Abstract
Nonlinear torsional models are used to analyze automotive transmission rattle problems and find solutions to reduce noise, vibration and dynamic loads. The torsional stiffness and inertial distribution of such systems show that the underlying mathematical problem is numerically stiff. In addition, the clearance nonlinearities in the gear meshes introduce discontinuous functions. Both factors affect the efficacy of time domain integration and smoothening functions are widely used to overcome computational difficulties and improve the simulation. In this paper, alternate smoothening functions are studied for their influence on the numerical solutions and their impact on global convergence and computation times. In particular, four smoothening functions (arctan, hyperbolic-cosine, hyperbolic-tan and quintic-spline) are applied to a five-degree-of-freedom generic torsional system with two backlash (clearance) elements. Each function is assessed via a global convergence metric across an excitation map (a design of experiment). Regions of the excitation map, along with multiple solutions, are studied and the implications to assessing convergence are critically examined. It is observed that smoothening functions do not lead to better convergence in many cases. The smoothening parameter needs to be carefully selected, or over-smoothened solutions may be found. The system studied is representative of a typical automotive rattle problem and it was found that benefits were limited from applying such smoothening functions.
Highlights
The vibro-impacts in the automotive driveline affect the perceived quality of a vehicle when a rattling noise is audible within the passenger compartment
In [7], arctan and quintic spline smoothening functions were applied to a generic automotive vibro-impact system and assessed for their effectiveness
A thorough examination of smoothening functions has been conducted for a generic automotive rattle problem
Summary
The vibro-impacts in the automotive driveline affect the perceived quality of a vehicle when a rattling noise is audible within the passenger compartment. When using ordinary differential equation solvers, such as Runge-Kutta routines, systems with discontinuous stiffness are typically handled two ways They either are set to solve explicitly until a discontinuity and stop with a restart after the parameter is set on the other side of the boundary, or alternatively integrate continuously across the boundary. It is common to “smoothen” the discontinuities with a continuously differentiable function [6], or “smoothening function” which approximates the absolute function and renders the discontinuity less severe This is expected to speed up the solution but the effect of such smoothening on the dynamic response or on the convergence of solutions has rarely been reported in the literature. The paper suggests a methodology that can be applied by researchers to carefully assess their formulation before proceeding with numerical studies of vibro-impact problems
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