## 1 Introduction  
Braking is applied frequently during the operation of cranes. For electric cranes, there are two kinds of braking, referred to as normal braking and emergency braking respectively in this paper. Normal braking is applied during normal operations and when moderate faults occur, such as overheating of the electrical system. However, emergency braking is also unavoidable in some situations. For example, electric power is cut off suddenly, over-current occurs in a circuit of a driving motor, the load handling device comes into contact with the lever of the counter-weighted limit switch, or the emergency stop button is pressed [1], and so on. A normal braking operation can be described as: at first the electrical braking is applied, then the mechanical brake is applied when the rotational speed of the motor gets to a specified low level, and subsequently the electric motor is switched off in less than a second. However, when emergency braking happens, the mechanism is stopped only by the mechanical brake, without the electrical braking process. High safety is required in all cranes, and therefore the mechanical brake is one of the critical devices to ensure the safe operation of cranes. It has been accepted that a braking process by a mechanical brake causes energy dissipation and wear of the friction pairs [2], and it is vital to maintain a steady braking capacity for brakes. Therefore, some fundamental principles have to be followed when choosing brakes [3]. According to the report of [PERSON] and [PERSON] [4], the braking capacity of mechanical brakes in operation is related to the thermal load of brakes, the friction work, the friction power at the beginning of braking, and metal particles diffused into the friction linings. The mean friction coefficient of the friction pair basically remains invariant within the normal range of thermal loads, but it decreases sharply when the friction pairs overheat [4; 5]. Meanwhile, [PERSON] et al. [6] demonstrated that there were quite different thermal characteristics for different types of materials of friction pairs of brakes. The safety of cranes is also much related to their dynamic responses during normal operations. [PERSON] et al. [7] analysed the dynamic factor of the rope of a floating crane during lifting based on flexible multi-body dynamics. [PERSON] et al. [8] carried out the dynamic analysis of a lattice mobile crane during hoisting controlled by a hydraulic secondary control system with speed regulation. [PERSON] [9] conducted a numerical simulation of a truck crane using a dynamic model considering the flexibility of the crane support system. [PERSON] et al. [10] studied the dynamic response of the steel structure of a tower crane induced by the movement of a spherical pendulum and a planar pendulum (the payload and the rope are regarded as a single pendulum). [PERSON] [11] presented a new dynamic model of slewing cranes with a large swaying angle of the payload during slewing. [PERSON] et al. [12] presented a dynamic model for luffing motion of a flexible knuckle boom crane. [PERSON] and [PERSON] [13] carried out a finite element dynamic analysis of the bridge of an overhead travelling crane while its traversing mechanism was in operation and introduced a kind of finite element subjected to a point load. [PERSON] [14] built a dynamics model of a forest crane that considered the flexibility of connections between the crane and the ground, the flexibility of links and their drives based on the rigid finite element method. [PERSON] et al. [15] implemented static and dynamic analyses of an offshore knuckle boom crane during the hoist acceleration or deceleration in a 3D dynamic model. [PERSON] and [PERSON] [16] applied screw theory to formulate the dynamic model of an offshore knuckle crane with passive rotary joints and piston actuators and obtained dynamic reaction forces of the joints during jib luffing motion.