We consider a mechanical system excited by a periodic external force. The main problem is to determine the best damping matrix to be able to minimize the system average displacement amplitude. ...Damping optimization usually includes optimization of damping positions and corresponding damping viscosities. Since the objective function is non-convex, a standard optimization approach requires a large number of objective function evaluations. We first propose a dimension reduction approach that calculates approximation of the average displacement amplitude and additionally we efficiently use a low rank update structure that appears in the external damping matrix. Moreover, an error bound which allows determination of appropriate approximation orders is derived and incorporated within the optimization method. We also present a theoretical error bound that allows determination of effective damping positions. The methodology proposed here provides a significant acceleration of the optimization process. The gain in efficiency is illustrated in numerical experiments.
The problem of finding a globally optimal
k
-partition of a set
A
is a very intricate optimization problem for which in general, except in the case of one-dimensional data, i.e., for data with one ...feature (
A
⊂
R
), there is no method to solve. Only in the one-dimensional case, there are efficient methods based on the fact that the search for a globally optimal
k
-partition is equivalent to solving a global optimization problem for a symmetric Lipschitz-continuous function using the global optimization algorithm DIRECT. In the present paper, we propose a method for finding a globally optimal
k
-partition in the general case (
A
⊂
R
n
,
n
≥
1
), generalizing an idea for solving the Lipschitz global optimization for symmetric functions. To do this, we propose a method that combines a global optimization algorithm with linear constraints and the
k
-means algorithm. The first of these two algorithms is used only to find a good initial approximation for the
k
-means algorithm. The method was tested on a number of artificial datasets and on several examples from the UCI Machine Learning Repository, and an application in spectral clustering for linearly non-separable datasets is also demonstrated. Our proposed method proved to be very efficient.
This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem (λ2(v)M+λ(v)D(v)+K)x(v)=0, ...where M, K are positive definite Hermitian n × n matrices and D(v) is an n × n Hermitian semidefinite matrix which depends on a damping parameter vector v=v1…vk∈R+k. With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters vi, which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments.
We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output ...of the system. We use a minimization criteria based on the
ℋ
2
system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates ‘interpolatory’ reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the
ℋ
2
system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.
We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for low-rank variation in the state matrix. Usual approaches for ...parametric model reduction typically involve exploring the parameter space to identify representative parameter values and the associated models become the principal focus of model reduction methodology. These models are then combined in various ways in order to interpolate the response. The initial exploration of the parameter space can be a forbiddingly expensive task. A different approach is proposed here that requires neither parameter sampling nor parameter space exploration. Instead, we represent the system response function as a composition of four subsystem response functions that are
non-parametric
with a purely parameter-dependent function. One may apply any one of a number of standard (non-parametric) model reduction strategies to reduce the subsystems independently, and then conjoin these reduced models with the underlying parameterization to obtain the overall parameterized response. Our approach has elements in common with the parameter mapping approach of Baur et al. (PAMM
14
(1), 19–22
2014
) but offers greater flexibility and potentially greater control over accuracy. In particular, a data-driven variation of our approach is described that exercises this flexibility through the use of limited frequency-sampling of the underlying non-parametric models. The parametric structure of our system representation allows for a priori guarantees of system stability in the resulting reduced models across the full range of parameter values. Incorporation of system theoretic error bounds allows us to determine appropriate approximation orders for the non-parametric systems sufficient to yield uniformly high accuracy across the parameter range. We illustrate our approach on a class of structural damping optimization problems and on a benchmark model of thermal conduction in a semiconductor chip. The parametric structure of our reduced system representation lends itself very well to the development of optimization strategies making use of efficient cost function surrogates. We discuss this in some detail for damping parameter and location optimization for vibrating structures.
The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems (λ2M+λD+K)x=0. We propose a new concept without ...linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference |∥X2*MX˜1∥F2−∥X2*MX1∥F2|, and for ∥X2*MX˜1−X2*MX1∥F, where the columns of X1=x1,…,xk and X2=xk+1,…,xn are linearly independent right eigenvectors and M is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for ∥sinΘ(X1,X˜1)∥F, where Θ is a matrix of canonical angles between the eigensubspaces X1 and X˜1,X1 is spanned by the set of linearly independent right eigenvectors of SDQEP and X˜1 is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.
Mixed control of vibrational systems Nakić, Ivica; Tomljanović, Zoran; Truhar, Ninoslav
Zeitschrift für angewandte Mathematik und Mechanik,
September 2019, 2019-09-00, 20190901, Letnik:
99, Številka:
9
Journal Article
Recenzirano
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.
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.
In this paper, we consider damping optimization in mechanical system excited by an external force. We use optimization criteria based on minimizing average energy amplitude and average displacement ...amplitude over the arbitrary time. As the main result we derive explicit formulas for objective functions. These formulas can be implemented efficiently and accelerate optimization process significantly, which is illustrated in a numerical example.
We consider a mechanical system excited by external force. Model of such a system is described by the system of ordinary differential equations: Mx¨(t)+Dẋ(t)+Kx(t)=fˆ(t), where matrices M,K (mass ...and stiffness) are positive definite and the vector fˆ corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. We introduce two criteria that allow damping optimization of mechanical system excited by an external force. Since in general a damping optimization is a very demanding problem, we provide a new formulas which have been used for efficient damping optimization. The efficiency of new formulas is illustrated with a numerical experiment.
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