Numerical methods for parametric model reduction in the simulation of disk brake squeal

Gräbner, Nils; Mehrmann, Volker; Quraishi, Sarosh; Schröder, Christian; Wagner, Utz von

doi:10.1002/zamm.201500217

Cited by 43 publications

(64 citation statements)

References 37 publications

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“…Here the input matrix 2 ∈ ℝ ×5 from (1) is such that with all other entries being equal to zero. First, in Figure 2 we illustrate the behaviour of the optimal parameters and the magnitude of the -mixed 2 norm of the system defined by (14). In more details, for given damping positions , ∈ {1, 2, … , } with < , we have optimized the viscosities and the optimal -mixed 2 norm is plotted.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Mixed control of vibrational systems

Nakić¹,

Tomljanović²,

Truhar³

2019

Z Angew Math Mech

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We consider new performance measures for vibrational systems based on the H2 norm of linear time invariant systems. New measures will be used as an optimization criterion for the optimal damping of vibrational systems. We consider both theoretical and concrete cases in order to show how new measures stack up against the standard measures. The quality and advantages of new measures as well as the behaviour of optimal damping positions and corresponding damping viscosities are illustrated in numerical experiments.

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Section: Numerical Experimentsmentioning

confidence: 99%

Mixed control of vibrational systems

Nakić¹,

Tomljanović²,

Truhar³

2019

Z Angew Math Mech

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“…The transition from stability to instability of the brake system is generally examined by finite element (FE) analysis of the system. In [7] FE models resulting for disk brakes are derived in form of second order differential equations…”

Section: Introductionmentioning

confidence: 99%

“…Mẍ + D(Ω)ẋ + K(Ω)x = f, (1. 7) In the absence of the circulatory effects, i.e., when N = 0, the system in (1.6) is a DH system and as a result it is Lyapunov stable and typically even asymptotically stable. However, small circulatory effects, i.e., perturbations by a nonsymmetric N of small norm, may result in instability.…”

Section: Introductionmentioning

confidence: 99%

Approximation of stability radii for large-scale dissipative Hamiltonian systems

2020

Self Cite

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A linear time-invariant dissipative Hamiltonian (DH) systemẋ = (J − R)Qx, with a skew-Hermitian J, an Hermitian positive semi-definite R, and an Hermitian positive definite Q, is always Lyapunov stable and under weak further conditions even asymptotically stable. In various applications there is uncertainty on the system matrices J, R, Q, and it is desirable to know whether the system remains asymptotically stable uniformly against all possible uncertainties within a given perturbation set. Such robust stability considerations motivate the concept of stability radius for DH systems, i.e., what is the maximal perturbation permissible to the coefficients J, R, Q, while preserving the asymptotic stability. We consider two stability radii, the unstructured one where J, R, Q are subject to unstructured perturbation, and the structured one where the perturbations preserve the DH structure. We employ characterizations for these radii that have been derived recently in [SIAM J. Matrix Anal. Appl., 37, pp. 2016 ] and propose new algorithms to compute these stability radii for large scale problems by tailoring subspace frameworks that are interpolatory and guaranteed to converge at a super-linear rate in theory. At every iteration, they first solve a reduced problem and then expand the subspaces in order to attain certain Hermite interpolation properties between the full and reduced problems. The reduced problems are solved by means of the adaptations of existing level-set algorithms for H∞-norm computation in the unstructured case, while, for the structured radii, we benefit from algorithms that approximate the objective eigenvalue function with a piece-wise quadratic global underestimator. The performance of the new approaches is illustrated with several examples including a system that arises from a finite-element modeling of an industrial disk brake.

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“…Regarding parameter‐dependent systems, it is important to mention the family of parametric model reduction methods, see, eg, previous works. () Although these approaches show similar characteristics to the LPV model reduction, the problem they address is fundamentally different. The parametric model reduction starts from a parameterized set of large‐scale LTI systems.…”

Section: Introductionmentioning

confidence: 99%

“…The latter step is very difficult in general, 16 because the independently reduced (transformed and projected) local, LTI systems have to be transformed into a consistent state-space representation.Regarding parameter-dependent systems, it is important to mention the family of parametric model reduction methods, see, eg, previous works. [17][18][19][20] Although these approaches show similar characteristics to the LPV model reduction, the problem they address is fundamentally different. The parametric model reduction starts from a parameterized set of large-scale LTI systems.…”

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confidence: 99%

Model reduction for LPV systems based on approximate modal decomposition

Luspay

Péni

Gozse

et al. 2017

Numerical Meth Engineering

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The paper presents a novel model order reduction technique for large-scale linear parameter varying (LPV) systems. The approach is based on decoupling the original dynamics into smaller dimensional LPV subsystems that can be independently reduced by parameter varying reduction methods. The decomposition starts with the construction of a modal transformation that separates the modal subsystems. Hierarchical clustering is applied then to collect the dynamically similar modal subsystems into larger groups. The subsystems formed from the groups are then independently reduced. This approach substantially differs from most of the previously proposed LPV model reduction techniques, since it performs manipulations on the LPV model and not on a set of linear time-invariant (LTI) models defined at fixed scheduling parameter values. Therefore the model interpolation, which is the most challenging part of most reduction techniques, is avoided. The applicability of the developed algorithm is thoroughly investigated and demonstrated by numerical case studies.

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Numerical methods for parametric model reduction in the simulation of disk brake squeal

Cited by 43 publications

References 37 publications

Mixed control of vibrational systems

Mixed control of vibrational systems

Approximation of stability radii for large-scale dissipative Hamiltonian systems

Model reduction for LPV systems based on approximate modal decomposition

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