Figures 2A,B are the ideal models for the conventional inerter and the SAI, respectively. The inertance and the output force of the conventional ball screw type inerter can be expressed as:

where b is the inertance of the inerter; F_b is the output force of the conventional inerter; J is the moment of inertia of the flywheel; p is the lead of the screw shaft of the ball screw; and x₁ and x₂ are the displacements at the two ends of the inerter.

According to equations (1) and (2), the ideal model of the conventional ball screw-type inerter is only related to the characteristics of the flywheel and the ball screw. The output force of the proposed SAI is dependent on the controllable force due to the viscosity change of the MR fluid. It can be expressed as:

where F is the output force of the SAI; F_MR is the controllable force due to the viscosity change of the MR fluid.

F_MR is expressed as:

where T_vis is the viscous torque produced by the MR fluid in field-off state (i.e., no applied current in the excitation coils) and T_MR is the field-dependent torque. T_vis and T_MR are respectively given by:

where η is the viscosity of the MR fluid in field-off state; τ_y is the shear yield stress of the MR fluid; R₁ is the outer radius of the flywheel; R₂ is the inner radius of the flywheel housing; n_d is the rotational speed of the flywheel; s_d is the width of the annular gap; L_d is the axial length of the flywheel; δ is the correction coefficient with consideration of the influence of the magnetic leakage and ξ is the coefficient of the effective area (i.e., the ratio of the effective area to the ideal area) (Bai et al., 2018).

The inertance is defined by the ratio of the output force of the inerter to the relative acceleration between its two ends. That is, the corresponding inertance can be obtained by the relationship between the output force and the relative acceleration of the inerter. The output force of the proposed SAI is controllable and related to the applied current. When under a certain relative acceleration, the desired inertance can be achieved by adjusting the applied current. The inertance obtained by adjusting the applied current is defined as the equivalent inertance b′ in this study.

Practically, the movement of the inerter should overcome a certain amount of inherent friction. Therefore, the output force F of the SAI in equation (3) and the equivalent inertance b′ can be rewritten as:

where f is the inherent friction force of the SAI and Δa is the relative acceleration at the two ends of the SAI.

The prototype of the SAI is fabricated to verify the objective mechanical properties. As shown in Figure 3, an experimental setup based on the servo hydraulic test equipment (Type: LFH-LFV3068, SAGINOMIYA Inc.) is established to test the mechanical properties of the SAI.

Figures 4A,B show the comparisons between the theoretical and the experimental results of the mechanical response of the SAI, when a 0.5 A excitation current and sinusoidal displacement excitations with a frequency of 0.25 Hz and the excitation amplitudes of 10 mm and 20 mm are applied, respectively. As shown in Figure 4A, the theoretical value of the output force of the SAI is larger than the experimental results. When the direction of the force changes, the experimental result presents a certain hysteresis and slight fluctuations. The generated hysteresis is due to the MR effect and the gap between the mechanical structures, including the gap between the screw and the nut and the gap between the SAI and the mounting fixture. The reason for the slight fluctuations is the elastic effect of the ball screw itself. Comparing Figures 4A,B, when the amplitudes of the displacement excitations are different, the output forces of the SAI are substantially the same, indicating that the amplitude of the displacement excitation exerts little influence on the output force of the SAI.

Figures 5A,B show the comparisons between the theoretical and the experimental results of the mechanical response of the SAI, when a 0.5 A current and sinusoidal displacement excitations with an amplitude of 20 mm and the different excitation frequencies of 0.5 Hz and 1.0 Hz are applied, respectively. As shown in Figure 5A, the theoretical value of the output force of the SAI is larger than the experimental value, but the peak experimental value is slightly larger than the theoretical one due to the elastic effect of the ball screw. Comparing Figures 5A,B, when the frequency of the displacement excitation is different, the output force of the SAI is almost constant, which indicates that the frequency of the displacement excitation has little effect on the output force of the SAI. However, the fluctuation phenomenon becomes more apparent as the frequency of the displacement excitation increases.

Figures 6A,B show the comparisons between the theoretical and the experimental results of the mechanical response of the SAI, when 0 A and 1.0 A currents and a sinusoidal displacement excitation with a frequency of 1.0 Hz and an amplitude of 20 mm are applied, respectively. As shown in Figure 6A, the theoretical value of the output force is basically the same as the change trend of the experimental result, but the fluctuation of the experimental result is obvious and covers almost the entire cycle of the motion. Comparing Figures 6A,B, the output force of the SAI is obviously affected by the applied current and increases with the increase of the applied current. Besides, the larger the applied current is, the larger the difference between the theoretical and the experimental results.

Based on the comparison of Figures 4–6, when under different amplitudes and frequencies of sinusoidal displacement excitations (i.e., the different excitation speeds), the maximal output force of the SAI is nearly unchanged, which is because of the pure shear working mode of the MR fluid in the SAI.

As shown in Figures 4–6, as the frequency of the sinusoidal displacement excitation increases, the fluctuation of the experimental result of the SAI increases, and the difference between the experimental and the theoretical values expands. Therefore, it can be deduced that the mechanical performance of the SAI is affected by nonlinear factors. The nonlinear factors affecting the output force of the SAI are mainly derived from the gap and the elastic effect of the ball screw and the inherent friction force, including the friction between the screw and the nut and the friction inside the bearing. The gap of the ball screw could be eliminated by preloading, so the effect of the gap will not be taken into account when establishing the nonlinear model. Figure 7 shows the nonlinear model of the SAI containing inherent friction force and the elastic effects of the ball screw (Wang and Su, 2008).

In the experimental tests, one end of the SAI is fixed, as shown in Figure 3, so x₃ = 0. According to Figure 7, the dynamic model of the SAI with nonlinearities is expressed by:

where k_s and c_s are the stiffness and damping of the ball screw, respectively.

The minimal inertance of the proposed SAI is set to 600 kg (i.e., the inherent inertance is 600 kg when the applied current is 0 A). When the SAI is excited by sinusoidal displacement excitations with low frequencies and no current is applied, the output force is dominated by the inherent friction force, and the inertia of the flywheel can be ignored. In this case, the output force of the SAI can be approximately regarded as the inherent friction force. In this study, a sinusoidal displacement excitation with a frequency of 0.1 Hz and an amplitude of 10 mm is applied to the SAI in field-off state and the inherent friction force f of the SAI is about 310 N.

Parameter identification of the nonlinear model of the SAI is conducted using experimental data under different operating conditions. The parameters k_s, c_s and F_MR are selected to be identified using the following criterion function with the least square method:

where f_th is the predicted value of the output force calculated by the nonlinear model of the SAI and f_exp is the experimental value of the output force of the SAI.

The stiffness k_s of the ball screw is 5,500 kN/m and the damping c_s is 7,000 Ns/m according to the parameter identification. The controllable force F_MR changes with the applied current, which is consistent with the actual situation. Figure 8 shows the comparison of the predicted value with the experimental value of the output force under the sinusoidal displacement excitation with an amplitude of 20 mm and a frequency of 1.0 Hz when the applied current is 0.5 A. It can be seen from Figure 8 that the predicted value of the output force agrees well with the experimental value. The effectiveness of the established nonlinear model of the SAI is therefore verified.

The established nonlinear model of the SAI can be used to achieve adjustment of the inertance. The SAI is separately subjected to an adjustment simulation of the 0–6,000 kg inertance under the condition of the lower and upper bound force (about 0–4 kN according to the test results) of the SAI. Figure 9 shows the simulation results under a sinusoidal displacement excitation with an amplitude of 20 mm and a frequency of 1.0 Hz when the expected inertance is from 0 to 6,000 kg. Since the friction factor must be considered, the inherent friction force is set to 0 N and 50 N, respectively. As shown in Figure 9, when the inherent friction force is 0 N, the adjustment of the equivalent inertance exhibits a segmented characteristic due to the existence of the phase difference between the velocity and the acceleration. The continuous adjustment of the equivalent inertance can be achieved when the controllable force works. When the controllable force fails, the equivalent inertance oscillates. In addition, both the output force and the controllable force of the SAI fluctuate at the corresponding positions, and the fluctuations are more intense as the equivalent inertance increases. When the friction force is 50 N, the fluctuation of the equivalent inertance is more obvious than that of the friction force of 0 N, but the influence on the controllable force and the output force is not very large.

The output force (inertance) of the SAI is related to the acceleration, while the controllable force is related to the velocity (according to Equations 4, 5). There is a 90° phase difference between the output force and the controllable force of the SAI, which may result in failure to achieve the desired inertance when adjusting the controllable force. However, the continuous adjustment of the equivalent inertance would be achieved if a force compensator (i.e., phase adjustment mechanism) could be provided.

The SAI with a compensator is proposed and shown in Figure 10. As shown in Figure 10, the compensator is composed of a semi-active damper in series with the rack pinion mechanism and in parallel with the SAI. Taking the case under sinusoidal displacement excitation as an example, the adjustment principle of the equivalent inertance is shown in Figure 11. It can be seen from the areas ① and ② in Figure 11 that the output force of the SAI cannot be continuously adjusted due to the phase difference between the relative velocity and the relative acceleration. The area ③ shows that the force generated by the compensator is always opposite to the direction of the controllable force, so that the controllable force that could not be achieved is compensated. As shown in the area ④, the output force of the SAI can be continuously adjusted to achieve continuous adjustment of the equivalent inertance.

The output force of the SAI with a compensator can be expressed as:

where F_com is the force generated by the compensator.

Figure 12 presents the simulation results under a sinusoidal displacement excitation with an amplitude of 20 mm and a frequency of 1.0 Hz when the expected inertance of the SAI with a compensator is from 0 to 6,000 kg. Since the friction could be offset by the compensator, the contrast under different inherent friction forces is not considered. It can be seen from Figure 12 that the continuous and steady adjustment of 0–6,000 kg of the equivalent inertance can be achieved. The controllable force, the force generated by the compensator and the output force produce fluctuations, and the fluctuations are more intense as the equivalent inertance increases. This is similar to the case whereby there is no compensator, as with Figure 9.

The equivalent inertance of the ball screw-type SAI, as shown in Figure 9, is segmented, and the inherent friction of the ball screw set will influence the controllability. To study the influence of the segmented equivalent inertance of the SAI and the inherent friction force on the vibration control, a single-degree-of-freedom vibration control system as shown in Figure 13 is established. The system model is given by:

where z₁ is the input displacement of the payload m; z₂ is the displacement of the payload m; k is the stiffness of the spring and c is the damping of the passive damper.

Figure 14 shows the acceleration response of the system under a sinusoidal displacement excitation with an amplitude of 50 mm and a frequency of 1.0 Hz (the resonant frequency of the passive spring-mass-damper system). The payload m, the stiffness k, and the damping c are 557 kg, 22 kN/m and 1,500 Ns/m, respectively. The inherent friction force is set as: f₁ = 310 N, f₂ = 110 N and f₃ = 10 N when the SAI has no compensator. As shown in Figure 14, the single-degree-of-freedom vibration system with the SAI has an over 4 times smaller acceleration amplitude than the conventional vibration system (i.e., the spring-mass-damper system). When the SAI is not connected to the compensator, the acceleration fluctuates due to the inherent friction force. The greater the inherent friction force, the more serious the fluctuation of the acceleration will be. However, the acceleration is still much smaller than the conventional vibration system. The acceleration of the payload of the system when the friction is 10 N basically agrees with the system with a compensator. In other words, the segmented equivalent inertance of the SAI will not influence the control performance of the SAI-based vibration control very much. This is similar to the analysis in reference (Wang and Su, 2008); the friction force of the inerter does have an adverse effect on the vibration control performance, but when the stiffness of the suspension is great, the overall performance of the system using the inerter with friction force is still superior to the conventional ones.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.