The influence of particle radius Rs, mass filling ratio δ, cavity length L, excitation frequency ω and amplitude A, and rotating speed n on the longitudinal vibration suppression of the system is reflected by the TEC and VRR. The TEC (including the energy consumption of collision and friction) comes from the simulation results, while the VRR comes from the bench test to verify the rationality of the simulation model and the reliability of the simulation data. According to the simulation and test results, the evaluation system of the internal relationship between the TEC and VRR was established.
The influences of the above parameters on the TEC and VRR are analyzed as follows:
Influence of particle radius on TEC and VRR. The influence of particle radius (RS=5, 6, 7, 8, 9 mm) on the TEC is shown in Fig. 6. The effect of different particle radius and rotating speed on VRR is shown in Fig. 7, and the vibration acceleration of the system at n = 90 r/min in Fig. 8.
As can be seen from Figs. 6 ~ 7, when Rs = 5 ~ 9 mm and n = 90 r/min, the TEC and VRR show obvious vibration suppression effect, and both decrease slightly with the increase of particle radius. At lower rotating speed, such as n = 30 r/min, particle radius has little influence on the VRR. Under different rotating speeds the VRR basically reaches the peak value at Rs = 6 mm, and then decreases with the increase of particle radius. In Fig. 8, at the dominant frequency of 5Hz, and the acceleration speed of particle system with different particle radius are lower than that of the cavity system, indicating that the vibration suppression effect is obvious.
Under a certain mass filling ratio, the larger the particle radius is, the larger the mass is, but the less the particle number is. The large mass particles have high friction and kinetic energy, which can increase the energy consumption of friction and collision. However, the reduction of the number of particles may reduce the contact probability between particle and surrounding particles or the cavity, which may affect the energy consumption. And vice versa. Therefore, the interaction of various influencing factors should be considered comprehensively to select the appropriate particle radius. In the case of constant particle radius, the influence of rotating speed on the VRR is complicated and involves the accumulation and motion state of particles, which will be discussed in detail in the following section (Influence of rotating speed on TEC and VRR.).
Influence of mass filling ratio on TEC and VRR. The influence of mass filling ratio (δ = 2.5%, 5%, 7.5%, 10%) on the TEC is shown in Fig. 9. The influence of different mass filling ratio and rotating speed on the VRR is shown in Fig. 10, and the vibration acceleration speed of the system at n = 90 r/min in Fig. 11.
It can be seen from Figs. 9 ~ 10 that collision energy consumption, friction energy consumption, TEC and VRR have similar variation trends (except n = 150 r/min), and they all reach peak values at δ = 5% and then show a downward trend. In Fig. 11, when δ = 2.5%, 5%, 7.5% and 10%, the VRR of the particle system is lower than that of the cavity system at the dominant frequency of 5Hz, showing obvious vibration suppression effects.
The mass filling ratio of particles can affects the intensity of collision and friction between particles and cavity. When the mass filling ratio is 2.5%, the total mass and number of particles are relatively little, the probability of collision and friction contact between particle and the wall is reduced so that the TEC is reduced. When mass filling rate rises to 7.5%, the number of particles and the number of accumulation layer are increased. Extruded by the gravity of the upper-layer particles, the clearance among the particles in middle-layer and low-layer is decreased, the movement space of particles is limited, both probability and intensity of collision and friction is declined. When the larger mass filling ratio is, the more obvious the variation trend of energy consumption of the system is, as a result, both collision and friction energy consumption of the system decreases more.
At the same rotating speed, higher or lower mass filling ratio is not conducive to play the role of the energy dissipation of particle damping. Although higher mass filling ratio can increases the total mass and quantity of particles and the static pressure between particles, it leads to the reduction of the movement space of particles, which is not conducive to the collision and friction between particles and wall or between particles. If the mass filling rate is lower, the number of particles is smaller, the momentum exchange between particles and the damper wall is reduced, and the energy consumption of particle damping is also greatly reduced.
Influence of excitation frequency on TEC and VRR. The influence of excitation frequency ( f = 3, 4, 5, 6, 7, 8 Hz) on the TEC is shown in Fig. 12. The influence of different excitation frequencies and rotating speed on VRR is shown in Fig. 13, and the vibration acceleration speed of the system at n = 90 r/min in Fig. 14.
From Figs. 12 ~ 13, it shows that at n = 90 r/min, the change trend of the energy consumption of both collision and friction of particles, the variation trend of both total energy consumption of system and vibration reduction ratio of the system are similar, and the trend do not change significantly with the increase of excitation frequency. It shows that the interaction intensity between particles and particles and between particles and cavity basically remains the same, but the friction energy consumption is slightly higher than the collision energy consumption.
It can be seen from Fig. 14 that the acceleration speed of the system with particle (RS=5 mm) near the dominant frequency of 5Hz is lower than that of the cavity system.
Influence of excitation amplitude on TEC and VRR. The influence of excitation amplitude (A = 1.5, 2, 2.5, 3, 3.5 mm) on the TEC is shown in Fig. 15. Under different excitation amplitudes and rotating speeds the VRR is shown in Fig. 16. Taking the excitation amplitude A = 3.5mm as an example, the system acceleration speed at n = 90 r/min is shown in Fig. 17 .
It can be seen from Figs. 15 ~ 16 that particle damping has obvious vibration reduction effect. At n = 90 r/min, with the increase of excitation amplitude, the collision energy consumption, friction energy consumption, the TEC and VRR have a similar variation trend, showing a slight upward trend. In Figure 17, at A = 3.5 mm, the acceleration speed of the particle system at the main frequency 5 Hz is lower than that of the cavity system, indicating obvious vibration suppression effect. When the rotating speed n is 30, 60, 90 r/min respectively, with the increase of excitation amplitude under the same speed, VRR hardly changes. The results show that though the variation of excitation amplitude can change the velocity and acceleration speed of cavity and particle, and increase the contact probability, collision and friction intensity between particle and the surfaces of barrel and two walls, but the influence degree is limited. Under the same excitation amplitude, the VRR increases with the enhancement of the rotating speed. However, when the rotating speed is increased to 120 r/min and 150 r/min respectively, the variation trend of VRR at two rotating speeds is opposite, and the VRR of the former is higher than that of the latter under the same excitation amplitude. It can be seen that the variation of rotating speed plays a more critical role in VRR, and it can changes the centrifugal force, stacking state and dropping motion state of particles.
Influence of cavity length on TEC and VRR. The influence of cavity length (L=25, 35, 45, 55, 65, 75 mm) on TEC is shown in Figure 18. Under different cavity lengths and rotating speeds the VRR is shown in Figure19, and the system acceleration speed at n = 90 r/min in Figure 20
As can be seen from Fig. 18, when the cavity length L = 25 ~ 45 mm, the collision energy consumption increases with the increase of the cavity length and reaches the maximum value at L = 45 mm. When the cavity length L is greater than 45 mm, the collision energy consumption decreases with the increase of the cavity length. The friction energy consumption increases with the increase of cavity length, but the increasing intensity decreases gradually. The TEC increases with the enhancement of cavity length and tends to be gentle.
In Fig. 19, the TEC increases first and then decreases gradually with the increase of cavity length. When n = 90 r/min, the variation trend of the VRR is basically similar to that of the TEC. However, when L > 45 mm, the variation trend of the VRR is slightly different, the former decreases slowly, and the latter gradually flattens out. The influence degree is closely related to the change of cavity length.
In Fig. 20, under different cavity lengths (L = 25 ~ 75 mm), the vibration acceleration speed of the particle system at n = 90 r/min is significantly lower than that of the cavity system.
Under the condition of a certain mass filling ratio, the variation of cavity length (L1 ~ L2) has an important effect on the stacking mode, motion space and motion state of particles (shown as in Fig. 21), thus affecting the TEC and VRR.
It can be seen from Fig. 21 that the cavity length affects the number of stacking layers of particles, the motion state and mechanical properties of upper, middle and lower layers of particles, and ultimately affects the energy consumption level of particles. With the increase of cavity length L (L1 ~ L2), the number of stacking layers of particles decreases, the distribution of particles is loose, the extrusion pressure among particles decreases, the gap among particles increases, the collision probability between particles and the front and rear walls decreases, the momentum exchange decreases, and the vibration reduction effect of the system is weak. As the clearance among particles increases, the moving space increases, and the friction contact probability between particles and particles and between particles and barrel wall surface also increases correspondingly, so the friction energy consumption of the system has been increasing.
Influence of rotating speed on TEC and VRR. Figure 22 shows the influence of rotating speed (n = 30, 60, 90, 120, 150 r/min) on the TEC. The analysis of the influence of rotating speed on the VRR has been involved in the influence of parameters such as particle radius, mass filling ratio, excitation frequency and excitation amplitude on the VRR. Now, taking the standard working condition (particle radius Rs = 5 mm in Fig. 7) as an example, Fig. 23 shows the influence of rotating speed (n = 30, 60, 90, 120, 150 r/min) on VRR.
As can be seen from Fig. 22, collision energy consumption, friction energy consumption and TEC increase first and then decrease with the increase of rotating speed. At n = 0 ~ 90 r/min, the energy consumption of collision and friction increases obviously, the former increases slightly more than the latter, and then presents a downward trend. In Fig. 23, the variation trend of VRR with rotating speed is similar to that of the TEC, reaching a maximum of 9.06% at n = 90 r/min.
Under different rotating speed, motion state and velocity vector of particles in damper are shown in Fig. 24.
Increasing the rotating speed (0 ~ 90 r/min) can enhance the extrusion, friction and collision between particles and the surface of the barrel, but when the particles lose balance under the action of centrifugal force, gravity, friction and surrounding pressure, the particles will roll down, slide along the damper wall or produce throwing motion. The friction and collision probability between particles and walls, between particles and barrel, and between particles and particles increases, and the TEC and VRR are enhanced. The motion state and velocity vector of particles are shown in Fig. 24(a).
However, when the rotating speed exceeds 90 r/min, the increase of centrifugal force of particles leads to some particles sticking close to the wall of the barrel, which reduces the drop motion of particles and the contact probability between particles, and the TEC begins to decline. When rotating speed of more than 120 r/min, particles attached to the wall of the barrel increase due to the further increase of the centrifugal force, at this time only the inner particles do fall movement, which leads to the friction and collision between particle and particle and between particles and wall reduced correspondingly, thus presents the downward trend of energy consumption. The motion state and the velocity vector of particles at n = 120 r/min is shown in Fig. 24 (b).
Comparison of simulation and test results
Taking rotating speed n = 90 r/min for instance, the simulation and test results of particle damping for vibration suppression show that the variation trends of both results are basically similar with particle radius, mass filling ratio, excitation frequency, excitation amplitude, cavity length and rotating speed. The influence of particle radius, mass filling ratio, cavity length and rotating speed on the system vibration suppression effect is greater, and the influence of excitation frequency and excitation amplitude weaker, as shown in Table 4.
As can be seen from Table 4, in the influencing parameter such as particle radius RS, mass filling ratio δ, excitation frequency f, excitation amplitude A, cavity length L, rotating speed n (except n = 150 r/min and Rs = 2.5), the variation trend of VRR is basically consistent with the TEC, which verifies the rationality of the simulation model and the reliability of simulation results and reveals internal relationship between VRR and TEC.