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DEM simulation of particle damping
Powder Technology 142 (2004) 154 – 165 www.elsevier.com/locate/powtec
DEM simulation of particle damping Kuanmin Mao a, Michael Yu Wang b,*, Zhiwei Xu c, Tianning Chen d a
College of Mechanical Engineering, Huazhong University of Science and Technology, Wuhan, China Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong, China c The State Key Laboratory for Smart Materials and Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China d College of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China b
Received 9 July 2003; received in revised form 22 April 2004; accepted 22 April 2004 Available online
Abstract Particle damping is a technique of providing damping with granular particles embedded within small holes in a vibrating structure. Particle-to-wall and particle-to-particle collisions arise under the vibrating motion of the structure. As a result, the structure and the particles will exchange momentum and thus dissipate kinetic energy due to frictional and in-elastic losses. The particle damping technology has been used successfully in many fields for vibration reduction. However, it is difficult to predict the damping characteristics due to complex collisions in the dense particle flow. In this paper, we utilize the discrete element method (DEM) for computer simulation of particle damping. By considering thousands of particles as Hertz balls, the discrete element model can describe the motions of these multi-bodies and determine the energy dissipation. We describe a DEM modeling system with an efficient collision detection algorithm for large-scale particle problems. The simulation system is validated by comparing with experimental results of a physical system. The DEM simulation system is further examined with examples for its computational complexity and effectiveness for different density parameters. It is concluded that the particle damping is a mix of two damping mechanisms of impact and friction. It is further shown that the relative significance of these damping mechanisms depends on a particular arrangement of the damper. D 2004 Elsevier B.V. All rights reserved. Keywords: DEM; Impact; Friction; Particle damping; Vibration
1. Introduction Passive vibration damping is often a preferable solution in most industrial applications of vibration control for the unconditional stability, robustness, and cost effectiveness [1]. Conventional passive damping techniques include viscoelastic material applications, frictional devices, impact or fluid dampers, tuned masses, and isolators. While viscoelastic materials are typically considered to be effective, viscoelastic materials, however, have a tendency to degrade, embrittle, and even disintegrate with time and in temperature changing environments. Particle vibration damping (PVD) is a combination of impact damping and friction damping [2,3]. It is a rela* Corresponding author. Department of Automation and ComputerAided Engineering, The Chinese University of Hong Kong, Shatin NT, Hong Kong. Tel.: +852-2609-8487; fax: +852-2603-6002. E-mail address: [email protected] (M.Y. Wang). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.04.031
tively simple concept where metal or ceramic particles or powders of small size ( f 0.05 – 0.5 mm in diameter) are placed inside cavities within or attached to the vibrating structure (see a schematic of Fig. 1). In contrast to viscoelastic materials which dissipate the stored elastic energy, particle damping treatment focuses on dissipation of the kinetic energy. Particle damping involves the potential of energy absorption and dissipation through momentum exchange between moving particles and vibrating walls, friction, impact restitution, and shear deformations. It is an attractive alternative in passive damping due to its conceptual simplicity, potential effectiveness over broad frequency range, temperature and degradation insensitivity, and very low cost [2– 7]. Particle damping is a derivative of single-mass impact damper that has been thoroughly studied on the influence of mass ratio, particle size, particle/slot clearance, excitation levels, and direction of excitation [8,9]. In the single-mass case, direct analyses exist and reveal design criterion for
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Fig. 1. A particle vibration damper (left) and its body contacts (right).
optimal efficiency based on reduction in system response. It is observed that a plastic ‘‘bean bag’’ filled with lead shot exhibited much greater damping effectiveness and ‘‘softer’’ impacts than a single lead slug of equal mass [6,8]. There have been some research efforts toward analytical or numerical models of the complicated phenomenon of multi-grain particle damping [4,6]. However, this has been considered as a specialty area and the literature is relatively scarce. It is generally understood that a dominant mechanism of particle damping is in the form of momentum transfer which occurs when the particles impact the cavity walls of the hosting structure [3,6]. The non-elastic impacts (or restitution) result in energy dissipation. This is similar to a singlemass impact damper, and the analogy permits the use of impact damper theory to roughly size the particle damper. However, in addition, particle damping dissipates energy due to inter-particle and particle – wall friction [6]. The relative effectiveness of these mechanisms changes based on a host of parameters such as particle size, density, and shape, frequency and amplitude of vibration [6,11]. Although the initial test results of the early 1990s substantiated the potential of particle damping [2,3,5], applications in the literature have been largely based only on heuristic guidelines [3]. Due to the complex interactions involved in particle damping, a comprehensive analysis methodology is not yet available [6]. Because of the very large number of particles in a typical particle damper, it appears that applications of concepts from particulate technologies on a particle-to-particle basis would offer the best analysis and design tools. In fact, molecular dynamics simulation (MDS) methods have been attempted to capture motions of particulates [10,14,15] and the technique appear promising in characterizing the particle damping behavior, although it typically is applied to rapid granular flows (or gas– solid flows [16]) involving moderate computational complexities. Recently, a discrete element method (DEM) has been used to perform limited studies of particle dampers [5,11]. This is a technique pioneered by Cundall and Strack [12] for the simulation of behavior of granular materials. The procedure is an explicit process with small time step iterations to determine resultant forces on any particle in the system. Over the years, the discrete element method has found a wide range of applications in various disciplines (see Refs. [15,20 –22]). In this paper, we describe a simulation approach to modeling particle vibration dampers (PVDs) with the discrete element method. We have developed an efficient
procedure for the dynamics of a large number of particles without excessive computing complexity. This allows us to perform large-scale simulations with realistic particle numbers (e.g., 1000) on highly reasonable time scales. This capability is especially important for designing PVDs for applications on a lightweight structural component such as wire/die bonder head arm [13]. It will facilitate the development of minimally invasive damping treatment with the use a minimal amount of damping particles and without any significant impact on the strength, stiffness, weight, and dynamic characteristics of the structure itself.
2. The discrete element simulation Discrete element simulation is a numerical scheme that involves following the motion of every particle and its host structure of a system, where each particle – particle and particle –structure contact is detected. Over small-step iterative computation cycles, complete loss of contacts, and formation of new contacts between the bodies take place. The contact forces resulting from each contact of a pair of bodies are calculated based on a contact force model. DEM is uniquely suited for the particle vibration damping problem. 2.1. The particle vibration damper The physical system of a simple particle vibration damper consists of a structural body with a vertical hole partially filled with metal particles of small size as shown in Fig. 2. The structural body is modeled as a single degree of freedom (SDOF) system with mass M, stiffness K, and intrinsic (e.g., material) damping coefficient C. The hole has diameter of D and a depth of H. The particles are typically made of a material of high density, for example, tungsten steel, for a better damping performance. The particles are often described as spheres for simplicity in discrete element simulation. The diameter of the spherical particles is typically less than 1/5 of the hole’s diameter and is usually in the range of 0.05 – 5 mm. Thus, it is not uncommon that a single hole may contain a large number of particles in an order of 1000 or even 10,000. 2.2. Contact forces As described in the Introduction, the fundamental mechanisms of the particle damping directly involve phys-
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Fig. 2. The model of a single degree of freedom system of a particle vibration damper.
ical contacts between particles and between particles and the cavity walls. In the DEM simulation, any contact between a pair of bodies (spheres and walls) is considered to be a distinct single-point contact (see Fig. 1). The bodies are considered to be rigid, however, the contacts are deformable. Every contact, either sphere – sphere or sphere – wall contact, is modeled using a contact law [16,17]. Selection of an appropriate contact model for inter-particle and particle –cavity impact and friction interactions is an important aspect of the modeling technique [20]. The simplest model is a linear model that uses a combination of a spring and a dashpot in both the normal and the shear directions. However, a linear model is known to be inaccurate in predicting the correct force – displacement relationship [20,21]. Various nonlinear models have been developed in the literature [16,17,20,21] and it is found that the nonlinear impact theory of Hertz is adequately suitable to metal – metal impact [16,21]. In this paper, the Hertz theory is used to describe the particle interaction. The contact model used in this work is described by a set of force – displacement relationships fn ¼ kn ðdn Þ þ cn d˙n
ð1Þ
fs ¼ ks ðds Þ þ cs d˙s ðif stickingÞ ¼ lfn ðif slidingÞ
ð2Þ
for the normal and tangential contact forces fn and fs, respectively, where dn and ds are the contact distances in normal and tangential directions, respectively. Small damping coefficients cn and cs are used for energy loss in the normal and tangential directions, respectively, while frictional dissipation is described by Coulomb’s friction coefficient l [15]. For metal – metal contact for spherical particles, Hertz theory of elastic contact yields the following normal force –displacement relationship kðdn Þ ¼
4 1 3 ER 2 dn2 3
ð3Þ
Fig. 3. A schematic of a particle vibration damper and its experimental setup.
where E is the equivalent elastic modulus and R is the equivalent contact radius of curvature. For convenience, we 1 refer to kn ¼ 43 ER2 as the contact stiffness [17]. 2.3. Particle and structure motions In the DEM simulation, the motions of each particle and the host structure are calculated through a cycling process of small time steps Dt. As the structure vibrates causing the walls of the hole to oscillate, each particle may collide with many other particle as well as the walls. All of these collisions will dissipate energy from the system, thus providing a damping effect on the host structure. The DEM makes the assumption that any body is affected only by its current contacts. This simplification substantially reduces the number of interactions per body that has to be resolved in order to update its motion. A calculation cycle proceeds as follows. First, the moving walls of the structure develop contact forces on each particle contact them. At each of the contacts, the normal and tangential components of the contact force are updated. The net force and moment on each particle involved is determined from all forces applied on the particle. Then, the incremental linear and angular accelerations of the particle are determined from the unbalanced force and moment acting on it. Using a central difference scheme, the linear and angular equations of motion of the particle can be integrated to obtain the incremental linear and
Table 1 Parameters of the PVD system in DEM simulation and experiments System parameter Mass M Stiffness K Damping C Hole diameter D Hole height H
Simulation parameter 0.0376 kg 470 N/m 0.1 N s/m 25.4 mm 7.04 mm
Normal stiffness kn Normal damping cn Tangential stiffness ks Tangential damping cs Friction coefficient l Sphere diameter d Sphere number n
360.0 kN/m 0.01 N s/m 330.0 kN/m 0.015 N s/m 0.55 0.88 mm 1246
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Table 2 Parameters of the PVD system in DEM simulation System parameter Mass M Stiffness K Damping C Hole diameter D Hole height H
Fig. 4. DEM simulation results of free vibration velocity responses with and without particles.
angular displacements. Therefore, in the selected Dt step, the structure walls as well as all of the particles of any contact are moved accordingly. As a result, in each time step, new contacts are generated and some of the existing contacts are broken. The cycle of contact calculation is repeated for the next time step. This incremental updating scheme was fully described in Ref. [12], and it has been widely tested in the literature [20 – 22], showing that it is reasonably accurate and stable when the time step Dt is properly chosen. For brevity, we shall omit the details of its formulation here. 2.4. Contact detection and tracking The DEM requires the determination of the contacts made by a particle with another particle or wall at any step in time. If a collision detection technique has to trace the path of each moving particle to perform contact checks with
Fig. 5. The specific damping capacity versus the dimensionless acceleration, obtained with this simulation study and the experiments in Ref. [4], respectively.
Simulation parameter 0.0376 kg 1410 N/m 0.1 N s/m 10 mm 24.64 mm
Normal stiffness kn Normal damping cn Tangential stiffness ks Tangential damping cs Friction coefficient l Sphere diameter d Sphere number n
360.0 kN/m 0.01 N s/m 330.0 kN/m 0.015 N s/m 0.55 0.88 mm 1246
all other bodies regardless of actual collisions, this would be extremely time-consuming when the number of particles is large. The success of the DEM simulation for a large-scale problem such as the particle damping simulation depends on an efficient contact detection scheme. To keep the virtual collision check for all pairs of n bodies in space, we need a calculation of computational complexity O(n2) at each step. Fortunately, there exist a number of efficient algorithms developed over the years in various disciplines (see, e.g., Refs. [23,24]). For hard sphere collision models, the most promising algorithm is the socalled ‘‘box’’ algorithm which has an O(log n) complexity on average per time step and the overall simulation complexity of O(n log n) [23]. This is the algorithm used in our implementation. The basic idea of the box algorithm is a subdivision of the space containing the sphere particles into a 3D grid of cubic cells. Within each cell, one or more spheres can be accommodated to move. The space subdivision lends itself to a method of localized contact detection method to determine nonempty cells intersecting one or more spheres. By keeping track of the path of every moving sphere and the list of the spheres intersecting each cell, the occurring times of the contacts can be calculated. The approach requires efficient data structures for keeping the nonempty cells and for handling the collision events [24]. For the large-scale problem of particle damping, this algorithm allows us to
Fig. 6. The displacement of the system mass obtained with the DEM simulation.
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avoid the contact detection process becoming the dominant cost of the simulation.
3. Simulation of particle damping In this section, we present the simulation results of a single degree of freedom system with a particle vibration damper by embedding metal particles in a closure attached to the end of a beam. Experimental tests of this system were reported in Ref. [4] with an experimental setup as illustrated in Fig. 3. Both the enclosure and the beam are made of 3003 aluminum alloy with Young’s modulus E of 70 GPa and density of 2.7 g/cm3. The mass of the enclosure is 15.2 g, and the diameter of its interior hole is 2.54 cm. Lead particles of 4.0 g were filled in the hole, resulting in a filling volume ratio
of 25%. The beam has a length of 25.4 cm, width of 3.18 cm, and height 0.23 cm. For the motion at the free end of the beam, the continuous beam is reduced to an equivalent single degree of freedom system with a natural frequency of 17.80 Hz when without the particles. The intrinsic material damping is measured as of a damping ratio of 0.12% [4]. The parameters of the system for the experiments in Ref. [4] and for our DEM simulation are listed in Table 1. Fig. 4 shows a DEM simulation result of the free vibration velocity of the mass enclosure for an initial condition of displacement of 15.7 mm and zero velocity. Velocity history for the case of the same system but without the particles is also plotted. It clearly shows a dramatic attenuation of the transient vibration due to the presence of particles. Comparing this simulation result with the experimental data of the same system presented in Ref. [4], we conclude
Fig. 7. Snapshots of the particles during the vibration of the damper.
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that the DEM simulation agrees with the experiments very well. Since we are not able to directly reproduce the results of [4] for a direct comparison, the reader is referred to Fig. 4 in that paper for the experimental results. Specifically, the experiments showed that the addition of particles would change the vibrating frequency of the system as predicted given their mass and the damping effect. The first natural frequency of the beam system without particles is at 17.80 Hz as predicted by the theoretical calculation. With the particles added into the damper attached to the end of the beam, the frequency was measured at 17.71 Hz [4]. A change in the frequency is also obtained by our DEM simulation as clearly shown in Fig. 4 between the calculated responses for the cases of with and without particles. The DEM prediction of the damped vibrating frequency is 16.88 Hz. While the effect of the particle damping is clearly shown in Fig. 4, the particle damping is highly nonlinear. Especially it depends on the amplitude of the vibration. Therefore, it is more appropriate to use the specific damping capacity to define the damping effect [4]. In the context of particle damping, the specific damping capacity is defined as w ¼ DT =T
ð4Þ
where DT is the kinetic energy converted into heat during one cycle of vibration, and T is the maximum kinetic energy during the cycle. The nonlinear nature of the damping is more evident when the specific damping capacity wi is shown as a function of the effective acceleration for each vibration cycle [4]. In Fig. 5, the computed results of specific damping capacity are given along with the experimental results reproduced from [4]. It is also convenient to define another dimensionless parameter, effective acceleration C, as C ¼ Ax2 =g
ð5Þ
with amplitude A and frequency x of the periodic vibration and gravity g [4].
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Fig. 9. The computing time for two different contact checking algorithms: with subdivision and without subdivision.
The DEM simulation reveals an important characteristic of particle vibration damping. The specific damping capacity depends highly on the dimensionless acceleration amplitude. Roughly speaking, there are three regimes [7,25]. At higher effective acceleration C from the beginning of the vibration, the specific damping is relatively low. The specific damping increases while the acceleration amplitude is reduced. Following Fig. 5 to yet smaller amplitudes, we find that a maximum in the specific damping capacity is reached. Around this point, the particle damping is remarkably high, reaching a highest level of specific damping capacity of approximately 40%. This high damping effect then reduces the acceleration amplitude rather quickly down to C c 1, i.e., when the maximal acceleration of the structure becomes comparable with gravity g. The particle damping effect nearly ceases to exist for C V 1, at which the acceleration of the structure is not sufficient enough for the particles to overcome the gravity to have any collisions.
4. Simulation performance analysis In this section, we present the performance of our DEM simulation system under varying particle number and density conditions in the particle motion simulation. Our numerical experiments focus on the computational time of the DEM simulation in contact detection and overall computation. The analysis results for various simulations described in this section were obtained on a personal computer with a 1.70-GHz Intel Pentium processor. We first consider a similar single degree of freedom system with parameters given in Table 2. The particle Table 3 Computing (execution) time (h) for different cell size
Fig. 8. The number of contacts and the computing time for every time step.
Cell size Computing time (h)
1.5 d 5.14
3d 14.1
Without subdivision 25.08
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Fig. 10. The computing time versus the particle number.
damper has a total of 1246 particles with a volume packing density of 50%, representing a system of dense flow inside the container hole. An initial condition of displacement of 15.7 mm and zero velocity is given to the system, and the motion of the system is simulated for 1.5 s as shown in Fig. 6. The displacement plot of the structure mass is similar to that of the example in the previous section of experimental validation. The damping effect of the particles is clearly shown to be significant.
In Fig. 7, a number of snapshots are given to show the motion of the structural mass and the particles contained in its hole. These snapshots illustrate the approximate damping regimes during the course of free vibration [7]. In the first regime approximately between 0.0 and 0.05 s, the damper container walls impact the particles transferring momentum to the particles. The second regime immediately follows the first region and lasts until 0.24 s. This regime has the most significant damping effect. The DEM simulation reveals that
Fig. 11. The computing time versus the hole’s size.
K. Mao et al. / Powder Technology 142 (2004) 154–165 Table 4 Parameters of the particle damping system for damping analysis System parameter Mass M Stiffness K Damping C Hole diameter D Hole height H Packing ratio p
Simulation parameter 0.0134 kg Mx2n 0.0875 N s/m 10 mm 12 mm 95%
Normal stiffness kn Normal damping cn Tangential stiffness ks Tangential damping cs Friction coefficient l Sphere diameter d Sphere number n
300.0 kN/m 0.02 N s/m 270.0 kN/m 0.018 N s/m 0.55 1.2 mm 386
the particles undergo a large number of collisions with significant relative speed, thus dissipating mechanical energy through both impact and dry friction. After this regime, the particles are nearly at rest relative to the container mass and little damping effect is observed. A time step Dt = 2.5 10 6 (s) is used in the DEM discrete integration scheme. This example represents a system of dense particle flow and the flow is highly transient. The motions of the particle relative to the structural mass last for less than 1 s. Within each time step of iteration, there are often substantially a large number of collisions between particles and between particles and the walls. Fig. 8 depicts the number of contacts for every time step. Within the period of 1.5 s, there are 1571 contacts on average. This phenomenon of dense particle flow is very different from the result in a dilute granular – gas flow [18]. Furthermore, after around t = 1.0 s when the particles nearly come to rest with respect to their container hole, the computational situation becomes extremely intense. The state of stationary contacts for nearly all bodies is represented in the DEM scheme by a resolution of repeated contacts at a high concentration in time. Fig. 8 also shows the total computing time for updating the motions of all bodies of the vibration system for every time step. The total computing time is directly proportional to the number of contacts detected within the time step. Therefore, the localized contact checking algorithm described in the previous section is an important element for the efficient implementation of the DEM simulation. The benefit of the localized contact detection algorithm is further illustrated in Fig. 9. For the simulation results shown in Fig. 7, the space of the container hole is divided into multiple cubic cells in the ‘‘box’’ algorithm with the side length of the cell equal to three times the sphere radius (i.e., l = 1.5d). As expected, without using the cell subdivision, the computing time of each time step would be much longer. In Fig. 9, the computing time is compared with that of the contact detection algorithm without using the cell subdivision, plotted for every time step of the DEM simulation. The computational efficiency is directly proportional to the size of the subdivision cell. As discussed in Refs. [23,24], the primary benefit of the subdivision cells is that they limit the number of collision calculations required for any given particle. A specific particle will be checked only with other particles in the same cell and in the cells adjacent (27 in total for three dimensions). All particles in nonadja-
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cent cells are ignored [24]. A drawback of this time-saving method is that collisions from particles or walls more than one cell away are not accounted for, thus resulting approximation in contact detection and motion calculation. Therefore, a trade-off has to be balanced. In our experience, the cell length l = 1.5d gives an acceptable level of accuracy while yields a practically reasonable computational complexity. Table 3 shows the total execution time for the above example for three different sizes of subdivision cells including without subdivision. Finally, we illustrate the computational complexity of the DEM simulation with respect to ‘‘particle density’’ of the damper. The density is simply defined as n/Vh, where n is the number of particles and Vh is the volume of the container hole. The same example of Table 2 is used here. In Fig. 10, we keep the hole’s size and the total mass of the particles the same while changing the number of particles and the sphere diameter. The total execution time and the mean number of collisions per time step are plotted for a number of cases of 300 – 1300 particles. As the number of particles increases, the computational effort clearly increases approximately linearly. This indicates that our algorithm behaves with a computational complexity of O(log n) per time step as predicted. Next, we keep the number of particles at 936, and vary the diameter of the container hole between 10 and 25 mm. As shown in Fig. 11, the execution time and the mean number of contacts would decrease as an increase in the hole’s diameter decreases the particle density in a quadratic inverse relation. This illustrates the said computational complexity of our simulation.
5. Distribution of energy dissipation In this section, we further examine the relative importance of the two energy dissipation mechanisms under a
Fig. 12. Various cumulative energy dissipations for 95% packing ratio. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
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Fig. 13. Various specific damping w for 95% packing ratio during t = 0.00 – 0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
Fig. 15. Various specific damping w for 50% packing ratio during t = 0.00 – 0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
different set of arrangements of the major parameters of the particle damper. This represents our effort towards a better and more complete understanding of the energy dissipation mechanisms in particle damping [11]. We utilize our DEM simulation system to calculate the dissipated energy by collisions between particles and between particle and container wall interfaces. The dissipated energy is divided into two classes due to impact and dryfriction respectively for each collision. When summarizing all the particles in collision, we obtain the system energy, the frictional energy dissipation, and the impact energy dissipation over the course of vibration attenuation of the free-vibrating system. The parameters of the particle damping system are listed in Table 4. In all the cases to be discussed, the particle mass remains the same of 10% of the structural mass M.
5.1. Energy dissipation versus packing ratio We first examine the effect of packing ratio of the particles. Under free vibration condition, the undamped natural frequency of the system is set at xn = 250 Hz, for three different packing ratios with H = 12, 24, and 48 mm, respectively, for three different packing ratios p = 95%, 50%, and 25%. In all cases, the initial displacement is 5 mm with zero initial velocity. The energy dissipations of the system due to impact, friction, and viscous damping are shown in Figs. 12 –17, respectively, for the three cases. In all three cases, the DEM simulation shows that both friction damping and impact damping play significant role. However, at different levels of height of the hole, each damping mechanism may have different relative contribution. At the high packing ratio of 95%, the container hole is
Fig. 14. Various cumulative energy dissipations for 50% packing ratio. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
Fig. 16. Various cumulative energy dissipations for 25% packing ratio. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
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Fig. 17. Various specific damping w for 25% packing ratio during t = 0.00 – 0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
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Fig. 19. Various cumulative energy dissipations at xn = 100 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
nearly packed full of the particles. The friction damping becomes more significant than the impact damping (Figs. 12 and 13). Intuitively, the small clearance between the particles and the container walls reduces the potential of strong collisions between the particle – particle and particle – wall contacts. For a higher clearance at p = 50%, the total energy dissipation remains nearly at the same level with that at p = 95%. In this case, however, the friction and impact damping mechanisms have reversed their roles with energy dissipated by impact being about 30% more than energy dissipated by friction (Figs. 14 and 15). For the even higher clearance at p = 25%, the impact energy dissipation is even more pronounced while the friction energy dissipation remains nearly the same (Figs. 16 and 17). Therefore, the total level of energy dissipation is higher. At this clearance level, the particle damper has the best performance for the particles have a best chance to shake and tumble inside the
hole, thus dissipating energy through the impact mechanism [25]. It should be noted from Figs. 12 – 17 that the bulk of the vibration energy is dissipated within a very short period of time approximately between t = 0.00 – 0.18 s for all the three cases. This again is a reflection of the nonlinear characteristics of the particle damping as discussed in Section 3.
Fig. 18. Various cumulative energy dissipations at xn = 20 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
Fig. 20. Various cumulative energy dissipations at xn = 250 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
5.2. Energy dissipation versus frequency Next, we examine the effect of vibrating frequency on the energy dissipation. The particle volumetric packing ratio is set at 50%, and the initial displacement is taken at 10 mm. The natural frequency of the system is set at four different levels, xn = 20, 100, 250, and 500 Hz. As shown in Figs. 18 –21, respectively, the friction dissipation is greater than the impact dissipation for the lower frequency cases of xn = 20 and 100 Hz. But when system
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numerical simulations of different physical and numerical parameters, including the number of particles, the hole’s size, the particle density, and the size of subdivision cells. The particle damping is a combination of the impact and the friction damping. It is further shown that the relative significance of these damping mechanisms depends on a particular arrangement of the damper. It is shown that our implementation is particularly efficient with a near linear dependency on the number of particles per time step. Thus, the DEM simulation can be readily applied as a practical tool for designing a PVD. It should complement experiments to furnish a deeper understanding of particle damping and to provide a comprehensive methodology for analysis and design of PVDs.
Fig. 21. Various cumulative energy dissipations at xn = 500 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
frequency is higher (at 250 and 500 Hz), the impact dissipation is greater than the friction dissipation. At xn = 20 Hz and x0 = 10 mm, the dimensionless acceleration of the system is initially C = x0xn2/g c 400. We found that when C < 400, the friction dissipation is greater than impact dissipation. This conclusion is consistent with the results of [10,25]. In the low frequency case, the friction dissipation is nearly twice greater than impact dissipation as shown in Fig. 18. However, at the much higher frequency of 500 Hz, the situation is nearly reversed with impact damping to be much more significant than the friction damping.
6. Conclusions We have presented a 3D DEM to simulate the motion of particles in a PVD. The metal particles are represented as spheres. For particle – particle and particle – wall contacts, a normal contact model and a tangential contact model are used to account for the coefficient of restitution and the dry friction. Both impact and friction are known to play an important role in the damping effect provided by the particles on their container structure. An efficient ‘‘box’’ algorithm is used for the detection of collisions in the dense particle flow during the course of vibration. With the DEM modeling tool, we performed numerical simulation for a number of examples of a single degree of freedom system. For a physical particle vibration damper with experimental results of a previous investigation, simulation results of the DEM modeling are found to have a good agreement with the experimental data. Furthermore, simulations provide information of particle motions within the container hole during three different regions and help explain their associated damping characteristics. The computational complexity of the DEM scheme is analyzed with
Acknowledgements The research work reported in this paper is sponsored in part by the Hong Kong Research Grants Council (project No. CUHK4196/01E), ASM Assembly Automation, a Croucher Chinese Visitorship of the Croucher Foundation of Hong Kong for Professor T. N. Chen, and the National Science Foundation of China (NSFC) under grant No. 59775019. The authors wish to express their sincere thanks to C.K. Liu, P.K. Choy, and G.P. Widdowson of ASM Assembly Automation for their valuable insight and support.
References [1] A.D. Nashif, D.I. Jones, J.P. Henderson, Vibration Damping, Wiley, New York, 1985. [2] H.V. Panossian, Structural damping enhancement via non-obstructive particle damping technique, ASME J. Vib. Acoust. 114 (1992) 101 – 105. [3] H.V. Panossian, An overview of NOPD: a passive damping technique, Shock Vib. 1 (6) (1991) 4 – 10. [4] R.D. Friend, V.K. Kinra, Particle impact damping, J. Sound Vib. 233 (1) (2000) 93 – 118. [5] S.S. Simonian, Particle beam damper, Proceedings of SPIE Conf. on Passive Damping, SPIE, vol. 2445, SPIE, San Diego, CA, 1995, pp. 149 – 160. [6] B.L. Fowler, E.M. Flint, S.E. Olson, Effectiveness and predictability of particle damping, Proceedings of SPIE Conf. on Damping and Isolation, Newport Beach, CA, March. [7] J.J. Hollkamp, R.W. Gordan, Experiments with particle damping, Passive Damping and Isolation, Proceedings of SPIE, vol. 3327, 1998 March, pp. 2 – 12, San Diego, CA. [8] N. Popplewell, S.E. Semergicil, Performance of bean bag impact damper for a sinusoidal external force, J. Sound Vib. 133 (2) (1989) 193 – 223. [9] A. Papalou, S.F. Masri, An experimental investigation of particle damper under harmonic excitation, J. Vib. Control 4 (1998) 361 – 379. [10] C. Saluena, T. Poschel, S.E. Esipov, Dissipative properties of vibrated granular materials, Phys. Rev. E59-4 (1999) 4422 – 4425. [11] T. Chen, K. Mao, X. Huang, M.Y. Wang, Dissipation mechanisms of non-obstructive particle damping using discrete element method, Proceedings of SPIE International Symposium on Smart Structures and Materials, Vol. 4331, Damping and Isolation, 2001 March, pp. 294 – 301, Newport Beach, CA.
K. Mao et al. / Powder Technology 142 (2004) 154–165 [12] P. Cundall, O. Strack, A distinct element model for granular assemblies, Geotechnique 29 (1979) 47 – 65. [13] P.K. Choy, C.K. Liu, W.H. Liao, Y. Wang, High speed pick and place apparatus, US Patent (pending), filing date July 2001. [14] C. Hogue, D. Newland, Efficient computer simulation of moving granular particles, Powder Technol. 78 (1) (1994) 51 – 66. [15] D.E. Wolf, Modeling and computer simulation of granular media, in: K.H. Hoffmann, M. Schreiber (Eds.), Computational Physics: Selected Methods – Simple Exercises – Serious Applications, Heidelberg, Springer, 1996. [16] X. Zhang, L. Vu-Quoc, Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions, Int. J. Impact Eng. 27 (2002) 317 – 341. [17] K.L. Johnson, Contact Mechanics, Cambridge Univ. Press, Cambridge, UK, 1985. [18] L.-S. Fan, C. Zhu, Principles of Gas – Solid Flows, Cambridge Univ. Press, Cambridge, UK, 1998. [19] R. Venugopal, R.K. Rajamani, 3D Simulation of charge motion in
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DEM simulation of particle damping Kuanmin Mao a, Michael Yu Wang b,*, Zhiwei Xu c, Tianning Chen d a
College of Mechanical Engineering, Huazhong University of Science and Technology, Wuhan, China Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong, China c The State Key Laboratory for Smart Materials and Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China d College of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China b
Received 9 July 2003; received in revised form 22 April 2004; accepted 22 April 2004 Available online
Abstract Particle damping is a technique of providing damping with granular particles embedded within small holes in a vibrating structure. Particle-to-wall and particle-to-particle collisions arise under the vibrating motion of the structure. As a result, the structure and the particles will exchange momentum and thus dissipate kinetic energy due to frictional and in-elastic losses. The particle damping technology has been used successfully in many fields for vibration reduction. However, it is difficult to predict the damping characteristics due to complex collisions in the dense particle flow. In this paper, we utilize the discrete element method (DEM) for computer simulation of particle damping. By considering thousands of particles as Hertz balls, the discrete element model can describe the motions of these multi-bodies and determine the energy dissipation. We describe a DEM modeling system with an efficient collision detection algorithm for large-scale particle problems. The simulation system is validated by comparing with experimental results of a physical system. The DEM simulation system is further examined with examples for its computational complexity and effectiveness for different density parameters. It is concluded that the particle damping is a mix of two damping mechanisms of impact and friction. It is further shown that the relative significance of these damping mechanisms depends on a particular arrangement of the damper. D 2004 Elsevier B.V. All rights reserved. Keywords: DEM; Impact; Friction; Particle damping; Vibration
1. Introduction Passive vibration damping is often a preferable solution in most industrial applications of vibration control for the unconditional stability, robustness, and cost effectiveness [1]. Conventional passive damping techniques include viscoelastic material applications, frictional devices, impact or fluid dampers, tuned masses, and isolators. While viscoelastic materials are typically considered to be effective, viscoelastic materials, however, have a tendency to degrade, embrittle, and even disintegrate with time and in temperature changing environments. Particle vibration damping (PVD) is a combination of impact damping and friction damping [2,3]. It is a rela* Corresponding author. Department of Automation and ComputerAided Engineering, The Chinese University of Hong Kong, Shatin NT, Hong Kong. Tel.: +852-2609-8487; fax: +852-2603-6002. E-mail address: [email protected] (M.Y. Wang). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.04.031
tively simple concept where metal or ceramic particles or powders of small size ( f 0.05 – 0.5 mm in diameter) are placed inside cavities within or attached to the vibrating structure (see a schematic of Fig. 1). In contrast to viscoelastic materials which dissipate the stored elastic energy, particle damping treatment focuses on dissipation of the kinetic energy. Particle damping involves the potential of energy absorption and dissipation through momentum exchange between moving particles and vibrating walls, friction, impact restitution, and shear deformations. It is an attractive alternative in passive damping due to its conceptual simplicity, potential effectiveness over broad frequency range, temperature and degradation insensitivity, and very low cost [2– 7]. Particle damping is a derivative of single-mass impact damper that has been thoroughly studied on the influence of mass ratio, particle size, particle/slot clearance, excitation levels, and direction of excitation [8,9]. In the single-mass case, direct analyses exist and reveal design criterion for
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Fig. 1. A particle vibration damper (left) and its body contacts (right).
optimal efficiency based on reduction in system response. It is observed that a plastic ‘‘bean bag’’ filled with lead shot exhibited much greater damping effectiveness and ‘‘softer’’ impacts than a single lead slug of equal mass [6,8]. There have been some research efforts toward analytical or numerical models of the complicated phenomenon of multi-grain particle damping [4,6]. However, this has been considered as a specialty area and the literature is relatively scarce. It is generally understood that a dominant mechanism of particle damping is in the form of momentum transfer which occurs when the particles impact the cavity walls of the hosting structure [3,6]. The non-elastic impacts (or restitution) result in energy dissipation. This is similar to a singlemass impact damper, and the analogy permits the use of impact damper theory to roughly size the particle damper. However, in addition, particle damping dissipates energy due to inter-particle and particle – wall friction [6]. The relative effectiveness of these mechanisms changes based on a host of parameters such as particle size, density, and shape, frequency and amplitude of vibration [6,11]. Although the initial test results of the early 1990s substantiated the potential of particle damping [2,3,5], applications in the literature have been largely based only on heuristic guidelines [3]. Due to the complex interactions involved in particle damping, a comprehensive analysis methodology is not yet available [6]. Because of the very large number of particles in a typical particle damper, it appears that applications of concepts from particulate technologies on a particle-to-particle basis would offer the best analysis and design tools. In fact, molecular dynamics simulation (MDS) methods have been attempted to capture motions of particulates [10,14,15] and the technique appear promising in characterizing the particle damping behavior, although it typically is applied to rapid granular flows (or gas– solid flows [16]) involving moderate computational complexities. Recently, a discrete element method (DEM) has been used to perform limited studies of particle dampers [5,11]. This is a technique pioneered by Cundall and Strack [12] for the simulation of behavior of granular materials. The procedure is an explicit process with small time step iterations to determine resultant forces on any particle in the system. Over the years, the discrete element method has found a wide range of applications in various disciplines (see Refs. [15,20 –22]). In this paper, we describe a simulation approach to modeling particle vibration dampers (PVDs) with the discrete element method. We have developed an efficient
procedure for the dynamics of a large number of particles without excessive computing complexity. This allows us to perform large-scale simulations with realistic particle numbers (e.g., 1000) on highly reasonable time scales. This capability is especially important for designing PVDs for applications on a lightweight structural component such as wire/die bonder head arm [13]. It will facilitate the development of minimally invasive damping treatment with the use a minimal amount of damping particles and without any significant impact on the strength, stiffness, weight, and dynamic characteristics of the structure itself.
2. The discrete element simulation Discrete element simulation is a numerical scheme that involves following the motion of every particle and its host structure of a system, where each particle – particle and particle –structure contact is detected. Over small-step iterative computation cycles, complete loss of contacts, and formation of new contacts between the bodies take place. The contact forces resulting from each contact of a pair of bodies are calculated based on a contact force model. DEM is uniquely suited for the particle vibration damping problem. 2.1. The particle vibration damper The physical system of a simple particle vibration damper consists of a structural body with a vertical hole partially filled with metal particles of small size as shown in Fig. 2. The structural body is modeled as a single degree of freedom (SDOF) system with mass M, stiffness K, and intrinsic (e.g., material) damping coefficient C. The hole has diameter of D and a depth of H. The particles are typically made of a material of high density, for example, tungsten steel, for a better damping performance. The particles are often described as spheres for simplicity in discrete element simulation. The diameter of the spherical particles is typically less than 1/5 of the hole’s diameter and is usually in the range of 0.05 – 5 mm. Thus, it is not uncommon that a single hole may contain a large number of particles in an order of 1000 or even 10,000. 2.2. Contact forces As described in the Introduction, the fundamental mechanisms of the particle damping directly involve phys-
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Fig. 2. The model of a single degree of freedom system of a particle vibration damper.
ical contacts between particles and between particles and the cavity walls. In the DEM simulation, any contact between a pair of bodies (spheres and walls) is considered to be a distinct single-point contact (see Fig. 1). The bodies are considered to be rigid, however, the contacts are deformable. Every contact, either sphere – sphere or sphere – wall contact, is modeled using a contact law [16,17]. Selection of an appropriate contact model for inter-particle and particle –cavity impact and friction interactions is an important aspect of the modeling technique [20]. The simplest model is a linear model that uses a combination of a spring and a dashpot in both the normal and the shear directions. However, a linear model is known to be inaccurate in predicting the correct force – displacement relationship [20,21]. Various nonlinear models have been developed in the literature [16,17,20,21] and it is found that the nonlinear impact theory of Hertz is adequately suitable to metal – metal impact [16,21]. In this paper, the Hertz theory is used to describe the particle interaction. The contact model used in this work is described by a set of force – displacement relationships fn ¼ kn ðdn Þ þ cn d˙n
ð1Þ
fs ¼ ks ðds Þ þ cs d˙s ðif stickingÞ ¼ lfn ðif slidingÞ
ð2Þ
for the normal and tangential contact forces fn and fs, respectively, where dn and ds are the contact distances in normal and tangential directions, respectively. Small damping coefficients cn and cs are used for energy loss in the normal and tangential directions, respectively, while frictional dissipation is described by Coulomb’s friction coefficient l [15]. For metal – metal contact for spherical particles, Hertz theory of elastic contact yields the following normal force –displacement relationship kðdn Þ ¼
4 1 3 ER 2 dn2 3
ð3Þ
Fig. 3. A schematic of a particle vibration damper and its experimental setup.
where E is the equivalent elastic modulus and R is the equivalent contact radius of curvature. For convenience, we 1 refer to kn ¼ 43 ER2 as the contact stiffness [17]. 2.3. Particle and structure motions In the DEM simulation, the motions of each particle and the host structure are calculated through a cycling process of small time steps Dt. As the structure vibrates causing the walls of the hole to oscillate, each particle may collide with many other particle as well as the walls. All of these collisions will dissipate energy from the system, thus providing a damping effect on the host structure. The DEM makes the assumption that any body is affected only by its current contacts. This simplification substantially reduces the number of interactions per body that has to be resolved in order to update its motion. A calculation cycle proceeds as follows. First, the moving walls of the structure develop contact forces on each particle contact them. At each of the contacts, the normal and tangential components of the contact force are updated. The net force and moment on each particle involved is determined from all forces applied on the particle. Then, the incremental linear and angular accelerations of the particle are determined from the unbalanced force and moment acting on it. Using a central difference scheme, the linear and angular equations of motion of the particle can be integrated to obtain the incremental linear and
Table 1 Parameters of the PVD system in DEM simulation and experiments System parameter Mass M Stiffness K Damping C Hole diameter D Hole height H
Simulation parameter 0.0376 kg 470 N/m 0.1 N s/m 25.4 mm 7.04 mm
Normal stiffness kn Normal damping cn Tangential stiffness ks Tangential damping cs Friction coefficient l Sphere diameter d Sphere number n
360.0 kN/m 0.01 N s/m 330.0 kN/m 0.015 N s/m 0.55 0.88 mm 1246
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Table 2 Parameters of the PVD system in DEM simulation System parameter Mass M Stiffness K Damping C Hole diameter D Hole height H
Fig. 4. DEM simulation results of free vibration velocity responses with and without particles.
angular displacements. Therefore, in the selected Dt step, the structure walls as well as all of the particles of any contact are moved accordingly. As a result, in each time step, new contacts are generated and some of the existing contacts are broken. The cycle of contact calculation is repeated for the next time step. This incremental updating scheme was fully described in Ref. [12], and it has been widely tested in the literature [20 – 22], showing that it is reasonably accurate and stable when the time step Dt is properly chosen. For brevity, we shall omit the details of its formulation here. 2.4. Contact detection and tracking The DEM requires the determination of the contacts made by a particle with another particle or wall at any step in time. If a collision detection technique has to trace the path of each moving particle to perform contact checks with
Fig. 5. The specific damping capacity versus the dimensionless acceleration, obtained with this simulation study and the experiments in Ref. [4], respectively.
Simulation parameter 0.0376 kg 1410 N/m 0.1 N s/m 10 mm 24.64 mm
Normal stiffness kn Normal damping cn Tangential stiffness ks Tangential damping cs Friction coefficient l Sphere diameter d Sphere number n
360.0 kN/m 0.01 N s/m 330.0 kN/m 0.015 N s/m 0.55 0.88 mm 1246
all other bodies regardless of actual collisions, this would be extremely time-consuming when the number of particles is large. The success of the DEM simulation for a large-scale problem such as the particle damping simulation depends on an efficient contact detection scheme. To keep the virtual collision check for all pairs of n bodies in space, we need a calculation of computational complexity O(n2) at each step. Fortunately, there exist a number of efficient algorithms developed over the years in various disciplines (see, e.g., Refs. [23,24]). For hard sphere collision models, the most promising algorithm is the socalled ‘‘box’’ algorithm which has an O(log n) complexity on average per time step and the overall simulation complexity of O(n log n) [23]. This is the algorithm used in our implementation. The basic idea of the box algorithm is a subdivision of the space containing the sphere particles into a 3D grid of cubic cells. Within each cell, one or more spheres can be accommodated to move. The space subdivision lends itself to a method of localized contact detection method to determine nonempty cells intersecting one or more spheres. By keeping track of the path of every moving sphere and the list of the spheres intersecting each cell, the occurring times of the contacts can be calculated. The approach requires efficient data structures for keeping the nonempty cells and for handling the collision events [24]. For the large-scale problem of particle damping, this algorithm allows us to
Fig. 6. The displacement of the system mass obtained with the DEM simulation.
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avoid the contact detection process becoming the dominant cost of the simulation.
3. Simulation of particle damping In this section, we present the simulation results of a single degree of freedom system with a particle vibration damper by embedding metal particles in a closure attached to the end of a beam. Experimental tests of this system were reported in Ref. [4] with an experimental setup as illustrated in Fig. 3. Both the enclosure and the beam are made of 3003 aluminum alloy with Young’s modulus E of 70 GPa and density of 2.7 g/cm3. The mass of the enclosure is 15.2 g, and the diameter of its interior hole is 2.54 cm. Lead particles of 4.0 g were filled in the hole, resulting in a filling volume ratio
of 25%. The beam has a length of 25.4 cm, width of 3.18 cm, and height 0.23 cm. For the motion at the free end of the beam, the continuous beam is reduced to an equivalent single degree of freedom system with a natural frequency of 17.80 Hz when without the particles. The intrinsic material damping is measured as of a damping ratio of 0.12% [4]. The parameters of the system for the experiments in Ref. [4] and for our DEM simulation are listed in Table 1. Fig. 4 shows a DEM simulation result of the free vibration velocity of the mass enclosure for an initial condition of displacement of 15.7 mm and zero velocity. Velocity history for the case of the same system but without the particles is also plotted. It clearly shows a dramatic attenuation of the transient vibration due to the presence of particles. Comparing this simulation result with the experimental data of the same system presented in Ref. [4], we conclude
Fig. 7. Snapshots of the particles during the vibration of the damper.
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that the DEM simulation agrees with the experiments very well. Since we are not able to directly reproduce the results of [4] for a direct comparison, the reader is referred to Fig. 4 in that paper for the experimental results. Specifically, the experiments showed that the addition of particles would change the vibrating frequency of the system as predicted given their mass and the damping effect. The first natural frequency of the beam system without particles is at 17.80 Hz as predicted by the theoretical calculation. With the particles added into the damper attached to the end of the beam, the frequency was measured at 17.71 Hz [4]. A change in the frequency is also obtained by our DEM simulation as clearly shown in Fig. 4 between the calculated responses for the cases of with and without particles. The DEM prediction of the damped vibrating frequency is 16.88 Hz. While the effect of the particle damping is clearly shown in Fig. 4, the particle damping is highly nonlinear. Especially it depends on the amplitude of the vibration. Therefore, it is more appropriate to use the specific damping capacity to define the damping effect [4]. In the context of particle damping, the specific damping capacity is defined as w ¼ DT =T
ð4Þ
where DT is the kinetic energy converted into heat during one cycle of vibration, and T is the maximum kinetic energy during the cycle. The nonlinear nature of the damping is more evident when the specific damping capacity wi is shown as a function of the effective acceleration for each vibration cycle [4]. In Fig. 5, the computed results of specific damping capacity are given along with the experimental results reproduced from [4]. It is also convenient to define another dimensionless parameter, effective acceleration C, as C ¼ Ax2 =g
ð5Þ
with amplitude A and frequency x of the periodic vibration and gravity g [4].
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Fig. 9. The computing time for two different contact checking algorithms: with subdivision and without subdivision.
The DEM simulation reveals an important characteristic of particle vibration damping. The specific damping capacity depends highly on the dimensionless acceleration amplitude. Roughly speaking, there are three regimes [7,25]. At higher effective acceleration C from the beginning of the vibration, the specific damping is relatively low. The specific damping increases while the acceleration amplitude is reduced. Following Fig. 5 to yet smaller amplitudes, we find that a maximum in the specific damping capacity is reached. Around this point, the particle damping is remarkably high, reaching a highest level of specific damping capacity of approximately 40%. This high damping effect then reduces the acceleration amplitude rather quickly down to C c 1, i.e., when the maximal acceleration of the structure becomes comparable with gravity g. The particle damping effect nearly ceases to exist for C V 1, at which the acceleration of the structure is not sufficient enough for the particles to overcome the gravity to have any collisions.
4. Simulation performance analysis In this section, we present the performance of our DEM simulation system under varying particle number and density conditions in the particle motion simulation. Our numerical experiments focus on the computational time of the DEM simulation in contact detection and overall computation. The analysis results for various simulations described in this section were obtained on a personal computer with a 1.70-GHz Intel Pentium processor. We first consider a similar single degree of freedom system with parameters given in Table 2. The particle Table 3 Computing (execution) time (h) for different cell size
Fig. 8. The number of contacts and the computing time for every time step.
Cell size Computing time (h)
1.5 d 5.14
3d 14.1
Without subdivision 25.08
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Fig. 10. The computing time versus the particle number.
damper has a total of 1246 particles with a volume packing density of 50%, representing a system of dense flow inside the container hole. An initial condition of displacement of 15.7 mm and zero velocity is given to the system, and the motion of the system is simulated for 1.5 s as shown in Fig. 6. The displacement plot of the structure mass is similar to that of the example in the previous section of experimental validation. The damping effect of the particles is clearly shown to be significant.
In Fig. 7, a number of snapshots are given to show the motion of the structural mass and the particles contained in its hole. These snapshots illustrate the approximate damping regimes during the course of free vibration [7]. In the first regime approximately between 0.0 and 0.05 s, the damper container walls impact the particles transferring momentum to the particles. The second regime immediately follows the first region and lasts until 0.24 s. This regime has the most significant damping effect. The DEM simulation reveals that
Fig. 11. The computing time versus the hole’s size.
K. Mao et al. / Powder Technology 142 (2004) 154–165 Table 4 Parameters of the particle damping system for damping analysis System parameter Mass M Stiffness K Damping C Hole diameter D Hole height H Packing ratio p
Simulation parameter 0.0134 kg Mx2n 0.0875 N s/m 10 mm 12 mm 95%
Normal stiffness kn Normal damping cn Tangential stiffness ks Tangential damping cs Friction coefficient l Sphere diameter d Sphere number n
300.0 kN/m 0.02 N s/m 270.0 kN/m 0.018 N s/m 0.55 1.2 mm 386
the particles undergo a large number of collisions with significant relative speed, thus dissipating mechanical energy through both impact and dry friction. After this regime, the particles are nearly at rest relative to the container mass and little damping effect is observed. A time step Dt = 2.5 10 6 (s) is used in the DEM discrete integration scheme. This example represents a system of dense particle flow and the flow is highly transient. The motions of the particle relative to the structural mass last for less than 1 s. Within each time step of iteration, there are often substantially a large number of collisions between particles and between particles and the walls. Fig. 8 depicts the number of contacts for every time step. Within the period of 1.5 s, there are 1571 contacts on average. This phenomenon of dense particle flow is very different from the result in a dilute granular – gas flow [18]. Furthermore, after around t = 1.0 s when the particles nearly come to rest with respect to their container hole, the computational situation becomes extremely intense. The state of stationary contacts for nearly all bodies is represented in the DEM scheme by a resolution of repeated contacts at a high concentration in time. Fig. 8 also shows the total computing time for updating the motions of all bodies of the vibration system for every time step. The total computing time is directly proportional to the number of contacts detected within the time step. Therefore, the localized contact checking algorithm described in the previous section is an important element for the efficient implementation of the DEM simulation. The benefit of the localized contact detection algorithm is further illustrated in Fig. 9. For the simulation results shown in Fig. 7, the space of the container hole is divided into multiple cubic cells in the ‘‘box’’ algorithm with the side length of the cell equal to three times the sphere radius (i.e., l = 1.5d). As expected, without using the cell subdivision, the computing time of each time step would be much longer. In Fig. 9, the computing time is compared with that of the contact detection algorithm without using the cell subdivision, plotted for every time step of the DEM simulation. The computational efficiency is directly proportional to the size of the subdivision cell. As discussed in Refs. [23,24], the primary benefit of the subdivision cells is that they limit the number of collision calculations required for any given particle. A specific particle will be checked only with other particles in the same cell and in the cells adjacent (27 in total for three dimensions). All particles in nonadja-
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cent cells are ignored [24]. A drawback of this time-saving method is that collisions from particles or walls more than one cell away are not accounted for, thus resulting approximation in contact detection and motion calculation. Therefore, a trade-off has to be balanced. In our experience, the cell length l = 1.5d gives an acceptable level of accuracy while yields a practically reasonable computational complexity. Table 3 shows the total execution time for the above example for three different sizes of subdivision cells including without subdivision. Finally, we illustrate the computational complexity of the DEM simulation with respect to ‘‘particle density’’ of the damper. The density is simply defined as n/Vh, where n is the number of particles and Vh is the volume of the container hole. The same example of Table 2 is used here. In Fig. 10, we keep the hole’s size and the total mass of the particles the same while changing the number of particles and the sphere diameter. The total execution time and the mean number of collisions per time step are plotted for a number of cases of 300 – 1300 particles. As the number of particles increases, the computational effort clearly increases approximately linearly. This indicates that our algorithm behaves with a computational complexity of O(log n) per time step as predicted. Next, we keep the number of particles at 936, and vary the diameter of the container hole between 10 and 25 mm. As shown in Fig. 11, the execution time and the mean number of contacts would decrease as an increase in the hole’s diameter decreases the particle density in a quadratic inverse relation. This illustrates the said computational complexity of our simulation.
5. Distribution of energy dissipation In this section, we further examine the relative importance of the two energy dissipation mechanisms under a
Fig. 12. Various cumulative energy dissipations for 95% packing ratio. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
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Fig. 13. Various specific damping w for 95% packing ratio during t = 0.00 – 0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
Fig. 15. Various specific damping w for 50% packing ratio during t = 0.00 – 0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
different set of arrangements of the major parameters of the particle damper. This represents our effort towards a better and more complete understanding of the energy dissipation mechanisms in particle damping [11]. We utilize our DEM simulation system to calculate the dissipated energy by collisions between particles and between particle and container wall interfaces. The dissipated energy is divided into two classes due to impact and dryfriction respectively for each collision. When summarizing all the particles in collision, we obtain the system energy, the frictional energy dissipation, and the impact energy dissipation over the course of vibration attenuation of the free-vibrating system. The parameters of the particle damping system are listed in Table 4. In all the cases to be discussed, the particle mass remains the same of 10% of the structural mass M.
5.1. Energy dissipation versus packing ratio We first examine the effect of packing ratio of the particles. Under free vibration condition, the undamped natural frequency of the system is set at xn = 250 Hz, for three different packing ratios with H = 12, 24, and 48 mm, respectively, for three different packing ratios p = 95%, 50%, and 25%. In all cases, the initial displacement is 5 mm with zero initial velocity. The energy dissipations of the system due to impact, friction, and viscous damping are shown in Figs. 12 –17, respectively, for the three cases. In all three cases, the DEM simulation shows that both friction damping and impact damping play significant role. However, at different levels of height of the hole, each damping mechanism may have different relative contribution. At the high packing ratio of 95%, the container hole is
Fig. 14. Various cumulative energy dissipations for 50% packing ratio. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
Fig. 16. Various cumulative energy dissipations for 25% packing ratio. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
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Fig. 17. Various specific damping w for 25% packing ratio during t = 0.00 – 0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
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Fig. 19. Various cumulative energy dissipations at xn = 100 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
nearly packed full of the particles. The friction damping becomes more significant than the impact damping (Figs. 12 and 13). Intuitively, the small clearance between the particles and the container walls reduces the potential of strong collisions between the particle – particle and particle – wall contacts. For a higher clearance at p = 50%, the total energy dissipation remains nearly at the same level with that at p = 95%. In this case, however, the friction and impact damping mechanisms have reversed their roles with energy dissipated by impact being about 30% more than energy dissipated by friction (Figs. 14 and 15). For the even higher clearance at p = 25%, the impact energy dissipation is even more pronounced while the friction energy dissipation remains nearly the same (Figs. 16 and 17). Therefore, the total level of energy dissipation is higher. At this clearance level, the particle damper has the best performance for the particles have a best chance to shake and tumble inside the
hole, thus dissipating energy through the impact mechanism [25]. It should be noted from Figs. 12 – 17 that the bulk of the vibration energy is dissipated within a very short period of time approximately between t = 0.00 – 0.18 s for all the three cases. This again is a reflection of the nonlinear characteristics of the particle damping as discussed in Section 3.
Fig. 18. Various cumulative energy dissipations at xn = 20 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
Fig. 20. Various cumulative energy dissipations at xn = 250 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
5.2. Energy dissipation versus frequency Next, we examine the effect of vibrating frequency on the energy dissipation. The particle volumetric packing ratio is set at 50%, and the initial displacement is taken at 10 mm. The natural frequency of the system is set at four different levels, xn = 20, 100, 250, and 500 Hz. As shown in Figs. 18 –21, respectively, the friction dissipation is greater than the impact dissipation for the lower frequency cases of xn = 20 and 100 Hz. But when system
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numerical simulations of different physical and numerical parameters, including the number of particles, the hole’s size, the particle density, and the size of subdivision cells. The particle damping is a combination of the impact and the friction damping. It is further shown that the relative significance of these damping mechanisms depends on a particular arrangement of the damper. It is shown that our implementation is particularly efficient with a near linear dependency on the number of particles per time step. Thus, the DEM simulation can be readily applied as a practical tool for designing a PVD. It should complement experiments to furnish a deeper understanding of particle damping and to provide a comprehensive methodology for analysis and design of PVDs.
Fig. 21. Various cumulative energy dissipations at xn = 500 Hz. (A) Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy dissipation; (E) remaining energy in the system.
frequency is higher (at 250 and 500 Hz), the impact dissipation is greater than the friction dissipation. At xn = 20 Hz and x0 = 10 mm, the dimensionless acceleration of the system is initially C = x0xn2/g c 400. We found that when C < 400, the friction dissipation is greater than impact dissipation. This conclusion is consistent with the results of [10,25]. In the low frequency case, the friction dissipation is nearly twice greater than impact dissipation as shown in Fig. 18. However, at the much higher frequency of 500 Hz, the situation is nearly reversed with impact damping to be much more significant than the friction damping.
6. Conclusions We have presented a 3D DEM to simulate the motion of particles in a PVD. The metal particles are represented as spheres. For particle – particle and particle – wall contacts, a normal contact model and a tangential contact model are used to account for the coefficient of restitution and the dry friction. Both impact and friction are known to play an important role in the damping effect provided by the particles on their container structure. An efficient ‘‘box’’ algorithm is used for the detection of collisions in the dense particle flow during the course of vibration. With the DEM modeling tool, we performed numerical simulation for a number of examples of a single degree of freedom system. For a physical particle vibration damper with experimental results of a previous investigation, simulation results of the DEM modeling are found to have a good agreement with the experimental data. Furthermore, simulations provide information of particle motions within the container hole during three different regions and help explain their associated damping characteristics. The computational complexity of the DEM scheme is analyzed with
Acknowledgements The research work reported in this paper is sponsored in part by the Hong Kong Research Grants Council (project No. CUHK4196/01E), ASM Assembly Automation, a Croucher Chinese Visitorship of the Croucher Foundation of Hong Kong for Professor T. N. Chen, and the National Science Foundation of China (NSFC) under grant No. 59775019. The authors wish to express their sincere thanks to C.K. Liu, P.K. Choy, and G.P. Widdowson of ASM Assembly Automation for their valuable insight and support.
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