Non-collocated vibration absorption using delayed resonator for spectral and spacial tuning -- analysis and experimental validation

TL;DR

This paper addresses non-collocated vibration absorption (NCVA) using a delayed resonator (DR) with position feedback. It develops a scalable analytical framework that operates directly on system matrices to design the resonant substructure without deriving full transfer functions and derives a practical solution for setting the DR gain and delay by satisfying $g e^{-j \omega \tau} = p(j \omega)$. The approach ensures a pair of complex-conjugate zeros on the imaginary axis at $\pm j\omega$ to suppress vibration at the target, while maintaining overall stability via spectral-abscissa criteria $\alpha_{OS}(g,\tau) < 0$ and $\alpha_{RS}(g,\tau) = 0$ for marginal substructure stability. The method is validated experimentally on a three-cart mechatronic setup, demonstrating nearly ideal vibration suppression for both collocated and non-collocated targets and revealing the practical challenges of stability margins and hardware speed. The work advances NCVA by confirming spatial tunability and scalability, with implications for large-scale structures, and identifies future directions in robustness, broader frequency ranges, and higher-rate hardware.

Abstract

Non-collocated vibration absorption (NCVA) concept using delayed resonator for in-situ tuning is analyzed and experimentally validated. There are two critical contributions of this work. One is on the scalable analytical pathway for verifying the concept of resonant substructure as the basis of the ideal vibration absorption. The second is to experimentally validate the spatial and spectral tunability of NCVA structures for the first time. For both novelties arbitrarily large dimensions of interconnected mass-spring-damper chains are considered. Following the state of the art on NCVA, control synthesis is performed over the resonant substructure comprising the delayed resonator and a part of the primary structure involved in the vibration absorption. The experimental validation of the proposed NCVA concept is performed on a mechatronic setup with three interconnected cart-bodies. Based on the spectral analysis, an excitation frequency is selected for which a stable vibration suppression can be achieved sequentially for all the three bodies, one collocated and two non-collocated. The experimental results closely match the simulations for complete vibration suppression at the targeted bodies, and thus validating the crucial spatial tunability characteristic as well as the traditional spectral tuning.

Figures (9)

• Figure 1: A general scheme of multi-body structure consisting of a series of linearly linked masses $m_1,\ \ldots,\ m_d$, being excited by an $\omega$-harmonic force $f$, together with an active absorber $m_a$. The structure can be split into: i) resonant substructure tuned by inner feedback $u(t)$ to resonate at frequency $\omega$, ii) target mass $m_n$ to be silenced, which is non-collocated with the absorber deployment at $m_p$, and iii) vibrating substructure.
• Figure 2: Mechatronic setup designed for experimental validation of non-collocated vibration absorption accompanied with the control implementation scheme.
• Figure 3: Spectral abscissas of the resonant substructure given by \ref{['eq:alphaRS']} (RS -- dash-dot line) and of the overall setup given by \ref{['eq:alphaOS']} (OS -- solid line) for branches $k=0$ and $k=1$.
• Figure 4: Amplitude frequency response of $P(\jmath \omega)$ by \ref{['eq:TF']}. Without any control in the top, with controllers to stop individual carts at frequency $\omega=4.20 \: \mathrm{Hz}$ in the middle and at frequency $\omega=8.30 \: \mathrm{Hz}$ in the bottom.
• Figure 5: Experimental results of absorbing vibration excited by the disturbance force $f$ given by \ref{['eq:general:excitation']}, with $F=3N$ and $\omega=4.20 \: \mathrm{Hz}$, applying the DR position feedback \ref{['eq:general:cDR_position']} with parameters: i) \ref{['eq:set1']} active at $t\in[15, 30]\mathrm{s}$ to silence $x_1$ (collocated), ii) \ref{['eq:set2']} active at $t\in[40, 55]\mathrm{s}$ to silence $x_2$ (non-collocated), and iii) \ref{['eq:set3']} active at $t\in[65, 80]\mathrm{s}$ to silence $x_3$ (non-collocated).

Theorems & Definitions (2)

• Proposition 1
• proof