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Effects of joint clearance on the dynamics of a partly compliant mechanism: Numerical and experimental studies
Mechanism and Machine Theory 88 (2015) 125–140
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Effects of joint clearance on the dynamics of a partly compliant mechanism: Numerical and experimental studies Selçuk Erkaya a,⁎, Selim Doğan b, Şaban Ulus a a b
Erciyes University, Engineering Faculty, Department of Mechatronics Engineering, 38039 Kayseri, Turkey Melikşah University, Engineering Faculty, Department of Mechanical Engineering, 38280 Kayseri, Turkey
a r t i c l e
i n f o
Article history: Received 8 December 2014 Received in revised form 29 January 2015 Accepted 21 February 2015 Available online 12 March 2015 Keywords: Joint clearance Compliant mechanism Bearing vibration Pseudo joint Multibody dynamics
a b s t r a c t Compliant mechanism has got at least one flexible member between conventional rigid links. It is a good choice for decreasing the number of movable joints and also their clearance effects. In articulated mechanisms, clearance is inevitable due primarily to the design, manufacturing and assembly processes or a wear effect. Also, it plays a crucial role and has a significant effect on the mechanism stability and the performances of whole system. In this study, both numerical and experimental investigations are carried out to analyze the effects of joint clearance on partly compliant and conventional articulated mechanisms. Bearings' and links' vibrations are considered to determine what is the main contribution of small flexural pivot on compliant mechanism having joint clearance? Five accelerometers have been located at different points to measure the vibrations on system during the mechanism motion. The results show that the joint clearance makes the mechanism performance worse. The flexibility of flexural pivot has a clear suspension effect to minimize the undesired outputs of joint clearance on mechanisms. Also, small-length flexural pivot is an important tool to prevent the separation between journal and bearing by constituting a force-closed kinematic pair in a joint with clearance. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Due to the advances in technologies, increased performance and cost reduction are particularly expected from mechanisms/ machines. Compliant mechanisms offer a great promise in providing new and better solutions to many mechanical design problems. An advantage of compliant mechanism is the potential for a dramatic reduction in the total number of parts required to accomplish a specified task [1,2]. Conventional rigid-body mechanisms consist of rigid links connected at movable joints. Compliant mechanisms have fewer movable joints relative to conventional mechanisms. That is a great advantage from the standpoint of joint clearance effects. Small clearances (tolerances) in the kinematic joints of mechanisms are necessary for assembly and mobility. In case of higher size, ideal lower pair joints have the potential of becoming higher pairs. This is due to the fact that the ideal surface between two contacting links really exhibits line or point contacts. This is known as a source of impact forces, and these forces not only create increasing vibration amplitude, but also reduce system reliability and stability. Joint clearance effects may also cause to motion accuracy loss and reduced service life. These effects may become more severe on high-speed and micro-mechanical systems, intelligent robots, and numerically controlled machine tools. Over the past few decades, a considerable amount of experimental and theoretical work has been implemented about only compliant or conventional mechanisms. However, the investigation of compliant mechanism having joint clearance is limited. Experimental investigation is nearly absent. Flores and his research group have valuable contributions about joint clearance in ⁎ Corresponding author. Tel.: +90 3522076666; fax: +90 3524375784. E-mail address: [email protected] (S. Erkaya).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.02.007 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
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Nomenclatures c Cmax dmax D E e FT FN fn I IDA Ki l32 n rB rJ t υT ν δ ζ μ
Radial clearance Maximum damping coefficient Positive real value for boundary penetration Hysteresis damping Young's modulus Clearance vector Tangential force component Normal force component Natural frequency Moment of inertia Intelligent Data Acquisition Stiffness coefficient Length of small flexural pivot Normal coordinate Bearing radius Journal radius Tangential coordinate Relative tangential sliding velocity in collision plane Poisson's coefficient Relative penetration depth Restitution coefficient Friction coefficient
thematic literature. They investigated clearance effects on conventional multibody mechanical systems. Dry contact including friction and lubrication effects between journal and bearing parts [3–6], different clearance sizes and joint types in 2D and 3D mechanism motions were considered in their case studies [7,8]. In order to quantify the wear phenomenon in clearance joints, a methodology was proposed [9]. Some computational studies were also implemented to increase the computational ability deal with the transitions between non-contact and contact situations in multibody dynamics [10]. Theoretical information about the effect of friction-induced vibration and contact mechanics on the maximum contact pressure and moment of artificial hip implants were also outlined [11]. Except for compliant theme, the effects of flexibility [12–17] and multiple clearance joints [18,19] were also considered. The effects of joint clearance on kinematics and dynamics of robot manipulator and conventional articulated mechanisms were performed [20,21]. In order to improve the precision of mechanism, optimization techniques were also introduced to decrease the deviations arising from joint clearances [22–24]. A theoretical study was implemented upon the trajectory optimization of a walking mechanism having revolute joints with clearance using adaptive network-based fuzzy inference system [25]. Artificial neural networks were used to model the vibration characteristics of mechanism having joint
(a)
(b)
(c)
Fig. 1. (a) Classic articulated slider-crank mechanism, (b) partly compliant slider-crank mechanism, and (c) pseudo-rigid-body model of compliant mechanism.
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Table 1 Geometric and material properties of mechanisms. Articulated
Compliant
Name
Length (mm)
Mass (kg)
Mass moment of inertia (kgmm )
Length (mm)
Mass (kg)
Mass moment of inertia (kgmm2)
Crank Connecting rod Small pivot Piston
150 564 – –
0.25 0.77 – 1.83
771.37 22,821 – 822.75
150 455 40/100a –
0.25 0.56 0.011 1.06
771.37 10,052.1 9.41 1090.52
a
2
Effective length/assembly length.
clearance [26]. Both theoretical and experimental studies about joint clearance were also presented [27–29]. A conventional slider–crank mechanism with tripod-ball sliding joint was considered as an example. Kinematics and dynamics outputs were investigated and compared with and without clearance. A quantitative analysis method was proposed by considering a planar mechanism [30]. Dynamic characteristics of multibody mechanical systems including revolute joints with clearance using a computational methodology were investigated. A partly/fully compliant mechanism having small flexural pivot(s) is a good choice to decrease the joint number. This output also leads to remove the undesired effects of clearance in joints. A case study is proposed to investigate the effects of joint clearance on a 2D partly compliant mechanism with clearance. A single-axis flexural pivot is used between connecting rod and piston links. Some experiments are constituted to investigate the outputs of joint clearance and the contribution of small flexural pivot for solving this problem. Vibration measurements are considered to realize the different motion modes between journal and bearing, and suspension effect of flexural pivot. This paper is organized as follows; model mechanisms and modeling of joint clearance are described in Section 2. Experimental test rig and measures are outlined in Section 3. Obtained results and discussions are summarized in Sections 4 and 5, respectively.
2. Model mechanisms A planar slider–crank mechanism was considered in the current study. Model mechanism includes two types of main motions: translation and rotation motions. Slider–crank mechanism converts these motions to each other. The classic type of model mechanism
(a)
(b)
Fig. 2. Modeled mechanisms for simulation: (a) classic articulated mechanism model, and (b) partly compliant mechanism model.
Fig. 3. Clearance vector and force components in a joint.
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Fig. 4. Coefficient of friction varying with slip velocity [31,33].
is shown in Fig. 1(a). The compliant slider–crank mechanism in a deflected position is outlined in Fig. 1(b). The pseudo-rigid-body model (PRBM) is also given in Fig. 1(c). One disadvantage of compliant mechanisms is the complexity of their analysis and design. In order to simplify their theoretical analysis, the pseudo-rigid-body model technique has been frequently used in the literature. This model can be considered as a bridge that connects rigid-body mechanism theory and compliant mechanism theory. It allows compliant mechanisms to be modelled as rigid-body mechanisms with rigid links and springs. Therefore, the resulting model is easily analyzed using common mechanism design methods [1]. PRBM can describe the behavior of compliant mechanisms in a high accuracy. In this model, geometry of small segment, especially its length, depends on the adjacent link's geometry. Due to the one dimensional rotation between connecting rod and piston links, single-axis flexural pivot is chosen in this study [2]. The beam's resistance to deflection is modeled using a torsional spring with spring constant K b. The torque required to deflect the torsional spring through angle Θ is [1] T ¼ Kb Θ
ð1Þ
where Θ is the difference between θ3 and θ30. θ3 is the angular variation of the connecting rod and θ30 is the initial value of connecting rod angle. It is assumed that the slider always remains in contact with the ground, the frictional resistance to slider motion is small/ negligible and the flexural segment is straight when undeflected, that is, θ30 = 0°. The spring constant Kb is found from the elementary beam theory [1] Kb ¼
Es fp Is fp
ð2Þ
l32
Fig. 5. Representation of experimental test rig.
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(a) (b)
(c)
(d)
Fig. 6. Material testing of small flexural pivot: (a) testing apparatus, (b) geometries of test piece, and (c) and (d) representation of test pieces before and after material testing.
where Esfp is the Young's modulus, Isfp is the moment of inertia, and l32 is the length of small flexural pivot. Kinematic and dynamic parameters of the model mechanisms are given in Table 1. In addition to an experimental study, the simulation test of model mechanism was built under the MSC. ADAMS software to compare and evaluate the experimental results. Fig. 2 gives the mechanism models in simulation software [31]. 2.1. Contact force and joint clearance model Clearance in a joint is necessary to provide a relative motion between neighbor links, as well as to permit the assembly of the mechanical parts. Radial clearance (c) is defined as the difference between journal and bearing radii, and it is modeled as a virtual, massless link. In the presence of clearance in a revolute joint, the two kinematic constraints lost, and two degrees of freedom consisting of the horizontal (x) and vertical (y) displacements of the journal center with respect to bearing center are added to the mechanism motion. These movements may lead to uncertainties in the mechanism motion. Three different types of motion between journal and bearing can be observed in a joint having clearance, that is, free-flight, impact and continuous contact modes. At the same time, free-flight and relative penetration affect the magnitude of clearance vector. Relative penetration depth (δ) between journal and bearing is given as δ ¼ e−c N0
ð3Þ
in which e is the magnitude of the clearance vector between the bearing and journal centers, and c is the radial clearance (Fig. 3).
(a)
(b)
Fig. 7. Joints having clearance, (a) 0.5 mm clearance, and (b) 0.85 mm clearance.
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Fig. 8. Block diagram of experimental measurement.
The magnitude of the clearance vector is expressed as |e| = (e2x + e2y )1/2. Here, ex and ey represent the relative displacements of the journal inside the bearing for x and x directions, respectively. When the journal and bearing are not in contact with each other, there is no contact forces associated with the journal-bearing. If the contact between the two bodies occurs, the contact–
Fig. 9. Contact force components between journal and bearing in crank-connecting rod joint having clearance.
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impact forces are modeled according to Hertzian contact theory (normal force, FN) together with Coulomb's friction law (tangential force, FT). FC ¼ 0 FC ¼ FN þ FT
for δ b 0 for δ ≥0
ð4Þ
When the relative penetration is equal or bigger than zero, the normal force is expressed as [4] ð3=2Þ
FN ¼ Kδ
þ Dδ
ð5Þ
where the first term represents the elastic force component and the second term explains the energy dissipation. K is the generalized stiffness parameter, and D is the hysteresis damping coefficient. The generalized stiffness parameter K depends on the geometry and physical properties of the contacting surfaces. The stiffness parameter K is calculated by [32] ri r j 4 K¼ ri þ r j 3 hi þ h j
!1=2 ð6Þ
where the material parameters hk are expressed as hk ¼
1−ν 2k Ek
ð7Þ
ðk ¼ i; jÞ
Fig. 10. Crank–frame and piston–frame bearing force components.
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ν and E are the Poisson's coefficient and the Young's modulus associated with each body. The radius of curvature rk is taken as positive for convex surfaces and negative for concave surfaces. The hysteresis damping coefficient is calculated as [4]
D¼
2 3=2 3 1‐ζ Kδ
ð8Þ
4v0
where ζ is the restitution coefficient and ν0 is the initial impact velocity. The normal force function in simulation software is given as [31]; FN ¼
8 <
3=2
Kδ
:0
þ STEPðδ; 0; 0; dmax ; Cmax Þ
dδ dt
for
δ ≥0
for
δb0
ð9Þ
where hysteresis damping coefficient is also outlined as; 8 <0 STEPðδ; 0; 0; dmax ; Cmax Þ ¼ Cmax ðδ=dmax Þ2 ð3‐2ðδ=dmax ÞÞ : Cmax
for for for
δ≤ 0 0 bδ bdmax δ ≥ dmax
ð10Þ
where dmax is a positive real value dealing with the boundary penetration. Cmax is the maximum damping coefficient. Friction force acts when contacting bodies tend to slide relative to each other. This force is tangential to the contact surface and is opposite to the sliding velocity. Friction force model is given as [33]: FT ¼ ‐μ ðυT ÞFN
υT jυ T j
ð11Þ
Fig. 11. Measured vibrations from compliant mechanism at 100 rpm running speed.
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where μ(υT) is the coefficient of dynamic friction. It is a function of relative tangential sliding velocity (υT) in the collision plane. The used function of dynamic friction coefficient in simulation software is given as;
μ ðυT Þ ¼
8 <
−μ d signðυt Þ ‐STEPðjυt j; υd ; μ d ; υs ; μ s Þsignðυt Þ : STEPðυt ; −υs ; μ s υs ; −μ s Þ
for for for
jυt jNυd υs ≤ jυt j≤υd jυt jb υs
ð12Þ
where μ s and μ d are static and dynamic friction coefficients, respectively. υs and υd denote the critical velocities of static and maximum dynamic frictions, respectively. Fig. 4 shows how the dynamic friction coefficient varies with slip velocity. 3. Experimental study An experimental test rig, as shown in Fig. 5, was set up to adapt the different geometries and materials of small flexural pivot to the different mechanisms. Experimental system was driven by a 1.5 kW AC electric motor, and Mitsubishi S500 frequency inverter was used to control the angular velocity. An encoder was adapted to the shaft of AC motor for defining the angular position of crank arm. Also, a flywheel was used to prevent the fluctuation of the angular velocity due to the impact forces at the joints with clearance. For comparison, same material characteristic in each mechanism's link was used and the analyses of mechanisms with and without clearances were implemented.
Fig. 12. Measured vibrations from compliant mechanism at 150 rpm running speed.
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The material of the small flexural pivot was selected as polypropylene and material characteristics were also determined before the experimental investigation (Fig. 6). The thickness and depth of single-axis flexural pivot having rectangular cross section were arranged as 4.15 mm and 30 mm, respectively. The length of the pseudo joint was determined as l32 = 40 mm. Obtained material characteristics of small flexural pivot; density was 9.0598 × 10−7 kg/mm3, and Young's modulus was 953 N/mm2. Due to the different motion types between journal and bearing in the joint, vibrations from different points were measured to define the system response as a reflection of contact– impact forces during the mechanism motion. In this study, revolute joints are considered at the link connections. 0.5 and 0.85 mm clearances are considered in the experimental investigation. For 0.5 mm clearance, the radii of journal and bearings are selected as 7 and 7.5 mm, respectively. In case of 0.85 mm clearance, these journal and bearing radii are selected as 7 and 7.85 mm, respectively. Fig. 7 gives the differences between journal and bearing diameters. 41× digital camera was used to scan the real values of clearance. In order to collect the vibration data, the first and second accelerometers were positioned at the x and y-directions of the crank–frame joint, respectively (see Fig. 5). The third accelerometer was positioned at the y-direction of the piston–frame joint. The fourth accelerometer was positioned at the x-direction on the piston link. And, the fifth accelerometer was positioned at the perpendicular direction on the connecting rod link. The piston link was at the top dead center for initial configuration. Block diagram for measuring configuration is given in Fig. 8. Brüel & Kjaer 3560B type intelligent data acquisition and Brüel & Kjaer 4514 type accelerometers were used in the experimental measurements [34]. Each sensor + magnet has 25 × 10− 3 kg mass. 4. Results In this study, the effects of joint clearance on compliant and articulated mechanisms were investigated by evaluation of mechanism vibrations. For experiments, artificial radial clearances as 0.5 and 0.85 mm were constituted in the revolute joints.
Fig. 13. Measured vibrations from compliant mechanism at 200 rpm running speed.
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A partly compliant mechanism was designed by using a pseudo joint, that is, small flexural pivot between connecting rod and piston links. Since the variations in physical geometries and material characteristics have a clear influence on the performance of any real system, identical geometric and material types, except for small flexural pivot, were used for compliant and articulated mechanisms with and without clearances. Therefore, similar inertial characteristics were ensured for each model mechanism. This is necessary for an exact comparison. For initial configuration, angular position of crank link was arranged as equal to zero for all experiments, that is, the piston link was at the top dead center. Steady state responses for model mechanisms were used for comparison. Three different scenarios for running speed were considered. Before the experimental tests, all parts of joints having clearance were washed out using an alcohol for removing the manufacturing remnants etc. In the numerical simulation, step size and restitution coefficient were considered as 0.001 s and 0.9, respectively. Dynamic friction coefficient was selected as 0.01. It is clear from the simulations that much more computation time, nearly upwards of three times, is necessary in the case of articulated mechanism with joint clearance. For meshing characteristic, tetrahedral element type was used for small-length flexural pivot in simulation. Mode shapes and natural frequencies of this pivot are given in Appendix A. Contact force in collision plane and bearing force components are outlined in Figs. 9 and 10 for 200 rpm running speed, respectively. As seen from the numerical simulation stage, impact type contact forces in small time intervals occur arising from joint clearance [18,28]. Force peaks in clearance joint also cause a proportional increase in the bearing forces. F21 and F41 denote the crank–frame and piston–frame bearing forces, respectively. These forces are main sources of vibration in related bearings. As seen from these figures, peak value and frequency of forces in compliant mechanism are smaller than that of classic articulated mechanism. Force peaks lead to vibration peaks and these peaks also make the mechanism vibration characteristics more chaotic. From the simulation software, mode shapes and natural frequencies for small-length flexural pivot are outlined in Appendix A.
Fig. 14. Measured vibrations from compliant and articulated mechanisms having 0.5 mm radial clearance at 200 rpm running speed.
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In the experimental stage, to show the different modes between journal and bearing during the mechanism motion, vibrations on main bearings and moving links were measured using five sensors. Figs. 11–13 give the measured vibrations of compliant mechanism for different clearance and running speed scenarios. By considering compliant mechanism with and without clearance, it is clear that the joint clearance leads to degradation of mechanism vibration performance. Both clearance value and running speed have important roles on system dynamics. Acceleration amplitudes increase, when the clearance value and/or running speed increase. Particularly, the results of first, second and third sensors outline some vibration peaks. It can be concluded as a reflection of impact mode between journal and bearing in a joint having clearance. When all results for compliant mechanism are investigated carefully, particularly for 200 rpm, the torsional spring character, that is, the equivalence of small flexural pivot, has a role to ensure the continuous contact between journal and bearing in a joint with clearance. This judgment is proved by evaluating the results of sensors 4 and 5. For comparison, measured vibrations of compliant and articulated mechanisms having 0.5 and 0.85 mm radial clearances are given in Figs. 14 and 15. While the articulated mechanism model has two joints with clearance, partly compliant mechanism has one clearance joint. The reducing number of joints can be evaluated as a positive effect to remove the potential clearance effects. Therefore, by using single or multi-axes small flexural pivot relative to mechanism motion necessity, compliant mechanisms can be evaluated instead of classic articulated mechanisms. As seen from Figs. 14 and 15, it is observed a clear difference between vibration results of two mechanisms. Pseudo joint, that is, small flexural pivot adds a suspension effect to the mechanism. In fact, the pseudo torsional spring decreases the free-flight motion between journal and bearing in a joint having clearance. In this motion, journal and the bearing are not in contact and journal moves freely within the bearing's boundaries. At the end of free-flight mode, contact between journal and bearing occurs. This also leads to impact phenomenon, and this phenomenon is the main sources of peaks in obtained graphs [6,16]. Decreases in the free-flight mode lead to decreasing impact in mechanism. This is concluded as a positive reflection of pseudo joint. When the obtained vibration from the fifth sensor, which is positioned on connecting rod link, is investigated, compliant mechanism with joint clearance has a periodical motion (Figs. 11–13). However, the classical articulated mechanism with
Fig. 15. Measured vibrations from compliant and articulated mechanisms having 0.85 mm radial clearance at 200 rpm running speed.
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joint clearance does not have an exact periodical character (see Figs. 14 and 15 and Appendix A) [28]. Partly compliant mechanism has fewer joint than classical articulated mechanism. This phenomenon removes the undesired effects of joint with clearance between connecting rod and piston links. For the other joint with clearance, that is, between crank and connecting rod links, the torsional spring behavior of pseudo joint also provides a continuous contact between journal and bearing parts of joint. Therefore, this behavior constitutes a force-closed kinematic pair effect between bearing and journal parts of joint with clearance. This force effect constraints the separation of bearing and journal during the mechanism motion as much as possible. In other words, torsional spring effect potentially converts the higher pair to the lower pair. On the other hand, additional freedoms in articulated mechanism arising from joint clearance lead to uncontrolled motion of moving links. During the mechanism motion, this also increases the free-flight and impact modes between journal and bearing parts in a clearance joint. 5. Conclusions Numerical and experimental studies were implemented to investigate the effects of joint clearance on dynamic characteristics of a partly compliant mechanism. For this aim, vibrations were measured from different points on mechanisms. Different scenarios for clearance values and running speeds in compliant or classical articulated mechanisms were studied. The evaluations of the proposed results are mainly summarized as follows; (i) Joint clearance leads to degradations in vibration characteristics of the mechanism relative to that of mechanism without clearance. Wear, fatigue etc. may be affected negatively due to these vibrations. Also, these make the mechanism's precision worse and the mechanism cannot fulfill the expected tasks exactly. (ii) If the clearance value and/or running speed increases, vibration amplitudes also increase. (iii) The pseudo torsional spring behavior of small-length flexural pivot in compliant mechanism provides a continuous contact between journal and bearing parts of joint having clearance. Therefore, the free-flight and impact modes decrease. This is a positive reflection of compliant mechanism to prevent the undesired effects of impact phenomenon arising from joint clearance. The spring effect converts the higher pair to the lower pair by constituting a force-closed kinematic pair effect between bearing and journal parts of joint with clearance. (iv) When the peak frequency and values are investigated for both mechanisms, the flexural pivot adds a suspension effect to the compliant mechanism. Particularly, sensor five is a proof for this evaluation. (v) Pseudo torsional spring effect of small-length flexural pivot can be used as an important tool to remove the additional freedom arising from joint clearance. This also prevents the uncertainties in the mechanism motion. In the next studies, different geometries and materials of small-length flexural pivot will be considered for different joint types in partly 2D and 3D compliant mechanisms having joint clearances. Acknowledgment The authors wish to express their thanks for financial support being provided by the Scientific Research Projects Coordination Unit of Erciyes University (FYL-2013-4350 and FBA-12-4111), in carrying out the experimental parts of this study. Appendix A Mode shapes and natural frequencies of small-length flexural pivot are given as follows,
Fig. A1. Mode shapes and natural frequencies of small-length flexural pivot.
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In case of different clearance values and running speeds, measured vibrations for partly compliant and articulated mechanisms are given in Figs. A2–A5.
Fig. A2. Measured vibrations from compliant and articulated mechanisms having 0.5 mm radial clearance at 100 rpm running speed.
Fig. A3. Measured vibrations from compliant and articulated mechanisms having 0.5 mm radial clearance at 150 rpm running speed.
S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125–140
Fig. A4. Measured vibrations from compliant and articulated mechanisms having 0.85 mm radial clearance at 100 rpm running speed.
Fig. A5. Measured vibrations from compliant and articulated mechanisms having 0.85 mm radial clearance at 150 rpm running speed.
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Effects of joint clearance on the dynamics of a partly compliant mechanism: Numerical and experimental studies Selçuk Erkaya a,⁎, Selim Doğan b, Şaban Ulus a a b
Erciyes University, Engineering Faculty, Department of Mechatronics Engineering, 38039 Kayseri, Turkey Melikşah University, Engineering Faculty, Department of Mechanical Engineering, 38280 Kayseri, Turkey
a r t i c l e
i n f o
Article history: Received 8 December 2014 Received in revised form 29 January 2015 Accepted 21 February 2015 Available online 12 March 2015 Keywords: Joint clearance Compliant mechanism Bearing vibration Pseudo joint Multibody dynamics
a b s t r a c t Compliant mechanism has got at least one flexible member between conventional rigid links. It is a good choice for decreasing the number of movable joints and also their clearance effects. In articulated mechanisms, clearance is inevitable due primarily to the design, manufacturing and assembly processes or a wear effect. Also, it plays a crucial role and has a significant effect on the mechanism stability and the performances of whole system. In this study, both numerical and experimental investigations are carried out to analyze the effects of joint clearance on partly compliant and conventional articulated mechanisms. Bearings' and links' vibrations are considered to determine what is the main contribution of small flexural pivot on compliant mechanism having joint clearance? Five accelerometers have been located at different points to measure the vibrations on system during the mechanism motion. The results show that the joint clearance makes the mechanism performance worse. The flexibility of flexural pivot has a clear suspension effect to minimize the undesired outputs of joint clearance on mechanisms. Also, small-length flexural pivot is an important tool to prevent the separation between journal and bearing by constituting a force-closed kinematic pair in a joint with clearance. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Due to the advances in technologies, increased performance and cost reduction are particularly expected from mechanisms/ machines. Compliant mechanisms offer a great promise in providing new and better solutions to many mechanical design problems. An advantage of compliant mechanism is the potential for a dramatic reduction in the total number of parts required to accomplish a specified task [1,2]. Conventional rigid-body mechanisms consist of rigid links connected at movable joints. Compliant mechanisms have fewer movable joints relative to conventional mechanisms. That is a great advantage from the standpoint of joint clearance effects. Small clearances (tolerances) in the kinematic joints of mechanisms are necessary for assembly and mobility. In case of higher size, ideal lower pair joints have the potential of becoming higher pairs. This is due to the fact that the ideal surface between two contacting links really exhibits line or point contacts. This is known as a source of impact forces, and these forces not only create increasing vibration amplitude, but also reduce system reliability and stability. Joint clearance effects may also cause to motion accuracy loss and reduced service life. These effects may become more severe on high-speed and micro-mechanical systems, intelligent robots, and numerically controlled machine tools. Over the past few decades, a considerable amount of experimental and theoretical work has been implemented about only compliant or conventional mechanisms. However, the investigation of compliant mechanism having joint clearance is limited. Experimental investigation is nearly absent. Flores and his research group have valuable contributions about joint clearance in ⁎ Corresponding author. Tel.: +90 3522076666; fax: +90 3524375784. E-mail address: [email protected] (S. Erkaya).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.02.007 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
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Nomenclatures c Cmax dmax D E e FT FN fn I IDA Ki l32 n rB rJ t υT ν δ ζ μ
Radial clearance Maximum damping coefficient Positive real value for boundary penetration Hysteresis damping Young's modulus Clearance vector Tangential force component Normal force component Natural frequency Moment of inertia Intelligent Data Acquisition Stiffness coefficient Length of small flexural pivot Normal coordinate Bearing radius Journal radius Tangential coordinate Relative tangential sliding velocity in collision plane Poisson's coefficient Relative penetration depth Restitution coefficient Friction coefficient
thematic literature. They investigated clearance effects on conventional multibody mechanical systems. Dry contact including friction and lubrication effects between journal and bearing parts [3–6], different clearance sizes and joint types in 2D and 3D mechanism motions were considered in their case studies [7,8]. In order to quantify the wear phenomenon in clearance joints, a methodology was proposed [9]. Some computational studies were also implemented to increase the computational ability deal with the transitions between non-contact and contact situations in multibody dynamics [10]. Theoretical information about the effect of friction-induced vibration and contact mechanics on the maximum contact pressure and moment of artificial hip implants were also outlined [11]. Except for compliant theme, the effects of flexibility [12–17] and multiple clearance joints [18,19] were also considered. The effects of joint clearance on kinematics and dynamics of robot manipulator and conventional articulated mechanisms were performed [20,21]. In order to improve the precision of mechanism, optimization techniques were also introduced to decrease the deviations arising from joint clearances [22–24]. A theoretical study was implemented upon the trajectory optimization of a walking mechanism having revolute joints with clearance using adaptive network-based fuzzy inference system [25]. Artificial neural networks were used to model the vibration characteristics of mechanism having joint
(a)
(b)
(c)
Fig. 1. (a) Classic articulated slider-crank mechanism, (b) partly compliant slider-crank mechanism, and (c) pseudo-rigid-body model of compliant mechanism.
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Table 1 Geometric and material properties of mechanisms. Articulated
Compliant
Name
Length (mm)
Mass (kg)
Mass moment of inertia (kgmm )
Length (mm)
Mass (kg)
Mass moment of inertia (kgmm2)
Crank Connecting rod Small pivot Piston
150 564 – –
0.25 0.77 – 1.83
771.37 22,821 – 822.75
150 455 40/100a –
0.25 0.56 0.011 1.06
771.37 10,052.1 9.41 1090.52
a
2
Effective length/assembly length.
clearance [26]. Both theoretical and experimental studies about joint clearance were also presented [27–29]. A conventional slider–crank mechanism with tripod-ball sliding joint was considered as an example. Kinematics and dynamics outputs were investigated and compared with and without clearance. A quantitative analysis method was proposed by considering a planar mechanism [30]. Dynamic characteristics of multibody mechanical systems including revolute joints with clearance using a computational methodology were investigated. A partly/fully compliant mechanism having small flexural pivot(s) is a good choice to decrease the joint number. This output also leads to remove the undesired effects of clearance in joints. A case study is proposed to investigate the effects of joint clearance on a 2D partly compliant mechanism with clearance. A single-axis flexural pivot is used between connecting rod and piston links. Some experiments are constituted to investigate the outputs of joint clearance and the contribution of small flexural pivot for solving this problem. Vibration measurements are considered to realize the different motion modes between journal and bearing, and suspension effect of flexural pivot. This paper is organized as follows; model mechanisms and modeling of joint clearance are described in Section 2. Experimental test rig and measures are outlined in Section 3. Obtained results and discussions are summarized in Sections 4 and 5, respectively.
2. Model mechanisms A planar slider–crank mechanism was considered in the current study. Model mechanism includes two types of main motions: translation and rotation motions. Slider–crank mechanism converts these motions to each other. The classic type of model mechanism
(a)
(b)
Fig. 2. Modeled mechanisms for simulation: (a) classic articulated mechanism model, and (b) partly compliant mechanism model.
Fig. 3. Clearance vector and force components in a joint.
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Fig. 4. Coefficient of friction varying with slip velocity [31,33].
is shown in Fig. 1(a). The compliant slider–crank mechanism in a deflected position is outlined in Fig. 1(b). The pseudo-rigid-body model (PRBM) is also given in Fig. 1(c). One disadvantage of compliant mechanisms is the complexity of their analysis and design. In order to simplify their theoretical analysis, the pseudo-rigid-body model technique has been frequently used in the literature. This model can be considered as a bridge that connects rigid-body mechanism theory and compliant mechanism theory. It allows compliant mechanisms to be modelled as rigid-body mechanisms with rigid links and springs. Therefore, the resulting model is easily analyzed using common mechanism design methods [1]. PRBM can describe the behavior of compliant mechanisms in a high accuracy. In this model, geometry of small segment, especially its length, depends on the adjacent link's geometry. Due to the one dimensional rotation between connecting rod and piston links, single-axis flexural pivot is chosen in this study [2]. The beam's resistance to deflection is modeled using a torsional spring with spring constant K b. The torque required to deflect the torsional spring through angle Θ is [1] T ¼ Kb Θ
ð1Þ
where Θ is the difference between θ3 and θ30. θ3 is the angular variation of the connecting rod and θ30 is the initial value of connecting rod angle. It is assumed that the slider always remains in contact with the ground, the frictional resistance to slider motion is small/ negligible and the flexural segment is straight when undeflected, that is, θ30 = 0°. The spring constant Kb is found from the elementary beam theory [1] Kb ¼
Es fp Is fp
ð2Þ
l32
Fig. 5. Representation of experimental test rig.
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(a) (b)
(c)
(d)
Fig. 6. Material testing of small flexural pivot: (a) testing apparatus, (b) geometries of test piece, and (c) and (d) representation of test pieces before and after material testing.
where Esfp is the Young's modulus, Isfp is the moment of inertia, and l32 is the length of small flexural pivot. Kinematic and dynamic parameters of the model mechanisms are given in Table 1. In addition to an experimental study, the simulation test of model mechanism was built under the MSC. ADAMS software to compare and evaluate the experimental results. Fig. 2 gives the mechanism models in simulation software [31]. 2.1. Contact force and joint clearance model Clearance in a joint is necessary to provide a relative motion between neighbor links, as well as to permit the assembly of the mechanical parts. Radial clearance (c) is defined as the difference between journal and bearing radii, and it is modeled as a virtual, massless link. In the presence of clearance in a revolute joint, the two kinematic constraints lost, and two degrees of freedom consisting of the horizontal (x) and vertical (y) displacements of the journal center with respect to bearing center are added to the mechanism motion. These movements may lead to uncertainties in the mechanism motion. Three different types of motion between journal and bearing can be observed in a joint having clearance, that is, free-flight, impact and continuous contact modes. At the same time, free-flight and relative penetration affect the magnitude of clearance vector. Relative penetration depth (δ) between journal and bearing is given as δ ¼ e−c N0
ð3Þ
in which e is the magnitude of the clearance vector between the bearing and journal centers, and c is the radial clearance (Fig. 3).
(a)
(b)
Fig. 7. Joints having clearance, (a) 0.5 mm clearance, and (b) 0.85 mm clearance.
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Fig. 8. Block diagram of experimental measurement.
The magnitude of the clearance vector is expressed as |e| = (e2x + e2y )1/2. Here, ex and ey represent the relative displacements of the journal inside the bearing for x and x directions, respectively. When the journal and bearing are not in contact with each other, there is no contact forces associated with the journal-bearing. If the contact between the two bodies occurs, the contact–
Fig. 9. Contact force components between journal and bearing in crank-connecting rod joint having clearance.
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impact forces are modeled according to Hertzian contact theory (normal force, FN) together with Coulomb's friction law (tangential force, FT). FC ¼ 0 FC ¼ FN þ FT
for δ b 0 for δ ≥0
ð4Þ
When the relative penetration is equal or bigger than zero, the normal force is expressed as [4] ð3=2Þ
FN ¼ Kδ
þ Dδ
ð5Þ
where the first term represents the elastic force component and the second term explains the energy dissipation. K is the generalized stiffness parameter, and D is the hysteresis damping coefficient. The generalized stiffness parameter K depends on the geometry and physical properties of the contacting surfaces. The stiffness parameter K is calculated by [32] ri r j 4 K¼ ri þ r j 3 hi þ h j
!1=2 ð6Þ
where the material parameters hk are expressed as hk ¼
1−ν 2k Ek
ð7Þ
ðk ¼ i; jÞ
Fig. 10. Crank–frame and piston–frame bearing force components.
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ν and E are the Poisson's coefficient and the Young's modulus associated with each body. The radius of curvature rk is taken as positive for convex surfaces and negative for concave surfaces. The hysteresis damping coefficient is calculated as [4]
D¼
2 3=2 3 1‐ζ Kδ
ð8Þ
4v0
where ζ is the restitution coefficient and ν0 is the initial impact velocity. The normal force function in simulation software is given as [31]; FN ¼
8 <
3=2
Kδ
:0
þ STEPðδ; 0; 0; dmax ; Cmax Þ
dδ dt
for
δ ≥0
for
δb0
ð9Þ
where hysteresis damping coefficient is also outlined as; 8 <0 STEPðδ; 0; 0; dmax ; Cmax Þ ¼ Cmax ðδ=dmax Þ2 ð3‐2ðδ=dmax ÞÞ : Cmax
for for for
δ≤ 0 0 bδ bdmax δ ≥ dmax
ð10Þ
where dmax is a positive real value dealing with the boundary penetration. Cmax is the maximum damping coefficient. Friction force acts when contacting bodies tend to slide relative to each other. This force is tangential to the contact surface and is opposite to the sliding velocity. Friction force model is given as [33]: FT ¼ ‐μ ðυT ÞFN
υT jυ T j
ð11Þ
Fig. 11. Measured vibrations from compliant mechanism at 100 rpm running speed.
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where μ(υT) is the coefficient of dynamic friction. It is a function of relative tangential sliding velocity (υT) in the collision plane. The used function of dynamic friction coefficient in simulation software is given as;
μ ðυT Þ ¼
8 <
−μ d signðυt Þ ‐STEPðjυt j; υd ; μ d ; υs ; μ s Þsignðυt Þ : STEPðυt ; −υs ; μ s υs ; −μ s Þ
for for for
jυt jNυd υs ≤ jυt j≤υd jυt jb υs
ð12Þ
where μ s and μ d are static and dynamic friction coefficients, respectively. υs and υd denote the critical velocities of static and maximum dynamic frictions, respectively. Fig. 4 shows how the dynamic friction coefficient varies with slip velocity. 3. Experimental study An experimental test rig, as shown in Fig. 5, was set up to adapt the different geometries and materials of small flexural pivot to the different mechanisms. Experimental system was driven by a 1.5 kW AC electric motor, and Mitsubishi S500 frequency inverter was used to control the angular velocity. An encoder was adapted to the shaft of AC motor for defining the angular position of crank arm. Also, a flywheel was used to prevent the fluctuation of the angular velocity due to the impact forces at the joints with clearance. For comparison, same material characteristic in each mechanism's link was used and the analyses of mechanisms with and without clearances were implemented.
Fig. 12. Measured vibrations from compliant mechanism at 150 rpm running speed.
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The material of the small flexural pivot was selected as polypropylene and material characteristics were also determined before the experimental investigation (Fig. 6). The thickness and depth of single-axis flexural pivot having rectangular cross section were arranged as 4.15 mm and 30 mm, respectively. The length of the pseudo joint was determined as l32 = 40 mm. Obtained material characteristics of small flexural pivot; density was 9.0598 × 10−7 kg/mm3, and Young's modulus was 953 N/mm2. Due to the different motion types between journal and bearing in the joint, vibrations from different points were measured to define the system response as a reflection of contact– impact forces during the mechanism motion. In this study, revolute joints are considered at the link connections. 0.5 and 0.85 mm clearances are considered in the experimental investigation. For 0.5 mm clearance, the radii of journal and bearings are selected as 7 and 7.5 mm, respectively. In case of 0.85 mm clearance, these journal and bearing radii are selected as 7 and 7.85 mm, respectively. Fig. 7 gives the differences between journal and bearing diameters. 41× digital camera was used to scan the real values of clearance. In order to collect the vibration data, the first and second accelerometers were positioned at the x and y-directions of the crank–frame joint, respectively (see Fig. 5). The third accelerometer was positioned at the y-direction of the piston–frame joint. The fourth accelerometer was positioned at the x-direction on the piston link. And, the fifth accelerometer was positioned at the perpendicular direction on the connecting rod link. The piston link was at the top dead center for initial configuration. Block diagram for measuring configuration is given in Fig. 8. Brüel & Kjaer 3560B type intelligent data acquisition and Brüel & Kjaer 4514 type accelerometers were used in the experimental measurements [34]. Each sensor + magnet has 25 × 10− 3 kg mass. 4. Results In this study, the effects of joint clearance on compliant and articulated mechanisms were investigated by evaluation of mechanism vibrations. For experiments, artificial radial clearances as 0.5 and 0.85 mm were constituted in the revolute joints.
Fig. 13. Measured vibrations from compliant mechanism at 200 rpm running speed.
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A partly compliant mechanism was designed by using a pseudo joint, that is, small flexural pivot between connecting rod and piston links. Since the variations in physical geometries and material characteristics have a clear influence on the performance of any real system, identical geometric and material types, except for small flexural pivot, were used for compliant and articulated mechanisms with and without clearances. Therefore, similar inertial characteristics were ensured for each model mechanism. This is necessary for an exact comparison. For initial configuration, angular position of crank link was arranged as equal to zero for all experiments, that is, the piston link was at the top dead center. Steady state responses for model mechanisms were used for comparison. Three different scenarios for running speed were considered. Before the experimental tests, all parts of joints having clearance were washed out using an alcohol for removing the manufacturing remnants etc. In the numerical simulation, step size and restitution coefficient were considered as 0.001 s and 0.9, respectively. Dynamic friction coefficient was selected as 0.01. It is clear from the simulations that much more computation time, nearly upwards of three times, is necessary in the case of articulated mechanism with joint clearance. For meshing characteristic, tetrahedral element type was used for small-length flexural pivot in simulation. Mode shapes and natural frequencies of this pivot are given in Appendix A. Contact force in collision plane and bearing force components are outlined in Figs. 9 and 10 for 200 rpm running speed, respectively. As seen from the numerical simulation stage, impact type contact forces in small time intervals occur arising from joint clearance [18,28]. Force peaks in clearance joint also cause a proportional increase in the bearing forces. F21 and F41 denote the crank–frame and piston–frame bearing forces, respectively. These forces are main sources of vibration in related bearings. As seen from these figures, peak value and frequency of forces in compliant mechanism are smaller than that of classic articulated mechanism. Force peaks lead to vibration peaks and these peaks also make the mechanism vibration characteristics more chaotic. From the simulation software, mode shapes and natural frequencies for small-length flexural pivot are outlined in Appendix A.
Fig. 14. Measured vibrations from compliant and articulated mechanisms having 0.5 mm radial clearance at 200 rpm running speed.
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In the experimental stage, to show the different modes between journal and bearing during the mechanism motion, vibrations on main bearings and moving links were measured using five sensors. Figs. 11–13 give the measured vibrations of compliant mechanism for different clearance and running speed scenarios. By considering compliant mechanism with and without clearance, it is clear that the joint clearance leads to degradation of mechanism vibration performance. Both clearance value and running speed have important roles on system dynamics. Acceleration amplitudes increase, when the clearance value and/or running speed increase. Particularly, the results of first, second and third sensors outline some vibration peaks. It can be concluded as a reflection of impact mode between journal and bearing in a joint having clearance. When all results for compliant mechanism are investigated carefully, particularly for 200 rpm, the torsional spring character, that is, the equivalence of small flexural pivot, has a role to ensure the continuous contact between journal and bearing in a joint with clearance. This judgment is proved by evaluating the results of sensors 4 and 5. For comparison, measured vibrations of compliant and articulated mechanisms having 0.5 and 0.85 mm radial clearances are given in Figs. 14 and 15. While the articulated mechanism model has two joints with clearance, partly compliant mechanism has one clearance joint. The reducing number of joints can be evaluated as a positive effect to remove the potential clearance effects. Therefore, by using single or multi-axes small flexural pivot relative to mechanism motion necessity, compliant mechanisms can be evaluated instead of classic articulated mechanisms. As seen from Figs. 14 and 15, it is observed a clear difference between vibration results of two mechanisms. Pseudo joint, that is, small flexural pivot adds a suspension effect to the mechanism. In fact, the pseudo torsional spring decreases the free-flight motion between journal and bearing in a joint having clearance. In this motion, journal and the bearing are not in contact and journal moves freely within the bearing's boundaries. At the end of free-flight mode, contact between journal and bearing occurs. This also leads to impact phenomenon, and this phenomenon is the main sources of peaks in obtained graphs [6,16]. Decreases in the free-flight mode lead to decreasing impact in mechanism. This is concluded as a positive reflection of pseudo joint. When the obtained vibration from the fifth sensor, which is positioned on connecting rod link, is investigated, compliant mechanism with joint clearance has a periodical motion (Figs. 11–13). However, the classical articulated mechanism with
Fig. 15. Measured vibrations from compliant and articulated mechanisms having 0.85 mm radial clearance at 200 rpm running speed.
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joint clearance does not have an exact periodical character (see Figs. 14 and 15 and Appendix A) [28]. Partly compliant mechanism has fewer joint than classical articulated mechanism. This phenomenon removes the undesired effects of joint with clearance between connecting rod and piston links. For the other joint with clearance, that is, between crank and connecting rod links, the torsional spring behavior of pseudo joint also provides a continuous contact between journal and bearing parts of joint. Therefore, this behavior constitutes a force-closed kinematic pair effect between bearing and journal parts of joint with clearance. This force effect constraints the separation of bearing and journal during the mechanism motion as much as possible. In other words, torsional spring effect potentially converts the higher pair to the lower pair. On the other hand, additional freedoms in articulated mechanism arising from joint clearance lead to uncontrolled motion of moving links. During the mechanism motion, this also increases the free-flight and impact modes between journal and bearing parts in a clearance joint. 5. Conclusions Numerical and experimental studies were implemented to investigate the effects of joint clearance on dynamic characteristics of a partly compliant mechanism. For this aim, vibrations were measured from different points on mechanisms. Different scenarios for clearance values and running speeds in compliant or classical articulated mechanisms were studied. The evaluations of the proposed results are mainly summarized as follows; (i) Joint clearance leads to degradations in vibration characteristics of the mechanism relative to that of mechanism without clearance. Wear, fatigue etc. may be affected negatively due to these vibrations. Also, these make the mechanism's precision worse and the mechanism cannot fulfill the expected tasks exactly. (ii) If the clearance value and/or running speed increases, vibration amplitudes also increase. (iii) The pseudo torsional spring behavior of small-length flexural pivot in compliant mechanism provides a continuous contact between journal and bearing parts of joint having clearance. Therefore, the free-flight and impact modes decrease. This is a positive reflection of compliant mechanism to prevent the undesired effects of impact phenomenon arising from joint clearance. The spring effect converts the higher pair to the lower pair by constituting a force-closed kinematic pair effect between bearing and journal parts of joint with clearance. (iv) When the peak frequency and values are investigated for both mechanisms, the flexural pivot adds a suspension effect to the compliant mechanism. Particularly, sensor five is a proof for this evaluation. (v) Pseudo torsional spring effect of small-length flexural pivot can be used as an important tool to remove the additional freedom arising from joint clearance. This also prevents the uncertainties in the mechanism motion. In the next studies, different geometries and materials of small-length flexural pivot will be considered for different joint types in partly 2D and 3D compliant mechanisms having joint clearances. Acknowledgment The authors wish to express their thanks for financial support being provided by the Scientific Research Projects Coordination Unit of Erciyes University (FYL-2013-4350 and FBA-12-4111), in carrying out the experimental parts of this study. Appendix A Mode shapes and natural frequencies of small-length flexural pivot are given as follows,
Fig. A1. Mode shapes and natural frequencies of small-length flexural pivot.
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In case of different clearance values and running speeds, measured vibrations for partly compliant and articulated mechanisms are given in Figs. A2–A5.
Fig. A2. Measured vibrations from compliant and articulated mechanisms having 0.5 mm radial clearance at 100 rpm running speed.
Fig. A3. Measured vibrations from compliant and articulated mechanisms having 0.5 mm radial clearance at 150 rpm running speed.
S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125–140
Fig. A4. Measured vibrations from compliant and articulated mechanisms having 0.85 mm radial clearance at 100 rpm running speed.
Fig. A5. Measured vibrations from compliant and articulated mechanisms having 0.85 mm radial clearance at 150 rpm running speed.
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