Suppression of bending waves in a beam using a tuned vibration absorber

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Abstract

This paper is concerned with the control of flexural waves in a beam using a tuned vibration absorber (TVA). The TVA may be located in the farfield or the nearfield of a time harmonic point force and it can be positioned to control both the downstream-transmitted power and that reflected upstream. Analytical and numerical investigations are presented. The effects of the incident evanescent wave on the optimal characteristics of the absorber are discussed when the TVA is located in the nearfield of the point force.

If the TVA is located in the nearfield, the power transmitted past the TVA depends on four independent tuning parameters: the absorber frequency, the mass ratio, the structural damping of the TVA and the distance between the TVA and the point force. If the incident nearfield wave is insignificant, then this distance becomes unimportant in determining the optimal characteristics of the TVA. The net power propagating upstream is due to the superposition of the wave reflected from the TVA and that input by the point force and depends on all four parameters.

Attention is focussed on finding the frequencies at which the minimum power is transmitted or the maximum power absorbed by the TVA. Experimental results are presented to validate the theoretical predictions.

Introduction

A wave propagates unchanged along a uniform beam unless it is incident on a discontinuity, where part of it is reflected and part is transmitted. This paper is concerned with how the transmission of flexural waves in a beam can be controlled using a single tuned vibration absorber (TVA) located in either the nearfield or the farfield of a point disturbance.

The use of a TVA to control a flexural wave on an infinite Euler–Bernoulli beam has been discussed previously. Mead [1] described how tuning the resonance frequency of an undamped absorber to the excitation frequency could pin the beam at the excitation position. Complete suppression of a flexural wave can be achieved by attaching a single undamped TVA and has been discussed by Clark [2] and Brennan [3] who modelled the TVA as point translational impedance in the farfield of a disturbance. More attention has recently been paid to adaptive-passive control in which the passive properties of the TVA can be adjusted to be optimal under changing conditions. Simplicity in design and a lower cost than active control are advantages of such an approach [4], [5], [6], [7].

The purpose of this paper is to expand on previous work, which only assumed the farfield condition, by discussing the behaviour of the TVA located in either the nearfield or farfield of a harmonic point force. New tuning parameters are determined and expressions derived for the tuned frequency. The cases of optimal energy absorption by the device or minimisation of the transmission of a propagating wave are considered. The presence of an incident evanescent wave, together with the incident propagating wave, affects the optimal characteristics of the TVA. The net upstream propagating wave is also considered. This is given by the superposition of the wave reflected by the TVA and the upstream wave generated by the point force, and depends on the location of the TVA together with its parameters.

The paper is set out as follows. Section 2 concerns the dynamic behaviour of the TVA and the way in which it affects wave transmission. The reflection and transmission matrices are found. The power transmitted downstream of the TVA is found in terms of four independent parameters: the ratio of the tuned frequency to the TVA frequency, the loss factor of the TVA, the mass of the TVA and the distance between the TVA and the source of disturbance. The maximum power that can be absorbed by the TVA is investigated as well as the optimum tuning parameters of the absorber. Section 3 is devoted to numerical examples, investigating the performance of the TVA and the optimum tuning parameters of the absorber for both nearfield and farfield cases. The TVA can be optimised either for minimum transmission or maximum energy absorption. Experimental validation of the theoretical predictions is reported in Section 4. Finally, conclusions are presented.

Section snippets

Dynamic behaviour of a TVA on a beam

The aim of this section is to investigate the reflection and transmission of waves at a TVA on a beam when the TVA is in either the nearfield or the farfield of a point force disturbance.

Numerical examples

The dependence of the power transmitted and absorbed on the TVA parameters is illustrated numerically in this section.

Experimental work

In this section, experimental measurements are compared with the theoretical and numerical predictions of 2 Dynamic behaviour of a TVA on a beam, 3 Numerical examples.

The absorber was made of a steel beam (1.7 mm×20.5 mm×80.4 mm) with blocks of brass (10.3 mm×10.2 mm×20 mm) attached at each end. The absorber was attached at its centre to a 6.4 mm×50.6 mm×5630 mm straight steel beam suspended at four points along its length. The ends of the beam were embedded in sand boxes to reduce reflections. The

Conclusions

This paper has presented theoretical and experimental investigations of the behaviour and the optimum tuning parameters of a TVA, which suppresses flexural waves in thin beams. The location of the TVA with respect to a point disturbance has been taken into account.

The reflection and transmission ratios for the TVA were derived and depend on four independent tuning parameters: the absorber frequency ωa, the mass ratio γ, the damping η, and the non-dimensional distance l/λa between the TVA and

Acknowledgements

The authors would like to acknowledge the financial support provided by the Engineering and Physical Sciences Research Council (EPSRC).

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