Nonlinear Characteristics of Three-parameter Fluid Viscous Damper with Considering of Principal Stiffness and Hydraulic Stiffness


 The three-parameter fluid viscous damper is used to isolate micro-vibration produced by control torque gyro (CMG) in satellite. In this paper, the damper is simplified by a single tube fluid viscous damper and two springs connected to the damping piston. With consideration of the principal stiffness of the bellows and the contraction and expansion effect of the damping orifice, the approximate analytical nonlinear model of the damper is established and verified by the computation fluid mechanics (CFD) method. Based on this analytical model, the displacement response of the damper and correction coefficient of hydraulic resistance are analyzed, the nonlinear characteristics in the frequency domain are also revealed. Furthermore, the energy consumptions of the nonlinear model and linear model are researched. The results show that the damper has an obvious amplitude at the first resonance peak, but not obvious at the second resonance peak. The vibration amplitude of the damping piston is only um level in the high-frequency domain. The correction coefficient of the hydraulic resistance at the resonance peak is much higher than other frequencies, which causes a significant nonlinear behavior. In addition, the energy consumption of the nonlinear model is larger than that of the linear model at the resonance peak, and the larger the resonance peak, the more obvious the phenomenon is. This means that the nonlinear damping can be used to further improve the suppression of the resonance peak of the three-parameter fluid viscous damper.


Introduction
The control torque gyro and reaction flywheel are the actuators on the spacecraft, whose output moment is used to control the attitude of the spacecraft. Due to the static and dynamic unbalance of the rotor, micro-vibration is easy to occur at high speed.
Micro-vibration has characteristics of small amplitude and high frequency [1]. If not suppressed, it will cause a serious influence on the high-precision payload on spacecraft. The research shows that the installation of a vibration damper on the propagation path of micro-vibration is an effective means to suppress micro-vibration [1].
At present, micro-vibration isolation devices are divided into passive vibration isolation devices, active vibration isolation devices, hybrid vibration isolation devices, and semi-active vibration isolation devices [1]. Among the passive vibration isolation devices, one kind of vibration damper is widely used, namely a three-parameter fluid viscous damper. This kind of vibration damper was first proposed in the 1990s for the vibration isolation of reaction flywheel on the Hubble Space Telescope, and the vibration isolation effect is perfect after on-orbit verification [2]. Later, it was widely used to isolate the vibration of the control moment gyroscope and reaction flywheel on conventional satellites. When a three-parameter fluid viscous damper is used to suppress micro-vibration, it is usually assembled as a vibration isolation platform to achieve six degrees of freedom vibration isolation. For example, the Fengyun-4 satellite uses three three-parameter fluid viscous dampers to constitute a vibration isolation platform to suppress the micro-vibration generated by the reaction flywheel. Through testing, the attenuation rate can reach more than 75% [3]. To further improve the performance of the damper, an elastic damper and a three-parameter fluid viscous damper can be installed in parallel. The main function of the elastic damper is to reduce the fundamental frequency of the system, to obtain better vibration isolation performance in the high-frequency band [4]. Besides, a three-parameter fluid viscous damper can also be combined with an active vibration isolator to realize hybrid vibration isolation, which will increase the performance and reliability of the system [5] [6].
The modeling methods of a three-parameter fluid viscous damper can be divided into two categories, one belongs to lumped parameter modeling method. The other belongs to the approximate analytical method. The lumped parameter modeling method is very simple in form and can also consider the influence of nonlinear damping, but it can not establish the relationship with the specific design parameters of the damper, so it can only be used for the design of stiffness coefficient and damping coefficient [2][3] [7]. The approximate analytical method can establish the relationship with the specific design parameters of the damper [9][10] [11]. The three-parameter fluid viscous damper uses the damping produced when the fluid flows through the damping orifice to mitigate the vibration energy. Due to the special configuration of damping orifice [12], the fluid will produce a contraction effect and expansion effect when flowing through the damping orifice, which will cause nonlinear damping and stiffness. For the convenience of modeling, most scholars did not consider the influence of nonlinear characteristics when using approximate analytical methods [3][4] [8] [11]. In describing the flow state in the damping orifice, they used the uncorrected hydraulic resistance, therefore the damping and hydraulic obtained are linear [13]. Shi [9] linearized the nonlinear damping according to the energy consumption, which is a linearized modeling method. Jiao's research points out that the reason for the nonlinear stiffness and damping of this damper is that the corrected hydraulic resistance is used when considering the contraction and expansion effect of damping orifice, and the nonlinear velocity term in the corrected hydraulic resistance leads to the nonlinear stiffness and damping [14]. The linear model is suitable for high viscosity fluid and the damping orifice has a large length-diameter ratio, but it is not accurate for low viscosity fluid.
However, the influence of the principal stiffness of bellows was not considered in his research, so it is impossible to evaluate the influence of nonlinear stiffness and damping on vibration isolation performance.
In general, there are still some unsolved problems in the researches of nonlinear characteristics of the three-parameter fluid viscous damper. Most scholars have adopted the linearization model, ignoring the influence of nonlinear stiffness and damping, and ignoring the effect of principal stiffness when considering the nonlinear behavior of this damper [15], so it is unable to accurately evaluate the influence of nonlinear characteristics on this damper. In this paper, an approximate analytical modeling

Modeling of three-parameter fluid viscous damper under displacement excitation and force excitation
The three-parameter fluid viscous damper can work in force excitation mode and displacement excitation mode. In this section, the model of these two modes will be derived.

Dynamic modeling of the damper under displacement excitation mode
The three-parameter fluid viscous damper can be described in Fig.1(a). The main components of the damper consist of two bellows and a damping piston. Bellows have two functions, one is to provide principal stiffness, the other is to accommodate silicone oil. The damping piston is used to produce damping. Micro-vibrations have small amplitude, usually in the range of 10 -6 g to 10 -3 g [16]. So the displacement amplitude of micro-vibration will be very small. Under the small displacement excitation, the displacement amplitude of the damping piston will be very small. At this time, the volume change of the cavity is very limited. Before modeling, we assumed that the volume of the cavity would not change, and the change of internal pressure is also limited. Therefore, this paper does not consider the fluid-structure coupling effect between bellows and fluid. The effective diameter of the bellows should be used to calculate the internal pressure of the fluid-filled bellows [17]. Therefore, the bellows can be replaced by a single tube, and the area of cross-section is equal to the effective area of the bellows. In addition, the bellows also plays the role of providing stiffness, so the spring element can be used to replace the stiffness of bellows. In this way, the damper can be simplified by a single tube fluid viscous damper with two springs connected to the damping piston. The simplified model is shown in Fig.1  . Because the liquid column and the damping orifice interact through friction, Rh F will also act on the damping piston. The three-parameter fluid viscous damper can work in displacement excitation mode, or force excitation mode. When the damping piston moves under the displacement excitation, the displacement amplitude of the damping piston does not change, but under different frequencies, the force required to produce the same amplitude displacement is quite different. This is mainly because the external force acting on the damping piston needs to overcome the pressure differential force inside the damper. This pressure differential force changes with frequencies [14], which leads to the changing external force. For constant amplitude force excitation, the displacement of the damping piston will vary with frequencies. This is mainly due to the difference in pressure at different frequencies. Under low-frequency excitation, the internal pressure of the damper is very small, part of the external excitation force is used to overcome the pressure differential force, and the other part is used to overcome the elastic force of the bellows. At this time, the damping piston has small displacement.
In the high-frequency domain, the internal pressure increases obviously, and almost all external excitation force needs to be used to overcome the pressure inside the damper, so the displacement of the damping piston will decrease.
By analyzing the forces acting on the three-parameter fluid viscous damper, it can be found that the forces transmitted to the base include the pressure differential force and the elastic force of the spring. The pressure differential force can be written as According to the conservation of flow, the next equation can be obtained.
It can be seen from the above equation that the corrected hydraulic resistance contains the first-order term of velocity, which indicates that the hydraulic resistance changes with the fluid velocity in the damping orifice.
Because the displacement of the damping piston is very small under the microvibration excitation, the volume of the cavity changes little, 1 2 0 V V V . According to equations (5), (6), and (7), the transfer function of pressure differential force relative to displacement can be obtained.

Dynamic modeling of the damper under force excitation
When the damper working under the sinusoidal force excitation, the solution of the pressure differential force is still consistent with the previous. However, the displacement of the damping piston is needed when establishing the transfer function of pressure differential force relative to displacement. Sinusoidal excitation force can be written as Flow rate Q can be written as According to equations (2) and (3), the transfer function can be obtained. ,, Make the denominator of equation (15) equal to zero to obtain characteristic roots.
Characteristic equation (16) has four roots. Therefore, the pressure differential force will be different due to the different forms of characteristic roots. Characteristic roots of the equation (16) depending on the damping orifice diameter, length, and viscosity of silicone oil. According to the different forms of characteristic roots, pressure differential force can be divide into two cases to discuss.
, because the hydraulic resistance contains the nonlinear term of velocity, the pressure differential force contains the nonlinear component.

Model validation
To verify the theoretical model established in the previous section, the CFD model  Table 1. The damping fluid is silicone oil. In the calculation process, we assumed that the viscosity of the silicone oil remains constant. Usually, the temperature of silicone oil will rise after the damper working for a long time, but the viscosity will decrease after the temperature rises, which will lead to the reduction of energy consumption and the temperature of silicone oil will further decrease. In addition, the damper itself will dissipate heat to the external environment, so it is a dynamic balance process. Therefore, the influence of temperature on viscosity is not considered in this paper. In the model, the external excitation force is sinusoidal and the amplitude is 48 N.

Analysis of hydraulic resistance correction coefficient
As has been mentioned, the hydraulic resistance correction coefficient can influence the nonlinear component in pressure differential force. In this section, the  Table 3.  6 shows the displacement of the damping piston at different damping orifice diameters. It can be seen from the figure that there is a resonance peak in the displacement curve. When the damping orifice diameter is 5 mm, the resonance peak occurs at 7 Hz, and the amplitude of damping piston displacement is 0.00416 m, which is much larger than the static displacement of the damping piston. As the damping orifice diameter decreases, the resonance peak decreases gradually. When the damping orifice diameter decreases smaller than 3 mm, the frequency of resonance peak increases obviously. When the damping orifice diameter decreases to 1 mm, there is a high-order resonance peak at 100 Hz, and the displacement of the damping piston is 7.81×10 -5 m. Fig.7 shows the hydraulic resistance correction coefficient at different damping orifice diameters. If the hydraulic resistance is not corrected, the coefficient of hydraulic resistance should be 1. The nonlinear term of velocity is included in the correction coefficient, which leads to nonlinear stiffness and damping. The larger the correction coefficient is, the more obvious the nonlinearity is. Corresponding to Fig.6, when the damping orifice diameter is 5 mm, the correction coefficient of hydraulic resistance increases significantly at the same resonance frequency, and the maximum value is 2.83, which indicates that the nonlinearity of the damper will be significant at 7 Hz. The reason is that when the damping piston resonates, it moves very fast, which will accelerate the flow speed in the damping orifice, resulting in a larger nonlinear term of velocity. With the decrease of damping orifice diameter, the correction coefficient at resonance frequency decreases gradually, but it increases at a higher-order resonance peak. When the damping orifice diameter is 1 mm and the frequency is 90 Hz, the correction coefficient is 1.52, which means the damper still has nonlinearity. In general, when the diameter of the damping orifice changes, the correction coefficient is only greater than 1 near the resonance peak but close to 1 at most other frequencies.
This is mainly due to the fast movement speed of the damping piston near the resonance peak. However, at higher frequencies, the correction coefficient is smaller because the movement amplitude of the damping piston is very small and the movement speed is relatively small. That means that for the three-parameter fluid viscous damper considering the principal stiffness, the nonlinearity of the damper will be significant near the resonance peak. the maximum displacement of the damping piston is only 2.12 × 10 -6 m. In the frequency band between the first and second resonance peaks, the displacement is nearly the same. Fig.9 shows the hydraulic resistance correction factor at different orifice lengths. In Fig.9, the correction coefficient curve also has a resonance peak at 7 Hz. When the damping orifice length is 30 mm, the correction coefficient can reach 2.83, which indicates the damper has significant nonlinearity. When the damping orifice length is 60 mm, the correction coefficient is only 1.53. The reason is that the larger the displacement resonance peak is, the faster the velocity of the fluid in the damping orifice will be, resulting in the larger nonlinear term of velocity and the larger correction coefficient. However, the increase of the correction coefficient is not obvious at the high-order resonance peak. This is mainly due to the displacement amplitude of the high-order resonance peak is only 10 -6 m, and the fluid velocity in the damping orifice is very small. This also indicates that the nonlinearity of the damper is obvious if the resonance peak is obvious. Fig.10 shows displacement of the damping piston at different viscosities. It can be seen from the figure that there is a significant first-order formant in the displacement curve, but the higher-order resonance peak is not obvious. Besides, the displacement curve is nearly the same above 20 Hz. When the viscosity is 100 cst, the displacement at the resonance peak is 4.16×10 -3 m. When the viscosity is 1000 cst, there is no obvious resonance peak in the displacement curve, which indicates that the damper has a significant suppression on the resonance peak. Fig.11 shows the hydraulic resistance correction coefficient at different viscosities. It can be seen that the correction coefficient reaches the maximum value at the resonance frequency. When the viscosity is 100 cst, the correction coefficient is 2.83, because the correction coefficient determines the strength of nonlinearity, it means that the nonlinearity at resonance peak will be significant. When the viscosity is 1000 cst, the correction coefficient is only 1.022. When the frequency is greater than 30 Hz, there exists a high-order resonance peak, but not obvious, so the correction coefficient is still close to 1 at most frequencies beyond 30 Hz, which indicates the nonlinearity is not obvious at these frequencies.

Nonlinear characteristics of pressure differential force and hysteresis loop
In this section, nonlinear characteristics of the pressure differential force and hysteresis loop at the resonance peak will be analyzed. Table 4 shows the frequencies and amplitudes of the first-order and the second-order resonance peak at different damping orifice diameters. Since there is no first-order resonance peak when the damping orifice diameter is 5 mm, the second-order resonance frequency and amplitude are not given and are represented by "None" in the table. Table 5 shows the frequencies and amplitudes of the first-order and the second-order resonance peak at different damping orifice lengths. Table 6 shows the frequencies and amplitudes of the first-order and the second-order resonance peak at different viscosities. Since there are no secondorder resonance peaks when damping orifice lengths are 30 mm and 35 mm, the secondorder resonance frequencies and amplitudes are not given, and are represented by "None". Fig.12 shows the pressure differential force at the first-order resonance peak.
If the uncorrected hydraulic resistance is used, the curve of pressure differential force should be sine or cosine curve. It can be seen from the enlarged diagram that the pressure difference force curve exists local distortion, which indicates there exists a nonlinear component in pressure differential force. The nonlinearity of the pressure differential force tends to decrease with the decrease of the damping orifice diameter. Fig.13 shows the pressure differential force at the second-order resonance peak. It can 1 10 100 be seen that the pressure differential force is a complete sine curve, and there is no local distortion, which indicates the nonlinearity of the damper at the second-order resonance peak is not obvious. The reason is that the damper has huge hydraulic stiffness in the high-frequency domain, which will hinder the movement of the damping piston, and resulting in a significant decrease of damping piston displacement. From Table 4, it can be seen that the displacement of the second-order resonance peak is only um, so the nonlinearity of the damper is not obvious at this time. Fig.14 shows the pressure differential force at the first-order resonance peak for different damping orifice lengths.
It can be seen from the local enlarged diagram that the curve exists distortion, indicating that there exist nonlinear components at this time. However, with the increase of orifice length, the nonlinearity of the damper decreases. Fig.15 shows the pressure differential force at the second-order resonance peak for the different orifice lengths. It can be seen from the figure that the curve is a complete sine curve, which indicates there is no nonlinear component in pressure differential force at this time. Fig.16 shows the pressure differential force at the first resonance peak for different viscosities. It can be seen from the local enlarged diagram that the curve is not a complete sine curve at this time, indicating that there is also a nonlinear component in pressure differential force.     .18 shows the hysteresis loop at the second-order resonance peak. It can be seen that the hysteresis loops of the two models are nearly the same, which is different from the hysteresis loop at the first-order resonance peak. That means the nonlinearity at the second-order resonance peak is not obvious. The main reason is that at the second-order resonance peak, the displacement amplitude of damping piston is only um level, the velocity of the fluid in the damping orifice will be smaller, which leads to a small nonlinear term of velocity in the corrected hydraulic resistance, so the nonlinearity of the damper is not obvious. Table 7 shows the energy consumption of the damper at the first-order and secondorder resonance peaks. It can be seen from the table that the energy consumption calculated by the nonlinear model at the first resonant peak is slightly larger than that calculated by the linear model, which indicates that the nonlinear characteristics can

Energy consumption
The energy consumption of the damper can reflect the performance of suppressing the resonance peak. In this section, the energy consumption of the damper is analyzed at different damping orifice diameters, lengths, and viscosities. The simulation parameters are shown in  This is mainly because when the resonance peak is large, the movement of the damping piston will be strong, and the nonlinear velocity term in the hydraulic resistance can not be ignored, more vibration energy will be consumed by nonlinear damping. However, the reason is opposite when has a small resonance peak. Besides, it can be seen from Fig.20 that when the orifice lengths are 60 mm and 80 mm, the second-order resonance peak appears, but the energy consumptions of the nonlinear model and linear model are nearly the same at the second resonance peak. It mainly because the displacement amplitude of the damping piston is very small at the second-order resonance peak, which leads to a small nonlinear velocity term in the hydraulic resistance, it makes the nonlinear model and linear model tend to be consistent.  But in the high-frequency domain, the energy consumption of nonlinear damping is slightly larger than linear damping. When viscosities are 200 cst, 600 cst, and 1000 cst, the energy consumption of nonlinear damping and linear damping is nearly the same both at the resonance peak and high-frequency domain. It can be concluded that the energy consumption of nonlinear damping at the first-order resonance peak is larger than that of linear damping at a low viscosity. This is mainly because of the high flow velocity in the damping orifice at low viscosity, which leads to large nonlinear velocity terms and nonlinear damping. When the viscosity is high, the velocity of the fluid in the damping orifice is relatively slow, which leads to a small nonlinear velocity term.
At this time, the energy consumption of nonlinear damping is nearly the same as linear damping.

Conclusion
In this paper, a nonlinear model of a three-parameter fluid viscous damper considering the principal stiffness and hydraulic stiffness is proposed. Based on this model, the displacement response and the hydraulic resistance correction coefficient of the damper are analyzed. Besides, the energy consumption of nonlinear damping and linear damping at different damping orifice diameters, lengths, and viscosities are compared. The results show that there exist first-order and second-order resonance peaks when considering the principal stiffness and hydraulic stiffness. Near the resonance peak, the correction coefficient of hydraulic resistance is greater than 1, while in the high-frequency domain, the correction coefficient of hydraulic resistance is almost 1. This means that the nonlinearity is obvious near the resonance peak.
Besides, in the region near the first-order resonance peak, the energy consumption of nonlinear damping is greater than that of linear damping. The larger the resonance peak is, the more obvious this phenomenon is. However, in the high-frequency domain, the energy consumption of nonlinear damping is almost the same as linear damping. It can be concluded that the resonance peak of the damper can be suppressed by increasing the nonlinear damping at the resonance peak without changing the isolation performance in the high-frequency domain. By choosing reasonable damping orifice parameters and fluid viscosities, suppression performance of resonance peak will be better. Besides, we have found that the energy consumption of nonlinear damping at the resonance peak is better than that of linear damping, but this advantage is not very obvious. In the future, through further research, the design parameters of the damper can be optimized, and the nonlinear stiffness and damping can be reasonably selected to achieve more ideal suppression performance at the resonance peak.