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. We present an algorithm which derives a formula for the trace of the solution X of the Lyapunov equation AX + 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 (λM + 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.