## 3 Mathematical Model of Linear Oscillating Loading System  
For the LOM is of excellent symmetry and linearity, it is available to assume that the LOM is of ideal relationship between output thrust force and input current. Hence, the electromagnetic force of the LOM can be written as:  
\[F_{\mathrm{e1}}=K_{\mathrm{e1}}i_{\mathrm{1}} \tag{1}\]  
where \(F_{\mathrm{e1}}\) is the electromotive force of the LOM to be tested, \(K_{\mathrm{e1}}\) is the force constant coefficient of the LOM, \(i_{\mathrm{1}}\) is input current.  
Analogously, the electromagnetic force function of the VCM can be written as:  
\[F_{\mathrm{e2}}=K_{\mathrm{e2}}i_{\mathrm{2}} \tag{2}\]  
where \(F_{\mathrm{e2}}\) is the output thrust force regarded as load force of the VCM, \(K_{\mathrm{e2}}\) is the force constant coefficient of the VCM, and \(i_{\mathrm{2}}\) is the input current.  
Assume that movers are rigid bodies and ignore deformation of the force sensor, the dynamic equation of the LOLS, based on the [PERSON] second law, can be written as:  
\[F_{\mathrm{e1}}-F_{\mathrm{e2}}=k\mathrm{x}+(\xi_{1}+\xi_{2})\mathrm{sgn}( \frac{dx}{dt})+(m_{1}+m_{2})\frac{d^{2}x}{dt^{2}} \tag{3}\]  
where \(k\) is the spring stiffness of the LOM; \(m_{1}\) is the mover mass of the LOM; \(m_{2}\) is the otal mass of the VCM mover and the sensor mechanism; \(\xi_{1}\) is the kinetic friction force of the LOM, and \(\xi_{2}\) is the kinetic friction force of the VCM.  
Similarly, the dynamic equation of the LOM can be written as:  
\[F_{\mathrm{e1}}-F_{\mathrm{s1}}=kx+\xi_{1}\mathrm{sgn}(\frac{dx}{dt})+(m_{1} +m_{2})\frac{d^{2}x}{dt^{2}} \tag{4}\]  
where \(F_{\mathrm{s1}}\) is force magnitude between the force sensor and the LOM.  
Approximately, the dynamic equation of the load mechanism can be written as:  
\[F_{\mathrm{e2}}-F_{\mathrm{s2}}=\xi_{2}\mathrm{sgn}(\frac{dx}{dt})+m_{2}\frac {d^{2}x}{dt^{2}} \tag{5}\]  
Figure 11: Different modes of force load.  
where \(F_{\rm s1}\) and \(F_{\rm s2}\) are the coupled forces of action and reaction. Thus, the force equivalency equation, based on the Newton third law, can be written as:  
\[F_{\rm s1}=F_{\rm s2}=F_{\rm s} \tag{6}\]  
As for the LOM, voltage is composed of three parts, i.e., resistance voltage, inductance voltage and electromotive force (EMF) voltage, so the voltage balance equation of the LOM can be written as:  
\[U_{1}=i_{1}R_{1}+L_{1}\frac{di_{1}}{dt}+K_{\rm e1}\frac{dx}{dt} \tag{7}\]  
where \(U_{1}\) is the input voltage of the LOM, \(R_{1}\) is the total resistance accorded with series and parallel mode of the LOM coils. \(L_{1}\) is the total inductance accorded with \(R_{1}\).  
Similarly, the voltage balance equation of the VCM can be written as:  
\[U_{2}=i_{2}R_{2}+L_{2}\frac{di_{2}}{dt}+K_{\rm e2}\frac{dx}{dt} \tag{8}\]  
where \(U_{2}\) is the input voltage of the VCM, \(R_{2}\) is the total resistance accorded with parallel mode of the VCM coils. \(L_{2}\) is the total inductance accorded with \(R_{2}\).  
In this paper, three typical types of force loads are set to be achieved, i.e., spring force, damping force and rectangular force. For \(F_{e2}\) is regarded as the load force, the relationship between \(x\) and \(i_{2}\) should be given as follows:  
\[i_{2}=f(x,\frac{dx}{dt}) \tag{9}\]  
If simulation force is spring force, the relationship between \(x\) and \(i_{2}\) can be written as:  
\[f(x,\frac{dx}{dt})=K_{\rm s}\cdot x \tag{10}\]  
\(K_{\rm s}\) is the spring stiffness of the spring simulation force. If the simulation force is a damping force, the relationship between \(x\) and \(i_{2}\) can be written as:  
\[f(x,\frac{dx}{dt})=\kappa\cdot\frac{dx}{dt} \tag{11}\]  
\(\kappa\) is damping coefficient of the damping simulation force.  
If the simulation force is a rectangular force, the relationship between \(x\) and \(i_{2}\) can be written as:  
\[f(x,\frac{dx}{dt})=K_{\rm r}\cdot\text{sgn}(\frac{dx}{dt}) \tag{12}\]  
\(K_{\rm r}\) is the gain coefficient of the rectangular simulation force.