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Mechanism and Machine Theory

Mechanism and Machine Theory 42 (2007) 393–408

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Dynamic analysis of a slider–crank mechanism with eccentric connector and planetary gears _ Uzmay Selc¸uk Erkaya, Sßu¨kru¨ Su, Ibrahim

*

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 2006 Available online 9 June 2006

Abstract In this study, the kinematic and dynamic analysis of a modified slider–crank mechanism which has an additional eccentric link between connecting rod and crank pin, as distinct from a conventional mechanism, are presented. This new extra link that may be called the eccentric connector transmits gas forces to the crank, and it also drives a planetary gear mechanism transmitting a great deal of driving forces to the output. In order to drive the planetary gear train, a pinion fixed to the eccentric connector is used. Consequently, the driving force is transmitted to crankshaft by means of two different ways. For the comparison, the dynamic analysis results of developed slider–crank mechanism have been evaluated with respect to that of a conventional slider–crank mechanism. As a result, although both the conventional and the modified slider–crank mechanisms have the same stroke and the same gas pressure in the cylinder, it is observed that the modified mechanism has a bigger output torque than that of the conventional mechanism. Ó 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 wankel engines. As a result of the investigations focused on mechanical and thermal design parameters in the internal combustion engines, mechanical strength, thermal efficiency, wear and surface quality of the elements have been improved during the 20th century. Consequently, it can be clearly seen that fuel consumption has been reduced and engine lifetime has been increased. In spite of these developments, any definite modification in the slider–crank mechanism has not been effected up to now. However, motion and force transmission characteristics 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 type of the crank shaper mechanism [1]. The influence of the cutting force on the motion of the mechanism was considered. The Lagrange equation was used and boundary values of the cutting force were obtained *

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

0094-114X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.04.011

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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 by using Hamilton’s principle, Lagrange multiplier, geometric constraints and partitioning method [2]. The formulation was expressed by only one independent variable. Dynamic responses between the experimental results and numerical simulations were compared to obtain the best dynamic modeling. Also, a new identification method based on the genetic algorithm was presented to identify the parameters of a slider–crank mechanism. Transient and steady state dynamic response of a class of slider–crank mechanisms which involves rigid members 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 were solved numerically to examine the resulting dynamical system. Koser has summarized kinematic performance analysis of a slider–crank mechanism based robot arm performance and dynamics [4]. The kinematic performance 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 its performance was much better than the best 2R robot arm. Dynamic analysis and vibration control of a flexible slider–crank mechanism driven by a servomotor have been studied by Fung and Chen [5]. To formulate the governing equations of the mechanism, geometric constraint at the end of a flexible connecting rod was derived and introduced into Hamilton’s principle. Chen and Huang have investigated the dynamic responses of flexible slider–crank mechanism by considering all the high order terms [6]. The non-linear equation of motion was solved by Newmark method. In the low-speed range, it was found that the dynamic responses predicted by non-linear and linear approaches (neglecting the high order terms) indeed made no significant difference. However, when the rotation speed increased up to about one-fifth of the fundamental bending natural frequency of the connecting rod, simplified linear approaches exhibited a noticeable error. So¨ylemez has summarized the classical transmission angle problem for slider–crank mechanisms which is the determination of the dimensions of planar slider–crank mechanisms with optimum transmission angle for given values of the slider stroke and corresponding crank rotation [7]. The complex algebra was used to solve that classical problem and the solution was obtained as the root of a cubic equation within a defined range. Another research about transmission angle was implemented by Shrinivas and Satish [8]. They have summarized importance of the transmission angle for most effective force transmission. In this sense, they investigated 4-, 5-, 6- and 7-bar linkages, 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 effect of slider–crank mechanism is obtained by a gas pressure arising from combustion of mixture consisting of fuel and air. The force corresponds to this pressure causes the piston to translate along the vertical axis and this action is transmitted to crank through connecting rod. The conventional mechanism which is widely used in internal combustion engines is a concentric slider–crank mechanism. This mechanism, as shown in Fig. 1, has one degree of freedom, that is, a constrained mechanism. Parameters used in the conventional mechanism are given in Table 1 in Appendix A. In considering the kinematic analysis of the slider–crank mechanism, it is necessary 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 the following form:  Spi

  1  2   ¼ l þ r cos h  k sin h 2 



ð1Þ 

where r * is the crank radius, l* is the connecting rod length, k ¼ rl 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 velocity can be found as   1      ð2Þ V pi ¼ r x21 sin h þ k sin 2h 2

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

395

y TDC B 4

β* BDC

S *pi

l* 3

A 2

θ*

r* x

O

Fig. 1. Concentric slider–crank mechanism.

By taking time-derivative of Eq. (2), the piston acceleration is given by api ¼ r x221½ cos h þ k cos 2h

ð3Þ

The purpose of dynamic analysis of the slider–crank mechanism is to determine the total output torque arising from resultant force (gas + inertia). In the mechanism, gas forces, known as driving effect, do not have constant value during the expansion stroke. So, the cylinder volume has to be expressed as a function of crank’s angular position considering the variation of gas forces h i pD2 V x ðh Þ ¼ V c þ l þ r  Spi ð4Þ 4  where V c 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  k V ð5Þ P e ðh Þ ¼ P 3 V x ðh Þ

where P3 is the pressure in the cylinder and V is the cylinder volume at the end of the compression stroke. k is the polytropic coefficient and usually taken to be equal 1.3 for diesel engines [9]. Gas forces can be expressed as a function of crank’s angular position in the following form: F g ¼ Api ½P e ðh Þ  P atm 

ð6Þ

where A pi is the piston sectional area and P atm 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 as active effects, are outlined in Fig. 2. Referring to Fig. 2(a), output torque on the crankshaft, arising from gas forces, is given by M gas ¼ rxF i32

ð7Þ

i F32

denotes the gas forces effect on the crank-pin center and can be expressed as a function of gas forces where in the following form: i ¼ F32

Fg cos b

ð8Þ

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

y

Fg

[– m a ]

y

F34ι

1

B

90+β

B 270

[– m a ]

B

B

1

*

4

y

x

4

F14ι

F14ιι

B

180

x

270+β*

Fg

F34ιι

aB a 31

β*

β*

l*

l*

[– m

3

G3

a G3

] y

3

A

A

x

90+β* δ x

A

*

A

2

270+β

*

r*

F32ιιι

ιι F32

ψ +β*

*

θ

G3

aG3

y

2

B

90

θ

F32ι

r* x

x

O

O

(a)

(b)

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 summation in the following form: X ii þ F iii F Inertia ¼ F32 ð9Þ 32

ii where F 32 denotes the inertial effect of the piston’s mass and F iii32 denotes the inertial effect of the connecting rod’s mass. These forces can be expressed, respectively

F ii32 ¼ 

ðmB aB Þ cos b

ð10Þ

where mB is the total mass on the piston-pin center. a B is the acceleration of the piston-pin center and equals to  in Eq. (3). api " #1=2 2    ðm a ÞðBG sinðw þ b Þ  l cos w sin b Þ þ ðI a Þ 2 G G 3 G 31 3 3 3 ð11Þ þ ððmG3 aG3 Þ cos wÞ F iii32 ¼ l cos b where mG3 is the mass of connecting rod and aG3 is the linear acceleration of connecting rod’s gravity center. Also, I G3 is the inertia moment and a 31 is the angular acceleration. aG3 and a 31 are given as a function of crank’s angular position in Appendix A. Output torque caused by resultant inertial force is defined by X M Inertia ¼ rx F Inertia ð12Þ

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397

From Eqs. (7) and (12), total output torque on the crankshaft can be written in the following form: M Total ¼ M gas þ M 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 rod and crank pin as distinct from conventional mechanism. The new extra link, may be called eccentric connector, transmits gas forces to the crank and also drives a planetary gear mechanism. In order to drive planetary gear train, a pinion fixed to the eccentric connector in a parallel plane is used. So, there are two transmission lines 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 connecting rod, eccentric connector, gear mechanism. When the motion characteristic of the mechanism in Fig. 3 is investigated carefully, a kinematic-based scheme in Fig. 4 is obtained. Referring to Fig. 4, it can be seen that the modified mechanism has one degree of freedom, that is, this model is a constrained mechanism. The eccentric connector makes a curvilinear translation because of chosen a particular gear ratio between pinion and ring gear. That is, it has no a relative motion with respect to crank pin (n31 = 0). Consequently, there is an identical freedom in the mechanism due to the specific motion of the eccentric 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 main reasons, one of them is gear ratio between pinion and ring gear and the other is curvilinear translation of the eccentric connector. As shown in Fig. 3, this ratio is also obtained from the planetary gear relationship among the crank arm, pinion and ring gear in the following form: n31  n21 r6 ; ðn31 ¼ 0Þ ð14Þ þ in ¼ n61  n21 rp For the position analysis of the modified mechanism, the position of piston-pin center with respect to crank rotation center is given by Spi ¼ rc cos h þ 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 of connecting rod and the cylinder axis. b = f(h) is given by h ei2 1 ð16Þ cos b ¼ 1  kc sin h þ kp  l 2 r where rp is the radius of pinion gear, e is the distance of eccentricity, kc ¼ rcl and kp ¼ lp . Substituting Eq. (16) into Eq. (15) yields     h ei 1 1 ð17Þ Spi ¼ l þ rc cos h  kc sin2 h  rp kc sin h þ kp þ e kc sin h þ kp  2 2l 2 Time-derivative of Eq. (17) yields the velocity of piston   1 V pi ¼ x21 rc sin h þ kc rc sin 2h þ kc rp cos h  kc e cos h 2

ð18Þ

From the differentiation of Eq. (18), the piston acceleration is defined by 2 ½rc cos h þ kc rc cos 2h  kc rp sin h þ kc e sin h api ¼ x21

ð19Þ

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

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

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

399

TDC

y

C 5

β

BDC

Spi

l 4

A θ

2

rc

3

rp

B x

O

e

Eccentric Connector (3) Planetary Gear (3) Ring Gear (6)

Gear (9) Gear (7) Gear (8) O1

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 difference between two forces, which one of them is transmitted from the connecting rod to the crank arm by the eccentric connector and the other is applied to ring gear by the pinion gear. This difference is approximately 41.5 mm. So, in the dynamic analysis of this mechanism, this distance has to be considered. The other important points about mechanism, as shown in Fig. 3, counterweight for eccentric connector (Part 3.1) provides that the gravity center of eccentric connector coincides with the crank-pin axis, 7th and 8th gears are joined each 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 expansion stroke. So, it is necessary that the cylinder volume, gas pressure and the gas forces have to be expressed as a function of crank’s angular position. For this purpose, by adapting the parameters, such as h, l, r c and Spi given in Eqs. (4)–(6), these expressions can be applied to the modified mechanism (the other parameters, such as Vc, D, P3 and V, are same values in that equations for each mechanism). Fig. 5 shows the effect of the gas forces 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 planetary gear mechanism to drive, the normal component also causes the crank to translate.

400

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

y

1

y

ι F45

90+β

5 C

270

x

F15ι

Fg

l D

4 ι M 61

ι F 36

360-(θ- ϕ) D

M 91ι

F ι32 2 A θ

rp 3

B

270+β

rc

ι F43

x

O

e

Planetary Gear (3) Ring Gear (6) Gear (9) Gear (7) Gear (8)

O1

ι ι M 71 = M 81

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, eccentric connector and crank arm links. Force equilibrium for the eccentric connector is given by F i43 þ F23i ¼ 0

ð20Þ

F i43

i

where force is exerted on eccentric connector by means of the connecting rod and also the force F 23 is exerted on eccentric connector by means of crank arm. Because of the special construction of the eccentric coni has to be taken instead of F i43 in Eq. (20). This force can nector and the planar difference, the component F 43 x be expressed as a function of gas forces in the following form: F i43x ¼ F g tan b

ð21Þ i

Output torque obtained by means of the direct transmission line (M 21 ) is given by i M i21 ¼ rc xF32

ð22Þ

i denotes the force which is exerted on the crank arm by means of the eccentric connector. By conwhere F 32 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 F g tan b As can be seen from Eqs. (21) and (23),

ð23Þ F i43x

and

F i32

forces are equal in magnitude and same direction.

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401

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 is given by i ¼0 MAi ¼ rp xF i43 þ rp xF63

ð24Þ

i F 63

From Eq. (24), force which is exerted on pinion gear by means of the ring gear can be obtained as a function of gas forces in the following form: Fg i F63 ð25Þ ¼ co...