We consider a second-order damped-vibration equation Mẍ+D(ε)ẋ+Kx=0, where M, D(ε), K are real, symmetric matrices of order n. The damping matrix D(ε) is defined by D(ε)=Cu+C(ε), where Cu presents internal damping and rank(C(ε))=r, where ε is dampers' viscosity. We present an algorithm which derives a formula for the trace of the Solution X of the Lyapunov equation AT X+XA=-B, as a function ε→Tr(ZX(ε)), where A=A(ε) is a 2n×2n matrix (obtained from M, D(ε),K) such that the eigenvalue problem Ay=λy is equivalent with the quadratic eigenvalue problem (λ2M+λD(ε)+K)x=0 (B and Z are suitably chosen positive-semidefinite matrices). Moreover, our algorithm provides the first and the second derivative of the function ε→Tr(ZX(ε)) almost for free. The optimal dampers' viscosity is derived as εopt=argmin Tr(ZX(ε)). If r is small, our algorithm allows a sensibly more efficient optimization, than standard methods based on the Bartels-Stewart's Lyapunov solver. © 2004 Elsevier B.V. All rights reserved.
Truhar, N. (2004). An efficient algorithm for damper optimization for linear vibrating systems using Lyapunov equation. Journal of Computational and Applied Mathematics, 172(1), 169–182. https://doi.org/10.1016/j.cam.2004.02.005