optimization of transmission angle for slider-crank mechanism with joint clearances

Struct Multidisc Optim (2009) 37:493–508DOI 10.1007/s00158-008-0243-6

RESEARCH PAPER

Optimization of transmission angle for slider-crankmechanism with joint clearances

Selçuk Erkaya · Ibrahim Uzmay

Received: 13 January 2007 / Revised: 20 July 2007 / Accepted: 9 February 2008 / Published online: 14 March 2008© Springer-Verlag 2008

Abstract In this study, kinematic analysis of a planarslider-crank mechanism having revolute joints withclearances was presented. Joint clearance was modelledas a massless virtual link, and Multi-Layered NeuralNetwork (MLNN) structure was used for approximat-ing the motion of this link with respect to the positionof input link. Training and testing data sets for theneural network were obtained from mechanism simula-tion using the ADAMS software. A genetic algorithmwas also used to optimize the design parameters forminimizing the deviations due to clearances. When twojoint clearances at crank-pin and piston-pin centerswere considered, the effects of these clearances onthe kinematic characteristics and transmission qualityof the mechanism were investigated using continuouscontact model between the journal and bearing at ajoint.

Keywords Kinematic analysis · Joint clearance ·Kinematic efficiency · Neural Network ·Genetic Algorithm

1 Introduction

Mechanisms consist of several links, considered rigidbodies and joints connecting these links in order to

S. Erkaya · I. Uzmay (B)Department of Mechanical Engineering, Erciyes University,38039 Kayseri, Turkeye-mail: [email protected]

S. Erkayae-mail: [email protected]

transmit motion or forces from one to adjacent link. Inmost cases, clearances occur in these joints due primar-ily to manufacturing tolerances, material deformationsand wearing after a certain working period. Excessivevalues of these clearances will cause to the impulsiveeffects of contact forces in the link connections, espe-cially in the high-speed mechanism, and this situationleads to decrease in mechanical performance.

In the past, many designers investigated the effectsof joint clearances on different mechanical systemsfrom those aspects. Dubowsky (1974) investigated theeffects of joint clearance in planar mechanisms. As anillustrative example, a slider-crank mechanism with onejoint clearance between crank and connecting rod con-nection was considered, and the impact model in linkconnection was assumed to analyze the system. Osmanet al. (1987) presented a procedure for analysis of mech-anism with bearing clearances, which mainly relied ondetermining clearance angles and their derivatives. Acam mechanism with two bearing clearances was con-sidered to illustrate the proposed procedure. Furuhashiet al. (1978a, b) and Morita et al. (1978a, b) presented ageneral approach for the dynamics of four-bar mecha-nisms, in which revolute joints with clearance, using thecontinuous contact model assumption that the journalwas always in contact with the bearing in each pair.The motion of the linkage for the angular directionsof joint clearances was analytically performed fromLagrangian function. Different numbers of joints withclearance were considered in their studies, and theeffects of these combinations, all joints with clearance,one, two or three joints with clearance etc., on the dy-namic characteristics of systems were investigated andcompared to each other. Flores et al. (2006) presenteda methodology for the analysis of mechanical systems

494 S. Erkaya, I. Uzmay

considering joints with clearance. In their study, mod-elling of revolute joints with clearance was constitutedby considering the dry contact model and lubricatedmodel. The proposed methodology was adapted to aslider-crank mechanism with one joint clearance be-tween piston and connecting rod. It was seen that theexistence of dry joint clearances caused high peaks onthe kinematic and dynamic characteristics of system.Tsai and Lai (2004) presented an effective method toanalyze the transmission performance of linkages withjoint clearances. Equivalent kinematic pairs were usedfor modelling the motion freedoms originated fromthe joint clearances. Geometrical properties of linkageswere used to derive the equations. Position analysis of aplanar four-bar mechanism, which all joints have clear-ance, was implemented by using loop-closure equationsas a numerical example. Vasiliu and Yannou (2001)presented a case-based approach (couples of trajecto-ries and dimensions of a given structure mechanism) tosynthesize the dimensions of a planar linkage using aneural network. The method was implemented in anintegrated pre-design platform and a four-bar linkagewas used as an example to illustrate the good quality ofthe synthesized solutions, for a tiny size of the network.Yıldırım et al. (2005) presented a Neural Networkscheme as a predictor for analyzing the transmissionangle of a slider-crank mechanism with eccentric con-nector. The neural network structure was feedforwardand five types learning algorithm were used for obtain-ing the best approximation. Mean square of the errorwas used as a performance criteria of neural schema.Erkaya et al. (2007) achieved kinematic and dynamicanalysis of a modified slider-crank mechanism whichhas an additional eccentric link between connecting rodand crank pin, as distinct from a conventional slider-crank mechanism. Their study was focused on the out-put torque of mechanism and several assumptions wereconsidered such as all the joints with no clearance etc.

The main objectives of this study are; (1) to consti-tute the mathematical expression for directions of jointclearances, (2) to determine the effects of these clear-ances on kinematic behaviour of a planar slider-crankmechanism, (3) to optimize the design parameters indesign stage of mechanism for maximizing the transmis-sion quality. For this purpose, this paper is organizedas follows; Section 2 describes the revolute joint clear-ance model. In Section 3, kinematic characteristics ofslider-crank mechanism with clearances are presented.Section 4 describes NN structure for modelling themotion of equivalent clearance vectors and objectivefunction for genetic algorithm approach. Results arepresented in Section 5. Finally, conclusion is outlinedin Section 6.

2 Joint clearance model

In a planar revolute joint, as shown in Fig. 1, jointclearance rc is defined as the difference between theradii of bearing and journal, rB and rJ ,respectively.

If there is no lubrication, the journal can move freelywithin the bearing until any contact between the twobodies takes place. When the journal impacts the bear-ing wall, normal and tangential contact forces occur.Also, if the friction is negligible, the direction of jointclearance vector coincides with the direction of normalforce at the contact point. When the continuous contactmode assumption between journal and bearing at eachjoint is considered, the clearances may be modelled asvectors which correspond to massless virtual links withthe lengths equal to joint clearance (Ting et al. 2000;Bengisu et al. 1986). As seen in Fig. 1, the equivalentclearance can be defined in the following form,

rc = rB − rJ (1)

Each joint clearance adds additional freedom to themechanism, and additional constraints are necessary toanalyze the system (Bengisu et al. 1986).

3 Kinematics of slider-crank mechanism

Slider-crank mechanism, widely used in the internalcombustion engines, converts the translatory motionof piston to rotary motion of crank. Engines used inmotor vehicles, in conformity with recent advances,work at higher speeds, and so dynamic effects suchas links’ inertias have a crucial role in kinematic anddynamic analyses and mechanical design of slider-crankmechanism. Consequently, excessive loads act at links’joints and cause to tolerances for joint clearances toreach beyond desired values. In this study, a concen-tric slider-crank mechanism with two joint clearances,as shown in Fig. 2, is considered and the effects ofthese clearances are investigated from the standpointof kinematic characteristics and kinematic transmissionquality of mechanism.

LB

rB

rJ

rc

BearingJournal

Fig. 1 Revolute joint with clearance

Optimization of transmission angle for slider-crank mechanism with joint clearances 495

Fig. 2 a Slider-crankmechanism model,b schematic representationof a slider-crank mechanismwith joint clearances, c vectorrepresentation of mechanism

Revolute Joint Model with

clearance

Crank

Piston

Connecting rod

a

L2 L3

θ2

G2

G3 c3θ

4GX

γ2 r2

γ3 r3

c G4

Ao x

L2

L3

θ2

2

3G2

G3

y

4

c3θ

b

μc

B=G4

B’

A

A’

Table 1 gives the link lengths for the modelmechanism.

In the kinematic analysis of the slider-crank mech-anism, it is necessary to determine the positions ofmass center for moving links and then their corre-sponding velocities and accelerations. So, in the caseof joint clearance these positions are derived fromthe vector representation of the mechanism in Fig. 2c,respectively.[

xcG2

ycG2

]= A0G2

[cos θ2

sin θ2

](2)

Table 1 Parametric values ofthe slider-crank mechanism Link Length

(mm)

Crank (2) 150Connecting rod (3) 564Piston (4) –

[xc

G3

ycG3

]= L2

[cos θ2

sin θ2

]+ r2

[cos γ2

sin γ2

]+ A′G3

[cos θc

3

sin θc3

]

(3)

[xc

G4

ycG4

]= L2

[cos θ2

sin θ2

]+ r2

[cos γ2

sin γ2

]+ L3

[cos θc

3

sin θc3

]

+ r3

[cos (π + γ3)

sin (π + γ3)

](4)

where the superscript c denotes the “value with clear-ance”, also, θ c

3 can be expressed as a function of θ2, γ 2

and γ 3 in the following form,

θc3= sin−1

[− L2 sin θ2 + r2 sin γ2 + r3 sin (π+γ3)

L3

](5)


Time-derivatives of the mass center positions formoving links yield the mass center velocities and accel-erations, respectively,

⎡⎣ x c

Gi

y cGi

⎤⎦ = ω2

⎡⎢⎢⎢⎣

∂xcGi

∂θ2

∂ycGi

∂θ2

⎤⎥⎥⎥⎦ +

3∑j=2

γ j

⎡⎢⎢⎢⎣

∂xcGi

∂γ j

∂ycGi

∂γ j

⎤⎥⎥⎥⎦ (6)

⎡⎣ xc

Gi

ycGi

⎤⎦=α2

⎡⎢⎢⎢⎣

∂xcGi

∂θ2

∂ycGi

∂θ2

⎤⎥⎥⎥⎦+ω2

2

⎡⎢⎢⎢⎢⎣

∂2xcGi

∂θ22

∂2 ycGi

∂θ22

⎤⎥⎥⎥⎥⎦+2ω2

3∑j=2

γj

⎡⎢⎢⎢⎢⎣

∂2xcGi

∂θ2∂γ j

∂2 ycGi

∂θ2∂γ j

⎤⎥⎥⎥⎥⎦

+3∑

j=2

γ j

⎡⎢⎢⎢⎣

∂xcGi

∂γ j

∂ycGi

∂γ j

⎤⎥⎥⎥⎦ +

3∑j=2

γ 2j

⎡⎢⎢⎢⎢⎣

∂2xcGi

∂γ 2j

∂2 ycGi

∂γ 2j

⎤⎥⎥⎥⎥⎦

+3∑

j=2

3∑k=2

γ jγk

⎡⎢⎢⎢⎣

∂2xGi

∂γ j∂γk

∂2 yGi

∂γ j∂γk

⎤⎥⎥⎥⎦ (k �= j) (7)

where ω2 and α2 denote the angular velocity and ac-celeration of the input link, respectively. i denote themoving link numbers and j denote the joint clearancenumbers. Also, angular velocity and acceleration ofconnecting rod with clearances are derived from (5),respectively,

θc3 = ω2

∂θc3

∂θ2+

3∑j=2

γ j∂θc

3

∂γ j(8)

θc3 = α2

∂θc3

∂θ2+ ω2

2∂2θc

3

∂θ22

+ 2ω2

3∑j=2

γ j∂2θc

3

∂θ2∂γ j+

3∑j=2

γ j∂θc

3

∂γ j

+3∑

j=2

γ 2j∂2θc

3

∂γ 2j

+3∑

j=2

3∑k=2

γ jγk∂2θc

3

∂γ j∂γk(k �= j)

(9)

3.1 Transmission angle of mechanism

Transmission angle (μ) is an important criteria formechanism design. It denotes the quality of motiontransmission in a mechanism and it is mainly usedto obtain the better results for various linkage appli-cations. A good choice for the transmission angle inmechanism design is to make its value maximum, thatis, 90˚ as a feasible value. It helps designers to determinethe “Best” among a family of possible mechanisms for

most effective force transmission. Mechanisms, whosetransmission angles having too much deviation from90˚, exhibit poor operational characteristics like noiseand jerk at high speed applications. Transmission anglein a slider-crank mechanism is defined as an anglebetween the connecting rod and the normal to thecylinder axis. Referring to Fig. 2b, transmission anglewith joint clearances is given as follows,

μc = cos−1

[L2 sin θ2 + r2 sin γ2 − r3 sin γ3

L3

](10)

4 Modelling of joint characteristics using neuralnetwork and objective function

The neural system, consists of many interconnectedidentical simple processing units called neurons ornodes, actually considers an unknown structure andadjusts the parameters of the unknown model by apply-ing optimization methods in order to minimize modelerrors. In this study, a multilayered feedforward NNstructure, which consists of an input layer, a series ofhidden layers and an output layer as shown in Fig. 3, isused for modelling the equivalent clearances, directionsof joint clearances during the mechanism motion. Eachlayer comprises linear or nonlinear neurons and eachindividual neuron sums its weighted inputs and yieldsan output through a nonlinear activation function witha bias threshold.

As seen from Fig. 3, the MLNN structure consists ofone input and output layers with one linear neuron andalso four hidden layers, which consist of ten nonlinearneurons in each layer. The neural network toolbox ofMATLAB (MATLAB ver 7.0 2005) was used to de-velop the proposed neural model, and tangent sigmoidactivation function was used in the nonlinear neuronstructure (see http://www.mathworks.com),

Tansig (x) = 2

1 + e−2x− 1 (11)

The values of the training and testing data were nor-malized between 0 and 1. Levenberg–Marquardt (LM)back-propagation learning algorithm was employed toupdate the weights of the neural model, that is, forminimizing the convergence errors between desiredand actual outputs of the neuron. For this purpose,mean square of the error (MSE) was used as perfor-mance measuring criteria of network structure. In orderto specify network accuracy in predicting the systemoutputs, designed network was examined in responseto inputs which were not used in the training step.

After modelling of equivalent clearances and direc-tions of joint clearances, a Genetic Algorithm (GA)

http://www.mathworks.com


Fig. 3 A feedforward NNstructure for modelling ofjoint variables

+1

Bias

Input Layer (I)

Hidden Layers (J,K,M,N)

Output Layer (O)

IJwNOw

+1

Bias

Clearance valuesand

directions

+1

Bias

.

.

.

.

.

.

Position variableof

input link

was used to optimize the design parameters in themechanism for making the transmission quality better,and it was performed on genetic algorithm toolboxof MATLAB (MATLAB ver 7.0 2005). Genetic al-gorithm technique searches the global minimum pointinstead of the local minima. It is not dependent on thestarting point such as the gradient based optimizationtechnique in ADAMS. So, this is an advantage forthis technique to find the global optimum minimumwithout falling into false minima, that is, local minima.A basic genetic algorithm comprises some main geneticoperators such as selection, reproduction, crossoverand mutation (Goldberg 1989). The selection functionchooses parents for the next generation. In this study,

the best individual and two individuals are selected withstochastic uniform function. The reproduction controlshow the next generation in the genetic algorithm iscreated. Two elite individuals in the current genera-tion are considered, and also crossover and mutationfractions are adjusted as 80% and 20% for the nextgeneration, respectively. In this study, an error basedobjective function (OF) is defined as follows,

Minimize f (x) =s∑

n=1

(μd

n − μan

)2

Subject to gk (x) ≤ 0xl ≤ xr ≤ xu (12)

Fig. 4 Schematicrepresentation ofNeural–Genetic diagram


Fig. 5 Equivalent clearancesand convergence errors forNN model

where μd and μa denote the desired (without clearance)and actual (with clearance) transmission angles, s is thenumber of considered points. gk(x) is the linear inequal-ity constraints, and xr denote the design variables whichconsist of link lengths (L2,L3), x1 and xu are the lowerand upper bounds of these variables, respectively. Theaim of this objective function is minimizing the trans-mission angle error arising from joint clearances inthe mechanism. So, this function can be expressed asthe sum of the squares of the error that defines thedeviation of each evaluated transmission angle from

the corresponding desired precision value. The neural–genetic scheme for determining of unmodelled joint pa-rameters and optimizing of design parameters is givenin Fig. 4. As shown, the first step of this schema con-sists of the neural network for modelling the clearancevalues and directions as a function of position variableof input link, and the second step is constituted byusing the genetic algorithm. In this step, an error basedobjective function is used for optimizing the trans-mission angle by adjusting the appropriate values oflink lengths.


5 Results

In this study, a concentric slider-crank mechanism mod-el in which two revolute joints with clearance was used.The radii of journal and bearing were considered 7 and8 mm, respectively. All the links and all the joints inmechanism were assumed to be rigid, and the runningspeed of the input link was considered as 600 rpm.Mass centers of links were assumed in the midpoint ofcorresponding links. ADAMS (MSC Software Corpo-ration (2005), a mechanical system simulation software,was used for modelling and simulating the mechanism,and Hertzian contact-impact definition between journal

and bearing was considered (Chen et al. 2003; MSCSoftware Corporation 2005). For this purpose, relatedcoefficient such as stiffness was calculated taken intoconsideration the geometry and physical properties ofthe contacting surfaces (Flores et al. 2006). Trainingand testing of the neural network for modelling theequivalent clearances and directions of joint clearancesas a function of input variable were performed usingappropriate software. Mean square of the error (MSE)was used for performance measuring criteria of MLNN,and 20000 iterations were implemented for trainingthe network weights. Each parameter was simulatedseparately by using the same MLNN structure. The

Fig. 6 Directions of jointclearances and convergenceerrors of NN model


Fig. 7 Time-derivatives ofjoint clearance directions inADAMS and NN models

variations of equivalent clearances and convergence er-rors for MLNN model with respect to the input variableare given in Fig. 5. As shown, in the beginning of themotion, the continuous contact assumption at the realmechanism condition is not valid. During this shortperiod, there is an unstable condition in the link con-nections with joint clearances. But after that duration,more stable condition occurs, and also the continuouscontact assumption is verified, that is, clearance vectorin each joint equals to difference between the radiiof journal and bearing. Also, it can be seen from NN

results in the same figure that there is a best approxima-tion, that is, the deviations between NN and ADAMSresults for two equivalent clearances are minimum.

Directions of the joint clearances and convergenceerrors of MLNN model with respect to the positionvariable of the input link are outlined in Fig. 6. Asshown, the NN results for two clearance directionsfollow the ADAMS outputs smoothly. Deviations be-tween NN and ADAMS results for the directions ofjoint clearances are approximately close to zero duringall the motion period.


10-8

10-6

10-4

10-2

100

0 5000 10000 15000 20000

Iteration

Err

or

Fig. 8 Performance measuring for MLNN for modelling theunknown joint characteristics

Time-derivatives of the joint clearance directionswith respect to the position variable of the input link aregiven in Fig. 7. The MLNN structure used for obtainingthe mathematical expressions exhibits the best conver-gence to the simulation results obtained from ADAMS.

MSE values for each parameter in the training stepof MLNN structure are outlined in Fig. 8. After the20,000 iterations achieved for each parameter, the ob-tained mean square of the errors as a performancemeasuring criteria of the proposed NN model equal to1.1980 × 10−5, 7.6899 × 10−6, 3.918 × 10−7, 3.3382×10−7, 4.6977 × 10−6, 1.3006 × 10−6, 2.5555 × 10−5

and 1.1352 × 10−6 for r2, r3, γ2, γ3, γ2, γ3, γ2 andγ3, respectively. These results are acceptable to modelthe equivalent clearances and directions of joint clear-ances as a function of input variable by using neuralnetwork. Also, it is concluded that these values are

Fig. 9 Position analysis ofconnecting rod mass center


successful reflections of the proposed MLNN structurefor determining of unmodelled parameters in jointswith clearance with respect to the input variable. Afterthe modelling of joint variables as a function of inputvariable by using NN, the obtained functions can beadapt to the mechanism kinematics to investigate theeffects of joints with clearance during the motion ofmechanism.

Kinematic equation of crank’s mass center with re-spect to the crank pivot in (2) comprises no clearance,therefore positions, velocities and accelerations of thiscenter with and without clearances are equal to eachother. (Kinematic equations of mass centers withoutclearances are given in Appendix 1). Position analysisof the connecting rod mass center includes the jointclearances effects of the crank-connecting rod and theconnecting rod-piston joints. The effect of the second

one is observed indirectly from the definition of θ c3.

These influences on the mechanism for two cases areshown in Fig. 9. As a result of joint clearance, thereare deviations between desired and actual positions ofthe connecting rod mass center for X and Y-directions.These deviations are approximately limited between±1 mm.

Velocity analysis of the connecting rod mass cen-ter for two cases is shown in Fig. 10. The deviationsbetween the desired and actual velocities of the con-necting rod mass center for X-direction are bigger inthe beginning of motion. These deviations for the Y-direction are relatively bigger than the other.

Acceleration analysis of connecting rod mass centeris shown in Fig. 11. As seen, there are big deviationsfrom the desired acceleration curve for each direction.Especially, deviations in the Y-direction are bigger than

Fig. 10 Velocity analysis ofconnecting rod mass center


that of the X-direction. It can be clearly seen that, thesedeviations occurred in the beginning of mechanismmotion, that is, between 0 and 30˚ of input variable.

Position analysis of piston mass center for two casesis given in Fig. 12. Since the position of the piston,(4), is dependent on the clearance vector of two joints,the deviation between the desired and actual positionschance in the range of ±2 mm.

Velocity analysis of piston mass center for two casesis shown in Fig. 13. It is clearly seen during the wholemechanism motion that, there are deviations betweendesired and actual velocities of piston mass center.

Acceleration analysis of piston mass center for twocases is shown in Fig. 14. As seen, there are excessive

deviations from the desired value in the beginning ofthe motion. Especially, these deviations lead to suddenchanges of inertial forces’ directions and these situa-tions will result in reducing the dynamic efficiency ofthe mechanism.

Transmission angle in the model mechanism, knownas a criteria for kinematic performance of the mech-anism, is tried to define for the cases of the jointclearance and without it, and the relevant results areoutlined in Fig. 15. Some little deviations in the valueof transmission angle are observed as a result of thejoint clearances. When these differences are higherthan feasible values, the transmission quality in suchrespect as motion, force and etc., becomes worse.

Fig. 11 Acceleration analysisof connecting rod mass center


Fig. 12 Position analysis ofpiston mass center

Fig. 13 Velocity analysis ofpiston mass center


Fig. 14 Acceleration analysisof piston mass center

Fig. 15 Transmission angleanalysis of mechanism


Fig. 16 Transmission angleerrors for desired-actual anddesired-optimizedmechanisms

By defining optimum design parameters using ge-netic algorithm, the error variations in the transmissionangle as the differences between the desired and actualvalues, and also between the desired and optimumvalues are given in Fig. 16. If the crank and connectingrod lengths are equal to 149,465 and 563,196 mm, re-spectively, the transmission angle error between the de-sired and optimized mechanisms is closer to zero thanthat of the desired and actual mechanisms. This errordecreases by 14.980% after the optimization process.

In addition to global optimizer such as genetic al-gorithm, local optimization technique was also used toadjust the optimum values of design variables for sixdifferent starting points. In the case of local optimiza-tion method, the transmission angle error decreases by14.804% using the values of 149,170 and 562,083 mmof the link lengths for the crank and connecting rod,respectively. If the same link parameters are chosen as147,287 and 555,001 mm for the crank and connectingrod, respectively, the decrease in that error is 13.716%.The number of function evaluations for the local andglobal optimizers occur as 17 and 32, respectively. Al-though the number of this function evaluation for theglobal optimizer is bigger to solve the problem than thatof the local optimizer, the decrease in the transmissionangle error for the global optimizer is also bigger thanthat of the local optimizer one’s. It is concluded that theoptimization problem has a lot of local minima, and thegenetic algorithm as a global optimizer can be used tosolve this problem with no need the starting points onthe searching space. The objective function constitutedfor investigated mechanism is very constrained due toonly two design parameters, that is, crank and connect-ing rod lengths. The proposed neural–genetic schemecan give the better results with the increasing number

of joints with clearance proportional to the numberof links.

6 Conclusion

In this study, joints with clearance at contact points asknown crank-pin center and piston-pin center are con-sidered, and their effects on the kinematic characteris-tics and transmission quality of the model mechanismare investigated. A neural–genetic scheme is proposedfor modelling the joint characteristics and minimizingthe undesired effects due primarily to clearances. So,the joint clearance is considered as a massless virtuallink and its motion equation with respect to the inputvariable is modelled using Neural Network. Trainingand testing data sets for the network weights are ob-tained from mechanism simulation using ADAMS. Inorder to designate network accuracy in predicting theoutputs, the MLNN model is examined in responseto input which are not used in the training step. Asseen from the results, equivalent clearances and di-rections of joint clearances are defined with minimumerrors in convergence to ideal value as a function ofinput variable. Also, Genetic Algorithm approach isused to optimize the design parameters for maximiz-ing the mechanism efficiency. For this purpose, anobjective function and corresponding constraints aredefined.

The direction variation of the joint clearance vector,which is closer to the input link, is agree with thevariation of position variable for the input link. Thetolerance at the joint clearance between connecting rodand piston is naturally affected by the previous joint


Fig. 17 Two different viewsof investigated slider-crankmechanism links

clearance, that is, the kinematic contribution of the pre-vious joint clearance is superposed to the kinematics ofthe successive joint. These effects are clearly seen fromthe values of position errors for connecting rod andpiston mass centers. The deviations obtained for the ve-locity and acceleration characteristics with and withoutclearances are relatively higher values in the beginningof the motion, that is, there is an unstable conditionduring this period. But and then, the continuous contactoccurs between journal and bearing and mechanismhas a more stable working condition relative to thebeginning of the motion. In stable working condition,these verify mainly the continuous contact model as-sumptions between journal and bearing in joints withclearance. The peaks in acceleration curves of the jointswith clearance have crucial effects on inertial forcesand their directions in mechanism. These effects makethe vibrations and noises greater and reduce the kine-matic and dynamic efficiencies of mechanism. By theoptimization of design parameters in mechanism, thetransmission angle error decreases by 14.980%. As a

practical implication, this result improves the kinematicquality and force transmission efficiency, and so me-chanical performance of mechanism. Therefore, me-chanical working condition becomes better. In futurestudies, dynamic analysis of system will be performedin light of these kinematic outputs, and effects ofjoints with clearance on mechanism dynamic will beinvestigated.

Acknowledgements This work is a part of the research projectFBT-07-53. The authors wish to express their thanks for financialsupport being provided by the Scientific Research Project Fundof Erciyes University, in carrying out this study.

Appendix

Front and side views of the examined mechanism’s linksare given, respectively, Fig. 17.

Dynamic parameters of slider-crank mechanism aregiven in Table 2,

Table 2 Parameters of modelmechanism Parameters Descriptions Values

I2 Moment of inertia of the crank 1.458 × 10−3 kg m2

I3 Moment of inertia of the connecting rod 2.482 × 10−2 kg m2

I4 Moment of inertia of the piston 4.30 × 10−4 kg m2

m2 Mass of the crank 0.3704 kgM3 Mass of the connecting rod 0.773 kgM4 Mass of the piston 0.4482 kg


Mass center positions without clearances for movinglinks are given, respectively,

[xG2

yG2

]= A0G2

[cos θ2

sin θ2

](13)

[xG3

yG3

]= L2

[cos θ2

sin θ2

]+ AG3

[cos θ3

sin θ3

](14)

[xG4

yG4

]= L2

[cos θ2

sin θ2

]+ L3

[cos θ3

sin θ3

](15)

Angular direction of connecting rod without clear-ances with respect to X-axis is given as follows,

θ3 = sin−1

[− L2 sin θ2

L3

](16)

Transmission angle without clearances in the mech-anism is given,

μ = cos−1

[L2 sin θ2

L3

](17)

Time-derivatives of mass center positions withoutclearances for moving links are given, respectively,

⎡⎣ xGi

yGi

⎤⎦ = ω2

⎡⎢⎢⎣

∂xGi

∂θ2

∂yGi

∂θ2

⎤⎥⎥⎦ (18)

⎡⎣ xGi

yGi

⎤⎦ = α2

⎡⎢⎢⎣

∂xGi

∂θ2

∂yGi

∂θ2

⎤⎥⎥⎦ + ω2

2

⎡⎢⎢⎢⎣

∂2xGi

∂θ22

∂2 yGi

∂θ22

⎤⎥⎥⎥⎦ (19)

Angular velocity and acceleration of connecting rodwithout clearances are given in the following form,

θ3 = ω2∂θ3

∂θ2(20)

θ3 = α2∂θ3

∂θ2+ ω2

2∂2θ3

∂θ22

(21)

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optimization of transmission angle for slider-crank mechanism with joint clearances

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