Experimental study on damping characteristics of the particle dampers with damping rod and soft inside wall

In the perpendicular simple sinusoidal excitation, the damping characteristics of the soft-wall particle damper with a damping rod were experimentally investigated by controlling the acceleration amplitude and vibration frequency of the damper motion and measuring the force and acceleration signals. The experimental results show that the loss power of the damper increases monotonically with the rise of the excitation acceleration amplitude, and the effective mass shows the change of “gentle decrease-critical point-decrease”. As the driving frequency rises, the loss power progressively decreases, and the effective mass does not change significantly. By analyzing the damping characteristics of different dampers, it shows that the damping effectiveness of the soft inside wall particle damper with a damping rod is obviously better.


Introduction
A soft inside wall particle damper with a damping rod is a passive vibration control technology [1].It utilizes the collision, extrusion, and friction between particles and the damping rod and soft wall surface during the vibration process of the system to convert the mechanical energy of the system into heat energy and sound energy, resulting in the damping effect, thus significantly improving the damping of the system structure.The damper has the advantages of small changes to the original structure, small additional mass, and a wide application range [6][7].
In this paper, the damping characteristics of a soft inside wall particle damper with a damping rod under vertical harmonic excitation are experimentally studied.The variation rules between the loss power, effective mass, loss factor, and excitation parameters of the damper are compared.

Computing method of damping characteristics
The steady-state energy flow method can be used to calculate the steady-state input power of the system [1].For harmonic excitation, the input power P in can be obtained from the following equation.
The loss factor η can be obtained by the following Equation [1].
The effective mass m of the particle damper can be obtained by the following Equation [1].
By using the above equation and the signal data measured in the experiment, the damping characteristics of the damper can be calculated.

Test equipment and measurement system
The test setup is shown in Figure 1, where carbon steel particles are loaded into a cylindrical rubber inside a wall container, and a damping rod is inserted into the particle body, from which the particles, the container, and the damping rod form a soft-wall particle damper with a damping rod [2-3].The relevant data are shown in Table 1.
Table 1.Specific parameters of particle containers and particles.

Item Parameter
Inside diameter(mm) 52 Height(mm) 50 Wall thickness(mm) 10 Particle diameter(mm) 0.7 Particle filling rate 70% Mass of 70% particles(g) 338.1 Damping rod diameter(mm) 14 Damping rod into the particle depth(mm) 26 The two ends of the impedance head are connected to the vessel and the electromagnetic shaker.The impedance head amplifies the force and acceleration signals through the built-in amplifier and then transmits them to the LMS.Test.Lab Signal Acquisition and Analysis System for feedback control of the excitation signals; they are amplified by the power amplifier and then transmitted to the electromagnetic shaker [4-5].

Damping characterization of a soft inside wall particle damper with damping rod
As can be seen in Figure 2, the effective mass is mainly affected by the incentive acceleration magnitude, while the variation in the excitation frequency is insignificant.Before the critical point (10 m/s 2 ), in the region of low excitation amplitude, the difference between the effective mass and static mass of the system is very small.With the increase of acceleration amplitude, the effective mass shows the change process of "gentle decrease-critical point-gradual decrease".The effective mass does not change significantly with frequency, which explains the phenomenon that the addition of damping particles to the structure does not change the intrinsic frequency of the structure.As can be seen from Figure 3, for different frequencies, the excitation force does work on the damper only when the acceleration amplitude is higher than a critical value (10 m/s 2 ).Only when the excitation acceleration reaches a certain intensity do the damping particles start to function; otherwise, the loss power of the damper is close to zero.When the acceleration is greater than a critical value, the loss power increases monotonically with the increase of excitation.Low loss power corresponds to low frequency and low amplitude region, and high loss power corresponds to high amplitude region.This law meets the requirement of applying particle damping to suppress intense vibration in engineering practice.As can be seen from Figure 4, at a certain frequency, the loss factor shows a "monotonous increasing-reaching peak-taper off" process as the acceleration is enhanced.When the acceleration is certain, the loss factor decreases slightly with the increase in frequency.In the low-frequency, high-vibration region, the loss factor has a higher value.When the excitation acceleration amplitude range is 22-37 m/s 2 , the loss factor of the damper is larger (greater than 0.5), with the loss factor peak value of 0.638.The loss factor can be changed through the law of change of the optimal working range of the particle damping, which is conducive to the design of the damper.

Effect of damping rod diameter on damping characteristics of dampers
To further explore the impact of rod diameter on dampers' damping characteristics, four different rod diameters (14 mm, 16 mm, 18 mm, and 20 mm) were tested.Similarly, under the vertical sinusoidal excitation, the damping lag angles of these particle dampers with damping rods were tested, and their damping characteristics were obtained by calculation.In order to facilitate a more intuitive comparison, the maximum lag angle at a certain frequency obtained from the test is compared and analyzed in this paper as the amplitude of the excitation acceleration varies.
From Figure 5, under a fixed rod diameter, as the incentive frequency is raised, the lag angle of the damper shows a trend of "increasing-reaching peak-slightly decreasing", and the changing trend in the absence of rods is basically the same as that in the presence of rods.Fixed incentive frequency: as the rod diameter increases, the lag angle of the damper mostly gradually increases, but the increase is small.This shows that the diameter of the damping rod has an effect on the damping characteristics of the damper, but the effect is not too large.Moreover, the peak value and increasing amplitude of the lag angle of the rubber inside wall damper are slightly greater than those of the hard inside wall damper.

Analysis of test data
The energy dissipation characteristics of soft-walled particle dampers with damping rods are realized by heat or acoustic radiation generated by inelastic collisions, friction, and crushing motions between particles, between particles and damper walls, and between particles and damping rods [8].
From this experiment, it can be seen that when the incentive acceleration magnitude is less than 10 m/s 2 , the particles of the soft inside wall particle damper with a damping rod are agglomerated together and in a state of agglomeration.At this time, the effective mass of the device is very small compared with the static mass, and the motion of the particles is similar to the overall motion.The dissipated energy is small.At the same time, the loss power and loss factor are both low.
After the incentive acceleration amplitude arrives at the marginal value, the motion of the particles intensifies.The surface particles begin to appear in a gas-like motion state, and the motion form of the particles begins to enter a fluidized state [2][3].
With the change of excitation conditions, the loss factor of this system increases from 0 to 0.638, the loss power increases from 0 W to 0.0673 W, and the effective mass decreases from 0.62 kg to 0.28 kg.Moreover, the structure has the best energy dissipation when the acceleration is in the range of 22-37 m/s 2 , with a smaller effective mass and a larger loss power.

Conclusion
(1) Only when the external excitation reaches a certain intensity do the particles begin to show their effects.There is an obvious turning mark phenomenon of the effective mass and power loss with the change of acceleration and frequency, and the turning mark of the soft inside wall particle damper with a damping rod appears earlier.
(2) The diameter of the damping rod has an effect on the damping characteristics of the damper, but the effect is not too large.Moreover, the peak value and increasing amplitude of the lag angle of the soft inside wall damper are slightly greater than those of the hard inside wall damper.
(3) Under the same excitation conditions, by analyzing the damping characteristics of different

Figure 2 .
Figure 2. Effective mass three-dimensional surface diagram.As can be seen from Figure3, for different frequencies, the excitation force does work on the damper only when the acceleration amplitude is higher than a critical value (10 m/s 2 ).Only when the excitation acceleration reaches a certain intensity do the damping particles start to function; otherwise, the loss power of the damper is close to zero.When the acceleration is greater than a critical value, the loss power increases monotonically with the increase of excitation.Low loss power corresponds to low frequency and low amplitude region, and high loss power corresponds to high amplitude region.This law meets the requirement of applying particle damping to suppress intense vibration in engineering practice.

Figure 3 .
Figure 3. Loss power three-dimensional Surface diagram.As can be seen from Figure4, at a certain frequency, the loss factor shows a "monotonous increasing-reaching peak-taper off" process as the acceleration is enhanced.When the acceleration is certain, the loss factor decreases slightly with the increase in frequency.In the low-frequency, high-vibration region, the loss factor has a higher value.When the excitation acceleration amplitude range is 22-37 m/s 2 , the loss factor of the damper is larger (greater than 0.5), with the loss factor peak value of 0.638.The loss factor can be changed through the law of change of the optimal working range of the particle damping, which is conducive to the design of the damper.

Figure 5 .
Figure 5.The variation of the maximum lag angle of the damper with the diameter of the rod.