An approach for large-scale gyroscopic eigenvalue problems with application to high-frequency response of rolling tires

Abstract

The transient dynamic response of rolling tires is of essential importance for comfort questions, e.g. noise radiation. Whereas finite element models are well established for stationary rolling simulations, it lacks computational methods for the treatment of the high frequency response. One challenge is the large mode density of tire structures that is up to the acoustic frequency domain and another lies on the physically correct description of rolling (gyroscopic) structures. Despite that the eigenvalue analysis of gyroscopic systems, described by complex-valued quadratic eigenvalue systems, seems to be well understood in general, specific problems arise for the computability of large scale three-dimensional tire models. In this presentation an overall computational strategy for the high frequency response of FE-tire models is outlined, where special emphasis is placed upon the efficient numerical treatment of the complex-valued eigenproblems for large scale gyroscopic systems. The practicability of the proposed approach will be demonstrated by the analysis of detailed finite element tire models. The physical interpretation of the computational results is also discussed in detail.Keywords Finite element method · Arbitrary Lagrangian Eulerian · Gyroscopic effects · Large scale quadratic eigenvalue problem · Rolling noise · Tire models M. Brinkmeier (B) · U. Nackenhorst

Tyre rolling kinematics and prediction of tyre forces and moments: part I – theory and method

“…This is the formulation with respect to a typical Downloaded by [Umeå University Library] at 16: 10 19 November 2014 test rig situation; while in the case of vehicle dynamics, v 0 must be replaced by v 0 z and v 0 y to cover the influence of the lateral rim centre point velocity. Similar research on kinematics using a reference configuration can be found in [20][21][22][23][24][25]. Large differences in previous studies are a result of the treatment of the coupled set of equations, as described in the following sections.…”

“…3. In the case of a rigid body (∂u 3 /∂χ 3 = 0, ∂u 3 /∂χ 2 = 0, and χ 2 = −R), the right-hand side of Equation (21) appears to be the longitudinal slip of the rigid body (v 0 − R )/V T . Furthermore, multiplication with a longitudinal contact stiffness and integration over the contact area lead to a result like those from simple brush models.…”

Semi-definite programming techniques for structured quadratic inverse eigenvalue problems

“…Numerical techniques for forward gyroscopic eigensystems have been discussed in [10, 22, 36], but the attention mostly is on the damping free case, i.e., C = G is skew-symmetric. A hybrid optimization method employing genetic algorithms and simulated annealing to identify bearing parameters of rotating machinery from bearing forces [4] is somewhat close to an inverse problem, but we have found no discussion on the gyroscopic inverse eigenvalue problem.…”

On the dynamics of rotating and rolling structures

“…The physical origins of these effects have been discussed in the previous sections on rather simple For the analysis of large-scale 3D structures by finite element methods great progress has been made recently in the context of rolling tire dynamics [7] and with emphasis on rolling noise simulations [5,6]. In those applications additional effects such as large elastic deformations and nonlinear as well as inelastic material properties have to be tackled, for which an arbitrary Lagrangian Eulerian (ALE) framework has been introduced [12].…”