## 2 Dynamic Model  
### Physical Model and Assumption  
In general, a real electric overhead travelling crane consists of a bridge, one or two trolleys and travelling mechanisms, as shown in Figure 1.  
Figure 1: Electric overhead travelling crane. The bridge, which is composed of two identical girders and two identical end carriages, is a welded thin-walled steel structure. The girders and end carriages, whose height and width are much smaller than the crane span, have high torsional rigidity. The cross sections of the girders and the end carriages are unchanged except in a very small range at their two ends. The maximum deflection of the girders is less than _L_/1000-_L_/750 (_L_ is the crane span) and is much larger than that of the end carriages. The bridge is supported on two crane rails by crane wheels with cylindrical tread. There is a certain curvature on the top surfaces of the rails and a side clearance from 20 mm to 30 mm between a rail and two flanges of all crane wheels. A trolley is made up of one or two lifting mechanisms, a traversing mechanism and a trolley frame. If there are two lifting mechanisms in a trolley, they are referred to as the main and auxiliary lifting mechanisms respectively. A typical lifting mechanism, as shown in Figure 2a, mainly consists of a 3-phase induction motor, a gear reducer, a mechanical brake (referred to as the service brake), a gear coupling with a brake drum (or a gear coupling without the brake drum, and a brake drum), a wire rope reeving system (including a rope drum, pulleys and one or two wire ropes) and a load handling device. The gear coupling is assembled on the high-speed shaft extension of the gear reducer and the shaft extension of the motor. To compensate for the misalignment between the axes of the motor shaft and the high-speed shaft of the reducer, a floating shaft is often adopted (item 5 in Figure 2b). The mechanical brake is used for not only holding the payload but also decelerating during braking. In general, when one service brake is used, the brake is assembled on a high-speed shaft extension of the gear reducer; when two service brakes are used, another brake may be assembled on another high-speed shaft extension of the gear reducer or one of the two motor shaft extensions, as shown in Figure 2b. The traversing mechanism may include the wheels, driving motors, reducers, brakes, shafts and couplings and so on. As the support structure of the lifting mechanism and the traversing mechanism, the trolley frame, a high order statically indeterminate steel structure, has high bending and torsional rigidity. In crane mechanisms, the moments of inertia of transmission shafts and rope sheaves are much smaller than those of other parts. The torsional deformation of the transmission shafts is much larger than those of other parts and the clearances between the transmission parts are small. The wire ropes are non-rotational, alternate lay wire ropes. That is, wires in each strand are arranged in the same helical direction while the strands are wound in opposite helical directions. When tensioning the wire rope, in theory, there is no rotation effect on the hung object and the wire rope itself is only subjected to a tensile force too. The payload undergoes only translational motion during lifting. Furthermore,  
Figure 2: A typical mechanical brake layout in lifting mechanism of an overhead crane. (**a**) The lifting mechanism without the floating shaft; (**b**) The lifting mechanism with the floating shaft. when lifting height is less than 50 m, the mass of the steel rope is small relative to the mass of the lifted heavy object; the elastic modulus of the steel rope varies within a certain small range although it is a nonlinear variable which depends on the rope load, lubrication, etc. In view of the above features of electric overhead cranes, some reasonable assumptions are made as follows:  
1. The two girders are treated as elastic Euler beams with the same small curvature, uniform damping and mass distributions. 2.