Numerical Modelling of the Vibro-acoustic Behaviour of a Vehicle Gearbox

Abstract

Nomenclature loaded distance between the inner and outer raceway A j groove curvature centre, mm radial component of , mm Ar j A j axial component of , mm Aa j A j B total curvature factor, dimensionless Bd unloaded distance between the inner and outer raceway groove curvature centre, mm c speed of sound, m/s C damping matrix, Ns/m d rolling element diameter, mm F load vector, N force on the spring which represents the gear meshF m stiffness, N mass moment of inertia with respect to direction i, kg m 2 Im i k wave number, m -1 gear mesh stiffness, N/m k g stiffness constant elements of the bearing stiffness k ij matrix, N/m K stiffness matrix, N/m load-deflection constant, N/m K cd gear mesh coupling stiffness matrix, N/m K e bearing stiffness matrix, N/m K m finite element matrix, N/m n K Mi gear mass, kg m i M mass matrix, kg resultant moment in the i direction, Nm M i p pressure, N/m 2 q displacement vector, m velocity vector, m/s . q acceleration vector, m/s 2 .. q r curvature radius of the internal ring (for ball roller) or primitive radius (for cylindrical roller), m curvature radius of the external ring of the ball bearing, m r e curvature radius of the internal ring of the ball bearing, m r i radial gap, m r L base circle radius of the i-th gear, m R bi driven gear pitch radius, m R bc driver gear pitch radius, m R bp resultant force in the i direction, N R i X main axis in the x-direction Y main axis in the y-direction Z main axis in the z-direction, or number of rolling elements Greek Symbols angular displacement in the i-direction, degrees i contact angle without load, degrees 0 resultant elastic deformation of the j-th ball or roller j element, mm axial displacement of the j-th roller element, mm a j translational displacement in the i-direction, mm i resultant elastic deformation of the j-th ball element, mm E radial displacement of the j-th roller element, mm r j resultant elastic deformation of the j-th roller ele-R ment, mmIn this article, a global vibro-acoustic method to model gearboxes, which is based on the finite element and boundary element methods, is presented. The final aim of the method is to investigate the vibration and noise transmitted to the gearbox structure casing, which originate from the excitation caused by the gear train, in order to predict the vibro-acoustic parameters. Thus, the mathematical formulae that allow the determination of generalised stiffness matrices are presented in terms of the bearing and gear elements. A numerical model of the geared axle system that allows the estimation of the bearing reactions due to the gear forces transmitted is developed. This model takes into account the influence of modifying the gears teeth profile. The finite elements and boundary meshes were devised and generated in order to represent the gearbox. These meshes were used for the estimation of the acoustic parameters and for vibro-acoustic predictions. † Member of the International Institute of Acoustics and Vibration (IIAV) (pp 61-72)

Criteria for Adequacy Estimation of Mathematical Descriptions of Physical Processes

“…It is note that authors of works on mathematical modeling concern seldom questions of adequacy of the constructed mathematical description of process to real measurements [2][3][4][5]. Sometimes, such adequacy is proved by the real facts; sometimes, authors refer to results of other authors; and sometimes, there have not been any arguments.…”

“…In the given work, the mathematical models of physical processes described only by the system of the ordinary differential equations will be examined [2,3]. Such idealization of real processes or dynamic systems is widely used in various areas for the description of control systems [11], as well as of mechanical systems with the concentrated parameters [5,12], economic processes [13], biological processes [14], ecological processes [15], etc.…”