Optimizing a damped system – a case study

Abstract

We consider a second-order damped-vibrational system described by the equation M ẍ+C(v) [xdot]+K x=0, where M, C(v), K are real, symmetric matrices of order n. We assume that the undamped eigenfrequencies (eigenvalues of (λ² M+K) x=0) ω₁, ω₂, …, ω _n , are multiple in the sense that ω₁=ω₂, ω₃=ω₄, …, ω _n−1=ω _n , or are given in close pairs ω₁ ≈ ω₂, ω₃ ≈ ω₄, …, ω _n−1 ≈ ω _n . We present a formula which gives the solution of the corresponding phase space Lyapunov equation, which then allows us to calculate the first and second derivatives of the trace of the solution, with no extra cost. It can serve for the efficient trace minimization.

Keywords:

• Lyapunov equation
• trace minimization
• dampers’ viscosity optimization
• multiple eigenfrequencies
• second-order vibrational system

2000 AMS Subject Classifications :

• 15A24
• 34D20
• 74G65
• 74G05
• 49L99