Particle Impact Dampers: Past, Present, and Future: History Edit

Particle damping, an effective passive vibration control technology, is developing dramatically at the present stage, especially in the aerospace and machinery fields. The aim of this paper is to provide an overview of particle damping technology, beginning with its basic concept, developmental history and research status all over the world. Furthermore, various interpretations of the underlying damping mechanism are introduced and discussed in detail. The theoretical analysis and numerical simulation, together with their pros and cons are systematically expounded, in which a discrete element method of simulating a multi-degree-of-freedom (MDOF) structure with a particle damper system is illustrated. Moreover, on the basis of previous studies, a simplified method to analyze the complicated non-linear particle damping is proposed, in which all particles are modeled as a single mass, thereby simplifying its use by practicing engineers. In order to broaden the applicability of particle dampers, it is necessary to implement the coupled algorithm of finite element method (FEM) and discrete element method (DEM). In addition, the characteristics of experimental studies on particle damping are also summarized. Finally, the application of particle damping technology in the aerospace field, machinery field, lifeline engineering and civil engineering are reviewed at length. As a new trend in structural vibration control, the application of particle damping in civil engineering is just at the beginning. The advantages and potential applications are demonstrated, whereas the difficulties and deficiencies in the present studies are also discussed. The paper concludes by suggesting future developments involving semi-active approaches that can enhance the effectiveness of particle dampers when used in conjunction with structures subjected to non-stationary excitation, such as earthquakes and similar non-stationary random excitations.

  1. INTRODUCTION

 

Particle damping technology is a form of an auxiliary-mass type vibration damper, wherein multiple metal, tungsten carbide, ceramic or other types of small particles are placed within the cavities of the vibrating structure, or the enclosures attached to the vibrating structure in order to mitigate the response of the primary structure [1,2]. As the primary structure vibrates, kinetic energy is significantly absorbed through the combined effects of particle-to-particle and particle-to-wall inelastic collisions and frictional losses, producing considerable damping to the primary structure [3-6]. Particle damping technology has been widely used due to its conceptual simplicity, moderate cost, good durability, and temperature insensitivity. Provided that the operating temperature is below the particles’ metal melting point, it could maintain normal operation. Furthermore, material such as tungsten powder can withstand the high temperatures of nearly 2,000 degrees Celsius. Additionally, properly-designed particle dampers can be effective over a broad frequency range [2,7-10]. Particle dampers are also suitable for employment in long-term harsh environments, such as high temperature, severe cold, oil contamination, etc., where other types of damping devices are no longer suitable or efficient, thus making the use of particle dampers a low-maintenance damping methodology. In addition, compared to the traditional passive damping technology, particle damping technology can be embedded or mounted without causing significant modifications in the mass or stiffness of the primary structure [11,12]. This vibration attenuation technology has been widely used in the aerospace and machinery fields, producing many kinds of industrial applications, for instance, the vibration suppression of cutting tools [13], engine turbine system in the space shuttle [2,14], antenna structures [10,15], etc. Furthermore, particle damping technology also shows its superiority in the vibration and noise control in lifeline engineering, such as wind turbines [16], power transmission towers [17,18], subsea jumpers [19,20], etc. It is worth mentioning that several papers have been conducted to study its application in reducing the vibrations of surgical and dental instruments for mechanical material ablation [21]. It has been found that particle damping technology could achieve excellent effectiveness of vibration attenuation. However, owing to its highly non-linear characteristics [22,23], its vibration attenuation mechanism has not been fully understood, and a comprehensive optimum design methodology has not been formed up to now [9].

Nowadays, structural vibration control approaches play a vital role in the earthquake and wind resistance fields, which mainly include passive control [24,25], active control [26,27], semi-active control [28,29] and hybrid control [30,31]. Passive control technology has been gaining wide acceptance. There are lots of successful paradigms [32,33], among which seismic isolation [34,35], viscous dampers [36], viscoelastic dampers [37], metallic dampers [38], friction dampers [39], tuned liquid dampers [40,41], tuned mass dampers [42-45], etc., are widely applied. However, seismic isolation techniques, such as laminated rubber bearing, may lose efficacy under near-fault earthquakes owing to the excessive horizontal displacement of the isolation bearing. Viscoelastic materials are sensitive to temperature, and therefore, they tend to lose efficacy and degrade in extremely high or low temperature. Although friction dampers could accommodate high temperature environments (such as turbine blades), their performance is closely related to the compact degree of the two interacting objects’ contact. Hence, their effectiveness will be reduced due to the change of surface condition. In addition, under various kinds of dynamic action, material properties tend to degrade and will induce fatigue effects. Tuned liquid dampers also face the dilemma that it is difficult to solve the leakage problems and apply them in severe environments (such as extreme temperature). Tuned mass dampers are effective in a small range around the resonance frequency, and they are sensitive to the working conditions [46]. Consequently, due to its special advantages, particle damping technology provides a wide range of applications and a good developing potential in civil engineering applications. While the particle damping technology has been widely used in the aerospace and machinery fields, its applications in civil engineering has not been explored thoroughly.

The aim of this paper is to review the state-of-the-art technologies of particle damping, acting as a full reference for interested readers in this research field. This paper is structured as follows: A brief review of the origin and major developments of particle damping technology is given in Section 2. The research results of the damping mechanism, theoretical analyses, and numerical simulation methods are introduced in detail in Section 3 and 4. In Section 5, the main results of experimental studies on particle damping are summarized. Applications of particle damping to various fields are discussed at length in Section 6. The new trend to utilize particle damping to achieve structural control in civil engineering, with its corresponding advantages and disadvantages, is also demonstrated. Some discussion of research needs and concluding remarks are presented in Section 7 and 8.

 

  1. BASIC CONCEPT AND DEVELOPMENT OF PARTICLE DAMPING

 

The concept of particle damping could be traced back to 1937, when Paget [47] was studying the vibration attenuation problem of the turbine blades, during which he invented the impact damper. The impact damper only involves a single particle, resulting in high noise levels and significant impact forces during the impact process, and it tends to become sensitive to the change in certain parameters (such as the excitation amplitude and the restitution coefficient). Later on, in 1945, Lieber and Jensen [48] proposed the concept of using a mass moving between two walls of a container to eliminate the vibration of mechanical systems, which evolved to the form of the impact damper. Because of a variety of flaws in a single-particle impact damper, its further application and development in more fields are limited. Hence, subsequent researchers have replaced the single particle with many smaller particles of equivalent total mass, thus resulting in the particle damper.

 

2.1 Basic traditional particle dampers

Based on the number of units and particles per unit, the traditional particle dampers can be grouped into four major types: the impact damper [48], the multi-unit impact damper [49,50], the particle damper [51,52] and the multi-unit particle damper [11,53], as shown in Figure 1. The development and evolution of these four fundamental types of particle dampers have broken the traditional single design concept, laying a solid foundation for the development of the subsequent multiform particle dampers.

 

 

 

 

 

 

 

 

 

 

             (a)                               (b)                                  (c)                                  (d)   

Figure 1. Schematic diagram of the traditional particle dampers: (a) Impact damper; (b) Multi-unit impact damper; (c) Particle damper; and (d) Multi-unit particle damper.

 

The multi-unit impact damper is the descendant of impact dampers. In 1969, Masri [50] obtained the exact solution for the steady-state motion of a multiple-unit impact damper attached to a sinusoidally excited primary system, verified by numerical simulation and experiments. He found that, compared to the impact damper of equivalent total mass, as to reducing vibration, especially reducing the noise generated during the damper’s impact process, properly designed multi-unit impact dampers are more effective. The increasing interest in the particle damper dates from 1985, when Araki [51] replaced the shock unit of the traditional impact damper containing a single particle with a bed of granular materials. He studied the effectiveness of this kind of impact damper for reducing the vibration of a single-degree-of-freedom (SDOF) system under simple harmonic excitation, and the effect of mass ratio and clearance on the performance of the damper was determined. The non-obstructive particle damper (NOPD), which is one of the most common types of particle dampers, was proposed by Panossian [2] in 1991 through the experiment of an aluminum beam. Such technology consists of making small diameter cavities at appropriate locations inside the vibrating structure, which is capable of yielding the maximum damping effect for any desired mode. The multi-unit particle damper was first put forward by Saeki [11] in 2005. He found that the damping performance of the particle damper depends on the size of the cavity. If the optimum size of the cavity is too large, from the standpoint of practical design, it will lead to excessive clearance and decrease the number of effective collisions between the particles and the wall, which has negative effects on the damping performance. Hence, Saeki divided the cavity into multiple small cavities with an appropriate number and the influence of the number and the size of the cavities on damping performance was studied.

 

2.2 Development of particle dampers

Based on the traditional particle dampers, several variants have evolved. Each variant has its own features so that it can be selected depending on the engineering requirements. According to the different aspects of the improvements made in the traditional particle dampers, these variants can be classified into three main categories: the configuration improved type, the material improved type and the combination type, as shown in Figure 2.

Figure 2. Particle dampers categorization.

 

2.2.1 The configuration improved type of particle dampers

The configuration improved type improves the traditional particle dampers by changing structural configurations, e.g., particles and cavities. It can optimize the mechanism of vibration attenuation and improve its characteristics in order to apply it to a wider range of working conditions. This type of variants mainly includes the bean bag impact damper [54], the piston-based particle damper [55,56], the beam-like impact damper [57], the linear particle chain impact damper [58], the tuned rolling-ball damper [16], the ball vibration absorber [59], etc. The bean bag impact damper (BBD) is also called the flexible restraint particle impact damper, which originated in the early 1980s. Popplewell et al. [54] adopted this kind of damping technology when he was studying the vibration attenuation problem of the boring bar. The bean bag impact damper wraps a certain amount of metal or non-metal micro particles with a soft bag which has a certain elastic resilience, putting it into a specific structural cavity instead of the rigid mass block of the traditional impact damper, as shown in Figure 3. Compared with the traditional impact damper, the bean bag impact

 
 

damper has the feature of no impulsive noise and a wide frequency band of vibration attenuation.

 

 
 
 
 

 

(a)

(b)

Figure 3. (a) Schematic diagram of BBD; (b) mechanical model of BBD.

 

The piston-based particle damper introduces damping pole on the basis of the non-obstructive particle damper, shown in Figure 4(a). When the main structure is vibrating, the particles and the damping pole collide, press and rub together, converting mechanical energy into thermal energy and acoustic energy, which generates the damping effect. Through experiments, Binoy et al. [60] found that the damping effect of such piston-based particle dampers is entirely unrelated to the temperature. Besides, it can still exert damping effect under the condition of small external excitation, whereas the impact damper usually cease to be effective in that case. Afterwards, on the basis of the piston-based particle damper under passive control, Binoy et al. [55] proposed to achieve semi-active control by means of magnetic field, shown in Figure 4(b). Regarding some specific structures, such as flexible mechanical arm and cantilever beam, they will generate a large instantaneous movement because of their flexibility when they are stopped suddenly during the process of fast running. In order to alleviate the damage into the structure caused by impulsive instantaneous vibration, Chen et al. [57] put forward the beam-like impact damper. Compared with the traditional impact damper, the beam-like impact damper can overcome the dependence of its damping performance on the direction. To enhance the dissipation of kinetic energy during the impact process, Gharib et al. [58] put forward a novel impact damper, called the linear particle chain impact damper (LPC impact damper), which was formed by a small-sized ball between each two large-sized balls, shown in Figure 4(c).

 

     

(a)

(b)

(c)

Figure 4. Schematic diagram of (a) the piston-based particle damper under passive control; (b) the piston-based particle damper under semi-active control by magnetic field; and (c) LPC impact damper.

 

2.2.2 The material improved type of particle dampers

The material improved type improves the materials of particles and cavities by employing novel materials or using traditional materials skillfully, in order to reduce noise levels, impact forces and increase plastic deformation, thus increasing the energy absorption. This type of variant mainly includes the buffered impact damper [61], the fine particle impact damper [62], the polymeric particle damper [63,64], the elastomer particle damper [65], the soft hollow particle damper [66], the metal swarf damper [67], the metallic wires particle damper [68-70] etc. Li et al. [61] covered the inner wall of the particle damper with thin rubber material, which formed the buffered impact damper. According to the experimental study, it was shown that rubber can alter the stiffness of the inner wall, leading to considerable improvements on the working performance of the dampers over a wide range of frequencies, especially decreasing the impact force and noise levels. Du [62] first proposed the concept of the fine particle impact damper in which a small quantity of fine particles are enrolled as damping agent, coupled with some steel spheres enrolled as impact partners. When the main structure is vibrating, the impact of steel spheres bring about plastic deformation of fine particles which are usually surrounded and covered around steel spheres, resulting in consuming the vibration energy of the main structure permanently. Darabi et al. [63] substituted relatively large particles with significant viscoelasticity for traditional rigid particles, thus forming the polymeric particle damper. He found that at relatively higher amplitude, the particles get into a convection region, in which the damping levels due to the combined effects of both friction and viscous are irrespective of the material properties.

Bustamante et al. [65] put forward the elastomer particle damper, which was formed by using elastomer as the material for the particles. The experimental results revealed that when the movement of particles reaches the fluidization point, the elastomer particles become optimally excited, thus leading to maximum damping. Michon et al. [66] replaced traditional rigid particles by soft hollow particles, in order to strengthen the properties of honeycomb structures in aerospace applications and alleviate the mass impact on the main structure, which formed the soft hollow particle damper. Experimental research indicated that this kind of particle dampers present good performance over a large frequency range with low impact on the main structure. Hussain et al. [67] proposed to substitute metal swarfs with rubber spheres and studied the damping characteristics of three metals (namely aluminum, stainless steel and mild steel) through experiments. The experiments found that metal swarfs can yield considerable damping in the main structure, across a wide range of excitation amplitudes and frequencies.

 

2.2.3 The combination type of particle dampers

The combination type combines the traditional particle damping technology with some existing technologies, which improves the damping performance in certain conditions and breaks through the bottleneck of particle dampers in engineering applications. This type of variant mainly includes the impact damper-tuned absorber combination [71-73], the particle tuned mass damper [74-76], the tuned liquid particle damper [77], etc.

The concept of integrating the impact damper with the tuned absorber was introduced first by Masri [78] in 1971 who proposed the use of the Dynamic Vibration Neutralizer (i.e., Tuned Mass Damper) with motion-limiting stops that reflect the physics of an impact damper interacting with the primary system when a motion threshold is exceeded. The study in Ref. [78] provided an exact analytical solution under steady-state conditions, as well as experimental verification demonstrating the potential advantages of such devices to provide effective damping at relatively low as well as high levels of the primary system response. Semercigil et al. [72] found that the addition of the impact damper can improve the vibration attenuation effect of the tuned vibration absorber significantly through experimental study. Chen et al. [71] put forward the tuned particle damper, which was achieved by a particle damper attached to the main structure with flexible supports, based on the operational principles of the classical dynamic vibration absorber, as shown in Figure 5. Compared with the particle damper with rigid supports and traditional dynamic vibration absorber, the tuned particle damper solves the ineffectiveness of conventional particle dampers at low vibration acceleration, especially, less than the acceleration of gravity (1g).

 

 
 
 
 
 
 
 
 

            

 
 
 
 

Particle damper

 
 

(a)

(b)

Figure 5. Mechanical model of (a) the classical dynamic vibration absorber; (b) the tuned particle damper.

 

Researchers have introduced particle damping technology into tuned mass dampers (TMDs), in order to take advantage of each other’s advantages and combine multiple damping mechanisms, for instance, tuning, impact, friction, etc. Under this combined energy-consuming mechanism, non-linear energy consuming system is formed, which can widen the frequency band of vibration attenuation, enhance robustness and heighten the control effects. This new device is called the particle tuned mass damper (PTMD). Lu et al. [74,75,79] introduced the PTMD to the vibration control field of high-rise buildings, and a series of shaking table tests and wind tunnel test have been conducted, during which the PTMD is suspended on the top of the primary structure by four strands with the same length. It has been shown that the PTMD has significant effect on the suppression of wind-induced and earthquake-induced vibration of high-rise buildings. Yan et al. [76] carried out the experimental research of shaking table tests for a 1/10-scale model bridge with this technology, and found that the damping effect and the robustness can be enhanced.

 

  1. DAMPING MECHANISM AND THEORETICAL ANALYSIS OF PARTICLE DAMPING

 

3.1 Damping mechanism of particle damping

The damping mechanism of particle damping mainly includes the energy consumption among particles and the impact energy dissipation between particles and the main structure. The method of using energy consumption among particles to reduce the oscillations of vibrating body has been used for several centuries, such as putting a bag filled with sands on the vibrating body, placing the particle material around the vibrating body, applying the sand-closed damping structure to the metal-cutting machine tool, by which the damping can be increased by 8~11 times [80], etc. The explanations about the mechanism of the damping among particles mainly include as follows: Kerwin [81] argued that particle material consumes the energy of the main structure by three approaches, which were: (i) the friction among the particles, (ii) nonlinear deformation of the contacting points among the particles, and (iii) the resonance of the particle material. Lenzi [82] stipulated that dry friction among the particles is the main mechanism of damping. Sun et al. [83] thought that damping is generated during the process in which acoustic energy radiated by structures is being consumed by sand particles. Single impact damper, with the rigid mass block as the impact body, is the typical representative of the impact damping theory, which is also the origin of the particle damper. The explanations about the mechanism of impact damping mainly include the following interpretations: Some scholars held the opinion that the mechanism of impact damping is a certain kind of energy loss which is produced based on the non-perfect elastic collision [84]. Popplewell [54] considered that the impact damping is mainly achieved by momentum exchange during the process of impact.

More specifically, regarding the particle dampers in different forms, many scholars have studied the balance between the different approaches of energy consumption, and the relationship between the damping mechanism and the design parameters. For example, Energy dissipation of particle dampers was predicted by Wong et al. [5] via the discrete element method. The results revealed that despite the fact that friction plays a leading role in energy dissipation, the actual friction coefficient has a little influence on energy consumption. Besides, Masmoudi et al. [85] discovered that the loss factor is unrelated to both the number of particles and the material properties, whereas it is only closely associated with the total mass of particles and the excitation amplitude. Chen et al. [86] elaborated the new damping mechanism of non-obstructive particle dampers based on the rheological behavior of damping particles. It was shown that the optimal damping performance of NOPDs comes from two aspects, and the vibration energy of the main system is dissipated through thermal energy and potential energy.

 

3.2 Nonlinear Energy Sink (NES)

Despite the fact that researchers around the world have done a lot of work, with regard to the damping mechanism of particle damping, no common explanation has been agreed on. From the broad theory of view, actually, particle damping technology belongs to the category of Nonlinear Energy Sink (NES) which is a vibration absorption technology studied enthusiastically in many fields, such as machinery [87], aerospace [88] and civil engineering [89]. There are resemblances between this type of nonlinear absorbers and tuned mass dampers (TMDs). Both of them reduce the response of the main structure by movement of the additional mass and should be placed at the position with large displacement (as for building structures, it usually is the top storey). The major difference between the two is that the resilience generated by TMD is linear, whereas the resilience generated by NES is nonlinear. Consequently, NES is different from the general linear oscillators which have single inherent frequency. The inherent frequencies of NES are variable, which tends to increase with the increase of vibration energy. Therefore, NES can resonate with multiple modes of the main structure within a wider frequency band. In other words, the goal of reducing the vibration for a wide range of frequencies can be attained by NES [90-92]. Based on the traditional tuned mass damper, Li and Song et al. [18-20] added the impact energy dissipation and put forward the pounding tuned mass damper. In the design, the vibration energy can be dissipated through heat generated by the impact or pounding between a tuned mass block and a delimiter covered with viscous-elastic material. Consequently, pounding TMD (i.e., a Dynamic Vibration Neutralizer with motion-limiting stops [78,93]) is also regarded as a kind of NES with good robustness. Besides, it has been successfully employed to the vibration control of subsea jumpers [19,20] and power transmission tower [18]. The mechanical models of different types of dampers are shown in Figure 6.

C

M

K

m

M

K

C

m

c(x)

K

M

C

m

(a)

(b)

(c)

c

k

M

C

K

m

C

 

M

K

c

k

m

k

C

K

M

c

m

(d)

(e)

(f)

k(x)

Figure 6. Mechanical model of (a) impact damper; (b) particle damper; (c) nonlinear energy sink; (d) tuned mass damper; (e) pounding tuned mass damper; and (f) particle tuned mass damper.

 

3.3 Theoretical analysis of particle damping

Regarding the theoretical analysis of particle damping, a lot of researchers took a SDOF system as the research object. Since the particle-particle and particle-wall collisions will cause abrupt changes of the motion parameters (such as velocity), its dynamic behavior shows a strong non-linearity. Therefore, only under certain conditions, can analytical solution be obtained, e.g., the main structure with single-particle dampers (impact dampers) under simple excitation (harmonic excitation or steady-state vibration) and assuming symmetrical collisions as twice per cycle.

 

3.3.1 Theoretical analysis of single-particle dampers (impact dampers)

Theoretical analysis of impact dampers first began with the work of Lieber et al. [48]. They considered each collision as a completely plastic collision. Grubin [94] introduced the elastic restitution coefficient of collision into analysis. He took account of the energy loss during the collision and established the theoretical model of the SDOF system attached with an impact damper under harmonic excitation. Masri [95] extended this assumption to the case of symmetric collisions twice per cycle. Furthermore, the non-linear control equations were adopted by Bapat [96] to analyze the vibration state of a SDOF system in case of colliding N times per cycle under harmonic excitation. Masri [50,95,97] deduced the analytical solution for the steady-state motion of the impact damper and the multi-unit impact damper attached to the sinusoidally excited main structure. Moreover, their stabilities were also analyzed. Bapat and Sankar [98,99] analyzed the influence of the coulomb friction. They plotted a table that shows the optimum gaps and the corresponding amplitude reduction when the impact damper is subjected to forced vibration [98]. Ema and Marui’s research [100] showed that the additional damping provided by the impact damper is due to the collisions of the impact mass with the primary system. In addition, the optimal damping is affected by the mass ratio and the clearance. The damping characteristics of a vertical impact damper under a wide range of frequencies and multiple amplitudes was studied by Duncan et al. [101] via numerical simulation.

 

3.3.2 Theoretical analysis of multi-particle dampers

Regarding multi-particle dampers, involving interactions between particles, it is difficult to acquire the analytical solution for such a system. One example of an exact analytical solution, under the assumption of steady-state response under harmonic excitation was derived and its stability boundaries were established by Masri [102]. Consequently, theoretical analysis of the nonlinear damping mechanism are very limited, and have to rely on experimental studies and numerical simulation. Mao et al. [6] simulated particle damping via the discrete element method. The simulation results demonstrated that the particle damper can provide considerable additional damping generated by the combined effects of impact damping and friction damping, which can produce a rapid decay in the amplitude of the main structure over a finite period of time. The characteristics of energy dissipation as for the vibrating particles were studied by Saluena et al. [103] through molecular dynamics simulations. They concluded that under different amplitudes and frequencies of external excitation, rates of energy loss correspond to three different regimes: namely solid, convective and gas-like regimes. Sánchez et al. [23] simulated the complicated dynamic interaction between the particles by the discrete element method. It was shown that with the increase of the excitation frequency, particle motion switches from a periodic motion to a chaotic motion. Ben et al. [104] extended the application of the Fourier-based Power Flow Method proposed by Yang [105] and developed a new experimental method, through which the loss factor of NOPDs can be measured quantitatively. Xiao et al. [106] presented the model of frictional energy dissipation and the model of collision energy dissipation based on the powder mechanics and the collision theory, respectively. The concept of ‘Effective Momentum Exchange’ that can be used to interpret the physical properties of the particles’ movement was proposed by Lu et al. [107-109]. This concept is also regarded as a comprehensive index to indicate the optimal damping performance.

 

  1. NUMERICAL SIMULATION OF PARTICLE DAMPING

 

As theoretical studies of multi-particle dampers are constantly enriched and evolving, researchers have worked out a series of simplified methods and numerical methods in order to simulate the non-linear effects generated by the particle-particle and particle-wall interactions. Papalou and Masri [52,110,111] simplified a multi-unit particle damper to an equivalent single-unit impact damper. Analogously, Friend and Kinra [7] wrapped all the mechanisms of energy dissipation into a new concept named ‘effective coefficient of restitution’ by means of modeling the multi-particle as a cohesive mass block. This coefficient was obtained by fitting the test data, and thereby a novel analytical method was formed. Moreover, Liu et al. [4] adopted equivalent viscous damping to simulate the non-linear characteristics of particle dampers, based on summarizing the corresponding experimental results. Xu et al. [112] put forward an empirical method for designing particle dampers, by which the relationship between the damping effect and each parameter was obtained through fitting the test data. Wu et al. [113] introduced multiphase flow theory of gas particles to the analysis of particle dampers and the corresponding theoretical model was presented. On the basis of that, Fang and Tang [114] further improved and completed the theoretical model to reduce analysis complexity and its computational cost. Recently, Wu et al. [115,116] utilized multiphase flow theory to the numerical analysis of vibration and acoustic radiation for rectangular plate with particle dampers by the aid of COMSOL multiphysics software package. Furthermore, the simulation methods, utilized to simulate the particle damping, also include restoring force method [117,118] , neural network method [119], molecular dynamics method [103], turbulence theory [120], etc. Despite the fact that these simplified models and experiment-based studies have gained substantial achievements, most of them belong to phenomena-based methods, which means that the conclusions drawn from these methods are difficult to extrapolate to other cases.

 

4.1 Discrete / distinct element method (DEM)

For the past few years, the discrete / distinct element method (DEM) has been widely introduced to the analysis of particle dampers [11,12,121-123]. DEM, proposed by Cundall [124] in 1971, is an effective method which is specially utilized to solve the non-continuum problems, such as the mechanical behavior of rock. Because DEM can account for particle-particle and particle-wall interactions, this method is quite suitable to analyze particle dampers quantitatively.

The DEM is a numerical analysis method that can analyze the mechanical behavior of discrete bodies via iterative solutions of step by step. In this method, discrete bodies are divided into a collection of numerous discrete elements whose movements are described by local contact laws and Newton’s equation of motion. It should be emphasized that DEM is based on the hypothesis that if the time step chosen is extremely small, during a single time step, disturbances cannot propagate from any particle further than its immediate neighbors. Hence, at all times, the forces acting on any particle are determined exclusively by its interaction with the adjacent particles, and then the behavior of the entire system can be captured in detail.

Figure 7(a) shows the model of a multi-degree-of-freedom (MDOF) structure with a particle damper on the top floor. The governing equation of the main structure can be written as

 

 

 

(1)

 

 

(2)

 

where M, C and K are mass, damping, and stiffness matrices, respectively; F is the contact force vector; E is the mass matrix induced by ground acceleration ; X is the relative displacement vector of each floor with respect to the ground and Fi is the contact force acting on the i-th floor by particles (i =1, 2, … , N), which is the linkage between the particles and the primary system.

The governing equation for a particle i at a certain moment can be written as

 

 

(3)

where  is the mass of particle i,  is the moment of inertia of particle i and  is the acceleration vector due to gravity;  is the position vector of the center of gravity of particle i,  is the angular displacement vector,  is the normal contact force between particle i and particle j (if particle i is in contact with container wall, then j denotes the wall), and  is the tangential contact force. The contact forces act at the contact point between particle i and particle j rather than the particle center, and they will generate a torque , causing particle i to rotate. For a spherical particle of radius ,  is given by , where  is the unit vector from the center of particle i to the center of particle j,  denotes the cross product, and  is the number of the particles in contact with particle i.

 

 
 

i

1

2

N-1

m

N

    

                                                  (a)                                                   (b)

Figure 7. (a) Model of a MDOF structure with a particle damper system, and (b) normal contact force model between a particle and the wall.

 

Many scholars have proposed various contact force models to determine the normal and tangential contact forces [125,126]. However, the linear contact model in the normal direction and the coulomb friction model in the tangential direction are widely adopted in the simulation study nowadays.

Figure 7(b) shows the linear contact model between a particle and the wall in the normal direction. Parameters ,  are the stiffness and natural frequency of the spring, respectively.  can be used to simulate a rigid barrier to any degree of accuracy, by a proper choice (  [127]). Parameters ,  are the damping coefficient and damping ratio of the damper, respectively.  can be used to simulate inelastic impacts, ranging from the completely plastic up to the elastic one, so that the value of any desired coefficient of restitution e can be adjusted by selecting the proper value for . Similarly, , ,  and  are the stiffness and natural frequency of the spring, the damping coefficient and damping ratio of the damper, respectively, in the inter-particle contact model along the normal direction. Hence, the normal contact force is expressed by

 

 

(4)

where  and  is the relative displacement and velocity of particle i with respect to particle j, and  is the distance from the center of particle i to the wall.

Considering Coulomb’s law of friction, the tangential contact force is expressed by

 

 

(5)

where  is the particle-particle or particle-wall friction coefficient, and  is the relative velocity of particle i with respect to particle j in the tangential direction.

According to the governing equation and the contact force model mentioned above, the procedure for calculating the response of a MDOF system provided with a particle damper using DEM can now be summarized as follows: First, determine the relative position of particle-to-particle and particle-to-wall. If , the contact force acting on this particle can be calculated from Equations (4) and (5), whereas if , no contact force will be generated. Second, if the contact forces acting on this particle exist, including particle-to-particle forces and particle-to-wall forces, then add them up. Third, the particle motion can be given by Equations (3). The above procedure is carried out for all the particles one by one. Finally, the contact force F acting on the primary system can be determined by summing all the contact forces between the particles and the wall. The equation of motion for the primary system, Equation (1) can be updated by using the contact force F.

It is worth mentioning that even though the DEM possesses the advantages of simple principle and can simulate the non-linear effect of particle dampers with high fidelity, many problems still exist mainly owing to its high requirements in computational resources. These problems mainly include how to build the contact force models that can describe the physical behaviors accurately, and how to determine the calculation parameters reasonably, such as stiffness coefficient in normal and tangential direction, damping coefficient and time step, and how to search and judge the contact status of particles effectively. Moreover, it is quite time-consuming to adopt DEM to simulate when the particle number is extremely large. All of these problems have a significant influence on the accuracy of the results [128].

 

4.2 Simplified analytical method

Considering the challenges in large calculations involving the DEM, Lu et al. [74] proposed a simplified analytical method, based on the correlational studies conducted by Papalou and Masri [52,110,111]. The essence of this method is to make all particles being equivalent to a single one, based on certain equivalence principles, to simulate the vibration effects of particle dampers. During the process of equivalent simulation, the collisions among the particles are neglected. However, the impact between particles and the wall tends to be the dominant factor that has a significant influence on the global response of the main structure. It should be noted that this simplified analytical method captures the most principal control force of particle dampers, so that it possesses a satisfactory degree of accuracy in practical applications, and can provide a useful reference for realistic applications in engineering projects.

Actually, this equivalent simplified method has a sound theoretical foundation, which is validated by scholars through numerical simulation and experimental studies. Sánchez et al. [129] has pointed out that when the number of particles is sufficiently large and the particles get into a state of dense lumped mass, the damping performance of particle dampers is insensitive to the material properties of the particles as well as the particle–particle interaction. This phenomenon was called as ‘universal response’, which not only explains the suitability of particle dampers working in harsh environments without having a specialized maintenance, but also greatly simplifies the design schemes of particle dampers. Moreover, Sánchez et al. [23] also simulated the complex dynamic interaction among the particles by DEM, and found that when the optimum damping is obtained, the particles tend to be dense. Interestingly, the dynamic characteristics of particle dampers in that case are similar to the impact damper with single particle.

Bannerman et al. [130] attached a particle damper to the free end of a spring blade and conducted a vibration control experiment under micro-gravity environment. It was shown that the effective energy dissipation is carried out under the collect-and-collide regime. Additionally, Sack et al. [131] performed a similar experiment with regard to steady state systems and drew the same conclusion.

The equivalent principles, based on the observation that a particle group is equivalent to a single particle, can be stated as follows:

1) The void volume of the multi-particle damper is equal to that of the single-particle damper.

2) The total mass of the particles in the multi-particle damper is equal to that of the single-particle damper.

3) The particles in both the multi-particle damper and the single-particle damper are spheres, and their densities are keep constant as ρ.

4) The container of the multi-particle damper is a cuboid, whereas the container of the single-particle damper is a cylinder, and its bottom diameter is equal to the diameter of an equivalent single particle.

Figure 8 shows the schematic diagram of the equivalent single-particle damper based on these principles. The parameters and corresponding symbols of the multi-particle damper and the equivalent single-particle damper are shown in table I.

 

Figure 8. The multi-particle damper and the equivalent single-particle damper.

 

Table I. The parameters and corresponding symbols of the multi-particle damper and the equivalent single-particle damper.

 

Physical significance

Symbolic representation

Multi-particle damper

Cavity size

(length, width, height)

 

Single particle mass

 

Total particle mass

 

Particle diameter

 

Equivalent single-particle damper

Particle mass

 

Particle diameter

 

Particle clearance

 

 

The simplified method can be carried out as follows:

    The volume occupied by particles in the multi-particle damper  can be obtained from Equation (6):

 

 

(6)

where N is the total number of particles and .

    The void volume of the multi-particle damper  can be obtained from Equation (7):

 

 

(7)

where  is the volume of the multi-particle damper and the volume ratio of particles to container, called the packing density of the multi-particle damper is .

The void volume of the single-particle damper  is

 

 

(8)

Based on the principle 1), by equating Equation (7) and (8), Equation (9) can be obtained:

 

 

(9)

The research of Hales [132] indicated that the volume ratio of close-packed spheres with equal radius in three dimensions should not exceed 0.74, which means . Based on the principle 2), that is, , the clearance of the equivalent single-particle damper can be obtained from Equation (10):

 

 

(10)

 

The key to the equivalent simplified simulation is the impact simulation between the particles and the wall, which is based on piecewise linearization, as shown in Figure 9.

 

 


 
 
 

 

 
 
 

(a)

(b)

Figure 9. (a) The form of function , and (b) function .

 

where y is the relative displacement of the simplified particle with respected to the container.  and  are the nonlinear functions that represent the relative displacement and velocity of the simplified particle with respected to the container during the collision process, which contribute to elastic force and damping force, respectively. To be more specific, when the particle moves within the length of the container d, there is no collision happening. Consequently, in this case, function  and  are equal to zero. However, the collision is generated when the particle touches the wall of the container ( ), so that function  and  exist with non-zero values. It should be noted that the penetration of the particle to the container wall during the impact process is rather small, and the continuous impact simulation can be implemented through this piecewise linear relationship.

The simplified method is illustrated by the example of a SDOF structure, on the top of which an equivalent single-particle damper is hung by rope. The schematic diagram is shown in Figure 10.

 

Main structure

Figure 10. The schematic diagram of the main structure with the equivalent single-particle damper.

 

The governing equation of the main structure can be expressed in matrix and written as equation (11):

 

 

(11)

where ,  and  are the mass, damping and stiffness matrices of the main structure, respectively; , ,  and  is the acceleration, velocity, displacement load vector, respectively.

    Assume the displacement of the shaking ground is , the displacement and mass of the main structure are  and , respectively; the displacement and mass of the container are  and , respectively; the displacement and mass of the simplified particle are  and , respectively. Then the governing equation of the main structure can be expanded as equation (12):

 

 

(12)

where  denotes the relative displacement of the simplified particle with respect to the container. The form of equation  and  which determine the nonlinear characteristics of the model, as shown in Figure 9.

 

4.3 The coupled algorithm of FEM and DEM

When particle dampers are employed in conjunction with MDOF structures, the simulation of particle dampers can be conducted by DEM, while the analysis of the main structure is usually performed via the finite element method (FEM). The contact force between the particles and the wall eventually acts on the main structure, and then the damping effect can be calculated. Hence, it is a natural trend to introduce the combination of FEM and DEM to the design of particle dampers applied in MDOF structures. Xia and Liu et al. [133] utilized particle damping to a rotating brake drum of the vehicle to control its noise and vibration. They implemented the design and calculation of particle dampers based on the coupled algorithm of FEM and DEM. In addition, Liu et al. [134] adopted the coupled algorithm to study the influence of distributed damping on the torsional vibration of a plate. Furthermore, Xia et al. [135] put forward the coupled algorithm for the rotating plate structure with particle damping. Although the combined algorithm of FEM and DEM has been applied in simple members, they lay a theoretic foundation for the extension of this coupled algorithm to much complex host structures.

 

  1. EXPERIMENTAL STUDY OF PARTICLE DAMPING

 

Due to the highly-nonlinear characteristic of particle damping, the theoretical analysis and numerical simulation have some limitations; consequently, it is difficult to establish a universal theoretical analysis method. On the other hand, many scholars carried out extensive experimental studies on particle damping. It can be summarized as following three broad approaches.

Approach #1:

The first is to carry out theoretical analysis of particle damping by virtue of experiments so as to explore the essence of the damping mechanism through experimental phenomenon. For example, Sadek et al. [136,137] studied the influence of gravity on the impact damper and found that the damper has better effect in zero gravity environment. Moreover, in the vicinity of resonance, asymmetric collisions twice per cycle is dominant. Cempel and Lotz [138] studied the vibration damping through filling particles in a container and found that the energy dissipation of the impact particle not only relies on the collision of internal particles, but also relates to the external collision (referring to the collision between the particles and the wall). In addition, the friction is also a factor that affects the energy dissipation of the impact particle. Hollkamp and Gordon [8] used metal and ceramic particles as the impacting bodies and put them into the structural cavities. It was found that when the main structure vibrates, the vibrational energy can be consumed by the collisions among the particles. The research of Xu et al. [139] focused on the contribution of shear friction induced by strain gradient along the length of the structure to damping. The experimental results indicated that particle dampers can provide significant additional damping over a broad frequency range, and the optimal damping might be obtained by utilizing multi-particles as the impact bodies given that the impact, friction and shear mechanisms are accounted for.

Approach #2:

The second is to validate the numerical results and investigate the damping performance of various forms of particle dampers utilized in different vibration objects as well as external excitations. For example, an experimental study of a vertical multi-particle impact damper under free excitation was performed by Trigui et al. [140], by which the influence of clearance and excitation intensity was studied. The damping effectiveness of a particle damper under centrifugal loads was analyzed by Daniel [141] via the experiment of a rotating cantilever beam. Jadhav et al. [142] compared the damping performance of the single-cell and multi-cell particle dampers by lots of experiments. Moreover, they established the models of the single-cell and multi-cell particle dampers using dimensional analysis method and validated their correctness by experiments. Zhang et al. [143] simplified the resilient bag in BBD (bean bag impact damper) to a particle-spring model based on DEM, which was verified by test data. The experiment of an L-shaped cantilever beam with a particle damper in the top free end was conducted by Wang et al. [144], through which the damping performance under horizontal-vertical excitations in the context of free decay was investigated. It was shown that the damping characteristics of the particle damper under horizontal-vertical excitations is similar to those under only vertical excitation. Additionally, Rongong et al. [145-147] carried out several experimental studies on the amplitude dependent behavior of particle dampers.

Approach #3:

The third is to study the influence of various design parameters on the vibration attenuation effect of the system, and provide guidance for the practical applications in engineering by means of the parametric analysis. For example, Veluswami et al. [148,149] used three types of material as the coating of strike plate inside the particle damper. It was concluded that soft materials with small restitution coefficient provide light damping during the process of resonance. Yokomichi et al. [150,151] and Saeki [12] studied the response of the particle damper under harmonic excitation and found that the impact body with greater mass tends to provide more additional damping to the main structure, whereas the impact body with lighter mass tends to exert damping effect rapidly at the beginning of the vibration. Furthermore, the optimal clearance was also determined. Yang [105] summarized a series of design curves to predict the damping characteristics of the particle damper. Li [152] conducted a series of experiments to study the properties of the impact damper attached to a MDOF structure. It was indicated that increasing the mass of the particles does not necessarily enhance the damping of the primary structure for all modes. Regarding the various design parameters of particle dampers, such as particle type, particle placement, packing ratio, etc., a series of tests were carried out by Hollkamp et al. [8]. Shaking table tests of a three-storey steel frame with a buffered particle damper [153,154], and a five-storey steel frame with a particle tuned mass damper [74] were carried out by Lu et al. Several design parameters, such as the auxiliary mass ratio, gap clearance, etc., were analyzed. Sathishkumar et al. [155] filled a boring bar with different metal particles and investigated the effects of certain particle parameters, such as particle size, density and hardness, on the surface roughness of machined surface.

In summary, through experimental study, the limitations of theoretical research owing to the nonlinear characteristics of particle damping have been overcome to a great extent. It is noteworthy that many theoretical methods were proposed based on the experiments, and through experimental phenomenon, the law of particle damping can be intuitively explored. Thus, the factors affecting the attenuation effect of particle damping can be summarized into two aspects, namely external and internal causes. To be specific, the external causes mainly include excitation amplitude and its frequency characterization. The internal causes mainly include the number, size and material of particles, container size, mass ratio of total particles to primary structure, group effect between particles, restitution coefficient of the particles, particle placement, the damping of primary structure, etc. However, at the present stage, the published experimental studies are mainly concentrated in mechanical and aerospace fields, whereas the experiments based on civil engineering are relatively scarce.

 

  1. APPLICATION OF PARTICLE DAMPING TECHNOLOGY

 

With the continuous development and evolution of particle damping technology, faced with a variety of vibration reduction problems in engineering fields, particle damping technology has taken full advantage of its favorable characteristics, which has led to many types of successful application. These engineering applications mainly focus on the aerospace field, machinery field, lifeline engineering, etc. For example, the protection for antenna structures [10,15] and printed circuit board [156] under shock-loading conditions, reducing the wind-induced vibration of lamp pole, chimney and some flexible structures [157], the suppression on the vibration of tennis rackets [158], electro-mechanical relays [33], aircraft [48], engine turbine system of space shuttle [2,14], fan and compressor blisks [159] and metal-cutting machine tools [13].

 

6.1 Particle damping applications in aerospace field

There are hundreds of applications in aerospace field lasting for more than 70 years. Lieber and Jensen [48] adopted impact damper to control the vibration of aircraft. They considered twice collision in each cycle and found that when the phase angle between impact mass and major structure is 180 degrees, the damping effect is best. Rocke and Masri [15] used an impact damper to attenuate the shock response of an antenna structure. Oledzki [160] utilized the impact damper to inhibit the vibration of a long tube on the light-weight spacecraft and adopted a rheological model for numerical simulation. The numerical results showed that resonance amplitude was weakened, which fitted well to the experimental results. Moore et al. [14] used the impact damper for high-speed rotor in the rocket engine turbine system in low temperature working condition, such as the main engine of the space shuttle. Experimental results revealed that it is feasible for the impact damper to inhibit the vibration of rotor-bearing system at a low temperature. Gibson [161], Torvik and Gibson [162] used the particle damper for spatial application, and found that the attenuation rate of the system response, and the minimum effective amplitude, are two major parameters in the damper design. Simonian [163] conducted experimental studies on employing particle dampers to reduce excessive vibration of cantilever beam type space structural subsystems. Veeramuthuvel et al. [156] proposed the novel use of particle damper capsule on a printed circuit board so as to suppress the vibration of printed circuit boards.

Honeycomb composite structures, commonly used in spacecraft, have great superiority in the installation of particle dampers. The damping particles with smaller mass can be directly placed in the cells of honeycomb without adding excessive weight. Liu et al. [164] filled the hollow glass microspheres into the honeycomb lattices and sealed it with face sheets, which formed the honeycomb composite beam. According to an experimental study, it was found that honeycomb composite beams can significantly increase damping, without causing significant modifications in the mass. Moreover, Panossian [165] introduced the NOPD into honeycomb composite beams and found that the beam with the optimal NOPD treatment has the lowest response amplitude through modal vibration tests. Furthermore, it is common for a spacecraft to suffer from large shock loads during its landing process, accompanied by rebound, swing and sideslip of the spacecraft. To reduce that adverse effects on spacecraft upon landing, Hara et al. [166] proposed the momentum exchange impact dampers (MEIDs) which can be classified into two types: the passive (PMEID) and active (AMEID) momentum exchange impact dampers, respectively, as shown in Figure 11. It was shown that the active momentum exchange impact damper can effectively reduce the responses of spacecraft upon landing. Additionally, based on using actuators in combination with passive elements, Kushida et al. [167] put forward the Hybrid Momentum Exchange Impact Damper (HMEID) and applied it into the robust landing gear system.

 

 

 

 
 
 
 
 
 
 
 
 
 
 

(a)

(b)

(c)

Figure 11. (a) Conceptual diagram of a MEID; (b) landing gear system with PMEID; and (c) landing gear system with AMEID.

 

6.2 Particle damping applications in machinery field

A great deal of applications in the machinery field have appeared since 1967, when Park [168] studied the response of mass-spring damper exposed to repeated impact effect. Those devices, like electric and pneumatic hammers, bin vibrators as well as pile drivers, are all driven by the repeated impact force. Thus, these devices usually generate strong vibration when they are working, whereas mass-spring damper makes it very convenient to control the vibration. Furthermore, Sims et al. [169] proposed that the chatter stability during a machining process can be improved by applying particle dampers. Fuse [170] utilized the impact damper, which generates the mutual collisions between the main system and the additional vibration system in opposite phases, to eliminate the adverse effects exerted by mechanical system resonance.

There are some applications used for mitigating the vibration and noise during the running course of office equipment. For example, Skipor [171] utilized the impact damper to the carrying cylinder of a web-fed printing press. Sato et al. [172] employed particle dampers to reduce the vibration in pantograph-support systems. Xu et al. [173] utilized the particle damper to substantially reduce the vibration and noise generated by the banknote processing machine. To control the vibration produced by metal processing, Aiba et al. [13] proposed that the variable-attractive-force impact damper can weaken the vibration produced in the process of face milling for low-rigidity workpieces. It is a quite effective way to restrain the vibration during the process of metal cutting by this kind of impact damper. Moreover, Sathishkumar et al. [155] employed particle damping to control the vibration of stationary boring bars, which are used for conducting boring operations on rotating parts, by filling them with various metal particles.

It has also been demonstrated that it is feasible for particle damping to reduce the vibration and noise of automobile [163]. For example, Hu et al. [174] first studied acoustic pressure reduction at a target point inside an enclosed cavity using particle dampers. In addition, they utilized it to control the acoustic radiation and vibration of the thin panel in the auto body. Xia et al. [133] applied the particle damper to attenuate the vibration of rotating brake drum. Regarding the vibration control towards some special structures of mechanical equipment, Chan et al. [175] conducted experimental studies on particle damping applied to a lightly damped bond arm in a die bonding machine. Additionally, Li et al. [176] introduced bean bag dampers to plate structure. Furthermore, Xiao et al. [177] pointed out that the tooth surfaces of transmission gears produce significant vibration and noise under centrifugal force, which causes negative influence on the service life of gears. Therefore, they filled the lightening holes inside the gears with different particle packing ratios and achieved beneficial damping effects.

 

6.3 Particle damping applications in lifeline engineering

Lifeline engineering is the basic engineering of facilities and systems that can maintain the survival of cities, and have significant influence on the national economy and people’s livelihood. Hence, lifeline engineering should have the ability to cope with harsh environments, such as earthquake and windstorm. Particle damping applications in lifeline engineering mainly include the vibration control of wind turbines [16], power transmission towers [17,18], subsea jumpers [19,20], oil-well or gas-well drilling [178], the vibration and noise reduction of high-speed rail wheels [179], etc.

To attenuate the dynamic response and prevent the fatigue failure of wind turbines, the tuned rolling-ball damper, which contains single or multiple balls rolling in a spherical container and is mounted on the top of wind turbines, was proposed by Chen et al. [16]. Furthermore, Zhang et al. [59] put forward a ball vibration absorber (BVA) to prevent offshore wind turbines from the destruction of vibration caused by earthquakes or combined wind-wave loads. The schematic diagram of BVA is shown in Figure 12(a). As an increasing number of longer-span and taller-span power transmission towers have been constructed recently, they are susceptible to dynamic excitations, such as wind, earthquake, and ice shedding. Pounding tuned mass damper was applied to mitigate the dynamic response of a power transmission tower under earthquake excitation (Song et al. [18]) and wind load (Tian et al. [17]).

Particle damping technology also has impressive applications in the oil and gas production. Subsea jumper is a flexible pipeline structure commonly used to transport the production fluids, which are usually a mixture of oil, gas and water, from the reservoir to the topside processing facilities. However, subsea jumper is susceptible to flow-induced vibration, which may cause pipeline break, production loss and even resonance risk [180]. To solve this problem, Song et al. [19,20] introduced an L-shaped pounding tuned mass damper installed at the middle of subsea jumper, as shown in Figure 12(b). The vibration energy can be dissipated as heat energy by pounding between a tuned mass block and a ring covered with viscous-elastic material. It was found that unlike TMD, the pounding tuned mass damper is maintenance-free and has better robustness. Soil-well or gas-well drilling may have detrimental effects on the life of the drill string elements and the surface equipment owing to the accompanying significant vibrations. Hence, Velichkovich et al. [178] applied a vibration-impact damper to change the dynamic conditions of the object by transporting the vibration energy from the object to the damper and enhancing the consumption of vibration energy.

 

   

(a)

(b)

Figure 12. (a) The schematic diagram of the ball vibration absorber; (b) L-shaped pounding tuned mass damper designed for a subsea jumper.

 

Nowadays, high-speed rail plays an important role in people’s daily life as it brings significant convenience and efficiency to our society. However, high-speed rail may generate noise radiation owing to the excessive vibration caused by the roughness of wheel and rail running surface [181]. Therefore, Guo [179] introduced the particle damping rings into the high-speed rail wheels and found that compared to the existing wheel product with solid rings, it is more effective to attenuate the radial and axial vibration of wheels with particle damping rings.

 

6.4 Particle damping applications in civil engineering

In order to ensure various kinds of structures staying safe and comfortable under wind load and earthquake action, numerous structural vibration control systems have been proposed over the years. The concept of structural control was first proposed by John Milne [182] over a hundred years ago, who built a house by wood and placed it on ball bearings to demonstrate that the structure could be isolated from earthquake shaking. Later, Yao [183] applied the modern control theory to civil structures in 1972, which marks the beginning of the intensive study of vibration control in civil engineering. During the same period, Kelly and Skinner et al. [184,185] presented an idea that the vibration energy of a structure can be dissipated through additional energy dissipation devices. Since then, energy dissipation technology has been widely applied in civil and architectural engineering, and shows great promise for realistic applications.

By now, passive control technology has already been developed and is comparatively mature in the field of civil engineering. As a type of passive control technology, it is a new tendency to apply particle damping to civil engineering and its development is just at the beginning. Consequently, there have been relatively rare practical engineering projects in the civil engineering field that incorporate particle dampers, which mainly include the following: Ogawa [186] utilized the impact damper for the pylon of the cable-stayed Fuchu Lake Bridge in Japan so as to attenuate wind-induced vibration of the pylon. Furthermore, another typical application is a tall building named the Parque Araucano building in Santiago, Chile [187]. A particle damper was installed on the 21st floor, and it performed very well during the 2010 offshore Maule, Chile earthquake.

 

6.4.1 Research and application status of particle damping in civil engineering

In recent years, the study of particle damping based on disaster prevention and reduction in civil engineering is still at the stage of experimental study and theoretical analysis. The application of particle damping to several basic structural elements, such as cantilever beams, stiffened plates, etc., provides worthy guidelines to the practical application in the full-scale building-like structures. For example, Masri et al. [188,189] utilized the impact damper to reduce the forced vibration of continuous beams and plates. Xia et al. [190] took a cantilever beam filled with particles as the research object and investigated the nonlinear variation of structural damping with the change of the particle parameters. In addition, the influence of distributed damping on the torsional vibration of a plate was studied by Liu et al. [134] through the coupled algorithm of FEM and DEM. Zhao et al. [191] carried out a preliminary experimental study on the damping performance of NOPD columns.

With the purpose of exploring the application of particle dampers in civil engineering more deeply, some scholars started to study whole buildings and other structures as the research objects. Lu [192] carried out the theoretical and experimental studies of frame structure with particle dampers under the excitations of different earthquake waves. Yan et al. [76] applied the tuned particle damper to the seismic control of continuous viaducts. Additionally, Papalou et al. [193] proposed that particle damping technology can be employed to protect ancient monuments (such as columns) from seismic damage by using multi-drum particle dampers. The multi-drum columns are made up of several drum-shaped cylinders, among which the damaged and missing ones can be replaced by the new ones with a hollow part containing particles. It is also common for cable-stayed bridges to suffer from vibrations induced by wind or rain–wind, and some novel techniques to solve this problem have been proposed [194,195]. Egger et al. [196] introduced an advanced form of the impact damper named distributed-mass impact damper, originated from the traditional impact damper. They applied this type of impact damper to a cable-stayed bridge in order to attenuate rain-wind induced vibrations.

 

6.4.2 The advantages of particle damping applied to civil engineering

Through theoretical and experimental studies performed by scholars all over the world, it has been demonstrated that the particle damping technology has great superiority in vibration attenuation control encountered in the civil engineering field.

First, particle damping possesses a wide frequency band of vibration attenuation, with effective vibration attenuation performance within the range of 0~6000 Hz [2], so it can be considered to be capable (if properly designed) to suppress earthquake, wind vibration and other low frequency vibrations faced by civil engineering structures. It can also mitigate the environmental vibrations caused by metro, high-speed rail and other working conditions.

Second, among the existing passive control technologies which are comparatively mature towards civil engineering structures, it is obvious that viscous damping is sensitive to temperature. Moreover, with regard to mechanical damping such as friction, its material properties tend to degrade and will exhibit fatigue effects under various kinds of dynamic action. Hence, the application of these passive control dampers in civil engineering has been restricted greatly. However, particle damping is almost unlimited by the ambient temperature. The problems such as material degradation and fatigue effect are also solved. Furthermore, as the damping performance won’t be reduced over time, its resonance peak value could be effectively restrained. In consideration of such characteristics of particle material, this kind of technology is especially suitable for structures under extreme field conditions, such as the vibration control of power transmission tower.

Third, the layout locations of the particles are very flexible. They could be attached to the external periphery of civil engineering structures as a TMD, and could also be embedded into structural members. Any interlayer and internal cavity is acceptable, without influence on structural usage, and without causing significant modifications in the main structure.

Fourth, the material of the particles can be found easily with low prices. Especially, some ordinary building materials, such as steel balls, sands, stones, etc., could be used. Consequently, the applicability of particle damping technology in civil engineering has a great potential that has not yet been fully exploited.

 

6.4.3 The challenges of particle damping applied to civil engineering

At present, many studies of the main structure with particle dampers in the technical literature are mainly focused on SDOF systems, or other structures generally used in the mechanical field, such as cantilevers, stiffened plates, etc., while accurate analysis for MDOF structures with particle dampers has not been carried out yet. In addition, fewer theoretical and experimental studies on the control effect of civil engineering structures with particle dampers in the case of earthquake and wind vibration have been performed. Moreover, the researches on the standardized and normalized design of particle dampers which could guide the engineering application have not been conducted. Hence, there are several challenges for particle damping technology being used in civil engineering, some of which can be outlined as follows:

First, in the aerospace and machinery fields, both the frequency and amplitude of excitation are relatively high, whereas in civil engineering, those tend to be relatively low. According to the current research results, it was found that under certain conditions, the reduction in the response of the main structure may be enhanced through increasing the excitation intensity. Apparently, under weak excitation conditions in civil engineering, the efficiency of particle damping technology might be confined to some extent. Hence, improved particle dampers would be the focus in future studies.

Second, the application objects in civil engineering are different from that in the aerospace and machinery fields, which means the mass and scale of the main structure are usually quite huge in civil engineering. Therefore, despite the fact that a large mass ratio generally can be utilized in the aerospace and machinery fields, the mass ratio should be strictly controlled in civil engineering. In general, the inertial mass takes up 0.5%~1.5% of the generalized mass of the main structure.

Third, Hollkamp et al. [8] proposed that the optimum arrangement form of the particles is that the particle group should be asymmetrically distributed at the maximum amplitude of each major mode of the structure (or members). However, from the comfort point of view, installing a particle damper at the top of a building can reduce the impact of the noise generated during the impact process. Consequently, it is still an open question as to which arrangement form of particle damping is most suitable to civil structures.

 

  1. DISCUSSION

 

While this survey paper has focused primarily, on the past and present of the field of particle damping, in the future, there will be some very promising avenues where the field will expand and develop to leverage ever improving capabilities in sensors, devices, microprocessors, and high-performance materials that open the door to incorporate more sophisticated approaches for enhancing the effectiveness of the class of nonlinear devices under discussion, when they are employed for vibration mitigation under rapidly changing dynamic loads (e.g., earthquakes). As known from basic physics, all passive devices have reduced effectiveness when used under transient (i.e., non-stationary) loads, compared to their performance under stationary loads whose spectral and amplitude characteristics are nearly constant. Consequently, there is a pressing need to develop in the future modified particle damping devises whose properties evolve (i.e., change with time) as the dynamic environment to which they are subject is changing, so as to continuously furnish the maximum amount of damping that they can provide. This is the motivation for considering the concept of semi-active particle dampers.

The idea of using semi-active control in the civil engineering field has been considered in the work of Housner et al. [182] since the use of full active control, which is very effective but infeasible for application to full-scale civil structures due to unrealistic energy demands. By contrast, semi-active control approaches use a very modest amount of energy (typically provided by a small battery) to actively control the critical parameters in a given damping device. This basic idea was evaluated in the work of Masri et al. [197] who developed a procedure to use nonlinear auxiliary dampers (operating as particle dampers) with adjustable motion-limiting stops. A mathematical model of the system to be controlled is not needed for implementing the control algorithm. The degree of the primary structure oscillation near each vibration damper determines the damper’s actively-controlled gap size and activation time. By using a tiny amount of control energy to adjust the damper parameters instead of directly attenuating the motion of the primary system, a significant improvement is achieved in the level of vibration attenuation, in comparison to what an optimally-designed passive impact damper can provide. A proof-of-concept experimental study showed that this semi-active impact damper approach has a great promise for extension to other applications in which the primary system is subjected to no-stationary excitation.

 

  1. CONCLUSION

 

This survey paper presents an overview of particle damping technology including a general literature review for all significant particle damping technologies that have appeared in the open literature up to now. Although many scholars have carried out various theoretical, numerical and experimental studies on particle damping in the recent decades, a common view of the energy dissipation mechanism of particle damping has not been proposed so far. The existing theoretical research cannot adequately explain the non-linear essence of particle damping mechanism. Moreover, numerical simulation methods are providing the capability to investigate three-dimensional effects via the Discrete Element Method (DEM). However, a simplified design method is more likely to be accepted by engineering practitioners working in the structural control field. Despite the fact that there have been a large number of experimental studies on particle damping, research studies on civil structures provided with particle dampers have been scarcely conducted. Therefore, although particle damping has a good application prospect in civil engineering, it still needs further studies, and has many challenging practical issues to be evaluated and resolved.

Based on the comprehensive overview of particle damping technology that is reported in this paper, the following conclusions can be drawn:

1) To enhance the operational performance of particle dampers, such as reducing noise levels, impact forces and increasing plastic deformation, new materials used in both the container and the particles, such as buffered material, viscoelastic polymer, elastomer, etc., need to be developed and utilized increasingly. Especially, when confronting tough engineering requirements, new materials are the preferred choice to deal with the problem.

2) Over the course of the study on particle damping technology, the nonlinearity and randomness of particle damping are two major difficulties. There have been no corresponding theories to explain these two complicated highly-nonlinear phenomena adequately up to now. However, there are obvious tradeoffs between the analytical accuracy and the computational efficiency. Hence, a simple analytical approach which can grasp the dominant influencing factors needs to be investigated, and a comprehensive and systematic design procedure based on that is very important for future wide applications and acceptance of particle dampers as a reliable methodology for structural control.

3) It is shown that particle damping technology has been widely used in the aerospace, mechanical, and electrical fields, whereas it is a relatively new trend in the structural vibration control of typical civil engineering structures. Regarding the civil structures with relatively low levels of vibration, the traditional particle dampers may no longer make full use of their advantages. Hence, it is necessary to study improved particle dampers that can be widely applied in civil engineering under typical dynamic environments.

4) Most analysis and design of civil structures were conducted based on the finite element method whose analysis theory and calculation method are relatively mature. However, the particle-particle and particle-wall collisions in particle dampers make the displacement of the particles become discontinuous with strong nonlinear characteristics. The design of civil structures should consider the contribution of particle dampers to the damping and mass of the main structure, and in turn, the design of particle dampers should also consider their influence on the vibration response of civil structures. Consequently, it is an inevitable trend to implement the coupled algorithm of FEM and DEM for realistic applications of extended civil structures whose vibrations are to be controlled.

5) While most of the material in this paper has focused exclusively on passive particle dampers, there is a great opportunity in the future to benefit from the field of mechatronics, advancements in microprocessors, and major improvements in available materials and components, to incorporate the concept of semi-active particle dampers (in various configurations such as the ones discussed above) in target structures whose response is to be controlled. In spite of the disadvantage of having to use control energy (only a very small amount is needed), major improvements can be achieved in optimizing the attenuation of the primary structure’s response, particularly under transient excitation (such as earthquakes). This can be accomplished through the use of the control energy to modify a critical parameter in the particle damper that has drastic effects in enhancing the beneficial effect of the incorporated particle damper.

While considerable efforts in computational studies and experiments are being devoted to the application aspects of the broad family of particle dampers from the vibration attenuation point of view, it is important to keep in mind that there is a need for parallel efforts on the analytical aspects of these devices, since a better understanding of their behavior can lead to more refined theoretical results which can subsequently lead to enhanced design of these highly nonlinear systems. A helpful reference with an extensive bibliography of publications related to impact dampers and vibro-impact systems is the work of Ibrahim [198] and Ibrahim et al. [199].

 

ACKNOWLEDGEMENTS

Financial support from the National Natural Science Foundation of China through grant 51478361 is highly appreciated. This work was also supported by the Fundamental Research Funds for the Central Government Supported Universities.

REFERENCE

  1. Fricke JR. Lodengraf damping - an advanced vibration damping technology. Sound and Vibration 2000; 34(7):22-27.
  2. Panossian HV. Structural damping enhancement via non-obstructive particle damping technique. Journal of Vibration and Acoustics 1992; 114(1):101-105.
  3. Fang X, Tang J, Luo H. Granular damping analysis using an improved discrete element approach. Journal of Sound and Vibration 2007; 308(1-2):112-131.
  4. Liu W, Tomlinson G R, Rongong J A. The dynamic characterisation of disk geometry particle dampers. Journal of Sound and Vibration 2005; 280(3-5):849-861.
  5. Wong CX, Daniel MC, Rongong JA. Energy dissipation prediction of particle dampers. Journal of Sound and Vibration 2009; 319(1-2):91-118.
  6. Mao K, Wang MY, Xu Z, Chen T. Simulation and characterization of particle damping in transient vibrations. Journal of Vibration and Acoustics, ASME 2004; 126(2):202-211.
  7. Friend RD, Kinra VK. Particle impacting damping. Journal of Sound and Vibration 2000; 233(1):93-118.
  8. Hollkamp JJ, Gordon RW. Experiments with particle damping. Smart Structures and Materials 2002: Damping and Isolation, San Diego, CA, 1998; 3327: 2-12.
  9. Fowler BL, Flint EM, Olson SE. Effectiveness and predictability of particle damping. Smart Structures and Materials 2000: Damping and Isolation, United States, Newport Beach, CA, USA, 2000; 3989: 356-367.
  10. Simonian SS. Particle beam damper. Proceedings of SPIE Conference on Passive Damping, San Diego, 1995; 2445: 149-160.
  11. Saeki M. Analytical study of multi-particle damping. Journal of Sound and Vibration 2005; 281(3-5):1133-1144.
  12. Saeki M. Impact damping with granular materials in a horizontally vibrating system. Journal of Sound and Vibration 2002; 251(1):153-161.
  13. Aiba T, Murata R, Henmi N, Nakamura Y. An Investigation on Variable-Attractive-Force Impact Damper and Application for Controlling Cutting Vibration in Milling Process. Journal of the Japan Society for Precision Engineering 1995; 61(1):75-79.
  14. Moore JJ, Palazzolo AB, Gadangi R, Nale TA, Klusman SA, Brown GV, Kascak AF. Forced response analysis and application of impact dampers to rotordynamic vibration suppression in a cryogenic environment. Journal of Vibration and Acoustics, ASME 1995; 117(3):300-310.
  15. Rocke RD, Masri SF. Application of a Single-Unit Impact Damper to an Antenna Structure. Shock and Vibration Bulletin 1969. No. 39, Part 34, 31-10.
  16. Chen J, Georgakis CT. Tuned rolling-ball dampers for vibration control in wind turbines. Journal of Sound and Vibration 2013; 332(21):5271-5282.
  17. Tian L, Gai X. Wind-induced vibration control of power transmission tower using pounding tuned mass damper. Journal of Vibroengineering 2015; 17(7):3693-3701.
  18. Zhang P, Song G, Li HN, Lin YX. Seismic Control of Power Transmission Tower Using Pounding TMD. Journal of Engineering Mechanics 2013; 139(10):1395-1406.
  19. Zhang P, Li L, Patil D, Singla M, Li HN, Mo YL, Song G. Parametric study of pounding tuned mass damper for subsea jumpers. Smart Materials and Structures 2016; 25(1):015028.
  20. Li H, Zhang P, Song G, Patil D, Mo Y. Robustness study of the pounding tuned mass damper for vibration control of subsea jumpers. Smart Materials and Structures 2015; 24(9).
  21. Heckel M, Sack A, Kollmer JE, Pöschel T. Granular dampers for the reduction of vibrations of an oscillatory saw. Physica A: Statistical Mechanics and its Applications 2012; 391(19):4442-4447.
  22. Wong C, Rongong J. Control of Particle Damper Nonlinearity. AIAA Journal 2009; 47(4):953-960.
  23. Sanchez M, Carlevaro CM. Nonlinear dynamic analysis of an optimal particle damper. Journal of Sound and Vibration 2013; 332(8):2070-2080.
  24. Casado CM, Diaz IM, de Sebastian J, Poncela AV, Lorenzana A. Implementation of passive and active vibration control on an in-service footbridge. Structural Control and Health Monitoring 2013; 20(1):70-87.
  25. Parulekar YM, Reddy GR. Passive response control systems for seismic response reduction: A state-of-the-art review. International Journal of Structural Stability and Dynamics 2009; 9(1):151-177.
  26. Hudson EJ, Reynolds P. Implications of structural design on the effectiveness of active vibration control of floor structures. Structural Control and Health Monitoring 2014; 21(5):685-704.
  27. Sun Z, Li B, Dyke SJ, Lu C, Linderman L. Benchmark problem in active structural control with wireless sensor network. Structural Control and Health Monitoring 2016; 23(1):20-34.
  28. Alqado TE, Nikolakopoulos G. Posicast control of structures using MR dampers. Structural Control and Health Monitoring 2016; 23(8):1121-1134.
  29. Hashemi SMA, Kazemi HH, Karamodin A. Localized genetically optimized wavelet neural network for semi-active control of buildings subjected to earthquake. Structural Control and Health Monitoring 2016; 23(8):1074-1087.
  30. Lin PY, Roschke PN, Loh CH. Hybrid base-isolation with magnetorheological damper and fuzzy control. Structural Control and Health Monitoring 2007; 14(3):384-405.
  31. Li L, Song G, Ou J. Hybrid active mass damper (AMD) vibration suppression of nonlinear high-rise structure using fuzzy logic control algorithm under earthquake excitations. Structural Control and Health Monitoring 2011; 18(6):698-709.
  32. Basu B, Bursi OS, Casciati F, Casciati S, Del Grosso AE, Domaneschi M, Faravelli L, Holnicki-Szulc J, Irschik H, Krommer M, Lepidi M, Martelli A, Ozturk B, Pozo F, Pujol G, Rakicevic Z, Rodellar J. A European association for the control of structures joint perspective. Recent studies in civil structural control across Europe. Structural Control and Health Monitoring 2014; 21(12):1414-1436.
  33. Casciati S, Chassiakos AG, Masri SF. Toward a paradigm for civil structural control. Smart Structures and Systems 2014; 14(5):981-1004.
  34. Xiong W, Zhang S-J, Jiang L-Z, Li Y-Z. Introduction of the convex friction system (CFS) for seismic isolation. Structural Control and Health Monitoring 2017; 24(1). doi: 10.1002/stc.1861.
  35. Fiore A, Marano GC, Natale MG. Theoretical prediction of the dynamic behavior of rolling-ball rubber-layer isolation systems. Structural Control and Health Monitoring 2016; 23(9):1150-1167.
  36. Halperin I, Ribakov Y, Agranovich G. Optimal viscous dampers gains for structures subjected to earthquakes. Structural Control and Health Monitoring 2016; 23(3):458-469.
  37. Tubaldi E. Dynamic behavior of adjacent buildings connected by linear viscous/viscoelastic dampers. Structural Control and Health Monitoring 2015; 22(8):1086-1102.
  38. Foti D, Diaferio M, Nobile R. Dynamic behavior of new aluminum-steel energy dissipating devices. Structural Control and Health Monitoring 2013; 20(7):1106-1119.
  39. Miguel LFF, Miguel LFF, Lopez RH. Robust design optimization of friction dampers for structural response control. Structural Control and Health Monitoring 2014; 21(9):1240-1251.
  40. Wang J-T, Gui Y, Zhu F, Jin F, Zhou M-X. Real-time hybrid simulation of multi-story structures installed with tuned liquid damper. Structural Control and Health Monitoring 2016; 23(7):1015-1031.
  41. Rozas L, Boroschek RL, Tamburrino A, Rojas M. A bidirectional tuned liquid column damper for reducing the seismic response of buildings. Structural Control and Health Monitoring 2016; 23(4):621-640.
  42. Fadel Miguel LF, Lopez RH, Fadel Miguel LF, Torii AJ. A novel approach to the optimum design of MTMDs under seismic excitations. Structural Control and Health Monitoring 2016; 23(11):1290-1313.
  43. Lu Z, Chen X, Li X, Li P. Optimization and application of multiple tuned mass dampers in the vibration control of pedestrian bridges. Structural Engineering and Mechanics 2017; 62(1):55-64.
  44. Miranda JC. Discussion of system intrinsic parameters of tuned mass dampers used for seismic response reduction. Structural Control and Health Monitoring 2016; 23(2):349-368.
  45. Rathi AK, Chakraborty A. Reliability-based performance optimization of TMD for vibration control of structures with uncertainty in parameters and excitation. Structural Control and Health Monitoring 2017; 24(1).
  46. Lu Z, Wang D, Li P. Comparison study of vibration control effects between suspended tuned mass damper and particle damper. Shock and Vibration 2014; 2014(2):1-7.
  47. Paget AL. Vibration in steam turbine buckets and damping by impacts. Engineering 1937; 143:305-307.
  48. Lieber P, Jensen D P. An acceleration damper: development, design and some applications. Transactions of the ASME 1945; 67:523-530.
  49. Nayeri RD, Masri, S. F., Caffrey, J. P. Studies of the performance of multi-unit impact dampers under stochastic excitation. Journal of Vibration and Acoustics 2007; 129(2):239-251.
  50. Masri SF. Analytical and experimental studies of multiple-unit impact dampers. Journal of the Acoustical Society of America 1969; 45(5):1111-1117.
  51. Araki Y, Yokomichi I, Jinnouchi Y. Impact Damper with Granular Materials : 4th Report, Frequency Response in a Horizontal System. Transactions of the Japan Society of Mechanical Engineers C 1986; 29(258):4334-4338.
  52. Papalou A, Masri S F. Performance of particle dampers under random excitation. Journal of Vibration and Acoustics, ASME 1996; 118(4):614-621.
  53. Masri SF. Periodic Excitation of Multiple-Unit Impact Dampers. Journal of the Engineering Mechanics Division 1970; 96:1195-1207.
  54. Popplewell N, Semercigil S E. Performance of the bean bag impact damper for a sinusoidal external force. Journal of Sound and Vibration 1989; 133(2):193-223.
  55. Shah BM, Nudell JJ, Kao KR, Keer LM, Jane Wang Q, Zhou K. Semi-active particle-based damping systems controlled by magnetic fields. Journal of Sound and Vibration 2011; 330(2):182-193.
  56. Bai X, Shah B, Keer LM, Wang QJ, Snurr RQ. Particle dynamics simulations of a piston-based particle damper. Powder Technology 2009; 189(1):115–125.
  57. Chen LA, Semercigil SE. A Beam-Like Damper for Attenuating Transient Vibrations of Light Structures. Journal of Sound and Vibration 1993; 164(1):53-65.
  58. Gharib M, Ghani S. Free vibration analysis of linear particle chain impact damper. Journal of Sound and Vibration 2013; 332(24):6254-6264.
  59. Zhang ZL, Chen JB, Li J. Theoretical study and experimental verification of vibration control of offshore wind turbines by a ball vibration absorber. Structure and Infrastructure Engineering 2014; 10(8):1087-1100.
  60. Shah BM, Pillet D, Bai X-M, Keer LM, Jane Wang Q, Snurr RQ. Construction and characterization of a particle-based thrust damping system. Journal of Sound and Vibration 2009; 326(3-5):489-502.
  61. Li K, Darby A P. A buffered impact damper for multi-degree-of-freedom structural control. Earthquake Engineering and Structural Dynamics 2008; 37(13):1491-1510.
  62. Du YC, Wang SL. Modeling the fine particle impact damper. International Journal of Mechanical Sciences 2010; 52(7):1015-1022.
  63. Darabi B, Rongong JA. Polymeric particle dampers under steady-state vertical vibrations. Journal of Sound and Vibration 2012; 331(14):3304-3316.
  64. Darabi B, Rongong JA, Zhang T. Viscoelastic granular dampers under low amplitude vibration. Journal of Vibration and Control 2016. doi:10.1177/1077546316650098.
  65. Bustamante M, Gerges SNY, Cordioli J, Martin OP, Weisbeck J, Ott M. Experimental study on some parameters that affect the performance of an elastomer particle damper. 21st International Congress on Acoustics, ICA 2013 - 165th Meeting of the Acoustical Society of America, Montreal, QC, 2013; 19.
  66. Michon G, Almajid A, Aridon G. Soft hollow particle damping identification in honeycomb structures. Journal of Sound and Vibration 2013; 332(3):536-544.
  67. Abbas H, Hai H, Rongong J, Xing YF. Damping performance of metal swarfs in a horizontal hollow structure. Journal of Mechanical Science and Technology 2014; 28(1):9-13.
  68. Lord C, Tang N, Rongong J. Damping of metallic wool with embedded rigid body motion amplifiers. Proceedings of 6th European Conference on Structural Control, Sheffield, 2016.
  69. Hong J, Chen L, Ma Y, Tomlinson GR, Rongong JA. Hysteretic properties of metal rubber particles. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2013; 227(4):693-702.
  70. Tang N, Rongong JA, Tomlinson GR. Nonlinear behaviour of tangled metal wire particle dampers. International Conference on Structural Engineering Dynamics (ICEDYN2015), Lagos, Portugal, 2015.
  71. Yao B, Chen Q, Xiang HY, Gao X. Experimental and theoretical investigation on dynamic properties of tuned particle damper. International Journal of Mechanical Sciences 2014; 80:122-130.
  72. Semercigil SE, Lammers D, Ying Z. A new tuned vibration absorber for wide-band excitations. Journal of Sound and Vibration 1992; 156(3):445-459.
  73. Semercigil SE, Collette F, Huynh D. Experiments with tuned absorber-impact damper combination. Journal of Sound and Vibration 2002; 256(1):179-188.
  74. Lu Z, Chen X, Zhang D, Dai K. Experimental and analytical study on the performance of particle tuned mass dampers under seismic excitation. Earthquake Engineering and Structural Dynamics 2017; 46(5):697-714.
  75. Lu Z, Wang D, Masri SF, Lu X. An experimental study of vibration control of wind-excited high-rise buildings using particle tuned mass dampers. Smart Structures and Systems 2016; 18(1):93-115.
  76. Yan W, Xu W, Wang J, Chen Y. Experimental Research on the Effects of a Tuned Particle Damper on a Viaduct System under Seismic Loads. Journal of Bridge Engineering 2014; 19(3):165-184.
  77. Dai K, Wang J, Mao R, Lu Z, Chen S-E. Experimental investigation on dynamic characterization and seismic control performance of a TLPD system. The Structural Design of Tall and Special Buildings 2017; 26(7):e1350.
  78. Masri SF. Theory of the Dynamic Vibration Neutralizer with Motion-Limiting Stops. Journal of Applied Mechanics, ASME 1972; 39(2):563-568.
  79. Lu Z, Wang D, Zhou Y. Experimental parametric study on wind-induced vibration control of particle tuned mass damper on a benchmark high-rise building. The Structural Design of Tall and Special Buildings 2017. doi: 10.1002/tal.1359.
  80. Dai D. Damping Technology for Vibration and Noise Control. Xi'an Jiaotong University Press, Xi'an 1986 (in chinese).
  81. Kerwin EM. Macro-mechanisms of damping in composite structures. Internal Friction Damping and Cyclic Plasticity, Baltimore, Md, 1965; 125-149.
  82. Lenzi A. The use of damping material in industrial machine, Ph.D. Thesis, University of Southampton, 1985.
  83. Sun JC, Sun H B. Predictions of total loss factors of structures Part II: loss factors of sand filled structure. Journal of Sound and Vibration 1986; 104(2):243-257.
  84. Deng W. Effects of impact damper and determination of its parameters. Journal of Mechanical Engineering 1964; 12(4):83-94 (in Chinese).
  85. Masmoudi M, Job S, Abbes MS, Tawfiq I, Haddar M. Experimental and numerical investigations of dissipation mechanisms in particle dampers. Granular Matter 2016; 18(3).
  86. Zhang K, Chen TN, Wang XP, Fang JL. Rheology behavior and optimal damping effect of granular particles in a non-obstructive particle damper. Journal of Sound and Vibration 2016; 364:30-43.
  87. Gourc E, Seguy S, Michon G, Berlioz A, Mann BP. Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink. Journal of Sound and Vibration 2015; 355:392-406.
  88. Bergeot B, Bellizzi S, Cochelin B. Passive suppression of helicopter ground resonance using nonlinear energy sinks attached on the helicopter blades. Journal of Sound and Vibration 2017; 392:41-55.
  89. Lu X, Liu Z, Lu Z. Optimization design and experimental verification of track nonlinear energy sink for vibration control under seismic excitation. Structural Control and Health Monitoring 2017. doi: 10.1002/stc.2033.
  90. Vakakis AF. Inducing Passive Nonlinear Energy Sinks in Vibrating Systems. Journal of Vibration and Acoustics 2001; 123(3):324-332.
  91. Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G, Nucera F, Tsakirtzis S, Panagopoulos PN. Passive non-linear targeted energy transfer and its applications to vibration absorption: A review. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 2008; 222(2):77-134.
  92. Roberson RE. Synthesis of a nonlinear dynamic vibration absorber. Journal of the Franklin Institute 1952; 254(3):205-220.
  93. Masri SF. Forced Vibration of a Class of Non-Linear Two-Degree-of-Freedom Oscillators. International Journal of Non-Linear Mechanics 1972; 7:663-674.
  94. Grubin C. On the theory of the acceleration damper. Journal of Applied Mechanics 1956; 23(3):373-378.
  95. Masri SF. General Motion of Impact Dampers. The Journal of the Acoustical Society of America 1970; 47(1B):229-237.
  96. Bapat CN. Periodic motion of an impact oscillator. Journal of Sound and Vibration 1998; 209(1):43-60.
  97. Masri SF. Effectiveness of two-particle impact dampers. Journal of the Acoustical Society of America 1967; 41(6):1553-1554.
  98. Bapat CN, Sankar S. Single unit impact damper in free and forced vibration. Journal of Sound and Vibration 1985; 99(1):85-94.
  99. Bapat CN, Sankar S. Multiunit impact damper—Re-examined. Journal of Sound and Vibration 1985; 103(4):457-469.
  100. Ema S, Marui E. A fundamental study on impact dampers. International Journal of Machine Tools and Manufacture 1994; 34(3):407-421.
  101. Duncan MR, Wassgren CR, Krousgrill CM. The damping performance of a single particle impact damper. Journal of Sound and Vibration 2005; 286(1-2):123-144.
  102. Masri SF. Motion and Stability of Two-Particle Single-Container Impact Dampers. Journal of Applied Mechanics 1967; 34(2):506-507.
  103. Saluena C, Poeschel T, Esipov SE. Dissipative properties of vibrated granular materials. Physical Review E 1998; 59(4):4422-4425.
  104. Ben Romdhane M, Bouhaddi N, Trigui M, Foltete E, Haddar M. The loss factor experimental characterisation of the non-obstructive particles damping approach. Mechanical Systems and Signal Processing 2013; 38(2):585-600.
  105. Yang MY, Lesieutre GA, Hambric SA, Koopmann GH. Development of a design curve for particle impact dampers. Noise Control Engineering Journal 2005; 53(1):5-13.
  106. Xiao WQ, Jin LN, Chen BQ. Theoretical Analysis and Experimental Verification of Particle Damper-Based Energy Dissipation with Applications to Reduce Structural Vibration. Shock and Vibration 2015.
  107. Lu Z, Masri SF, Lu X. Studies of the performance of particle dampers attached to a two-degree-of-freedom system under random excitation. Journal of Vibration and Control 2011; 17(10):1454-1471.
  108. Lu Z, Masri S F, Lu X L. Parametric studies of the performance of particle dampers under harmonic excitation. Sturctural Control and Health Monitoring 2011; 18(1):79-98.
  109. Lu Z, Lu X L, Masri S F. Studies of the performance of particle dampers under dynamic loads. Journal of Sound and Vibration 2010; 329(26):5415-5433.
  110. Papalou A, Masri S F. Response of impact dampers with granular materials under random excitation. Earthquake Engineering and Structural Dynamics 1996; 25(3):253-267.
  111. Papalou A, Masri S F. An experimental investigation of particle dampers under harmonic excitation. Journal of Vibration and Control 1998; 4(4):361-379.
  112. Xu ZW, Chan K W, Liao W H. An empirical method for particle damping design Shock and Vibration 2004; 11(5-6):647-664.
  113. Wu CJ, Liao W H, Wang M Y. Modeling of granular particle damping using multiphase flow theory of gas-particle. Journal of Vibration and Acoustics 2004; 126(2):196-201.
  114. Fang X, Tang J. Granular Damping in Forced Vibration: Qualitative and Quantitative Analyses. Journal of Vibration and Acoustics 2006; 128(4):489-500.
  115. Wang DQ, Wu CJ. Vibration Response Prediction of Plate with Particle Dampers Using Cosimulation Method. Shock and Vibration 2015.
  116. Wang DQ, Wu CJ. A novel prediction method of vibration and acoustic radiation for rectangular plate with particle dampers. Journal of Mechanical Science and Technology 2016; 30(3):1021-1035.
  117. Liu W, Tomlinson G, Worden K. Nonlinearity study of particle dampers. Proceedings of the 2002 International Conference on Noise and Vibration Engineering, ISMA, Leuven, 2002; 280: 495-499.
  118. Chen Q, Worden K, Rongong J. Characterisation of particle dampers using restoring force surface technique. 6th International Conference on Structural Dynamics-EURODYN 2005, Paris, FRANCE, 2005; 1-3: 1785-1790.
  119. Tanrikulu AH. Application of ANN techniques for estimating modal damping of impact-damped flexible beams. Advances in Engineering Software 2009; 40(10):986-990.
  120. Cui Z, Wu J, Chen H, Li D. A quantitative analysis on the energy dissipation mechanism of the non-obstructive particle damping technology. Journal of Sound and Vibration 2011; 330(11):2449-2456.
  121. Mao KM, Wang MY, Xu ZW, Chen TN. DEM simulation of particle damping. Powder Technology 2004; 142(2-3):154-165.
  122. Lu Z, Lu X, Jiang H, Masri SF. Discrete element method simulation and experimental validation of particle damper system. Engineering Computations 2014; 31(4):810-823.
  123. Wong C, Spencer A, Rongong J. Effects of Enclosure Geometry on Particle Damping Performance. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California, 2009.
  124. Cundall PA. A computer model for simulating progressive large scale movements in blocky rock systems. Proceedings of the Symposium of the International Society of Rock Mechanics, Nancy, France, 1971; 1: 11-18.
  125. Elperin T, Golshtein E. Comparison of different models for tangential forces using the particle dynamics method. Physica A: Statistical and Theoretical Physics 1997; 242(3-4):332-340.
  126. Renzo AD, Maio, F P D. Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chemical Engineering Science 2004; 59(3):525-541.
  127. Masri SF. Steady-State Response of a Multidegree System with an Impact Damper. Journal of Applied Mechanics 1973; 40(1):127-132.
  128. Zhou X, Xu W, Niu X, Cui Y. A review of distinct element method researching progress and application. Rock and Soil Mechanics 2007; (S1):408-416 (in Chinese).
  129. Sanchez M, Rosenthal G, Pugnaloni LA. Universal response of optimal granular damping devices. Journal of Sound and Vibration 2012; 331(20):4389-4394.
  130. Bannerman MN, Kollmer JE, Sack A, Heckel M, Mueller P, Pöschel T. Movers and shakers: Granular damping in microgravity. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2011; 84(1).
  131. Sack A, Heckel M, Kollmer JE, Zimber F, Poschel T. Energy Dissipation in Driven Granular Matter in the Absence of Gravity. Physical Review Letters 2013; 111(1).
  132. Hales TC. The sphere packing problem. Journal of Computational and Applied Mathematics 1992; 44(1):41-76.
  133. Xia Z, Liu X, Shan Y. Application of particle damping for vibration attenuation in brake drum. International Journal of Vehicle Noise and Vibration 2011; 7(2):178-194.
  134. Tan D, Liu X, Shan Y. Simulation of Applying Particle Damper to Torsion Vibration of Plate. Journal of System Simulation 2011; 23(8):1594-1597 (in Chinese).
  135. Xia Z, Wen H, Liu X. Dynamic characteristics of a rotating plate structure with particle damping. Journal of Vibration and Shock 2014; 33(9):61-65,88 (in Chinese).
  136. Sadek MM, Mills B. Effect of gravity on the performance of an impact damper, part 1: steady-state motion. Journal of Mechanical Engineering Science 1970; 12(4):268-277.
  137. Sadek MM, Williams C J H, Mills B. Effect of gravity on the performance of an impact damper, part 2: stability of vibrational modes. Journal of Mechanical Engineering Science 1970; 12(4):278-287.
  138. Cempel C, Lotz G. Efficiency of vibrational energy dissipation by moving shot. Journal of Structural Engineering 1993; 119(9):2642-2652.
  139. Xu ZW, Wang M Y, Chen T N. Particle damping for passive vibration suppression: numerical modelling and experimental investigation. Journal of Sound and Vibration 2005; 279(3-5):1097-1120.
  140. Trigui M, Foltete E, Abbes MS, Fakhfakh T, Bouhaddi N, Haddar M. An experimental study of a multi-particle impact damper. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2009; 223(9):2029-2038.
  141. J. Els DN. Damping of Rotating Beams with Particle Dampers: Experimental Analysis. AIAA Journal 2011; 49(10):2228-2238.
  142. Jadhav TA, Awasare PJ. Enhancement of particle damping effectiveness using multiple cell enclosure. Journal of Vibration and Control 2016; 22(6):1516-1525.
  143. Zhang C, Chen TN, Wang XP, Li YG. Discrete element method model and damping performance of bean bag dampers. Journal of Sound and Vibration 2014; 333(23):6024-6037.
  144. Wang YR, Liu B, Tian AM, Tang W. Experimental and numerical investigations on the performance of particle dampers attached to a primary structure undergoing free vibration in the horizontal and vertical directions. Journal of Sound and Vibration 2016; 371:35-55.
  145. Rongong J, Tomlinson G. Amplitude Dependent Behaviour in the Application of Particle Dampers to Vibrating Structures. 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Austin, Texas, 2005.
  146. Wong C, Rongong J. Macromodel Characterisation and Application of Particle Dampers to Vibrating Structures. 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, RI, United States, 2006.
  147. Wong CX, Daniel MC, Rongong JA. Prediction of the amplitude dependent behaviour of particle dampers. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, Hawaii, 2007; 4: 4167-4182.
  148. Veluswami MA, Crossley F R E. Multiple impacts of a ball between two plates, Part 1: Some experimental observations. Journal of Engineering for Industry, ASME 1975; 97(3):820-827.
  149. Veluswami MA, Crossley F R E, Horvay G. Multiple impacts of a ball between two plates. Part 2: Mathematical modeling. Journal of Engineering for Industry, ASME 1975; 97(3):835-838.
  150. Yokomichi I, Araki Y, Jinnouchi Y, Inoue J. Impact damper with granular materials for multibody system Journal of Pressure Vessel Technology 1996; 118(1):95-103.
  151. Yokomichi I, Muramatsu H, Araki Y. On shot impact dampers applied to self-excited vibrations. International Journal of Acoustics and Vibration 2001; 6(4):193-199.
  152. Li K, Darby A P. Experiments on the effect of an impact damper on a multiple-degree-of-freedom system. Journal of Vibration and Control 2006; 12(5):445-464.
  153. Lu Z, Lu X L, Lu W S, Masri S F. Experimental studies of the effects of buffered particle dampers attached to a multi-degree-of-freedom system under dynamic loads,. Journal of Sound and Vibration 2012; 331(9):2007-2022.
  154. Lu Z, Lu X, Lu W, Masri SF. Shaking table test of the effects of multi-unit particle dampers attached to an MDOF system under earthquake excitation. Earthquake Engineering and Structural Dynamics 2012; 41(5):987-1000.
  155. Sathishkumar B, Mohanasundaram KM, Kumar MS. Impact of Particle Damping Parameters on Surface Roughness of Bored Surface. Arabian Journal for Science and Engineering 2014; 39(10):7327-7334.
  156. Veeramuthuvel P, Sairajan KK, Shankar K. Vibration suppression of printed circuit boards using an external particle damper. Journal of Sound and Vibration 2016; 366:98-116.
  157. Akl FA, Butt AS. Application of impact dampers in vibration control of flexible structures. NASA Johnson Space Center, National Aeronautics and Space Administration (NASA)(American Society for Engineering Education (ASEE) Summer Faculty Fellowship Program, 1 15 P, Washington, DC, 1995.
  158. Ashley S. New racket shakes up tennis. Mechanical Engineering 1995; 117(8):80-81.
  159. Kielb R, Macri FG, Oeth D, Nashif AD, Macioce P, Panossia H, Lieghley F. Advanced damping systems for fan and compressor blisks. 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Cleveland, United States, 1998.
  160. Oledzki A. New kind of impact damper — from simulation to real design. Mechanism and Machine Theory 1981; 16(3):247-253.
  161. Gibson, B.W. Usefulness of Impact Dampers for Space Applications. Air Force Institute of Technoligy, Wright–Patterson AFB, OH. School of Engineering, Report: AFIT/GA/AA/83M–2, 147, 1983.
  162. Torvik PJ, Gibson W. Design and effectiveness of impact dampers for space applications. Design Engineering Division, ASME 1987; 5:65-74.
  163. Simonian S. Particle Damping Applications. 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs, California, 2004.
  164. Liu W, Ewing MS. Particle damping of composite honeycomb beams by the power input method. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, Hawaii, 2007; 4: 4183-4191.
  165. Panossian H. Optimized Non-Obstructive Particle Damping (NOPD) treatment for composite honeycomb structures. 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Newport, RI, 2006; 10: 7353-7362.
  166. Hara S, Ito R, Otsuki M, Yamada Y, Kubota T, Hashimoto T, Matsuhisa H, Yamada K. Momentum-exchange-impact-damper-based shock response control for planetary exploration spacecraft. Journal of Guidance, Control, and Dynamics 2011; 34(6):1828-1838.
  167. Kushida Y, Hara S, Otsuki M, Yamada Y, Hashimoto T, Kubota T. Robust landing gear system based on a hybrid momentum exchange impact damper. Journal of Guidance, Control, and Dynamics 2013; 36(3):776-789.
  168. Park WH. Mass-Spring-Damper Response to Repetitive Impact. Journal of Manufacturing Science and Engineering 1967; 89(4):587-596.
  169. Sims ND, Amarasinghe A, Ridgway K. Particle dampers for workpiece chatter mitigation. Manufacturing Engineering Division, ASME 2005; 16(1):825-832.
  170. Fuse T. Prevention of resonances by impact damper. Seismic Engineering - 1989: Design, Analysis, Testing, and Qualification Methods, New York, NY, United States, Honolulu, HI, USA, 1989; 182: 57-66.
  171. Skipor E, Bain L J. Application of impact damping to rotary printing equipment. Journal of Mechanical Design 1980; 102(2):338-343.
  172. Sato T, Takase M, Kaiho N, Makino T, Tanaka K. Vibration reduction of pantograph-support system using an impact damper (influence of curve track). Proceeding of International Conference on Noise and Vibration Engineering, ISMA, Leuven, Belgium, 2002; 1669-1676.
  173. Xu ZW, Wang MY, Chen TN. A particle damper for vibration and noise reduction. Journal of Sound and Vibration 2004; 270(4-5):1033-1040.
  174. Hu L, Shi Y, Yang Q, Song G. Sound reduction at a target point inside an enclosed cavity using particle dampers. Journal of Sound and Vibration 2016; 384:45-55.
  175. Chan KW, Liao WH, Wang MY, Choy PK. Experimental studies for particle damping on a bond arm. Journal of Vibration and Control 2006; 12(3):297-312.
  176. Li W, Zhu D, Huang X. Study on the Damping Performance of Flexible Restraint Particle Impact Damper Applied to Plate Structure. Noise and Vibration Control 1998; 4(1):2-5 (in Chinese).
  177. Xiao WQ, Huang YX, Jiang H, Lin H, Li JN. Energy dissipation mechanism and experiment of particle dampers for gear transmission under centrifugal loads. Particuology 2016; 27:40-50.
  178. Velichkovich AS, Velichkovich SV. Vibration-impact damper for controlling the dynamic drillstring conditions. Chemical and Petroleum Engineering 2001; 37(3-4):213-215.
  179. Guo S. Research on the Characteristics of Vibration and Noise Reduction of Particle Damping Rings Installed on High-speed Rail Wheels, M.Sc. Thesis, Harbin Institute of Technology, 2016 (in Chinese).
  180. Lu Y, Liang C, Manzano-Ruiz JJ, Janardhanan K, Perng Y-Y. Flow-Induced Vibration in Subsea Jumper Subject to Downstream Slug and Ocean Current. Journal of Offshore Mechanics and Arctic Engineering 2016; 138(2):021302-021302-021310.
  181. Yang X, Shi G. The Effect of Slab Track on Wheel/Rail Rolling Noise in High Speed Railway. Intelligent Automation and Soft Computing 2014; 20(4):575-585.
  182. Housner GW, Bergman LA, Caughey TK, Chassiakos AG, Claus RO, Masri SF, Skelton RE, Soong TT, Spencer BF, Yao JTP. Structural control: past, present, and future. Journal of Engineering Mechanics, ASCE 1997; 123(9):897-971.
  183. Yao JTP. Concept of structural control. Journal of the structural Division, ASCE 1972; 98(7):1567-1574.
  184. Kelly JM, Skinner R L, Heine A J. Mechanics of energy absorption in special devices for use in earthquake-resistant structures. Bulletin of New Zealand National Society for Earthquake Engineering 1972; 5(3):63-88.
  185. Skinner RI, Kelly J M, Heine A J. Hysteretic dampers for earthquake-resistant structures. Earthquake Engineering and Structural Dynamics 1974; 3(3):287-296.
  186. Ogawa K, Ide T, Saitou T. Application of impact mass damper to a cable-stayed bridge pylon. Journal of Wind Engineering and Industrial Aerodynamics 1997; 72(1-3):301-312.
  187. Naeim F, Lew M, Carpenter LD, Youssef NF, Rojas F, Saragoni GR, Adaros MS. Performance of tall buildings in Santiago, Chile during the 27 February 2010 offshore Maule, Chile earthquake. The Structural Design of Tall and Special Buildings 2011; 20(1):1-16.
  188. Masri SF, Kahyai K. Steady-State Motion of a Plate With a Discontinuous Mass. International Journal of Non-Linear Mechanics 1974; 9:451-462.
  189. Masri SF. Forced Vibration of a Class of Nonlinear Dissipative Beams. Journal of the Engineering Mechanics Division 1973; 99:669-683.
  190. Xia ZW, Shan YC, Liu XD. Experimental research on particle damping of cantilever beam. Journal of Aerospace Power 2007; 22(10):1737-1741 (in Chinese).
  191. Zhao L, Liu P, Lu Y-y. Experimental investigation on damping characteristics of NOPD columns. Journal of Vibration and Shock 2009; 28(8):1-5 (in Chinese).
  192. Lu Z. Numerical Simulation and Performance Analysis of Particle Dampers, Ph.D. Thesis, Tongji University, 2011 (in Chinese).
  193. Papalou A, Strepelias E, Roubien D, Bousias S, Triantafillou T. Seismic protection of monuments using particle dampers in multi-drum columns. Soil Dynamics and Earthquake Engineering 2015; 77:360-368.
  194. Izzi M, Caracoglia L, Noe S. Investigating the use of Targeted-Energy-Transfer devices for stay-cable vibration mitigation. Structural Control and Health Monitoring 2016; 23(2):315-332.
  195. Zhou P, Li H. Modeling and control performance of a negative stiffness damper for suppressing stay cable vibrations. Structural Control and Health Monitoring 2016; 23(4):764-782.
  196. Egger P, Caracoglia L, Kollegger J. Modeling and experimental validation of a multiple-mass-particle impact damper for controlling stay-cable oscillations. Structural Control and Health Monitoring 2015; 23(6):960-978.
  197. Masri SF, Miller RK, Dehghanyar TJ, Caughey TK. Active Parameter Control of Nonlinear Vibrating Structures. Journal of Applied Mechanics 1989; 56(3):658-666.
  198. Ibrahim R. Vibro-Impact Dynamics: Modeling, Mapping and Applications. Springer Verlag, 2009.
  199. Ibrahim R, Babitsky VI, Okuma M. Vibro-Impact Dynamics of Ocean Systems and Related Problems. Springer-Verlag Berlin Heidelberg, 2009.