# Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance
## 3 Natural Vibrations of a Linear System  
Occasionally, publications on robot vibrations present resonance curves that show the dependence of the resonance frequencies on the position of the robot arm [31; 32]. They are prepared on the basis of modal experiments in which the vibration is excited by an impact hammer pulse. The robot in the selected position is treated as a linear object; thus, bifurcation phenomena are not analysed. In order to be able to compare nonlinear models with such basic analyses, simplified robot motion equations were first considered. The influence of nonlinear elements, damping and excitation was ignored in order to determine the natural frequencies of the arm in selected configurations. Therefore, taking into account the above-mentioned simplifications, Equation (3) was reduced to the following form:  
\[\mathbf{M}(\mathbf{\alpha},\mathbf{\beta})\tilde{\mathbf{\alpha}}+\mathbf{K}\mathbf{\alpha}=0. \tag{28}\]  
as indicated in [30], the natural frequencies of the arm can be determined from the following equation:  
\[det\left(-\omega_{0}^{2}M_{0}+\mathbf{K}\right)=0 \tag{29}\]  
where \(\omega_{0}\)--natural frequency of the robot; \(\mathbf{M}_{0}=\mathbf{M}(\mathbf{\alpha}=\mathbf{0},\mathbf{\beta})\)--an inertia matrix in which the influence of the \(\alpha_{i}\) angles is neglected and which is constant at a given position of the  
\begin{table}
\begin{tabular}{c c c} \hline
**Coefficient** & **Unit** & **Value** \\ \hline \(I_{A}^{(1)}\) & kg\(\cdot\)m\({}^{2}\) & 0.55 \\ \(I_{S2}^{(2)}\) & kg\(\cdot\)m\({}^{2}\) & 0.0009 \\ \(m_{2}\) & kg & 50.58 \\ \(l_{1}\) & m & 0.63 \\ \(l_{2}\) & m & 0.1 \\ \(k_{1}\) & N\(\cdot\)m/rad & 555,000 \\ \(k_{2}\) & N\(\cdot\)m/rad & 128,000 \\ \(d_{1}\) & N\(\cdot\)m\(\cdot\)s/rad & 0.9 \\ \(d_{2}\) & N\(\cdot\)m\(\cdot\)s/rad & 0.9 \\ \hline \end{tabular}
\end{table}
Table 1: Coefficient values used in robot model.  
arm defined by the coordinate vector \(\mathbf{\beta}\). Taking into account the forms of the inertia and stiffness matrices defined by Equations (19) and (22), the following two natural frequencies (\(\omega_{0\{1\}}\) and \(\omega_{0\{2\}}\)) of the robot arm were determined:  
\[\omega_{0\{1\}}=\sqrt{\frac{\left(a_{1}k_{2}+2a_{2}k_{2}\cos\beta_{2}+a_{3}k_{1} \right)\mp\sqrt{\left(a_{1}k_{2}+2a_{2}k_{2}\cos\beta_{2}+a_{3}k_{1}\right)^{2 }-4\left(a_{1}a_{3}-a_{3}^{2}-a_{2}^{2}\cos^{2}\beta_{2}\right)k_{1}k_{2}}}{2 \left(a_{1}a_{3}-a_{3}^{2}-a_{2}^{2}\cos^{2}\beta_{2}\right)}}. \tag{30}\]  
it is clear from solution (30) that the natural frequencies depend on the position of link 2 relative to link 1, expressed by the angle \(\beta_{2}\). The dependence of the natural frequencies on the angle \(\beta_{2}\) is shown in Figure 3. In addition, the natural frequencies are expressed in Hz using the transformation \(f_{0\{1,2\}}=\omega_{0\{1,2\}}/(2\pi)\).  
The natural frequencies of the robot arm vibrations depend strongly on its position, and the changes are symmetrical with respect to the so-called straight arm configuration (\(\beta_{2}=0\)). The simulation results are in agreement with the results of the experimental research presented in [30]. On this basis, it can be assumed that the mathematical model has suitable properties for simulation studies of vibration phenomena.