XXI ICTAM, 15–21 August 2004, Warsaw, Poland NONLINEAR VIBRATIONS OF GEAR DRIVES Vladimír Zeman * , Miroslav Byrtus * , Michal Hajžman * * Department of Mechanics, Faculty of Applied Sciences, University of West Bohemia in Pilsen, Univerzitní 8, CZ-30614 Plze ˇ n, Czech Republic Summary The contribution presents the modal synthesis method of the mathematical modelling of the gear drive nonlinear vibrations caused by internal excitation generated in gear meshings. Especially undesirable vibrations characterized by discontinuity of mesh gear can be caused by kinematic transmission errors and time dependent meshing stiffnesses in case of small static load. These gear drive impact motions are explained by direct time–integration method and using time series, phase trajectories and Poincaré map. INTRODUCTION The presented original modal synthesis method is based on the system decomposition into subsystems, modelling of linearized uncoupled subsystems by FEM, discretization of linear or nonlinear couplings between subsystems, modelling of gyroscopic effects of the rotating subsystems and a assembling of the condensed mathematical model of the system. The condensed mathematical model of the complex coupled system is created by means of spectral m Λ j and modal m V j submatrices corresponding to the lower master mode shapes of the mutually uncoupled and undamped subsystems (see [1], [2]). The methodology described in this contribution allows to model very complex systems with complicated structure and nonlinear couplings. MODELLING OF GEAR DRIVES Let the gear drive (system) be decomposed into rotating shafts with gears (subsystems j =1,...,N − 1) and a housing (subsystem j = N ). The shafts are joined together by gear couplings z =1, 2,...,Z . The shafts are joined with housing by means the rolling–element bearings. The motion equation of the system, which is decomposed into N subsystems, in the space of their generalized coordinates q j (t) can be written in the matrix form M j ¨ q j (t)+(B j + ω j G j )˙ q j (t)+ K j q j (t)= f E j (t)+ f C j , j =1, 2,...,N, (1) where the mass, damping and stiffness matrices M j , B j , K j of the mutually uncoupled subsystems are symmetric and gyroscopic matrix G j of the subsystem rotating by angular velocity ω j (ω N =0) is skew–symmetric. The vector f E j (t) describes external forced or kinematic excitation. The interaction between the subsystems in the configuration space q(t)=[q j (t)] of dimension n = ∑ n j can be expressed by global coupling force vector f C =[f C j ] in the form f C (t, q, ˙ q)= −B B ˙ q(t) − K B q(t)+ Z  z=1 c z F z (t, q, ˙ q) , (2) where B B and K B are damping and stiffness matrices of linearized bearing couplings. Vectors c z (see [1]) and forces F z (t, q, ˙ q) transmitted by gearings correspond to gear coupling z. The elastic parts of these forces are expressed as nonlinear functions of gearing deformations d z = −c T z q(t)+Δ z (t) , (3) where Δ z (t) is the kinematic transmission error on the gear mesh line. In order to make a condensed mathematical model of the system the forces F z (t, q, ˙ q) will be written in the form F z (t, q, ˙ q)= k z (t)d z + b z ˙ d z + f z (d z ,t) , z =1, 2,...,Z, (4) where k z (t) are time dependent meshing stiffnesses, b z are viscous meshing damping parameters and nonlinear functions f z (d z ,t) express the influence of an interruption of the mesh gear. CONDENSED MODEL The model (1) can be transformed by means of modal submatrices m V j ∈ R nj ,mj of the uncoupled undamped subsystems into the condensed form using relations q j (t)= m V j x j (t) for j =1, 2,...,N and m j <n j ¨ x(t)+(D + ω 0 G + V T B B V + V T B G V )˙ x(t)+(Λ + V T K B V + V T K G (t)V )x(t)= = V T [f E (t)+ f G (t)+ Z ∑ z=1 c z f z (d z ,t)] . (5) Mechanics of 21st Century - ICTAM04 Proceedings