directory

Chapter 1 Introduction 1

1.1Overview1

1.2 Damper technology for civil engineering 2

1.2.1 Direct energy dissipation damper 2

1.2.2 Tuned damper 4

1.2.3 Particle damper 5

1.3 Research theory of granular materials9

1.3.1 Application of granular materials in industry9

1.3.2 Research history of particulate materials10

1.3.3 Interparticle forces11

1.3.4 Research methods for particle dampers12

1.4Discrete element method14

1.4.1 The basic process of discrete element method calculations14

1.4.2 Hard and soft ball models15

1.4.3 Contact model16

1.5 The main research work and content arrangement of this paper21

1.5.1 The main research work in this paper21

1.5.2 Arrangement of the contents of this document22

Chapter 2 Simulation of a Particle Damper-Cable System Based on the Discrete Element Method 23

2.1Introduction23

2.2 Simulation model of rootless particle damper 23

2.2.1 Application background of rootless particle dampers23

2.2.2 Discrete element method software selection24

2.2.3 Modeling26

2.2.4 Calculation example27

2.3 Energy dissipation analysis of rootless cable dampers27

2.3.1 Vibration intensity27

2.3.2 Fill rate 30

2.3.3 Particle size32

2.3.4 Collision recovery factor34

2.3.5 Sliding friction coefficient 35

2.3.6Conclusion36

2.4 Vibration damping effect of rootless cable damper37

2.4.1 Cable-damper system model construction37

2.4.2 Damper damping effect 39

2.4.3 Parametric analysis41

2.4.4Conclusion44

2.5 Conclusion 45 of the chapter

Chapter 3 Micromechanical Models of Particle Damping 47

3.1Introduction47

3.2 Coulomb friction damping model47

3.3 Viscous damping model49

3.4 Single-particle model50

3.4.1Vibration process analysis51

3.4.2 Category friction damping situations57

3.4.3 Effect of inter-particle interactions60

3.4.4 Energy consumption of single-particle model61

3.5 Single-particle model check65

3.6Conclusion70

Chapter 4 Application of the Single Particle Model to the Analysis of Two Dampers 71

4.1Introduction71

4.2 Application on floor tile type particle dampers71

4.2.1 Floor tile type particle damper concept71

4.2.2Model building72

4.2.3Results and discussions74

4.2.4 Comparison of energy dissipation calculation results with discrete elements76

4.2.5Conclusion79

4.3 Application on rootless particle dampers79

4.3.1 Establishment of the model of the rootless cable damper 79

4.3.2 Model validation81

4.3.3 Conclusion89

4.4 Conclusions of this chapter90

Chapter 5 Conclusions and Prospects 91

Acknowledgments 93

Ref. 95

Appendix A: Three Cases in the Single Particle Model99

Curriculum vitae, academic papers and research results published during the study period 103

Chapter 1 Introduction

1.1 Overview

In recent years, with the continuous innovation of science and technology and the continuous updating of construction goals in the new era, China's civil building construction theory and construction technology have made great progress and rapid development. In the landmark and important projects, the building faces extremely challenging technical challenges with higher and longer goals. Among them, the vibration problem has a great impact on the safety, durability, comfort and economy of the whole structure, so vibration control has become one of the key issues that must be considered and solved in contemporary construction.

As a new type of damping technology, particle damping has attracted extensive attention in recent years. Figure 1.1 shows the search for the key words "particle damper" and "granular da" in the webof sciencemper", "particledamping", and "granulardamping". As you can see, since 2015, the amount of attention this damping technique has received has suddenly increased. On the one hand, this reflects the increasing demand for structural vibration reduction, as well as the demand of scientific researchers and engineers for new vibration reduction methods. On the other hand, it reflects the rapid development of computer technology, because the modeling and simulation of this technology has a large amount of calculation, and has high requirements for the running speed and storage capacity of the computer.

Figure 1.1 The annual publication of literature related to particle damping in Web ofScience

Particle damping technology uses the friction and impact between tiny particles filled in a limited enclosed space in the vibrating body to dissipate the vibration energy of the system. It mainly relies on the collision and friction between particles and between particles and containers to convert kinetic energy into heat energy release, so as to achieve the effect of shock absorption. At the same time, because the movement of the particles and the container is not synchronized, the damper also has a certain tuning effect.

This chapter will mainly present the theoretical background required for the work of this paper. Firstly, the common dampers in buildings are classified and introduced by whether they consume energy or not, and the particle dampers are the research objects of this paper, and at the same time, because of their damping benefits

From energy consumption, but also from tuning, so it is introduced separately. Then, according to the history, mechanical model, and research methods of granular materials, the relevant theories are sorted out to understand this new material more clearly. Then, focusing on the modeling and simulation methods adopted in this paper, the discrete element method and the contact model that will be used in the following chapters are introduced. Finally, from the perspectives of research work and arrangement, the structure of the following chapters is introduced.

1.2 Damper technology for civil engineering

There are many kinds of dampers in the field of civil engineering, and the application scenarios of various types of dampers are also different due to different mechanisms. The direct energy-dissipating damper converts the kinetic energy of the system into thermal energy through friction or material deformation, so as to achieve the purpose of energy dissipation and vibration reduction; The tuned damper mainly achieves vibration damping by changing the resonance characteristics of the structure; As a new type of damper, the efficiency of particle damper comes not only from collision and friction energy dissipation, but also from tuning. This section will classify these dampers and their typical cases, with a focus on particle dampers.

1.2.1 Direct energy dissipation dampers

Friction dampers

Friction dampers use sliding friction between two surfaces to achieve energy dissipation. The hysteresis curves are mostly rectangular and similar to the Coulomb friction. It has the advantage of being insensitive to changes in frequency, oscillating revolutions, and temperature, and has good fatigue resistance [1].

Figure 1.2 Friction damper [2].

Metal material dampers

The damper uses a metal material with strong plastic deformation ability to achieve energy dissipation. Because of its good durability and economy, it is widely used in engineering. Its initial application in engineering dates back to 1980 in New Zealand [3]. Metallic dampers do not change their properties at temperatures from -70°C to 30°C, making them more resistant to changes in the environment and weather.

ADAS (AddedDampingandStiffness) damping devices are characterized by their ability to provide more damping energy dissipation to the structure and increase the stiffness of the structure. It is composed of an X-shaped steel plate, and the principle is to pass through the plastic of the steel plate

The shape is bent and deformed to absorb energy. This damping device has been used in a variety of structures, such as the ADAS damper installed on the 29th floor of a steel building in Naples [3].

Figure 13 ADAS dampers [1].

Viscoelastic dampers

Viscoelastic dampers typically use polymer or rubber materials that absorb energy when shear deformities. Figure 1.4 shows a viscoelastic damping device. Shear deformation occurs when structural vibrations cause relative motion between the external steel flange and the center plate, resulting in energy dissipation [4].

Figure 1.4Viscoelastic damper [5].

These damping devices are particularly effective for high- and low-frequency vibrations and can be used to protect buildings that are exposed to strong winds or moderate earthquakes.

It has also been shown that this viscoelastic damping device can not only increase the damping of the system, but also improve the stiffness of the structure, thereby increasing the natural frequency of the structure [6].

However, the properties of viscoelastic damping materials are affected by frequency and ambient temperature, although studies have shown that their properties remain relatively constant when the deformation does not exceed 20% [7]. Therefore, devices using this damping principle cannot be used where the temperature cannot be continuously controlled [4].

Viscoelastic dampers were first used in 1969 on the twin towers of the World Trade Center in New York to withstand wind loads. In Japan, this damper was also used in the Seavans Tower in Tokyo (1991) and

Gookuoku Hyogo Distribution Center (1998), which has a remarkable seismic effect [6].

Viscous dampers

Viscous dampers are widely used in the aerospace sector (satellites) and are characterized by linear viscous damping over a wide frequency band and are insensitive to temperature [8].

The principle of this damper vibrates the structure of the piston to deform the highly viscous fluid (such as silica gel) filled in the cavity, thereby converting mechanical energy into heat energy to achieve the purpose of energy dissipation.

If the filled fluid is a purely viscous fluid (Newtonian fluid), the force of the damper is proportional to the piston velocity.

Figure 15 Viscous dampers

1.2.2 Tuned dampers

Tuned liquid dampers

A Tuned LiquidDamper is a passive damping device that dissipates energy through the viscous action of a fluid. The results of Fujino et al. (1992), Wakahara et al. (1992), Sun (1991), Wakahara et al. (1989) showed that tuned liquid dampers were effective in reducing simple harmonic vibration and windEffectiveness in terms of vibration [3].

The first applications of this damping device were the steel truss towers of Nagasaki Airport and the wind-resistant design of the 149,4-meter-high New Yokohama Prince Hotel.

Figure 1.6 TLD of Shin-Yokohama Prince Hotel

Another type of tuned liquid damper is the TunedLiquid ColumnDamper (TLCD), which is constructed as a liquid-filled tube through which fluid passes through a hole Dissipate energy. The frequency of action of the damper can be adjusted by changing the length of the liquid column, and the damping size can be adjusted by adjusting the hole.

A study by Tamura et al 1995 showed that the displacement of the structure with a tuned liquid column damper installed at a wind speed of 20 m/s can be reduced by 35% [9]. The study by Sadek et al. [10] demonstrated the effectiveness of such dampers in reducing seismic vibrations. They studied the optimal parameters of the tuned liquid column damper, and the results showed that both displacement and acceleration could be reduced by more than 47% at a mass ratio equal to 4%.

Tuned mass dampers

Tuned Mass Damper (TMD) is now widely used in passive control and is used in many applications, such as automotive, aviation, marine and engineering structures.

TMD is mostly used in the field of civil engineering for flexible structures such as bridges and skyscrapers. The common denominator of these structures is that they are very sensitive to incentives such as traffic, wind, and earthquakes. Its advantages are simple, effective, and convenient. TMD is reliable for suppressing both simple harmonic vibrations and wind-induced vibrations. Its efficiency depends on the mass ratio, frequency ratio, and damping ratio of the AMA [4].

There are two main forms of TMD, one is a spring-mass structure, such as a vertical TMD installed under the deck of the Millennium Bridge in London to suppress vibrations generated by pedestrian load excitation. The other is the pendulum structure, the most famous application case of which is the 730-ton pendulum in the Taipei 101 Tower.

Figure 1. 7 TMD damper for Taipei 101 Building (Wikipedia: Taipei 101).

1.2.3 Particle dampers

Particle dampers use the friction and collision between tiny particles filled in a limited enclosed space in a vibrating body to convert the kinetic energy of the structure into heat energy dissipation. At the same time, the research in this paper also shows that the tuning effect in the damping mechanism of particle damper cannot be ignored, so the single classification of particle damper is emphatically introduced.

1.2.3.1 Development history of particle dampers

The predecessor of the particle damper is the collision damper. When the structure vibrates, the collision damper achieves the purpose of dampening the vibration through the collision force of the mass, which attracted attention as early as 1937. However, its high force and high noise limit its application. Later researchers improved the collision damper by combining the single inside the collision damper

masses are replaced with small and numerous particle groups, resulting in particle dampers.

Subsequently, in order to reduce the noise of the crash damper, in 1988 it was decided by Popplewell and Semercigil A bean bagimpact damper was proposed [11]. Pod-type particle dampers are characterized by the fact that the collision unit inside the damper is the pellet in a flexible container. Compared to collision dampers, it has the advantage of being less noisy and damping over a wider frequency range.

Particle dampers in general were proposed by Araki in 1995. It was first proposed by Araki as an "impactdamperwithgranularmaterial" [12]. It was then referred to as a particle damper by S. F. Masri [13].

1.2.3.2 Types of particle dampers

In the development process of particle dampers, on the basis of the original principle of vibration reduction, scholars have proposed improvement methods for the material and structure of particle dampers, or combine particle dampers with other dampers to meet different engineering needs.

By varying the size of the cavities and the number of separations, Saeki proposed a multi-unit particle damper [14]. He believed that too large a void would reduce the effective number of collisions between the particles and the structure, which would be detrimental to the damping effect of the damper, so he divided the void into multiple compartments. The influence of the size and number of separations on the damping effect was studied.

(a) Crash dampers (b) Pod dampers

(c) Multi-unit particle damper (d) Chain particle damper Figure 1. 8 Various types of particle dampers [15].

Gharib proposed a linear parti cle chain impact damper by adding smaller particles to the particles[15]. His research found that small particles surrounded by large particles under external excitation can produce more collisions, which in turn will produce more energy.

Some scholars have also changed the shape of the particles and studied the vibration damping effect of non-spherical particles and hollow particles.

In addition to improving the construction of the damper, scholars have also tried to improve it using different materials

Noise performance, vibration damping performance of particle dampers, etc.

Li et al. used rubber material to cover the inner wall of the container, which not only greatly improved the damping effect of the particle damper over a wide range of frequencies, but also reduced its impact and noise. Such a damper is called a bufferedimpact damper [16]. In addition to the original steel particles in the particle damper, the fine particle impact damper is filled with tiny deformable particles, so that under the effect of collision and friction, the tiny particles can be shaped and deformed, permanently consuming energy, which was proposed by Du Yancheng [17]. Polymer particle dampers were proposed by Darabi et al. [18]. He replaced traditional metal particles with highly viscoelastic ones. Experiments show that the energy dissipation efficiency of particle damping is not sensitive to the stiffness of the material, but the energy dissipation of particles with small stiffness is mainly concentrated in low frequencies. Experiments with Bustamante were carried out using the elastomeric material EEC-163001 and compared with steel pellets and polyethylene pellets. The results show that the damping effect of EEC-163001 is significantly due to the other two materials [19]. Abbas et al. used metal chipsinstead of spherical particles, using aluminum, stainless steel, and mild steel to make the pellets [20].

In addition to improving the structure and material of the particle damper itself, scholars have also tried to combine the particle damper with other types of damper to form a new type of damper with advantages in specific aspects.

Shah applied particle damping technology to semi-active vibration control by adding a magnetic field to his particle-based thrustdamping system [ 21]。

More scholars are combining particle dampers with TMD. This concept first appeared in collision dampers. In 1992, Semercigil et al. combined a collision damper with a tuned mass damper to create a new tuned crash damper [22]. His research has shown that the addition of collision dampers can improve the tuning of vibration absorbers

damping properties.

Figure 1. 9 Tuning the collision damper [22].

In 2003, Junling Chen proposed tuned rolling‐ball damper [23]. This is a combination of a tuned mass damper and a particle damper. He replaced the tuned mass with 3-12 particles, and achieved good results in the control of wind-induced vibration of the structure. Lu et al. suspended the particle damper on a slender cable to form a pendulum structure, and proved that the damper has a significant effect on suppressing the seismic response and wind-induced vibration of high-rise structures through shaking table experiments and wind tunnel experiments [24].

Figure 110 Tuning the Rolling Ball Damper [23].

1.2.3.3 New particle dampers

In terms of the structure of particle dampers, TPDs (tuned particle dampers), which combine particle dampers and TMDs, have been attracting a lot of attention in recent years. ZhengLu uses TPD for wind resistance in high-rise buildings, and his aeroelastic wind tunnel test shows that compared with the experimental model of TMD, the TPD damper has a good and stable wind vibration control effect [25]。 Similarly, TPD can be applied to seismic control of five-layer steel frames with good results [26]. Kai Zhang's experimental results show that TPDs have a wider operating frequency than particle dampers [27]. Compared to the damping efficiency of TMD, the spikes of TPD are more gentle. Shilong Li added a tuning structure to the particle damper, suggesting that the tuned mass damper can be used as a motion amplifier to improve the energy dissipation capacity of the particle damper when the local displacement or acceleration level of the structure is low [28].

At the same time, some new particle damper structures have also been proposed. MohamedGharib proposed a new linear particle chain impact damper [15]. It consists of a linear arrangement of two sizes of free-moving particles, with two restraint stops. When the structure is excited, the smaller particles collide with the larger particles several times, causing each collision to dissipate more energy. YanchenDu proposed a spring-supported fine particle impact damper [ 17]。 The cavity is connected to the structure with a spring with a high recovery coefficient, and the cavity is a fine particle impact damper with a low recovery coefficient, which is a comprehensive integration of elastic deformation damping and plastic collision damping.

Since 2013, our research group has carried out research on particle dampers, during which we have published invention patents such as rootless cable dampers, pod rain line dampers suitable for cable-stayed cables, and floor tile particle dampers suitable for horizontal vibration suppression of high-rise buildings.

The rootless cable damper scheme sets a cavity cylindrical container on the cable, and fills the cavity of the cylinder wall with particles, and uses the collision and friction between the particles, the particles and the container wall, as well as the asynchronous movement of the particles, to realize the damping effect on the movement of the container and the cable. Since this energy-dissipating mechanism controls the vibration by the damping particles themselves and the interaction between the particles and the cabin wall, it is different from the traditional dampers that require a fulcrum to provide a reaction force to counter the vibration of the cable, and can be arranged at any position required for vibration control without the limitation of the height of the damper installation [29].

The pod rain line type damper suitable for the cable-stayed cable is tightly wound on the cable-stayed cable, and the outer part is tightly wound on the cable-stayed cable as the cable-stayed rain line, plays the purpose of destroying the cable rain line, effectively reduces the wind and rain vibration of the cable. Particles of different sizes are evenly distributed inside, which simultaneously increases the damping in the cable and reduces other forms of vibration [30].

Floor tile particle damper suitable for horizontal vibration suppression of high-rise buildings provides a floor tile type particle damper suitable for horizontal vibration suppression of high-rise buildings, which can not only give full play to the characteristics of energy consumption and seismic resistance of particle dampers, but also do not occupy the space of high-rise buildings and can be installed at any height of high-rise buildings. Its shape is that multiple floor tile particle dampers are assembled with each other and used as floor tiles for the floor slab of high-rise buildings. The interior is divided into subcompartments, which are filled with metal-damped particulate matter [31].

1.2.3.4 The status and value of particle dampers in the field of civil engineering

However, particle damping technology has made a lot of progress in engineering applications in the mechanical and aerospace fields, and is gradually being introduced into vibration control applications of civil engineering structures. As early as 1997, Ogawa, Ide had already applied large-particle impact damping technology to suspension bridge pylons to control wind vibration [32]; In the 2010 Chile earthquake, a high-rise building with a particle damper was put to the test[33]. In recent years, Yan has demonstrated that tuned particle dampers can effectively reduce the vibration response of continuous viaducts under seismic loads through shaking table experiments and finite element analysis of continuous viaducts [34]. Papalou, Strepelias, which filled the monument with grains, had a good damping effect [35]. Lu Zheng experimentally demonstrated that particle dampers can also perform well in wind and earthquake resistance in high-rise buildings [25] [26]. Egger applies the principle of collision damper to the vibration damping of cable-stayed cables, placing a steel wire in a cavity with a larger diameter, and the steel wire collides with the cavity during vibration to play a role in vibration damping. Its 1:1 model experiments proved the effectiveness of the scheme [36].

Particle damping technology is expected to bring benefits to the vibration control of the structure in the following two aspects: first, because the damper container can be an arbitrary structural cavity, it does not occupy the structural space and does not lose the usability of the structure; Second, because particle damping can provide good vibration damping effect in a wide frequency range, and its damping effect is not sensitive to vibration direction, it is expected to realize multi-modal and multi-directional vibration control of structures. At the same time, the particle damping technology also has the advantages of good durability, high reliability and low temperature resistance.

1.3 Research theory of granular materials

1.3.1 Application of granular materials in industry

Granular materials have shown important practical value in a variety of industries and are widely used in various industries [37].

mining

The process of obtaining raw materials through mining faces a number of scientific problems related to granular materials: the open pit

The stability of the mine wall, the stability of the mine under explosive load, the stability of material accumulation, the wind erosion of raw materials under open-pit accumulation, the problem of conveyor belts, the problem of raw materials transported by railway, road and sea and land, and the working efficiency of the stone crusher are studied.

civil engineering

The concrete material widely used in civil engineering is composed of cement, and its production process includes extraction, mixing, firing and curing. Even the concrete material itself is a saturated granular material. The study of its fluidity, viscosity and resistance is closely related to the study of granular materials.

The relationship between buildings and soil has attracted wide attention in the field of civil engineering and geotechnical. As a kind of granular material existing in nature, the study of the characteristics of soil under load is closely related to the foundation design of various structures.

Earthworks, including the construction of dams and artificial islands, are inseparable from the study of the mechanical properties of granular materials.

Energy Engineering

The process of fossil energy extraction is inseparable from the study of the characteristics of particulate materials in the underground: oil drilling, anchoring and safety of offshore development platforms, and large-scale hydraulic fracturing processes in gas extraction.

At the same time, there are many problems related to the research of particulate materials in the process of fossil energy treatment: the basic design of offshore water and wind power plants, the storage of carbon dioxide in geological formations, and the extraction and extraction of fossil fuel utilization waste.

Processing

There are many granular materials in the agro-processing industry: grains, fruits, feed, sugar, starch, etc. The storage and transportation of these products require the study of the properties of the granular material.

There are also a wide variety of granular materials in the pharmaceutical industry: the active ingredients and excipients in the pills are ground, mixed and compressed in the powder state.

In the chemical industry, the study of particulate materials is mostly focused on the chemical interaction with particles. In the machinery industry, the conveying of granular materials and the forming of powder materials are more common.

1.3.2 History of research on particulate materials

The use and understanding of particulate materials can be traced back to the beginning of human construction with soil resources, so it is difficult to define the beginning of research on particulate materials. Charles-AugustinCoulomb (1736-1806), who is generally considered to have been the founder of the study of granular materials, was best known for his 1773 paper "The Application of the Statical Maximum and Minimum Theorems to Architecture." research" [38]. Coulomb was a French military engineer whose research focused on the stability of slopes, the stability of stone mills, and the stability of stone arches. He also derived the Coulomb friction theorem, which is widely used in granular materials, from engineering.

MichaelFaraday (1791-1867) studied the behavior of granular materials under vertical loads in 183 1. In contrast to Coulomb's research, Faraday focused on the kinetic analysis of granular materials. the

The study eventually extended to chaotic phenomena in physics [38].

In 1857, WilliamRankin (1829-1872) theoretically studied the Coulomb friction theorem, and thus proposed the theory of passive failure and active failure of soil commonly used in modern soil mechanics [39].

In 1895, an engineer named Jasssen proposed the arching effect of granular materials when they were stacked. The phenomenon was refined by JohnWilliam Strutt Rayleigh in 1906. This theorem is now widely used in material storage engineering [40].

Between 1940 and 1970, RalphBagnold (1896-1990) published several studies on the phenomenon of sand and soil flow. In 1954, he proposed a dimensionless number—the Bagnolde number—to define the different states of granular material as it flows on a slope [40].

The reason for the slow progress of research on particulate materials is not only because of their different properties from traditional engineering materials, but also because of the difficulties faced by scientists when trying to model them [38]. The modeling of traditional engineering materials is based on rigorous mathematical models, so the models of materials are highly applicable and easy to promote. For example, Hooke's law, which is common in solid mechanics. This law allows engineers to describe the compressive expansion properties of a material using a simple dimensionless parameter, which can be moved to more complex materials such as non-Newtonian fluids with some variations. However, none of these laws apply to granular materials where there is no continuity at all.

However, there is no universal constitutive model for particulate materials. Therefore, the research on this material is based on some local laws, practical experience and experimental phenomena. Each of these laws has its own conditions (geometry, flow state, specific material, etc.). What works in some scenarios doesn't always work in new scenarios.

1.3.3 Interparticle forces

For particles with a scale greater than 100 μm, the interaction is dominated by collision and friction.

Based on the energy consumption or not, the collision between particles can be divided into elastic collision and inelastic collision. For elastic collisions, the kinetic energy is conserved before and after impact; If a portion of kinetic energy is converted into other forms of energy (heat, deformation, or wave energy) during a collision, such a collision is called an inelastic collision. Isaac. Isaac Newton defined the collision recovery coefficient e:

v1,t − v2, t = e. (v1,0 − v2,0 ) (1. 1)

where v1t and v2t are the velocities of the two objects after collision, v 10 and v2 0 is the velocity of the two objects before they collide.

The Newtonian collision recovery coefficient is a property of the material, and for elastic collisions, the coefficient is 1. In practice, no material has a collision recovery coefficient of 1: there is no ideal elastic collision that does not consume energy.

There is an elastic deformation during the collision. The collision force is related to the material and geometric parameters of both objects.

The model of the oblique collision process is also related to the friction between the particles. The Coulomb friction model contains three theorems defined [41]:

1) The friction is proportional to the weight of the solid placed on the horizontal plate;

2) For a given weight, the friction is independent of the contact area, only the roughness of the contact surface;

3) For most materials, the coefficient of friction is independent of the sliding velocity.

There are two important coefficients in Coulomb's friction model: the static friction coefficient and the dynamic friction coefficient. The former gives the conditions under which friction occurs and the limits of friction. If the lateral force exceeds this limit, a slip phenomenon occurs. The magnitude of the frictional force between two objects is obtained from the coefficient of dynamic friction. From the third clause of Coulomb's theorem, it can be seen that the friction coefficient is independent of the sliding velocity.

In summary, the interaction between particles is a combination of collision and friction mechanisms. To put it simply, when the particles come into contact, they will produce both normal and lateral forces, the former from the collision model and the latter from the friction model.

For fine particles (less than 10μm in diameter), the cohesive forces between the particles, such as electrostatic force, van der Waals force, capillary force, etc., cannot be ignored [41].

The electrostatic force is directly proportional to the amount of electricity between the two particles and inversely proportional to their distance from each other. For particles larger than 100μm in diameter, the force is negligible due to the large distance between the particles.

The van der Waals force is the electromagnetic force between particles at the molecular level. For single pair of particles, the force magnitude is sensitive to the change in distance between the particles, but for the particle population, the van der Waals force is more robust.

Capillary forces occur in the swarm of unsaturated particles. The particles are attracted to each other due to the presence of a liquid film. When two particles with capillary force move away from each other due to an external force, the presence of a liquid film between the particles will resemble a spring, bringing the two particles closer to each other.

However, the size of the filler in the particle damper used in civil engineering is in the centimeter range, so the cohesion between the particles can not be considered in the analysis.

1.3.4 Research methods for particle dampers

1.3.4.1 Parsing methods

Due to the nonlinearity and complexity of the particle damper, the study of the analytical model mostly uses the simplified model to describe the damping characteristics of the particles: assuming that all the particles are moving mass points, a series of modifications are made to the mold of the impact damper to establish the vibration reduction model of the particle damper.

Based on the assumption that "particles collide with the container wall twice per cycle", Masri derives the equation of motion of a single particle in the simple harmonic motion of the structure [42]. Bapat extends this assumption to mean that in simple resonant oscillation, particles collide with containers N times per cycle [43]. Based on this assumption, Masri developed an analytical model of a single particle damper. Bapat and Sankar used this model to analyze the optimal void size and its damping effect by considering the effects of friction [44].

Collisions that occur during particle motion cause abrupt changes in the equations of motion. For single-particle dampers, researchers can still specify the path of particles by assuming the moment when the particles collide with the container, and obtain the equation of motion of the single-particle damper. However, in the case of multi-particle dampers, the relative position of the particles and the collision between the particles

The situation is difficult to generalize to hypotheses, so it is difficult to describe the state of motion of particles in the presence of multiple particles at the same time. So far, there has been no breakthrough in the derivation of theoretical models for multi-particle dampers.

Masri derives a theoretical model of a damper with two particles under the assumption of a steady-state response of simple harmonic excitation [45]. Subsequently, Gapat established the equation of motion of the multi-element particle damper based on the results of the single-particle damper analysis. In his research, it has been found that when the number of particles is large, the theoretical solution of the equation cannot be obtained, and it needs to be calculated with the help of programs [46].

In the subsequent research on multi-particle dampers, most of the researchers used numerical simulation and experimental methods

1.3.4.2 Numerical simulation method

S. E. Olson used molecular dynamics methods to model the damping behavior of particle dampers and analyzed the damping mechanism [47]. This modeling method treats each particle as a point of mass and calculates the trajectory of each point of mass. Within each time step, the calculated interaction forces between the particles and the net force acting on each particle can be displayed. The particles only exert a force effect on the adjacent particles. This method is mostly used for modeling at the atomic or molecular level.

Discrete element modeling is widely used in the study of particulate media. It is even more common in the simulation of vibration damping of particle dampers. This method is derived from molecular dynamics models and was originally proposed by Cundall and Strack [48].

In the calculation of the model established by the discrete element method, the Newtonian equation of motion is used to track the trajectory of each particle. A suitable contact model is used to calculate the interaction forces between the particles. The main difference between this method and the molecular dynamics method is that the particles analyzed have shape, size, and rotational degrees of freedom, so other interactions between the particles can be analyzed, such as rotation, friction, etc.

The Monte Carlo method [49] uses a sequence of random numbers to simulate processes, such as the decay of particles and the transport of particles in a medium. This approach can also be used to solve deterministic problems that are not directly random with the help of probabilistic models. It makes the solution of the problem equal to the parameters of a hypothetical statistical model, and a random sequence of numbers is used to construct a sample of this statistical model, and the statistical estimate of this parameter can be derived.

Although the model does not simulate the motion process of individual particles, it has the advantage of short time and high efficiency.

As an algorithm with strong nonlinear mapping ability, neural network is also used in computational modeling of motion sensing particle media. [50] Li et al. used the LM algorithm of BP neural network to model the particle damping model on mass ratio, clearance, recovery coefficient, friction coefficient, particle diameter and vibration level, and used 20 samples to train the model to achieve good accuracy.

The advantage of this method is that the results are very consistent with the experimental data, but the internal rules of the model are "black box", which makes it difficult to give physical meaning to the model, and the results of this model are not universal, and are only suitable for prediction of cases similar to the training sample.

Experimental methods are still used as a reference for judging the correctness of the results of other methods, including shaking table experiments and wind tunnel experiments [25] [26].

1.4 Discrete Element Method

Among the many numerical calculation methods for the study of particulate materials, the discrete element method is widely used due to the following three advantages:

i. Simple model setup. The calculations of the discrete element method are based solely on Newton's second theorem of motion

ii.Strong operability. In Eq. 1.1, the external force Σ- term can be added to various contact forces and external forces according to the purpose of the study

Forces, such as: gravity, viscous force, frictional force, etc

iii. The results are intuitive and easy to understand. The discrete element method calculates the trajectory of each unit, which can completely restore the velocity, displacement, force and other information in the process of imitation, which is difficult to obtain from experiments.

In this paper, this numerical method will also be used to study particle dampers.

1.4.1 The basic process of discrete element method calculation

The discrete element method was proposed by Cundal and Strack in the 1970s and was originally used to analyze rock mechanics [48]. Subsequently, this method has been widely used to study the dynamic behavior of granular materials, such as sand, powder, and grain. The concept is to use contact and friction mechanics, Newton's second law of structure to describe the trajectory of all particles in a system. The calculation process is mainly divided into the following steps:

Figure 1.11 below illustrates the process of the DEM algorithm in detail. The discrete element method calculates the conversion of kinetic energy to thermal energy in a system and analyzes a variety of phenomena due to particle motion.

Figure 111 Discrete element method calculation process

1.4.2 Hard Ball vs. Soft Ball Model

When modeling granular materials using the discrete element approach, there are two modeling methods: one based on the assumption that the elements are not deformable (the hard sphere model) and one based on the assumption that the elements are deformable (the soft sphere model).

In the hard ball model, the particles are assumed to be non-deformable, and the collisions between the particles are concentrated at one point. Each impact collision can be divided into two phases: in the first stage, a shock wave is generated due to instantaneous collision compression, and when the compression reaches its maximum, the collision progresses to the second stage, which is the recovery phase. The collision time is very short in this model (instantaneous collision), the impact force is large, and there is no rigid body movement of the particles at the moment of collision [51]. In this hard-sphere model, the equations of motion of the particle elements can be written from the conservation of momentum and kinetic energy. Among them, the energy loss due to the collision is expressed by the introduction of the collision recovery coefficient [51] [52]. The collision recovery coefficient can be obtained based on the change in velocity, or it can be obtained based on the change in energy before and after the collision.

The hard sphere model is mostly used for fast-moving particle flows with specific velocity values [53]. At the same time, the actual should

It is not necessary to explicitly express the collision force between particles, which can greatly reduce the calculation time [51] [54]. The hardball model has been used in research by several scholars [55] [56]. However, one of the assumptions of this model—that particles collide in pairs—limits its application. Another drawback of this model is the inability to introduce cohesive forces into the model [57]. As a result, the hardball model is not the most common model in discrete element simulations.

The hypothesis of the soft-sphere model was first proposed by Cundall and Strack [48]. At the same time, they are also pioneers in the numerical calculation of discrete elements. Compared with the non-deformable assumption, the assumption of discrete element variability is closer to the actual phenomenon. In the soft sphere model, every contact between elements, including normal and tangential directions, can be simulated by the spring and damper models. Springs are used to simulate contact stiffness, and dampers are used to simulate the energy losses that occur in collisions. The collision force between the particles is calculated by the amount of overlap that occurs during the collision. In the soft sphere model, the amount of deformation of a particle can be considered to be extremely small relative to its size (the wavelength of the elastic wave is small relative to the particle size). Unlike the hard ball mold, the time required for each collision is not zero.

This model is more suitable for dense particle systems. However, the computation time using this model is longer than that of the hardball model, so many scholars have also worked to simplify the model to reduce the time required for model computation [58] [59] [60] [61].

The respective characteristics of the two models are summarized in Table 1. It can be seen that the essential difference between the two is that the hard-ball model uses the conservation theorem to establish the element equations, while the soft-sphere model uses a phenomenon-based contact model to simulate the interaction between particles.

Considering the applicability of the model and the modeling assumptions of the supporting software, the soft sphere model will be used in this paper. It is also the most commonly used discrete element model [53].

Table 11 Comparison of the soft sphere model and the hard sphere model

1.4.3 Contact Model

1.4.3.1Cundall model

The Cundall and Strack models [48] are considered to be the basis for the soft-sphere model. The contact model used can be summarized in Figure 1.12.

Figure 112 Hertz-Mindlin contact model [62].

Suppose two spheres with r and rj are and, and their centers are c and cj, respectively. As shown in Figure 1.13, the points ci, c and c are in a straight line, and the direction defined by the line is the collision normal (c, c ), denoted as n-→. with

, a straight line that passes through the point C and is tangent to both spheres is defined as the collision tangential direction, which is also perpendicular to n-→

Straight.

Figure 113 Collision coordinate system [38].

In the coordinate system defined by n-→, the force of the particle acting on the particle can be recorded as the vector sum of the normal force f- and the tangential force f-:

f--=fn . n-→ +f. (1.3)

From this, the normal overlap between particles can be defined δn and tangential overlap δt. where the normal overlap can be expressed by the geometric relation as:

However, for the collision coordinate system represented in Figure 1.13, the amount of tangential overlap is not related to the particle position.

Therefore, Cundall uses tangential overlap velocity to represent the amount of tangential overlap. The expression includes the translational velocities v and v of the two particles, as well as the rotational velocities w and w.

d.t =

When the two particles just start to collide, the tangential overlap is δt = 0. This allows the tangential overlap to be calculated by integrating δt. As can be seen, the effect of loading history needs to be taken into account in the calculation of tangential overlap quantities.

In the Cundall and Strack models, the damped spring system between the particles is linear, so the normal and tangential forces can be expressed as:

f- = —kn δnn-→ — in (-

f- = —kt δt + yt (-

where kn and yn are the normal contact stiffness and contact damping (the corresponding kt and yt are tangential stiffness and damping ），

- is the relative velocity at which particles come into contact with each other, and is expressed as:
-→ -→ -→ v = v — v	+ rj - — ri-	(1. 8)

Finally, Cundall and Strack took the model even better by adding the concept of friction to tangential collisions. It uses the Coulomb friction theorem, which defines the upper limit of the tangential force. When the tangential force exceeds this upper limit, the tangential motion becomes friction. This upper limit is defined as a linear function of the normal force:

|ft |lim = μ. fn (1. 9)

In summary, the contact model adopted by Cundall and Strack can be summarized as follows:

f--=fn . n-→ +f(1.(Art. 10)

thereinto

When δn<0 when δn≥0

(1. 11)

1.4.3.2 Hertz-Mindlin 模型

The Hertz-Mindlin model is one of the most commonly used discrete element contact models [63], and it is also the contact model used in the discrete element approach in this paper.

Hertz model: normal force

Elastic deformation and frictionlessness are the basic assumptions of Hertz's normal force model [64]. At the same time, the Hertz model assumes that the contact surface of any two objects is elliptical, as shown in Figure 1.14.

(a) The shape of the contact surface when any two objects collide (b) The distribution of pressure on the contact surface 114 Hertz model [65].

For a collision between two spheres, Hertz proved that the area of the contact surface is a circle with a half-diameter a[64], where a is

thereinto

is the equivalent radius of the collision;

Andeq=

is the equivalent modulus of elasticity of the collision.

Similar to the Cundall and Strack models, normal overlap is introduced

Combined with the above equation, the normal contact force can be obtained as

thereinto

is the normal stiffness of the collision.

In contrast to the Cundall and Strack models, the normal forces of the Hertz model are no longer linearly related to the overlapping quantities.

Mindlin Model: Tangential Force

For the tangential force of inter-particle collision, Mindlin and Deresiewcz [63] proposed a relatively complete tangential force model. This section illustrates its model in the simplest case, where the normal force is constant.

Suppose ftk+1 is the tangential force after the collision, ftk is the tangential force before the collision, and Δδt is the change of tangential overlap during the collision. Hence the tangential stiffness can be expressed as:

It can be seen that the tangential stiffness varies with the change in the tangential force ftk. Assuming that ftlim is the upper limit of tangential force loading, it is thus possible to distinguish the values of ftk in four cases:

When ftk is loading, and |ftk |≤|ftlim |, you can write tangential stiffness:

kt = k0

When ftk is loading, and |ftk |>|ftlim |, you can write tangential stiffness:

kt = k0

When ftk is unloading, and |ftk |≤|ftlim |, you can write tangential stiffness:

kt = k0

When ftk is unloading, and |ftk |>|ftlim |, you can write tangential stiffness:

kt = k0

Mindlin also proves at the same time

k0 = 8a

where v and G are the Poisson's coefficient and shear modulus of the particles.

The change in tangential force with the amount of tangential overlap is shown in Figure 1.15. It can be seen that when the tangential force is less than the tangential force limit, its expression is related to the loading path.

Figure 115. When the normal force is constant, the tangential force changes with the tangential overlap [66].

The advantage of the Mindlin model is that it completely describes the changes in the tangential force during loading and unloading, and can truly reflect the magnitude of the collision force [66].

1.5 This paper mainly focuses on the research work and content arrangement

1.5.1 The main research work in this paper is the main work

Particle damping technology is a kind of damping technology that has gradually emerged in the field of structural vibration resistance in recent years, which has the characteristics of stability and high efficiency. However, due to its nonlinearity and discontinuity, numerical simulation methods have become one of the main research methods, and the discrete element method is the most commonly used simulation method in the field of granular materials. In this paper, we will first use this method to discover the variables that affect the damping performance of particles and their influencing rules. The specific method is to use the principle modeling software to establish a model of particle damper, to simulate the loading of a single damper and the damper-structure system respectively, select the variables to be studied for a large number of simulation calculations, and summarize the laws from the results. Then, with the help of the experience obtained in the simulation, the theoretical damping force model of the particle damper is established, the model is applied to different dampers, and the applicability of the theoretical model is verified by comparing with the discrete element simulation results, and the use conditions of the model are summarized.

Although the single-particle model proposed in this paper is two-dimensional and regular-shaped, it can be applied to three-dimensional floor tile particle dampers and rootless cable dampers through certain equivalence principles. At the same time, the model can reflect the variation law of particle damping with load, material and geometric size, which can be used to improve the optimal design of particle damper

for theoretical basis.

1.5.2 Arrangement of the content of this document

The main contents of this article are arranged as follows:

The first chapter is the introduction, which mainly introduces the common dampers and highlights the development history and common forms of particle dampers. At the same time, the research status and commonly used calculation methods of granular materials are given. Finally, a further detailed introduction is given to the discrete element method to be used in this paper, and an introduction to the commonly used contact model is given.

The main content of the second chapter is to establish the discrete element simulation model of the rootless cable damper, and on this basis, to study the law of the change of energy dissipation of the particle damper. The coupling modeling of the discrete element method and the multi-body dynamics method is used to further analyze the performance of the damper in reducing the vibration of the cable.

Chapter 3 proposes a single-particle model to study the damping force of a particle damper, and analyzes how each variable changes the energy dissipation of the damper from the perspective of a theoretical model. The model is compared with the common Coulomb friction model and viscous damping model, and the characteristics and advantages of the model are summarized.

Chapter 4 focuses on applying the model from Chapter 3 to two types of particle dampers: the floor tile particle damper and the rootless cable damper. The equivalent modeling method of the actual damper is given. At the same time, compared with the calculation results of the discrete element method, the applicability of the single-particle model is verified, and the application conditions of the model are summarized on this basis.

Chapter 5 is a summary of the work of this paper and a prospect for further research.

Chapter 2: Simulation of a Particle Damper-Cable System Based on the Discrete Element Method

2.1 Introduction

When faced with a damper system composed of multiple particles, it will be very difficult to study the damping performance by theoretical analysis method due to the many factors affecting the damping effect of particles, the complex mechanism, the strong nonlinearity, and the mutual influence between various effects [67]. At the same time, the test method is expensive, and it is not possible to directly use the test method to conduct aimless trial and error research in the early exploration without understanding the performance of the particle damper. A more effective and feasible method is to use numerical simulation experiments to conduct exploratory studies to obtain the performance characteristics of particle dampers, and then design specific schemes for key problems [6, 8].

At present, numerical and experimental methods are mainly used in the research and analysis of multi-particle dampers. This is due to the fact that granular dampers have strong nonlinear characteristics, and there is no mature theory to describe their characteristics. Among the existing numerical simulation methods, the discrete element method is widely used because of the advantages of simple model and strong operability. The discrete element method holds that when the time step is small enough, any discrete element is only subjected to the forces of its neighbors, which, in combination with Newton's second law, can be calculated step by step, and the interaction forces between the elements can be described by different contact modes.

In this chapter, we will analyze a special type of particle damper, the rootless cable damper, using the discrete element method. The damper overcomes the shortcomings that the general cable damper can only be installed at the anchor end of the cable, and can greatly improve the efficiency of the damper.

Firstly, the discrete element model of the rootless cable damper will be carried out by using the commercial discrete element software EDEM, and the energy consumption efficiency of the particle damper under different load conditions will be analyzed by using the established model. Secondly, the coupling of discrete element method and multi-body dynamics method is used to establish a cable-damper system model to analyze the damping performance of the damper under three mechanisms: collision, friction and tuning. At the same time, the change law of energy dissipation efficiency and vibration damping performance was analyzed according to the design parameters of the damper, including filling rate, particle size, collision recovery coefficient and friction coefficient, so as to provide a direction for follow-up research.

2.2 Simulation model of rootless particle damper

2.2.1 Application background of rootless particle damper

The construction of extra-long span cable-stayed bridges presents many new challenges for bridge engineers. One of them is the vibration damping problem related to cable-stayed cables, which are prone to vibration due to their low internal damping.

The vibration control methods of cable-stayed cables include active control, passive control, and active and passive joint control. Among them, the passive control method is widely used because of its simplicity and ease of implementation, especially the passive control measure for installing dampers. Due to its convenient construction and remarkable effect, it has become the most widely used method of vibration control of cable-stayed cables. However, existing damper technologies, such as high-damping rubber dampers, hydraulic dampers, and viscous shear dampers, all need to be connected to the fixed point, and in practice can only be installed near the anchor end of the main beam of the cable [69]. The results show that the maximum damping ratio that can be obtained by the cable-stayed cable after attaching the damper is closely related to the installation height of the damper [70], which limits the restriction of the installation height and the vibration damping effect of the damper on the cable.

In order to solve this problem, a rootless cable damper scheme that can be installed at any height of the cable was proposed in Ref. [29]. The scheme sets the cavity cylinder container on the cable, fills the inside of the cylinder cavity container, and uses the collision and friction between the particles and the container wall, as well as the asynchronous movement of the particles, to realize the vibration damping effect of the container and the cable movement, as shown in Figure 2.1 and Figure 2.2. In the figure, 1 is the left chamber, 2 is the right chamber, 3 is the damping particle, 7 is the connecting bolt, and 8 is the central elastic sleeve [29]. This damper is not limited by the installation height, which is conducive to improving the vibration damping effect of the cable.

Figure 21 Schematic diagram of a rootless particle damper [29].

Fig. 2.2 Cross-sectional view of a rootless particle damper [29].

2.2.2 Discrete element method software selection

There are many computational tools based on the above logic for researchers, such as LMGC90, LIGGGHTS,

Esys-Particle, YADE, PFC and EDEM, etc. Although they all share the same basic principles, the accuracy of the calculations varies due to differences in the actual details.

The LMGC90 uses Non Smooth Contact Dynamics as a discrete element modeling method. This method does not take into account the overlap between particles, employs the laws of irregular interactions (threshold law, sudden change in velocity, etc.), implicitly integrates the kinetic equations, and solves using the display integral (Cundall model). This approach is suitable for modeling a large number of particles [72], as well as for masonry structures. It should be noted that there are various interaction laws in the software: friction, cohesion, reinforcement connection, etc., and attention should be paid to the selection of models when using them.

LIGGGHTS (LAMMPSImprovedfor GeneralGranularand Granular HeatTransfer Simulations) is a free software written in C++ , which is an advanced version of LAMMPS for applications in granular materials. Improvements have been made in granular materials and heat transfer in particles. The software is developed by Kristen ADeveloped by ChristophKloss of the Doppler Laboratory for particle flow modeling (physics) at Johannes Kepler University in Austria.

ESyS-Particle is a numerical modeling software based on the discrete element method, which is widely used to model large deformation problems, particle flows, and rock fragmentation processes. The software runs on a Linux system and has high requirements for the computing performance of the computer.

The YADE calculation code (YetAnotherDynamicEngine) [73] was developed by the 3SR laboratory in Grenoble to perform three-dimensional numerical models. Since the software has a large number of users and is open-source, the functionality of the codecode is constantly evolving. The calculation part is written in C++, and the pre- and post-processing can be created and debugged in Python.

The PFC (Particle flow code) software was developed by the ITASCA group. It uses the discrete element square method to model the motion and interaction of spherical particles. The software can work in either 2D or 3D mode. In the calculation of the PFC, the kinetic equations are solved at each time step, instead of saving the results and reusing the matrix as in the implicit scheme. Therefore, while each solution takes only a fraction of a second, the computation time for a system composed of multiple particles is quite long. So, the calculation time is usually one of the limitations of this software.

EDEM is a high-performance commercial software developed on the basis of a discrete element approach to simulate the flow of bulk materials. EDEM can quickly and accurately model and analyze the behavior of loose materials. EDEM not only has a user-friendly interface compared to other software, but also provides a coupling interface for bi-directional simulations that can be coupled with CFD software and multibody dynamics (MBD) software, allowing for multiphysics simulations.

Table 2.1 describes the advantages and disadvantages of source and commercial software for use as a reference when selecting discrete element software.

Table 21 Comparison of open source discrete element software and commercial discrete element software

Considering the superior conditions of calculation speed and allowable coupling, EDEM is selected to carry out the numerical simulation tool of rootless damper.

2.2.3 Model building

In this paper, the discrete element method is used to simulate the particle motion, and the numerical model of the particle damper is established by using the commercial software EDEM. In order to improve the speed of numerical simulation, the particle damper scheme mentioned above is simplified to a certain extent. Ignoring the partition inside the particle damper, the container is not separated inside, forming a monolithic chamber. The thickness of the outer wall of the container is not considered, and the corresponding quality is directly assigned to the container. The numerical model of the damper is shown in Figure 2.3.

Table 2.2 Particle damper parameters

During the calculation, the damper is placed horizontally, considering the effect of gravity, and the excitation direction is vertical.

2.2.4 Computational Studies

In this section, five sets of examples are set for the vibration amplitude, filling rate, particle radius, collision recovery coefficient and friction coefficient, and the load frequencies are 0.5, 1, 2, 5, 10 and 20 Hz in each group of examplesThe damper response. Table 2.2 shows the parameter values of each study.

Table 23. Working condition setting

In Table 2.2, the first row is the basic default parameter, and the rest of the examples are based on a variable to explore the energy dissipation characteristics of the rootless particle damper.

2.3 Energy consumption analysis of rootless cable damper

2.3.1 Vibration Intensity

The vibration intensity includes the frequency and amplitude of the simple harmonic vibration. Fig. 2.4 and Fig. 2.5 show the relationship curves of the energy consumption per cycle with frequency and amplitude of the particle damper, respectively. The frequency of the simple harmonic vibration varies between [0.5,1,2,5,10, 20]Hz, and the amplitude varies between [50,100,200,300,400 500]mm

A combination of frequency and amplitude can solve the energy dissipation of the damper per unit period, and a total of 36 sets of data can be obtained.

Figure 2. 4 Relationship between energy consumption and vibration frequency per cycle Figure 2. 5. The relationship between energy consumption and vibration amplitude per cycle

As can be seen from Figure 2.4, the energy dissipation per unit cycle of the particle damper increases rapidly with the increase of frequency. As can be seen from Figure 2.5, although the amplitude has a positive effect on the energy dissipation, the rate at which the energy dissipation increases with the amplitude is smaller than the rate at which the energy dissipation increases with frequency.

Therefore, in order to improve the readability of the graph, the dimensionless energy dissipation efficiency η of the damper is defined, and its value is the ratio of the energy dissipation per cycle to the maximum kinetic energy in the cycle. For dampers that do simple harmonic vibrations:

h=

Figures 2.6 and 2.7 illustrate the effects of frequency and amplitude on energy efficiency, respectively.

Figure 2. 6 Curves of energy dissipation efficiency and vibration frequency

Figure 2. 7 Curve of energy dissipation efficiency and vibration amplitude

As can be seen from Figure 2.6, the energy consumption efficiency first increases with the vibration frequency and then tends to be stable. In other words, although the damper has a higher energy dissipation value at high frequencies, the kinetic energy of the system is higher and the energy dissipation efficiency of the damper remains stable. At lower amplitudes, high frequencies of 20Hz can even reduce energy efficiency.

Figure 2.7 illustrates the three forms of the effect of vibration amplitude on energy efficiency:

When the vibration frequency is low, the energy consumption efficiency is very low, almost zero, and the energy consumption efficiency does not change with the amplitude. This is due to the fact that the vibration intensity is too low to reach the threshold for the relative motion of the particles to the container

There is almost no energy conversion. At a frequency of 0.5 Hz, the structure starts to consume energy only when the amplitude reaches 400 mm.

When the vibration frequency is 1Hz, the vibration efficiency slowly rises with the increase of amplitude, and begins to decrease after reaching a peak at 400mm.

When the vibration frequency exceeds 1Hz, the vibration efficiency decreases with the amplitude.

Figure 2.8 shows the fluctuation plot of energy efficiency in the amplitude-frequency plane. The upper left corner of the plane indicates that the vibration frequency and vibration amplitude are small, and the vibration intensity is small. The lower right corner indicates a high vibration intensity. The colors in the diagram represent the energy efficiency. Gray tends to be the area with high energy efficiency, blue area is the area with low energy efficiency, and orange area is the area with medium energy efficiency.

Figure 2. 8 Energy dissipation efficiency as a function of amplitude and frequency

It can be seen that the area with the highest energy efficiency is concentrated in the lower left of the plane, that is, the area with less amplitude but higher frequency. The areas with the lowest energy efficiency are concentrated in the lower frequency area, the 0.5-1 Hz region and the amplitude above 400 mm.

The particle damper can dissipate energy in two forms: collision and friction, and the proportion of collision energy dissipation and friction energy dissipation in the total energy consumption can reflect the movement state of particles to a certain extent. Figure 2.9 shows the fluctuation of the proportion of collision energy dissipation in the amplitude-frequency plane. The colors in the diagram represent the proportion of the total energy consumed by the collision. The blue area is the small area with a proportion of 0.6-0.7, the gray area is the large area with a proportion of 0.8-0.9, and the orange area is the medium area.

Figure 2.9 The proportion of collision energy dissipation in total energy dissipation vs. amplitude vs. frequency

It can be seen that the area with a small proportion of collision energy dissipation exists above the plane, that is, the vibration frequency is close to 1 Hz. The area with a large proportion of collision energy dissipation extends from the lower left corner of the plane to the center of the plane at an angle of 45 degrees, that is, the area with small amplitude but large frequency. Compared with Figure 2.8, it is found that the region with high energy efficiency is consistent with the region with a high proportion of collision energy consumption. This shows that the high proportion of collision energy consumption is a manifestation of the high energy consumption efficiency of the system.

(a) The vibration frequency is 1 Hz, and (b) the vibration frequency is 20 Hz, Figure 210. The distribution of energy dissipation varies with amplitude

As can be seen from Figure 2.10, the proportion of collision energy dissipation at high frequencies is generally higher than that at low frequencies. However, this ratio fluctuates very little, varying between 60% and 90%. Overall, collision energy consumption accounts for more than 50% of the total energy consumption, which is higher than friction energy consumption.

2.3.2 Fill rate

The ratio of the total particle volume to the volume of the cavity inside the damper is defined as the particle filling rate within the particle damper. The expression for the particle fill rate is:

where n is the number of particles; d is the particle diameter; V is the volume of the container, and D1 and D2 are the diameter of the outer ring and the diameter of the inner ring of the damper cross-section, respectively. The higher the number of particles, the smaller the container volume and the greater the filling rate. When the particles are not filled, the filling rate is 0. The filling rate is greatest when the container is filled with particles. However, due to the voids in the arrangement of the particles, the filling rate cannot reach 1, and in this case the maximum filling rate is 0.514. Figure 2.11 shows the grains with different filling rates

Particle dampers.

Figure 211. Particle dampers under different filling rates

Calculate the energy efficiency of the particle damper at each fill rate under a load of 0.5-20Hz, while keeping all other parameters constant. Figure 2.12 shows the effect of frequency on energy dissipation efficiency at different filling rates and the effect of filling rate on energy dissipation efficiency at different frequencies, respectively.

(a) Influence curves of frequency on energy consumption efficiency under different filling rates

(b) Influence curves of fill rate on energy consumption efficiency at different frequencies

Figure 212 Calculation results of case 2

As can be seen from Figure 2.12(a), the energy efficiency increases with frequency for particle dampers with low fill rates (0.073-0.367). For high fill rates (0.441), the energy efficiency increases with frequency, and begins to decrease when the frequency is higher than 10 Hz. This suggests that in the case of low fill rates, the increased vibration intensity can make the relative movement of particles within the container more intense. Especially when the fill rate is only 0.073, the positive effect of high frequencies is extremely significant. With high filling, the effect of vibration intensity is reversed. This is because the space in which the particles can move freely in the container is limited, and the high frequency does not bring about strong relative motion, but on the contrary, the kinetic energy of the system increases and the energy consumption efficiency decreases.

As can be seen from Figure 2.12(b), the energy efficiency remains low when the frequency is low and the vibration intensity of the system is low

level, and the change in fill rate had no significant effect on it. When the vibration frequency is higher than 1Hz, the energy dissipation efficiency increases with the filling, and then decreases after reaching a maximum value. The fill rate corresponding to the highest energy consumption efficiency of the system is called the optimal fill rate. For 2 Hz, 5 Hz, 10 Hz, 20 Hz loads, the optimal values are 0.147, 0.220, 0.220, 0.294, 0.294, respectively. It can be seen that the higher the load frequency, the greater the optimal filling rate.

2.3.3 Particle size

Particle radius is one of the important parameters in the contact model. The larger the particle radius, the greater the damping coefficient cn1ct1 of the particle-to-particle collision [74], and the greater the damping force at the same collision velocity. At the same time, the stiffness coefficient kn1kt1 of the collision between the particles also increases with the larger radius of the particles [74], that is, the elastic force becomes larger during the collision process, and the damping force component in the force balance equation becomes smaller. And with the same fill rate, a larger particle radius means fewer particle counts, which in turn reduces the number of particle-to-particle collisions. These effects also apply to particle-container collisions. It is difficult to judge which of these three effects is more important, so in this paper, the effect of particle size on the vibration damping effect of the damper is explored by parametric analysis experiments. In order to make the particle size measurement dimensionless, the diameter ratio is defined as the ratio of the particle diameter to the difference between the radius of the torus

The diameter ratio is changed between 0.45-0.91, and the corresponding particle diameter is 20, 24, 28, 32, 36, 40mm, which can change the particle arrangement from 2 layers to 1 layer arrangement on the damper section, as shown in Figure 2.13.

2 layers of granules, 1 layer of granules

Figure 213. Schematic diagram of the change in diameter ratio

The energy efficiency of the particle damper at 0-20 Hz was calculated by changing the diameter ratio and the number of particles at the same time to keep the total particle mass constant, as shown in Figure 2.14.

(a) The effect of diameter on energy dissipation efficiency under high-frequency loads

(b) The effect of diameter on energy dissipation efficiency under low-frequency loads

Figure 214. Calculation results of case 3

As can be seen from Figure 2.14(a), when the frequency is higher than 2Hz, the variation law of energy dissipation efficiency with diameter is relatively consistent at different frequencies. When the diameter ratio is 0.45, the energy consumption efficiency is the highest. With the increase of diameter, the energy consumption efficiency of the diameter ratio of 0.54 shows a significant trough, and then continues to increase slowly, and the energy consumption efficiency decreases slightly again when the diameter ratio reaches 0.91. The effect of diameter on energy efficiency at low frequencies can be seen in Figure 2.14(b): as the diameter increases, the efficiency decreases gradually, with a sudden increase at a diameter ratio of 0.82, and then decreases again.

The steep drop in energy efficiency at a diameter ratio of 0.54 at high frequencies occurs because when the diameter of the particles is slightly greater than 0.5 times the inner diameter of the container, the particles in the container are just unable to be placed in two rows side by side from a cross-sectional point of view. In this state, when vibrating, the movement of particles is prone to "blockage". Figure 2.15 is a screenshot of the diameter ratio of 0.45 and the diameter ratio of 0.54 at the same time, blue represents small particle velocity and slow movement, and red represents large particle movement speed. It can be found that at a diameter ratio of 0.54, some particles move slowly, and their relative positions inside the container are basically unchanged. The energy dissipation of these particles through collision and friction is almost negligible, and the number of effective particles in the system is much less than the number of particles. At the same time, these immobile particles also block the movement space of the active particles, which makes the vibration damping effect of the system drop sharply. When the diameter ratio is 0.45, there is no "blockage".

0.45 0.54

Figure 215. Screenshot of particle velocity when the diameter ratios are 0.45 and 0.54 respectively, which can also be seen from the proportion of collision energy dissipation Figure 2.16.

Fig. 2.16 Collision energy dissipation as a function of diameter ratio and frequency

In Figure 2.16, it is evident that at a diameter ratio of 0.55 and a diameter ratio of 0.82, the proportion of collision energy dissipation decreases at a trough and a peak, respectively. To a certain extent, it can be reflected that due to the blocking effect, the overall movement of particles is slow when the diameter ratio is 0.55. Conversely, since this effect is avoided, the ratio in diameter is 0At 45 and 0.82, dark blue areas appear in the figure, indicating intense particle movement.

2.3.4 Collision recovery factor

The collision recovery coefficient e is used to measure the amount of energy lost during a collision.

∆E =

The smaller the value, the more the velocity decays during the collision and the greater the energy dissipation during the collision. But less velocity also means that the particles move more gently, which is not conducive to more frequent collisions between particles and between particles and containers.

At the same time, the collision recovery coefficient between particles and particles and the collision recovery coefficient between particles and containers are changed to 0.001, 0.2, 0.4, 0.6, 0.8, 1, and the energy consumption efficiency of the damper in the range of 0.5-20Hz is calculated, and the results are shown in Figure 2.17

show.

Figure 2. 17 Effect of collision recovery coefficient on energy efficiency at different frequencies

As can be seen from Figure 2.17, the energy efficiency decreases steadily as the collision recovery coefficient decreases. And for any load frequency, when the collision recovery coefficient is reduced to 1, the energy consumption efficiency is almost reduced to 0.

Figure 2.18 shows the contour of the proportion of collision energy dissipation as a function of the collision recovery coefficient and frequency.

Figure 2. 18 Collision energy dissipation as a function of collision recovery coefficient and frequency

Figure 2.18 From the bottom left to the top right, the proportion of collision energy dissipation gradually decreases, and the higher the frequency, the smaller the number of collision recovery systems, and the greater the proportion of collision energy dissipation. Also in conjunction with Figure 217 It can be seen that the energy consumption efficiency is also higher at this time. From this example, it can also be shown that the high proportion of collision energy consumption is a manifestation of the high energy consumption efficiency of the system.

2.3.5 Sliding friction coefficient

For spherical particles, the coefficient of sliding friction controls the maximum tangential force when it collides. The greater the coefficient of sliding friction, the greater the tangential force that can occur when the particles collide, and the more energy is dissipated. When the sliding friction coefficient is 0, the tangential deformation of particles does not consume energy during collision. When the particle motion is relatively gentle, the tangential force of the collision is small

Its value does not exceed the sliding friction, and changing the sliding friction coefficient does not affect the damping force of the particle damper. When the particle motion is relatively violent, the collision tangential force is limited by the sliding friction force, and increasing the sliding friction coefficient will increase the tangential damping force. However, the increase of the damping force of a single collision will reduce the velocity of particle motion, which is not conducive to the energy dissipation of subsequent collisions. Therefore, the influence of the coefficient of sliding friction on the damping effect of the particle damper needs to be analyzed parametrically

Experimentally determined.

At the same time, the sliding friction coefficient between particles and particles and the sliding friction coefficient between particles and containers are changed to 0, 0.25, 0.5, 0.75, 1, and the energy consumption efficiency of the damper in the range of 0.5-20Hz is calculated, and the results are shown in Figure 2.19.

Figure 2. 19 Effect of friction coefficient on energy efficiency at different frequencies

As can be seen from Figure 2.19, the energy efficiency is highest when the friction coefficient is 0, regardless of the load frequency. This shows that the friction energy dissipation does not play a key role in the vibration damping of the cable particle damper. Figure 2.20 shows the proportion of energy dissipated by collision as a function of friction coefficient and frequency. It can be seen that when the friction coefficient increases, the proportion of collision energy dissipation decreases, corresponding to the decrease in energy dissipation efficiency in Figure 2.19.

Figure 2. 20 Variation of the proportion of energy dissipated by collision as a function of friction coefficient and frequency

However, it is only when the coefficient of friction increases from 0 to 0.25 that the benefit of sacrificing frictional energy is most apparent. When the coefficient of friction varies between 0.25-1, the damper energy efficiency barely changes for a given excitation. This shows that if you want to increase energy consumption by reducing friction, you need to greatly reduce the friction to close to zero, which is difficult to achieve in practice.

2.3.6 Conclusion

In this section, a simple harmonic load is applied to the rootless cable damper, and the energy dissipation under different conditions is calculated by using the discrete element method, and the energy dissipation efficiency of the damper is evaluated by the ratio of total energy dissipation to maximum kinetic energy as the energy dissipation efficiency, and the following conclusions are obtained

l The energy dissipation efficiency increases first and then decreases slightly with the increase of frequency, and the optimal frequency with the highest corresponding efficiency decreases with the increase of amplitude. When the amplitude is 50mm, the optimal frequency is 10Hz; At an amplitude of 500mm, the optimal frequency is 1Hz.

l For the load of 0.5Hz and 1Hz, the energy dissipation efficiency increases with the increase of amplitude; For loads with higher frequency rates, the energy efficiency decreases as the amplitude increases.

l With the increase of filling rate, the energy consumption efficiency increases first and then decreases. The optimal filling rate is related to the vibration frequency, and the higher the load frequency, the greater the optimal filling rate.

When the particle size is slightly greater than half of the width of the ring or slightly less than the width of the ring, the particle movement is blocked, the proportion of energy consumption of particle collision is reduced, and the energy consumption efficiency decreases sharply.

l The collision recovery coefficient increases, that is, the proportion of kinetic energy converted into heat energy in the collision process decreases, and the energy dissipation efficiency decreases almost linearly.

l The friction energy dissipation does not play a key role in the vibration damping of the cable particle damper, and the energy consumption efficiency is the highest when the friction coefficient is 0. However, since friction is difficult to completely eliminate in practice, the method of improving energy efficiency by reducing the coefficient of friction has little benefit.

l The high proportion of collision energy dissipation is a manifestation of the high energy consumption efficiency of the system, which is reflected in the analysis of the influence of vibration intensity, particle size, collision recovery coefficient and friction coefficient.

2.4 Vibration damping effect of rootless cable damper

In the previous section, the energy dissipation characteristics of the damper under simple harmonic load were analyzed by the discrete element method. However, the amount of energy dissipated does not fully reflect the damping performance of the damper. In this section, we will further establish the damper-cable system and explore the effectiveness of particle dampers in cable damping.

In terms of specific implementation, the discrete element software EDEM is used to establish the rootless cable particle damper model, and the multi-body dynamics software ADAMS is used to establish the cable model, and then the two are coupled through the E ALink plug-in. In the simulation process, in each time step, the force and moment of the particle damper calculated by EDEM to the container are transmitted to the cable consolidation point through the consolidation point, and the displacement of the consolidation point calculated in ADAMS is also returned to EDEM, so as to realize the data exchange of force and displacement during the interaction between the particle and the structure, and the tension is establishedSystem-level simulation model of a cable-particle damper.

2.4.1 Model construction of cable-damper system

In the cable-damper system considered in this section, the cable is placed horizontally, simply supported at both ends, and a particle damper is installed at the midpoint, as shown in Figure 2.21, without considering the sag and internal damping of the cable. The parameters of the cable are shown in Table 2.3.

Figure 2. Schematic diagram of the 21 cable damper system

Table 2.4 Cable parameters

The multi-body dynamics simulation software ADAMS was used to establish the cable model. To simplify the model, ignore the inner cable damping. Such an overlook is acceptable because the cable damping ratio is small, usually between 0.1 and 0.15% [75]. The model consists of 100 1m long cylindrical rigid body elements. A certain stiffness coefficient and tension connection are set between the elements, and the parameter calculation method is referred to Ref. [76], and the calculation results are shown in Table 2.4. Three translational degrees of freedom for two nodes in the fixed model simulates hinge supports. The consolidation point with the particle damper is located 0.5L from the end of the cable.

The first three modes of the cable are critical, as the main requirement of the cable damping system is to provide sufficient damping for the first three or four cable modes [75]. Therefore, it is important to verify that the cable model generates the desired modalities.

This section uses classical string theory to simplify a cable to a tensioned string that is hinged at both ends. The undamped natural frequencies of the cable can be obtained [77].

where i is the order of frequency, L is the length, T is the tension in the cable, and μ is the unit mass of the cable. The calculated first three mode shapes of the cable are shown in Figure 2.22.

Table 2. 5 Cable Model Element Connection Parameters

Fig. 2. 22 Lasso first three mode shapes

Table 2.5 shows the deviation between the first three natural vibration frequencies obtained by tension string theory and the natural oscillation frequencies of the model.

Table 2.6 Deviation of the calculation of the first three natural frequencies of the cable

It can be seen that the frequencies calculated by the numerical model are consistent with the analysis results with a deviation of less than 5%. Cause

This model is reasonable.

2.4.2 Damper damping effect

In the simulation test of the damper-cable system, sinusoidal excitation is used, and the excitation acts at a distance of 0.2L from the end of the cable. By numerical method, the vertical response at the maximum position of the cable response was calculated. The level of structural vibration is measured by the root mean square value of acceleration, and the expression for root mean square of acceleration is:

aRMS =

where a is the acceleration value of each data point, and n is the number of data points for calculation. In this paper, 5 s are calculated for each working condition, and data is recorded every 1×10-3 s. Define the normalized vibration response η in different studies as

where a represents the root mean square of acceleration with the particle damper structure attached, and aRMS represents the addition of the original structure

Root mean square velocity. The smaller the normalized vibration response, the better the damping.

(a) Acceleration time history diagram (b) Normalized response time history diagram Figure 2. 23 Vibration damping efficiency of particle dampers

In order to explore the temporal effect of the damping effect, the normalized oscillatory response of the damper was calculated by using Eq. 2.7 at regular intervals. Considering that the first-order period of the cable is 0.772s, the calculation interval is taken as 0.8s, that is, n=800, and 400 data before and after each calculation point are taken to calculate aRMS sumaMS。 Figure 2.23(b) shows the normalized vibration response curves calculated at 0.4s, 1.2s, 2.0s, 2.8s, 3.6s, and 4.2s. It can be seen that the damping effect of particle damper in the late stage of vibration is significantly better than that in the early stage of vibration. This shows that there is a certain time lag in the damping effect of this damper. After 1 s, the normalized vibration response can be reduced to less than 0.7, and the normalized vibration response in the 4.2s is 0.40, which is 0.4s0.5 times the normalized vibration response.

Adjust the frequency of the input frequency from 1Hz to 20Hz, fill 100 particles with a radius of 20mm in the particle damper, and adjust the magnitude of the excitation frequency force at the same time to make the cable amplitude 100mm Around. Figure 2.24(a) shows the normalized vibration response curve of the structure when it is stimulated sinusoidally from 1 Hz to 20 Hz. It can be seen that the particle damper has a good damping effect on the excitation between 1Hz and 20Hz, and the damping capacity increases with the increase of the excitation frequency. When the excitation frequency reaches 20 Hz, the acceleration normalized vibration response can reach 0.725.

At the same time, the influence of excitation intensity on the performance of the particle damper was investigated, and 100 particles with a radius of 20mm were filled into the particle damper, and the excitation of different amplitudes of 20Hz was applied, and the amplitudes were 50mm, 100mm, 200mm, 300mm, and 500mm, respectively. Fig. 2.24(b) shows the normalized vibration response as a function of the excitation intensity, and it can be seen that the normalized vibration response hardly changes with the excitation intensity in the range of 50mm-500mm.

The test results show that the particle damper has a good damping effect in a wide frequency band for the cable under sinusoidal excitation, and the damping effect is little affected by the vibration amplitude.

Through the numerical simulation study of the above cable-damper system, it is illustrated that the particle damper is used in the cable

(a) Effect of vibration frequency (b) Effect of vibration amplitude Figure 2. 24 Curves of the normalized response of the cable under different loads

2.4.3 Parameter analysis

When the damper design parameters are changed, the vibration damping effect obtained by the above numerical test may change, this section will explore the vibration damping effect of the damper system by changing the damper parameters, in order to find the influence law between the design parameters and the vibration reduction effect. In order to obtain the maximum number of cycles in the limited simulation time (1 second of simulation requires 4 hours of computational time), a 20Hz excitation is selected in this section to calculate the subsequent parametric analysis examples. The typical time history of cable displacement and acceleration is shown in Figure 2.25.

(a) Displacement time history (b) Acceleration time history diagram 2. Typical response of a 25 20Hz load under a cable

1

0,8

0,4

0,2

0

0 0,1 0,2 0,3 0,4 0,5

ξ

(a) Effect of fill rate

0,8

0,6

0,4

0,2

0

0,4 0,5 0,6 0,7 0,8 0,9 1,0

λ

(b) Effect of particle size

1

0,8

0,4

0,2

0

0 0,2 0,4 0,6 0,8 1

e

1

0,8

0,4

0,2

0

0 0,2 0,4 0,6 0,8 1

μsf

(c) Effect of crash recovery coefficient (d) Effect of friction coefficient Figure 2. 26 Influence of damper design parameters

The number of particles was changed as shown in Figure 2.11, and the response of the structure at the excitation frequency of 20 Hz and the excitation amplitude of 100 mm was calculated when the filling rate was 0.073, 0.147, 0.220, 0.294, 0.367, 0.441, and 0.514. Figure 2.26(a) shows the curve of the structural response at each filling rate. It can be seen from the figure that the vibration damping effect of the particle damper increases first and then decreases with the increase of the filling rate, and reaches the optimal when the filling rate is 0.367. This is because when the number of particles is small, with the increase of filling rate, the number of particles increases, and the collision and friction energy dissipation between particles also increases. However, as the number of particles continues to increase, the particles accumulate and the number of particles participating in the movement decreases, the overall movement of the particles slows down, and the momentum exchange with the container also decreases, so the vibration damping effect is reduced.

Figure 2.26(b) shows the normalized vibration response curves with λ of 0.455, 0.545, 0.636, 0.727, 0.818, and 0.909. As can be seen from the figure, when λ is 0.727, 0.818 and 0.909, the normalized oscillation response is 0.7, 0.71 and 0.73, respectively. This shows that when the filling rate is the same, the smaller particle size can obtain a better vibration damping effect. This indicates that with the increase of particle radius, the effect of increasing the elastic coefficient and the decrease of the number of collisions are more significant than the effect of increasing the damping coefficient, so that the collision damping force becomes correspondingly smaller. But the smaller the particle half-diameter, the better. When the particle size ratio λ is 0.636 and 0.545, the normalized vibration response increases sharply, and the vibration damping effect decreases sharply. This is because when the diameter of the particles is slightly greater than 0.5 times the inner diameter of the container, the particles in the container are just not in a state where two rows cannot be placed side by side. In this state, when vibrating, the movement of particles is prone to "blockage". Fig. 17(a) is case 7, λ = 0.545 when 1In the screenshot of the 938s, blue represents the minimum particle velocity of 0.01m/s, and red represents the maximum particle velocity of 1.17m/s. It can be found that when a blockage occurs,

Most of the particles move at a low speed and their relative position is basically the same. The energy dissipation of these particles through collision and friction is almost negligible, and the number of effective particles that play a role in the system is far less than the number of particles. At the same time, these immobile particles also hinder the movement space of the active particles, which makes the vibration damping effect of the system drop sharply. When the particle radius continues to decrease to λ=0.455, the blockage disappears, as shown in Figure 2.27(b), the minimum velocity of the particles in blue is 0.05m/s, and the maximum velocity of the particles in red is 0.05m/s 2.46m/s, all particles can move freely in the container, and the damper damping effect is restored.

(a) λ=0.545 (b)λ=0.455

Figure 2. Screenshot of 1.938s under 27 case 7 different λ conditions

According to Table 2.2 Example 4, from 0 to 1, the collision recovery coefficient between particles and between particles and between particles and containers is changed with a step size of 0.2, and the excitation frequency is 20Hz and the excitation amplitude is 100mm The damping effect of particle dampers. Figure 2.26(c) shows the normalized vibration response with E at a particle diameter of 40 mm and a filling rate of 0.367.

It can be seen that the damping effect increases first and then decreases with the increase of the collision recovery coefficient during the collision. This is due to the increase of the collision recovery coefficient, the rebound velocity of the particles after collision becomes larger, the particle movement is more intense, the number of collisions increases, the total collision energy consumption is increased, and the vibration damping effect is improved. However, if the collision recovery coefficient continues to increase, the energy consumption reduction effect of a single collision will exceed the effect of increasing the number of collisions, the total collision energy consumption will be reduced, and the vibration damping effect will be weakened. It can be seen that the optimal collision recovery coefficient is larger, around 0.8.