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

Ninoslav Truhar, Krešimir Veselić

TL;DR

The work addresses damping optimization in discrete mechanical systems resembling rod/string vibrations by formulating two initial‑data‑independent criteria: total average energy and total average displacement. It shows that both criteria are equivalent to trace minimization of Lyapunov equations with different right‑hand sides, enabling efficient computation of optimal damper parameters. For a single damper in a discrete mass–spring chain, the optimal damper position depends only on the number of dominant eigenfrequencies $s$ and is independent of the ambient dimension $n$, with the energy criterion yielding linear scaling in $n$ and the displacement criterion yielding cubic scaling in $n$ for the optimal trace. Numerical experiments validate the theoretical results across large $n$ and various spectral configurations, illustrating how optimal damping placement shifts between center and edges depending on whether the focus is on calming low‑ versus high‑frequency components. Overall, the paper provides a tractable, Lyapunov‑based framework for damping optimization with clear asymptotic behavior and practical guidance for selecting damper position and viscosity from spectral considerations.

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.

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

TL;DR

The work addresses damping optimization in discrete mechanical systems resembling rod/string vibrations by formulating two initial‑data‑independent criteria: total average energy and total average displacement. It shows that both criteria are equivalent to trace minimization of Lyapunov equations with different right‑hand sides, enabling efficient computation of optimal damper parameters. For a single damper in a discrete mass–spring chain, the optimal damper position depends only on the number of dominant eigenfrequencies and is independent of the ambient dimension , with the energy criterion yielding linear scaling in and the displacement criterion yielding cubic scaling in for the optimal trace. Numerical experiments validate the theoretical results across large and various spectral configurations, illustrating how optimal damping placement shifts between center and edges depending on whether the focus is on calming low‑ versus high‑frequency components. Overall, the paper provides a tractable, Lyapunov‑based framework for damping optimization with clear asymptotic behavior and practical guidance for selecting damper position and viscosity from spectral considerations.

Abstract

This paper investigates two optimization criteria for damping optimization in a multi-body oscillator system with arbitrary degrees of freedom (), 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 . Consequently, the optimal damping position is determined solely by the number of dominant eigenfrequencies and the optimal viscosity, independent of the dimension , offering efficient damping optimization in discrete systems. The paper concludes with numerical examples illustrating the presented theoretical framework and results.

Paper Structure

This paper contains 11 sections, 3 theorems, 167 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Optimization criteria
  3. Optimization criterion -- average total energy
  4. Optimization criterion -- average total displacements
  5. Vibrations of rod/string and discrete mass--spring model
  6. Optimal dampers' position -- discrete mass spring mechanical system
  7. Optimal position of one damper for the discrete mass--spring model
  8. Optimal trace for the average total energy
  9. Optimal trace for the total average displacement
  10. Numerical illustration of the main results
  11. Conclusion and outlook

Key Result

Proposition 1

The matrix $\hat{X}$ is symmetric positive definite.

Figures (4)

  • Figure 1: The $n$-mass vibration chain with one damper
  • Figure 2: The optimal trace as a function of positions; damping of the whole spectrum.
  • Figure 3: The optimal trace as a function of positions; damping of the part of the spectrum.
  • Figure 4: The optimal trace as a function of positions; damping of the part of the spectrum.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2