linear coefficients nec-essary were found to be of 0 (10). The effect of the magnitude of a, y» T), 6 on the approximate solution of Equation (3.1) has not been examined. Figure V shows numerical results for the indicated input values. The s t a b i l i t y analysis confirms that the middle amplitudes of C„ and H are unstable, and that the other amplitudes are stable. The results demonstrate the system's a b i l i t y to model the be-haviour of CT reasonably well within lock-in (as i t was designed to). The frequency and phase variations remain a problem, however, as to a f i r s t order approximation they are independent of C and thus do not reflect jumps in amplitude which the system produces. The behaviour of the pre-dicted cylinder amplitude i s clearly a problem as well. The predicted results indicate that an extension to seventh order non-linearity in C' results in only marginal improvement of the system behaviour, while introducing further complications i n doing so. 3.2 COMBINATION-OSCILLAT Currie and Oey (9) proposed that the double amplitude response could be accounted for by the existence of different solutions to the system of Equations (2.1) and (2.2) for harmonic, or combination-type forms of solution; that i s , whether X and C are assumed to be of form given by Equation (2.3), or as shown below (combination-type) 9. X = sin Qr + sin aye (3.2) C L = C H sin («x + cpR) + C p sin (oyx + ^ They draw comparisons between the coupled-oscillator system and the forced o s c i l l a t i o n of the van der Pol equation. Actual results of a detailed analysis have yet to be published. Experimental evidence supports a combination-oscillation form of solution over a range of U)0 adjacent to the lock-in region (Figure I'll, region B). There is no evidence for a solution of this form within the lock-in region, however. A study was carried out to see whether or not a solution of this form could r e a l i s t i c a l l y account for one of the amplitudes within lock— in,-or the system behaviour outside of i t . The actual analysis i s i n -cluded i n Appendix C. A s t a b i l i t y analysis was not carried out, as the approximations which are required i n order to combine the K-B method with a combination-oscillation form of solution are not at a l l obvious. Figure VI il l u s t r a t e s the important numerical results for the indicated input values. The phase and frequency variations for and $ are identical to those for the harmonic case and thus have not been shown. Away from the neighbourhood of to0 = 1, the forced cylinder response at oy i s negligible, thus A^ , and cf>F have not been shown as well. The re-sults demonstrate the p o s s i b i l i t y of the existence of a combination-type o s c i l l a t i o n within lock-in. Unfortunately, the analysis predicts a solution vali d only within lock-in, and a complicated C behaviour over this range - C i s predicted to have components of approximately equal Li magnitude at frequencies of Q and o j . 10. It would appear that the governing equations as formulated are not capable of accommodating a combination-type solution. 11. If one assumes the cylinder motion to be governed by Equation (2.1), and that within lock-in X and C may be approximated by; Equation (2.3), then by substituting for X and C i n Equation (2.1) and applying the principle of harmonic balance, the following result may be obtained: Since a i l the quantities on the right-hand-side of the equation are known or are measurable, the apparent structural damping during vortex-induced cylinder o s c i l l a t i o n may be calculated. These calculated values are then to be compared with the value measured in s t i l l - a i r (which is the value given by Feng). Table I summarizes the experimental results and the calculated 2 3 r a t i ° ( 2 3 ^ ) ' w h e r e C 2 3 o ) i s the wind-off structural damping. The effective structural damping appears to depend on cylinder o s c i l l a t i o n amplitude as well wind speed. as Wo H 2 3 2 3 o • .98 .03 .45 4° .57«-> 1.7 1.1 1.06 .11 .8 2 «-KL4 9° .3 •«-* 2 . 2 1.5 1 . 1 2 . 2 1 1.5 1 0 +-+16 11° 1.7 2.8 1.9 1 . 2 1 .48 .3 1.91 .5 37 -e-^-59 37 1 0 2 4 -w 5.6 4 2.7 a 2 3 c a = . 0 0 2 : = . 0 0 2 = .97 TABLE I Effective Structural Damping During Vortex-Induced Oscillat xon. 1 2 . It i s clear that any model which f a i l s to take this effect into account w i l l have l i t t l e chance of success i n predicting experimental behaviour. It i s proposed that the effective structural damping 'be. approximated by a relationship of form: 2 3 = 2 g 0 ( 1 + fu)02 A J J ) 2 The 0Jo and provide a dependence of system damping on the wind force acting on the cylinder, and cylinder displacement respectively. One would expect the constant f to depend on the experimental configuration. An appropri-ate value can be calculated from the experimental results as follows: too 2 3 2 3 o f . 9 8 . 0 3 1 . 1 3 . 5 1 . 0 6 . 1 1 1 . 5 4 . 0 1 . 1 2 . 2 1 1 . 9 3 . 4 1 . 2 1 . 4 8 . 3 4 2 . 7 4 . 3 3 . 9 TABLE II Damping Parameter Determination A value of f = 4 would seem to be indicated. The modified equation governing cylinder response then i s X " + 2 3 0 ( 1 + fw 0 2 A J J ) X ' + X = ato D 2 C L ( 3 . 3 ) In order to assess the effect of the proposed variable damping term, the system of Equations ( 2 . 2 ) and ( 3 . 3 ) has been solved approximately, assuming harmonic and combination-type forms of solution for X and C ^ . A s t a b i l i t y analysis has been carried out for the harmonic solution and i s included 13. i n Appendix D. (i) Harmonic Solution Within the lock-in range, assume X and C to be given by Li Equation (2.3). If one substitutes for X and C into Equations (2.2) and Li (3.3) and neglects terms in A^, C^, <f>^  and higher harmonics, the following system of equations can be obtained by applying the principle of harmonic balance: aw 0 2 C R cos (pH = (1 - f t 2 ) 2 2 aco0 C R sin <J>H = A^ , ftB0 (1 + fco0 A^) o : / ' " " . . ^ " ' , ' 1 - ^ ' . ' - 1 . . . . ( 3 . 4 ) (0J o 2 - ft2) . , , (1 - ft2 ) b A H - v- s—~ sin <p - cos A — * p„ = — aw0ft H H • 2 HH a co0C to0 H where B c 5 230 To proceed, i t i s necessary to make an assumption concerning the frequency behaviour ft(co0) (which i s close to 1 throughout the lock-in region). Intro-duce where |X| = 0(1) and make the assumption that 14. 1 - ft2 = X B„ - X 2 ^ — S XB0 fi, fi = 1 both of which are reasonable, since Bc then, one obtains =0(10 ). From Equation (3^4) ao)0 X B . ao) 0 2 C R sin ^ S y ^ l + f 0) o 2 A^) ^ o C ° S *H + S ± n *H ( 1 rH H ' = 0 0)0 (3.5) -A- sin * - cos * ( 1 ~ ^  > S u)0 H where A = 0)o - 1 From Equations (3.5.1 and 2) tan a = 1 + f"° AH n. -\ 2 , Br aw, X 2 + (1 + f W o 2 A ^ ) 2 ... (3.6) From Equations (3.5.3 and 4) X 2 = (1 + fo) 0 2 A H) no) ^ - - a + fa).' y . (3.7) . , _ ab where n = —— Bo Substituting for A in Equation (3.6.2) aw0 n u » Substituting for tan (f> in Equation ( 3 . 5 . 3 ) — - + < 1 + f0)o cxwr ^ } (1 - ) = 0 2 w0 then substituting for A and p (from Equations(3.7 and 8)) one (_A_) 2 -ml _ ( i + fo) 0 2 A H ) ) cta)0 A (!•:+ fa>„ A J J ) [ 1 - ^ A ^ (1 + F A ) O 2 A J J ) where C, = f — - ) — i -1 V ~2) nA which can be expanded to yield 0 - SX \ 7 + 8 2 ^ + where + g 7 ' A H + 8 8 ^ g ; L = C 1 2 (f 0) o 2)' 8 , = 3 C 2 (f0) o 2) 8 3 = 3C1 f W°2 g 4 = C^ 2 - 2 C 1 (fw o 2) 5 1 o 16. H = - 2 c i 2 A 2 g 7 = f u„ (1 + ( — ) ) ' aw0 ' The seventh order polynomial i n A^ can be solved approximately as a function of 0)o and the input parameters (n,b, C L , f ) . Once the roots A^ have been 2 1 2 determined, values C can be determined from Equation(3. 8), and A from i B 1 Equation (3.7). The sign of A^^ (and thus = 1 - A -y) can be determined 2 by substituting for C„ and tan <p'„ i n Equation (3.5.3). 1 1 Figure VII shows the results of such an analysis for the indicated input values. The results demonstrate the system's a b i l i t y to.generate multiple amplitudes i n A^, C , (pR and ft with varying to0. The p o s s i b i l i t y of producing a hysteresis effect exists as the upper branch of A ^ o O i s valid for ft < 1 only, and the two lower branches for ft > 1 only. The principle result of the s t a b i l i t y analysis (Appendix D) i s that the middle branch of Ay.(u)0) i s unstable, while the upper and lower branches are stable. The arrows on Figure VII incorporate this information i n describing possible behaviour for increasing or decreasing w0• Although there are s t i l l remaining d i f f i c u l t i e s with the amplitudes of X and C , and with trends i n the phase angle for ft > 1, the inclusion of the variable damping term has resulted i n a significant improvement i n model performance within the lock-in range. ( i i ) Combination-Oscillation Solution If one assumes X and C^ to be given by Equation (3.2), then substituting into Equations (2.2) and (3.3) and neglecting terms such as A^, (pp, higher harmonics and combination tones and f i n a l l y applying the principle of harmonic balance, one obtains the following system of equations: 2 2 ato0 C p cos (f>F = (1 -a>F ) 2 2 aco0 C p sin <j>F = Ap o^B. (1 + fw0 j^) ato 0 2 C H cos (J>h = A H (1 - ft2) a t o „ ... (3.10) (C0o :) aoj 0 toT cos cJ)F + sin (pF 0J„ 1- (Sr> ( P F (if) + 2 P H > = 0 (Wo " W p . « o . o L S ± N ^ F - G O S * F o 2 2 1 - C—) (P* + 2P„) H ' 3 A F au) 0 C„ 2 2 (Wo - ST) , . . , ~ ~ o ^ f t c o s + S l n .ft + 2 P : 1 ( P H ^ F ( f t " ) ) = 0 2 2 (u 0 - ST) ato 0 f t sin <J>H - cos <J>H J AH ato0C H Next introduce a 2 2 1 8 . 2 ^ 2 = o "H OX00 ft then from Equations ( 3 . 1 0 . 5 and 6) ? b a 0 J o tan cp CT ( 1 + tan <j> ) = F 2 F' ~ ~ ~ 5 C 3- 1 1) B 0a OJ F ( 1 + f u. V bAp bat0o sin cp^  which uses — — — = : from Equation ( 3 . 1 0 . 2 ) . To pro-ceed, i t i s necessary to make assumptions concerning the frequency behaviour ft (too) (which i s close to 1 ) and co„(co0) (which i s close to U)0). Assume that w = 0Jo, then from Equations ( 3 . 1 0 . 1 and 2) . t a n * 3 ( 1 + ^ o 2 V 1 - too 2 2 thus, tan <J> < < 1 for to0 away from the immediate neighbourhood of i0o = 1 . From Equation ( 3 . 1 1 ) then 2 where A = 0)o - 1 Substituting for O and tan d) i n Equation ( 3 . 1 0 . 5 ) yields P F ( ? ) 2 + 2 P H ^ ( 1 - £ O I ^ > < ^ ) 2 . . - . . ( 3 . 1 2 . 1 ) From Equation ( 3 . 1 0 . 7 ) PH + Vft^)2 = ( 1 + Otoo'tan * > ^ 2 •••• ( 3- 1 2- 2) H then solving for p„ from Equations ( 3 . 1 2 . 1 and 2) From Equations (3.10.7 and 8) ( a „ p r - r - — — — - 7 — - .) tan 6 , + a = o H aft (1 + fca0 2 O YH H X B introduce ft = 1 - —^- and assume that 1 - ft2 = X B 0 - X 2 B 2 = X B F then examine ft, ft2 = 1 2 2 E 0 ) / - ftZ ~, A H ao)0 ft aw 0 Q -D „ 1 + f 0 ) o 2 A t a n V E r 3 ^ ( i + f -o 2 v = x — -Substituting for a and tan (J) i n Equation (3.14), one obtains H H X 2 = ( > U ) ° 2 ? - ] ) (1 + fto G 2 A ) 2 A (1 + fto/ A H) From Equations (3.10.1 and 2) 2 2 2 (aw 0 2) C R 2 = A ^ Bo 2 ( (1 - ft2) +• (1 + f too2 Ag) ) 2 „ 2 # , 2 . „ . . 2 2 A H " B 0 " ( X ' + (1 + fto 0" A J J ) ; ) 2 then substituting for X : 2 h 2 C „ " = " 2 (1 + f a ) , A J J ) nto0 A or 20. PH = A H 2 2 I 2 . ( 1 + f w ° 2 V nco0 C A (3.16) Equating (3.13.1) and (3.16), and substituting for tan <j> and X one obtains H . nton A(l + fw 0 ^ 1) aio0 / 3b , 2 ,., , _ 2 . v nA (C co0 ) i-"o + 2 nBQ0Jo - 1) cxA which can be expanded to yield 0 = g l A H 7 + g 2 A H 6 + . . . +gg Y (3.17) where & 1 = 9 C 1 2 (fto 0 2) g 2 = 27 C^ 2 ( f W o 2 ) g 3 = 27 C x 2 (ft0 o 2) g 4 5 9 C 1 2 + 6 C± C 2 (fco 0 2) g 5 5 12 C 2 foj0 H E 6 c i C2 *7 * < C2 2 + > 2 1 . L 0 r - 2 n B 0 toq °2 ~ a A ~ 1 The seventh order polynomial i n can be solved in a manner similar to 2 Equation (3.9). Once the roots have been determined, values for 1 2 i can be determined from Equation (3.16), and A. from Equation (3.15).. The A. B o sign of X. (and thus ft^ = 1 — ) can be determined by substituting for 2 2 C and tan cp i n Equation (3.13.1). C i s then given by Equation (3.13.2). H. H. h . 1 1 x and Aj, and (p^, from Equations (3.10.1 and 2). Figure VIII shows the results of such an analysis for the i n -dicated input values. Since the forced cylinder response at to is negligible F away from the neighbourhood of toD - 1, A^ , and cp^  have not been shown. The results demonstrate the system's a b i l i t y to generate a combination-type solution valid only at the extremes of the resonance region, and r e a l i s t i c behaviour of C for toQ < 1.15 or toe > 1.38. These features are both character-i s t i c of vortex-induced cylinder o s c i l l a t i o n . There i s no solution for 1.15 < to0 < 1.28 as C i s imaginary r . 1 over this range. There i s no solution for to0 < 1.05 as the results are i n -valid in the neighbourhood of toQ = 1. The inclusion of the variable damping term i n the d i f f e r e n t i a l equation governing cylinder displacement appears to allow for the r e a l i s t i c accommodation of a combination-oscillation form of solution. This has the effect of extending the range of applicability of the coupled-oscillator model outside of the lock-in region. 22. 4. DISCUSSION Several changes in form of the governing equations of Hartlen and Currie's original coupled-oscillator model for vortex-induced os-c i l l a t i o n have been suggested and examined. Various forms of solution to the modified equations and the question of their s t a b i l i t y have been investigated as well. Predicted results have been compared with exper-imental information, i n order to obtain a measure of their usefulness. The results of this work show the application of a combination-o s c i l l a t i o n form of solution to Hartlen and Currie's original model, and the extension to a seventh order non-linearity in C ' to be unproductive. A positive contribution has been made, however, with the inclusion of an effective structural damping term dependent on wind speed and cylinder displacement. The modified governing equations then produce a hysteresis mechanism within the lock-in region (harmonic solution), and r e a l i s t i c system behaviour outside of lock-in (combination-oscillation form of solution). The inclusion of a variable structural damping term (which i s consistent with experimental evidence) has the effect of improving trends i n the coupled-oscillator model performance, and extending i t s range of applicability. It i s proposed that the results are encouraging enough to warrant further investigation of this form of non-linearity. 23. REFERENCES 1. Ferguson, N., "The Measurement of Wake and Surface Effects on the Sub-critical Flow Past a Circular Cylinder at Rest and,in Vortex-Excited Oscillation", M.A.Sc. Thesis, U.B.C, 1965. 2. Feng, C.C, "The Measurement of Vortex Induced Effects in Flow Past Stationary and Oscillating Circular and D-section Cylinders", M.A.Sc. Thesis, U.B.C, 1968. 3. Parkinson, G.V., "Mathematical Models of Flow-Induced Vibrations", Symposium on Flow-Induced Structural Vibrations, Karlsruhe, August 1972. 4. Hartlen, R.T., Baines, W.D., and Currie, I.G., "Vortex-Excited Oscillation of a Circular Cylinder", UTME - TP 6809, November 1968. 5. Parkinson, G.V., "Wind-Induced Instability of Structures", Phil Trans. Roy. Soc. Lond. A, 269, 1971, 395 - 409. 6. Stoker, J.J., "Non-linear Vibrations i n Mechanical and E l e c t r i c a l Systems", Interscience Publishers, Inc., New York, 1950. 7. Minorsky, N., "Non-linear Oscillations", van Nostrand, 1962. 8. Landl, R., "Theoretical Model for Vortex-Excited Oscillations", •International Symposium Vibration Problems i n Industry, Keswick, England. Apr i l 1973. 9. Currie, I.G., Leutheusser, H.J. and Oey, H.L., "On the Double-Amplitude Response of Circular Cylinders Excited by Vortex Shedding", Proc. CANCAM '75, Fredericton, New Brunswick, May, 1975. 24. APPENDIX A Hartlen and Currie's original system of d i f f e r e n t i a l equation-soluti on by the methods of van der Pol and K-B Governing System \ A . l ...» A. 2 X" + 230X' + X = au) 0 2 C 3 C " - aa) 0C + (CT') + u)„2 CL = bX' J-1 Li u)0 L L (i) Solution after van der Pol Assume X = A^ sin fir C L = CH S ± n + V then,,.substituting for X and C i n Equation A . l and neglecting terms such as A^, <p^, and higher harmonics, one obtains the following system of equations after applying the principle of harmonic balance: 2 2 au)0 C R cos <j>H = (1 - ft ) 2 a<jj 0 C H sin <p = B 0fi ... A.3 2 2 2 O0) o fl c o s * H + S l n ^H ( 1 - t > PH } = ° sin A - cos <f> (1 - ( 7 — ) pTI) = owe fl H " " VH V J" KH' aco c CL H where B 0 — 230 25. C C L 0 V 3 Y _ C 2 P H V - ) From Equations A.3.1 and 2 , B Q ft tan (p H 1 - f t 2 2 ao) 2 2 •1 + C o t 2 <f> n From Equations A.3.3.. and 4 ... A. 4 2 2 ft 1 - n 1 + G o t 2 <j> H C 2 = ( C too ) 2 ( 1 + ( t o p 2 - ft2) ) H L 0 ^ '•" ctto0 ft tan * n , _ ab where n = — ( i i ) Solution by the K-B method (to ascertain the s t a b i l i t y of the approximate solutions to Equation A.1). Rewriting Equation A.l X ' ' + X = atoo 2 C - | * ( C J ' - Oto0 c ' + X- c ; 3 + W o 2 C T ) L b L L to0 L L ... A.5 assuming X = (x) sin (x + 8 H (x)) .. A.6 X ' = A J J (x) cos (x + e R (x)) 26. which implies that (x) sin(T + e R (x)) + ^ (T) 6 r (x) cos(x + 6 H (x)) = 0 or 6 ' (x) = - ^ C O sin(x + 6 H (x)) ^ (x) cos(x+ 9 R (x)) then multiplying Equation A.5 by X', one can determine A. 7 V ( T ) - ( a C L - ¥ ( C L * ' " ^ C L + t C L > 3 + ^ W C ° S ^ + 8 H ( T ) ) and from Equation A.7 e ' (x) = / 2 „ Bc (ato 0 C L - ^ (C- - aco0 C L + J - C' L 3 + 0 ) o 2 c L ) ) sin(x + 6 H (x)> Since a, B c are 0 (10~ ) ^ ~ 2TT 2lT 2TT •» i 2 B (a w0 CT - Tf- ( • • • )) cos ip dip L D A. 9 2TT 2 g (aw0 C L -'-jp ( ... )) sin ip dip where ip = x + 8. H If one assumes that C = CTT(x) sin ( X + cp (x)) L H H and that C', cp*, are 0 (10~ ), then from Equation A. 9 n. n a C H H 2 1 2 . ato - (w o + - (1 - " ) ) sinC + — - (1 -n n 2 .) cos C 27, r -a C, 3' * H 2 (w D 2 + ^  (1 - w a 2)) cos C + ( i - s i n c too ... A.10 where £ = 9 R - <J>H 2 2 C„ C = H ._ H H ~ 4. a ~ C 2 3 y L o Stationary solutions to Equations A.10 exist for = 0 i n which case K, = K, , C„ = C and 6 ' = - K which implies 8 = -K x. To f i r s t H H H . H H H H H . S s S order approximation £ must not be a function of X, thus <j• = 8 - Z, s H H s s s = - K„ X - r where r f r CO . In which case one obtains H ^s s s s P H - ( w 0 2 + • £ ( ! - w 0 2) ) sin ? + ^JSifl: ( ! _ _|) Cos C = 0 n s n 2. s Wo ... A.11 0 1 0 PH 2K ^ ( W o 2 + f- ( 1 - w 0 2) ) cos ? + ^  ( 1 - ~ § ) sin K = —V"5-0 s Two further equations are obtained by requiring that the station-ary solutions satisfy Equation A . l . Substituting for X and C T: X = AJJ. sin (x + e R ) = A J J sin (1 - K R)x E ^  sin fix ... A.12 C L = C H sin (x + cp h ) E C. I sin ( (1 - K )x - C ) = C sin (ftx - X, ) xi 11 s H s where ft = 1 - K H provides at0o C H sin £ g = - B„ ft A^ S £ 2 2 aco0 Cfl cos Cg = (1 - ft ) s s Solving Equations A.11 and A.13 one obtains 28. ... A. 13 2 1 - n 1 + cot C s 2 2 = rau s_ BO H (1 + cot £ ) s A.-14 c H 2 - (Co,.2) S tan aojc (1 + a>0 (n - 1)) where tan £ i - f t which can be shown to be identical to the results obtained by the method 2 ~ of van der Pol (Equation A. 4) provided that ft, ft = 1. To determine the s t a b i l i t y of a particular solution, one need d A H examine the sign of - j — only in the neighbourhood of the root A^ . From H s the expression for A^ (Equation A.10.1) one can determine d A i d C H r ^ a a P H ^ = ( — — cos C) dA H dAjj C R n u 0 ^ thus 2 9 . -aa PH d C, s H cos r - T - 7 -s d h n as. A. 1 5 From Equation A . 1 4 . 2 d CH _ , B^Jl , 2 . . L 2 . A ____ = ( } ( l + C o t O acoc H The s t a b i l i t y c r i t e r i o n i s d A' < 0 Stable > 0 Unstable d C H Examining Equation A . 1 5 , since 7 - 7 - , a, a , p , n, co0, are a l l positive quantities, then the question of s t a b i l i t y i s decided by - cos? or - cos ( Thus ( - C ) < T / 2 •> w i l l be stable ( • - 5 ) > n / 2 - > A J J w i l l be unstable 30. APPENDIX B Extension to 7th order non-linearity in C' - solution by the methods of van der Pol and K-B. Governing System X'* + Bo X' + X = aw 0 2 C CL* - ™ ° C L + ^ ( C L ) 3 - ^ 3 ( C V 5 + A 5 ( C L ) 7 + C L = b X ' 0 J o 0 J o B.l (i) Solution after van der Pol Assume X = Ag sin ftx • • • B . 2 C L = C R sin (fix + cbR) Substituting for X and C into Equation B.l, and applying the principle of harmonic balance one obtains: 2 2 aco0 C R cos cpH = (1 - ft ) 2 ,;aco0 C R sin o>R = A R B0ft ... B.3 (co02 - ft2) cos cp + a toaft n i y r 2 5 n > ft,4 4 6 (n ) 6 r 6 i c ^ 6 - n ^ 4a ~ 2 H 8 a w7 H ~ 64 a w 7 CH 3 s l n *H " 0 C0 o 2 2 (COQ ft ) . , 9 P^ - s i n <p„ -ao)0 ft H ! _ 31 ( A ) 2 2 31. I H rJL)2 c 4 _ 15 6 _ft 6 6 8 a ^ w0 ' 64 a ^ W/ U H cos A = H aa)0C Equations B .3 .3 and 4 are obtained by assuming that C* = C ft cos (ftr + A ) L H ri ( C L ) 3 " I fi3 c o s (flT + V (C[) 5 = | C H 5 ft5 cos (ftx + ( J ) H ) ( C L ) 7 = If C H ? FI? COS (fiT + V From Equation B .3 one can determine: H tan B0ft 1 - ft' r 2 ( \ 2 H BGft (1 + cot <|>H) B.4 Wo ft' 1 — n 1 + cot A H 6 8 ]1 rWp_. 2 4 48 j_ . u^ 6 ' 7 ; L H + 35 6 ft; * | ( " Q 2 - " 3 > 6 6w0 fttan A H = 0 In order to provide for a double amplitude response within the lock-in region, three real roots of the cubic polynomial in C must exist. H For a particular value of ft (and thus c o 0 ) within the region, values of 32. and C are available from experimental data and provide two max min equations for the determination of the non-linear coefficients. The ) i s made in order to establish "unstable choice of a third root (CT Y Ct •g, ~f so that a single real root of predetermined amplitude (C ) i s lo predicted outside of the lock-in region. This requires the selection of an appropriate value of 6 as well. Once the various parameters have 2 been specified, Equation B.4.4 can be solved for C (wc) by standard H, 1 methods. ( i i ) Solution by the K-B method Rewrite Equations B.l X' ' + x = a W„ CT -L b V v +k C C L , ) 3 .... B.5 , „ 3 < V >  + ~ i ' V 7 + " ' 2 ° i W„ W0 assume X = (T) sin (T + 0 H (x)) X' = (T) cos (X + 9 R ( T ) ) then proceeding in a manner identical to that introduced i n Appendix A ( i i ) one obtains a C T ~ H I 2 - (Wo2 +~ (1 - Wo2)) s i n - C - r - ^ G (C„) cos £ n n n -a C 0. H H ( w 02 + - (1 - w 0 2)) cosX + — G (C„) sin X n n n B.6 33. where t, = - (p r 2 C 4 C 6 r ( c ) = 1 . 3 i c x . + I n J L _ 3 5 6 S i _ G K BJ 4a 2 8 a 4 64 a i.6 0) o W 0 0) o Stationary solutions to Equation B.6 exist for A R' =0 i n which case A R = s C = C H H s 0 * = - K which implies 0 = - K x H H H H s To a f i r s t order approximation, £ must not be a function of X, thus A = 9 H - ^ s = - K H T - ? s W h e r e C s ^ s ( T ) s s In which case one obtains 9 1 9 OUJ ( 0 ) o Z + j- (1 - to/)) s i n r + ^  G (C ) cos r = 0 XI o XJL Jtl o ' s ' 2 K < u. 2 + ± ( i - a,.2)) cos ? s + ^ c (cH \ . i n c . - <r± • • •  B" 7 s H It i s required as well that the stationary solution satisfy Equation B.l for X = A sin (x + 01T ) E A sin (1 - K„)x E A sin fix H H 11 n n C L = C H sin (X + <pH ) E C R sin ( (1 - K^x - r, g) E C R sin (fix - ? g) s s s s 3 4 . which provides atoQ C u sinC = - B 0ft A a s H s s a0J o 2 C u cos £ = A u (1 -ft 2) ri S n S S B.8 From Equations B.7 and 8 one obtains: 2 1 1 - n 1 + cot r, 9 M 2 2 H H ^Bft ' . 2 , S (1 + cot £ ) B.9 CH 7 ° 6 LH + 35 6 u° CH 64 . 6 35 W ° = 0 where tan £ -ftB 1 - ft' which can be shown to be identical to the results obtained by the method 2 4 6 ~ of van der Pol (Equation B .4 ) provided that ft, ft , ft , ft = 1 . To determine the s t a b i l i t y of a root A^ , examine the sign of d V From Equation B . 6 . 1 one has that A^' = F (C R), therefore AH = AH d V d F d CH  d A H " d C H d A H which can be evaluated from Equations B.6.1 and 9.2. The criterion of s t a b i l i t y being 35. < 0 stable > 0 unstable 36. APPENDIX C Combination-oscillation solution applied to Hartlen and Currie's original system (solution by the method of van der Pol). Assume X = A^ sin fix + Ap sin to^ x ... C.l C L = ° H S± n (fiT + V + C F S L N ( W F T + V' then substituting for X and C into Equation A . l and neglecting terms such as A ', (f> ' , higher harmonics and combination tones, the following results H r are obtained after applying the principle of harmonic balance: aw 0 2 C F cos cf>F = Ap (1 - Up2) ao>0 c sin i * F = Ap, B 0 2 2 ao)c C R cos A H = A R (1 - fi ) ato, 0 C R sin 4>H = ^  B 0 C.2 / 2 2. (w0 - Wp ) aw0. w„ cos <p_ + sin <J) 2 2 1 ( P F CnO + 2 P H > 2 2 (Wo - Wp^) S ^ T T sin Ap - cos cpi CO,/ cy 2 ( P F (TT> + 2 PH ) aa)Q CT 2 2 (LOO - ft ) i , > i ato0 ft C O S + S l n *H ft 2 W F 2 = 0 < 2 o 2^ ( l0o - ft ) . , , aa)0 ft S l n ~ c o s ft 2 U F 2 ao)0 C H. where p = 2 2 H _4 a C J 3 y L ° 2 2 F 4 a ~ VC. 3 Y Note that Equations C.2.5-7 assume that 3 3. 0) 'V* ~i  a C F <rcos <V + V o w 2 °F <<T> + 2 C H + T ft C„ cos (ftx:+ cb ) ? O W F 2 CH + 2 C F V > From Equations C.2.5 and 6 one obtains tOo 1 - n 1 + cot CO • O L CO 1 + 2 2 ao)0 Wp tan cPF • • * C • 3 From Equations C.2.7 and 8 2 COo ft 1 -• n 1 + cot cb. H to„ P H + 2 p F < f i T > = <^> 1 +. 2 2 (Mo - « ) au)0ft tan <J> H J 38. From Equations C.2.1-4 tan <p. t O p B F 2 1 - w„ 2 2 2 .a to„ . F - ( B D U > F (1 + cot cf> ) tan <p = fiB0 1 - JT 2 2 N B 0 n ( i + cot <{>H) If one assumes that fi ? 1 and to = 0)o then Equations C.3 can solved for p„ „, A.T „, cp „ as functions of w0> a,y and r\. fi(to0) and H, r i i , r H, r to (to0) are given by the appropriate roots of Equations G. 3.3, and• 1. F 39. APPENDIX D Variable structural damping - solution by the K-B method. r''" Governing system Assume X" + B 0 (1 + f03 o 2 y X' + X = aw 0 2 Cj Y 2 CT " - au>0 CT ' + -1- (CT ') + a>0 CT = b X' X = Ag (x) sin(x + 9 R (x)) ... D.l X' A J J (x) cos (x + 9 R (x)) C L = C R sin (x + <|>H (x)) Then proceeding in a manner identical to that introduced i n Appendix A ( i i ) one obtains V 2 1 2 2 sin £ (u) o - - (1 + f0)o ^ (0Jo - 1)) 2 _ OU) „ ( 1 " PH ) : - cos C (1 + f 0)o A^) — i _ H 2 2 1 2 cos C ( aj0 - - (1 + fw0 A^) (u>0 - 1)) 2 . * «o 0 ( 1 " V + sin £ (1 + fw0 Ay) — 0)o2 D.2 4 0 . Stationary solutions to Equation D.2 exist for A J J ' = 0 i n which case A ^ = CH = CH and 8 ' = - K which implies 8 = - K T. To a f i r s t order approximation H H H H s £ must not be a function of T, thus A s H H ^s K T - r where H s £ ^ £ g (x). Two further equations are obtained by requiring that the stationary solution satisfy Equation D . l . l for X = A H S ± n ( 1 ~ K H ) T = *H sin fir C L = CH S ± n ( ( 1 " K H ) T " C s } = CH S l n ( f i T " C s } s s The expressions which are derived for A , C , fi and £ are H H s S s identical to those obtained by the method of van der Pol (Equations(3.6-9)) where - K - \BC H To determine the s t a b i l i t y of a root A ^ , examine the sign of From Equation D.2.1 one has that A ^ ' = F ( A ^ , C R, 0, therefore AH = \ d V = 9 F 3 F d CH 3F d £ _ d A H _ 9 A H ' C H d A H H d A H . D.3 41. The p a r t i a l derivatives can be obtained from Equation D.2.1, and the exact differentials from Equations (3.5.2) and (3.8). The criterion foi s t a b i l i t y being d V a. < 0 stable > 0 unstable 42. FIGURE I: Experimental Results For Vortex-Induced Oscillation of a Circular Cylinder (Feng) 43. FIGURE II: Schematic Diagram of Experimental Configuration 45. FIGURE IV: Theoretical Predictions from Hartlen and Currie's Original Model 47. FIGURE VI: Theoretical Predictions for Combination-Oscillation Solution Applied to Hartlen and Currie's Original Model 120 48. 1.2H Q H -80 1-40 •8H C / o /3 = .001 a = .0022 C. = .5 b =1.24 a = . 5 5 { = 4 ~V5 1.8 -.8 -.4 fr-2 FIGURE VIS = Theoretical Predictions ot the Effect of Variable Structural Damping — Harmonic Solution FIGURE VIII: Theoretical Predictions of the Effect of Variable Structural Damping—Combination-Oscillation 

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