Integrated design of system structure and delayed resonator towards efficient non-collocated vibration absorption

TL;DR

This work tackles non-collocated vibration absorption by integrating structural design with a delayed resonator (DR) to ensure full absorption at the target location while controlling fatigue and energy use. It develops a phasor-based model of a serial mass-spring-damper chain, derives DR parameter relations via $Q(\omega)$, and formulates a constrained nonlinear optimization to minimize fatigue-related elastic energies $W_{i,\max}$ and DR power $P_{\max}$ under stability and physical constraints. The approach is validated experimentally on a 3-mass setup and numerically on a 5-mass system, showing that optimized structural parameters can substantially reduce fatigue risk and actuation energy without sacrificing absorption performance. The results demonstrate the practical viability of jointly optimizing structure and DR in non-collocated configurations, with implications for fatigue resistance and energy efficiency in engineering systems.

Abstract

The problem of non-collocated vibration absorption by a delayed resonator is addressed with emphasis on system fatigue resistance and energy efficiency of control actions. The analysis is performed for a system consisting of an arbitrary large series of flexibly linked single-degree-of-freedom masses. For the stage where the vibration of the target mass is fully absorbed by the non-collocated resonator, key forces, motion amplitudes and potential energies across the system structure are assessed. Next, a complete parameter set of the resonator gain and delay is derived, and the actuation force and power needed by the resonator for the full vibration absorption is determined. The derived quantities are utilized in forming an optimization problem to balance minimal risk of fatigue across the system structure and power needed by the resonator, under the closed loop stability and parameter constraints. Next to the gain and delay of the resonator, selected structural parameters of the system are used as variables in the constrained nonlinear optimization problem. Experimental and numerical case studies are included to demonstrate benefits of the proposed integrated structural and control design.

Figures (8)

• Figure 1: System configuration and its structural decomposition.
• Figure 2: Experimental setup with three flexibly linked carts actuated by the voice coils
• Figure 3: Left column --- objectives, maxima of potential elastic energies in links $W_{i, max}$ as a function of $m_3$ (top), maxima of power as a function of $m_3$ and $m_a$. Right column --- non linear inequalities as a function of $m_3$ and $m_a$, spectral abscissa (top) and maxima of potential elastic energy in absorber link (bottom). Red polygons define structural constraints (linear inequalities), small red dots represent considered grid with $0.025kg$ steps. Thick red line represents stability boundary, i.e. $\alpha(m_3, m_a) = 0$. Plane graphs contain marker for nominal (circle) and optimized (star) solutions.
• Figure 4: Left --- Measured transients of potential elastic energies in links $i=1, \dots, 4$ Right --- Measured transients (from top to bottom): position of target mass $m_2$, control force $u$, potential elastic energy in absorber link $W_a$, absolute value of power $|P$. For all plots: Nominal parameters (blue) and optimized parameters (green). Passive regime for $t\in[0, 2.5)\, \mathrm{s}$; active regime $t\in[2.5, 10]\, \mathrm{s}$.
• Figure 5: Spectrum distribution of the system with the DR, represented by DDAE \ref{['eq:stability:DDAE']} with both nominal parameters (blue, +), and optimized parameters (green, x). Left --- Large scale view; Right --- Rightmost, stability determining roots with visualized value of the spectral constraint $\xi_{\alpha}=-0.2$ (red, dashed).

Theorems & Definitions (9)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• Corollary 1
• Remark 1
• Proposition 3
• proof