by DJ Thompson · 2007 · Cited by 7 — 2.2 Vibration absorbers. Mass-spring or mass-spring-damper systems are widely used to control the response of resonant structures, see Mead (2000), Nashif et al
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UNIVERSITY OF SOUTHAMPTON INSTITUTE OF SOUND AND VIBRATION RESEARCH DYNAMICS GROUP The theory of a continuous damped vibration absorber to reduce broad-band wave propagation in beams by D.J. Thompson ISVR Technical Memorandum No: 968 January 2007 Authorised for issue by Professor M.J. Brennan Group Chairman © Institute of Sound & Vibration Research
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Acknowledgements The analysis described here was carried out in part during a period of sabbatical leave in September 2005. The author is grateful to Trinity College, Cambridge for providing a visiting scholarship, to the Engineering Department of the University of Cambridge who hosted him during this period and to Dr Hugh Hunt in particular. ii
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CONTENTS Page Acknowledgements ii Contents iii Abstract v List of symbols vi 1. Introduction 1 2. Background 2 2.1 Techniques for reduction of vibration 2 2.2 Vibration absorbers 2 2.3 Application to waves in beams 4 2.4 Distributed absorbers 5 2.5 Broadband absorbers 6 3. Beam on elastic foundation 7 3.1 Undamped case 7 3.2 Effect of damping 8 4. Beam with attached continuous absorber 10 4.1 Frequency-dependent stiffness 10 4.2 Undamped absorber 12 4.3 Damped absorber 14 4.4 Motion of absorber mass 16 5. Approximate formulae 17 5.1 Approximate formulae for the decay rate far from the tuning frequency 17 5.2 Approximate formulae for the decay rate in the blocked zone 18 5.3 Bandwidth of absorber 19 6. Multiple tuning frequencies 22 7. Absorber applied to beam on elastic foundation 24 8. Two-layer foundation viewed as an absorber 28 9. Results for other wave types 32 9.1 Non-dispersive waves 32 9.2 Timoshenko beam 34 10. Practical application to railway track 36 iii
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List of symbols A Cross-sectional area of beam E Young™s modulus I Second moment of area of beam L Perimeter length of the beam cross-section S Surface area of beam Wrad Radiated sound power c0 Speed of sound in air k Wavenumber in the beam (real part) k0 Wavenumber of the unsupported beam at 0ka Wavenumber of the unsupported beam at akb Bending wavenumber in the unsupported beam kl Longitudinal wavenumber in the unsupported beam ‘ Mass per unit length of absorber am’bm Mass per unit length of beam ‘ Mass per unit length of intermediate mass in two layer support sms Stiffness of foundation per unit length s1 Stiffness of upper foundation layer per unit length s2 Stiffness of lower foundation layer per unit length sa Stiffness of absorber per unit length u Longitudinal displacement of beam v Vibration velocity of beam w Bending displacement of beam x Distance along beam Decay rate of wave in beam (dB/m) Wavenumber in the beam (imaginary part) Frequency bandwidth of absorber Increment of frequency Damping loss factor of foundation 1 Damping loss factor of upper foundation layer 2 Damping loss factor of lower foundation layer vi
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a Damping loss factor of absorber b Damping loss factor of beam b,eq Equivalent damping loss factor of beam due to absorber Ratio of absorber mass to beam mass 0 Density of air Radiation ratio of beam Angular frequency 0 Cut-off frequency of beam on elastic foundation a Tuning frequency of absorber b Upper frequency of absorber stop band c Mid frequency of absorber stop band Damping ratio vii
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1. Introduction Structural wave propagation in beam structures can lead to unwanted noise transmission and radiation. The particular application providing the motivation for the present work is a railway track (Thompson et al. 2003, Jones et al. 2006) but many other examples exist, such as piping systems for fluids or gases (see e.g. Clark 1995, de Jong 1994), or beam-like components which are present in structures such as bridges, cranes and buildings. Such beam systems are often very long and may be characterised in the audio frequency range in terms of propagating waves rather than modal behaviour. Whereas geometrical attenuation plays a significant role in two- and three-dimensional structures, in a one-dimensional structure there is no attenuation with distance apart from the effect of damping or discontinuities. Thus, in lightly damped uniform beams, structural waves may propagate over large distances and noise may be transmitted far from its source, to be radiated as sound by the beam itself or by some receiver structure. To reduce the total noise radiated by a vibrating beam, the spatial attenuation must be increased. The use of a damped mass-spring absorber system applied continuously on a beam is studied here. The purpose of introducing such a system is to attenuate structural waves over a broad frequency range, and for arbitrarily located excitation. A specific application of such a system to a railway track is discussed by Thompson et al. (2007) where, by sufficiently increasing the attenuation of vibration along the rail in a broad frequency band, the radiated noise from the track has been reduced by around 6 dB. The focus in this report is on determining the effects of the various parameters controlling the behaviour of a continuous absorber attached to a beam and deriving simple formulae for this behaviour. After a discussion of the background to the problem, a simple model of a beam on an elastic foundation is first considered. The decay rates of waves in the beam and the effects of the support are illustrated. Using this as a basis, the analysis is extended to an unsupported beam to which a continuous absorber is attached, the absorber being treated as a frequency-dependent complex support stiffness. Approximate formulae are then derived for the effects of the absorber, illustrating simply the influence of mass and damping. The use of multiple tuning frequencies is also considered in order to widen the bandwidth of the absorber. It is then shown that the damping effect of an absorber system attached to a supported beam can be approximated by adding the separate spatial attenuations from the supported beam and the 1
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