## 1 Introduction  
Mechanism systems are the core functional modules in mechanical products, which are mainly used for the transmission of motion and force. Through the design of mechanism structure, the specified input motion form can be transformed into reciprocating motion, rotary motion, intermittent motion and other target motion forms to complete the displacement and speed conversion. Mechanisms are also capable of changing the magnitude and direction of force to enable mechanical products to perform complex operations, such as lifting heavy objects, amplifying and reducing force, etc. Mechanism systems are widely used in aerospace machinery systems, automobiles, industrial robots and other important mechanical equipment [1; 2; 3]. With the increase in product requirements for high speed, high precision and heavy loads, the requirements for mechanisms in terms of accuracy, stability, durability and safety are constantly increasing. If failures occur, such as excessive clearance wear and component stiffness degradation, the mechanisms may fail with serious consequences. For example, the fault in a satellite connection mechanism or a failure of the synchronization mechanism may affect the normal separation or deployment of the satellite and prevent the satellite from working properly [4]. The normal operation of the aircraft landing gear mechanism has a very important impact on the safe operation of the aircraft and passenger safety. Therefore, it is very necessary to carry out the reliability analysis of the mechanism, identify the key mechanism failure, analyze the impact of failure, and improve the service life and reliability of mechanisms. Since the mechanism systems need to meet the requirements of motion accuracy and force transmission during the working process, the failures to be considered in the reliability analysis of the mechanisms are often related to the elements affecting the velocity and displacement of the mechanism motion as well as the force applied to them. In general, the failure factors that affect the reliability of a mechanism include the clearances of the kinematic joints, the machining dimensional errors of the linkage lengths of the members that make up the mechanism, the stiffness of the linkages, the mounting position errors, the operating loads, and the environmental stresses. The failure modes involved in these factors include fatigue failure, wear failure, corrosion failure, and so on. It should be noted that the failure factors such as clearance of the kinematic joints, defective material of the members, etc. cause the nonlinear characteristics of the systems. Such nonlinear characteristics under the combined influence of random loads, component stiffness, manufacturing and installation errors, etc. cause difficulties in analyzing the stochastic dynamics and time-varying reliability of the systems. In order to solve the above difficulties, a large number of researchers have carried out numerous analyses of the dynamics of mechanisms with faults [5; 6; 7; 8]. [PERSON], [PERSON] and [PERSON] developed a method for analyzing the dynamics of a partially compliant mechanism with a joint clearance [9]. The results show that the joint clearance leads to chaotic behavior in the kinematic and dynamic output of the mechanism. The flexibility effect of small pivot and the force-closure kinematic joint between the ball and the socket in the joint clearance reduce the chaotic phenomena caused by the clearance in the output of the mechanism. [PERSON], [PERSON] and [PERSON] investigated the dynamic behavior of a slider crank mechanism with clearance faults [10]. The dynamic equations of the slider crank mechanism with clearance were established by equating the rotating link with clearance to a virtual light mass rod and utilizing the Lagrangian method. The results show that the clearance has a small effect on the displacement and velocity response of the slider crank mechanism, but has a significant effect on the acceleration response of the mechanism.