## 3 Simulation Parameters  
To better understand the behaviour of cranes during the braking process, the dynamic responses during braking of the lifting mechanism which is equipped with two service brakes of a 32-ton overhead travelling crane are simulated. Before braking, the system is considered to be in a quasi-steady state and the electric motor of lifting mechanism is operating at a specified speed \(n_{0}\). The two service brakes, which are assembled on components \(J_{4}\) and \(J_{6}\) respectively, are assumed to be identical and actuated at the same time. Hereafter, the two service brakes assembled on components \(J_{4}/J_{6}\) are referred to as brake 4/6, and \(n_{0}\) is referred to as the initial speed.  
According to the test report from manufacturers and those in [4], the dynamic friction coefficient of the friction pairs of the mechanical brake used in cranes is relatively invariable when the brake has a  
Figure 4: The flow chart of simulation algorithm for braking process of lifting mechanism.  
sufficient thermal capacity, and the difference between the static and dynamic friction coefficients is quite small, hence Coulomb friction with equal static and dynamic friction coefficients is adopted in this paper. Considering the mechanical and electro-magnetic delay, and the pressure variation on friction pairs in the initial braking stage, the variation of the braking capacity with time of the mechanical brake is idealized as the function shown in Figure 5. The braking capacity of the mechanical brake is the maximum friction torque between the brake linings and the brake drum.  
In Figure 5, the origin of the time axis is the instant when the motor and the actuator of mechanical brake are switched off, and when the payload starts to get out of control. \(t_{\rm bs}\) is the instant when the braking capacity of the brake starts to increase from zero. \(t_{\rm ba}\) is the time interval when the braking capacity of the brake increases from zero to its maximum value. \(M_{\rm blim}\) is the maximum braking capacity of the brakes converted to the corresponding quantity in reference to the axis of the rope drum.  
The parameters of a 32-ton crane used in the simulation of two basic braking processes are given in Table 1. The data about the brake is from its manufacturer. The data about the steel rope is from the [39]. Some other date is calculated by the authors based on the machine elements of this actual crane. In subsequent analyses, a few parameters will be varied while other parameters remain unchanged when discussing their effects.  
In Table 1, \(n_{\rm e}\) is the rated rotational speed of motor used for driving the lifting mechanism. \(\xi_{\rm g}\) and \(\xi_{\rm w}\) are the damping ratios corresponding to \(c_{\rm g}\) and \(c_{\rm w}\), respectively. \(l_{00}\) and \(l_{0\rm b}\) are the free lengths of the rope hanging from the rope drum when the load-handling device is at its lower limit position and when the basic braking scenarios start, respectively. \(y_{\rm o3b}\) is the \(Y\) coordinate of the centres of wheels O\({}_{3}\) and O\({}_{3}^{\prime}\) when wheels O\({}_{4}\) and O\({}_{4}^{\prime}\) are located on the most unfavourable sections of the girders, respectively.