# New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots
## 4 Numerical Simulation
### Performance Evaluation for Percussive Riveting
The robot configurations corresponding to Points \(P_{1}\) to \(P_{4}\) are also plotted in Figure 10. To demonstrate the vibration of the robot joints due to the percussive impact force, the time and frequency responses of the tool displacement are analyzed for Point \(P_{1}\) when the percussive impact is along the \(-X\) direction (see Figure 10). The frequency response function for the tool displacement due to a force along the--X direction can be obtained from (7), and the magnitude of the frequency response function with different frequencies are plotted in Figure 11. If the dynamic loading is approximated as an expansion of Fourier series as (26). Then, the amplitude of the tool vibration depends on the Fourier coefficients and the frequency response function at the corresponding frequencies (\(\omega_{v}\), \(2\omega_{v}\), \(3\omega_{v}\) as shown in Figure 11). It can be seen that the first two Fourier series have larger influence on the tool vibration than the other Fourier series. In addition, although the riveting force is along the \(-X\) direction, the tool displacement in the \(Z\) direction has a larger amplitude than those in the other two directions. Furthermore, the time response of the tool displacement can be computed by solving (1) and (3). The resulting tool displacement along different directions are plotted in Figure 12 for five periods of the external force (about 0.5 s). Again, the tool displacement in the \(Z\) direction has the largest amplitude among the three directions. Figure 10: Potential operation workspace (in red). Figure 9: Periodic impact force: (**a**) time history; (**b**) Fourier coefficients  
Now, we proceed to analyze the potential risk of the tool vibration when the percussive force acts in different directions on the robot end-effector. Once the maximum displacement-force frequency response ratio \(\eta_{\text{vmax}}\) is given, then the maximum amplitude of TCP displacement due to the \(k\)th Fourier series of the periodic impact force \(\Delta E_{vmax,k}\) can be easily calculated using (24). For example, the values of \(\Delta E_{vmax,k}\) are plotted over the potential workspace when \(x\) is 0.5, 0.6, 0.7, 0.8, 0.9 and 1 m in Figure 13a,b, for \(k=\pm\) 1 and \(\pm\) 2 respectively. It can be found that the distributions of \(\Delta E_{vmax,k}\) are symmetric about the \(\alpha xz\) plane. When only considering the Fourier series with \(k=\pm 1\), the amplitude of the TCP displacement can be as high as \(2\times 10^{-3}\) m in certain positions where the first natural frequency of the robot (related to deflections of Joint 2 and Joint 3) is close to \(\omega_{v}\). When only considering the Fourier series with \(k=\pm 2\), the amplitude of the TCP displacement can be as high as \(2.3\times 10^{-3}\) m in the positions where the second natural frequency (related to the deflection of Joint 1) is close to \(2\omega_{v}\). Figure 11: Frequency response function of tool displacement at \(P_{1}\) under a force along—X direction. Figure 12: Time responses of tool displacement at \(P_{1}\) under a percussive force along—X direction. It is also observed that when \(k>4\), \(k\omega_{p}\) is beyond the robot's natural frequencies, and the values of \(\Delta E_{vmax,k}\) are much smaller than those for \(k=\pm 1\), \(\pm 2\), \(\pm 3\) and \(\pm 4\). Figure 13: Maximum amplitudes of TCP displacement due to the \(k\)th Fourier series: (**a**) \(k=\pm 1\); (**b**) \(k=\pm 2\). A detailed investigation on the eigenvectors of matrix \(A_{v}(k\omega_{v})\) can give force components under which the maximum amplitude of TCP displacement due to the \(k\)th Fourier series occurs. The evaluation of \(\Delta E_{vmax,k}\) calculated from (24) is conservative in this case, since the three force components in the riveting force (25) have identical phases which can be insufficient to meet the requirement for the phases of the components in the eigenvectors of \(A_{v}(k\omega_{v})\).