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

We consider a second-order damped-vibration equation Mx¨+D(e)x˙+Kx=0, where M, D(e), K are real, symmetric matrices of order n. The damping matrix D(e) is defined by D(e)=C u +C(e), where C u presents internal damping and rank(C(e)) = r, where e is dampers' viscosity. We present an algorithm which derives a formula for the trace of the solution X of the Lyapunov equation ATX + XA = -B, as a function e → Tr(ZX(e)), where A = A(e) is a 2n × 2n matrix (obtained from M, D(e),K) such that the eigenvalue problem Ay = λy is equivalent with the quadratic eigenvalue problem (λ2M+λD(e)+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 e → Tr(ZX(e)) almost for free. The optimal dampers' viscosity is derived as e opt = argmin Tr(ZX(e)). If r is small, our algorithm allows a sensibly more efficient optimization, than standard methods based on the Bartels-Stewart's Lyapunov solver.