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# Analytical Solutions for DVA Optimization Based on the Lyapunov Equation

[+] Author and Article Information
Dong Du1

251 HuaNing Road, MinHang District, Shanghai City, People's Republic of Chinawideface@sjtu.edu.cn, doodon@163.com

1

Corresponding author.

J. Vib. Acoust 130(5), 054501 (Aug 12, 2008) (10 pages) doi:10.1115/1.2948373 History: Received June 27, 2006; Revised April 13, 2008; Published August 12, 2008

## Abstract

A novel method is proposed to obtain the optimum configuration of dynamic vibration absorber (i.e., DVA) attached to an undamped or damped primary structure. The performance index, i.e., the quadratic integration, includes two types of controls, i.e., velocity and displacement controls of the primary mass. Based on the Lyapunov equation, the performance indices are simplified into matrix quadratic forms. With the help of the Kronecker product and matrix column expansion, the closed-form solutions of optimum parameters for undamped primary structure are finally presented. Moreover, in some cases, the method can produce perturbation solutions in simple forms for damped primary structure. Especially, from these solutions, the classical optimum $H2$ designs under external force or base acceleration excitation can be derived out.

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## Figures

Figure 4

Type (II) under ρ=−1: (a) the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; (b) the velocity transient response of the primary structure under the corresponding initial state for various mass ratio μ

Figure 1

The analytical models of the DVA-structure system (a) under external force excitation; (b) under base acceleration excitation; (c) with nonzero initial velocities, and (d) with nonzero initial displacements

Figure 2

Type (II) under ρ=5: (a) the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; (b) the velocity transient response of the primary structure under the corresponding initial state for various mass ratio μ

Figure 3

Type (II) under ρ=0: (a) the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; (b) the velocity transient response of the primary structure under the corresponding initial state for various mass ratio μ

Figure 8

Type (I) under (a) ρ=3, (b) ρ=0, and (c) ρ=−2: the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; the displacement transient response of the primary structure under the corresponding initial state for various mass ratio μ

Figure 9

Type (IV) under (a) η=3, (b) η=0, and (c) η=−2: the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; the velocity transient response of the primary structure under the corresponding initial state for various mass ratio μ

Figure 5

Type (III) under η=5: (a) the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; (b) the displacement transient response of the primary structure under the corresponding initial state for various mass ratio μ

Figure 6

Type (III) under η=0: (a) the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; (b) the displacement transient response of the primary structure under the corresponding initial state for various mass ratio μ

Figure 7

Type (III) under η=−1: (a) the tuning ratio and damping ratio of DVA versus the primary structure’s damping ratio; (b) the displacement transient response of the primary structure under the corresponding initial state for various mass ratio μ

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