# Vibration Isolation of a Surveillance System Equipped in a Drone with Mode Decoupling
## 2 Theories for System Identification
### Mathematical Model of a Multi-DOF System  
The mechanical response of a multi-DOF system consists of responses in each single-DOF coordinate. This study assumes the mechanical system to be a holonomic system, to describe the behavior of vibration with a superposition principle. The EOM of a multi-DOF system can be derived using [PERSON]'s method [14]. [PERSON]'s equation for a viscous damped system is defined as follows:  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{k}}\right)-\frac{\partial L }{\partial q_{k}}+\frac{\partial R}{\partial\dot{q}_{k}}=0,\;k=1,2,\ldots,n, \tag{1}\]  
where \(L\), \(R\), and \(q_{k}\) denote the Lagrangian, the Rayleigh's dissipation function, and generalized coordinates, respectively. The Lagrangian of \(L\) is defined as \(T-V\), where \(T\) and \(V\) denote the kinetic and potential energy, respectively. The mechanical systems can be modeled by exciting the payload with the motion of a drone (base). For the sake of brevity, a two-DOF system is used in this study to explain the coupling characteristics and its EOM instead of a multi-DOF system. A two-DOF problem can be modeled in a base-excited system, as shown in Figure 1. The kinetic energy, potential energy, and Rayleigh's dissipation can be expressed as  
\[T=\frac{1}{2}{m\dot{q}_{1}}^{2}+\frac{1}{2}{m\dot{q}_{2}}^{2}. \tag{2}\]  
\[V=\frac{1}{2}{k_{11}}({q_{1}}-{q_{1b}})^{2}+\frac{1}{2}{k_{12}}({q_{1}}-{q_{1b }})({q_{2}}-{q_{2b}})+\frac{1}{2}{k_{21}}({q_{2}}-{q_{2b}})({q_{1}}-{q_{1b}})+ \frac{1}{2}{k_{22}}({q_{2}}-{q_{2b}})^{2}. \tag{3}\]  
\[R=\frac{1}{2}{c_{11}}({\dot{q}_{1}}-{\dot{q}_{1b}})^{2}+\frac{1}{2}{c_{12}}({ \dot{q}_{1}}-{\dot{q}_{1b}})({\dot{q}_{2}}-{\dot{q}_{2b}})+\frac{1}{2}{c_{21}} ({\dot{q}_{2}}-{\dot{q}_{2b}})({\dot{q}_{1}}-{\dot{q}_{1b}})+\frac{1}{2}{c_{22 }}({\dot{q}_{2}}-{\dot{q}_{2b}})^{2}, \tag{4}\]  
where \(m\) denotes the mass of the payload, \(k_{ij}\) is the stiffness in the \(ij\)-coordinate, \(c_{ij}\) is the damping coefficient in \(ij\)-coordinate, \(q_{r}\) is the r-coordinate directional motion of the payload, and \(q_{rb}\) is the r-coordinate directional motion of a base. Equations (2)-(4) are substituted into Equation (1) to derive the EOM as follows:  
\[-\frac{\delta L}{\delta q_{1}}=\ k_{11}({q_{1}}-{q_{1b}})+\frac{1}{2}{k_{12}} ({q_{2}}-{q_{2b}})+\frac{1}{2}{k_{21}}({q_{2}}-{q_{2b}}), \tag{5}\]  
\[\frac{\delta L}{\delta{\dot{q}_{1}}}={m\dot{q}_{1}}, \tag{6}\]  
\[\frac{d}{dt}\left(\frac{\delta L}{\delta{\dot{q}_{1}}}\right)={m\ddot{q}_{1}}, \tag{7}\]  
\[\frac{\partial R}{\partial{\dot{q}_{1}}}={c_{11}}({\dot{q}_{1}}-{\dot{q}_{1b} })+\frac{1}{2}{c_{12}}({\dot{q}_{2}}-{\dot{q}_{2b}})+\frac{1}{2}{c_{21}}({ \dot{q}_{2}}-{\dot{q}_{2b}}), \tag{8}\]  
\[{m\ddot{q}_{1}}+{c_{11}}({\dot{q}_{1}}-{\dot{q}_{1b}})+{k_{11}}({q_{1}}-{q_{1b }})+\left(\frac{1}{2}{c_{12}}+\frac{1}{2}{c_{21}}\right)({\dot{q}_{2}}-{\dot{q }_{2b}})+\left(\frac{1}{2}{k_{12}}+\frac{1}{2}{k_{21}}\right)({q_{2}}-{q_{2b} })=0. \tag{9}\]  
Additionally, the EOM at a generalized coordinate, \(q_{2}\), can be derived as,  
Figure 1.0  
The stiffness term is assumed as follows:  
Figure 1.1  
Figure 1.2 A free body diagram of (**a**) general and (**b**) coupled two-degree-of-freedom system. Coordinates are mutually independent in a general two-DOF system (Figure 1a). Therefore, \(k_{ij}=c_{ij}=0\), if \(i\
eq j\). Hence, the EOM for a general two-DOF system is derived as  
\[\left[ \begin{array}{cc}m&0\\ 0&m\end{array} \right]\left[ \begin{array}{cc}\ddot{q}_{1}\\ \ddot{q}_{2}\end{array} \right]+\left[ \begin{array}{cc}c_{11}&0\\ 0&c_{22}\end{array} \right]\left[ \begin{array}{cc}\dot{q}_{1}\\ \dot{q}_{2}\end{array} \right]+\left[ \begin{array}{cc}k_{11}&0\\ 0&k_{22}\end{array} \right]\left[ \begin{array}{cc}q_{1}\\ q_{2}\end{array} \right]=\left[ \begin{array}{cc}c_{11}&0\\ 0&c_{22}\end{array} \right]\left[ \begin{array}{cc}\dot{q}_{1b}\\ 0&k_{22}\end{array} \right]\left[ \begin{array}{cc}q_{1b}\\ q_{2b}\end{array} \right]. \tag{12}\]  
It implies that each coordinate is decoupled--independent--from the other.