Stability and vibration analysis of a complex flexible

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Journal home page http www sciencedirect com science journal 10075704. Stability and vibration analysis of a complex flexible rotor bearing system. Communications in Nonlinear Science and Numerical Simulation Volume 13 Issue 4 July. 2008 Pages 804 821,C Villa J J Sinou and F Thouverez. Stability and vibration analysis of a complex flexible rotor bearing system. C Villa J J Sinou and F Thouverez, Laboratoire de Tribologie et Dynamique des Syst mes UMR CNRS 5513. cole Centrale de Lyon 69134 Ecully Cedex France, This paper presents the non linear dynamic analysis of a flexible unbalanced rotor supported by ball. bearings The rolling element bearings are modeled as two degree of freedom elements where the. kinematics of the rolling elements are taken into account as well as the internal clearance and the. Hertz contact non linearity In order to calculate the periodic response of this non linear system the. harmonic balance method is used This method is implemented with an exact condensation strategy. to reduce the computational time Moreover the stability of the non linear system is analyzed in the. frequency domain by a method based on a perturbation applied to the known harmonic solution in. the time domain, Keywords rolling bearings non linear dynamic stability harmonic balance method bearing. clearance Hertz contact,1 Introduction, In the last two decades a lot of research efforts has been devoted to study the stability and non.
linear dynamic analysis of flexible rotor bearings Effectively one of the most important. mechanical elements to take into account is bearings due to their large influence on the dynamic. behavior of rotating machinery Bently et al 1 Ehrich 2 Harris 3 and Vance 4 Rolling. elements bearings fluid film bearings and gas bearings are the three major types bearings that are. currently used Gas bearings operate without noise and are not subjected to wear However self. excited vibrations may occur due to the loss of damping properties of the gas film So a complete. non linear analysis and stability of flexible rotors supported by gas journal bearings are essential to. estimate the range of their applications Wang et al 5 and Czo czy ski et al 6 Then fluid film. bearings are often used due to the important damping effect on rotors and a long life limit. with good lubrication and squeeze film dampers are widely utilized in aircraft turbine in order to. reduce amplitudes of the rotor while passing through critical speeds However the multiple solution. response of flexible rotor supported on fluid film bearing is a typical non linear phenomenon that. needs to be carefully undertaken in order to avoid worse design Inayat Hussain 7 and C S Zhu. Nowadays rolling element bearings is commonly used for aircraft engine and many types of. rotating machinery In contrast to fluid film bearings rolling element bearings allow to rotors to be. more stable and are convenient to use It is well known that this mechanical system may drastically. influence the dynamic behavior of rotating system So the vibration analysis of flexible rotor with. rolling element bearings is of great importance for the design of rotating systems Moreover one of. the most important problem is the presence of non linearities due for example to the internal radial. clearance the Hertzian ball race contact and the associated non linear restoring forces. Then there are many techniques that have been employed for designing rotating system and for. obtaining the dynamic responses of rotor with non linearities One of the methods available to. discretize the equations of motion of a elastic body is the finite element method Specifically in. rotordynamics one of the first works dealing with finite elements is the one of Nelson and. McVaugh 9 where rotatory inertia axial loads and gyroscopics moments are considered Later. Zorzi and Nelson 10 showed how to take into account the damping of the rotating parts Several. works followed these pioneer works Genta 11 Hashish and Sankar 12 Ku 13 zg ven and. zkan 14 showing the maturity and efficiency of the finite element method for rotor dynamics. Moreover due to the complexity of non linear systems and to save time there are many methods. that have been developed in order to simplify and to reduct in the non linear equations The most. popular methods for approximating the non linear responses of systems are the harmonic balance. methods where the non linear solution is assumed to be a truncated Fourier series harmonic. balance method These numerical methods are well known and have been commonly used to solved. non linear problems in the fields of mechanical engineering However various alternative. approaches may be used in order to obtain the non linear response and stability of the flexible. bearing rotor system We refer the interested reader to 15 16 for an extensive overview of. alternative approaches, So this paper will firstly presents the rolling bearing model with radial clearance and Hertz contact. Secondly the harmonic balance method with a condensation strategy will be developed in order to. obtain the periodic solution of the non linear bearing rotor Then the stability analysis is carried out. in the frequency domain using a method based on a perturbation in the time domain applied to the. known harmonic solution, Finally numerical tests and results of a rotor with one disk and two bearing elements will be. investigated The non linear response the stability analysis and the evolution of the non linear. behavior of the rolling elements will be undertaken. 2 The rolling bearing model, A numerical model of a rotor bearing system may have a very complex behavior when the rolling. bearing model tries to incorporate the non linearities found in such a component With these special. rolling bearing models non linear phenomena such as quasiperiodic motions chaotic motions. jumps and super harmonics have been observed Childs 17 Kim and Noah 18 Karpenko et al. 19 Tiwari et al 20 Harsha et al 21, In this work a rolling bearing model were the restoring forces are calculated based on the kinematics. of the rolling elements is adopted The rolling bearing is represented in figure where Ri is the. internal groove radius Ro is the external groove radius j is the angular position of the jth rolling. element and is the constant angular speed of the rotor The external ring of the bearing is fixed. to the stator and the internal ring is fixed to the rotor The rolling bearing has a radial clearance. and the dynamical effects on the rolling elements gyroscopic moments and centrifuge forces are. not considered The rolling elements are kept with constant angular spacing as a result of the use of. a retainer or cage The cage speed or the speed of rotation of the set of rolling elements around the. origin is Tiwari et al 20, This rolling bearing model can generate a parametric excitation due to the combination of a constant.
lateral force like the weight of the rotor and the variation of the angular position of the rolling. elements However this excitation is not considered here in that the main excitation of the rolling. bearing system An unbalance force which is more important than this parametric effect will be. considered The rolling bearing has N b rolling elements and the position of the j th one can be. calculated in function of the rotating speed as,j j 1 cage t j 1 N b 2. At the position j the relative displacement between the rotor and the stator is. j xr x s sin j z r z s cos j j 1 N b 3, where x r and y r are the coordinates of the rotor and x s and y s are the coordinates of the stator. If the relative displacement j is bigger than the radial clearance there appears some level of. compression at the rolling element As a result a local restoring force occurs Otherwise the. restoring force is null This can be stated as,Q j K C j j. The global restoring force exerted by the rolling bearing over the rotor is then. F X Q j sin j,F Z Q j cos j,3 Harmonic balance formulation and reduction. To obtain the periodic solutions of the non linear rotor system with the bearing element previously. described the harmonic balance method is one of the most interesting methods 22 24 The. principle idea of the method is to impose an harmonic solution with unknown coefficients and with. the same period of the excitation After inserting this solution into the equation of motion of the. system the resulting harmonic terms are balanced and the unknown coefficients can be found This. method can be used to find approximate analytical solutions for small number of degrees of freedom. dof systems like a mass spring damper system with a cubic stiffness non linearity However. when the system contains more complex non linearities or a significant number of dof the amount. of symbolic manipulations can be disencouraging To overcome this problem a special. implementation of the harmonic balance method is available the AFT alternating frequency time. technique 22 23 and 25 To present the method the following equation of motion is considered. M X D X K X F NL X X t 6, where M is the mass matrix D is the damping plus gyroscopic matrices K is the stiffness.
matrix X is the displacement vector F NL X X t is the vector containing all the efforts. acting on the system is the rotating frequency and t is the time The overdot means time. differentiation The number of degrees of freedom in this discretization is r For simplicity. F NL X X t will be written as F, Assuming that the harmonic excitation causes a harmonic response X t can be written as a. Fourier series up to the m th term,X t B 0 Bk cos k t Ak sin k t 7. The forcing term F can also be written as a Fourier series expansion. F C 0 C k cos k t S k sin k t 8, The Fourier series representation of the displacement 7 and forcing 8 are inserted into the. equation of motion 6 and the terms of same frequency are balanced For the constant terms the. balance leads to, For the i th sine term the result of the balance is. K i M A i D B S,Finally for the i th cosine term,i D Ai K i M B i C i.
Gathering together all the harmonics the following system of equations of order 2m 1 r is. This can be recognized as a nonlinear algebraic system of equations since the Fourier coefficients of. the forcing terms are implicit functions of the Fourier coefficients of the displacement In this sense. Equation 12 can be written as,H Z Z b Z 14, where the entities Z and b Z can be readily identified H Z defines the residue and. is imposed to zero To establish the implicit relation between forcing and displacement coefficient. that is the vector b Z the AFT strategy is employed This strategy starts with an estimation of. the Fourier coefficients used to build the Z vector This coefficients are used to synthesize the. displacement X t which in turn is used to evaluate the nonlinear forces in the time domain A. Fourier analysis of these forces gives the coefficients C 0 C i and S i. Ci F cos i t dt,Si F sin i t dt, These coefficients are used to build the vector b Z Knowing the procedure to construct the. vector b Z the response curves of the dynamical system are found by calculating the zeros of the. equation 14 given range of This can be accomplished with the aid of a nonlinear system of. equations solver like the Newton Raphson 26 or Broyden 27 methods Figure 2 illustrates the. computational method, When a given system as a significant number of degrees of freedom but only a few of then are. related to nonlinear efforts it is recommended to use some kind of condensation From now on. these degrees of freedom will be called nonlinear degrees of freedom in contrast with the linear. degrees of freedom which are not directly linked to the nonlinear efforts The main idea of the. condensation is to solve the algebraic nonlinear system of equations only for the nonlinear degrees. of freedom letting the others be determined later by a linear transformation In this way a. significantly smaller number of degrees of freedom means less computational effort The. condensation scheme used here is the one presented by Han and Chen 24 and it will be outlined in. the following paragraphs, The nonlinear degrees of freedom are stored at the end of the displacement vector If there are p. linear degrees of freedom and q nonlinear degrees of freedom with p q r the equation of. motion 6 is written as, Following the harmonic balance procedure the balance equation for the constants terms is.
From equation 19 it is possible to write B0q in function of B0q. K qq K qp K pp 1 K pq B 0q C 0q K qp K pp 1 C 0p 20. To proceed with the condensation for the sine and cosine terms it is needed to define the following. Qi K i M Ri i D, Then from the i th block of the equations 10 and 11 one can write. From equation 22 it is possible to write,T i Bi U i 23. T i Ai W i 24,T i Q i Ri Qi 1 Ri,U i C i Ri Q i 1 S i 25. W i S i Ri Q i C i, Using the same procedure applied to equation 19 one obtains. T iqq T iqp T ipp 1 T ipq Biq U iq T iqp T ipp 1 U ip 26. T iqq T iqp T ipp 1 T ipq Aiq W iq T iqp T ipp 1 W ip 27. Considering the m harmonics and the equations 20 26 and 27 it is possible to obtain. K qq K qp K pp 1 K pq B 0q 1,C 0q K qp K pp C 0p,T iqq T iqp T ipp 1 T ipq 0.
T iqq T iqp T ipp T ipq,W iq T iqp T ipp 1 W ip,U iq T iqp T ipp 1 U ip. Equation 28 is a non linear system of equations of order 2m 1 q When compared with the. 2m 1 r order of the system given by the equation 14 it becomes clear the performance. Stability and vibration analysis of a complex flexible rotor bearing system Cristiano Villa Jean Jacques Sinou Fabrice Thouverez To cite this version Cristiano Villa Jean Jacques Sinou Fabrice Thouverez Stability and vibration analysis of a com plex flexible rotor bearing system Communications in Nonlinear Science and Numerical Simulation Elsevier 2008 13 4 pp 804 821 10 1016

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