# Simulation Analysis and Experimental Verification of the
## 3 Principle  
A simplified model is used in the analysis of structural dynamics. The MSHML is regarded as the boundary, the MP is equivalent to the concentrated mass, and the system is simplified to a single-degree-of-freedom vibration system. Its response is the forced vibration under the basic excitation [15]. A damped forced vibration mechanical model is adopted to establish the dynamic model as shown in Figure 3.  
Preload is applied on the MP, the static equilibrium position is as the origin and the right direction is as positive. The amplitude is \(B\), the vibration frequency is \(\omega\), the mass of  
Figure 3: Equivalent model of vibration system. Figure 2: The structure of the MSHML. the MP is \(m\), the displacement excitation changed by time is \(x_{f}(t)\), and the vibration rule is expressed as:  
\[x_{f}(t)=Be^{i\omega t} \tag{1}\]  
The harmonic excitation force can be written as:  
\[-m\ddot{x}_{f}=mB\omega^{2}e^{i\omega t} \tag{2}\]  
The displacement of the MP relative to the cabinet is \(x_{1}\), the damping ratio is \(\zeta\), and the natural frequency is \(\omega_{1}\), so the differential equation of motion is shown in Equation (3). \[\ddot{x}_{1}+2\zeta\omega_{0}\dot{x}_{1}+\omega_{0}^{2}x_{1}=B\omega^{2}e^{i \omega t} \tag{3}\]  
The relative motion amplitude is \(A_{1}\), phase is \(\theta_{1}\) and amplitude amplification factor is \(\beta_{1}=A_{1}/B\), and Equation (4) can be obtained. \[x_{1}=\beta Be^{i(\omega t-\theta_{1})} \tag{4}\]  
The frequency ratio is \(s=\omega/\omega_{0}\), the relation of the parameters \(\beta_{1}\) and \(\theta_{1}\) between the excitation frequency is given below:  
\[\beta_{1}(s)=\frac{s^{2}}{\sqrt{\left(1-s^{2}\right)^{2}+\left(2\zeta s \right)^{2}}} \tag{5}\]  
\[\theta_{1}(s)=\arctan(\frac{2\zeta s}{1-s^{2}}) \tag{6}\]  
The absolute displacement \(x\) of the MP is equal to the sum of the relative displacement of \(x_{1}\) and the displacement of the base \(x_{f}\). \[x=\beta_{1}Be^{i(\omega t-\theta_{1})}+Be^{i\omega t}=(\beta_{1}+e^{i\theta_{ 1}})Be^{i(\omega t-\theta_{1})} \tag{7}\]  
By using Equations (5) and (6), it can be deduced:  
Figure 3:0  
Let the absolute amplitude be \(A\), and substitute Equation (9) into Equation (8) to get:  
Figure 3:1  
where the phase difference q is as below:  
Figure 3:2  
The simple harmonic inertia force generated by the MP can be written:  
Figure 3:3  
The displacement response \(x\) in the vibration process can be obtained by differential Equation (9). According to Equation (11), the inertial force \(F\) in the vibration process can be solved. The preload on the left side is \(F_{1}\), and the preload on the right side is \(F_{2}\), and in the static state, \(F_{1}=F_{2}\). In the first \(1/4\) period of sinusoidal vibration, the MSHML starts to move to the right side. Due to the influence of inertia, the MP pressure is on the left side of the cabinet, at this time \(F_{1}=F_{2}+F\). As the vibration continues, the MP will continue to pressure on the left,so \(F_{1}\) will increase and \(F_{2}\) will decrease. If the preload \(F_{2}\) is too small, the situation of \(F_{1}=0\) will occur, and at this time it is in the critical state of contact. Under the action of inertia, the MP will continue to move to the left, pressing the left beam, \(F_{2}\) will continue to increase, and it will cause the separation between the right locking screw and the MP. When the sinusoidal vibration in the first stage of two quarters to three quarters of a period, displacement starts to move in the opposite direction, producing the opposite acceleration, and \(F_{2}\) decreases gradually. When the locking device on the right side starts to contact with the MP, there will be impact generated, and as pulse signal \(F_{1}\) will increase gradually. Similarly, the effect of \(F_{2}=0\) and the impact of the next contact must occur during continuous exercise. With the reciprocating vibration, the periodic impact response will occur between the MP and the lead screw, which will affect the reliability of the product.