# Dynamics Analysis of a Variable Stiffness Tuned Mass Damper Enhanced by an Inerter
## 2 Modeling  
A primary structure with a variable stiffness tuned mass damper and an inerter (VS-TMD-I) is shown in Figure 1. The primary structure is a whole-spacecraft system consisting of a satellite and an adaptor. It is simplified as a two degrees of freedom (DOF) model that has been verified in the previous experiments [42]. The satellite is simplified as mass \(m_{1}\), linear stiffness \(k_{1}\), and damping \(c_{1}\), and the adaptor is simplified as mass \(m_{2}\), linear stiffness \(k_{2}\), and damping \(c_{2}\). The proposed TMD is composed of a positive stiffness spring \(k_{3}\), a flexion steel plate exhibiting nonlinear stiffness \(k_{4}\), a damper \(c_{4}\), and an inerter. The mass and the inertance of the inerter are \(m_{\rm b}\) and \(b\), respectively. Generally, the inertance of an inerter is 60-240 times its mass [31]. In this work, the ratio is selected as 100, namely \(b=100\)\(m_{\rm b}\). The nonlinear stiffness of the flexion steel plate is induced from the buckling effect and thus can be negative. The stiffness can be adjusted by changing the length of the plate. A piezoelectric actuator is adopted to press the plate and change its length. The piezoelectric actuator has the advantages of large driving force and fast response. It is regarded as a strain-induced actuator, and the relation between the output strain and the input voltage is assumed to be linear. A PI controller is selected to control the actuator. The displacement of the satellite (\(m_{1}\)) is detected by a sensor, as the input signal of the controller. Voltage is computed by the controller and sent to the actuator. Displacement excitations are applied on the base, denoted by \(x_{\rm e}\). The whole-spacecraft mainly experiences harmonic excitations induced by the structural vibration and impulse excitations during launch. Thus, the displacement responses of the satellite (\(x_{1}\)) and the adaptor (\(x_{2}\)) are investigated under harmonic and impulse excitations. The governing equations of the equivalent model with the VS-TMD-I can be written as:  
\[ \begin{array}{l}m_{1}\ddot{x}_{1}+k_{1}(x_{1}-x_{2})+c_{1}\big{(}\dot{x}_{1}- \dot{x}_{2}\big{)}\ =\ 0\\ m_{2}\ddot{x}_{2}+k_{1}(x_{2}-x_{1})+c_{1}\big{(}\dot{x}_{2}-\dot{x}_{1}\big{)} +k_{2}(x_{2}-x_{\rm e})+c_{2}\big{(}\dot{x}_{2}-\dot{x}_{\rm e}\big{)}+c_{3} \big{(}\dot{x}_{2}-\dot{x}_{3}\big{)}\\ +\Big{(}k_{3}-\frac{2k_{4}}{L}X_{0}\Big{)}(x_{2}-x_{3})\ =\ 0\\ 100m_{\rm b}\big{(}\dot{x}_{3}-\ddot{x}_{\rm e}\big{)}+c_{3}\big{(}\dot{x}_{3}- \dot{x}_{2}\big{)}+\Big{(}k_{3}-\frac{2k_{4}}{L}\Big{)}\big{(}x_{3}-x_{2}\big{)} \ =\ 0\\ \end{array}  \tag{1}\]  
where \(x_{1}\), \(x_{2}\), and \(x_{3}\) express the displacements of \(m_{1}\), \(m_{2}\), and \(m_{\rm b}\), respectively. \(X_{0}\) is the output displacement of the piezoelectric actuator. The output displacement is assumed to be linear to the input voltage \(U\), and thus  
\[X_{0}\ =\ k_{\rm T}U \tag{2}\]  
Figure 1: Equivalent model of the primary structure with VS-TMD-I. where \(k_{\rm T}\) is the linear coefficient dependent from the piezoelectric material. The input voltage is decided by the PI controller, namely \(U~{}=~{}Px_{1}+i\dot{x}_{1}\). \(P\) and \(I\) are the coefficients of the controller. The displacement excitation can be expressed as \(x_{\rm e}\) and  
\[x_{\rm e}~{}=~{}f_{\rm e}\cos(\omega t) \tag{3}\]  
where \(t\) is the time. \(f_{\rm e}\) and \(\omega\) are the amplitude and the frequency of the displacement excitation, respectively. The dimensionless parameters are introduced as:  
\[