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Experimental modal analysis using undamped control for high damping system

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Abstract

This study proposes a methodology to identify the modal damping ratio of a highly damped structure. Specifically, we explain the change in the modal damping ratio of the object itself by using an existing novel multiple excitation testing method with velocity feedback (FB) control to counteract the damping force, and we propose a methodology to identify the original modal damping ratio of the object based on the methodology of modal analysis. In this methodology, the relation equation between the modal damping ratio and control gain is derived. In the numerical and experimental validations, the modal damping ratio is identified by applying velocity FB excitation to a multi-degree-of-freedom system, and it is confirmed that the original modal damping ratio of the target structure can be identified from the frequency response function after damping reduction. Therefore, the proposed method is expected to improve experimental modal analysis by facilitating the accurate identification of the modal damping ratios of the vibration mode wherein the resonance peak does not appear clearly.

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Abbreviations

\(m\) :

Mass

\(c\) :

Viscous damping coefficient

\(k\) :

Spring constant

\(x\) :

Displacement

\(\dot{x}\) :

Velocity

\(\ddot{x}\) :

Acceleration

\(f\) :

External force

\(F\) :

Magnitude of external force

\(\omega\) :

Angular frequency

\(t\) :

Time

\({\omega }_{n}\) :

Natural angular frequency

\(\zeta\) :

Damping ratio

\(\Delta \zeta\) :

Damping reduction parameter

\({\zeta }^{^{\prime}}\) :

Damping ratio after damping reduction

\(u\) :

Velocity feedback excitation force

\(d\) :

Velocity feedback control gain

\(N\) :

Number of degrees of freedom

\(n\) :

Degree of freedom and mode order

\({u}_{n}\) :

Velocity feedback excitation force in n degrees of freedom

\({d}_{n}\) :

Velocity feedback control gain in n degrees of freedom

\(\mathbf{M}\) :

Mass matrix

\(\mathbf{C}\) :

Damping matrix (tridiagonal matrix)

\(\mathbf{K}\) :

Stiffness matrix

\(\mathbf{f}\) :

External force vector

\(\mathbf{x}\) :

Displacement vector

\(\dot{\mathbf{x}}\) :

Velocity vector

\(\ddot{\mathbf{x}}\) :

Acceleration vector

\({\upvarphi}_{n}\) :

n-Th natural vibration mode vector (\(n\)-th mode)

\({\varvec{\Phi}}\) :

Modal matrix

\(\mathbf{I}\) :

Unit matrix

\({\zeta }_{n}\) :

n-Th modal damping ratio (\(n\)-th mode)

\({\zeta }_{n}\mathrm{^{\prime}}\) :

n-Th modal damping ratio after damping reduction

\({\mathbf{C}}_{\mathrm{m}}\) :

Modal damping matrix (diagonal matrix with \({2{\zeta }_{n}\omega }_{n}\))

\({{\varvec{\Omega}}}^{2}\) :

Diagonal matrix with \({\omega }_{n}^{2}\)

\(\mathbf{D}\) :

Diagonal matrix with velocity feedback control gain

\({\mathbf{D}}_{\mathrm{m}}\) :

Modal control gain matrix

\({D}_{nn}\) :

n-Th diagonal component of modal control gain matrix

\(\mathbf{q}\) :

Modal displacement vector

\(\dot{\mathbf{q}}\) :

Modal velocity vector

\(\ddot{\mathbf{q}}\) :

Modal acceleration vector

\(\lambda\) :

Eigenvalue

\(\mathbf{Q}\) :

Eigenvector

\(\mathrm{T}\) :

Symbol for transpose

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Acknowledgements

The authors would like to thank Hiroki Nakao, a student at The University of Shiga Prefecture, for assisting with the experiments performed in this study.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku-Cho, Toyohashi-City, Aichi, 441-8580, Japan

    Daiki Tajiri, Masami Matsubara & Shozo Kawamura

  2. Department of Mechanical Systems Engineering, The University of Shiga Prefecture, Hassaka-Cho, Hikone-City, Shiga, 2500522-8533, Japan

    Takashi Tanaka

  3. Graduate School, Department of Mechanical Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku-Cho, Toyohashi-City, Aichi, 441-8580, Japan

    Taiki Goto

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  1. Daiki Tajiri
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Correspondence to Daiki Tajiri.

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Tajiri, D., Tanaka, T., Matsubara, M. et al. Experimental modal analysis using undamped control for high damping system. Arch Appl Mech 93, 2947–2964 (2023). https://doi.org/10.1007/s00419-023-02419-y

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