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An analytical particle damping model

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   JOURNAL OFSOUND AND VIBRATION www.elsevier.com/locate/jsvi Journal of Sound and Vibration 264 (2003) 1155–1166 An analytical particle damping model Steven E. Olson* University of Dayton Research Institute, 300 College Park, Dayton, OH 45469, USA Received 28 November 2001; accepted 20 August 2002 Abstract Particle damping is a passive vibration control technique where multiple auxiliary masses are placed in acavity attached to a vibrating structure. The behavior of the particle damper is highly non-linear and energydissipation, or damping, is derived from a combination of loss mechanisms. These loss mechanisms involvecomplex physical processes and cannot be analyzed reliably using current models. As a result, previousparticle damper designs have been based on trial-and-error experimentation. This paper presents amathematical model that allows particle damper designs to be evaluated analytically. The model utilizes theparticle dynamics method and captures the complex physics involved in particle damping, includingfrictional contact interactions and energy dissipation due to viscoelasticity of the particle material. Modelpredictions are shown to compare well with test data. r 2003 Elsevier Science Ltd. All rights reserved. 1. Introduction Active and passive damping techniques are common methods of attenuating the resonantvibrations excited in a structure. Active damping techniques are not applicable under allcircumstances due, for example, to power requirements, cost, environment, etc. Under suchcircumstances, passive damping techniques are a viable alternative. Various forms of passivedamping exist, including viscous damping, viscoelastic damping, friction damping, and impactdamping. Viscous and viscoelastic damping usually have a relatively strong dependence ontemperature. Friction dampers, while applicable over wide temperature ranges, may degrade withwear. Due to these limitations, attention has been focused on impact dampers, particularly forapplication in cryogenic environments or at elevated temperatures.Particle damping technology is a derivative of impact damping with several advantages. Theliterature typically distinguishes particle damping from impact damping based on the number and ARTICLE IN PRESS *Tel.: +937-229-3022; fax: +937-229-4251. E-mail address:  olsonse@udri.udayton.edu (S.E. Olson).0022-460X/03/$-see front matter r 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0022-460X(02)01388-3  sizes of the auxiliary masses (or particles) in a cavity. As shown in the idealized single-degree-of-freedom system in Fig. 1, impact damping usually refers to only a single (somewhat larger)auxiliary mass in a cavity, whereas particle damping is used to imply multiple auxiliary masses of small size in a cavity.Particle dampers significantly reduce the noise and impact forces generated by an impactdamper and are less sensitive to changes in the cavity dimensions or excitation amplitude. Studiesconducted over recent years have demonstrated the effectiveness and potential application of particle dampers to reduce vibration in a space shuttle main engine liquid oxygen inlet tee [1] andto attenuate the resonant vibrations of antennae [2]. In recent years, a tennis racquetincorporating particle damping has even been introduced [3]. The behavior of particle dampersis highly non-linear with energy dissipation, or damping, derived from a combination of lossmechanisms. These loss mechanisms involve complex physical processes and cannot be analyzedreliably using current models. As a result, previous particle damper designs have been based ontrial-and-error experimentation. A mathematical model has been developed which enables particledamper designs to be evaluated analytically. The model utilizes the particle dynamics method andcaptures the complex physics involved in particle damping. 2. Model development The granular material in particle dampers is unique in that the material can display behaviorsimilar to a solid, liquid, or gas, depending on the amount of energy contained in the material.Various techniques of modelling granular materials, including the particle dynamics method, havebeen proposed. The particle dynamics method is a method similar to that used to study moleculardynamics, where individual particles are modelled and their motions tracked in time. Theprocedure is an explicit process with sufficiently small time steps taken such that during a singletime step, disturbances cannot propagate from any particle further than its immediate neighbors.As a result, at any given time, the resultant forces on any particle are determined exclusively by itsinteraction with the particles with which it is in contact. This feature makes it possible to followthe non-linear interaction of a large number of particles without excessive memory or the need foran iterative procedure.The utility of the particle dynamics method is based on the ability to simulate contactinteractions using a small number of parameters that capture the most important contact ARTICLE IN PRESS Fig. 1. Idealized single-degree-of-freedom system with (a) impact damper and (b) particle damper. S.E. Olson / Journal of Sound and Vibration 264 (2003) 1155–1166  1156  properties. Interaction forces between the individual particles and the cavity walls are calculatedbased on force–displacement relations. Thus, one of the critical aspects for developing an accuratemathematical model is the selection of appropriate force–displacement relations to account forthe forces created due to particle–particle and particle–cavity impacts.In the mathematical model, it is assumed that the particle dampers consist of spherical particlesof a single material. Consider a typical impact of two spherical particles,  i   and  j  ;  with radii  R i   and R  j  ;  with the particle centers separated by a distance,  d  ij  ;  as shown in Fig. 2. These two particlesinteract if their approach,  a ;  is positive. The approach can be defined as a  ¼ ð R i   þ  R  j  Þ   d  ij  :  ð 1 Þ In this case, the colliding spheres are subject to the contact force: ~ F F   ¼  F  N    ~ nn N  þ  F  S    ~ nn S  ;  ð 2 Þ where  F  N  and  F  S  are the normal and shear forces and  ~ nn N  and  ~ nn S  are the unit vectors in thenormal and shear directions, respectively, for a given sphere. The opposing sphere experiencesequal forces in the opposite direction.For purely elastic contacts, expressions for the normal force can be found from Hertz’s theoryof elastic contact [4,5]. For the case of two contacting spheres with identical properties, a circularcontact area with radius,  a ;  results. Hertz’s expression for the normal force becomes F  N  ¼  23 RE  ð 1    u 2 Þ a 3 ;  ð 3 Þ ARTICLE IN PRESS Fig. 2. Typical particle–particle impact parameters S.E. Olson / Journal of Sound and Vibration 264 (2003) 1155–1166   1157  where R  ¼ ð R i  R  j  Þð R i   þ  R  j  Þ ð 4 Þ and  E   and  u  are the elastic modulus and the Poisson ratio of the spheres, respectively. Theapproach and contact circle radius are related as a  ¼  a 2 R :  ð 5 Þ Hertz’s expression is for two contacting spheres, but also holds for two impacting spheresprovided that the duration of the collision is long compared with the first fundamental mode of vibration in the spheres.Typically, particle–particle impacts are not purely elastic and energy is dissipated during theimpact event. For accurate damping predictions, it is important to incorporate this dissipationinto the model. The enduring types of contacts which occur in the particle damper preclude theuse of a coefficient of restitution or similar parameter. However, coefficient of restitution studieshave demonstrated that energy is dissipated due to the viscoelastic behavior of the sphere material[6]. A three-parameter, generalized Maxwell model is used to represent the viscoelastic material,such that the relaxation function,  C ð t Þ ;  takes the form C ð t Þ ¼  E  0  þ  E  1 e  t = t 1 :  ð 6 Þ Earlier work [6] indicates that, for hard metals and plastics, the dissipation due to the deviatoricand dilatational strains are of similar magnitudes. As a result, the relaxation function is notbroken into separate deviatoric and dilatational components or, equivalently, it is assumed thatthe Poisson ratio remains constant.Ideally, the dissipative portion of the normal force would be expressed in terms of a viscoelasticformulation of Hertz’s theory. Lee and Radok [7] have shown that, as long as the contact area isincreasing, a simple relation for the normal force can be derived by replacing the elastic modulusin Hertz’s relation (Eq. (3)) with the relaxation function for the sphere material. Substituting therelaxation function into Hertz’s relation and recognizing that the contact radius is also a functionof time, the total normal force (i.e., the normal force due to the combined elastic and dissipativecomponents) at any time can be expressed as F  N  t ð Þ ¼  23 R 11    u 2 ð Þ Z   t 0 C  t    t 0    dd t 0  a t 0   3 d t 0 :  ð 7 Þ Substituting the relation in Eq. (5) for the contact radius, and the relation in Eq. (6) for therelaxation function yields F  N  ð t Þ ¼  R 1 = 2 ð 1    u 2 Þ Z   t 0 X ni  ¼ 0 E  i  e ð t  t 0 Þ = t i  a ð t 0 Þ 1 = 2 ’ a ð t 0 Þ d t 0 ;  where 1 t 0 ¼  0 :  ð 8 Þ For greater utility, it would be beneficial to express Eq. (8) in incremental form such that statevariables can be used to describe the loading at any time. By considering the individualcontributions of each of the viscoelastic terms to the total normal force, the total force at any time ARTICLE IN PRESS S.E. Olson / Journal of Sound and Vibration 264 (2003) 1155–1166  1158

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