## 1 Introduction
[PERSON] [17] presented a dynamic model of an overhead crane during lifting mechanism operation, which incorporated two steel rope system models, namely the Kelvin-Voigt model and Bouc-Wen model, to consider the influence of rope system parameters on dynamic factors due to the inertia effect of the cargo mass during the lifting mechanism operation. To ensure the safety and efficiency of installation operations of a jack-up crane vessel, [PERSON] et al. [18] built a fully coupled aero-hydro-soil-elastic-mechanical dynamic model for the vessel, and studied the important dynamic responses of the vessel during installation of an offshore wind turbine blade. [PERSON] et al. [19] analysed the system response of a mobile crane during operations in a windy environment using a coupled kinematic and dynamic model. Furthermore, motion control strategies for cranes cannot be developed without a proper dynamic analysis [20; 21; 22; 23; 24; 25]. Dynamic responses under unusual circumstances are the other factors related to the safety of cranes. [PERSON] et al. [26] investigated the vertical seismic response of a 1/8-scale model of an overhead crane and carried out simulation of a nonlinear dynamic model of the crane excited by catastrophic ground motion with gap elements at the crane wheels/tracks interface. To simulate three-dimensional nonlinear seismic responses of container cranes, [PERSON] et al. [27] presented a model including the contact problem between the wheels and rails of cranes based on multi-body dynamics formulation. [PERSON] et al. [28] carried out experimental and numerical investigations of the earthquake response of crane bridges. [PERSON] et al. [29] compared the effects of three kinds of boundary conditions, which were used to respectively model the contact between wheels and rails as pin supports, gap elements and friction contact elements, on seismic responses of the container crane. Today, emergency braking of cranes has recently emerged as a big concern [5; 30; 31]. [PERSON] et al. [5] prepared the higher friction brake shoe material used for the emergency brake of an ultra-deep coal mine hoist and verified its tribological behaviour by means of the test designed by the uniform experimental design method. [PERSON] et al. [30] proposed a control method for emergency braking of the traversing mechanism of overhead travelling cranes, which considered the payload swing suppression. [PERSON] [31] proposed a new emergency braking scheme for a coal handling gantry hoist to prevent high loads from appearing on the drive chains of the hoist, which was equipped with an additional safety brake on the rope drum. [PERSON] et al. [32] investigated the stress-strain state of the steel frame of a gantry crane under the horizontal constrained forces of rails caused by the difference of the braking torques of the brakes located on the two sides of the gantry crane when braking. Although emergency braking is a kind of unusual operation, during which the components of cranes may be subjected to the most unfavourable loads, especially for those controlled by speed control systems, because the dynamic loads during normal operations are now less serious after the adoption of the frequency control, electrical braking and command shaping technology [33; 34; 35]. Therefore, dynamic loads during emergency braking are added as critical design loads to ensure load-bearing capacity of components of cranes in the latest edition of design codes on cranes, but the dynamic effect is considered only crudely by an amplification factor [36; 37]. However, the problems that design engineers care about most, the characteristic of dynamic responses under emergency braking operations and pertinent influence factors, remain unclear. Given that the failure of the lifting mechanism is more dangerous than that of the travelling or traversing mechanism, accidents induced by braking of lifting mechanisms using mechanical brakes are not uncommon, and the test of the emergency braking is unrealistic or infeasible during product design.