engleski

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

We consider a second-order damped-vibration equation M 1x + D() ̇x + Kx = 0, where M ; D() ; K are real, symmetric matrices of order n. The damping matrix D() is de7ned by D() = Cu + C(), where Cu presents internal damping and rank(C()) = r, where  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  → Tr(ZX()), where A = A() is a 2n × 2n matrix (obtained from M , D() ; K) such that the eigenvalue problem Ay = y is equivalent with the quadratic eigenvalue problem ( 2M + D()+K)x =0 (B and Z are suitably chosen positive-semide7nite matrices). Moreover, our algorithm provides the 7rst and the second derivative of the function  → Tr(ZX()) almost for free. The optimal dampers’ viscosity is derived as opt = argmin Tr(ZX()). If r is small, our algorithm allows a sensibly more e$cient optimization, than standard methods based on the Bartels–Stewart’s Lyapunov solver

Damped vibration ; Lyapunov equation ; Optimization of dampers’ viscosities

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