A comprehensive survey of the analytical, numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints

TL;DR: A comprehensive survey of the literature of the most relevant analytical, numerical, and experimental approaches for the kinematic and dynamic analyses of multibody mechanical systems with clearance joints is presented in this review.

About: This article is published in Mechanism and Machine Theory.The article was published on 2018-04-01 and is currently open access. It has received 271 citations till now.

Summary

1. Introduction

• Mechanisms, as multibody systems, are made of several mechanical components and inter connections, which can be classified into two major groups, namely links; i.e., bodies with a convenient geometry; and kinematic joints, which introduce some kinematic constraints or restrictions on the relative motion between adjacent links [1].
• Therefore, in order to achieve the required performances of the mechanisms with some tolerances or clearance joints, it is quite important to quantify the effects of the joint clearances on these systems’ dynamic responses [4].
• In general, some energy is dissipated in the process of contact/impact, and hence a hysteresis is formed representing the progression of the contact force during the contact period.
• For the lubricated joint case, the hydrodynamic theory for dynamically loaded journal-bearings is used to compute the forces generated by lubrication action.

2.1. Planar revolute joint with clearance

• In 1967, Chace [88] highlighted the lack of investigation on the performance of mechanisms with clearance joints.
• Three different modes of journal motion inside the bearing can be considered, namely the contact or following mode, the free flight mode and the impact mode [11, 26, 94, 100, 143], as it is depicted in Fig.
• The contact force model proposed by Lankarani and Nikravesh [145] can account for both the elastic and damping effects.
• The contact force model proposed by Lankarani and Nikravesh (L-N) has extensively been utilized to study the dynamics of mechanisms with planar revolute clearance joints.
• The numerical results obtained by the proposed model were validated by the experimental results obtained by Flores et al. [202].

2.2. Spatial revolute joint with clearance

• The problem of the dynamic modeling of multibody systems with planar revolute clearance joints has been extensively investigated over the last decades.
• When the journal and bearing are in contact with each other, local deformations take place at the contact area and, consequently, contact-impact forces characterize the interaction between the bodies.
• Deck and Dubowsky [113] theoretically and experimentally investigated the dynamic behavior of mechanical systems with flexible links and clearance joints.
• Virlez et al. [254] developed a spatial revolute clearance joint model, which accounted for the clearance, the misalignment and the friction effects.
• Most of the works described above utilized the well known L-N contact force model [145].

2.3. Spherical joint with clearance

• The fundamental aspects on the modeling of multibody systems with spherical clearance joints are revisited here.
• Figure 11 illustrates the three motion modes of the rigid ball inside the socket; namely: the permanent contact or following mode, the free flight mode, and the impact mode.
• This approach was validated using the finite element method.
• Zhang et al. [15] analyzed the nonlinear dynamics behavior of spherical joints with clearance.
• This formulation has been extensively analyzed and compared with the classical Hertz contact force model and validated with FEM data.

2.4. Translational joint with clearance

• The most relevant works on the modeling of planar translational joints with clearance are summarized and examined.
• These contact forces can be evaluated according to the continuous contact force proposed together with the dissipative friction force model selected [145].
• These authors also showed how the slider motion in a translational clearance joint depends on the geometry, speed and mass distribution.
• The algorithm can guarantee the exact satisfaction of the complementarity condition at position level, which means that no penetration is allowed, and at the velocity level.

3.1. Planar revolute joint with lubrication

• It is known that in most engineering applications, the mechanical joints are designed to operate with some lubricant fluid, as in the case of the well-established journal-bearings.
• These force components can be obtained by integration of the pressure field around half domain π; i.e., the pressure distribution is integrated only over the positive region by setting the pressure in the remaining portion equal to zero [334-338].
• Equations (27) through (31), for infinitely-long journal-bearings, present the relationship between the journal center motion and the fluid reaction force on the journal.
• More recently, Machado et al. [65] comparatively studied the effects of three HD lubrication models, clearance sizes, and crank speeds and lubricant viscosity on the dynamic responses of a planar rigid slider-crank mechanism with a lubricated revolute joint between the connecting-rod and slider.
• The lubrication models were deducted according to the average Reynolds equation, and solved with finite element method to derive the hydrodynamic forces according to the motion of the system.

3.2. Spatial revolute joint with lubrication

• The fundamental aspects related to the modeling of spatial revolute joints with lubrication are revisited, namely for the hydrodynamic (HD) and elastohydrodynamic (EHD) cases.
• In their work, the rigid body attitude and displacement were described by using Euler parameters and Cartesian position coordinates, respectively.
• The results showed that there were obvious changes in film pressure distribution, the highest film pressure, film thickness distribution, the least film thickness, and the misalignment moment when misalignment took place.
• Choi et al. [371] developed an EHD lubricated cylindrical joint model.
• The reported outcomes reported were compared and verified with the data produced with commercial code ADINA.

3.3. Spherical joint with lubrication

• In contrast with the planar revolute joint, there are very few works for modeling and simulating spherical joint with lubrication in the context of multibody system dynamics.
• Wang et al. [377] investigated the tribological aspects of spherical bearings with complex spherical-based geometry with the purpose of modeling and simulating friction and lubrication at this type of mechanical joint.
• Based on the work developed by Goenka [378], Tian et al. [277] proposed a HD lubricated spherical joint for flexible multibody systems, which is schematically represented in Fig. 22 where the Reynolds’ equation is represented in spherical coordinate system r-θ-φ.
• In the work by Tian et al. [277], the flexible bodies were meshed by the fully parameterized beam elements of ANCF.
• In their work when the eccentricity ratio was close to zero the pressure was neglected.

3.4. Hybrid/transition joints models

• This section provides a description of hybrid formulations that allow the study of the transition between the dry and the lubricated joint models.
• To smooth the force transition between the lubricated and dry contact, a force transition model from hydrodynamic lubrication forces to dry contact forces was proposed by Flores et al. [100], which can ensure continuity in the joint reaction force.
• In the mode 2, the journal and bearing wall are in contact, thus the contact force between the journal and the bearing is modeled with the continuous contact force model represented by Eq. (8).
• It should be noted that the clearance used for the pure squeeze force model is not c but it is c+et1 instead.
• Due to the difficulty in choosing the preselected parameters, the Flores’s transition model [100] lacks general applicability in mechanisms with different clearance sizes.

4.1. Simple journal-bearing systems

• The subject of the representation of real physical mechanical joints has attracted the attention of significant number of researchers, and a large number of theoretical and experimental works on the dynamics of multibody mechanical systems with clearance joints has been published [118].
• Some of the most relevant experimental investigations on mostly simple journal-bearing are examined.
• Norton et al. [387] discussed the bearing forces as a function of mechanical stiffness and the vibration isolation in an experimental setup consisting of a four bar mechanism.
• They also observed that the larger the journal misalignment resulted from shaft deformation, the more obvious effect was seen on lubrication performance of journal-bearing.
• More recently, Yan et al. [251] developed a spatial revolute joint modeling and a test setup considering the journal misalignment, in which a group of contact force models were employed to describe different contact-impact phenomena.

4.2. Mechanisms with clearance joints

• Over the last decades, a large number of works have been proposed based on various methodologies for modeling mechanisms with clearance joints, as described in detail in Section 3.
• In 1971, Wilson [299] presented a detailed description of both analytical and experimental approaches for a slider-crank mechanism, which allowed for the investigation of the influence of the clearance at the slider joint.
• Earles and Kilicay [397] also experimentally investigated the dynamic response of mechanisms with clearance joints.
• The theoretical results were compared with the experimental ones and showed good qualitative agreement.
• They also predicted contact loss in a mechanism with multiple clearance joints.

4.3. Mechanisms with flexible links

• Some of the works that combine the influence of clearance joints and flexibility of the links are revisited.
• Dubowsky and Young [409] supported the idea that the flexibility of the links could reduce the level of impact when mechanisms included clearance joints.
• The authors also proposed an approach to account of flexibility of the links on the mechanisms’ response when incorporating clearance joints.
• In turn, Erkaya and Uzmay [411] also designed a test rig for the slider-crank mechanism with two clearance revolute joints.
• Figure 39 shows the experimental apparatus of the actual mechanical system developed.

4.4. Other experiments with clearance joints

• Over the last years, some experimental works on clearance joints that involve other aspects have been published.
• For this, the connecting rod was modeled with lumped masses.
• Liu and co-authors [417] designed and built an experimental apparatus to compare and validate different contact force models for the case of revolute joints with clearance.
• The applied force was measured using a force sensor.
• They concluded that when compared to the experimental observations, the Lankarani-Nikravesh (L-N) contact force model was found to provide the best correlation.

5.1. Wear in mechanisms with clearance joints

• According to the standard DIN 50320, wear can be defined as “the progressive loss of material from the surface of a solid body due to mechanical action; i.e., the contact and relative motion against a solid, liquid or gaseous counter body” [422].
• According to their work, the contact force at each time increment is firstly calculated using the L-N contact model [145] and then used to evaluate the contact pressure through a finite element analysis.
• The results were also validated against those from actual experiments.
• Using the hybrid contact force model proposed by Bai and Zhao [200] and the Archard’s wear model, Bai et al. [434, 435] also predicted the wear of a planar revolute clearance joint of a rigid four-bar mechanism.
• The Archard’s wear model was adopted to calculate the wear amount of contact surface.

5.2. Optimization and control of mechanisms with clearance joints

• The dynamic behavior of the mechanisms with clearance joints is in general quite sensitive to small changes of parameters, namely the clearance size and the friction coefficient.
• Farahanchi and Shaw [301] studied the dynamic performance of the slider-crank mechanism with a slider clearance.
• Rhee and Akay [447] used a discontinuous contact force model to analyze a four bar mechanism with one revolute clearance joint.
• They showed that, depending on the friction coefficient and clearance size, the system could exhibit periodic or chaotic behavior.
• The kinematics relations of the clearance joint elements were introduced as the constraint conditions of the objective function (potential energy) into the optimization process.

5.3. Uncertainty of mechanisms with clearance joints

• Large majority of previous studies on the dynamic modeling and analysis of mechanisms with clearance joints are based on the dynamics equations with deterministic system parameters.
• There are two main types of methods to describe the uncertain parameters [470-472], namely the probabilistic methods and the non-probabilistic methods.
• Pang et al. [473] proposed a modeling approach that allowed for the analysis of accuracy reliability and sensitivity of planar mechanisms with multi-factors such as clearances, manufacturing and assemble tolerances among others.
• In their work, the dynamic equations of the flexible mechanisms were established using the impact-pair model of clearance connection and the kineto-elastodynamics (KED) method, and the link lengths, pin radius, and radial clearance size were assumed to be the random parameters with normal distribution.
• The dynamic equations were solved using the Newmark algorithm, and the Monte Carlo method was employed to analyze the effects of parameter uncertainty on the system’s motion accuracy.

5.4. Link flexibility in mechanisms with clearance joints

• Some works on the effect of flexibility of the links on the performance of mechanisms with clearance joints has also been mentioned in the previous sections.
• They demonstrated that the links flexibility tends to reduce the clearance joint stresses.
• Schwab and his co-authors [343] investigated the dynamic response of mechanisms with clearance joints and link flexibility as well.
• The normal contact force in the joints with clearance was evaluated using a nonlinear spring-damper model, and the friction effect was considered using the Coulomb friction model.
• Both analytical and experimental models were considered.

6. Concluding remarks and future challenges

• The main objective of the presented work was to provide a general and comprehensive overview of the methodologies dealing with the modeling and analysis of realistic mechanical systems or mechanisms, which include imperfect or clearance joints; i.e., joints in which the effects of clearance, friction, and lubrication are taken into account.
• The planar and spatial revolute joints, spherical joints, and translational joints were each addressed individually.
• These topics included wear phenomena in clearance joints, optimization and control of mechanism that incorporate clearance joints, uncertainty of mechanical systems with clearance joints, and effect of link flexibility.
• There are also the solid or semi-solid lubricants used in the mechanism joints.
• The numerical modeling and simulation of mechanisms with clearance joints is still quite a challenge faced by the researchers in this field.

Figures

Fig. 27 Test rig utilized at the University of Minho to investigate simple journal-bearings [390].

Fig. 13 Different scenarios for the slider motion inside the guide: (a) no contact; (b) one corner in contact with the guide; (c) two adjacent corners in contact with guide; (d) two opposite corners in contact with guide [83]

Fig. 21 A geared quick-return mechanism supported by four EHD lubricated spatial revolute joints [374]

Fig. 14 (a) Non contact situation; (b) penetration between the slider corner A and the guide surface [83]

Fig. 15 (a) Contact between a spherical surface and a plane; (b) contact between two plane surfaces [83]

Fig. 42 Journal surface divided into several sectors to conduct wear analysis [429].

Fig. 6 Typical representation of a spatial revolute joint with clearance [83].

Fig. 7 Four different possible scenarios for the journal motion relative to the bearing: (a) no contact; (b) one point contact; (c) line contact; (d) two contact points in opposite sides [239].

Fig. 11 Modes of the ball motion inside the socket [83]

Fig. 33 Experimental test rig and its slider-crank mechanism with an adjustable radial clearance at the revolute joint between the slider and the connecting rod [202].

Fig. 32 The test rig of a double pendulum with/without a clearance joint [319].

Fig. 17 (a) Sommerfeld’s boundary conditions; (b) Gümbel’s boundary conditions [83].

Fig. 3 Revolute joint with clearance, in which the clearance size is exaggerated for clarity [4].

Fig. 16 Cross section of a smooth dynamically loaded journal-bearing [83].

Fig. 43 ABAQUS bearing finite element model [438].

Fig. 22 Schematic representation of a lubricated spherical joint [277]

Fig. 18 The generic configuration of a lubricated spatial revolute joint: (a) The lubricated spatial revolute joint with journal misalignment (b) The cross section of the lubricated spatial revolute joint [358].

Fig. 26 The transition model proposed by Li et al. [383].

Fig. 31 The four-bar linkage test setup, with the mirrors mounted on the rocker by Tasora et al. [402].

Fig. 30 Test rig of a four-bar linkage with a clearance at the coupler-rocker joint [399].

Fig. 44 A representation of subdomains on the cup and head surface [439].

Frequently asked questions

Q1. What are the contributions in "A comprehensive survey of the analytical, numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints" ?

A comprehensive survey of the literature of the most relevant analytical, numerical, and experimental approaches for the kinematic and dynamic analyses of multibody mechanical systems with clearance joints is presented in this review. The main assumptions procedures and conclusions for the different methodologies are also examined and compared. 

Q2. What future works have the authors mentioned in the paper "A comprehensive survey of the analytical, numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints" ?

For additional future work in the field of modeling and analysis of mechanical systems with imperfect joints, the following challenge subjects are still in need of further investigation: ( 1 ) Modeling mechanisms with multiple clearance joints, and with a variety of joints such as prismatic and universal joints, by proposing efficient numerical algorithms to simulate the complex dynamical systems ; ( 2 ) Implementation of the methodologies presented in this work to more complex mechanical systems, such as automotive systems where the compliance between contacting surfaces may be described using different contact force laws. The effect of clearance size is also worthy of further investigation, especially the coupling effects of clearance size, flexibility and friction ; ( 3 ) Flexibility of components, joint clearance, and uncertainty generally exist in multibody systems simultaneously. Dynamic analyses and optimization of complex flexible multibody systems with clearance joints and uncertainty is still a challenge subject which need to be further studied ; ( 4 ) Including of the effect of roughness and geometric imperfections of the contacting surfaces, and friction laws, including stick-slip conditions ; ( 5 ) Extending the models for the flexible and soft bodies or joints themselves in order to obtain a more accurate picture of the local body or joint deformations, impact forces, and accelerations experienced during the contact periods, and examining the intrarelation between link flexibility and joint clearance on the dynamic response a system ; ( 6 ) Carrying out an extended experimental investigation on multibody systems with lubricated joints and flexible bodies, in order to better characterize the existing models and help in identifying parameters and coming up with new models ; ( 7 ) Development of new types of joints, models, and formulations based on contact between complex shape surfaces, which are of fundamental importance in describing many real physical models such as the human skeletal biomechanical systems ; ( 8 ) Investigating new monolithic schemes to simulate dynamics of clearance joints exposed to multi-domain loads, such as temperature, heat flux, and electromagnetic force. 

Q3. Why did the velocity and acceleration curves obtained by the proposed method be smooth?

Because of the non-penetration assumption, the velocity and acceleration curves obtained by the proposed method were quite smooth. 

Q4. What are some examples of dynamically loaded journals?

Typical examples of dynamically loaded journal-bearings include the crankshaft bearings in combustion engines, and high-speed turbines bearings supporting dynamic loads caused by unbalanced rotors [325-329]. 

Q5. How did they find that the revolute clearance joint was closer to the driving crank?

through simulating the mechanism with two revolute clearance joints, these authors [133] found that compared to the connection clearance joint far from the driving crank, the revolute clearancejoint close to the driving crank would generate the larger contact forces, which would further lead to large wear rate of the revolute clearance joint close to the driving crank. 

Q6. How did Bai and Zhao predict the wear of a planar rigid slider-crank mechanism?

Using the hybrid contact force model proposed by Bai and Zhao [200] and the Archard’s wear model, Bai et al. [434, 435] also predicted the wear of a planar revolute clearance joint of a rigid four-bar mechanism. 

Q7. What is the way to smooth the force transition between the lubricated and dry contact?

To smooth the force transition between the lubricated and dry contact, a force transition model from hydrodynamic lubrication forces to dry contact forces was proposed by Flores et al. [100], which can ensure continuity in the joint reaction force. 

Q8. What are the challenges of modeling and analysis of mechanical systems with imperfect joints?

For additional future work in the field of modeling and analysis of mechanical systems with imperfect joints, the following challenge subjects are still in need of further investigation:(1) Modeling mechanisms with multiple clearance joints, and with a variety of joints such as prismatic and universal joints, by proposing efficient numerical algorithms to simulate the complex dynamical systems; 

Q9. What was the synthesis procedure used to include the friction effects at the clearance joints?

a simple friction force model based on the Coulomb’s law was utilized to include the friction effects at the clearance joints.