## 7 Calculation Example  
Based on the actual crane operating conditions, the simulation experiment parameters are given: \(m_{1}=1000\) kg, \(m_{2}=1000-10,000\) kg (including the spreader), \(l=1-8\)\(m\), and \(\varphi_{0}=5^{\circ}\). By calculation, the system frequency is \(\omega_{n}=4.6-14.7\) Hz. Based on Equation (11), as the length \(l\) of the rope decreases, the system frequency will increase when the weight is lifted. So, the system frequency can avoid the frequency of natural wind. To illustrate the effectiveness of the linear simplification involved in moving from Equation (8) to Equation (9), simulation verification results are presented in Figure 5. Figure 5 shows the dynamic response process for the load swaying angle when there is no hydraulic anti-swaying system (\(k=0\)), and Figure 5 shows the dynamic response process for the load swaying angle when the hydraulic anti-swaying system is implemented (\(k\
eq 0\)). Figure 5 indicates that the linear simplification is reasonable and effective. Figure 5: Dynamic response process for the load swaying angle. (**a**) \(k=0\). (**b**) \(k\
eq 0\). Based on Equations (9) and (10), when \(M\) and \(l\) are known, \(k\) significantly influences the attenuation of the swaying angle, as shown in Figure 6. Figure 6 shows that a larger \(k\) value is conducive to the attenuation of the swaying angle. However, according to Equation (7), a larger \(k\) value means that the hydraulic system would provide a larger oil pressure. Figure 7 shows the influence of the damping coefficient of the under-damping hydraulic anti-swaying system on the attenuation of the swaying angle. Equation (15) and Figure 7 indicate that the swaying angle of the under-damping hydraulic anti-swaying system experiences an oscillatory attenuation process during which the amplitude changes with time. When the damping coefficient, \(\xi\), is smaller, the oscillation is more obvious, indicating that the container oscillates more significantly, which is not allowed in engineering practice. Meanwhile, the entire oscillatory attenuation process experienced by the swaying angle is always contained between a pair of envelope surfaces, \(\pm\frac{0.085}{\sqrt{1-\xi^{2}}}e^{-12.85t}\), as shown in Figure 8. Figure 8 shows that a larger damping coefficient, \(\xi\), leads to a more rapid convergence of the envelope surfaces to zero, indicating that the oscillation in Figure 7 is weaker and that the swaying angle attenuation is faster, so the anti-swaying effect of the anti-swaying system is better. Figure 6: Influence of \(k\) on the attenuation process of the swaying angle. Figure 7: Anti-swaying effect of the under-damping anti-swaying system. According to Equation (17), when \(\Delta\) and \(\varphi_{0}\) are known, the attenuation time, \(t_{s}\), of the swaying angle is determined by the damping coefficient, \(\xi\), and the length of the hoisting rope. Figure 9 illustrates this conclusion. Figure 9 also shows that when \(\xi\) increases, \(t_{s}\) decreases from fast to slow. If \(\xi\) is small, then \(t_{s}\) increases as \(l\) increases; if \(\xi\) is large, then \(t_{s}\) remains nearly constant as \(l\) changes, indicating that when \(\xi\) is large, the anti-swaying effect of the system has a certain robustness to the changes in \(l\). Figure 8: A pair of envelope surfaces of the under-damping anti-swaying system. Figure 9: Influence of the damping coefficient and the rope length on the attenuation time. Figure 10 shows the influence of the hoisting rope length, \(l\), of the critical-damping hydraulic anti-swaying system on the anti-swaying effect. Figure 10 indicates that the critical-damping system has a good anti-swaying effect, and its anti-swaying effect is less dependent on the changes in the hoisting rope length, \(l\). Figure 11 shows the changes in the swaying angle with time for the over-damping hydraulic anti-swaying system as the damping coefficient, \(\xi\), changes.