# Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance
## 4 Bifurcation Analysis of Forced Vibrations  
Firstly, the simulation results of the influence of the forcing frequency as a bifurcation parameter on the vibration characteristics are presented. The calculations were carried out using \"Matcont 7.4\" software [33; 34]. The research used a mathematical model given by Equation (3), taking into account Equations (18)-(25) and (27) and the parameters given in Table 1. The following forcing vector was used:  
\[\mathbf{u}= \begin{bmatrix}u_{e1}\sin(2\pi f_{e1}t)\\ 0\end{bmatrix} , \tag{31}\]  
where \(u_{e1}\)--forcing amplitude for link 1; \(f_{e1}\)--forcing frequency for link 1. During the tests, the forced vibrations of the system were analysed at a variable forcing frequency in the range of \(f_{e1}\in\langle 0,80\rangle\) Hz. As the direction of the change in the frequency value affects the behaviour of the system, tests were carried out for increasing and decreasing frequencies (so called forward and backward sweep). In this case, the vector of external forces \(F_{ext}\) is equal to 0. The tests were carried out for many arm configurations defined by  
Figure 3: Robot natural frequencies as a function of the \(\beta_{2}\) angle. the angle \(\beta_{2}\in\langle 0,150\rangle\) deg. The obtained amplitude-frequency characteristics of link 1 for selected positions of the robot arm are shown in Figure 4. Figure 4a shows the amplitude-frequency characteristics for the angle \(\beta_{2}\) of 0 rad (0 deg), 0.5236 rad (30 deg), 1.0472 rad (60 deg), 1.5708 rad (90 deg), 2.0944 rad (120 deg) and 2.6180 rad (150 deg), with areas of instability marked by a red line. The graphs show that, similar to the natural frequencies in a linear system (Section 3), in a nonlinear system the resonance zones move closer together as the absolute value of angle \(\beta_{2}\) increases. In addition, the phenomenon of the dependence of the vibration amplitude on the direction of the frequency change is visible, resulting in a slope of the characteristics in the resonance zones. For small \(\beta_{2}\) angles, a stiff resonance curve is obtained around the first natural frequency (Figure 4b), which for large \(\beta_{2}\) transits into a mixed softening-hardening curve. In contrast, for the resonance around the second natural frequency (Figure 4c), soft characteristics are obtained. As the angle \(\beta_{2}\) increases, the situation changes: the curves shift towards lower frequencies and become stiff characteristics. In the range of negative values of \(\beta_{2}\), the characteristics are identical to those for positive values; therefore, the results for the angle \(\beta_{2}<0\) are not presented. This is physically justified, as from the point of view of the behaviour of the dynamic system it does not matter whether the arm configuration is \"elbow up\" or \"elbow down\". In addition, the characteristics for link 2 are qualitatively identical to the characteristics for link 1 shown in Figure 4 and are therefore not included in this paper. The influence of the forcing amplitude as a bifurcation parameter on the vibration characteristics was also investigated. The tests used the forcing vector in the form of Equation (25) and assumed the forcing frequency \(f_{e1}=22.5\) Hz and a variable forcing amplitude. The tests were carried out for selected arm configurations defined by the \(\beta_{2}\) angle. Figure 5 shows selected results obtained for the selected forcing frequency and various arm positions. The characteristics show that the oscillations of the link can have different amplitudes depending on the forcing amplitude. The areas shown with red lines are areas of instability. In the next study, the characteristics of the system were determined in the case of a force applied to the end-effector of the robot. It was assumed that the end-effector takes successive positions on a straight line parallel to the \(x\) axis (Figure 6). A harmonic excitation was applied with a direction parallel to the \(x\) axis:  
\[F_{ext}= \begin{bmatrix}F_{ext\_x}\sin(2\pi f_{x}t)\\ 0\end{bmatrix}