Numerical Resolution Of The ContactVibrations Under Harmonic Loads

Abstract

Nonlinear vibrations, induced between two elastic bodies in Hertzian contact, are studied. The purpose of this paper is to present a numerical resolution with the implicit temporal integration Newmark method including the use of the Newton Raphson algorithm to deal with the non-linear stiffness. The nonlinear character of the frequency response, as shown by precedent authors, is described numerically. The normal contact deflection under harmonic loads are smaller than in static conditions. Loss of contact may occur for a couple of materials with high elastic modulus. Applications to high precision engineering are enhanced. INTRODUCTION Elastic bodies in Hertzian contact vibrate as rigid masses on the nonlinear contact spring (1) where k is a constant dependent upon the local radii of curvature and the elastic constants of the matting surfaces. The resolution of this kind of problem has been undertaken previously by Nayak [1] with reasonable analytical approximations. Gray and Johnson [2] formulated models and measurements of the rolling contact vibration. Arnov, D'souza, KalpaKjian and Shareef [3] have conducted pin-on-disk experiments to measure contact noise induced parameters. Finally, in a recent paper Hess and Soom [4] proposed a perturbation technique known as the method of multiple scales to compute the contact vibrations under harmonic loads. SYSTEM MODEL Figure 1 shows two noise sources encountered in high precision mechanics : roller guideways (a) and ball bearings (b). The dynamic model studied by Hess & al. (c) [4]. A body of mass M is in contact with a flat surface. The contact is represented by a nonlinear stiffness and a linear viscous damper. Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 62 Contact Mechanics a Rolling guideway b Roller bearing c System model Figure 1 The contact is loaded by the weight, Mg, and an external load P = Po ( 1+ Y cos Qt ) with constant and simple harmonic components. The normal displacement of the mass, y, is refered to its static equilibrium position. The governing equation of motion during contact when summing the forces exerted on the mass is M y + C y k (yo y) 3/2 = -PO (i + y cos £2t) Mg , (2) where yo is the static contact compression given by yo = ((Po + Mg) / k) 2/3 . The stiffness constant k has been calculated from Hertz formula for a spheric surface in contact with a flat plate (see eq.7). To be solved, equation (2) may be written as M z + C z + K(z) z = PO (l + 7cos Qt) + Mg , (3) z = yo y > 0 , to prevent the loss of contact ; where