## 5 Discussions
### Time History of Rope Force and Bending Moment of Girder  
Obviously, the heavier the payload and the bigger \(M_{\rm blim}\), the bigger the maximum value and the variation range of the rope load. When the position of the trolley is at \(y_{\rm o3}=11.305\) m and \(y_{\rm o3}=0.245\) m respectively, the comparison of the time histories of the relative rope force (the ratio of the instant rope force with the gravity of the lifted object) during braking processes of the lifting mechanism equipped with dual-brakes, whose braking capacity increases from 630 Nm (\(M_{\rm blim}\)= 51,471 Nm) to 1250 Nm (\(M_{\rm blim}=102,125\) Nm), is shown in Figure 14.  
Figure 12: Time histories of torques acting on high-speed links during braking for \(n_{0}=0.25n_{\rm e}\). **(a)** The torque of the shaft connecting components \(J_{3}\) and \(J_{4}\) at full load; **(b)** The torque of the shaft connecting components \(J_{4}\) and \(J_{5}\) at full load; **(c)** The torque of the shaft connecting components \(J_{5}\) and \(J_{6}\) at full load; **(d)** The torque of the shaft connecting components \(J_{3}\) and \(J_{4}\) at 75% rated load; **(e)** The torque of the shaft connecting components \(J_{4}\) and \(J_{5}\) at 75% rated load; **(f)** The torque of the shaft connecting components \(J_{5}\) and \(J_{6}\) at 75% rated load. Figure 13: Dynamic responses when brake 4 cannot work vs. those in basic scenarios. **(a)** Friction power in the brake assembled on component \(J_{6}\) during lowering braking; **(b)** Displacement of the payload during lowering braking; **(c)** The torque of the shaft to connect component \(J_{4}\) and \(J_{5}\) during lifting braking; **(d)** The torque of the shaft to connect component \(J_{5}\) and \(J_{6}\) during lifting braking. Moreover, the initial position of the payload and the initial speed of the lifting mechanism influence the maximum value of the rope force in each braking process as well, and the relationships are nonlinear as shown in Figure 15. The initial position of the payload is given by parameters \(l_{0}\) and \(y_{\text{o3}}\). As shown in Figure 15, when changing \(y_{\text{o3}}\), the maximum rope force takes its biggest value at \(y_{\text{o3}}=0.245\) m (the limit position of the trolley in the crane span range). The variation range of the maximum rope force decreases when the trolley approaches the middle of the girder. In the view of the traditional design theory on cranes, the global peak of maximum rope force occurs when the trolley is in the middle of the girders and a payload is braked during lowering or is hoisted suddenly from its support surface. However, in this simulation, it is found that the global peak of the rope force may occur during braking of the lifting mechanism at full speed in hoisting direction when the trolley is in its limit position at the crane span end. In order to understand better the effect of the initial speed on the maximum rope force, the results at \(y_{\text{o3}}=0.245\) m are reorganized to Figure 16 from Figure 15 (the relationship between the maximum rope force and \(l_{0}\) at different \(n_{0}\) when \(y_{\text{o3}}=0.245\) m). It is found that the effect of the initial speed is completely different under braking processes of hoisting and lowering a payload. The maximum rope forces are very sensitive to the initial speed during braking processes of hoisting a payload; however, they are not so during braking processes of lowering a payload. It is further observed that the maximum rope forces during braking of a hoisting payload do not increase monotonously with \(n_{0}\) under the same \(l_{0}\). Note that, when \(n_{0}=0.25n_{\text{e}}\) and \(0.1n_{\text{e}}\), the lifting mechanism does not get braked by the mechanical brakes until after changing its direction of movement, and the braking torques still does not reach the maximum capacity when braking ends. Therefore, maybe the above results are jointly resulted in by the magnitude of the braking torque when braking ends and the motion states of the system when braking starts.