## 4 Time Domain Dynamic Equations for the Anti-Swaying System  
### Dynamic Analysis of the Anti-Swaying System  
In general, when there is no wind load, the influences of the trolley and the crane on the load swaying are the same. Therefore, for the sake of convenience, only the trolley-crane coupling structure was considered in this study. As mentioned previously, the installation distance, \(h_{0}\), of the two-trolley frame anti-swaying fixed pulleys (8) is much smaller than the transverse size, \(h_{1}\), of the trolley frame, that is, \(h_{0}<<h_{1}\). Therefore, it can be assumed that the four anti-swaying ropes, a, b, c, and d, are suspended at a point P through the trolley frame fixed pulleys; the projection of the suspension point P on the spreader is the intersection point Q, and point Q is the intersection of the rectangular diagonals determined by the spreader fixed pulley group (4). Suppose the load sways to the right with an angle of \(\varphi\) (rad). A schematic diagram of the simplified structure of the crane and trolley-loadsystem is shown in Figure 3. When the load does not swing, that is, when \(\varphi=0\), the spatial layout of the anti-swaying wire ropes a, b, c, and d is shown in Figure 4, where \(\theta_{1}\) represents the angle between the anti-swaying wire ropes and the mounting plane angle of the anti-swaying fixed pulleys. It is clear that \(\theta_{1}\) depends on the height of the load, that is, the length of the hoisting rope when the load does not swing but is in its equilibrium position. \(\theta_{2}\) is determined by the installation positions of the anti-swaying fixed pulleys on the spreader. \(e\) and \(f\) are determined by the installation positions of the anti-swaying fixed pulleys. \(e\) represents the installation distance of the anti-swaying fixed pulleys in the running direction of the trolley, referred to as the longitudinal distance of the anti-swaying fixed pulleys. \(f\) is the installation distance of the anti-swaying fixed pulleys in the direction perpendicular to the running direction of the trolley, referred to as the transverse distance of the anti-swaying fixed pulleys. The units of \(e\) and \(f\) are both \(m\). \(G(t)\) represents the sum of the tensions in the two tensioning side wire ropes. \(F_{x}\) is the sum of the horizontal components of the forces the two tensioning side wire ropes exert on the load. The units of \(G(t)\) and \(F_{x}\) are both N. It is clear that \(G(t)=F_{x}\), and \(F_{x}\) is also the force exerted by the cylinder connected to the tensioning side anti-swaying wire ropes. Figure 4: Schematic diagram of the spatial layout of the anti-swaying wire ropes. Figure 3: Schematic diagram of the simplified structure of the crane trolley–load system. In addition, for the convenience of the study, the following simplifications and approximate hypotheses were made:  
1. The load swaying is similar to the swinging of a single pendulum and is longitudinal swinging when the influences of starting and braking are considered. The load swaying angle is \(\varphi\) and does not exceed \(5^{\circ}\), namely, \(\varphi\leq 5^{\circ}\);
2. According to the basic requirement of the hydraulic system, the oil pressure in the high-pressure circuit of the hydraulic system changes with changes in the load-swaying speed. The horizontal component, \(F_{x}\), of the force that the anti-swaying wire ropes exert on the load is approximately proportional to the load-swaying speed. During swaying, the anti-swaying wire ropes on the tensioning side are affected by \(F_{x}\). Considering the structural characteristics of the hydraulic anti-swaying system and the lag factor caused by hydraulic damping, the force that the slack side wire ropes exert on the load is not considered. Considering that the load swaying angle, \(\varphi\), is smaller than both \(\theta_{1}\) and \(\theta_{2}\), it can be assumed that the force that the anti-swaying wire ropes exert on the load is constant;
3.