Revolute joints with clearance in multibody systems

Abstract

A computational methodology for dynamic analysis of multibody mechanical systems with joint clearance is presented in this work. Clearances always exist in real joints in order to ensure the correct relative motion between the connected bodies being the gap associated to them a result of machining tolerance, wear, and local deformations. Clearance at different joints is the source for impact forces, resulting in wear and tear of the joints, and consequently the degradation of the system performance. The model for planar revolute joints is based on a thorough geometric description of contact conditions and on a continuous contact force model, which represents the impact forces. It is shown that the model proposed here lead to realistic contact forces. These forces correlate well with the joint reaction forces of an ideal revolute joint, which correspond to a null joint clearance. The application to the analysis of a simple planar multibody system illustrates the use of the different models proposed.

Introduction

In general dynamic analysis of multibody mechanical systems it is assumed that the kinematic joints are ideal or perfect, that is, clearance, local deformations, wear, and lubrication effects are neglected. However, in a real mechanical kinematical joint a gap is always present. Such clearance is necessary to allow the relative motion between the connected bodies and to permit the components assemblage as well. For instance, in a journal–bearing joint there is a radial clearance allowing for the relative motion between the journal and the bearing. This clearance is inevitable due to the machining tolerances, wear, material deformations, and imperfections. The presence of such joint gaps leads to degradation of the performance of mechanical systems in virtue of the impact forces that take place. Not only these impact forces dissipate energy but they are also a source for vibrations and noise.

The general purpose computational tools used for design and analysis of mechanical systems, such as ADAMS [1] or DADS [2], have a wide number of modeling features that require the description of rigid or flexible bodies for which geometry, mass, center of mass, moment of inertia, and other relevant properties are defined. These codes also provide a large library of kinematic joints that constrain relative degrees of freedom between connected bodies. The kinematic joints available in these commercial programs are represented as ideal joints, i.e., there are no clearances or deformations in them.

The subject of the representation of real joints draw the attention of a large number of researchers that produced several theoretical and experimental works devoted to the dynamic simulation of mechanical systems with joint clearances [3], [4], [5], [6], [7], [8]. Some of these works focus on the planar systems in which only one kinematic joint is modeled as clearance joint [6], [7]. Bengisu et al. [8] presented a study of a four-bar mechanism with multiple joints with clearances. Some researchers also include the influence of the flexibility of the bodies in the dynamic performance of multibody systems besides the existence of gaps in the joints [5], [9]. Based on different clearance models, Claro and Fernandes [10] presented a qualitative study of the performance of a slider-crank mechanism with a nonperfect revolute clearance joint.

While an ideal revolute joint imposes permanent kinematic constraints to the systems, a revolute joint with clearance has to be dealt with a different approach, for instance using force constraints [6]. For planar systems the existence of a radial clearance in a revolute joint removes the two kinematic constraints associated with the ideal revolute joint, and, hence, two extra degrees of freedom are introduced. Thus, the journal can move freely inside the bearing boundaries. When the journal reaches the bearing wall an impact takes place and contact forces control the dynamics of the joint.

In general, there are three main modeling strategies for mechanical systems with revolute clearance joints, namely, the massless link approach [11], [12], [13], the spring–damper approach [3], [4], [14], and the momentum exchange approach [6], [15]. In the massless link approach, the presence of clearance at a joint is modeled by adding a link of zero mass that has a constant length equal to the radial clearance, as shown in Fig. 1. The result is a mechanism that has an additional degree of freedom, when compared with the system that has the ideal joint. In the spring–damper approach, the clearance is modeled by introducing a spring–damper element, represented in Fig. 2, which simulates the surface elasticity. This model does not represent the physical nature of energy transfer during the impact process. Moreover, there is a real difficulty in quantifying the parameters of the spring and damper elements. In the third model, shown in Fig. 3, the journal–bearing elements are considered as two colliding bodies and the contact forces control the dynamics of the clearance joint.

The existence of impacts in the joint lead to the appearance of high level of contact forces during dynamic analysis. The difference in radius between the bearing and the journal, which defines the radial clearance size, is directly associated to the model of contact forces that develop during the motion of the system. In the first two models, the clearance is replaced by an equivalent component, which tries to simulate its behavior as closely as possible. The third model is more realistic, as it allows for the contact force models to develop as a function of the elasticity properties of contacting surfaces and it takes into account the dissipation of energy during the impact process.

The modeling of the impact in multibody systems is well described by two types of methods, namely continuous and discontinuous approaches [16], [17]. Within the continuous approaches the methods commonly used are the continuous force model, which is in fact a penalty method, and the unilateral constraint methodology, based on complementary approaches [18]. The continuous contact force model represents the forces arising from the collisions, assuming that the force is a continuous function of deformation. In this model, when contact is detected, a force perpendicular to the plane of collision is applied. The contact force model can be linear, as in the Kelvin–Voigt model, or nonlinear, as represented by the Hertz law. For long impact durations this method is effective and accurate in so far as the instantaneous contact force is evaluated and introduced into the equations of motion of the system. In the second continuous approach, when contact is detected a kinematic constraint is introduced in the system equations. Such constraint is maintained while the reaction forces are compressive, and removed when the impacting bodies rebound from contact [19].

The second type of methods, the discontinuous model, assumes that the impact occurs instantaneously and that no change of the system configuration occurs during contact. The integration of equations of motion is halted at the time of impact and a momentum balance is performed to calculate the post impact velocities of the system components. The restitution coefficient is employed to quantify the dissipation energy in the process. This method is relatively efficient, however, the unknown duration of impact limits its application, mainly for long impact duration in which the system configuration changes significantly and the assumption of instantaneity of impact is no longer valid [20].

The main emphasis of this work is on the modeling revolute clearance joints in multibody mechanical systems. The contact between the journal and the bearing is modeled by using a continuous impact force model. The impact forces are then introduced into the system's equations of motion in order to analyze the dynamic behavior of the system.

In order to demonstrate the use of the methodology described throughout this work, an application to a planar slider-crank mechanism in which the revolute joint between the connecting rod and slider has a controlled clearance is presented.