华侨大学陈振川全球求助翻译英语论文_(7).pdf

50 The Fifth International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY Huazhong University of Science and Technology Wuhan P R CHINA 27 28 August 2009 2009 Huazhong Universiti of Science and Technology Press NATURAL FREQUENCIES OF NONLINEAR TRANSVERSE VIBRATION OF AXIALLY MOVING BEAMS IN THE SUPERCRITICAL REGIME Hu Dinga Li Qun Chen a b a Shanghai Institute of Applied Mathematics and Mechanics Shanghai 200072 China b Department of Mechanics Shanghai University Shanghai 200436 China Email dinghu3 ABSTRACT Natural frequencies are investigated for transverse vibration of axially moving beams in the supercritical ranges In the supercritical transport speed regime the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions The transverse motion can be governed by a nonlinear partial differential equation or a nonlinear integro partial differential equation For motion about each bifurcated solution those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform The natural frequencies are investigated for the beams via the 8 term Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary differential equations under the simple support boundary Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency and the two nonlinear models predict qualitatively the same tendencies of the natural frequencies with the changing parameters Quantitative comparisons demonstrate that results of the 4 term Galerkin method for the natural frequency for axially moving beams in the supercritical range are with rather high precision KEYWORDS axially moving beam nonlinearity supercritical natural frequencies galerkin method I INTRODUCTION Axially moving beams are extensively investigated because they can model many engineering devices such as elevator cables aerial cable tramways belt saws power transmission band If the large amplitude motion is concerned the geometric nonlinearity should be taken into consideration In 1965 Mote 1 presented the fundamental frequency approximations via the Galerkin method and supported by the experiment 2 Wickert and Mote 3 studied a classical vibration theory for the traveling string and the traveling beam where natural frequencies and modes associated with free vibration serve as a basis for analysis Wickert 4 used a perturbation method to investigated the natural frequency of an axially moving beam under the simple support boundary conditions in the supercritical regime z 5 investigated natural frequencies of an axially moving beam in contact with a small stationary mass under pinned pinned or clamped clamped boundary conditions Chen and Yang 6 gave the first two frequencies of axially moving elastic and viscoelastic beams on simple supports with torsion springs Kong and 51 Parker 7 combined perturbation techniques for algebraic equations and phase closure principle to derive approximate natural frequencies of an axially moving beam Ghayesh and Khadem 8 computed natural frequency for the first two modes in free non linear transverse vibration of an axially moving beam in which rotary inertia and temperature variation effects have been considered All of above literatures the natural frequency of axially moving beams were calculated from linear governing equation of transverse vibration There are two kinds of nonlinear transverse models of axially moving beams One is a nonlinear integro partial differential equation used by Wickert 4 Pellicano and Zirilli 9 Ravindra and Zhu 10 Chakraborty and Mallik 11 Chakraborty Mallik and Hatwal 12 Pellicano and Vestroni 13 Parker and Lin 14 Chen and Yang 15 Ding and Chen 16 The other is a nonlinear partial differential equation used by z Pakdemirli and Boyaci 17 Marynowski 18 Chen and Yang 15 Chen Huang and Sze 19 Chen and Yang 6 Zhang and Song 20 and Ghayesh and Khadem 8 Ding and Chen 16 Because there has no evidence to favor any models both models are treated in the investigation It should be remarked that the literatures on axially moving materials in the supercritical ranges are rather limited In reference 4 Wicket noticed that the straight equilibrium configuration becomes unstable and bifurcates in multiple equilibrium positions of a translating beam are analogous to those in a buckled beam problem from the nonlinear integro partial differential equation and for each non trivial equilibrium solutions the governing equation is cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform and the natural frequency of an axially moving beam is calculated by perturbation method Hwang and Perkins 21 22 investigated the effect of an initial curvature due to supporting wheels and pulleys on the bifurcation and stability of equilibrium in the supercritical speed regime they underlined the system sensitivity to initial imperfections Ravindra and Zhu 10 investigated a symmetric pitchfork bifurcation for a parametrically excited as the axial velocity of the beam is varied beyond a critical value Pellicano and Vestroni 23 focused on exploring the dynamics of a travelling beam subjected to a transverse load with simple supported via the Galerkin method when its main parameters vary in the supercritical velocity range Pakdemirli and z 24 calculated the natural frequencies values of axially moving beams under clamped clamped boundary conditions in the supercritical regime from the governing equation in the subcritical range they found the complex frequency values non zero imaginary parts i e the beam is unstable at those velocities To address the lacks of research in those aspects the present investigation focus on the natural frequency of an axially moving beam in the supercritical speed range based on the Galerkin truncation to the full governing equations II GOVERNING EQUATIONS AND EQUILIBRIA Consider a uniform axially moving beam travels at the uniform constant transport speed between two boundaries separated by distance l with density cross sectional area A moment of inertial I initial tension P0 and Young s modulus E Assume that the deformation of the beam is confined to the vertical plane For a slender beam the linear moment curvature relationship is sufficiently accurate The fixed axial coordinate x measure the distance from the left boundary The transverse displacement v x t related to a spatial frame The beam is subjected to no external loads The nonlinear the partial differential equation and the integro partial differential equation for transverse motion of axially moving elastic beam can be cast into the dimensionless form 6 xxxxxxxxxxttt vvkvkvvv 2 3 1 2 2 2 1 2 f 2 1 and xvvkvkvvv xxxxxxxxxxttt d 2 1 1 2 1 0 2 2 1 2 f 2 2 where a comma preceding x or t denotes partial differentiation with respect to x or t and The dimensionless variables and parameters as follows 2 0 f 0 1 0 2 0 lP EI k P EA k P A Al P tt l x x l v v 3 52 In the present investigation only the boundary conditions of the beam is simply supported at both ends are considered as follows 0 1 0 0 1 0 tvtvtvtv xxxx 4 Equilibrium solutions xv of equation 1 and 2 satisfy 0 2 3 1 22 1 2 4 2 f vvkvk 5 and 0 d 2 1 1 2 f 1 0 22 1 2 vkvxvk 6 Introduce the equispaced mesh grid with space step h 0 1 2 j xjhjL hl L 7 Denote the function values i x v at xi as i v The trivial configuration0 0 v is always an equilibrium solution The numerical schemes are presented for equation 5 and 6 for pairs of non trivial equilibrium solutions i v in the supercritical regime via centered difference approximations to the time space and mixed partial derivatives to implement the simply supported boundary conditions 4 The non trivial equilibrium solutions of nonlinear equations 5 and 6 can be solved using iterative schemes 222 f 1111 2 1 2 2112 2 f 126 314 4 4 hk hvvvvkhvvvvk v iiiiiiii i 8 and 222 f 2 11 2 1 2 2112 2 f 126 2 1 1 4 4 hk hvvIntkvvvvk v iiiiii i 9 where N j jj LL h vv h vv h vvh Int 1 2 11 2 1 2 01 2 2 2 10 Over the supercritical range there exists the two non trivial solutions 1 v and 1 v Substituting txvxvtxv in equation 1 and 2 and omitting nonlinear terms of the equations yield vvvkvkvvkvv xxxxxxxxttt 2 3 1 2 2 1 2 f 2 2 1 2 11 and 2 11 2222 1f1 00 1 2 1d d 2 ttxtxxxxxxx vvkvx vk vk vv vx 12 III NATURAL FREQUENCIES OF NONLINEAR MODELS In the present investigation both the trial and weight functions are chosen as eigenfunctions of a linear non translating beam under the boundary condition 4 namely suppose that the solution to equations 11 and 12 take the form 1 sin 1 2 n j j v x tqtj xjn 13 where tqj are sets of generalized displacements of the beam After substituting equation 13 into equations 11 and 12 the Galerkin procedure leads to the following set of second order ordinary differential equations 53 0 111 n j j p ij n j j p ij n j j p ij tqKtqCtqM 14 and 0 111 n j j i ij n j j i ij n j j i ij tqKtqCtqM 15 where 1 0 1 sin sin d 2 pi ijijij MMj xi xx 16 22 1 0 4 2cos x sin d 0 pi ijij ijijij CCjji xx ij 17 1 2 2222222 f11 0 3 1 sin 3 cos sin d 2 p ij Kkjk vjj xk v vjj xi xx 18 11 2 222222 ff 00 1 sin 1 4 sin cos cosd sin d i ij Kjkjj xkxjj xxx i xx 19 Equations 14 and 15 can be written in matrix vector form as MqCqKq0 20 where T 12 n q qq q 21 If tq defined as it te Qq 22 Eq 20 yield 2 i0 MGK Q 23 Non triviality of solutions to Eq 23 requires its determinant of coefficients to be zero therefore the natural frequencies can be obtained from 2 i0 MGK 24 In the computations n 2 4 and 8 that is to say the natural frequency is numerical calculated based on 2 4 and 8 term Galerkin truncation Here only show the results of txvxvtxv as the results via txvxvtxv are the completely same In the present investigation beams are assumed to move with a supercritical axial speed The results of 2 term Galerkin truncation 4 term Galerkin truncation and 8 term Galerkin truncation for equations 11 and 12 with the flexural stiffness kf 0 8 and the nonlinear coefficient k1 100 for the first natural frequency are shown in figure 1 where the dash dot lines are for 2 term the solid lines are for 4 term and the dot lines are for 8 term Galerkin truncation Fig 1 demonstrate that the 4 term Galerkin truncation predict the almost same results as 8 term Galerkin method while the 2 term ones bigger than them and the difference increase with the growth of axial speed 54 2345 0 5 10 15 Natural frequency axial speed kf 0 8 2345 0 5 10 15 Natural frequency kf 0 8 axial speed a The first natural frequency Eq 11 b The second natural frequency Eq 12 Fig 1 Comparison between 2 term 4 term and 8 term Galerkin truncation results 12345 5 0 5 10 15 axial speed Natural frequency kf 0 4 kf 0 6kf 0 8 kf 1 12345 5 0 5 10 15 kf 0 4 kf 0 6kf 0 8 kf 1 axial speed Natural frequency a The first natural frequency Eq 11 b The second natural frequency Eq 12 Fig 2 Natural frequencies versus axial speed and flexural stiffness Fig 2 illustrates the dependence of the first natural frequency on the axial speed for fixed nonlinear coefficient k1 100 and four different kf values for equations 11 and 12 Fig 3 shows the effects of the nonlinear coefficient with kf 0 8 on the first natural frequencies for equations 11 and 12 In figure 3 the dots lines and the solid lines respectively stand for the first natural frequency to k1 100 and k1 200 The comparisons indicate that nonlinear coefficient has little effects on the natural frequency 2345 0 5 10 15 Natural frequency axial speed kf 0 8 2345 0 5 10 15 Natural frequency kf 0 8 axial speed a The first natural frequency Eq 11 b The second natural frequency Eq 12 Fig 3 The effects of the nonlinear coefficient on the natural frequencies changing with axial speed The results of equations 11 and 12 are quantitatively contrasted in the case with k1 100 Fig 4 show the comparison in which the solid lines and dash dot lines represent the natural frequencies of equations 11 and 12 for kf 0 6 kf 0 8 and kf 1 The numerical results demonstrate that two nonlinear 55 models quantitatively predict certain differences with the changing flexural stiffness kf and the difference increase with the axial speed and the natural frequencies of equation 11 bigger than equation 12 2345 0 5 10 15 Natural frequency axial speed kf 0 6 2 53 03 54 04 55 0 0 5 10 15 Natural frequency axial speed kf 0 8 345 0 5 10 15 Natural frequency axial speed kf 1 a kf 0 6 b kf 0 8 c kf 1 Fig 4 Natural frequencies versus axial speed and flexural stiffness for Eqs 11 and 12 IV CONCLUSIONS This paper is devoted to natural frequency of axially moving beams in the supercritical regime The non trivial equilibrium equations of the nonlinear integro partial differential equation and the nonlinear partial differential equation are solved via the finite difference scheme For motion about each bifurcated solution those equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform The natural frequencies are computationally studied for the linear equations correspondingly those standard form via the Galerkin method under the simple support boundary The investigation leads to the following conclusions 1 In the supercritical regime the nonlinear coefficient has little effects on the natural frequency 2 The 4 term Galerkin method for the natural frequency for axially moving beams in the supercritical range yields rather accurate results 3 The two models predict the tendencies of the natural frequencies with the changing flexural stiffness 4 The nonlinear partial differential equation leads to the larger natural frequency V ACKNOWLEDGMENTS This work was supported by the National Outstanding Young Scientists Fund of China No 10725209 the National Natural Science Foundation of China No 10672092 Shanghai Municipal Education Commission Scientific Research Project No 07ZZ07 and Shanghai Leading Academic Discipline Project No S30106 References 1 Jr C D Mote A study of band saw vibrations Journal of the Franklin Institute 276 430 444 1965 2 Jr C D Mote and S Naguleswaran Theoretical and experimental band saw vibrations ASME Journal of Engrg Indus 88 151 156 1966 3 J A Wickert and Jr C D Mote Classical vibration analysis of axially moving continua ASME Journal of Applied Mechanics 57 738 44 1990 4 J A Wickert Non linear vibration of a traveling tensioned beam International Journal of Non Linear Mechanics 27 503 517 1992 5 H R z Natural frequencies of axially travelling tensioned beams in contact with astationary mass Journal of Sound and Vibration 259 445 456 2003 6 L Q Chen and X D Yang Vibration and stability of an axially moving viscoelastic beam with hybrid supports European Journal of Mechanics A Solids 25 996 1008 2006 7 L Kong and R G Parker Approximate eigensolutions of axially moving beams with small flexural stiffness Journal of Sound and Vibration 276 459 469 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band saw vibratio