Damping optimization of discrete mechanical systems -- rod/string model

TL;DR

This paper develops a theoretical framework for damping optimization in multi-body oscillator systems, showing how to minimize energy and displacement through trace minimization of Lyapunov solutions, with explicit formulas for optimal damping.

Contribution

It introduces a novel approach linking damping optimization to trace minimization of Lyapunov solutions and provides explicit formulas for optimal damping parameters in discrete systems.

Findings

Optimal damping position depends on eigenfrequencies, not system dimension

Minimal trace can be expressed as linear or cubic functions of system size

Numerical examples validate the theoretical results

Abstract

This paper investigates two optimization criteria for damping optimization in a multi-body oscillator system with arbitrary degrees of freedom ($n$), resembling string/rod free vibrations. The total average energy over all possible initial data and the total average displacement over all possible initial data. Our first result shows that both criteria are equivalent to the trace minimization of the solution of the Lyapunov equation with different right-hand sides. As the second result, we prove that in the case of damping with one damper, for the discrete system, the minimal trace for each criterion can be expressed as a linear or cubic function of the dimension $n$. Consequently, the optimal damping position is determined solely by the number of dominant eigenfrequencies and the optimal viscosity, independent of the dimension $n$, offering efficient damping optimization in discrete systems. The paper concludes with numerical examples illustrating the presented theoretical framework and results.