Pseudo Spectral Methods Applied to Problems in Elasticity (original) (raw)
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After decades of investigating brake noise using advanced tools and methods, brake squeal remains a major problem of the automotive industry. The Finite Element Analysis (FEA) method has long been used as a means of reliable simulation of brake noise, mainly using the Complex Eigenvalue Analysis (CEA) to predict the occurrence of instabilities resulting in brake noise. However it has been shown that CEA often over-predicts instabilities. A major improvement for CEA proposed in this study is tuning the model with an accurate level of damping. Different sources of damping are investigated and the system components are tuned using Rayleigh damping method. Also, an effective representative model for the brake insulator is proposed. The FEA model of the brake system tuned with the damping characteristics highlights the actual unstable frequencies by eliminating the over-predictions. This study also investigates effectiveness of a hybrid Implicit-Explicit FEA method which combines frequency domain and time domain solution schemes. The time/frequency domain co-simulation analysis presents time-domain analysis results more efficiently. Frictional forces are known as a major contributing factor in brake noise generation. A new brake pad design is proposed, addressing the frictional forces at the disc-pad contact interface. This concept is based on the hypothesis that variation of frictional coefficient over the radius of the brake pad is effective in reducing the susceptibility of brake squeal.
Acta Polytechnica, 2020
This paper deals with the evaluation of eigenvalues of a linear damped elastic two-degrees-of-freedom system under a non- onservative loading. As a physical interpretation of a proposed mathematical model, a simplified disk brake model is considered. A spectral analysis is performed to predict an eigenvalues bifurcation, known as the Krein collision, leading to double eigenvalues, one of them having a positive real part causing a vibration instability of the mechanical systems. This defective behaviour of eigenvalues is studied with respect to a magnitude of non-conservative Coulomb friction force, through the variation of the friction coefficient. The influence of a proportional versus general damping on the system stability is further analysed. The generalized non-symmetric eigenvalue problem calculation is employed for spectral analyses, while a modal decomposition is performed to obtain a time-domain response of the system. The analyses are compared with an experiment.
Multi Scale Modelling of Friction Induced Vibrations at the Example of a Disc Brake System
Applied Mechanics
Friction induced vibrations such as brake squealing, or juddering are still challenging topics in product engineering processes. So far, this topic was particularly relevant for the automobile industry because they were the main market for disc brake systems. However, since mobility habits change, disc brake system are more often to be found on bikes or e-scooters. In all of these systems, vibrations are excited in contacts on the micro scale but affect the user comfort and safety on the macro scale. Therefore, the aim of this cross-scale method is to analyze a system on a micro scale and to transfer the excitation mechanisms on a macro scale system. To address both scales, the current work presents a finite element model on the micro scale for the determination of the coefficient of friction, which is transferred to the macro scale and used in a multi-body simulation. Finally, a finite element modal analysis is conducted, which allowed us to evaluate the brake system behavior on ba...
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In this paper we analyze a mixed displacement-pseudostress formulation for the elasticity eigenvalue problem. We propose a finite element method to approximate the pseudostress tensor with Raviart-Thomas elements and the displacement with piecewise polynomials. With the aid of the classic theory for compact operators, we prove that our method is convergent and does not introduce spurious modes. Also, we obtain error estimates for the proposed method. Finally, we report some numerical tests supporting the theoretical results.
IMA Journal of Numerical Analysis, 2018
We present a priori and a posteriori error analyses of a virtual element method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyse a variational formulation relying only on the solid displacement and propose an H1(Omega)H^{1}(\Omega )H1(Omega)-conforming discretization by means of the VEM. Under standard assumptions on the computational domain, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal–order error estimate for the eigenfunctions and a double order for the eigenvalues. Since the VEM has the advantage of using general polygonal meshes, which allows efficient implementation of mesh refinement strategies, we also introduce a residual-type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests that allow us to assess the performance of this approach.
Shock and Vibration
During the past decades, the problem of friction-induced vibration and noise has been the subject of a huge amount of works. Various numerical simulations with finite elements models have been largely investigated to predict squeal events. Although a nonlinear analysis is more predictive than Complex Eigenvalues Analysis, one of the main drawbacks of the time analysis is the need of large computational efforts. In view of the complexity of the subject, this approach appears still computationally too expensive to be used in industry for finite element models. In this study, the potential of a new reduced model based on a double modal synthesis (i.e., a classical modal reduction via Craig and Bampton plus a condensation at the frictional interface based on complex modes) for the prediction of self-excited vibrations of brake squeal is discussed. The effectiveness of the proposed modal reduction is tested on a finite element model of a simplified brake system. It will be shown that num...