## Natural Frequencies and Modes of Vibration
### Modes of Vibration Due to Elastic Deformability
\(L\) is the Lagrange's function (which is the difference between kinetic and potential energy). The components of the velocity of the system's center of mass can be calculated by neglecting second order terms:  
\[(L_{v}+s_{h})\cos\Bigl{(}\theta_{h}+\frac{\pi}{2}\Bigr{)}+L_{h}cos(\varphi)- \Bigl{(}L_{v}+s_{f}\Bigr{)}\cos\Bigl{(}\theta_{f}+\frac{\pi}{2}\Bigr{)}-L_{h}=0 \tag{1}\]2  
The kinetic energy according to [PERSON]'s theorem is:  
\[E_{k}=\frac{1}{2}m_{top}\  
The expansion of the [PERSON]'s equations leads to the equations of the free undamped vibrations of the robot in matrix form:  
\[\left[ \begin{array}{ccc}L_{\sigma}^{2}m_{top}&0&0\\ 0&m_{top}&0\\ 0&0&I_{top}\end{array} \right]\left\{ \begin{array}{ccc}\hat{\theta}\\ \bar{\hat{\sigma}}\\ \bar{\hat{\sigma}}\end{array} \right\}+\left[ \begin{array}{ccc}2k_{\theta}-m_ {tot}gL_{\sigma}&0&-2k_{\theta}\\ 0&2k_{\theta}&0\\ -2k_{\theta}&0&2k_{\theta}+0.5L_{h}^{2}k_{s}\end{array} \right]\left\{ \begin{array} []{c}\theta\\ s\\ \varphi\end{array} \right\}=\left\{ \begin{array}{c}0\\ m_{tot}g\\ 0\end{array} \right\} \tag{15}\]  
The analysis of these equations shows that the translational vibration of the top mass along the direction of the legs, which is due to axial compliance, is completely uncoupled from the rotational vibrations involving coordinates \(\theta\) and \(\varphi\). There is a coupling between rotation \(\theta\) (torsion rotation of the linkage) and rotation \(\varphi\) (pitch rotation of the spine). Actually, with the values of \(k_{s}\) and \(k_{\theta}\) corresponding to straight legs, the stiffness of the legs (\(0.5\ L_{h}^{2}k_{s}\)) that opposes to the pitch rotation of the spine (\(\varphi\)) is very large and rotation \(\varphi\) is very small. Therefore, the modes of vibration corresponding to the torsion rotation of the linkage (\(\theta\)) and pitch rotation of the spine (\(\varphi\)) are nearly uncoupled. Figure 3 shows the modes of vibration of the robot walking on the ground. Since the three modes are nearly uncoupled, simplified equations can be used to estimate the natural frequencies of the vibrational modes:  
\[\omega_{\theta}=\sqrt{\frac{2k_{\theta}-m_{top}gL_{\sigma}}{L_{\sigma}^{2}m_{ top}}} \tag{16}\]  
\[\omega_{s}=\sqrt{\frac{2k_{L}}{m_{top}}} \tag{17}\]  
\[\omega_{\varphi}=\sqrt{\frac{0.5L_{h}^{2}k_{s}}{I_{top}}} \tag{18}\]  
Figure 3: Modes of vibration on the ground: (**a**) twist mode, (**b**) hopping mode, and (**c**) pitch mode. The equations for the torsion and heave frequencies are equal to the ones reported in Reference [25]. With the mass and stiffness properties of the present robot (\(k_{\theta}\) = 0.17 Nm/rad and \(k_{L}\) = 4.3\(\times\)10\({}^{6}\) N/m), the difference between exact and approximate values of \(\omega_{\theta}\) and \(\omega_{\varphi}\) is less than 0.1%. The natural frequencies of the three modes are: \(\omega_{\theta}\) = 108.3 rad/s, \(\omega_{s}\) = 38,487.3 rad/s, and \(\omega_{\varphi}\) = 72,654.4 rad/s. Since the torsion mode has the lowest natural frequency, the eccentric mass can excite this mode more easily than the other modes.