GALLOPING OF SMALL ASPECT RATIO SQUARE CYLINDER

Transcription

1 OL. NO. JANUARY 5 ISSN Asian Research Publishing Network (ARPN). All rights reserved. GALLOPING OF SMALL ASPET RATIO SQUARE YLINDER A. N. Rabinin and. D. Lusin Facult of Mathematics and Mechanics Saint-Petersburg State Universit 8 Universit Ave. St. Petersburg Russia a.rabinin@spbu.ru ABSTRAT A mathematical model of galloping is considered in a quasi-stead approximation. Normal force aerodnamic coefficients of square clinder are measured in the subsonic wind tunnel at different angles of attack α. The clinder aspect ratio is. The same clinder is tested with end plates as well. A new function for the approximation of the aerodnamic coefficients dependence on the angle of attack is used. Krlov-Bogoliubov method is applied. Mathematical model allows predicting the critical air velocit at which oscillations occur. The amplitude of the oscillations can be calculated as a function of flow velocit. It appeared that the end plates significantl change the aerodnamic coefficients at low angles of attack. The critical air velocit reduces. The inflection point appears on the dependence (α). Results of mathematical simulation are verified in the wind tunnel experiments. Square clinder is suspended on two springs across the flow generated in the wind tunnel working section. The tension of the spring is measured with semiconductor tensoconverter. The output signal from tensoconverter is recorded in a computer file for further processing. In a separate experiment the elastic damping sstem is measured. The calibration of instruments allows determine the amplitude of stead oscillations of the clinder. The results of calculation and experiment are in good agreement for all the stable modes of oscillation. Kewords: galloping aspect ratio wind tunnel mathematical model. INTRODUTION Galloping is self-excited oscillations of bluff bodies in the wind flow caused b a specific dependence of the aerodnamic force on the angle of attack. Galloping can result in destruction of structures. The quasi-stead mathematical model of bluff bod galloping was developed in the middle of th centur. One of the first studies of modeling was published in the paper of Parkinson and Brooks []. A more complete model of the galloping of elasticall fixed bluff bod in the air flow was proposed b Parkinson and Smith []. In these papers the galloping of prisms with a square cross-section was considered. Later Novak [3] studied the clinder with a rectangular cross-section. In the last decade a number of studies revealed the aeroelastic instabilit of man bodies having different cross section [ - 8]. Barrero-Gil et al. [9] considered the galloping of square clinders in cross flow at low Renolds numbers using finite element method. Hémon and Santi [] performed the experiments with a flexible clinder clamped at both ends. In all the papers mentioned above bodies of infinite aspect ratio were considered. However the bodies of small aspect ratio also can oscillate under the action of a fluid flow. The example of such bod is the rope wa cabin []. In the present paper we stud the transverse galloping of elasticall fixed square clinder of small aspect ratio. MATHEMATIAL MODEL Let axis z be normal to the base of the clinder. The clinder can move onl along the axis. The incoming flow has a constant velocit v and is directed along the axis x. The clinder has a mass m a length L (along z) a width H and a height H (along the axes of x and respectivel). The displacement from the equilibrium position of the clinder is. The clinder is held with an elastic holder. The stiffness of a holder is k and viscous damping force of the holder is equal r d/dt. This force is alwas directed against the velocit of the clinder d/dt. Quasi-stead assumption is adopted according to which the aerodnamic coefficients depend onl on the angle of attack. The normal component of the aerodnamic force acting along the axis is s v /. r Here densit of the medium s = HL face area of the clinder - the normal force coefficient v r relative velocit of the clinder. This velocit is the sum of the own clinder velocit ( d/dt) and the free stream velocit with opposite sign (- v ). Thus the relative velocit and the tangent of the angle of attack are calculated as follows: d d v r v tan. v dt v dt The aspect ratio λ of the clinder is defined to be the ratio of the length L of the clinder to the width H: λ= L/H. The equation of transverse motion of the clinder is given b d d v r m r k s. () dt dt oefficient of aerodnamic force can be approximated b polnomial function of the angle of attack or tangent of the angle of attack [ 8 9 ]. However for the small aspect ratio clinders as shown below this polnomial approximation describes well the 3

2 OL. NO. JANUARY 5 ISSN Asian Research Publishing Network (ARPN). All rights reserved. experimentall determined dependence С (tan α) onl for small angles of attack. In the paper [3] we propose the following approximation of the dependence (tan) : B B tan i A (tan) i i3... B B tan if tan w if w tan w if w tan. The Equation () of the clinder motion can be transformed to a dimensionless form b choosing m / k as the unit of time and r m / s k as the unit of length: d Y Y d dy d r mk where Y and τ are dimensionless transverse displacement of the clinder the dimensionless flow velocit and dimensionless time: r Y s m k r v s t mk. The second order differential equation can be transformed to a sstem of two first-order equations: dy Z d dz Y Z. d For the solution of Equation () the Krlov- Bogoliubov method [] was applied. The method is based on the averaging principle when the exact differential equation of the motion is replaced b its averaged version. Let suppose that () (3) () Y cos( ) Z sin( ) (5) where ρ and φ are the slowl varing amplitude and phase of oscillation. After substitution the expressions (5) to the sstem Equation () the first equation of the sstem should be multiplied b cos (τ + φ) the second one should be multiplied b - sin (τ + φ). The summation of these two equations after averaging leads to differential equation for slowl varing amplitude ρ of the oscillations. The equation for phase φ one can obtain b the same wa but first equation of Equation () should be multiplied b - sin (τ + φ) the second one should be multiplied b - cos (τ + φ). Thus the first approximation of Krlov- Bogoliubov method [] leads to a sstem of differential equations for slowl varing amplitude and phase: d A 3 A 3 3 if w d A A d B B S d w w B R S A S 3 w 3 3w A3 R S 5 w 5 5w A5 3 R S 8 8 if w d d where S w w 3 w R arcsin. In this paper a clinder with a square crosssection is considered but one can appl the mathematical model to the oscillations of bluff bodies of different shapes including asmmetrical. If the bod is not smmetrical the even expansion coefficients for the normal force are not equal to zero but after the application of the Krlov-Bogoliubov procedure corresponding terms would be equal to zero. Application of the Krlov-Bogoliubov method is possible if the aerodnamic force and the resistance force are smaller than the elastic force. In this case the oscillation of the bod is closed to the harmonic one. The characteristic time of change of the oscillation amplitude is much greater than the oscillation period. The first formula of Equation (6) implies that the solution ρ = becomes unstable when the dimensionless flow velocit exceeds the critical value of / A. Thus to describe the oscillation of the clinder it is necessar to measure the dependence of the aerodnamic normal force coefficient on the angle of attack and determine the parameters w A i B B and B. EXPERIMENTAL DETERMINATION OF THE AERODYNAMIAL OEFFIIENTS AND MATHEMATIAL SIMULATION OF OSILLATIONS The clinder with square cross section was taken. linder aspect ratio was λ =. The clinder was tested with end plates as well. The experiment was conducted in the closed return wind tunnel with open working section. Renolds number Re was equal to. 5. linder was rigidl fixed in the aerodnamic balance. Measurements of (6) 35

3 OL. NO. JANUARY 5 ISSN Asian Research Publishing Network (ARPN). All rights reserved. aerodnamic forces were conducted at angles of attack from - 3 o and 3 o. For clinder without end plates satisfactor normal force approximation could be obtained b putting A 3 =. Figure- shows the dependence of the approximation on tan α for clinder without end plates. Equation (7) describes the dependence of the aerodnamic coefficients of the angle of attack. Figure-. Aerodnamic coefficients for clinder λ = with end plates. Experimental data and approximation tan if tan tan 57.83tan 5 89 tan if tan tan if tan.5. (8) Figure-. Aerodnamic coefficients for clinder λ = without end plates. Experimental data and approximation..8 5.tan tan if tan tan 7 tan if tan.3 if tan.3. The presence of end plates leads to the appearance of an inflection point on the curve (tan α) in the range of angles of attack tan α < w. It is impossible to obtain a satisfactor approximation taking the coefficient A 3 =. Figure- illustrates the effect of the end plates which are disks attached to the ends of the clinders. The diameter of end plate is equal to 3.63 H. End plates change the dependence on tan α for small angles of attack. Without disks the values of the normal component of the aerodnamic force at low angles of attack are small but normal force acting on the clinder with end plates significantl differs from zero. After joining the endplates absolute value of the coefficient A dramaticall increases (see Eq.8). The dependence of (tan α) for small angles is similar to the dependence obtained in the work of Parkinson and Smith []. (7) One can see that in both tested cases there is the range of angles of attack in which the sign of the angle of attack is opposite to. In this range the directions of the aerodnamic force and the velocit of the clinder coincide. Thus the aerodnamic force does positive work and the vibrational energ of the sstem increases. The presence of such a range is the cause of galloping. Figure- and Figure- show that aerodnamic normal force at α = is not equal to zero. Probabl this phenomenon is caused b errors of manufacturing. Dependence of amplitude of oscillation ρ on dimensionless velocit presented on Figure-3. If the flow velocit exceeds the critical value zero solutions ρ = of Equation (6) becomes unstable. For the clinder without end plates the oscillation amplitude increases abruptl. Equation (6) ma have one two or three solutions with constant amplitude of oscillation. In the last two cases one solution is unstable. There is a hsteresis region characterized b the existence of three solutions one of solution is unstable. For clinder without end plates the critical velocit is the right border of the hsteresis range. The presence of end plates leads to appearance of an inflection point on the dependence of (tan α). The appearance of an inflection point as it was shown in [8] changes the dependence of the amplitude of the steadstate oscillation on the free stream velocit (see Figure-3). The critical velocit decreases the hsteresis of flow velocities decreases too. The left boundar of the hsteresis range becomes greater than the critical velocit. linder equipped with the end plates can vibrate with small amplitudes at a slight excess of the critical flow velocit. 36

4 OL. NO. JANUARY 5 ISSN Asian Research Publishing Network (ARPN). All rights reserved. Figure-3. The dependence of the dimensionless oscillation amplitude ρ on the dimensionless flow velocit. The calculation results and experimental data. - calculation the clinder without end plates - calculation the clinder with end plates 3 - experiment with a clinder without end plates with different damping 5 6- experiment with a clinder with end plates with different damping. EXPERIMENTAL ERIFIATION Despite the fact that different authors have frequentl used the classical model of aeroelastic galloping it certainl should be verified b comparison with the experimental results. Aerodnamic tests were performed in a subsonic wind tunnel. linder was manufactured from wood. Its dimensions were L =.6 m H =.6 m. linder was suspended on two springs across the flow generated in the wind tunnel working section (see Figure-). Figure-. Scheme of the experiment in a wind tunnel. - semiconductor tensoconverter - confusor 3 - spring - clinder. The threads let the clinder to move onl along the arc of a circle with radius of two meters approximatel along single direction normal to the flow velocit. The forces acting on the clinder were measured using a semiconductor integral tensoconverter S-5 located between the lower spring and rigidl fixed holder. Tensoconverter is designed for measurement of the forces within the range from to 5 N. The tensoconverter output signal fed an analog-to-digital converter L-5 performed as a P card. A program written in Pascal language controls the measurement process. The measurement results had been recorded in a file for later processing. alibration of the instrument was carried out b loading the known weights which were placed on top face of the clinder. In addition the displacement of the clinder caused b the known load was measured. Laser beam spot was established on the surface of the clinder. After loading a known weight the spot of the laser beam moved over the surface of the clinder. The measured displacement of spot is used to calculate the stiffness of the sstem. It should be noted that in this experiment the friction force plaed the role of damping force. To increase the frictional force we inserted a snthetic fibrous material in the springs. The oscillation amplitude was calculated using the experimental force data. To find the damping coefficient experiment was performed in the absence of airflow. In a series of experiments which were carried out at different flow velocities some modes of stead-state oscillation were discovered. At the same time in some cases the excitement of oscillation was realized b putting the initial moment to the clinder. In other cases the oscillations developed from equilibrium position. The experiment was carried out with the clinder without end plates and with the same clinder equipped with the end plates. The results were compared with calculations according a mathematical model of galloping. The calculation results and the experimental data are presented in Figure-3. As one can see on the graph the results of calculation and experiment are in good agreement for all the stable modes of oscillation. ONLUSIONS Based on a mathematical model of galloping we analzed the stationar oscillations of a square clinder. Experimentall determined aerodnamical coefficients are used in the model. linders of small aspect ratio cannot oscillate with small amplitude at a slight excess of the critical flow velocit. The critical flow velocit for the clinders of low aspect ratio is at the right border of the hsteresis range. The clinder equipped b end plates that prevent air flowing through the ends changes the dependence of the normal force on the angle of attack at low angles of attack. The critical velocit decreases and hsteresis range of flow velocities decreases. The left boundar of the hsteresis region is greater than the critical velocit. The clinder equipped with the end plates can vibrate with small amplitudes at a slight excess of the critical flow velocit. These results were verified in experiments with elasticall fixed clinders in the wind tunnel. 37

5 OL. NO. JANUARY 5 ISSN Asian Research Publishing Network (ARPN). All rights reserved. AKNOWLEDGEMENTS This work was supported b a grant of Saint- Petersburg State Universit 6... REFERENES [] Parkinson G.. and Brooks N. P. 96. On the aeroelastic instabilit of bluff clinders. Journal of Applied Mechanics. 8: [] Parkinson G.. and Smith J. D. 96. The square prism as an aeroelastic non-linear oscillator. Quarterl Joutnal of Mechanics and Applied Mathematics. 7: [3] Novak M Aeroelastic galloping of prismatic bodies. ASE Journal of Engineering Mechanics Division. 95: 5-. cross-wind influence. Mathematical and omputer Modelling of Dnamical Sstems. 3(): [] Thompson J.M.T. 98. Instabilities and catastrophes in science and engineering. hichester: John Wile and Sons. [3] Lusin.D. and Rabinin A.N.. Investigation of aspect ratio of the prism on its aerodnamic characteristics and the vibration amplitude during prism galloping. estnik Sankt-Peterburgskogo Universiteta. Seria No. : 39-5 (In Russian). [] Bogoliubov N. N. and Mitropolski Y. A. 96. Asmptotic Methods in the Theor of Non-Linear Oscillations. New York: Gordon and Breach. [] Alonso G. Meseguer J. and Perez-Grande I. 5. Galloping instabilities of two-dimensional triangular cross-section bodies. Experiments in Fluids. 38: [5] Alonso G. and Meseguer J. 6. A parametric stud of the galloping stabilit of two-dimensional triangular cross-section bodies. Journal of Wind Engineering and Industrial Aerodnamics. 9: [6] Alonso G. Meseguer J. and Perez-Grande I. 7. Galloping stabilit of triangular cross-sectional bodies: A sstematic approach. Journal of Wind Engineering and Industrial Aerodnamics. 95: [7] Alonso G. Meseguer J. and alero E. 9. An analsis on the dependence on cross section geometr of galloping stabilit of two-dimensional bodies having either biconvex or rhomboidal cross sections. European Journal of Mechanics. B / Fluids. 8: [8] Barrero-Gil A. Sanz-Andres A. and Alonso G. 9. Hsteresis in transverse galloping: The role of the inflection points. Journal of Fluids and Structures. 5: 7-. [9] Barrero-Gil A. Sanz-Andres A. and Roura M. 9. Transverse galloping at low Renolds numbers. Journal of Fluids and Structures. 5: 36-. [] Hémon P. Santi F.. On the aeroelastic behavior of rectangular clinders in cross-flow. Journal of Fluids and Structures. 6: [] Petrova R.. Hoffmann K. and Liehl R. 7. Modelling and simulation of bicable ropewas under 38