Available online 12 March 2015

articulated mechanisms, clearance is inevitable due primarily to the design, manufacturing and

Keywords:

es, inc

l surface between twoforces not only createalso cause to motion-mechanical systems,

Mechanism and Machine Theory 88 (2015) 125140

Contents lists available at ScienceDirect

Mechanism and Machine Theoryintelligent 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.size, ideal lower pair joints have the potential of becoming higher pairs. This is due to the fact that the ideacontacting links really exhibits line or point contacts. This is known as a source of impact forces, and theseincreasing vibration amplitude, but also reduce system reliability and stability. Joint clearance effects mayaccuracy loss and reduced service life. These effects may become more severe on high-speed and micromachines. Compliant mechanisms offer a great promise in providing new and better solutions to manymechanical design problems.An advantage of compliant mechanism is the potential for a dramatic reduction in the total number of parts required to accomplish aspecied task [1,2]. Conventional rigid-body mechanisms consist of rigid links connected at movable joints. Compliant mechanismshave fewer movable joints relative to conventional mechanisms. That is a great advantage from the standpoint of joint clearanceeffects. Small clearances (tolerances) in the kinematic joints ofmechanisms are necessary for assembly andmobility. In case of higherDue to the advances in technologiExperimental investigation is nearly absen

Corresponding author. Tel.: +90 3522076666; fax:E-mail address: serkaya@erciyes.edu.tr (S. Erkaya).

http://dx.doi.org/10.1016/j.mechmachtheory.2015.02.000094-114X/ 2015 Elsevier Ltd. All rights reserved.reased performance and cost reduction are particularly expected from mechanisms/1. Introductionassembly processes or a wear effect. Also, it plays a crucial role and has a signicant effect onthe mechanism stability and the performances of whole system. In this study, both numericaland experimental investigations are carried out to analyze the effects of joint clearance on partlycompliant and conventional articulated mechanisms. Bearings' and links' vibrations areconsidered to determine what is the main contribution of small exural pivot on compliantmechanism having joint clearance? Five accelerometers have been located at different points tomeasure the vibrations on system during the mechanism motion. The results show that thejoint clearance makes the mechanism performance worse. The exibility of exural pivot has aclear suspension effect to minimize the undesired outputs of joint clearance on mechanisms.Also, small-length exural pivot is an important tool to prevent the separation between journaland bearing by constituting a force-closed kinematic pair in a joint with clearance.

2015 Elsevier Ltd. All rights reserved.Joint clearanceCompliant mechanismBearing vibrationPseudo jointMultibody dynamicsAccepted 21 February 2015Effects of joint clearance on the dynamics of a partly compliantmechanism: Numerical and experimental studies

Seluk Erkaya a,, Selim Doan b, aban Ulus aa Erciyes University, Engineering Faculty, Department of Mechatronics Engineering, 38039 Kayseri, Turkeyb Melikah University, Engineering Faculty, Department of Mechanical Engineering, 38280 Kayseri, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 December 2014Received in revised form 29 January 2015

Compliantmechanism has got at least one exible member between conventional rigid links. It isa good choice for decreasing the number of movable joints and also their clearance effects. In

j ourna l homepage: www.e lsev ie r .com/ locate /mechmtt. Flores and his research group have valuable contributions about joint clearance in

+90 3524375784.

7

Nomenclatures

c Radial clearanceCmax Maximum damping coefcientdmax Positive real value for boundary penetrationD Hysteresis dampingE Young's moduluse Clearance vectorFT Tangential force componentFN Normal force componentfn Natural frequencyI Moment of inertiaIDA Intelligent Data AcquisitionKi Stiffness coefcientl32 Length of small exural pivotn Normal coordinaterB Bearing radiusrJ Journal radiust Tangential coordinate

126 S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125140thematic literature. They investigated clearance effects on conventional multibody mechanical systems. Dry contact includingfriction and lubrication effects between journal and bearing parts [36], different clearance sizes and joint types in 2D and 3Dmechanism motions were considered in their case studies [7,8]. In order to quantify the wear phenomenon in clearance joints,a methodologywas proposed [9]. Some computational studies were also implemented to increase the computational ability deal

T Relative tangential sliding velocity in collision plane Poisson's coefcient Relative penetration depth Restitution coefcient Friction coefcientwith the transitions between non-contact and contact situations in multibody dynamics [10]. Theoretical information about theeffect of friction-induced vibration and contact mechanics on the maximum contact pressure and moment of articial hipimplants were also outlined [11]. Except for compliant theme, the effects of exibility [1217] and multiple clearance joints[18,19] were also considered. The effects of joint clearance on kinematics and dynamics of robot manipulator and conventionalarticulatedmechanismswere performed [20,21]. In order to improve the precision of mechanism, optimization techniques werealso introduced to decrease the deviations arising from joint clearances [2224]. A theoretical study was implemented upon thetrajectory optimization of a walking mechanism having revolute joints with clearance using adaptive network-based fuzzyinference system [25]. Articial 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.

clearance [26]. Both theoretical and experimental studies about joint clearance were also presented [2729]. A conventionalslidercrank mechanism with tripod-ball sliding joint was considered as an example. Kinematics and dynamics outputs wereinvestigated and compared with and without clearance. A quantitative analysis method was proposed by considering a planarmechanism [30]. Dynamic characteristics of multibody mechanical systems including revolute joints with clearance using acomputational methodology were investigated.

A partly/fully compliant mechanism having small exural pivot(s) is a good choice to decrease the joint number. This output alsoleads to remove the undesired effects of clearance in joints. A case study is proposed to investigate the effects of joint clearance on a 2Dpartly compliant mechanism with clearance. A single-axis exural pivot is used between connecting rod and piston links. Some ex-periments are constituted to investigate the outputs of joint clearance and the contribution of small exural pivot for solving thisproblem. Vibrationmeasurements are considered to realize the differentmotionmodes between journal and bearing, and suspensioneffect of exural pivot. This paper is organized as follows; model mechanisms and modeling of joint clearance are described inSection 2. Experimental test rig and measures are outlined in Section 3. Obtained results and discussions are summarized inSections 4 and 5, respectively.

2. Model mechanisms

A planar slidercrank mechanism was considered in the current study. Model mechanism includes two types of main motions:translation and rotationmotions. Slidercrankmechanism converts thesemotions to each other. The classic type ofmodelmechanism

Table 1Geometric and material properties of mechanisms.

Articulated Compliant

Name Length (mm) Mass (kg) Mass moment of inertia (kgmm2) Length (mm) Mass (kg) Mass moment of inertia (kgmm2)

Crank 150 0.25 771.37 150 0.25 771.37Connecting rod 564 0.77 22,821 455 0.56 10,052.1Small pivot 40/100a 0.011 9.41Piston 1.83 822.75 1.06 1090.52

a Effective length/assembly length.

127S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125140(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.

is showmodel

On

deecthroug

Fig. 4. Coefcient of friction varying with slip velocity [31,33].

128 S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125140T Kb 1

where is the difference between 3 and 30. 3 is the angular variation of the connecting rod and 30 is the initial value of connectingrod angle. It is assumed that the slider always remains in contact with the ground, the frictional resistance to slider motion is small/negligible and theexural segment is straightwhen undeected, that is, 30= 0. The spring constant Kb is found from the elementarybeam theory [1]

Kb Es fpIs fp

l322tion is modeled using a torsional spring with spring constant Kb. The torque required to deect the torsional springh angle is [1]ical analysis, the pseudo-rigid-bodymodel technique has been frequently used in the literature. This model can be considered asa bridge that connects rigid-body mechanism theory and compliant mechanism theory. It allows compliant mechanisms to bemodelled as rigid-body mechanisms with rigid links and springs. Therefore, the resulting model is easily analyzed using com-mon mechanism design methods [1]. PRBM can describe the behavior of compliant mechanisms in a high accuracy. In thismodel, geometry of small segment, especially its length, depends on the adjacent link's geometry. Due to the one dimensionalrotation between connecting rod and piston links, single-axis exural pivot is chosen in this study [2]. The beam's resistance ton in Fig. 1(a). The compliant slidercrank mechanism in a deected position is outlined in Fig. 1(b). The pseudo-rigid-body(PRBM) is also given in Fig. 1(c).e disadvantage of compliant mechanisms is the complexity of their analysis and design. In order to simplify their theoret-Fig. 5. Representation of experimental test rig.

Clemechamasslconsismechajournatime, fbearin

in whi

(a)(b)

(c)

129S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125140arance in a joint is necessary to provide a relative motion between neighbor links, as well as to permit the assembly of thenical parts. Radial clearance (c) is dened as the difference between journal and bearing radii, and it is modeled as a virtual,where Esfp is the Young's modulus, Isfp is the moment of inertia, and l32 is the length of small exural pivot. Kinematic and dynamicparameters 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 tocompare and evaluate the experimental results. Fig. 2 gives the mechanism models in simulation software [31].

2.1. Contact force and joint clearance model

Fig. 6.Material testing of smallexural pivot: (a) testing apparatus, (b) geometries of test piece, and (c) and (d) representation of test pieces before and aftermaterial testing.(d)ess link. In the presence of clearance in a revolute joint, the two kinematic constraints lost, and two degrees of freedomting of the horizontal (x) and vertical (y) displacements of the journal center with respect to bearing center are added to thenismmotion. Thesemovements may lead to uncertainties in the mechanismmotion. Three different types of motion betweenl and bearing can be observed in a joint having clearance, that is, free-ight, impact and continuous contact modes. At the sameree-ight and relative penetration affect themagnitude of clearance vector. Relative penetration depth () between journal andg is given as

ec N0 3

ch 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.

The magnitude of the clearance vector is expressed as |e| = (ex2 + ey2)1/2. Here, ex and ey represent the relative displacementsof the journal inside the bearing for x and x directions, respectively. When the journal and bearing are not in contact with eachother, there is no contact forces associated with the journal-bearing. If the contact between the two bodies occurs, the contact

Fig. 8. Block diagram of experimental measurement.

130 S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125140Fig. 9. Contact force components between journal and bearing in crank-connecting rod joint having clearance.

impact forces are modeled according to Hertzian contact theory (normal force, FN) together with Coulomb's friction law (tangentialforce, FT).

FC 0 for b 0FC FN FT for 0

4

When the relative penetration is equal or bigger than zero, the normal force is expressed as [4]

FN K 3=2 D 5

where the rst term represents the elastic force component and the second term explains the energy dissipation. K is the generalizedstiffness parameter, and D is the hysteresis damping coefcient. The generalized stiffness parameter K depends on the geometry andphysical properties of the contacting surfaces. The stiffness parameter K is calculated by [32]

K 43 hi hj rir j

ri r j

!1=26

where the material parameters hk are expressed as

hk 12kEk

k i; j 7

131S. Erkaya et al. / Mechanism and Machine Theory 88 (2015) 125140Fig. 10. Crankframe and pistonframe bearing force components.

and E are the Poisson's coefcient and the Young'smodulus associatedwith each body. The radius of curvature rk is taken as positivefor convex surfaces and negative for concave surfaces. The hysteresis damping coefcient is calculated as [4]

D 3 12

K3=2

4v08

where is the restitution coefcient and 0 is the initial impact velocity. The normal force function in simulation software is givenas [31];

FN K3=2 STEP ;0;0;dmax;Cmax

ddt

for 00 for b 0

8