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Article

Research on Thermal Dissipation Characteristics Based on the Physical Laws of Forced Vibration in Granular Assemblies

by
Shengwu Zhang
1,
Wei Xu
1,
Chiming Wang
2,*,
Yuanchao Zhang
3,
Yanan Li
2 and
Jianwei Cheng
1
1
National Key Laboratory on Ship Vibration & Noise, Institute of Noise & Vibration, Naval University of Engineering, Wuhan 430030, China
2
School of Computer and Information Engineering, Xiamen University of Technology, Xiamen 361024, China
3
National Key Laboratory of Science and Technology on Underwater Acoustic Antagonizing, Zhanjiang 524000, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(13), 5864; https://doi.org/10.3390/app14135864
Submission received: 29 April 2024 / Revised: 17 June 2024 / Accepted: 2 July 2024 / Published: 4 July 2024
(This article belongs to the Section Acoustics and Vibrations)
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Abstract

:
Particle damping technology is applied in vibration and noise reduction because of its good broadband vibration reduction effect. The energy transfer and loss between particles are keys to the role of damping. This paper investigates the relationship between the thermal energy dissipation caused by the collision of particles and the input energy. The temperature rise characteristics under different vibration states are studied. The results show significant differences in the thermal dissipation characteristics of granular assemblies for different vibration states. Under equivalent excitation amplitudes, the frequency increases, and the thermal loss of the particles increases. At the same frequency, the excitation amplitudes increase, and the thermal loss of the particles decreases. Granular assemblies in strong vibrational states, such as a gas-like state, have intense vibrations and apparent temperature-increasing effects. However, in this vibration state, the input energy to the particles is considerable, and the thermal loss accounts for a small proportion of the total energy. In states such as solid-like states, micro-vibrational states, and intermediate vibrational states, the particles interact closely, and the input energy for the particles is small. Additionally, the movement of the particles is not intense, and the thermal loss accounts for a large proportion of the total energy. The thermal loss of the particles also shows a frequency variation characteristic. According to the different temperature rise characteristics of the particles, the proportion of thermal energy consumption is analyzed. The research shows that the proportion of thermal energy consumption is not more than 70%, so there are other forms of energy consumption in the vibration reduction and energy consumption of the particles.

1. Introduction

Particle damping technology can better solve the problem of damping performance degradation caused by thermal aging, creep, and brittle cracking and suppress the vibration and noise of large structures under complex and harsh working conditions. Therefore, it has a broad application prospect in engineering [1,2]. In vibration and noise reduction, the damping characteristics and energy dissipation law of particle dampers are affected by the vibration state of internal particles [3]. However, the damping characteristics of particle damping technology are highly nonlinear in the application process [4], which makes it difficult to form a relatively complete theory so far, and its internal energy dissipation mechanism also limits the better application of particle damping technology.
The particle damping energy dissipation mechanism is mainly manifested as energy transfer and energy dissipation. From the energy analysis of the system, the total nonlinear kinetic energy, the total rotational kinetic energy, the total elastic energy stored in the deformation of the particle, the potential energy of the spring, and the kinetic energy of the moving structure are all involved in the energy transfer. The sum of this instantaneous energy is the residual energy of the particle system. The total energy of friction dissipation and acoustic energy dissipation in the inelastic collision process is the cumulative energy generated by energy dissipation. The dissipated energy is converted into thermal energy dissipation and acoustic energy dissipation, which cannot be transferred back to the system. After energy is transferred to the particles, it is dissipated by friction and inelastic collisions between the particles. The relative importance of friction and collision depends on the particles’ physical properties and excitation conditions. Among them, the thermal energy dissipation between particles is considered the main factor in determining vibration energy dissipation. However, there is no effective energy loss relationship between vibration input and heat dissipation.
During forced vibration, particles show a wide range of motion states and complex kinetic properties. However, energy conversion, consumption, and transfer during particle aggregate vibration have yet to be unified [5]. The various engineering applications of particle aggregates have promoted research on their energy dissipation mechanisms. Meyer studied the energy loss of particles under forced vibration conditions and established design principles for particle applications [6,7,8]. Mao analyzed the collision and friction between particles using the discrete element method and discussed the energy consumption forms of particles [9,10]. Using the discrete element method, Yin explored the dissipation capabilities of particles with various motion states [11]. Zhang conducted experimental studies on the energy loss process of particle aggregates during forced vibration and obtained changes in the kinetic and potential energy of particles during vibration [12,13]. The research on the energy dissipation of particle aggregates focuses on the changes in kinetic energy and potential energy caused by particle collision and friction. The recovery coefficient is an essential basis for establishing different dissipation models. Models are established based on the change in recovery coefficient caused by collisions between particles. These models make it difficult to describe the relationship between particle collisions and thermal energy consumption.
Fowler et al. [14] divided the loss mechanism of the particle damper into an external mechanism and an internal mechanism. The external mechanism is mainly the friction and collision between the discrete body and the container wall, while the internal mechanism is mainly the friction and collision between the discrete bodies. Xu et al. [15] believe that the energy dissipation mechanism of particle dampers is mainly related to the highly nonlinear phenomenon of friction and impact. Simonian [16] considered that the structural vibration caused by the outside world will lead to the particle–particle and particle–container wall interaction placed in it. Esipov and Poschel [17] studied the collective dissipation characteristics of vibrating granular materials by molecular dynamics simulation and found that the gas-like dissipation capacity was the highest.
Regarding thermal losses generated by particle aggregates, Hunt studied the heat exchange behavior of small particles. Tsotsas proposed a particle–particle heat transfer model to optimize the heat transfer process in the discrete element method [18,19]. Haydar and Chaudhuri also studied the thermal transport of particles in rotating drums using the discrete element method [20,21]. It is understood that thermal losses constitute a vital energy dissipation method in particle aggregate vibration. However, there needs to be a clear explanation of the process of thermal losses or their corresponding proportion [22,23]. The discrete element model focuses on the analysis of the kinetic energy, potential energy, and recovery coefficient of the particles. The energy dissipation problem is transformed into the problem of viscous damping caused by the contact between particles. The study of energy dissipation characteristics focuses on the collision between microscopic particles. The discrete element model also includes analysis and research on the transfer of internal heat energy. However, only the energy transfer inside the particle body is analyzed, and the outward dissipation energy is not considered. This leads to the conversion of the study of the internal heat energy of the particles by the discrete element method to the study of the heat transfer model between the particles [24,25]. Simulating the energy conversion and dissipation in particle vibration reduction is complex, and it is impossible to show the internal energy conversion and dissipation process directly.
The organization of this paper is as follows. In Section 2, a discrete element vibration energy characteristics model is established to analyze the energy distribution characteristics and collision rules inside the particles under different excitation conditions. These characteristics reflect the energy transfer process inside the particles. In Section 3, a mathematical and physical model of thermal energy dissipation of particle aggregates is established to study the internal thermal energy dissipation effect. In Section 4, the temperature rise characteristics and internal temperature rise distribution of particles under different excitation conditions are measured by experimental methods, and the distribution of thermal energy consumption under different excitations is obtained. It is found that the proportion of heat dissipation energy is less than 70% in the energy dissipation effect of particles. Therefore, in the process of energy dissipation of particles, there will be other forms of energy dissipation, and not all the input energy will be dissipated in the form of heat energy.

2. Analysis of the Vibration Energy Characteristics of Particles

The particle body will show different vibration characteristics under different excitation conditions, so it is necessary to select the excitation. The discrete element method can be used to obtain the forced vibration process inside the particle body and the internal kinetic energy of the particle assembly during the vibration process. The vibration energy in the particle body is finally dissipated by other forms of energy, such as heat energy.
The excitation frequency range is 5–300 Hz, and the excitation intensity is 0–10. The excitation amplitudes is characterized by reduced acceleration. The expression is Γ = A ω 2 / g , where A is the vibration amplitude, ω is the vibration angular frequency, and the g is the gravity acceleration. In this excitation range, the particles generally have five typical vibration states: solid-like, micro-vibration state, medium vibration state, strong vibration state, and gas-like state. The internal vibration state and energy transfer process of the particles under different excitations are obtained by the discrete element method, as are the internal velocity distribution and kinetic energy distribution of the particles under different vibration conditions. The velocity distribution of the particles reflects the energy transfer process and energy distribution characteristics inside the particles after excitation. The particles generate velocity and collision friction through interaction and generate thermal energy consumption and other energy consumption forms through collision and friction. Therefore, the discrete element simulation software EDEM2021 is used to establish the discrete element simulation analysis model of particle thermal energy consumption, as shown in Figure 1. The particle mass is 557 g, the particle size is 2 mm, the filling rate is 80%, and the particle container is a Φ 72 × 37.5 mm cylindrical container. The setting of discrete element simulation parameters is shown in Appendix A.
Figure 2 shows the velocity distribution characteristics inside the granular assemblies over a period of time. Generally, at 0.5 T, the particles inside the particles vibrate violently, and the velocity distribution is obvious. The velocity distribution of the upper, middle, and lower layers of the particle body is selected when the vibration period is 0.5 T. The velocity distribution of the upper, middle, and lower layers of the particles in each picture can be directly characterized by color, and the color corresponds to the velocity distribution range in the side view.
When the reduced acceleration is 0.8 and the excitation frequency is 50 Hz, the vibration process inside the particle and the velocity distribution of the particle are shown in Figure 2a. Under this excitation, the velocities of different particles inside the particle are similar, and almost no relative motion is generated.
When the reduced acceleration is 1.2 and the excitation frequency is 50 Hz, as shown in Figure 2b, the micro-vibration of some particles appears inside the particles, which shows that some particles in the particles produce different velocities from the surrounding particles.
When the reduced acceleration is 2.5 and the excitation frequency is 50 Hz, as shown in Figure 2c, more particles show a larger velocity distribution under excitation, which also means that a large amount of collision and friction will occur in the inner part of the particles.
When the reduced acceleration is 2.5 and the excitation frequency is 25 Hz, as shown in Figure 2d, the particle body is in a strong vibration state, and the particle body as a whole shows an impact effect on the bottom of the container. At the same time, the internal particle body is affected by the container wall and the particle body so far during the impact process, and the speed changes dramatically, and the speed is significantly higher than other vibration forms.
When the reduced acceleration is 6 and the excitation frequency is 50 Hz, as shown in Figure 2e, the particles are in a gas-like state and have a strong excitation effect. The particle velocity distribution has a stratification effect, and the energy is transferred from the bottom to the top, and the particle velocity increases correspondingly.
Figure 3 shows the relationship between total particle kinetic energy, total particle rotational kinetic energy, and the total number of collision times in a cycle under different excitation conditions. The number of collisions reflects the intensity of collisions in a period between particles. The ‘force chain’ is constantly generated between particles and breaks, and the corresponding energy is continuously dissipated. The changes in translational kinetic energy and rotational kinetic energy reflect that the particles will produce collision and friction under different excitation conditions, and the translational kinetic energy between the particles is dominant. Through the discrete element analysis, it can be seen that the particles will show different energy distributions under different vibration states, and the distribution of this energy will affect the thermal energy dissipation of the particles.

3. Mathematical and Physical Model of Thermal Energy Dissipation in Particle Assembly

The basic modes of heat transfer include conduction, convection, and radiation [26,27]. Without considering the kinetic energy generated by particle oscillation on a long-term scale, the heat Q H absorbed by the particle during forced vibration Δ t can be calculated as follows:
Q H = Q p + Q A + Q i
In this case, Q p is the heat transfer due to thermal convection in the internal part of the particle during vibration, Q A is the heat transfer between the surface of the particle in the container and the air, and Q i is the heat radiation from the particle. Q p is the heat transfer due to thermal convection in the internal part of the particle for solid conduction, which can be obtained using the following formula:
Q p = c m Δ T
In this case, c is the specific heat capacity of the particle, m is the total mass of the particle, and Δ T is the change in temperature.
According to Newton’s cooling law, in a closed container containing the particle, natural convection of air is a heat transfer method with no forced airflow. Q A is the convective heat transfer rate of air:
Q A = h A Δ T
h is the coefficient of convection heat transfer used here, and A is the surface area of the object exposed to the moving fluid.
According to the Stefan–Boltzmann law, the heat radiation of a particle can be determined by the following formula:
Q i = σ A Δ T 4
In this case, the constant σ is the Stefan–Boltzmann constant, with a value of σ = 5.67 × 10 − 8   W / m 2 ⋅ K 4 . In the experiment, adding insulation layers can reduce the consumption of heat radiation, and since Q i < < Q p , the thermal radiation of the particle can be ignored.
The total energy Q during the forced vibration of a particle is the sum of the energy of the forced vibration in each period. Among them, the experiment can record the variation curve of the reduced acceleration of the particle body with time under different forced vibration states. Because the reduced acceleration curve of the particle damper in each cycle is similar (see Appendix B), the reduced acceleration in multiple cycles is integrally averaged. The average speed of the particle damper in a cycle under different vibration states is calculated. The total energy of each cycle in the forced vibration time is as follows:
Q = ∑ 1 n 1 2 m v 2
where n is the total number of all periods in the forced vibration time. The ratio of thermal energy dissipation generated by the particle during forced vibration to its total energy is as follows:
α = Q H Q

4. Granular Assembly Frequency Variation Thermal Loss Testing Device

Section 2 discusses the interaction of particles in different vibration states during forced vibration, which affects the energy transfer and dissipation between particles. In order to study the thermal energy dissipation of particles under forced vibration, experiments are carried out for further analysis. The research on the thermal energy consumption of particles is mainly based on the change law of temperature after the particles are input by external energy to reflect the conversion process of internal energy. The thermal loss law of granular assemblies is illustrated in Figure 4. In the experiment, infrared thermal imaging thermometers are used to test and record the surface temperature of the particles. High-precision temperature meters with two temperature probes are used to test and record the temperature of the particles. The particle size of 2 mm is taken as the research object. The material of the particles is non-magnetic 304 stainless steel particles; the filling rate of the particles is 80%; and the mass of the particles is 557 g. During the experiment, the signal generator generates a single-frequency, fixed-amplitude sinusoidal excitation through the exciter. The excitation intensity is generally described by reduced acceleration. The reduced accelerations of the experiment are 0.8, 1.2, 2.5, and 6, respectively. The excitation frequencies are 25 Hz, 50 Hz, 75 Hz, 100 Hz, 150 Hz, 200 Hz, 250 Hz, and 300 Hz. According to the discrete element analysis in Section 2, it can be seen that the particle assembly will show different vibration states under different excitations. However, at the above research frequencies, the particle assembly generally appears in five typical vibration states: solid-like vibration, micro-vibration, medium-vibration, strong-vibration, and gas-like vibration. These vibration states correspond to the vibration of the internal particles gradually becoming severe. The B&K data acquisition system is configured to observe the first peak of the vibration signal in the output signal, triggering the vibration data recording. The excitation time is 2 h of continuous excitation at each frequency with reduced acceleration. The infrared temperature sensor and the contact temperature sensor are used to measure the particles’ surface, middle, and bottom temperatures for 1 h and 2 h, respectively. In the experiment, a single-frequency sinusoidal signal and a constant reduced acceleration are used for excitation. When measuring the temperature, it is necessary to stop the excitation and open the upper cover of the container. First, the infrared temperature sensor is used to measure the surface temperature of the particles. Two contact high-precision thermometers are used to measure the particles deeply inside and at the bottom of the particles at the same time. The measurement time is 1 min. After the end of the measurement, the upper cover plate is quickly closed, and the particle body is continuously excited. The infrared temperature sensor takes a non-contact measurement, so its position is fixed. Two temperature sensors are used to fix the contact probe through the fixed bracket, and the temperature of the particles’ middle and bottom layers is measured simultaneously. The outer side of the particle container is coated with thermal insulation material to reduce heat transfer from the particle to the outside and from the external environment to the inside. The experiment was carried out in a room with a constant temperature.

5. Analysis of Thermal Loss Results for Granular Assemblies

5.1. Analysis of Thermal Uplift Characteristics of Granular Assemblies under Typical Vibration Conditions

In this study, the temperature changes in 2 mm granular assemblies under five typical vibration conditions, namely, the solid-like state, micro-vibrational state, intermediate vibrational state, strong vibrational state, and gas-like state, are tested. The classification of the forced vibration state of the particles is shown in Appendix B. After being subjected to forced vibration, the particles consume energy through collisions and friction. It is generally believed that a large amount of energy dissipates as heat energy, but there is little research on this process. The changes in the surface temperature of the particles under the five typical vibration conditions over a period of 2 h are shown in Figure 5. The figure is the enlarged figure of the local surface temperature distribution of the particles taken by the infrared thermometer and the histogram of the temperature distribution of the particles in the image. The temperature distribution of the local surface of the particle is enlarged, and the color tends to be bright yellow at the higher temperature. The local surface temperature distribution amplification diagram of the particle is used to reflect the temperature distribution characteristics of the local temperature distribution amplification diagram of the particle.
At longer time scales, after being subjected to forced vibration, part of the energy in the particles is converted into heat through collisions and friction. Different motion states have varying effects on temperature increases.
When the excitation frequency is 50 Hz and the reduced acceleration Γ is 0.8, the granular assemblies are in a solid-like state. From the infrared thermal images of the surface temperature changes and the corresponding temperature distribution diagrams, it can be seen that although the particles are subjected to 2 h of excitation, there is no significant change in the surface temperature of the particles. In the solid-like state, there is almost no vibration in the particles; at this time, the friction and collision effects inside the particles are not obvious, so the temperature change is also not obvious.
When the excitation frequency is 50 Hz and the reduced acceleration Γ is 1.2, the granular assemblies are in the micro-vibration state. At this time, there is no significant temperature change in the particles. During the micro-vibration process, some particles vibrate locally and rotate slightly at their positions, but this does not cause a significant change in the overall temperature.
When the excitation frequency is 50 Hz and the reduced acceleration Γ is 2.5, the granular assemblies are in an intermediate vibrational state. For a longer period of excitation, the temperature of the particles increases slightly. According to the temperature distribution diagram of the intermediate vibrational state, there is a slight increase in the temperature of the particles. In the intermediate vibrational state, a large number of particles have small movements, and under continuous excitation, a certain amount of heat is generated in the particles.
The granular assemblies are in a strong vibrational state when the excitation frequency is 25 Hz and the reduced acceleration Γ is 2.5. At this time, the motion of the particles is intense, and there is violent friction between the entire particle and the container wall. Collisions and friction also exist between internal particles. This makes the temperature-increasing effect of the particles obvious. During forced vibration, there is a large amount of heat generated in a strongly vibrating state, and the temperature increase is mainly concentrated on the contact surfaces of each particle. As shown in the corresponding pixel temperature statistics diagrams for each window, the temperature increase is transmitted through interaction between particles, making the temperature distribution more uniform and covering a wider range.
When the excitation frequency is 50 Hz and the reduced acceleration Γ is 6, the particle aggregates are in a gas-like state. Compared with the strong vibrational state, at this time, there is an intensification of the particle vibration in the outer layer of particles. Moreover, there are violent frictional interactions between internal particles. The temperature-increasing effect of individual particles on the outer layers is obvious.
The temperature change curve of particles in different motion states at the same frequency is shown in Figure 6. The change in the temperature curve shows that for particles in different motion states, the corresponding temperatures of the upper, middle, and lower layers change. Under continuous excitation, the temperature changes in particles with the same vibration state are similar. Particles in the strong vibrational state and gas-like state have obvious temperature-increasing effects, while those in the solid-like and intermediate vibrational states do not show obvious temperature-increasing effects, and particles in the middle vibrational state have a specific temperature-increasing effect.

5.2. Analysis of Frequency-Varying Heating Characteristics of Granular Assemblies

The temperature change in the 2 mm granular assemblies for different frequencies and reduced accelerations is analyzed. The excitation frequency interval is 50 Hz, with reduced accelerations Γ of 0.8, 1.2, 2.5, and 6. The granular assembly container’s surface, middle layer, and bottom layer temperatures are also measured for approximately 1 min. As shown in Figure 7, there is a general heat loss phenomenon in the particle aggregate, and there are differences in the heat loss effects of the internal particles. The temperature changes show that the heating effect of the granular assemblies is evident at low frequencies and large, reduced accelerations. With increasing frequency, the heating effect of the granular assemblies gradually weakens. This is because as the frequency increases, the vibration states of the particles in the aggregate mainly transition from the micro-vibration and intermediate vibrational states, and the collision and friction between particles are not obvious. Interparticle interactions are the key to generating heat loss, so the heating effect of the granular assemblies decreases. When the reduced acceleration increases and the corresponding amplitude increases, the vibration states of the particles gradually transition from the solid-like, micro-vibrational, and intermediate vibrational states to strong vibrational and gas-like states. The interaction between particles becomes more intense, and the heating effect of the granular assemblies becomes evident. The temperature distribution of the internal particles shows a gradually weakening trend from the bottom to the surface. During the first hour of forced vibration, the particles in the strong vibration and gas-like states experience significant temperature increases, while those in the micro-vibrational and intermediate vibrational states experience slow temperature growth. When the vibration time continues to increase to 2 h, the temperature increase in the particles in the strong vibrational and gas-like states decreases, while the particles in the micro-vibrational and intermediate vibrational states continue to experience a significant temperature increase. This reflects a limit to the forced vibration heating effect of the granular assemblies, that is, at this point, the input energy and the temperature dissipated to the environment are in a dynamic equilibrium state.

5.3. Analysis of Frequency-Varying Heat Loss Characteristics of Particles

The collision and friction between particles convert and consume energy. The experimental process of the thermal energy consumption of particles shows that the heat produced by collision and friction is a long-term energy consumption mode. Only through a longer period of collision and friction can significant thermal energy dissipation be observed. The appearance of thermal energy dissipation is closely related to the internal motion state of particles. When the particle’s vibration state is in a solid-like or micro-vibrational state, less heat is produced. When the particle is in the intermediate vibrational state, strong vibrational state, or gas-like state, the particle movement is active, and the collision and friction are obvious. With the increase in time, a large amount of thermal energy will be generated. Simultaneously, the increased heat is concentrated in the interior of the particle, accumulating temperature continuously. The changes in the excitation frequency and amplitude change the motion state inside the particle, which reflects the interaction force between particles and the energy consumption process of particles.
According to Equation (6), the frequency-variable heat loss characteristics of a 2 mm particle are as shown in Figure 8:
Figure 8 shows the change in the thermal loss percentage of a forced-vibration particle as a function of frequency and reduced acceleration under vibration for 1 h and 2 h. As the frequency increases, the thermal loss of the particle increases. Similarly, as the reduced acceleration increases, the thermal loss of the particle decreases. In this vibration state, although particles in strong vibrational states and gas-like states are intensely vibrating and have significant heating effects, the input energy to the particle is large and is mainly converted into the internal kinetic energy and potential energy of the particle. Later, frictional collisions between particles further generate thermal losses, but they are chiefly stored in the form of kinetic energy and potential energy to maintain the vibration state of the internal particle. Therefore, the thermal loss percentage is relatively small. When particles are in solid-like, micro-vibrational, and intermediate vibrational states, there is tight interaction between particles. The input energy is generally converted into the potential energy of the particle and kinetic energy resulting from micro-vibration and micro-rotation, resulting in thermal losses. At this time, the input energy is small, and the motion of the particle is not intense, so the thermal loss percentage is high. The thermal loss percentage does not exceed 70% during the energy consumption process of particle vibration. A large amount of energy may be stored in the form of the internal kinetic energy and potential energy of the particle or dissipated efficiently.

6. Conclusions and Prospects

Based on the investigation of the thermal loss percentage and mechanism of forced-vibration particles with a focus on 2 mm particles, the following conclusions can be drawn:
(1) Differences in temperature changes exist for different vibration states of granular assemblies. The temperature-increasing effect of particles is significant in strong and gas-like vibration states. There is a certain temperature increase effect over a long time scale when particles are in micro-vibrational and intermediate vibrational states. In contrast, when particles are in solid-like states, the temperature increase effect is not obvious even under prolonged excitation. The temperature increase characteristics of particles are related to their vibration states, which reflect the strength of collisions and frictional interactions between particles.
(2) The forced-vibration heating characteristics of particles are related to frequency and amplitude. As the frequency increases, the heating effect of particles decreases, while as the amplitude increases, the heating effect of particles strengthens.
(3) For the same reduced acceleration, as frequency increases, the thermal loss of particles increases. At the same frequency, as the reduced acceleration increases, the thermal loss of particles decreases. Particles in strong and gas-like vibration states vibrate intensely and exhibit significant heating effects. However, because of the high input energy, the thermal loss percentage is relatively small. Particles in micro-vibrational and intermediate vibrational states have weaker vibrations and lower input energy, which results in higher thermal loss percentages due to intense particle collisions and frictional interactions.
In summary, granular assemblies mainly dissipate input energy through collision and friction during forced vibration, which is reflected in an increase in the temperature of particles. Through experimentation, it was found that particles do experience an increase in temperature under forced vibration conditions, but the corresponding thermal loss percentage does not significantly increase. This suggests that there may be more effective ways for particles to dissipate energy during energy consumption, such as sound loss caused by collisions or other forms of energy loss.

Author Contributions

S.Z., W.X., C.W. and Y.Z. proposed the idea. S.Z., W.X. and J.C. completed the experiments. S.Z., C.W., Y.Z. and Y.L. wrote the paper. All of the authors analyzed the data, discussed the results, and reviewed the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by National Major Special Basic Research grant number J2019-Ⅱ-0013-0033, the Basic Strengthening Plan Key Basic Research grant number 2020-JCJQ-ZD-204, and the Discrete Mass Damping Nonlinear Energy Dissipation Mechanism and Application Research grant number No. 1227020353.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A. Discrete Element Vibration Energy Characteristics Model

When the adhesion of the granular surface is not considered, the normal force between the granules is generally used. Normal contact force F c is the soft ball contact model, which is a soft ball dry contact model based on the improved discrete element method [28,29]. Deformation will occur, and the normal contact force will be obtained F n :
F n = 4 3 E * R * δ n 3 2
where R * is the equivalent radius; E * is the equivalent elastic modulus, given by the following formula; and δ n is the normal overlap.
1 R * = 1 R 1 + 1 R 2
1 E * = 1 − v 1 2 E 1 + 1 − v 2 2 E 2
The energy loss in this process is mainly based on the research of Tsuji et al. [30]. The normal damping force F n d is represented in the following equation:
F n d = 2 5 6 ζ k n m * v n r e l →
where ζ is the damping ratio, v n r e l → is the normal component of the relative velocity of the two particles, m * is the equivalent mass, and k n is the normal stiffness; its expression is as follows:
1 m * = 1 m j + 1 m i − 1
β = − ln e ln 2 e + π 2
k n = 2 E * R * δ n
where m i , m j is the mass of the two collision particles. The particles selected in this experiment are particles of the same material and volume. Therefore, the mass of the particles that collide is the same.
The tangential contact force model of the particle is based on the Mindlin–Deresiewicz contact theory. The calculation of the tangential contact force is as follows:
F t = − k t δ t
where δ t is the tangential overlap and k t is the tangential stiffness, derived from Equation (A9):
k t = 8 G * R * δ n
In Equation (A9), G * is the equivalent shear modulus is as follows:
G * = 2 − v 1 G 1 − 2 − v 2 G 2
where G 1 and G 2 are the shear modulus of the two particles, which is determined by the elastic modulus and Poisson’s ratio of the material.
G 1 = E 1 2 ( 1 + v 1 ) , G 2 = E 2 2 ( 1 + v 2 )
Based on the study of Tsuji et al. [30], the tangential damping force can be defined as follows:
F t d = 2 5 6 ζ k t m * v t r e l →
where v t r e l → is the tangential component of the relative velocity of the two particles.
In the forced vibration friction model, the particle is in contact with other particles or container walls and is subjected to vertical positive pressure F n and horizontal thrust F t . In the discrete element method, the shareholder friction of the particle body is considered to apply a moment to the surface of the particle as follows:
τ i = − μ r F n R i ω i
where μ r is the rolling friction coefficient, R i is the distance from the particle contact point to the center of mass, and ω i is the unit angular velocity vector of the object at the contact point.
The forced vibration model of the granular assemblies adopts the Hertz contact theory model, and the normal contact force adopts the soft ball contact model as the soft ball dry contact model of the improved discrete element method.
Table A1. Characteristic parameters of granular assemblies forced vibration model.
Table A1. Characteristic parameters of granular assemblies forced vibration model.
ParametersValues
Density of granular material, ρ p ( kg / m 3 ) 7800
Elastic modulus of granular materials, E p ( Pa ) 2.06 × 10 11
Poisson’s ratio of the granular materials, υ p 0.3
Particle size, D ( mm ) 2
Material density of the granular container, ρ c ( kg / m 3 ) 1190
Elastic modulus of granular container material, E c ( Pa ) 1.204 × 10 9
Poisson’s ratio of granular container material, υ c 0.37
Restitution coefficient, e 0.92
Static friction coefficient, μ s 0.3
Rolling friction coefficient, μ r 0.01
Time step, Δ t ( s ) 1 × 10 − 6
Gravity acceleration, g ( m / s 2 ) 9.79

Appendix B. Granular Vibration Transition Curves and Vibration State

Based on the traditional interaction between particles, the vibration forms of granular flows can be classified into three types: (1) quasistatic granular flow, which exhibits solid-like behavior; (2) collisional or rapid granular flow, resembling liquid behavior; and (3) dense flow, showing gas-like behavior [31,32,33,34,35]. Under different excitation conditions, granular materials exhibit non-equilibrium energy dissipation phenomena and display rich and complex dynamic behaviors. While there have been numerous studies on the vibrational forms of granular materials, there is currently no widely accepted classification system. However, there is still much to be understood regarding the vibration patterns and behavior of granular assemblies. Based on extensive experimental research, scholars believe that granular materials exhibit diverse vibrations within different particle containers, including solid-like states [36,37], fluidization [38,39,40], bouncing bed behavior [41], convection [42,43], the Leidenfrost effect [44,45,46,47,48], and buoyancy-driven convection [49,50].
In this paper, it is difficult to stimulate the violent motion state inside the particle body under this excitation condition. Therefore, in the vibration division of the particle body, according to the impact of the particle body on the container, it is further refined under its typical vibration state, and it is classified into five kinds of forced vibration states, which are the solid-like state, micro-vibrational state, intermediate vibrational state, strong vibrational state, and gas-like state. Figure A1 shows the variation of different vibration states of particles with different frequencies and reduced acceleration amplitudes. Figure A2 shows the curve of reduced acceleration under a typical vibration state.
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Figure A1. Granular vibration transition curves and vibration state.
Figure A1. Granular vibration transition curves and vibration state.
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Figure A2. Reduced acceleration curves corresponding to different vibration states.
Figure A2. Reduced acceleration curves corresponding to different vibration states.
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Figure 1. Discrete element method of granular assemblies.
Figure 1. Discrete element method of granular assemblies.
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Figure 2. Velocity distribution characteristics inside the granular assemblies. (a) the reduced acceleration is 0.8 and the excitation frequency is 50 Hz. (b) the reduced acceleration is 1.2 and the excitation frequency is 50 Hz. (c) the reduced acceleration is 2.5 and the excitation frequency is 50 Hz. (d) the reduced acceleration is 2.5 and the excitation frequency is 25 Hz. (e) the reduced acceleration is 6 and the excitation frequency is 50 Hz.
Figure 2. Velocity distribution characteristics inside the granular assemblies. (a) the reduced acceleration is 0.8 and the excitation frequency is 50 Hz. (b) the reduced acceleration is 1.2 and the excitation frequency is 50 Hz. (c) the reduced acceleration is 2.5 and the excitation frequency is 50 Hz. (d) the reduced acceleration is 2.5 and the excitation frequency is 25 Hz. (e) the reduced acceleration is 6 and the excitation frequency is 50 Hz.
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Figure 3. Total particle kinetic energy, total particle rotational kinetic energy, and total number of collision times under different excitation conditions.
Figure 3. Total particle kinetic energy, total particle rotational kinetic energy, and total number of collision times under different excitation conditions.
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Figure 4. Schematic diagram of granular assemblies thermal dissipation characteristics.
Figure 4. Schematic diagram of granular assemblies thermal dissipation characteristics.
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Figure 5. Distribution of surface temperature of granular assemblies in different vibration states. (a) Solid-like state (the reduced acceleration is 0.8 and the excitation frequency is 50 Hz). (b) Micro-vibrational state (the reduced acceleration is 1.2 and the excitation frequency is 50 Hz). (c) Intermediate vibrational state (the reduced acceleration is 2.5 and the excitation frequency is 50 Hz). (d) Strong vibrational state (the reduced acceleration is 2.5 and the excitation frequency is 25 Hz). (e) Gas-like state (the reduced acceleration is 6 and the excitation frequency is 50 Hz).
Figure 5. Distribution of surface temperature of granular assemblies in different vibration states. (a) Solid-like state (the reduced acceleration is 0.8 and the excitation frequency is 50 Hz). (b) Micro-vibrational state (the reduced acceleration is 1.2 and the excitation frequency is 50 Hz). (c) Intermediate vibrational state (the reduced acceleration is 2.5 and the excitation frequency is 50 Hz). (d) Strong vibrational state (the reduced acceleration is 2.5 and the excitation frequency is 25 Hz). (e) Gas-like state (the reduced acceleration is 6 and the excitation frequency is 50 Hz).
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Figure 6. Temperature variations of granular assemblies in different vibration states. (a) Temperature variations in different vibration states of 25 Hz. (b) Temperature variations in different vibration states of 50 Hz.
Figure 6. Temperature variations of granular assemblies in different vibration states. (a) Temperature variations in different vibration states of 25 Hz. (b) Temperature variations in different vibration states of 50 Hz.
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Figure 7. Frequency-varying temperature distribution of granular assemblies. (a) Temperature measurement location diagram. (b) Temperature change in granular assemblies for 1 h vibration. (c) Temperature change in granular assemblies for 2 h vibration.
Figure 7. Frequency-varying temperature distribution of granular assemblies. (a) Temperature measurement location diagram. (b) Temperature change in granular assemblies for 1 h vibration. (c) Temperature change in granular assemblies for 2 h vibration.
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Figure 8. Frequency-varying thermal loss percentage characteristics of granular assemblies.
Figure 8. Frequency-varying thermal loss percentage characteristics of granular assemblies.
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Zhang, S.; Xu, W.; Wang, C.; Zhang, Y.; Li, Y.; Cheng, J. Research on Thermal Dissipation Characteristics Based on the Physical Laws of Forced Vibration in Granular Assemblies. Appl. Sci. 2024, 14, 5864. https://doi.org/10.3390/app14135864

AMA Style

Zhang S, Xu W, Wang C, Zhang Y, Li Y, Cheng J. Research on Thermal Dissipation Characteristics Based on the Physical Laws of Forced Vibration in Granular Assemblies. Applied Sciences. 2024; 14(13):5864. https://doi.org/10.3390/app14135864

Chicago/Turabian Style

Zhang, Shengwu, Wei Xu, Chiming Wang, Yuanchao Zhang, Yanan Li, and Jianwei Cheng. 2024. "Research on Thermal Dissipation Characteristics Based on the Physical Laws of Forced Vibration in Granular Assemblies" Applied Sciences 14, no. 13: 5864. https://doi.org/10.3390/app14135864

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