## 3 Results and Discussion
### Analysis and Design of Optimal Compensate Criterion
'Simulation' in 'Matlab' was used here for exact residual vibration under different combinations of the frequency ratio and amplitude ratio. For the system of Equation (6), we set its natural frequency and damping ratio as \(\omega_{sp}=\sqrt{0.9}\) and \(\zeta_{sp}=0.01/\omega_{sp}\), where the subscript'sp' means mass-spring system. The amplitude value of sinusoidal disturbance \(A\) is set to be 1. In this way, only amplitude ratio and frequency ratio are required. It should be noted that the parameter used here is idealized. The detailed relevance of criterion design and damping will not be considered too much here. We may study those problems in the future. The compensation effect may not notable or excessive if the two ratios are too small or too big. The range of the amplitude ratio is selected from 0.3 to 1 while the range of the frequency ratio is selected from 0.3 to 3. The tip vibration is chosen to be the standard to show the effectiveness of disturbance torque suppression. Simulation results are exhibited in Figures 2-5.  
the same frequency \(\omega_{f}\). Resonance appeared when \(\lambda=1\). Another kind of excitation is the remaining harmonics of square force. Frequency of the second order harmonic almost meets the fundamental natural frequency of system when \(\lambda=0.3\). As a result, resonance will appear again in the frequency ratio-domain. For each frequency ratio, there is always an amplitude ratio where the residual vibration reaches the minimum. For most frequency ratios, the best compensation effect appeared near the amplitude ratio \(\beta=0.7\). The result for frequency ratio \(\lambda=1\) is a typical case. For every frequency ratio, the fundamental harmonic of the square force compensates the sinusoidal force near \(\beta=0.7\). Those facts illustrated that \(\beta=0.7\) is the optimal option for most feasible frequency ratios. Figure 3a,b provides the accelerations with different combinations of the amplitude and frequency ratio. From the results, we can find that the variation laws are the same as those for the displacement results. Resonances are obvious at \(\lambda=1\) and \(\lambda=0.3\). Beyond that, values vary homogeneously at different frequency ratios. The condition under which residual vibrations always have a minimum value for each frequency ratio is also distinct, especially for frequency ratios larger than 1. The minimum values are at about the centre of the amplitude ratio range. To show the results more clearly, Figures 4 and 5 show the comparisons between different ratios. Figure 3: Acceleration amplitude results in two forms: (**a**) In surface representation; (**b**) In marker representation. Figure 2: Displacement amplitude results in two forms: (**a**) In surface representation; (**b**) In marker representation. Figure 4a,b are parts of the results shown for the frequency ratio-domain. Four equally distributed amplitude ratios are selected. Data in two figures have the same variation laws. Situations when amplitude ratio \(\beta=0.7\) are the best as their vibrations are the smallest for almost all the frequency ratios. That is because the sinusoidal disturbance is properly counteracted by the fundamental harmonic of the step force. Resonance is conspicuous at frequency ratio \(\lambda=0.3\) and \(\lambda=1\). Figure 5a,b are parts of the results shown for the amplitude ratio-domain. Each curve has its own minimum, but not all the minimums are located near \(\beta=0.7\). For frequency ratios greater than 1, the minimum appeared near \(\beta=0.7\). For frequency ratios less than 1, the minimum amplitude ratios are not fixed. Floating windows in Figure 5a,b illustrate that for some frequency ratios, their minimum may appear near \(\beta=0.2\). However, the corresponding frequency ratios do not frequently occur in practical engineering. Results indicate that there is always an optimal combination of the amplitude and frequency ratio for disturbance compensation.