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The FEM analysis and approximate model for cylindrical joints with clearances
Mechanism and Machine Theory
Mechanism and Machine Theory 42 (2007) 183–197
www.elsevier.com/locate/mechmt
The FEM analysis and approximate model for cylindrical joints with clearances Cai-Shan Liu a
a,*
, Ke Zhang a, Rei Yang
b
Department of Mechanics and Engineering Science, State Key Laboratory of Turbulence and Complex Systems, Peking University, Beijing 100871, China b Chinese Academy of Space Technology, Beijing 100086, China Received 21 January 2005; received in revised form 26 January 2006; accepted 28 February 2006 Available online 11 May 2006
Abstract The clearances existing in the joints of mechanisms are the mainly sources for vibrations and noise in mechanical systems. The accuracy of the contact model is the important factor influencing the simulating results of the dynamic responses of the mechanism. For cylindrical joints with clearance, the Hertz model and the Persson model are the common methods for describing the contact behavior of the clearance. However, the numerical results based on the FEM analysis show that both of them have limitations in application. The Hertz model is available only in the case that there is a large clearance with a small normal load. The Persson model can be applied only in the case that there is a small clearance. By introducing some appropriate assumptions and analyzing the FEM numerical results, such as the contact area, the pressure distribution, and the maximum sustainable load, an approximate model for the contact problem of cylindrical joints with clearances is developed through modeling the pin as a rigid wedge and the elastic plate as a simple Winkler elastic foundation. The approximate model presented in this paper provides a more direct and effective formula to describe the properties of the contact in the cylindrical joint with clearances. 2006 Elsevier Ltd. All rights reserved. Keywords: Clearance; Conformal contact; FEM; Hertz theory; Persson theory
1. Introduction Clearances existing in different joints are inevitable due to the machining tolerances, wear, material deformation, and imperfections. The presence of such joint gaps can lead to degradation of the stability and accuracy of mechanisms movement. Meanwhile, the impact forces induced by joint clearances not only dissipate the energy of the system, but also are the mainly sources for vibrations and noise. The subject of studying the influences of joint clearances on the movement of mechanism draws the attention of many researchers in the past couple of years, not only in the field of mechanism dynamics, but also in multibody dynamics [1,2]. *
Corresponding author. Tel.: +86 10 627 561 77. E-mail address: [email protected] (C.-S. Liu).
0094-114X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.02.006
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The cylindrical joint is the most common type used in many mechanisms. Unlike the ideal case, a cylindrical joint with clearance has to be dealt with a different approach to consider the interaction of the impact taking place in the interior of the joint. In general, there are two main modeling strategies for mechanical system with cylindrical clearance joints, the addition-constraint approach [3,4], and the contact force approach [5–8]. In the addition-constraint approach, the clearance at a joint is modeled by adding a link with zero mass that has a constant length equal to the radial clearance. The contact force approach is more familiar and realistic since it can consider the impact process between two colliding bodies at the joint gaps, and the detailed information of the contact forces can be got by dynamic simulation. A good review about the studying of the mechanical system with revolute clearance joints can be found in [9]. Once the contact forces approach is adopted, the contact model becomes the important factor influencing the precision of the simulation results. Hertz theory is one of the most common methods establishing the relationship between the contact force and penetration displacement. Dubowsky et al. [10,11] adopted Hertz theory to deal with the contact problem of revolute joints with clearance, and presented ‘‘an impact pair’’ model to represent the interaction force at the contact surfaces. Bauchau et al. [12,13] have provided the kinematical description for different types of joints and established a systematic analyzing model through considering the non-linear factors, such as friction, clearances and lubrication existing in joints. However, Hertz theory is available only in solving the contact problem that the geometric shape of contact bodies is non-conformal, the properties of conformal contact at the contact surfaces of cylindrical joints consequentially result in the limitation of Hertz model in solving the contact problem of cylindrical joints with clearances. The finite element methods and boundary element methods are the most efficient tools for solving contact problems. Sassi and Desvignes [14] developed a semi-numerical method to solve the three-dimensional frictionless conformal contact problems. Loc et al. [15,16] presented a relationship between the contact force and the penetration displacement for the collisions of two elastic spheres by using finite element method. Rokach [17] studied the connection between the indentation and the contact force for a beam specimen pressed by a rigid cylindrical indenter based on a mixed analytical and numerical approach. Lim and Strong [18] analyzed the problem of oblique elastic–plastic impact between two rough cylinders by using FEM code. The influences of different contact parameters, such as elastic module ratio and Poisson’s ratios etc., on the contact stress around spherical and cylindrical inclusions are studied by Knight et al. [19]. The simulation parameters selected in FEM software may greatly influence the precision of the simulation results. Seieried et al. [20] studied the radial impact of a steel sphere colliding with a half-circular aluminum plate, and provided the comparison between the numerical results using FEM and experiment results using advanced measurement instruments. The FEM computed results can agree well with the experimental results if the simulation parameters are chosen appropriately. Yet, applying these numerical methods, FEM or BEM, directly into the multibody system is very difficult as these numerical methods often spend many computation resources, which may be unacceptable in solving a complex multibody system. So searching for an approximate force–displacement relationship properly describing the contact behavior becomes very important in dealing with the contact problem in multibody system. The simple formula, P = kf(d), is often used to represent the dynamic behavior of the joints with clearance in the software packages of multi-body dynamic analysis, such as ADAMS [21] and DADS [22] etc., where k is contact stiffness, P is the external load applied on each contact body, d is the normal penetration displacement. Obviously, the selection of the stiffness coefficient in this model strongly depends on experiment results, numerical simulations or theoretical analysis for a particular contact problem [23]. Even though it is difficult to get a general closed form solution for the conformal contact problem, some researchers have made many efforts to search for the approximate theoretical solutions for this contact problem. Johnson [24] had found the solution of the pressure distribution for the contact body with planar symmetric profiles or axi-symmetric profiles, but his solution is too obscure to apply, and only covers some special cases. Johnson [24] also provided a solution for the pressure distribution when two contact bodies have the identical material. Hussain [25] derived a different closed form solution for the case of that two contact bodies have zero clearance and elastic similar materials. Recently, Ciavarella and Decuzzi [26,27] developed the Persson’s theory and provided a completely closed form solution for the contact problem of cylindrical joints with clearance. However, all these theoretical results are based on a series of assumptions for some special
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cases, and the normal force–displacement relations, which are more useful for solving the clearance contact problem in multibody dynamic analysis, are not expressed clearly. Both Hertz model and Person model are the theoretical results based on some assumptions. In this paper, FEM is used to investigate the precisions of these methods. The numerical results show that both of them have limitations in application. The Hertz model is available only in the case that there is a large clearance with a small normal load. The Persson model can be applied only in the case that there is a small clearance. For studying the influence of the size of the joint clearance on the relation between contact load and penetration displacement, FEM models with different size of the clearances and different load are established. The numerical results are analyzed which then are used as guidelines for modeling the approximate formula of the normal force–displacement. Based on the Winkler elastic foundation theories and the assumption of pressure distributions based on FEM results, an approximate model for the cylindrical joint with clearance is developed, which is not only easy to be implemented in the analysis of multibody system, but more precise than the Hertz model and Persson model. 2. Hertz model Referring to Fig. 1, the contact problem of the cylindrical joints with clearance can be modeled as a planar stress contact problem of a pin in a hole in an infinite plate. R1, R2 refer to the radius of the pin and the hole in plate, respectively; DR = R2 R1 represents the radial clearance of joints. Both the pin and the plate are made of isotropic material; Ei, vi (i = 1, 2) are the Young’s modulus and Poisson’s ratio; / and w are the angles which represent the points on the interface departure from the centric line, respectively. e is the semi-angle of contact corresponding to the contact arc. The key to solve contact problems is to get the area of contact interface and the stress distribution. In Hertz theory [24], the contact area is approximated as a plane with a long strip of width 2a lying parallel to the axis of the cylinder, which can be considered as a limit of ellipse contact. The pressure distribution p(x) can be expressed in terms of semi-width a and the maximum pressure p0 at the center of the contact width. pðxÞ ¼ p0 ð1 x2 =a2 Þ1=2
ð1Þ
where x is the distance of the point departure from the center of the contact plane. The relation of the compressive load per unit axial length P and the semi-width a would be
Fig. 1. Cylindrical joints with clearance.
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P ¼ pa2 E =4R
ð2Þ
where E1 ¼ E11 þ E12 , R1 ¼ R11 R12 The maximum pressure p0 is written by 2P ¼ p0 ¼ pa
1=2 PE pR
ð3Þ
Defining the clearance of cylindrical joint DR = R2 R1, the semi-width a can be expressed as a2 ¼ 4PR1 R2 =pE DR
ð4Þ
Unlike the contact problem of three-dimensional bodies, the magnitude of the maximum elastic compression in two-dimensional contact bodies cannot be calculated directly from the contact stresses. The shape and size of the contact bodies themselves, as well as the way in which they are supported, may influence the relationship between the external load and the magnitude of the maximum elastic compression. Concerning the contact problem of cylindrical joints with clearance, the internal stress fields of two contact body can be regarded as planar stress fields if the length of the cylindrical joint is long enough. The magnitude of the maximum elastic compression is composed of two parts: one of that is the elastic compression of the pin d1 caused by the Hertz pressure distribution at the contact area, the other part is the compression of the plate d2 caused by the counterforce on the contact surfaces. Based on Ref. [24], the first part d1 is expressed as P 4R1 d1 ¼ 2 ln 1 ð5Þ pE1 a The compression of a half-space relative to a point at a depth d below the center of a contact pressure distribution, which corresponding to d2, is given by 1 2d d2 ¼ P 2 ln v2 ð6Þ pE2 a By taking d = R2, the compression of the elastic half-space can be approximately replaced as follows: 1 4R2 d2 ¼ P 2 ln 1 pE2 a
ð7Þ
According to Ref. [24], the error between Eqs. (6) and (7) does not exceed 10%, then, we can use Eq. (7) to represent the normal compression of the elastic half-space. If the material of the two contact bodies are identical, E1 = E2 = E*, the total maximum compression can be written as the following formula based on Eqs. (6) and (7). 1 4R1 R2 d ¼ d1 þ d2 ¼ 2P ln 1 ð8Þ pE a2 Substituting Eq. (4) into Eq. (8), the expression for the relationship between the total maximum compression displacement and the external load can be stated as 2P pE DR 1 ð9Þ d ¼ ln pE P Some authors adopted the relation in (9) to represent the contact behavior at the cylindrical clearance joints. However, the conditions of applying Hertz theory are that the shapes of two contact bodies should be nonconformal and the contact surface should be a plane. Concerning the cylindrical clearance joints, the contact area will grow rapidly and become comparable with the radius of the pin as the external load increase. So Eq. (9) must have limitations when it is applied into solving the contact problem of the cylindrical joint with clearance.
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3. Persson’s theory Ref. [26,27] developed the Persson theory and presented a completely closed form solution for the contact problem of cylindrical joints with clearance. The following assumptions are introduced in their research. (1) The coupling between the radial displacement and the tangential displacement at contact region is neglected. (2) The relation between the radial displacements at contact area and the maximum normal penetration satisfies the geometric rigidity condition of a rigid body contact. Referring to Fig. 2, points at the two surfaces S1 and S2 which come into contact experience both radial and tangential displacement uri, uhi (i = 1, 2). As described in the above section, / and w are the angles which represent the points on the interface departure from the centric line. e is the semi-angle of contact corresponding to the contact arc. Based on above assumptions, there exits the following geometric relations: ðR2 þ ur2 Þ ðR1 þ ur1 Þ ¼ ðDR þ dÞ cos /
ð10Þ
ur2 ur1 ¼ d cos / DRð1 cos /Þ
ð11Þ
i.e. Defining the following non-dimensional parameters, load parameter E1DR/P, Persson’s first material parameter g = E1/E2, Persson’s second material parameter k = (1 v1) g(1 v2). The auxiliary variables are defined as follows: y ¼ tanð/=2Þ; t ¼ tanðw=2Þ; b ¼ tanðe=2Þ; p 2ð1 þ gÞ k 1g k a¼ ; b¼ k¼ 2 1þg 1þg 2ð1 þ gÞ
ð12Þ
The relationship between load parameter and semi-angle is expressed in [26] as E DR ða 1Þ½logðb2 þ 1Þ þ 2b4 þ 2 4b ¼ 2 2 P pð1 þ aÞ pð1 þ aÞðb þ 1Þb
ð13Þ
If the two contact bodies with identical materials, then a = 0 and b = 0 according to Eq. (12), and then Eq. (13) can be further simplified as E DR 2 ½logðb2 þ 1Þ þ 2b4 ¼ P pðb2 þ 1Þb2
Fig. 2. The geometric description of the cylindrical joint with clearance.
ð14Þ
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Due to the fact that DR and d are much less than R1 or R2, Let us evaluate the limit of Eq. (14) as b = tan(e/ 2) e/2 1 and a e R1 R2e, then E DR 2 ½logðb2 þ 1Þ þ 2b4 2 4R2 R1 ¼ lim 2 2 2 b!0 P pa2 pðb þ 1Þb pb
ð15Þ
Obviously, Eq. (15) has the same form as Eq. (4) which is based on the Hertz theory. So we can say that the Persson model will approach to the solution of the Hertz theory when the contact semi-angle is very small. Based on the boundary condition of the contact region ur2 ur1 = 0, the relationship between contact semiangle e and the penetration displacement d can be expressed as follows: cosðeÞ ¼
DR DR þ d
ð16Þ
Combining Eq. (16) with Eq. (14), the relation between the contact load P and the penetration displacement d can be calculated through numerical methods. It should be noticed that the geometric relations expressed in Eq. (16) is available only when the radial clearance is very small and the contact semi-angle is large enough. If this model is applied into the case of large clearance, it will definitely result in a large error because the geometric rigidity relation expressed in Eq. (16) is not satisfied in this situation. 4. Finite element analyse for cylindrical joints with clearance 4.1. Modeling of the cylindrical joints with clearances in FEM There are mainly two methods in modeling and simulation for the normal contact problem in the FEM code: one that is the penalty method; one that is the Lagrange methods. The penalty method can fulfill the impenetrability condition only approximately by introducing a penalty factor. But the simulation results strongly depend on the chosen penalty factor. A rough estimation for the choice of the penalty factor can be found in [20]. In Lagrange methods, the impenetrability conditions are introduced by Lagrange multipliers which have the physical meaning of contact forces. This method provides an exact compliance of the impenetrability. However, the computation time of this method may be very long since the non-linear system equations are difficult to solve. A combination of penalty and Lagrange method is the combined Lagrange–penalty method. Here, the Lagrange multipliers are not treated as unknowns but they are computed iteratively until a predefined acceptable penetration tolerance is not exceeded within a time step. The advantage of the combined Lagrange–penalty method is that the penetrations are controlled directly by the penetration tolerance and not indirectly by the penalty factor as in the penalty method. In [20], the influence of different simulation parameters is studied by using the sphere to rod impact problem as test example for FE approach. It pointed out that the penalty factor is crucial for a correct simulation even if the penalty method is much more efficient than the combined Lagrange–penalty method in computation time. Concerning the contact problem of cylindrical joints with clearance, the combined Lagrange–penalty method is adopted in this paper to ensure the impenetrability condition although it needs much more computation time. Since the cylindrical joints with clearance only sustain the unilateral contact forces in the movement of mechanism, the upper half of the cylinder beyond the central plane can be regarded as a rigid body. The analytical model is referred to Fig. 3. The normal load P is applied on the rigid plane AB passing through the center of cylinder and only moving along the z-direction. Considering the conditions of that the contact materials are identical and linear elastic and without friction in the contact interface, we introduce the following computation strategy to simulate the contact process: firstly, a certain penetration displacement d is applied on the rigid plane AB in the FEM model of the cylindrical joint with clearance; secondly the contact load P can be determined by summing the constraint forces acting on AB plane according to the finite element numerical results.
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Fig. 3. The analytical model.
In order to get accurate FEM numerical results to represent the contact behavior of the cylindrical joint with clearance, we divide the pin into four regions and the plate into three regions. The mesh of each region is generated by using automatic generation mesh command in the FEM software. Referring to Fig. 4, the finite element mesh of cylindrical joints contains in all 15,400 quad8 planar stress element. The contacting material is steel with Young’s modulus E = 2.0 · 105 MPa and Poisson’s ratio l = 0.3, the radius of hole R2 = 100 mm. For studying the influence of different clearances and different loads, we select the following model parameters in FEM analyses: the sizes of the clearance DR are selected in all 15 different values, 5, 4, 3, 2, 1, 0.9, 0.8, . . . , 0.2, 0.1 mm, respectively. The penetration displacements d are selected in all 10 different values, 0.02, 0.04, . . . ,0.18, 0.2, 0.02 mm, respectively. The combination of these two parameters generates in all 150 different models. 4.2. Analysis of FEM results Figs. 5 and 6 show the normal contact force P against the normal penetration displacement in DR = 1 mm and DR = 0.1 mm, respectively. The solid-line curves in the figures represent the Hertz relation of Eq. (9), and the dots in the figures represent the numerical results based on FEM simulation.
Fig. 4. The FE mesh.
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Fig. 5. The relation between contact load and the penetration displacement in DR = 1 mm.
Fig. 6. The relation between contact load and the penetration displacement in DR = 0.1 mm.
If the contact load is not big enough, it can be seen from Fig. 5 that the Hertz relation of Eq. (9) can agree well with the FEM results in DR = 1 mm. However, the Hertz curves will gradually departure from the FEM results as the increase of the normal load. For observing how the difference between the Hertz curves and FEM results changes as the size of the clearance decreases, Fig. 6 shows the numerical results in DR = 0.1 mm. It is obvious that the Hertz relations will departure from the FEM results even if the load is very small. The reason of inducing the above phenomena can be attributed to that the Hertz assumption is unavailable when the radial clearance is very small. For each pair of the parameters (DR, d), the dimensionless load parameter EDR/P can be obtained through FEM calculations. Meanwhile, the Persson’ solution can be calculated based on Eqs. (14) and (16). In different radial clearance DR and different normal displacement d, Fig. 7 shows the relationship between the dimensionless load parameter EDR/P and the parameter b = tan(e/2).
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Observe Fig. 7, we can see that the FEM results closely approach to the Persson solution in b P 0.6, but they will gradually departure from each other as the contact semi-angle decrease (b 6 0.6). This illustrates that the Persson’s theory is available only when the contact angle is bigger enough. If the contact angle is small, then the geometrical relationship expressed in Eq. (11) will not satisfied, and the basic assumption of the Persson theory will be violated. The relationship of dimensionless pressure distribution Rp(y)/P against the contact angle w in DR = 0.1 mm is exhibited in Fig. 8. We firstly set the semi-angle of contact e = 30, 60, 80, respectively. Based on Eq. (16), we can determine the penetration displacement corresponding to each contact semi-angle. Then pressure distribution can be got based on FEM analysis. It is obvious that the contact region is not completely consistent with the initial value of the contact semi-angle especially in the case of small value of the contact semi-angle.
Fig. 7. The relation between dimensionless load parameter and the contact semi-angle based on Persson model.
Fig. 8. The relation between the dimensionless pressure distribution and the contact angle.
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5. The approximate compliance model for cylindrical joints with clearance The difficult of solving the contact problem of cylindrical joint with clearance is that the profiles of the contact bodies cannot be adequately represented by their radii of curvature at the point of first contact. According to Ref. [24], the elastic foundation model can provide simple approximate solutions in the complex situations. In order to get a reasonable simple contact compliance model for this problem, the method of the Winkler elastic foundation model is used in this paper, and some assumptions are made firstly. (1) The pin is equivalent to a rigid wedge, and the contact pressure distribution in z-direction along the profile of the rigid wedge is ellipsoidal as given by the Hertz theory. (2) The contact boundary between the equivalent rigid wedge and elastic foundation satisfies the geometric relation Eq. (16). (3) The plate is modeled by a simple Winkler elastic foundation with depth R2 and stiffness k, which rests on a rigid base and is compressed by the rigid wedge. The assumption (1) and (2) determine the profile of the equivalent rigid wedge and the shape of the contact pressure distribution, the assumption (3) is used to determine the maximum contact pressure between the rigid wedge and elastic foundation. The analytical model for the cylindrical joints with clearance is illustrated in Fig. 9. The semi-angle b of the wedge is determined by the contact semi-angle e, and tan b = 1/tan(e/2). According to the assumption (2), the relationship between contact semi-angle e and the penetration displacement d of the equivalent rigid wedge at the contact boundary is determined by Eq. (16). Once the profiles of the rigid wedge are determined, the contact shape and the size are also determined. The contact pressure distribution in z-direction along the profile of the rigid wedge is expressed as 2 1=2
pz ðxÞ ¼ pz0 ½1 ðx=lÞ
ð17Þ
where x is the distance along the wedge direction between the point on the profile and the vertex of the wedge, l is the length of the chord, pz0 is the maximum contact pressure at the vertex. Since DR is much less than R1 and R2, the relationship between l and e can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l ¼ R2 ð1 cos eÞ þ sin e2 ¼ 2R2 sinðe=2Þ
Fig. 9. The simple model for cylindrical joints with clearance.
ð18Þ
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There is no interaction between the springs of the Winkle elastic foundation, i.e. shear between adjacent elements of the foundation is ignored. The contact pressure at any point depends only on the displacement at that point, thus pz ¼ ðE=R2 Þuz
ð19Þ
where E is the elastic modulus of the foundation, uz is the normal elastic displacements of the foundation. Since the displacement of the elastic foundation at the vertex of the wedge is approximately half of the total maximum normal penetration d based on FEM numerical results, i.e. uz0 = d/2, and the materials of the two contact bodies are identical, i.e. E = E*, the linear relation between pz0 and the penetration d can be expressed as follows: py0 ¼ E d=2R2
ð20Þ
For verifying the correctness of Eq. (20), Fig. 10 shows the curves of the maximum contact pressure at the vertex of the rigid wedge against the penetration displacement based on FEM results. It is obvious that the compliance of the point at the vertex can be well modeled by above expression. According the condition of equilibrium, we can obtain: Z 2 py ðxÞdx ¼ P ð21Þ DC
Substituting Eqs. (17), (18), and (20) into (21), the relationship of load P and rigid body displacement can be simply expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d ð22Þ P ¼ pdE 2 2ðDR þ dÞ By introducing the following non-dimensional load parameter DRE*/P and k = sin(e/2), Eq. (16) can be expressed as d ¼ DR
2k 1 2k2
ð23Þ
Introducing Eq. (23) into Eq. (22), the relationship between the non-dimensional load parameter and the contact semi-angle can be written by 1 2k2 DRE ¼ P pk3
Fig. 10. The maximum pressure at different clearance and different normal penetration displacement through FEM analyses.
ð24Þ
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6. The comparison of different models for cylindrical joints with clearance Referring to Fig. 11, the relation between the dimensionless load parameter EDR/P and contact semi-angle k agrees well with the FEA results. The maximum error between them is less than 2.0% even in the case of contact semi-angle k 0.2, corresponding to a very small contact area of the cylindrical joints with a large clearance. When k is greater than 0.3, the results of above two methods are almost the same. This illustrates that the approximate model proposed in the paper can effectively solve the contact problem of the cylindrical joints with clearance in a wider range as compared with the Hertz theory and the Persson theory. The relations between the normal load P and the normal penetration displacement d obtained from different model with different normal clearance DR and dmax = 0.2 mm are shown in Fig. 12. Referring to Fig. 12(a), which represents the normal load versus normal displacement curves in the case of DR = 1 mm, it can be observed that the curve produced by the Persson model deviates largely from the FEM results even in a very small normal load, on the other hand, the curve produced by Hertz model agrees well with the FEM results. The reason is the geometric rigidity relation Eq. (16) is unavailable when the clearance is larger enough, but the Hertz assumption that the contact area is nearly a plane is satisfied in this case. Although the geometric relation Eq. (16) is applied in the approximate model based on elastic foundation theory, the curve given by Eq. (22) also agrees well with the FEM results. This illustrates that the model presented in this paper can effectively represent the contact behavior of the cylindrical joints with joints even in the case of large clearance. In the case of DR = 0.5 mm, We can observe from Fig. 12(b) that the curve obtained by Hertz model is near the FEM results only when the normal load P is less than 5000 N. With the increase of the normal load, the curve obtained by Hertz theory begins to departure from the FEM results. Meanwhile, the curve obtained by Persson theory is still away from FEM results in this case. This shows that the contact area does not persist in a plane with the decrease of the size of the clearance, and the geometric relation Eq. (16) is not satisfied in this case. Although the clearance has been changed, the curve obtained by the new model proposed in this paper still agrees well with the FEM results as seen in Fig. 12(b). Referring to Fig. 12(c) for the case of DR = 0.2 mm, the curve obtained by Hertz theory agrees with FEM results only when the normal load P is less than 2000 N, and will rapidly deviate from the FEM results with the increase of the normal load. The reason is the equivalent radii at the contact area may exceed the admissible range of [0, R1] as clearance is very small. The curve obtained by Persson theory still shows a difference as compared with the FEM results. Meanwhile, the approximate model still agrees well with the FEM results regardless of the variation of the size of the clearance.
Fig. 11. The relation between dimensionless load parameter and semi-angle based on the approximate model.
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Fig. 12. The relations between the normal load and the penetration displacement obtained from different model with different normal clearance.
In the case of DR = 0.1 mm, it can be seen from Fig. 12(d) that the Hertz theory is effective only for a very small normal load, and the curve obtained from Persson results is much closer to the result of FEM than before. This further explains that the geometric relation Eq. (16) is available only in the case of very small clearance. Likewise, the new model is still available in this case. 7. Conclusion This paper has studied the relation between the contact load and the penetration displacement for cylindrical joints with clearance. Using FEM code, the contact area, the pressure distribution, and the maximum sustainable load for clearance joint are presented based on the numerical results. The Hertz theory and the Persson theory are the common method used to represent the contact behavior of the cylindrical joint with clearance, but both of them are not always available, which are strongly depend on the size of the clearance, as compared with the FEM numerical results. The Hertz model is effective only in the condition of that the clearance is large enough and the normal load is very small. The Persson model can be applied only in the case that there is a small clearance. This can be attributed to the different assumptions adopting in the Hertz model and the Persson model. Hertz theory is established based on the plane assumption of the contact region, so the contact between two bodies in the cylindrical joint can satisfy the assumption of Hertz theory only in
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the case of that the clearance is larger enough and the load is very small. Persson theory is developed based on the geometrical rigidity constrain equations, so the contact between two bodies can satisfy the condition of the Persson theory only in the situation of that the clearance is small enough. By introducing some appropriate assumptions, a new approximate model for solving this contact problem has been presented in this paper. The contact problem of the cylindrical joint is simplified as a rigid wedge pressed into an elastic half-space plate, and a simple formula to describe the relation between the contact load and the penetration displacement is proposed. A comparison between the results obtained using the simple model and the FEM results shows that the new model more effective than the Hertz theory and the Persson theory. Acknowledgements The support of the National Science Foundation of China (10272002; 60334030) and the Foundation of Engineering Research Institute of Peking University (204035) is gratefully acknowledged. References [1] G. Gilardi, I. Sharf, Literature survey of contact dynamics modeling, Mechanism and Machine Theory 37 (2002) 1213–1239. [2] W. Schiehlen, Multibody system dynamics: roots and perspectives, Multibody System Dynamics 1 (1997) 149–188. [3] R.S. Haines, A theory of contact loss at resolute joints with clearance, Journal of Mechanical Engineering and Science 22 (3) (1980) 129–136. [4] K.L. Ting, J.D. Zhu, D. Watkins, The effects of joint clearance on position and orientation deviation of linkages and manipulators, Mechanism and Machine Theory 35 (2000) 391–401. [5] P. Ravn, A continuous analysis method for planar multibody systems with joint clearance, Multibody System Dynamics 2 (1988) 1– 24. [6] Y. Khulief, A. Shabana, A continuous force model for the impact analysis of flexible multi-body systems, Mechanism and Machine Theory 22 (1987) 213–224. [7] H. Lankarani, P. Nikravesh, A contact force model with hysteresis damping for impact analysis of multi-body systems, Journal of Mechanical Design 112 (1990) 369–376. [8] H. Lankarani, P. Nikravesh, Continuous contact force models for impact analysis in multibody systems, Nonlinear Dynamics 5 (1994) 193–207. [9] J. Ambrosio, P. Flores, Revolute joints with clearance in multibody systems, Computers and Structures 82 (2004) 1359–1369. [10] S. Dubowskey, F. Freudenatein, Dynamic analysis of mechanical systems with clearances, part 1: formulation of dynamic mode, and part 2: dynamic response, ASME Journal of Engineering for Industry 93 (1971) 305–316. [11] J.F. Deck, S. Dubowsky, On the limitations of predictions of the dynamic response of machines with clearance connections, ASME Journal of Mechanical Design 116 (1994) 833–841. [12] O.A. Bauchau, C.L. Bottasso, Contact conditions for cylindrical, prismatic, and screw joints in flexible multibody systems, Multibody System Dynamics 5 (2001) 251–278. [13] Olivier A. Bauchau, Jesus Rodriguez, Modeling of joints with clearance in flexible multibody systems, International Journal of Solids and Structure 39 (2002) 41–63. [14] M. Sassi, M. Desvignes, A seminumerical method for three-dimensional frictionless contact problems, Mathematical and Computing Modelling 28 (1998) 413–425. [15] Vu-Quoc Loc, Z.H. Xiang, L. Lee, Normal and tangential force–displacement relations for frictional elasto-plastic contact of spheres, International Journal of solids and Structure 38 (2001) 6455–6489. [16] Z. Xiang, Vu-Quoc Loc, Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions, International Journal of Impact Engineering 27 (2003) 317–341. [17] I.V. Rokach, On the accurate determination of contact compliance for impact test modeling, International Journal of Solids and Structures 40 (2003) 2715–2729. [18] C.T. Lim, W.J. Strong, Oblique elastic–plastic impact between rough cylinders in plane strain, International Journal of Engineering Science 37 (1999) 97–122. [19] M.G. Knight, L.A. Lacerda, L.C. Wrobel, J.L. Henshall, Parametric study of the contact stresses around spherical and cylindrical inclusions, Computational Materials Science 25 (2002) 115–121. [20] R. Seieried, B. Hu, P. Eberhard, Numerical and experimental investigation of radial impacts on a half-circular plate, Multibody System Dynamics 9 (2003) 265–281. [21] R.R. Ryan, ADAMS—Multibody System Analysis Software, Multibody Systems Handbook, Springer-Verlag, Berlin, 1990. [22] R.C. Smith, E.J. Haug, DADS—Dynamic Analysis and Design System, Multibody Systems Handbook, Springer-Verlag, Berlin, 1990. [23] A.L. Schwab, J.P. Meijaard, P. Meijers, A comparison of revolute joint clearance models in the dynamic analysis of rigid and elastic mechanical systems, Mechanism and Machine Theory 37 (2002) 895–913.
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[24] K.L. Johnson, Contact Mechanics, Cambridge University Press, London, 1992. [25] N. Hussain, Exact solution of certain dual series for indentation and inclusion problems, International Journal of Engineering Science 7 (1969) 1149–1161. [26] M. Ciavarella, P. Decuzzi, The state of stress induced by the plane frictionless cylindrical contact 1: the case of elastic similarity, International Journal of Solids and Structures 38 (2001) 4507–4523. [27] M. Ciavarella, P. Decuzzi, The state of stress induced by the plane frictionless cylindrical contact. 2: the general case (elastic dissimilarity), International Journal of Solids and Structures 38 (2001) 4523–4533.
Mechanism and Machine Theory 42 (2007) 183–197
www.elsevier.com/locate/mechmt
The FEM analysis and approximate model for cylindrical joints with clearances Cai-Shan Liu a
a,*
, Ke Zhang a, Rei Yang
b
Department of Mechanics and Engineering Science, State Key Laboratory of Turbulence and Complex Systems, Peking University, Beijing 100871, China b Chinese Academy of Space Technology, Beijing 100086, China Received 21 January 2005; received in revised form 26 January 2006; accepted 28 February 2006 Available online 11 May 2006
Abstract The clearances existing in the joints of mechanisms are the mainly sources for vibrations and noise in mechanical systems. The accuracy of the contact model is the important factor influencing the simulating results of the dynamic responses of the mechanism. For cylindrical joints with clearance, the Hertz model and the Persson model are the common methods for describing the contact behavior of the clearance. However, the numerical results based on the FEM analysis show that both of them have limitations in application. The Hertz model is available only in the case that there is a large clearance with a small normal load. The Persson model can be applied only in the case that there is a small clearance. By introducing some appropriate assumptions and analyzing the FEM numerical results, such as the contact area, the pressure distribution, and the maximum sustainable load, an approximate model for the contact problem of cylindrical joints with clearances is developed through modeling the pin as a rigid wedge and the elastic plate as a simple Winkler elastic foundation. The approximate model presented in this paper provides a more direct and effective formula to describe the properties of the contact in the cylindrical joint with clearances. 2006 Elsevier Ltd. All rights reserved. Keywords: Clearance; Conformal contact; FEM; Hertz theory; Persson theory
1. Introduction Clearances existing in different joints are inevitable due to the machining tolerances, wear, material deformation, and imperfections. The presence of such joint gaps can lead to degradation of the stability and accuracy of mechanisms movement. Meanwhile, the impact forces induced by joint clearances not only dissipate the energy of the system, but also are the mainly sources for vibrations and noise. The subject of studying the influences of joint clearances on the movement of mechanism draws the attention of many researchers in the past couple of years, not only in the field of mechanism dynamics, but also in multibody dynamics [1,2]. *
Corresponding author. Tel.: +86 10 627 561 77. E-mail address: [email protected] (C.-S. Liu).
0094-114X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.02.006
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The cylindrical joint is the most common type used in many mechanisms. Unlike the ideal case, a cylindrical joint with clearance has to be dealt with a different approach to consider the interaction of the impact taking place in the interior of the joint. In general, there are two main modeling strategies for mechanical system with cylindrical clearance joints, the addition-constraint approach [3,4], and the contact force approach [5–8]. In the addition-constraint approach, the clearance at a joint is modeled by adding a link with zero mass that has a constant length equal to the radial clearance. The contact force approach is more familiar and realistic since it can consider the impact process between two colliding bodies at the joint gaps, and the detailed information of the contact forces can be got by dynamic simulation. A good review about the studying of the mechanical system with revolute clearance joints can be found in [9]. Once the contact forces approach is adopted, the contact model becomes the important factor influencing the precision of the simulation results. Hertz theory is one of the most common methods establishing the relationship between the contact force and penetration displacement. Dubowsky et al. [10,11] adopted Hertz theory to deal with the contact problem of revolute joints with clearance, and presented ‘‘an impact pair’’ model to represent the interaction force at the contact surfaces. Bauchau et al. [12,13] have provided the kinematical description for different types of joints and established a systematic analyzing model through considering the non-linear factors, such as friction, clearances and lubrication existing in joints. However, Hertz theory is available only in solving the contact problem that the geometric shape of contact bodies is non-conformal, the properties of conformal contact at the contact surfaces of cylindrical joints consequentially result in the limitation of Hertz model in solving the contact problem of cylindrical joints with clearances. The finite element methods and boundary element methods are the most efficient tools for solving contact problems. Sassi and Desvignes [14] developed a semi-numerical method to solve the three-dimensional frictionless conformal contact problems. Loc et al. [15,16] presented a relationship between the contact force and the penetration displacement for the collisions of two elastic spheres by using finite element method. Rokach [17] studied the connection between the indentation and the contact force for a beam specimen pressed by a rigid cylindrical indenter based on a mixed analytical and numerical approach. Lim and Strong [18] analyzed the problem of oblique elastic–plastic impact between two rough cylinders by using FEM code. The influences of different contact parameters, such as elastic module ratio and Poisson’s ratios etc., on the contact stress around spherical and cylindrical inclusions are studied by Knight et al. [19]. The simulation parameters selected in FEM software may greatly influence the precision of the simulation results. Seieried et al. [20] studied the radial impact of a steel sphere colliding with a half-circular aluminum plate, and provided the comparison between the numerical results using FEM and experiment results using advanced measurement instruments. The FEM computed results can agree well with the experimental results if the simulation parameters are chosen appropriately. Yet, applying these numerical methods, FEM or BEM, directly into the multibody system is very difficult as these numerical methods often spend many computation resources, which may be unacceptable in solving a complex multibody system. So searching for an approximate force–displacement relationship properly describing the contact behavior becomes very important in dealing with the contact problem in multibody system. The simple formula, P = kf(d), is often used to represent the dynamic behavior of the joints with clearance in the software packages of multi-body dynamic analysis, such as ADAMS [21] and DADS [22] etc., where k is contact stiffness, P is the external load applied on each contact body, d is the normal penetration displacement. Obviously, the selection of the stiffness coefficient in this model strongly depends on experiment results, numerical simulations or theoretical analysis for a particular contact problem [23]. Even though it is difficult to get a general closed form solution for the conformal contact problem, some researchers have made many efforts to search for the approximate theoretical solutions for this contact problem. Johnson [24] had found the solution of the pressure distribution for the contact body with planar symmetric profiles or axi-symmetric profiles, but his solution is too obscure to apply, and only covers some special cases. Johnson [24] also provided a solution for the pressure distribution when two contact bodies have the identical material. Hussain [25] derived a different closed form solution for the case of that two contact bodies have zero clearance and elastic similar materials. Recently, Ciavarella and Decuzzi [26,27] developed the Persson’s theory and provided a completely closed form solution for the contact problem of cylindrical joints with clearance. However, all these theoretical results are based on a series of assumptions for some special
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cases, and the normal force–displacement relations, which are more useful for solving the clearance contact problem in multibody dynamic analysis, are not expressed clearly. Both Hertz model and Person model are the theoretical results based on some assumptions. In this paper, FEM is used to investigate the precisions of these methods. The numerical results show that both of them have limitations in application. The Hertz model is available only in the case that there is a large clearance with a small normal load. The Persson model can be applied only in the case that there is a small clearance. For studying the influence of the size of the joint clearance on the relation between contact load and penetration displacement, FEM models with different size of the clearances and different load are established. The numerical results are analyzed which then are used as guidelines for modeling the approximate formula of the normal force–displacement. Based on the Winkler elastic foundation theories and the assumption of pressure distributions based on FEM results, an approximate model for the cylindrical joint with clearance is developed, which is not only easy to be implemented in the analysis of multibody system, but more precise than the Hertz model and Persson model. 2. Hertz model Referring to Fig. 1, the contact problem of the cylindrical joints with clearance can be modeled as a planar stress contact problem of a pin in a hole in an infinite plate. R1, R2 refer to the radius of the pin and the hole in plate, respectively; DR = R2 R1 represents the radial clearance of joints. Both the pin and the plate are made of isotropic material; Ei, vi (i = 1, 2) are the Young’s modulus and Poisson’s ratio; / and w are the angles which represent the points on the interface departure from the centric line, respectively. e is the semi-angle of contact corresponding to the contact arc. The key to solve contact problems is to get the area of contact interface and the stress distribution. In Hertz theory [24], the contact area is approximated as a plane with a long strip of width 2a lying parallel to the axis of the cylinder, which can be considered as a limit of ellipse contact. The pressure distribution p(x) can be expressed in terms of semi-width a and the maximum pressure p0 at the center of the contact width. pðxÞ ¼ p0 ð1 x2 =a2 Þ1=2
ð1Þ
where x is the distance of the point departure from the center of the contact plane. The relation of the compressive load per unit axial length P and the semi-width a would be
Fig. 1. Cylindrical joints with clearance.
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P ¼ pa2 E =4R
ð2Þ
where E1 ¼ E11 þ E12 , R1 ¼ R11 R12 The maximum pressure p0 is written by 2P ¼ p0 ¼ pa
1=2 PE pR
ð3Þ
Defining the clearance of cylindrical joint DR = R2 R1, the semi-width a can be expressed as a2 ¼ 4PR1 R2 =pE DR
ð4Þ
Unlike the contact problem of three-dimensional bodies, the magnitude of the maximum elastic compression in two-dimensional contact bodies cannot be calculated directly from the contact stresses. The shape and size of the contact bodies themselves, as well as the way in which they are supported, may influence the relationship between the external load and the magnitude of the maximum elastic compression. Concerning the contact problem of cylindrical joints with clearance, the internal stress fields of two contact body can be regarded as planar stress fields if the length of the cylindrical joint is long enough. The magnitude of the maximum elastic compression is composed of two parts: one of that is the elastic compression of the pin d1 caused by the Hertz pressure distribution at the contact area, the other part is the compression of the plate d2 caused by the counterforce on the contact surfaces. Based on Ref. [24], the first part d1 is expressed as P 4R1 d1 ¼ 2 ln 1 ð5Þ pE1 a The compression of a half-space relative to a point at a depth d below the center of a contact pressure distribution, which corresponding to d2, is given by 1 2d d2 ¼ P 2 ln v2 ð6Þ pE2 a By taking d = R2, the compression of the elastic half-space can be approximately replaced as follows: 1 4R2 d2 ¼ P 2 ln 1 pE2 a
ð7Þ
According to Ref. [24], the error between Eqs. (6) and (7) does not exceed 10%, then, we can use Eq. (7) to represent the normal compression of the elastic half-space. If the material of the two contact bodies are identical, E1 = E2 = E*, the total maximum compression can be written as the following formula based on Eqs. (6) and (7). 1 4R1 R2 d ¼ d1 þ d2 ¼ 2P ln 1 ð8Þ pE a2 Substituting Eq. (4) into Eq. (8), the expression for the relationship between the total maximum compression displacement and the external load can be stated as 2P pE DR 1 ð9Þ d ¼ ln pE P Some authors adopted the relation in (9) to represent the contact behavior at the cylindrical clearance joints. However, the conditions of applying Hertz theory are that the shapes of two contact bodies should be nonconformal and the contact surface should be a plane. Concerning the cylindrical clearance joints, the contact area will grow rapidly and become comparable with the radius of the pin as the external load increase. So Eq. (9) must have limitations when it is applied into solving the contact problem of the cylindrical joint with clearance.
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3. Persson’s theory Ref. [26,27] developed the Persson theory and presented a completely closed form solution for the contact problem of cylindrical joints with clearance. The following assumptions are introduced in their research. (1) The coupling between the radial displacement and the tangential displacement at contact region is neglected. (2) The relation between the radial displacements at contact area and the maximum normal penetration satisfies the geometric rigidity condition of a rigid body contact. Referring to Fig. 2, points at the two surfaces S1 and S2 which come into contact experience both radial and tangential displacement uri, uhi (i = 1, 2). As described in the above section, / and w are the angles which represent the points on the interface departure from the centric line. e is the semi-angle of contact corresponding to the contact arc. Based on above assumptions, there exits the following geometric relations: ðR2 þ ur2 Þ ðR1 þ ur1 Þ ¼ ðDR þ dÞ cos /
ð10Þ
ur2 ur1 ¼ d cos / DRð1 cos /Þ
ð11Þ
i.e. Defining the following non-dimensional parameters, load parameter E1DR/P, Persson’s first material parameter g = E1/E2, Persson’s second material parameter k = (1 v1) g(1 v2). The auxiliary variables are defined as follows: y ¼ tanð/=2Þ; t ¼ tanðw=2Þ; b ¼ tanðe=2Þ; p 2ð1 þ gÞ k 1g k a¼ ; b¼ k¼ 2 1þg 1þg 2ð1 þ gÞ
ð12Þ
The relationship between load parameter and semi-angle is expressed in [26] as E DR ða 1Þ½logðb2 þ 1Þ þ 2b4 þ 2 4b ¼ 2 2 P pð1 þ aÞ pð1 þ aÞðb þ 1Þb
ð13Þ
If the two contact bodies with identical materials, then a = 0 and b = 0 according to Eq. (12), and then Eq. (13) can be further simplified as E DR 2 ½logðb2 þ 1Þ þ 2b4 ¼ P pðb2 þ 1Þb2
Fig. 2. The geometric description of the cylindrical joint with clearance.
ð14Þ
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Due to the fact that DR and d are much less than R1 or R2, Let us evaluate the limit of Eq. (14) as b = tan(e/ 2) e/2 1 and a e R1 R2e, then E DR 2 ½logðb2 þ 1Þ þ 2b4 2 4R2 R1 ¼ lim 2 2 2 b!0 P pa2 pðb þ 1Þb pb
ð15Þ
Obviously, Eq. (15) has the same form as Eq. (4) which is based on the Hertz theory. So we can say that the Persson model will approach to the solution of the Hertz theory when the contact semi-angle is very small. Based on the boundary condition of the contact region ur2 ur1 = 0, the relationship between contact semiangle e and the penetration displacement d can be expressed as follows: cosðeÞ ¼
DR DR þ d
ð16Þ
Combining Eq. (16) with Eq. (14), the relation between the contact load P and the penetration displacement d can be calculated through numerical methods. It should be noticed that the geometric relations expressed in Eq. (16) is available only when the radial clearance is very small and the contact semi-angle is large enough. If this model is applied into the case of large clearance, it will definitely result in a large error because the geometric rigidity relation expressed in Eq. (16) is not satisfied in this situation. 4. Finite element analyse for cylindrical joints with clearance 4.1. Modeling of the cylindrical joints with clearances in FEM There are mainly two methods in modeling and simulation for the normal contact problem in the FEM code: one that is the penalty method; one that is the Lagrange methods. The penalty method can fulfill the impenetrability condition only approximately by introducing a penalty factor. But the simulation results strongly depend on the chosen penalty factor. A rough estimation for the choice of the penalty factor can be found in [20]. In Lagrange methods, the impenetrability conditions are introduced by Lagrange multipliers which have the physical meaning of contact forces. This method provides an exact compliance of the impenetrability. However, the computation time of this method may be very long since the non-linear system equations are difficult to solve. A combination of penalty and Lagrange method is the combined Lagrange–penalty method. Here, the Lagrange multipliers are not treated as unknowns but they are computed iteratively until a predefined acceptable penetration tolerance is not exceeded within a time step. The advantage of the combined Lagrange–penalty method is that the penetrations are controlled directly by the penetration tolerance and not indirectly by the penalty factor as in the penalty method. In [20], the influence of different simulation parameters is studied by using the sphere to rod impact problem as test example for FE approach. It pointed out that the penalty factor is crucial for a correct simulation even if the penalty method is much more efficient than the combined Lagrange–penalty method in computation time. Concerning the contact problem of cylindrical joints with clearance, the combined Lagrange–penalty method is adopted in this paper to ensure the impenetrability condition although it needs much more computation time. Since the cylindrical joints with clearance only sustain the unilateral contact forces in the movement of mechanism, the upper half of the cylinder beyond the central plane can be regarded as a rigid body. The analytical model is referred to Fig. 3. The normal load P is applied on the rigid plane AB passing through the center of cylinder and only moving along the z-direction. Considering the conditions of that the contact materials are identical and linear elastic and without friction in the contact interface, we introduce the following computation strategy to simulate the contact process: firstly, a certain penetration displacement d is applied on the rigid plane AB in the FEM model of the cylindrical joint with clearance; secondly the contact load P can be determined by summing the constraint forces acting on AB plane according to the finite element numerical results.
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Fig. 3. The analytical model.
In order to get accurate FEM numerical results to represent the contact behavior of the cylindrical joint with clearance, we divide the pin into four regions and the plate into three regions. The mesh of each region is generated by using automatic generation mesh command in the FEM software. Referring to Fig. 4, the finite element mesh of cylindrical joints contains in all 15,400 quad8 planar stress element. The contacting material is steel with Young’s modulus E = 2.0 · 105 MPa and Poisson’s ratio l = 0.3, the radius of hole R2 = 100 mm. For studying the influence of different clearances and different loads, we select the following model parameters in FEM analyses: the sizes of the clearance DR are selected in all 15 different values, 5, 4, 3, 2, 1, 0.9, 0.8, . . . , 0.2, 0.1 mm, respectively. The penetration displacements d are selected in all 10 different values, 0.02, 0.04, . . . ,0.18, 0.2, 0.02 mm, respectively. The combination of these two parameters generates in all 150 different models. 4.2. Analysis of FEM results Figs. 5 and 6 show the normal contact force P against the normal penetration displacement in DR = 1 mm and DR = 0.1 mm, respectively. The solid-line curves in the figures represent the Hertz relation of Eq. (9), and the dots in the figures represent the numerical results based on FEM simulation.
Fig. 4. The FE mesh.
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Fig. 5. The relation between contact load and the penetration displacement in DR = 1 mm.
Fig. 6. The relation between contact load and the penetration displacement in DR = 0.1 mm.
If the contact load is not big enough, it can be seen from Fig. 5 that the Hertz relation of Eq. (9) can agree well with the FEM results in DR = 1 mm. However, the Hertz curves will gradually departure from the FEM results as the increase of the normal load. For observing how the difference between the Hertz curves and FEM results changes as the size of the clearance decreases, Fig. 6 shows the numerical results in DR = 0.1 mm. It is obvious that the Hertz relations will departure from the FEM results even if the load is very small. The reason of inducing the above phenomena can be attributed to that the Hertz assumption is unavailable when the radial clearance is very small. For each pair of the parameters (DR, d), the dimensionless load parameter EDR/P can be obtained through FEM calculations. Meanwhile, the Persson’ solution can be calculated based on Eqs. (14) and (16). In different radial clearance DR and different normal displacement d, Fig. 7 shows the relationship between the dimensionless load parameter EDR/P and the parameter b = tan(e/2).
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Observe Fig. 7, we can see that the FEM results closely approach to the Persson solution in b P 0.6, but they will gradually departure from each other as the contact semi-angle decrease (b 6 0.6). This illustrates that the Persson’s theory is available only when the contact angle is bigger enough. If the contact angle is small, then the geometrical relationship expressed in Eq. (11) will not satisfied, and the basic assumption of the Persson theory will be violated. The relationship of dimensionless pressure distribution Rp(y)/P against the contact angle w in DR = 0.1 mm is exhibited in Fig. 8. We firstly set the semi-angle of contact e = 30, 60, 80, respectively. Based on Eq. (16), we can determine the penetration displacement corresponding to each contact semi-angle. Then pressure distribution can be got based on FEM analysis. It is obvious that the contact region is not completely consistent with the initial value of the contact semi-angle especially in the case of small value of the contact semi-angle.
Fig. 7. The relation between dimensionless load parameter and the contact semi-angle based on Persson model.
Fig. 8. The relation between the dimensionless pressure distribution and the contact angle.
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5. The approximate compliance model for cylindrical joints with clearance The difficult of solving the contact problem of cylindrical joint with clearance is that the profiles of the contact bodies cannot be adequately represented by their radii of curvature at the point of first contact. According to Ref. [24], the elastic foundation model can provide simple approximate solutions in the complex situations. In order to get a reasonable simple contact compliance model for this problem, the method of the Winkler elastic foundation model is used in this paper, and some assumptions are made firstly. (1) The pin is equivalent to a rigid wedge, and the contact pressure distribution in z-direction along the profile of the rigid wedge is ellipsoidal as given by the Hertz theory. (2) The contact boundary between the equivalent rigid wedge and elastic foundation satisfies the geometric relation Eq. (16). (3) The plate is modeled by a simple Winkler elastic foundation with depth R2 and stiffness k, which rests on a rigid base and is compressed by the rigid wedge. The assumption (1) and (2) determine the profile of the equivalent rigid wedge and the shape of the contact pressure distribution, the assumption (3) is used to determine the maximum contact pressure between the rigid wedge and elastic foundation. The analytical model for the cylindrical joints with clearance is illustrated in Fig. 9. The semi-angle b of the wedge is determined by the contact semi-angle e, and tan b = 1/tan(e/2). According to the assumption (2), the relationship between contact semi-angle e and the penetration displacement d of the equivalent rigid wedge at the contact boundary is determined by Eq. (16). Once the profiles of the rigid wedge are determined, the contact shape and the size are also determined. The contact pressure distribution in z-direction along the profile of the rigid wedge is expressed as 2 1=2
pz ðxÞ ¼ pz0 ½1 ðx=lÞ
ð17Þ
where x is the distance along the wedge direction between the point on the profile and the vertex of the wedge, l is the length of the chord, pz0 is the maximum contact pressure at the vertex. Since DR is much less than R1 and R2, the relationship between l and e can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l ¼ R2 ð1 cos eÞ þ sin e2 ¼ 2R2 sinðe=2Þ
Fig. 9. The simple model for cylindrical joints with clearance.
ð18Þ
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There is no interaction between the springs of the Winkle elastic foundation, i.e. shear between adjacent elements of the foundation is ignored. The contact pressure at any point depends only on the displacement at that point, thus pz ¼ ðE=R2 Þuz
ð19Þ
where E is the elastic modulus of the foundation, uz is the normal elastic displacements of the foundation. Since the displacement of the elastic foundation at the vertex of the wedge is approximately half of the total maximum normal penetration d based on FEM numerical results, i.e. uz0 = d/2, and the materials of the two contact bodies are identical, i.e. E = E*, the linear relation between pz0 and the penetration d can be expressed as follows: py0 ¼ E d=2R2
ð20Þ
For verifying the correctness of Eq. (20), Fig. 10 shows the curves of the maximum contact pressure at the vertex of the rigid wedge against the penetration displacement based on FEM results. It is obvious that the compliance of the point at the vertex can be well modeled by above expression. According the condition of equilibrium, we can obtain: Z 2 py ðxÞdx ¼ P ð21Þ DC
Substituting Eqs. (17), (18), and (20) into (21), the relationship of load P and rigid body displacement can be simply expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d ð22Þ P ¼ pdE 2 2ðDR þ dÞ By introducing the following non-dimensional load parameter DRE*/P and k = sin(e/2), Eq. (16) can be expressed as d ¼ DR
2k 1 2k2
ð23Þ
Introducing Eq. (23) into Eq. (22), the relationship between the non-dimensional load parameter and the contact semi-angle can be written by 1 2k2 DRE ¼ P pk3
Fig. 10. The maximum pressure at different clearance and different normal penetration displacement through FEM analyses.
ð24Þ
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6. The comparison of different models for cylindrical joints with clearance Referring to Fig. 11, the relation between the dimensionless load parameter EDR/P and contact semi-angle k agrees well with the FEA results. The maximum error between them is less than 2.0% even in the case of contact semi-angle k 0.2, corresponding to a very small contact area of the cylindrical joints with a large clearance. When k is greater than 0.3, the results of above two methods are almost the same. This illustrates that the approximate model proposed in the paper can effectively solve the contact problem of the cylindrical joints with clearance in a wider range as compared with the Hertz theory and the Persson theory. The relations between the normal load P and the normal penetration displacement d obtained from different model with different normal clearance DR and dmax = 0.2 mm are shown in Fig. 12. Referring to Fig. 12(a), which represents the normal load versus normal displacement curves in the case of DR = 1 mm, it can be observed that the curve produced by the Persson model deviates largely from the FEM results even in a very small normal load, on the other hand, the curve produced by Hertz model agrees well with the FEM results. The reason is the geometric rigidity relation Eq. (16) is unavailable when the clearance is larger enough, but the Hertz assumption that the contact area is nearly a plane is satisfied in this case. Although the geometric relation Eq. (16) is applied in the approximate model based on elastic foundation theory, the curve given by Eq. (22) also agrees well with the FEM results. This illustrates that the model presented in this paper can effectively represent the contact behavior of the cylindrical joints with joints even in the case of large clearance. In the case of DR = 0.5 mm, We can observe from Fig. 12(b) that the curve obtained by Hertz model is near the FEM results only when the normal load P is less than 5000 N. With the increase of the normal load, the curve obtained by Hertz theory begins to departure from the FEM results. Meanwhile, the curve obtained by Persson theory is still away from FEM results in this case. This shows that the contact area does not persist in a plane with the decrease of the size of the clearance, and the geometric relation Eq. (16) is not satisfied in this case. Although the clearance has been changed, the curve obtained by the new model proposed in this paper still agrees well with the FEM results as seen in Fig. 12(b). Referring to Fig. 12(c) for the case of DR = 0.2 mm, the curve obtained by Hertz theory agrees with FEM results only when the normal load P is less than 2000 N, and will rapidly deviate from the FEM results with the increase of the normal load. The reason is the equivalent radii at the contact area may exceed the admissible range of [0, R1] as clearance is very small. The curve obtained by Persson theory still shows a difference as compared with the FEM results. Meanwhile, the approximate model still agrees well with the FEM results regardless of the variation of the size of the clearance.
Fig. 11. The relation between dimensionless load parameter and semi-angle based on the approximate model.
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Fig. 12. The relations between the normal load and the penetration displacement obtained from different model with different normal clearance.
In the case of DR = 0.1 mm, it can be seen from Fig. 12(d) that the Hertz theory is effective only for a very small normal load, and the curve obtained from Persson results is much closer to the result of FEM than before. This further explains that the geometric relation Eq. (16) is available only in the case of very small clearance. Likewise, the new model is still available in this case. 7. Conclusion This paper has studied the relation between the contact load and the penetration displacement for cylindrical joints with clearance. Using FEM code, the contact area, the pressure distribution, and the maximum sustainable load for clearance joint are presented based on the numerical results. The Hertz theory and the Persson theory are the common method used to represent the contact behavior of the cylindrical joint with clearance, but both of them are not always available, which are strongly depend on the size of the clearance, as compared with the FEM numerical results. The Hertz model is effective only in the condition of that the clearance is large enough and the normal load is very small. The Persson model can be applied only in the case that there is a small clearance. This can be attributed to the different assumptions adopting in the Hertz model and the Persson model. Hertz theory is established based on the plane assumption of the contact region, so the contact between two bodies in the cylindrical joint can satisfy the assumption of Hertz theory only in
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the case of that the clearance is larger enough and the load is very small. Persson theory is developed based on the geometrical rigidity constrain equations, so the contact between two bodies can satisfy the condition of the Persson theory only in the situation of that the clearance is small enough. By introducing some appropriate assumptions, a new approximate model for solving this contact problem has been presented in this paper. The contact problem of the cylindrical joint is simplified as a rigid wedge pressed into an elastic half-space plate, and a simple formula to describe the relation between the contact load and the penetration displacement is proposed. A comparison between the results obtained using the simple model and the FEM results shows that the new model more effective than the Hertz theory and the Persson theory. Acknowledgements The support of the National Science Foundation of China (10272002; 60334030) and the Foundation of Engineering Research Institute of Peking University (204035) is gratefully acknowledged. References [1] G. Gilardi, I. Sharf, Literature survey of contact dynamics modeling, Mechanism and Machine Theory 37 (2002) 1213–1239. [2] W. Schiehlen, Multibody system dynamics: roots and perspectives, Multibody System Dynamics 1 (1997) 149–188. [3] R.S. Haines, A theory of contact loss at resolute joints with clearance, Journal of Mechanical Engineering and Science 22 (3) (1980) 129–136. [4] K.L. Ting, J.D. Zhu, D. Watkins, The effects of joint clearance on position and orientation deviation of linkages and manipulators, Mechanism and Machine Theory 35 (2000) 391–401. [5] P. Ravn, A continuous analysis method for planar multibody systems with joint clearance, Multibody System Dynamics 2 (1988) 1– 24. [6] Y. Khulief, A. Shabana, A continuous force model for the impact analysis of flexible multi-body systems, Mechanism and Machine Theory 22 (1987) 213–224. [7] H. Lankarani, P. Nikravesh, A contact force model with hysteresis damping for impact analysis of multi-body systems, Journal of Mechanical Design 112 (1990) 369–376. [8] H. Lankarani, P. Nikravesh, Continuous contact force models for impact analysis in multibody systems, Nonlinear Dynamics 5 (1994) 193–207. [9] J. Ambrosio, P. Flores, Revolute joints with clearance in multibody systems, Computers and Structures 82 (2004) 1359–1369. [10] S. Dubowskey, F. Freudenatein, Dynamic analysis of mechanical systems with clearances, part 1: formulation of dynamic mode, and part 2: dynamic response, ASME Journal of Engineering for Industry 93 (1971) 305–316. [11] J.F. Deck, S. Dubowsky, On the limitations of predictions of the dynamic response of machines with clearance connections, ASME Journal of Mechanical Design 116 (1994) 833–841. [12] O.A. Bauchau, C.L. Bottasso, Contact conditions for cylindrical, prismatic, and screw joints in flexible multibody systems, Multibody System Dynamics 5 (2001) 251–278. [13] Olivier A. Bauchau, Jesus Rodriguez, Modeling of joints with clearance in flexible multibody systems, International Journal of Solids and Structure 39 (2002) 41–63. [14] M. Sassi, M. Desvignes, A seminumerical method for three-dimensional frictionless contact problems, Mathematical and Computing Modelling 28 (1998) 413–425. [15] Vu-Quoc Loc, Z.H. Xiang, L. Lee, Normal and tangential force–displacement relations for frictional elasto-plastic contact of spheres, International Journal of solids and Structure 38 (2001) 6455–6489. [16] Z. Xiang, Vu-Quoc Loc, Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions, International Journal of Impact Engineering 27 (2003) 317–341. [17] I.V. Rokach, On the accurate determination of contact compliance for impact test modeling, International Journal of Solids and Structures 40 (2003) 2715–2729. [18] C.T. Lim, W.J. Strong, Oblique elastic–plastic impact between rough cylinders in plane strain, International Journal of Engineering Science 37 (1999) 97–122. [19] M.G. Knight, L.A. Lacerda, L.C. Wrobel, J.L. Henshall, Parametric study of the contact stresses around spherical and cylindrical inclusions, Computational Materials Science 25 (2002) 115–121. [20] R. Seieried, B. Hu, P. Eberhard, Numerical and experimental investigation of radial impacts on a half-circular plate, Multibody System Dynamics 9 (2003) 265–281. [21] R.R. Ryan, ADAMS—Multibody System Analysis Software, Multibody Systems Handbook, Springer-Verlag, Berlin, 1990. [22] R.C. Smith, E.J. Haug, DADS—Dynamic Analysis and Design System, Multibody Systems Handbook, Springer-Verlag, Berlin, 1990. [23] A.L. Schwab, J.P. Meijaard, P. Meijers, A comparison of revolute joint clearance models in the dynamic analysis of rigid and elastic mechanical systems, Mechanism and Machine Theory 37 (2002) 895–913.
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