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1、 20The Fifth International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY Huazhong University of Science and Technology, Wuhan, P.R. CHINA. 27-28 , August, 2009 2009 Huazhong Universiti of Science and Technology Press A SURVEY ON FREQUENCY-DEPENDENT DISSIPATION OF SOFT MATTER VIBRAT
2、ION AND ACOUSTICS Wen Chen* Xiaodi Zhang a Institute of Soft Matter Mechanics, Department of Engineering Mechanics, Hohai University (*Email:) ABSTRACT The classical damping models of vibration and acoustic wave propagation in lossy medium are only applicable to the two particular cases of the frequ
3、ency- independent and frequency-squared dependent energy dissipation. However, many experimental and field measurements show that the energy dissipation in viscoelastic soft matters, also known as complex fluids, obeys the arbitrary power law frequency dependence. In recent decades, lots of efforts
4、are made on developing appropriate models to describe such “anomalous” attenuation, for instance, frequency domain methods, adaptive proportional damping model, multiple relaxation models, time fractional derivative model, Szabos time convolutional integral model, modified Szabos model, and fraction
5、al Laplacian wave model. All these models are phenomenological in nature and do not necessarily reflect the physics mechanism underlying arbitrary power law attenuation. In this survey paper, we will review the above-mentioned power law frequency-dependent attenuation models and compared their merit
6、s and demerits and consequently present the perspective in this area of research and engineering applications. KEYWORDS power law frequency-dependent dissipation, vibration, acoustics, soft material, attenuation, damping I. INTRODUCTION When acoustic wave travels through the lossy medium or damped v
7、ibration occurs, the output energy never equals the input energy, for there is always thermal consumption of energy 1. Many experimental and field measurements show that the energy dissipation of vibration and acoustic wave obeys an empirical power law of frequency-dependence. ( )()( )exS xxS x +=,
8、0= (1) where is the angular frequency, S the pressure, x the wave propagation distance, () the attenuation coefficient , 0 and can be arbitrary real non-negative constants and the value of ranges from 0 to 2 and is a characteristic parameter of material of interest. Water and many metals have =2, na
9、mely, frequency-squared dependency. However, it is widely noted that unlike ideal solids and Newtonian fluids, parameter of viscoelastic soft matters is often a real number in between 0 and 2 and their energy dissipation is thus also called “anomalous” attenuation 2-4. For example, of most biomateri
10、als and sediments is in between 1 and 1.7. Acoustic and vibration attenuation in viscoelastic soft 21matter play an important role in many engineering fields, e.g. mechanical engineering, civil engineering, aeronautical engineering etc. Thus, the research in this area has drawn great attention in re
11、cent decades. The classical time-domain models of damped vibration can describe only the two extreme cases of the frequency-independent and frequency-squared dependent attenuation. For instance, the classical Rayleigh proportional damping model 5, which is commonly used in structural vibration analy
12、sis, is a typical viscous damping model. Its damping coefficient is a weighted sum of stiffness and inertia. This implies that its energy dissipation depends on a weighted combination of the square-frequency dependence and frequency independence 5. If the damping coefficient is independent of stiffn
13、ess, the Rayleigh model degenerates to frequency-independent attenuation. On the other hand, without inertia term, the Raleigh model describes a square-frequency dependent dissipation process. It is found that the Rayleigh model can not accurately represent arbitrary frequency-dependent damping beha
14、viors of majority of real-world materials. In particular, when the excitation is from a broad band source, the accuracy of the Rayleigh model is in general not desirable 10. On the other hand, the classical viscous models of dissipative acoustic wave propagation are also confined to the frequency-in
15、dependent and frequency-squared dependent absorption. If the lossy medium is supposed to be viscous, the wave equation can be given by simply adding a linear viscous attenuation term as described below 6 20222 01cSSStct+= (2) where denotes the Laplacian, c0 represents the sound speed, the constant i
16、s the attenuation coefficient. In this case, the acoustic attenuation is frequency-independent. If viscous force is introduced into the constitutive relationship of pressure and density, we can get the classical thermoviscous wave equation 20222 0()1cSSStct+= (3) where is the collective thermoviscous coefficient. The thermoviscous wave equation governs the sound
