An efficient algorithm for damper optimization for linear vibrating systems using Lyapunov equation

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. sent an algorithm which derives a formula for the trace of the solution X of the Lyapunov equation ATX+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. timal 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–Stewartʹs Lyapunov solver.