TECHNICAL PAPER Vibration performance evaluation of planar flexible multibody systems with joint clearance Saeed Ebrahimi 1 • Esmaeil Salahshoor 1 • Shapour Moradi 2 Received: 24 February 2017 / Accepted: 6 July 2017 Ó The Brazilian Society of Mechanical Sciences and Engineering 2017 Abstract Clearances are necessary in assemblage of mechanisms to allow the relative motion between the members. This clearance is due to machining tolerances, wear, material deformations, and imperfections, and it can worsen mechanism performance such as precision and vibration. As a new study in this topic, the effect of joint stiffness on the variation of instantaneous natural fre- quencies and mode shapes of a flexible four-bar mecha- nism with a clearance between coupler and follower is studied in this paper. To model the clearance, the contin- uous contact approach is used. The Lankarani’s and Nik- ravesh’s continuous contact force model is used to model the contact force arising from contact between journal and bearing. Finite element method is used to determine the instantaneous natural frequencies and their corresponding mode shapes. The stiffness of the clearance is modeled as a linear spring added to the assembled stiffness matrix. To validate the clearance model in rigid mechanism, the dynamic response is compared with the results in the lit- erature. To show the validity of the formulation which calculates the instantaneous natural frequencies, two methods are used and compared with each other in the case no clearance exists. The results show that taking the joint stiffness into account has a considerable effect on the instantaneous natural frequencies and their corresponding mode shapes of a flexible multibody system. Keywords Clearance  Joint stiffness  Continuous contact approach  Finite element method  Instantaneous natural frequencies 1 Introduction In multibody systems, kinematic joints are generally assumed to be without clearance. On the contrary, there is a gap between journal and bearing in real mechanical joints. This clearance exists due to machining tolerances, wear, material deformations, and imperfections, and can worsen mechanism performance such as precision and vibration due to impact forces between journal and bearing. Clearance has attracted a vast investigation by researchers. Generally, three main types of dry clearance models could be found in the literature, namely, the massless link approach [1–6], the spring–damper approach [7–9], and the continuous contact approach [10–15]. Due to using the continuous contact approach in this paper, this model is explained here in more detail. In this approach, the clearance is modeled as two colliding bodies and the impact–contact force between them controls the dynamic behavior of the system, as shown in Fig. 1. The impacts in the clearance joint lead to high contact force and conse- quently high acceleration. This model is more realistic than two other approaches as it considers the contact forces as a function of surface elasticity in addition to taking into account the energy dissipation during impact. Generally, Technical Editor: Ka ´tia Lucchesi Cavalca Dedini. & Saeed Ebrahimi ebrahimi@yazd.ac.ir Esmaeil Salahshoor esalahshoor@stu.yazd.ac.ir Shapour Moradi moradis@scu.ac.ir 1 Faculty of Mechanical Engineering, Yazd University, Yazd, Iran 2 Faculty of Mechanical Engineering, Chamran University, Ahwaz, Iran 123 J Braz. Soc. Mech. Sci. Eng. DOI 10.1007/s40430-017-0855-0