Dynamic analysis of a slider–crank mechanism with eccentric connector and planetary gears

Mechanism

Mechanism and Machine Theory 42 (2007) 393–408

www.elsevier.com/locate/mechmt

andMachine Theory

Dynamic analysis of a slider–crank mechanism with eccentricconnector and planetary gears

Selcuk Erkaya, S�ukru Su, _Ibrahim Uzmay *

Erciyes University, Engineering Faculty, Department of Mechanical Engineering, 38039 Kayseri, Turkey

Received 15 November 2005; received in revised form 24 March 2006; accepted 11 April 2006Available online 9 June 2006

Abstract

In this study, the kinematic and dynamic analysis of a modified slider–crank mechanism which has an additional eccen-tric link between connecting rod and crank pin, as distinct from a conventional mechanism, are presented. This new extralink that may be called the eccentric connector transmits gas forces to the crank, and it also drives a planetary gear mech-anism transmitting a great deal of driving forces to the output. In order to drive the planetary gear train, a pinion fixed tothe eccentric connector is used. Consequently, the driving force is transmitted to crankshaft by means of two differentways. For the comparison, the dynamic analysis results of developed slider–crank mechanism have been evaluated withrespect to that of a conventional slider–crank mechanism. As a result, although both the conventional and the modifiedslider–crank mechanisms have the same stroke and the same gas pressure in the cylinder, it is observed that the modifiedmechanism has a bigger output torque than that of the conventional mechanism.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Slider–crank mechanism; Kinematic and dynamic analysis; Eccentric connector; Epicyclic gear mechanism; Torque output

1. Introduction

In general, conventional slider–crank mechanism is used in the internal combustion engines except for wan-kel engines. As a result of the investigations focused on mechanical and thermal design parameters in the inter-nal combustion engines, mechanical strength, thermal efficiency, wear and surface quality of the elements havebeen improved during the 20th century. Consequently, it can be clearly seen that fuel consumption has beenreduced and engine lifetime has been increased. In spite of these developments, any definite modification in theslider–crank mechanism has not been effected up to now. However, motion and force transmission character-istics have been improved by means of studies on kinematics and dynamics of mechanism.

Cveticanin and Maretic have summarized dynamic analysis of a cutting mechanism which is a special typeof the crank shaper mechanism [1]. The influence of the cutting force on the motion of the mechanism wasconsidered. The Lagrange equation was used and boundary values of the cutting force were obtained

0094-114X/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechmachtheory.2006.04.011

* Corresponding author. Tel.: +90 3524375832; fax: +90 3524375784.E-mail address: [email protected] (_I. Uzmay).

mailto:[email protected]

394 S. Erkaya et al. / Mechanism and Machine Theory 42 (2007) 393–408

analytically and numerically. Ha et. al. have derived the dynamic equations of a slider–crank mechanism byusing Hamilton’s principle, Lagrange multiplier, geometric constraints and partitioning method [2]. The for-mulation was expressed by only one independent variable. Dynamic responses between the experimentalresults and numerical simulations were compared to obtain the best dynamic modeling. Also, a new identifi-cation method based on the genetic algorithm was presented to identify the parameters of a slider–crank mech-anism. Transient and steady state dynamic response of a class of slider–crank mechanisms which involves rigidmembers but compliant supporting bearings have been investigated by Goudas et al. [3]. In their research,both the driving and the resisting loads were expressed as a function of the crank’s angular position.Lagrange’s equations were also used to derive the non-linear equations of motion. These equations weresolved numerically to examine the resulting dynamical system. Koser has summarized kinematic performanceanalysis of a slider–crank mechanism based robot arm performance and dynamics [4]. The kinematic perfor-mance of the robot arm was analyzed by using generalized Jacobian matrix. It was obtained that the slider–crank mechanism based robot arm had almost full isotropic kinematic performance characteristics and itsperformance was much better than the best 2R robot arm. Dynamic analysis and vibration control of a flexibleslider–crank mechanism driven by a servomotor have been studied by Fung and Chen [5]. To formulate thegoverning equations of the mechanism, geometric constraint at the end of a flexible connecting rod wasderived and introduced into Hamilton’s principle. Chen and Huang have investigated the dynamic responsesof flexible slider–crank mechanism by considering all the high order terms [6]. The non-linear equation ofmotion was solved by Newmark method. In the low-speed range, it was found that the dynamic responsespredicted by non-linear and linear approaches (neglecting the high order terms) indeed made no significantdifference. However, when the rotation speed increased up to about one-fifth of the fundamental bending nat-ural frequency of the connecting rod, simplified linear approaches exhibited a noticeable error. Soylemez hassummarized the classical transmission angle problem for slider–crank mechanisms which is the determinationof the dimensions of planar slider–crank mechanisms with optimum transmission angle for given values of theslider stroke and corresponding crank rotation [7]. The complex algebra was used to solve that classical prob-lem and the solution was obtained as the root of a cubic equation within a defined range. Another researchabout transmission angle was implemented by Shrinivas and Satish [8]. They have summarized importance ofthe transmission angle for most effective force transmission. In this sense, they investigated 4-, 5-, 6- and 7-barlinkages, spatial linkages and slider–crank mechanisms.

2. Dynamic analysis of conventional slider–crank mechanism

Slider–crank mechanism converts the translatory motion of piston to rotary motion of crank. Driving effectof slider–crank mechanism is obtained by a gas pressure arising from combustion of mixture consisting of fueland air. The force corresponds to this pressure causes the piston to translate along the vertical axis and thisaction is transmitted to crank through connecting rod. The conventional mechanism which is widely used ininternal combustion engines is a concentric slider–crank mechanism. This mechanism, as shown in Fig. 1, hasone degree of freedom, that is, a constrained mechanism. Parameters used in the conventional mechanism aregiven in Table 1 in Appendix A. In considering the kinematic analysis of the slider–crank mechanism, it isnecessary to determine the displacement of the slider and then its corresponding velocity and acceleration.

For this purpose, displacement of piston can be defined as a function of crank’s angular position in thefollowing form:

S�pi ¼ l� þ r� cos h� � 1

2k� sin2 h�

� �ð1Þ

where r* is the crank radius, l* is the connecting rod length, k� ¼ r�

l� and h* denotes the crank’s angular position.If the piston displacement is derived in time assuming the angular velocity to be constant, the piston velocitycan be found as

V �pi ¼ �r�x21 sin h� þ 1

2k� sin 2h�

� �ð2Þ

x

TDC

4

*r

*l3

y

B

*piS

BDC

β*

A

θ*

O

2

Fig. 1. Concentric slider–crank mechanism.

S. Erkaya et al. / Mechanism and Machine Theory 42 (2007) 393–408 395

By taking time-derivative of Eq. (2), the piston acceleration is given by

a�pi ¼ �r�x221 cos h� þ k� cos 2h�½ � ð3Þ

The purpose of dynamic analysis of the slider–crank mechanism is to determine the total output torque arisingfrom resultant force (gas + inertia). In the mechanism, gas forces, known as driving effect, do not have con-stant value during the expansion stroke. So, the cylinder volume has to be expressed as a function of crank’sangular position considering the variation of gas forces

V xðh�Þ ¼ V c þ l� þ r� � S�pi

h i pD2

4ð4Þ

where Vc is the cylinder clearance volume, S�pi is the piston displacement and D is the cylinder bore diameter.Gas pressure during the expansion stroke is given by

P eðh�Þ ¼ P 3

VV xðh�Þ

� �k

ð5Þ

where P3 is the pressure in the cylinder and V is the cylinder volume at the end of the compression stroke. k isthe polytropic coefficient and usually taken to be equal 1.3 for diesel engines [9]. Gas forces can be expressed asa function of crank’s angular position in the following form:

F g ¼ Api½P eðh�Þ � P atm� ð6Þ
where Api is the piston sectional area and Patm is the atmospheric pressure. In order to determine joint forces,dynamic force analysis has to be completed considering gas forces and inertial forces. These forces, known asactive effects, are outlined in Fig. 2.
Referring to Fig. 2(a), output torque on the crankshaft, arising from gas forces, is given by

Mgas ¼ rxF i32 ð7Þ

where F i32 denotes the gas forces effect on the crank-pin center and can be expressed as a function of gas forces

in the following form:

F i32 ¼ �

Fg

cos b�ð8Þ

y

xA

270+β*

ι32F

*r

*l

(a)

x

4

3

A

β*

gF

B1

y

θ*

O

2

ι14F

90+β*

y

270

gF

ι34F

x

[ ]33 GG– m a

ιι14F

ιι34F

y

x90

[ ]BB– m a

31a

*l

*r

x

O

(b)

[ ]BB– m a

3Ga

+βψ *

Ba

4

3

A

β*

B

1

y

θ*2

G3

xδ

ιιι32F

y

90+β*ιι

32F

A

270+β*

180

Fig. 2. The gas (a) and inertial forces (b) for conventional mechanism.


As stated schematically in Fig. 2(b), the resultant inertial force on the point A is given as a vectorial summa-tion in the following form:
X
FInertia ¼ F ii32 þ F iii

32 ð9Þ

where F ii32 denotes the inertial effect of the piston’s mass and F iii

32 denotes the inertial effect of the connectingrod’s mass. These forces can be expressed, respectively

F ii32 ¼ �

ð�mBaBÞcos b�

ð10Þ

where mB is the total mass on the piston-pin center. aB is the acceleration of the piston-pin center and equals toa�pi in Eq. (3).

F iii32 ¼

ð�mG3aG3ÞðBG3 sinðwþ b�Þ � l� cos w sin b�Þ þ ð�IG3

a31Þl� cos b�

� �2

þ ðð�mG3aG3Þ cos wÞ2

" #1=2

ð11Þ

where mG3is the mass of connecting rod and aG3

is the linear acceleration of connecting rod’s gravity center.Also, IG3

is the inertia moment and a31 is the angular acceleration. aG3and a31 are given as a function of

crank’s angular position in Appendix A. Output torque caused by resultant inertial force is defined by

M Inertia ¼ rxX

FInertia ð12Þ


From Eqs. (7) and (12), total output torque on the crankshaft can be written in the following form:

MTotal ¼Mgas þM Inertia ð13Þ

3. Analysis of modified slider–crank mechanism

3.1. Elements of developed mechanism

Modified slider–crank mechanism, as shown in Fig. 3, has an additional extra link between connecting rodand crank pin as distinct from conventional mechanism. The new extra link, may be called eccentric connec-tor, transmits gas forces to the crank and also drives a planetary gear mechanism. In order to drive planetarygear train, a pinion fixed to the eccentric connector in a parallel plane is used. So, there are two transmissionlines in this new system. One of them called direct transmission line consists of the way of connecting rod,eccentric connector, crank arm and the other one called indirect transmission line consists of connectingrod, eccentric connector, gear mechanism.

When the motion characteristic of the mechanism in Fig. 3 is investigated carefully, a kinematic-basedscheme in Fig. 4 is obtained.

Referring to Fig. 4, it can be seen that the modified mechanism has one degree of freedom, that is, thismodel is a constrained mechanism. The eccentric connector makes a curvilinear translation because of chosena particular gear ratio between pinion and ring gear. That is, it has no a relative motion with respect to crankpin (n31 = 0). Consequently, there is an identical freedom in the mechanism due to the specific motion of theeccentric connector. Parameters used in the modified mechanism are given in Table 2 in Appendix B.

3.2. Kinematics of modified slider–crank mechanism

In the modified mechanism, ring gear rotates 1/2 times at the speed of the crank arm because of two mainreasons, one of them is gear ratio between pinion and ring gear and the other is curvilinear translation of theeccentric connector. As shown in Fig. 3, this ratio is also obtained from the planetary gear relationship amongthe crank arm, pinion and ring gear in the following form:

þ r6in

rp

¼ n31 � n21

n61 � n21

; ðn31 ¼ 0Þ ð14Þ

For the position analysis of the modified mechanism, the position of piston-pin center with respect to crankrotation center is given by

Spi ¼ rc cos hþ l cos b ð15Þ
where rc is the crank radius, l is the connecting rod length and b is defined as the angle between the line ofconnecting rod and the cylinder axis. b = f(h) is given by
cos b ¼ 1� 1

2kc sin hþ kp �

el

h i2

ð16Þ

where rp is the radius of pinion gear, e is the distance of eccentricity, kc ¼ rc

l and kp ¼ rp

l . Substituting Eq. (16)into Eq. (15) yields

Spi ¼ lþ rc cos h� 1

2kc sin2 h

� �� rp kc sin hþ 1

2kp

� �þ e kc sin hþ kp �

e2l

h ið17Þ

Time-derivative of Eq. (17) yields the velocity of piston

V pi ¼ �x21 rc sin hþ 1

2kcrc sin 2hþ kcrp cos h� kce cos h

� �ð18Þ

From the differentiation of Eq. (18), the piston acceleration is defined by

api ¼ �x221½rc cos hþ kcrc cos 2h� kcrp sin hþ kce sin h� ð19Þ

Fig. 3. Modified slider–crank mechanism; 3D view (a), side view (b) and schematical representation (c) [10].


PlanetaryGear (3)

O1

piS

x

BDC

TDC

l

crpr B

O

5

y

β

A

e

θ

C

23

4

Gear (9)

Gear (8)

Gear (7)

Ring Gear (6)

Eccentric Connector (3)

Fig. 4. Scheme of modified mechanism.


3.3. Dynamics of modified slider–crank mechanism

In the modified mechanism, eccentric connector has three joints with such other links as the crank arm,connecting rod and ring gear. When the connecting rod line is taken to be reference, there is a plane differencebetween two forces, which one of them is transmitted from the connecting rod to the crank arm by the eccen-tric connector and the other is applied to ring gear by the pinion gear. This difference is approximately41.5 mm. So, in the dynamic analysis of this mechanism, this distance has to be considered. The other impor-tant points about mechanism, as shown in Fig. 3, counterweight for eccentric connector (Part 3.1) providesthat the gravity center of eccentric connector coincides with the crank-pin axis, 7th and 8th gears are joinedeach other as a unit link and 9th gear is joined to the crankshaft by means of wedge.

3.3.1. Output torque obtained by means of gas forces

Gas forces, as considering driving effect in this mechanism, do not have constant value during the expan-sion stroke. So, it is necessary that the cylinder volume, gas pressure and the gas forces have to be expressed asa function of crank’s angular position. For this purpose, by adapting the parameters, such as h, l, rc and Spi

given in Eqs. (4)–(6), these expressions can be applied to the modified mechanism (the other parameters, suchas Vc, D, P3 and V, are same values in that equations for each mechanism). Fig. 5 shows the effect of the gasforces on the mechanism links schematically.

Gas forces pushing the piston exert the force, F i43, on eccentric connector at point B. This force has two

different components orthogonal to each other. As one of them which parallel to cylinder axis causes the plan-etary gear mechanism to drive, the normal component also causes the crank to translate.

O1

ι61M

ι91M

ιι8171 MM =

D

ι32F

ι36F360-(θ- ϕ)

D4

pr

270+βι

43F

1

C

B

x

l

PlanetaryGear (3)

gF

5

y

e

O

θ2 A 3

cr

ι15F

90+β

270

gF

ι45F

y

x

Gear (9)

Gear (8)

Gear (7)

Ring Gear (6)

Fig. 5. Schematic representation of the gas forces effect on the modified mechanism.


3.3.1.1. Output torque obtained by means of direct transmission line. This line consists of connecting rod, eccen-tric connector and crank arm links. Force equilibrium for the eccentric connector is given by

F i43 þ F i

23 ¼ 0 ð20Þ
where F i
43 force is exerted on eccentric connector by means of the connecting rod and also the force F i23 is ex-

erted on eccentric connector by means of crank arm. Because of the special construction of the eccentric con-nector and the planar difference, the component F i

43xhas to be taken instead of F i

43 in Eq. (20). This force canbe expressed as a function of gas forces in the following form:

F i43x¼ �Fg tan b ð21Þ

Output torque obtained by means of the direct transmission line (M i21) is given by

M i21 ¼ rcxF i

32 ð22Þ
where F i
32 denotes the force which is exerted on the crank arm by means of the eccentric connector. By con-sidering Newton’s law, it is clear that this force is equal in magnitude and opposite in direction to F i

23

F i32 ¼ �Fg tan b ð23Þ

As can be seen from Eqs. (21) and (23), F i43x

and F i32 forces are equal in magnitude and same direction.


3.3.1.2. Output torque obtained by means of indirect transmission line. This line consists of connecting rod,eccentric connector and gear mechanism.

Referring to Fig. 6, moment equilibrium for the eccentric connector with respect to crank-pin center isgiven by

M iA ¼ rpxF i

43 þ rpxF i63 ¼ 0 ð24Þ

From Eq. (24), F i63 force which is exerted on pinion gear by means of the ring gear can be obtained as a func-

tion of gas forces in the following form:

F i63 ¼ �

Fg

cos uð25Þ

where u is the pressure angle between pinion and ring gears. Moment expression for the ring gear is given by

M i61 ¼ r6in

xF i36 ð26Þ

where r6inis the inner diameter of the ring gear and F i

36 force is exerted on the ring gear by means of the piniongear. It is equal in magnitude and opposite in direction to F i

63 force in Eq. (25). For obtaining the output tor-que by way of the indirect line (M i

91), it can be used the gear ratio among the ring, 7th, 8th and 9th gears. Thistorque can be expressed in the following form:

M i91 ¼ � r9

r8

� �� r7

r6out

� �M i

61 ð27Þ

From Eqs. (22) and (27), total output torque on the crankshaft, caused by gas forces, can be written in thefollowing form:
X
Mgas ¼M i21 þM i

91 ð28Þ

Planetary Gear (3)

Gear (9)

Gear (8)

Gear (7)

Ring Gear (6)

O1

y

x

ι63F 180-(θ- ϕ)

ι61M

ι91M

ιι8171 MM =

pr

270+βι

43F

B

e

O

θ2 A

3

cr

D

Fig. 6. Schema of the moment equilibrium for pinion gear.


3.3.2. Output torque arising from inertial forces

For a given configuration, Fig. 7 shows the inertial forces and the moments of inertia on the mechanismschematically.

If the joint B is considered as a point that driving effects arising from gas forces and inertial effects are trans-mitted to the crankshaft, resultant forces at this points can be classified as vertical and horizontal components.As the horizontal components of inertial forces drive the crankshaft via direct line, resultant vertical forcesreach to the output through gear mechanism, that is, indirect line. Resultant effect on the crankshaft of inertialforces is sum of these components

M21Inertia¼M ii

21 þM iii21 þM iv

21 þMv21 þMvi

21 þMvii21 þMviii

21 ð29ÞM91Inertia

¼M ii91 þM iii

91 þM iv91 þMv

91 þMvi91 þMvii

91 þMviii91 ð30Þ

where M i21 and M i

91, (i = ii, iii, . . . , viii), are the moments of inertia on the crank due to inertial effects of mech-anism’s links, and relevant expressions are given in Appendix B. From Eqs. (29) and (30), the total moment ofinertia is expressed in the following form:

M Inertia ¼M21InertiaþM91Inertia

ð31Þ

PlanetaryGear (3)

O1

[ ]vιιG– I 7187

α

8171

8171

ααωω

==

6161,αω

[ ]vιιιG– I 919

α

[ ]vιG– I 616

α

9191,αω

prBa

[ ]vGG– m

33a

3Ga

ψ+β

θ

G4 [ ] ιιιGG– m

44a

4Ga

Ca

[ ] ιιCC– m a

4

1

C

B

x

l

5

y

β

e

O

A≡G3

cr

μ[ ]ιv

BB– m a

2

Gear (9)

Gear (8)

Gear (7)

Ring Gear (6)

Fig. 7. Schematic representation of the inertial force effects on the modified mechanism.


As seen from Eqs. (28) and (31), total output torque on the crankshaft, consists of gas and inertial forces, canbe defined in the following form:

MTotal ¼Mgas þM Inertia ð32Þ

4. Conclusion and discussion

In this study, in order to compare two different mechanisms in kinematic and dynamic respects, the samestroke and gas force are considered and the rotation speed of 2500 rpm is also used for both mechanisms. It isassumed that unbalance of rotating masses is balanced using counterweight. Frictional effects of link jointsand gravitational effects are ignored. The total output torques of conventional slider–crank mechanism,defined by Eq. (13), and the new mechanism with eccentric connector, given in Eq. (32), are shown in Fig. 8.

As seen from Fig. 8, although the driving force is the same for two mechanisms, a certain difference foroutput torque values is observed. Increasing in the output torque of modified mechanism arises from newpower transmission line consisting of eccentric connector and planetary gear mechanism. Relatively slowmotion of piston at the beginning of power stroke also compensates disadvantage of dead position.

In the case of conventional slider–crank mechanism, when piston is at top dead center, gas forces reachtheir maximum values. But any torque effect can not be obtained for a small time interval due to this deadposition, that is, torque output decreases. This situation does not occur in the modified mechanism with eccen-tric connector owing to structural arrangement. In the beginning of the power stroke, that is, when the drivingforce has its maximum value, force transmission is possible for both power transmission lines. Therefore, asopposed to conventional mechanism, torque output is directly proportional to driving force for the proposedmechanism. The reason of rapidly decrease in torque output of the new system is a decrease in gas pressure asa result exhaust phenomena. The output torque values for the mechanism with eccentric connector are higherthan that of classical mechanism, but it should be noted that mechanical losses arising from additional powertransmission line are not considered.

As a result of Eqs. (12) and (31), output torques resulting from inertial effects are summarized in Fig. 9.Inertial torque in the developed mechanism, as expected, has bigger equivalent work or energy value than

that of classical mechanism. This result can be evaluated as a natural effect of mechanism link’s inertias inwhich the mechanism with an eccentric connector has much more number of moving links than the conven-tional mechanism.

0 60 120 180 240 300 360-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

Crank’s angular position [Degree]

Tor

que

[N

m]

ModifiedConventional

Fig. 8. Total output torque obtained by resultant force for each mechanism.

0 60 120 180 240 300 360-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35Modified

Conventional


Tor

que

[N

m]

Fig. 9. Output torque obtained by inertial forces for each mechanism.


In the case of the proposed slider–crank mechanism, torque outputs resulting from gas forces, obtainedfrom Eqs. (22), (27) and (28), are given in Fig. 10.

It is obvious that a large amount of total torque output is transmitted by the way of epicyclic gear train.For the case of the new mechanism, in Fig. 11, torque outputs are classified in two groups according to

power transmission lines, respectively.As seen from this figure, a certain amount of total torque is transmitted via gear mechanism. Consequently,

if mechanical loses of epicyclic gear trains in the proposed mechanism with eccentric connector are considered,it can be said that the system efficiency decreases.

0 30 60 90 120 150 180-5

0

5

10

15

20

25

30

35

MGear

MTotal

MCrank


Tor

que

[N

m]

Fig. 10. Output torque obtained by gas forces for modified mechanism.

0 60 120 180 240 300 360-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

MGear

MTotal

MCrank


Tor

que

[N

m]

Fig. 11. Total output torque obtained by resultant force for modified mechanism.


Acknowledgements

This work is a part of the research project DPT-05-07. The authors wish to express their thanks for financialsupport being provided by the State Planning Organization, Republic of Turkey and also Mr. Hasan BasriOZDAMAR which has the patent right of the proposed modified mechanism.

Appendix A. Parameters and equations for conventional mechanism (Table 1)

Linear and angular accelerations of the connecting rod’s gravity center in Eq. (11) can be given in the fol-lowing form: 2

TableParam

Param

r*

l*

x21

mB

AG3

mG3

IG3

aG3¼ �x2

21

AG3 2r�ðl�Þ2 cos h� þ 2ðr�Þ2l� cos 2h��

þ ðr�Þ3 cos h�ðcos 2h� � 5 sin2 h�Þh i

2ðl�Þ2 ðr�Þ2 þ AG23 þ 2r�AG3

cos h� 2ðl�Þ2�ðr�Þ2 sin2 h�ð Þ2ðl�Þ2 � r� sin2 h�

l�

� �� 1=2

6664

�AG2

3 2r�ðl�Þ2 sin h� þ ðr�Þ3 sin h�½2� 3 sin2 h�� þ 2ðr�Þ2l� sin 2h�h i2

4ðl�Þ4 ðr�Þ2 þ AG23 þ 2r�AG3 cos h� 2ðl�Þ2�ðr�Þ2 sin2 h�

2ðl�Þ2

� �� r� sin2 h�

l�

h ih i3=2

37775

a31 ¼ �r�x2

21 sin h�i þ ð2 cos h� þ k� cos 2hÞj½ � � l�x231ðcos b�i þ sin b�jÞ

l�ðsin b�i � cos b�jÞ
1eters used in the conventional mechanism
eters Descriptions Values

Crank radius 50 mmLength of connecting rod 269 mmAngular velocity of crank arm 261.6 rad/sDynamically equivalent mass in piston-pin center 2.0933 kgDistance between cran-pin center and connecting rod gravity center 97 mmDynamically equivalent mass in connecting rod gravity center 0.5131 kgMoment of inertia of the connecting rod 0.01612 kg m2

Stroke 100 mm


Appendix B. Parameters and equations for modified mechanism (Table 2)

Linear acceleration of the connecting rod’s gravity center can be expressed in the following form:

TableParam

Param

rc

l

rp

e

ux21

mC

CG4

BG4

mG4

IG4

mB

mG3

r6in

r6out

IG6

r7

r8

IG(7–8)

r9

IG9

aG4¼ x2

21

rcBG4 BG4rc cos 2h� 2rpBG4 sinhþ 2eBG4 sinh� 2rcl cos 2hþ 2rpl sinh�

2l2rG4

"

�rcBG4 2el sinh� 2rpl sinhþ 2l2 coshþ r2

c coshðcos 2h� 5 sin2 hÞ�

þ rcrp rG4 sinhþA1 cosh½ �2l2rG4

þrcBG4 r2

p coshþ e2 coshþ rcrp sin 2h� rce sin 2h� 2erp cosh� rcBG4 cos 2hþ 2rpBG4 sinhh i

2l2rG4

�rcBG4A1 BG4rc sin 2hþ 2rpBG4 cosh� 2eBG4 cosh� 2rcl sin 2h� 2lrp cosh

� 2l2r2

G4

�rcBG4A1 2el cosh� 2rpl cosh� 2l2 sinh� r2

c sinhð2� 3 sin2 hÞ þ r2p sinh

h i2l2r2

G4

�rcBG4A1 e2 sinh� rcrp cos 2hþ rce cos 2h� 2erp sinh� rcBG4 sin 2h� rpBG4 coshþ 2eBG4 cosh

� 2l2r2

G4

#

where A1 and rG4can be given, respectively

A1 ¼rc cos hBG2

4½rc sin hþ e� � rcl cos hBG4½2rc sin h� eþ 2rp� þ rcrpl2 cos h

l2rG4

�rc sin hBG4 2l2 þ r2

cð2� 3 sin2 hÞ � r2p � e2 þ 2erp þ 2rcBG4 cos h

h iþ r2

c cos 2hBG4½rp � e�

2l2rG4

rG4¼ rc sin hþ rp � BG4 sin b �2 þ rc cos hþ BG4 cos b

�2h i1=2

2eters used in the modified mechanism

eters Descriptions Values

Crank radius 50 mmLength of connecting rod 269 mmRadius of pinion gear 50 mmDistance of eccentricity 50 mmPressure angle 20o

Angular velocity of crank arm 261.6 rad/sDynamically equivalent mass in piston-pin center 2.7309 kgDistance between piston-pin center and connecting rod gravity center 184 mmDistance between point B and connecting rod gravity center 85 mmDynamically equivalent mass in connecting rod gravity center 0.5681 kgMoment of inertia of the connecting rod 0.0488 kg m2

Dynamically equivalent mass in point B 2.1343 kgTotal mass in eccentric connector gravity center 9.5602 kgInner radius of ring gear 100 mmOuter radius of ring gear 110 mmMoment of inertia of the ring gear 0.2133 kg m2

Radius of gear 7 55 mmRadius of gear 8 82.5 mmMoment of inertia of the gear (7–8) 0.0273 kg m2

Radius of gear 9 82.5 mmMoment of inertia of the gear 9 0.0284 kg m2

Stroke 100 mm


Angular acceleration of the connecting rod’s gravity center can be expressed in the following form:

a41 ¼aG4

sinðw� kÞBG4

2 þ BI412 � 2BG4BI41 cos b

� 1=2

where w can be given, respectively,

w ¼ arctanrc sin hþ rp � BG4 sin b

rc cos hþ BG4 cos b

� �

Linear acceleration for point B of the modified mechanism can be given in the following form:

aB ¼ �x221

2rcrp sin h r2c þ r2

p þ 2rcrp sin hh i

þ r2cr2

p cos2 h

2 r2c þ r2

p þ 2rcrp sin hh i3=2

264

375

Linear acceleration of eccentric connector’s gravity center can be given in the following form:

aG3¼ aB cos l

where l can be expressed, respectively

l ¼ arctanrc cos h

rc sin hþ rp

� �

Angular accelerations for elements of the epicyclic gear mechanism can be given, respectively

a61 ¼aB cos l cos h

r6in

a71 ¼ a81 ¼ �aB cos l cos hr6out

r7r6in

a91 ¼ aB cos l cos hr8r6out

r9r7r6in

The moments of inertia in Eq. (29) can be given in the following form:

M ii21 ¼ �ð�mCaCÞrc cos h tan b

M iii21 ¼

rc cos hl cos b

ð�mG4aG4ÞðCG4 sinðwþ bÞ þ l cos w sin bÞ � ð�IG4

a41Þ�

M iv21 ¼

ð�mBaBÞrc sin h cos b½rcr8r6out cosðlþ hÞ þ r9r7r6insin l�

rcr8r6out sinðbþ hÞ � r9r7r6incos b

þ ð�mBaBÞrc sin h sin l

� �

� ð�mBaBÞrc cos h sin b½�rcr8r6out cosðlþ hÞ � r9r7r6insin l�

rcr8r6out sinðbþ hÞ � r9r7r6incos b

þ ð�mBaBÞrc cos h cos l

� �

Mv21 ¼ �

ð�mG3aG3Þr2

c cos h sinðbþ hÞcos b

r6inr7r9

r6out r8

� �� rc sinðbþ hÞ

� ð�mG3aG3Þrc cos h

Mvi21 ¼ �ð�IG6

a61Þ tan b cos hrc

r6in

Mvii21 ¼ ð�IGð7–8Það71�81ÞÞ tan b cos h

r6out rc

r7r6in

Mviii21 ¼ �ð�IG9

a91Þ tan b cos hrcr8r6out

r9r7r6in


The moments of inertia in Eq. (30) can be given in the following form:

M ii91 ¼ �ð�mCaCÞ

r6inr7r9

r6out r8

� �

M iii91 ¼ ð�mG4

aG4Þ cos w cos b

r6inr7r9

r6out r8

� �

M iv91 ¼

ð�mBaBÞ cos bð�rcr8r6out cosðlþ hÞ � r9r7r6insin lÞ

�r9r7r6incos bþ rcr8r6out sinðbþ hÞ � ð�mBaBÞ sin l

� �r9r7r6in

r8r6out

� �

Mv91 ¼ �

ð�mG3aG3Þrcr6in

r7r9 cos h cos b cos urcr6out r8 cos u sinðbþ hÞ þ r6in

r7r9 cos b

Mvi91 ¼

ð�IG6a61Þr9r7

r8r6out

Mvii91 ¼ �

ð�IGð7–8Það71�81ÞÞr9

r8

� �Mviii

91 ¼ ð�IG9a91Þ

References

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Vibration 289 (4–5) (2006) 1019–1044.[3] I. Goudas, I. Stavrakis, S. Natsiavas, Dynamics of slider–crank mechanisms with flexible supports and non-ideal forcing, Nonlinear

Dynamics 35 (2004) 205–227.[4] K. Koser, A slider crank mechanism based robot arm performance and dynamic analysis, Mechanism and Machine Theory 39 (2)

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motor drive, Journal of Sound and Vibration 214 (4) (1998) 605–637.[6] J.S. Chen, C.L. Huang, Dynamic analysis of flexible slider–crank mechanisms with non-linear finite element method, Journal of

Sound and Vibration 246 (3) (2001) 389–402.[7] E. Soylemez, Classical transmission angle problem for slider–crank mechanisms, Mechanism and Machine Theory 37 (4) (2002) 419–

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Documents

Dynamic analysis of a slider–crank mechanism with eccentric connector and planetary gears