Optimization of damping positions in a mechanical system

Kanno, Yoshihibo; Puvača, Matea; Tomljanović, Zoran; Truhar, Ninoslav

doi:10.21857/y26kec33q9

Cited by 4 publications

(7 citation statements)

References 28 publications

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“…We consider the mechanical system shown in Figure 1. Similar examples were considered in [3,5,8,9,13,15]. In all our examples the mass oscillator contains two rows of d masses that are grounded from one side, while on the other side masses are connected to one mass which is then grounded.…”

Section: Examplesmentioning

confidence: 90%

“…This criterion has many benefits and it was investigated widely in the last two decades. More details can be found, e.g., in [3,5,8,9,[13][14][15]. Moreover, this criterion can be extended to the case where we consider Multiple-Input Multiple-Output systems that appear in the control theory in many applications, e.g., in paper [16] authors consider mixed control performance measure that includes also the total average energy into account.…”

Section: Introductionmentioning

confidence: 99%

“…However, since the optimization of damping positions is a very hard problem, in this work we will concentrate only on the optimization of viscosities. This can be then applied for different damping positions or it can be used within the algorithm that optimizes damping positions, but we also emphasize that such algorithms are typically heuristic algorithms, see, e.g., [15]. Moreover, the objective function (which is in our case the total average energy) that needs to be minimized is a non-convex function that includes viscosity parameters and damping positions.…”

Section: Introductionmentioning

confidence: 99%

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Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

2022

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We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O(kn2) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.

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Section: Examplesmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

2022

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“…In damping optimization where 𝐶(𝑣) is present, avoiding undesirable frequency bands can be achieved either by choosing damping positions (by optimizing matrix 𝐺) or by viscosity values (by optimizing 𝑣 ∈ ℝ 𝑟 + ) or doing both simultaneously. Computing the optimal damping positions is a very challenging problem and there is no efficient algorithm for it, though some heuristics can be found in, for example, Kanno et al [18]. One approach to determining optimal damping positions is "direct" brute force, where all possible damping configurations are considered and viscosities are optimized for each configuration.…”

Section: Introductionmentioning

confidence: 99%

Fast optimization of viscosities for frequency‐weighted damping of second‐order systems

et al. 2022

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We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques.First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain the stability of the system. These frameworks also allow us to weight, which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems (QEPs) that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one (DPR1) complex symmetric matrices, which we leverage by showing how each QEP can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new approach are verified and illustrated on several numerical examples.

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“…In damping optimization where C(v) is present, avoiding undesirable frequency bands can be achieved by either choosing damping positions (by optimizing matrix G) or by damping viscosities (by optimizing v ∈ R r + ) or doing both simultaneously. Computing the optimal damping positions is a very challenging problem and there is no efficient algorithm for it, though some heuristics can be found in, e.g., [22]. One approach to determining optimal damping positions is "direct" brute force, where all possible damping configurations are considered and viscosities are optimized for each configuration.…”

Section: Introductionmentioning

confidence: 99%

Fast optimization of viscosities for frequency-weighted damping of second-order systems

Stor¹,

Mitchell²,

Tomljanović³

et al. 2021

Preprint

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We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain stability of the system. These frameworks also allow us to weight which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one complex symmetric matrices, which we leverage by showing how each quadratic eigenvalue problem can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new methods are verified and illustrated on several numerical examples.

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Optimization of damping positions in a mechanical system

Abstract: This paper deals with damping optimization of the mechanical system based on the minimization of the so-called "average displacement amplitude". Further, we propose three different approaches to solving this minimization problems, and present their performance on two examples.

Cited by 4 publications

References 28 publications

Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

Fast optimization of viscosities for frequency‐weighted damping of second‐order systems

Fast optimization of viscosities for frequency-weighted damping of second-order systems

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