Fast computation of optimal damping parameters for linear vibrational systems (original) (raw)
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Estimation of optimal damping for mechanical vibrating systems
This paper is concerned with the efficient algorithm for dampers' and viscosity optimization in mechanical systems. Our algorithm optimize simultaneously the dampers' positions and their viscosities. For the criterion for optimization we use minimization of the average total energy of the system which can be done by the minimization of the trace of the solution of the corresponding Lyapunov equation. Efficiency of the algorithm is obtained by new heuristics for finding the optimal dampers' positions and for the approximation of the trace of the solution of the Lyapunov equation.
Optimizing a damped system – a case study
International Journal of Computer Mathematics, 2011
We consider a second order damped-vibrational system described by the equation Mẍ + C(v)ẋ + Kx = 0, where M, C(v), K are real, symmetric matrices of order n. We assume that the undamped eigenfrequencies (eigenvalues of (λ 2 M + K)x = 0) ω 1 , ω 2 ,. .. , ω n , are multiple in the sense that ω 1 = ω 2 , ω 3 = ω 4 ,. .. , ω n−1 = ω n , or are given in close pairs ω 1 ≈ ω 2 , ω 3 ≈ ω 4 ,. .. , ω n−1 ≈ ω n. We present a formula which gives the solution of the corresponding phase space Lyapunov equation, which then allows us to calculate the first and second derivatives of the trace of the solution, with no extra cost. This one can serve for the efficient trace minimization.
On linear vibrational systems with one dimensional damping II
Integral Equations and Operator Theory, 1990
We are interested in the quadratic eigenvalue problem of damped oscillations where the damping matrix has dimension one. This describes systems with one point damper. A generic example is a linear n-mass oscillator fixed on one end and damped on the other end. We prove that in this case the system parameters (mass and spring constants) are uniquely (up to a multiplicative constant) determined by any given set of the eigenvalues in the left half plane. We also design an effective construction of the system parameters from the spectral data. We next propose an efficient method for solving the Ljapunov equation generated by arbitrary stiffness and mass matrices and a one dimensional damping matrix. The method is particularly efficient if the Ljapunov equation has to be solved many times where only the damping dyadic is varied. In particular, the method finds an optimal position of a damper in some 60n 3 operations. We apply this method to our generic example and show, at least numerically, that the damping is optimal (in the sense that the solution of a corresponding Ljapunov equation has a minimal trace) if all eigenvalues are brought together. We include some perturbation results concerning the damping factor as the varying parameter. The results are hoped to be of some help in studying damping matrices of the rank much smaller than the dimension of the problem.
An Efficient Approximation for Optimal Damping in Mechanical Systems
International Journal of Numerical Analysis and Modeling, 2017
This paper is concerned with an efficient algorithm for damping optimization in mechanical systems with a prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent to the minimization of the trace of the corresponding Lyapunov equation. Thus, the prescribed structure in our case means that a mechanical system is close to a modally damped system. Although our approach is very efficient (as expected) for mechanical systems close to modally damped system, our experiments show that for some cases when systems are not modally damped, the proposed approach provides efficient approximation of optimal damping.
Derivative of eigensolutions of nonviscously damped linear systems
AIAA Journal, 2002
Derivatives of eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters are presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy any orthogonality relationship. It is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems. Numerical examples are provided to illustrate the derived results.
Optimal design of viscoelastic dampers using eigenvalue assignment
Earthquake Engineering & Structural Dynamics, 2004
In this study a procedure for determining the optimum size and location of viscoelastic dampers is proposed using the eigenvalue assignment technique. Natural frequencies and modal damping ratios, required to realize a given target response, are determined ÿrst by the convex model. Then the desired dynamic structural properties are realized by optimally distributing the damping and sti ness coecients of viscoelastic dampers using non-linear programming based on the gradient of eigenvalues. This optimization method provides information on the optimal location as well as the magnitude of the damper parameters. The proposed procedure is applied to the retroÿt of a 10-story shear frame, and to a three-dimensional structure with an asymmetric plan. The analysis results conÿrm that the responses of model structures retroÿtted by the proposed method correspond well with the given target response.
Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2022
This article proposes an approach to resolve the partial eigenvalue assignment output feedback control problem for second-order damped vibration systems. The matrices are prescribed in advance and highly depend on controllability conditions, and the system's observability eigenvalues are assigned. In addition, the real value spectral decomposition P (λ) is exploited to establish conditions so that the feedback gain matrices do not spill over eigenvalue assignment. Thus, the numerical method presented applies to the active vibration control design of multiple inputs and outputs of functional engineering structures. However, it should be noted that the proposed algorithm can present complex computational problems if the mass matrix of the system is almost singular, as it involves the inverse calculation of the mass matrix. Two theorems were presented using Sylvester equations. Three algorithms were implemented based on the Sylvester equation, and examples were presented with their conclusions.
Iterative methods for non-classically damped dynamic systems
Earthquake Engineering & Structural Dynamics, 1994
Non-classically damped structural systems do not easily lend themselves to the modal superposition method because these systems yield coupled second-order differential equations. In this paper, a variety of new computationally efficient iterative methods for determining the response of such systems are developed. The iterative approaches presented here differ from those presented earlier in that they are computationally superior and/or are applicable to the determination of the responses of broader classes of structural systems. Numerical examples, which are designed to evaluate the efficacy of these schemes, show the vastly improved rates of convergence when compared to earlier iterative schemes.