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Determining link parameters using genetic algorithm in mechanisms with joint clearance
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Mechanism and Machine Theory 44 (2009) 222–234
Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Determining link parameters using genetic algorithm in mechanisms with joint clearance _ Selcßuk Erkaya, Ibrahim Uzmay * Erciyes University, Engineering Faculty, Department of Mechanical Engineering, 38039 Kayseri, Turkey Received 3 April 2007; received in revised form 22 January 2008; accepted 4 February 2008 Available online 11 March 2008
Abstract This paper presents an investigation of joint clearance influences on the mechanism path generation and transmission angle. Joint clearance was treated as a massless virtual link and mathematical expression of its motion was obtained by using Lagrange’s equation. Genetic Algorithm (GA) approach was used to describe the direction of the joint clearance relative to input link’s position and also to implement the optimization of link parameters for minimizing the error between desired and actual paths due to clearance. Four-bar path generator was used as an illustrative example. The main advantages of the proposed approach are its simplicity of implementation and its fast convergence to optimal solution, with no need of deep knowledge of the searching space. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Optimal mechanism synthesis; Joint clearance; Transmission quality; Genetic algorithm; Evolutionary technique
1. Introduction Planar mechanisms with revolute joints are widely used in machines to ensure their functions. Mechanism links exert various forces via joints to each other. Some clearances in these joints are inevitable due to tolerances and defects arising from design and manufacturing process or wearing after a certain working period. In the case of excessive joint clearances, contact forces generate impulsive effect, and this situation causes mechanical losses and vibratory running condition. It is well known that the performance measure for a mechanism is usually referred to as the ability of reaching the desired position or orientation precisely. Link dimension tolerances and also the joint clearances make the performance of mechanisms worse. In the past, many designers have investigated the effects of joint clearance from those aspects. Ting et al. [1] have presented an approach to identify the worst position and direction errors due to the joint clearance of linkages and manipulators. Joint clearance was modelled as a small link with the length equals to one half of the clearance. A geometrical model was used in their method to assess the output position or
*
Corresponding author. Tel.: +90 3524375832; fax: +90 3524375784. _ Uzmay). E-mail address: [email protected] (I.
0094-114X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2008.02.002
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direction variation, to predict the limit of position uncertainty, to determine the maximum clearance, and hence to assure the safety or precision of a mechanism. Also, that model could be used directly in any single closed loop linkage or manipulator and could be extended to some multi loop linkages. Tsai and Lai [2] have presented an effective method to analyze the transmission performance of linkages with joint clearances. Equivalent kinematic pairs were used for modelling the motion freedoms originated from the joint clearances. Geometrical properties of linkages were used to derive the equations. Position analysis of a planar four-bar mechanism was implemented by using loop-closure equations as a numerical example. Shi et al. [3] have presented a probabilistic model for the deviation between desired and actual paths generated by a coupler point, and a robust synthesis procedure of the path generating mechanism was implemented. Both the structural and mechanical errors were incorporated and a four-bar mechanism was used as a numerical example in their approach. Cabrera et al. [4] have presented solution methods for the optimal synthesis of planar mechanisms. Optimization procedure was defined by using genetic algorithm and a suitable objective function. Synthesis of a four-bar mechanism was used to test the proposed method. Another study about application of genetic algorithm technique to the mechanism synthesis has been presented by Kunjur and Krishnamurty [5]. In their study, synthesis of a four-bar path generator mechanism was implemented to illustrate the application of GA technique to the mechanism synthesis. Error between actual and desired paths was considered as a goal function, and it was expressed as the sum of the squares of the error at each point in the path. Furuhashi et al. [6,7] have presented a general approach for the dynamics of four-bar linkages with joint clearances using the continuous contact model based on the assumption that the pin is always in contact with the socket in each pair. The motion of the linkage for the contact angles was analytically treated from Lagrangian function. Direction of the joint clearance with respect to input position was derived by using Runge–Kutta–Gill method considering the Lagrange‘s equation. In this paper, the effect of the joint clearance on path generation and transmission quality of a four-bar mechanism is investigated. A genetic algorithm approach, based on evolutionary techniques and an appropriate goal function, is presented to describe the direction of the joint clearance with respect to input position taking into account continuous contact model between pin and socket. By using genetic algorithm technique, optimization of the mechanism links parameters is also performed to minimize the path errors between desired (without clearance) and actual (with clearance) mechanisms. For this purpose, this paper is organized as follows; Section 2 describes the joint clearance and relevant link’s model. In Section 3, synthesis of four-bar path generator is presented. Section 4 describes Genetic algorithm approach and objective functions. Results are presented in Section 5. Finally, conclusion is outlined in Section 6. 2. Joint clearance In this study, it is assumed that mechanism links are connected to each other by revolute joints with clearance. Joint clearance, as shown in Fig. 1, can be defined as the difference of the radii of the pin and socket at a joint. When the continuous contact model assumption between pin and socket at each joint is considered, the clearances may be modelled as vectors which correspond to massless virtual links with the lengths equal to joint clearance [1,8]. If the friction is negligible, the directions of these vectors coincide with the normal of the contact force between pin and socket. Each joint clearance adds additional freedom to the mechanism
a
b jth link
rj r ri ith link
Fig. 1. (a) Joint clearance model and (b) equivalent clearance link.
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and, supplementary constraints are necessary to perform kinematic and dynamic analysis of the mechanism. When the radii of pin and socket are denoted by ri and rj, respectively, the equivalent clearance vector is given as follows: r ¼ rj ri :
ð1Þ
3. Synthesis of four-bar path generator Dimensional synthesis of planar linkages for path generation consists of determining the dimensions of the linkage in accordance with requirements related to the desired path [9]. Also, it involves the minimization of structural error which is subject to a set of size and geometric constraints such as Grashof’s rules. A four-bar mechanism, as shown in Fig. 2, is considered as an example to determine the effect of joint clearance between crank and coupler link on path generation and transmission quality. Link parameters of the four-bar mechanism are given in Table 1. Kinematic equations are derived from the vector configuration of the mechanism, shown in Fig. 2c, as follows:
a
b
c
P θc4
y
β δ
A
G3
3 μc
L3
2 A0
θ3c L4
A’ r2
G4
L2
θ1 L1
1
B0
γ2
L4
A
4 G2 θ2
θc4
L3
B
θ3c
A’
B
L2
θ2
x A0
L1
θ1 B0
Fig. 2. (a) Four-bar mechanism model, (b) schematic representation of a four-bar mechanism with clearance and (c) vector representation of mechanism.
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Table 1 Parametric values of the four-bar mechanism Link
Material
Length (mm)
Fixed (1) Crank (2) Coupler (3) Follower (4)
Steel Steel Aluminium Aluminium
400 100 360 240
c
c
L1 eih1 þ L2 eih2 þ r2 eic2 þ L3 eih3 þ L4 eih4 ¼ 0:
ð2Þ
By separating equation (2) into its real and imaginary parts and using trigonometric relations, hc3 and hc4 can be expressed as a function of h2 and c2, respectively: " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# B B2 4AC c ; ð3Þ h3 ¼ 2 arctan 2A L1 L2 cos h2 r2 cos c2 L3 cos hc3 hc4 ¼ arccos ; ð4Þ L4 where the superscript c denotes the ‘‘value with clearance”. A, B and C terms are given, respectively A ¼ ðL1 þ L3 Þð2L2 cos h2 þ 2r2 cos c2 Þ þ 2L2 r2 cosðh2 c2 Þ þ 2L1 L3 þ L21 þ L22 þ L23 L24 þ r22 ; B ¼ 4L3 ðL2 sin h2 þ r2 sin c2 Þ; C ¼ ðL3 L1 Þð2L2 cos h2 þ 2r2 cos c2 Þ þ 2L2 r2 cosðh2 c2 Þ 2L1 L3 þ L21 þ L22 þ L23 L24 þ r22 : As shown in Fig. 2b, the position of the coupler point (P(x, y)) relative to the crank pivot (A0) is given with and without joint clearance, respectively 2 c3 2 3 2 3 2 3 Px cosðhc3 þ bÞ cos h2 cos c2 4 5 ¼ L2 4 5 þ r2 4 5 þ A0 P 4 5; ð5Þ c Py sinðhc3 þ bÞ sin h2 sin c2 2 4
Px
3
2
5 ¼ L2 4
cos h2
3
2
5 þ AP 4
3 5;
ð6Þ
sinðh3 þ bÞ
sin h2
Py
cosðh3 þ bÞ
where P cx ; P cy denote the x- and y-coordinate values for the path of coupler point in considering the joint clearance, and Px, Py denote the x- and y-coordinate values for the path of coupler point without joint clearance. Due to motion transmission line from crank link to follower link, joint clearance between crank and coupler connection has an important role on path generation of point P. If the crank pivot (A0) is taken as the reference point, the mass center positions for moving links are given as follows: 2 c 3 2 3 xG2 cos h2 4 5 ¼ A0 G 2 4 5; ð7Þ y cG2 sin h2 2 4 2 4
xcG3 y cG3 xcG4 y cG4
3
2
5 ¼ L2 4
cos h2
3
2
5 þ r2 4
sin h2 3
2
5 ¼ L2 4
cos h2 sin h2
cos c2
2
3
5 þ A 0 G3 4
sin c2 3
2
5 þ r2 4
cos c2 sin c2
3
2
5 þ L3 4
cosðhc3 þ dÞ sinðhc3
cos hc3 sin hc3
3
3 5;
ð8Þ
þ dÞ 2
5 þ BG4 4
cos hc4 sin hc4
3 5:
ð9Þ
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Transmission angles for the model mechanism without and with clearance are defined as follows [10]: 2 L þ L22 L23 L24 2L1 L2 cos h2 l ¼ arccos 1 ; ð10Þ 2L3 L4 ! L21 þ A0 A0 2 L23 L24 2L1 A0 A0 cosðh2 wÞ c l ¼ arccos ; ð11Þ 2L3 L4 where A0 A0 and w are given, respectively A0 A0 ¼ ½L22 þ r22 þ 2L2 r2 cosðh2 c2 Þ1=2 ; " # 02 2 2 1 L2 þ A0 A r2 w ¼ cos : 2L2 A0 A0
ð12Þ ð13Þ
If the mechanism links are assumed to be rigid, direction variable of clearance vector can be derived by using Lagrange’s equation in the following form [6,7,11]: d oT oT oU oDF þ þ ¼ 0; ð14Þ dt o_c2 oc2 oc2 o_c2 where T, U and DF denote the kinetic energy, the potential energy and the dissipation function, respectively. The above mentioned terms are given as follows: T ¼ U¼
4 4 1X 1X 2 2 2 I i ðh_ ci Þ þ mi ½ð_xcGi Þ þ ð_y cGi Þ ; 2 i¼2 2 i¼2 4 X
mi gy cGi ;
ð15Þ ð16Þ
i¼2
DF ¼
4 1X 1 C hi ðh_ ci Þ2 þ C c2 c_ 22 ; 2 i¼2 2
ð17Þ
where i designates the link number. If the above terms are substituted into Eq. (14), Lagrange’s equation can be established in the following form: 4 X oxc oy c oy c ohc ohc ð18Þ hci i þ mi €xcGi Gi þ €y cGi Gi þ gmi Gi þ C hi h_ ci i þ C c2 c_ 2 ¼ 0: I i€ oc2 oc2 oc2 oc2 oc2 i¼2 4. Optimization procedure for path generation in mechanism with clearance Two objective functions (OF) are defined, firstly, to determine the direction of the joint clearance and secondly, to minimize the error between the desired and actual positions of point P. As seen from Eq. (18), the motion equation of mechanism has nonlinear character. Genetic algorithm approach is implemented to solve this equation, and so to determine the direction of joint clearance as a function of position variable of input link. This equation is considered as the first objective function, and it is expressed in the following form: ! 4 c c X oxcGi oy cGi oy cGi c ohi c c c ohi € _ þ mi €xGi þ €y Gi þ gmi þ C hi hi þ C c2 X ; I i hi Minimize F 1 ðX Þ ¼ f oX oX oX oX oX i¼2 Subject to
hj ðX Þ ¼ 0; xl 6 x j 6 x u ;
ð19Þ
xj 2 X ; where hj(x) are the equality constraints which are determined depending on parametric relations between joint clearance direction and input variable. X is a vector comprising the design variables corresponding to joint
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clearance direction, that is, c2 ; c_ 2 and €c2 xl and xu are the lower and upper bounds of the design variables, respectively. In path generation problems, the coupler point has to track a given path with minimum error. Objective function is a measure of the error between the desired and actual paths, and is usually expressed as the sum of the squares of the error at each point of the path [5]. For minimizing the error between desired and actual positions of point P, the second objective function is given as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N h i u1 X 2 2 ð20Þ ðP dxk P axk Þ þ ðP dyk P ayk Þ Minimize F 2 ðxÞ ¼ t N k¼1 Subject to
gm ðxÞ 6 0; xl 6 x m 6 x u ;
where P dxk ; P dyk are x- and y-coordinate values of the kth point in the desired path and P axk ; P ayk are the x- and ycoordinate values of the kth point in the actual path. The inequality constraints (gm(x)) consist of Grashof’s rule, that is, crank-rocker condition. xm denote the independent design variables, and they consist of link lengths (Li) and structural angle (b). Design variables for the second objective function are given in the following form: T
x ¼ bL1 L2 L3 L4 bA0 P c :
ð21Þ
5. Results
180
600
150
500
120
400
90
300
60
200
30
100
0
0
-30
0
60
120
180
240
300
-100 360
Input Position [Degree] (
):
γ 2 [Degree], (
):
γ2
[Rad/sec]
Fig. 3. Direction of joint clearance with respect to position of input link.
Radian/second
Degree
In the present paper, a theoretical model is used for the case of single clearance in a four-bar mechanism with rigid links, and 600 rpm is considered as running speed. ADAMS [12], a mechanical system simulation software, is used to model and simulate the mechanism, and also to verify the analytical results. The determining of the direction variable of the joint clearance and the optimum link parameters ðL1 ; L2 ; L3 ; L4 ; b; A0 P Þ in the model mechanism with joint clearance are realized using genetic algorithm on MATLAB software. Therefore,
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the variation of joint clearance direction with respect to position variable of input link, which is obtained from the first objective function using genetic algorithm, is outlined in Fig. 3. If the convergence rate of the energy terms in Eq. (18) to zero is considered as a measure of performance criteria for Genetic algorithm to solve the nonlinear motion equation, that is, to determine the joint clearance direction, the convergence error in this equation by using the obtained solution set is given in Fig. 4. As shown, maximum and average errors are 1.5 105 and 4.3907 107, respectively. It can be said that these results are acceptable in determining appropriate design parameters. Two case studies are implemented for observing the influences of joint clearance on the path generation and transmission angle. In the first study, joint clearance value is taken to be equal to 1 mm, and in the second one, this value is considered as 2 mm. In order to minimize the path error, the second objective function for the genetic algorithm application is developed. As a result, the original and optimized values of design parameters are given in Table 2. If the kinematic equations (Eqs. (5) and (6)) are solved for two different clearance values, the obtained path configurations for desired (without clearance), actual (with clearance) and optimized (optimized with clearance) mechanisms are outlined in Figs. 5 and 6, respectively. As shown in these figures, the path error reaches the largest value when the mechanism goes into dead-center positions. The path errors obtained for two different joint clearance values in the original and optimized mechanisms are given in Fig. 7. As shown, these errors for original mechanism are bigger than that of the optimized mechanism. When error values for x-
1.6 1.4
1.2
Error (x10-5)
1
0.8 0.6
0.4
0.2 0
0
60
120
180
240
300
360
Input Position [Degree] Fig. 4. Performance measure of GA for Lagrange equation.
Table 2 Optimized design variables for different joint clearance values Design variables
L1 (mm) L2 (mm) L3 (mm) L4 (mm) b (degree) A0 P ðmmÞ
Original values
400 100 360 240 20.3319 305.07
Optimum values with joint clearance r2 = 1.0 mm
r2 = 2.0 mm
399.91846 100.01917 358.89966 239,98779 20.37210 304.24561
399.90625 100.03125 357.82812 239.60156 20.48950 303.41797
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229
302
Right Dead-center position
300
Y-coordinate [mm]
280 286 266
280
260
240
196
220
Left Dead-center position
200 182 88
180 80
100
120
140
160
180
108
200
220
240
260
280
X-coordinate [mm] (
) : Desired Path (
) : Actual Path
(
) : Optimized Path
Fig. 5. Desired, actual and optimized path configurations for the joint clearance of 1 mm.
320
302
300
Right Dead-center position
Y-coordinate [mm]
280
260
286 266
280
240
196
220
Left Dead-center position
200 182
180 80
100
120
140
160
180
88
108
200
220
240
260
280
X-coordinate [mm] (
) : Desired Path (
) : Actual Path
(
) : Optimized Path
Fig. 6. Desired, actual and optimized path configurations for the joint clearance of 2 mm.
and y-directions of each mechanism are evaluated separately, it is clearly seen that path errors for the joint clearance of 1 mm decrease by 78.46% and 75.72% in x- and y-directions, respectively. Also, in the case of 2 mm clearance, these errors decrease by 78.45% and 75.69% in each directions, respectively. The transmission angles under two different clearance conditions during the whole motion cycle of the mechanism are shown in Figs. 8 and 9, respectively. As seen from related figures, the deviation of the trans-
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230
1 mm joint clearance (X-direction)
3 1.5 0 -1.5 -3 0
60
120
180
240
300
360
300
360
300
360
300
360
1 mm joint clearance (Y-direction) 3 1.5
Path error [mm]
0 -1.5 -3
0
60
120
180
240
2 mm joint clearance (X-direction)
3 1.5 0 -1.5 -3 0
60
120
180
240
2 mm joint clearance (Y-direction)
3 1.5 0 -1.5 -3 0
60
120
180
240
Input Position [Degree] ( (
) : Path error between desired and actual mechanisms ) : Path error between desired and optimized mechanisms
Fig. 7. Path errors for two joint clearance values.
mission angle from the desired value is larger when the joint clearance value increases. Also, it can be clearly seen that the biggest deviations of transmission angle from the desired value occurred at the extreme positions. The difference between desired and actual transmission angles, and also the deviation of the desired transmission angle relative to the optimized ones, as shown in Fig. 10, can be assigned as a measure of performance for the second optimization. As seen form Fig. 10, transmission angle errors between desired and optimized mechanisms are smaller than those of the other. These errors approximately decrease by 55.56% and 42.77% for 1 and 2 mm joint clearances, respectively.
6. Conclusion In this study, assuming that the presence of joint clearance between crank and coupler links, the effect of this factor on mechanism path generation and transmission quality is investigated. Joint clearance is considered as a massless virtual link and mathematical expression of its motion is derived by using Lagrange’s equation. An objective function is defined, and parametric relations between input variable and joint clearance direction are considered as the constraints for solving nonlinear differential equation by using genetic algorithm approach. So, the direction of joint clearance relative to input variable is obtained. Also, to minimize the deviations in the path generation and transmission angle arising from joint clearance, that is, for obtaining
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231
115 110
Transmission Angle [Degree]
105 100 95 90 112
85 80 75 70 102 130
65
230
60 55
0
60
120
180
240
300
360
Input Position [Degree] (
) : Desired µ
(
) : Actual µ
(
) : Optimized µ
Fig. 8. Transmission angle variation for the joint clearance of 1 mm.
115 110
Transmission Angle [Degree]
105 100 95 90
112
85 80 75 70 102 130
65
230
60 55
0
60
120
180
240
300
360
Input Position [Degree] (
) : Desired µ
(
) : Actual µ
(
) : Optimized µ
Fig. 9. Transmission angle variation for the joint clearance of 2 mm.
the desired values with minimum error, a second objective function is defined and optimization of the link parameters in mechanism is implemented. Crank-rocker situation corresponding to Grashof’s rule is considered as a geometric constraint to obtain the best approximation for the desired values.
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1 mm joint clearance
µ error [Degree]
1.5 1 0.5 0 -0.5 -1 -1.5
0
60
120
300
360
300
360
2 mm joint clearance
1.5
µ error [Degree]
180 240 Input Position [Degree]
1 0.5 0 -0.5 -1 -1.5
0
60 ( (
120
180 240 Input Position [Degree]
) : µ error between desired and actual mechanisms ) : µ error between desired and optimized mechanisms
Fig. 10. Transmission angle errors for two joint clearances.
As seen from the results, if the values of equivalent joint clearances are bigger, the mechanism is subjected to higher deviations from the desired characteristics. After the optimization process, the position errors of the point P for the joint clearance of 1 mm decrease by 78.46% and 75.72% in x- and y-directions, respectively. In the case of 2 mm joint clearance, these errors decrease by 78.45% and 75.69% in each directions, respectively. Also, transmission angle errors approximately decrease by 55.56% and 42.77% for two cases of joint clearance. These results can be seen as a measure of success of optimization strategy, constraints and inequalities. The proposed approach is a versatile method to carry out the optimum synthesis of planar mechanisms, and it can be easily adapted to different mechanisms having joints with clearance. Acknowledgements This work is a part of the research project FBA-07-06. The authors wish to express their thanks for financial support being provided by the Scientific Research Project Fund of Erciyes University, in carrying out this study. Appendix A Mechanism dynamic parameters used in Lagrange’s equation are given in Table A.1. Front and side views of the examined mechanism’s links are given in Fig. A.1, respectively. Partial differentiations of mass center positions for moving links and links’ position variables (hc3 and hc4 ) with respect to direction of joint clearance are given, respectively, 2 c 3 2 c 3 oxGi oxGi " # 6 oc 7 6 oh2 7 _xcGi 6 27 7 ¼ x2 6 ðA:1Þ 7; 4 oy c 5 þ c_ 2 6 4 oy cGi 5 y_ cGi Gi oh2 oc2
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Table A.1 Dynamic parameters for four-bar mechanism Parameters
Descriptions
Values
I2 I3 I4 m2 m3 m4 g x2 C hi C c2
Moment of inertia of the crank Moment of inertia of the coupler Moment of inertia of the follower Mass of the crank Mass of the coupler Mass of the follower Acceleration of gravity Angular velocity of input link Damping coefficient [11] Damping coefficient [11]
5.12 104 kg m2 8.85 103 kg m2 1.58 103 kg m2 0.121 kg 1.048 kg 0.071 kg 9.81 m/s2 62.83 rad/s 0.2 106 kg m s/rad 0.2 106 kg m s/rad
Fig. A.1. Two different views of investigated four-bar mechanism links.
2
"
€xcGi €y cGi
#
3
3 2 2 c 3 o xGi o2 xcGi 2 oxc 3 6 Gi 2 7 6 oh oc 7 6 oc2 7 7 6 oh oc2 6 2 27 6 2 7 2 7 ¼ x22 6 7 þ €c2 4 oy c 5 þ c_ 22 6 2 c 7; 6 2 c 7 þ 2x2 c_ 2 6 2 c 5 4 4 o yG 5 Gi o y o y 4 Gi 5 Gi i oc2 2 oh2 oc2 oc22 oh2 o2 xcGi
2
ðA:2Þ
ohc ohc h_ cz ¼ x2 z þ c_ 2 z ðz ¼ 3; 4Þ; oh2 oc2
ðA:3Þ
o2 hc o2 hcz ohc o2 hc € hcz ¼ x22 2z þ 2x2 c_ 2 þ €c2 z þ c_ 22 2z ðz ¼ 3; 4Þ: oh2 oc2 oc2 oc2 oh2
ðA:4Þ
References [1] K.W. Ting, J. Zhu, D. Watkins, The effects of joint clearance on position and orientation deviation of linkages and manipulators, Mechanism and Machine Theory 35 (2000) 391–401. [2] M.J. Tsai, T.H. Lai, Kinematic sensitivity analysis of linkage with joint clearance based on transmission quality, Mechanism and Machine Theory 39 (2004) 1189–1206. [3] Z. Shi, X. Yang, W. Yang, Q. Cheng, Robust synthesis of path generating linkages, Mechanism and Machine Theory 40 (2005) 45–54. [4] J.A. Cabrera, A. Simon, M. Prado, Optimal synthesis of mechanisms with genetic algorithms, Mechanism and Machine Theory 37 (2002) 1165–1177.
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[5] A. Kunjur, S. Krishnamurty, Genetic algorithms in mechanism synthesis, Journal of Applied Mechanisms and Robotics 4 (1997) 18– 24. [6] T. Furuhashi, N. Morita, M. Matsuura, Research on dynamics of four-bar linkage with clearances at turning pairs (1st Report, General theory of continuous contact model), Bulletin of the JSME 21 (1978) 518–523. [7] N. Morita, T. Furuhashi, M. Matsuura, Research on dynamics of four-bar linkage with clearances at turning pairs (2nd Report, Analysis of crank-level mechanism with clearance at joint of crank and coupler using continuous contact model), Bulletin of the JSME 21 (1978) 1284–1291. [8] M.T. Bengisu, T. Hidayetog˘lu, A. Akay, A theoretical and experimental investigation of contact loss in the clearances of a four-bar mechanism, Transactions of the ASME – Journal of Mechanisms, Transmissions, and Automation in Design 108 (1986) 237–244. [9] F.T.S. Marin, A.P. Gonzalez, Global optimization in path synthesis based on design space reduction, Mechanism and Machine Theory 38 (2003) 579–594. [10] S.S. Balli, S. Chand, Transmission angle in mechanisms (Triangle in mech), Mechanism and Machine Theory 37 (2002) 175–195. [11] B. Feng, N. Morita, T. Torii, A new optimization method for dynamic design of planar linkage with clearances at joints-optimizing the mass distribution of links to reduce the change of joint forces, Transactions of the ASME – Journal of Mechanical Design 124 (2002) 68–73. [12] MSC.ADAMS 2005 r2, Automatic Dynamic Analysis of Mechanical Systems, MSC Software Corporation.
Mechanism and Machine Theory 44 (2009) 222–234
Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Determining link parameters using genetic algorithm in mechanisms with joint clearance _ Selcßuk Erkaya, Ibrahim Uzmay * Erciyes University, Engineering Faculty, Department of Mechanical Engineering, 38039 Kayseri, Turkey Received 3 April 2007; received in revised form 22 January 2008; accepted 4 February 2008 Available online 11 March 2008
Abstract This paper presents an investigation of joint clearance influences on the mechanism path generation and transmission angle. Joint clearance was treated as a massless virtual link and mathematical expression of its motion was obtained by using Lagrange’s equation. Genetic Algorithm (GA) approach was used to describe the direction of the joint clearance relative to input link’s position and also to implement the optimization of link parameters for minimizing the error between desired and actual paths due to clearance. Four-bar path generator was used as an illustrative example. The main advantages of the proposed approach are its simplicity of implementation and its fast convergence to optimal solution, with no need of deep knowledge of the searching space. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Optimal mechanism synthesis; Joint clearance; Transmission quality; Genetic algorithm; Evolutionary technique
1. Introduction Planar mechanisms with revolute joints are widely used in machines to ensure their functions. Mechanism links exert various forces via joints to each other. Some clearances in these joints are inevitable due to tolerances and defects arising from design and manufacturing process or wearing after a certain working period. In the case of excessive joint clearances, contact forces generate impulsive effect, and this situation causes mechanical losses and vibratory running condition. It is well known that the performance measure for a mechanism is usually referred to as the ability of reaching the desired position or orientation precisely. Link dimension tolerances and also the joint clearances make the performance of mechanisms worse. In the past, many designers have investigated the effects of joint clearance from those aspects. Ting et al. [1] have presented an approach to identify the worst position and direction errors due to the joint clearance of linkages and manipulators. Joint clearance was modelled as a small link with the length equals to one half of the clearance. A geometrical model was used in their method to assess the output position or
*
Corresponding author. Tel.: +90 3524375832; fax: +90 3524375784. _ Uzmay). E-mail address: [email protected] (I.
0094-114X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2008.02.002
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direction variation, to predict the limit of position uncertainty, to determine the maximum clearance, and hence to assure the safety or precision of a mechanism. Also, that model could be used directly in any single closed loop linkage or manipulator and could be extended to some multi loop linkages. Tsai and Lai [2] have presented an effective method to analyze the transmission performance of linkages with joint clearances. Equivalent kinematic pairs were used for modelling the motion freedoms originated from the joint clearances. Geometrical properties of linkages were used to derive the equations. Position analysis of a planar four-bar mechanism was implemented by using loop-closure equations as a numerical example. Shi et al. [3] have presented a probabilistic model for the deviation between desired and actual paths generated by a coupler point, and a robust synthesis procedure of the path generating mechanism was implemented. Both the structural and mechanical errors were incorporated and a four-bar mechanism was used as a numerical example in their approach. Cabrera et al. [4] have presented solution methods for the optimal synthesis of planar mechanisms. Optimization procedure was defined by using genetic algorithm and a suitable objective function. Synthesis of a four-bar mechanism was used to test the proposed method. Another study about application of genetic algorithm technique to the mechanism synthesis has been presented by Kunjur and Krishnamurty [5]. In their study, synthesis of a four-bar path generator mechanism was implemented to illustrate the application of GA technique to the mechanism synthesis. Error between actual and desired paths was considered as a goal function, and it was expressed as the sum of the squares of the error at each point in the path. Furuhashi et al. [6,7] have presented a general approach for the dynamics of four-bar linkages with joint clearances using the continuous contact model based on the assumption that the pin is always in contact with the socket in each pair. The motion of the linkage for the contact angles was analytically treated from Lagrangian function. Direction of the joint clearance with respect to input position was derived by using Runge–Kutta–Gill method considering the Lagrange‘s equation. In this paper, the effect of the joint clearance on path generation and transmission quality of a four-bar mechanism is investigated. A genetic algorithm approach, based on evolutionary techniques and an appropriate goal function, is presented to describe the direction of the joint clearance with respect to input position taking into account continuous contact model between pin and socket. By using genetic algorithm technique, optimization of the mechanism links parameters is also performed to minimize the path errors between desired (without clearance) and actual (with clearance) mechanisms. For this purpose, this paper is organized as follows; Section 2 describes the joint clearance and relevant link’s model. In Section 3, synthesis of four-bar path generator is presented. Section 4 describes Genetic algorithm approach and objective functions. Results are presented in Section 5. Finally, conclusion is outlined in Section 6. 2. Joint clearance In this study, it is assumed that mechanism links are connected to each other by revolute joints with clearance. Joint clearance, as shown in Fig. 1, can be defined as the difference of the radii of the pin and socket at a joint. When the continuous contact model assumption between pin and socket at each joint is considered, the clearances may be modelled as vectors which correspond to massless virtual links with the lengths equal to joint clearance [1,8]. If the friction is negligible, the directions of these vectors coincide with the normal of the contact force between pin and socket. Each joint clearance adds additional freedom to the mechanism
a
b jth link
rj r ri ith link
Fig. 1. (a) Joint clearance model and (b) equivalent clearance link.
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and, supplementary constraints are necessary to perform kinematic and dynamic analysis of the mechanism. When the radii of pin and socket are denoted by ri and rj, respectively, the equivalent clearance vector is given as follows: r ¼ rj ri :
ð1Þ
3. Synthesis of four-bar path generator Dimensional synthesis of planar linkages for path generation consists of determining the dimensions of the linkage in accordance with requirements related to the desired path [9]. Also, it involves the minimization of structural error which is subject to a set of size and geometric constraints such as Grashof’s rules. A four-bar mechanism, as shown in Fig. 2, is considered as an example to determine the effect of joint clearance between crank and coupler link on path generation and transmission quality. Link parameters of the four-bar mechanism are given in Table 1. Kinematic equations are derived from the vector configuration of the mechanism, shown in Fig. 2c, as follows:
a
b
c
P θc4
y
β δ
A
G3
3 μc
L3
2 A0
θ3c L4
A’ r2
G4
L2
θ1 L1
1
B0
γ2
L4
A
4 G2 θ2
θc4
L3
B
θ3c
A’
B
L2
θ2
x A0
L1
θ1 B0
Fig. 2. (a) Four-bar mechanism model, (b) schematic representation of a four-bar mechanism with clearance and (c) vector representation of mechanism.
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Table 1 Parametric values of the four-bar mechanism Link
Material
Length (mm)
Fixed (1) Crank (2) Coupler (3) Follower (4)
Steel Steel Aluminium Aluminium
400 100 360 240
c
c
L1 eih1 þ L2 eih2 þ r2 eic2 þ L3 eih3 þ L4 eih4 ¼ 0:
ð2Þ
By separating equation (2) into its real and imaginary parts and using trigonometric relations, hc3 and hc4 can be expressed as a function of h2 and c2, respectively: " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# B B2 4AC c ; ð3Þ h3 ¼ 2 arctan 2A L1 L2 cos h2 r2 cos c2 L3 cos hc3 hc4 ¼ arccos ; ð4Þ L4 where the superscript c denotes the ‘‘value with clearance”. A, B and C terms are given, respectively A ¼ ðL1 þ L3 Þð2L2 cos h2 þ 2r2 cos c2 Þ þ 2L2 r2 cosðh2 c2 Þ þ 2L1 L3 þ L21 þ L22 þ L23 L24 þ r22 ; B ¼ 4L3 ðL2 sin h2 þ r2 sin c2 Þ; C ¼ ðL3 L1 Þð2L2 cos h2 þ 2r2 cos c2 Þ þ 2L2 r2 cosðh2 c2 Þ 2L1 L3 þ L21 þ L22 þ L23 L24 þ r22 : As shown in Fig. 2b, the position of the coupler point (P(x, y)) relative to the crank pivot (A0) is given with and without joint clearance, respectively 2 c3 2 3 2 3 2 3 Px cosðhc3 þ bÞ cos h2 cos c2 4 5 ¼ L2 4 5 þ r2 4 5 þ A0 P 4 5; ð5Þ c Py sinðhc3 þ bÞ sin h2 sin c2 2 4
Px
3
2
5 ¼ L2 4
cos h2
3
2
5 þ AP 4
3 5;
ð6Þ
sinðh3 þ bÞ
sin h2
Py
cosðh3 þ bÞ
where P cx ; P cy denote the x- and y-coordinate values for the path of coupler point in considering the joint clearance, and Px, Py denote the x- and y-coordinate values for the path of coupler point without joint clearance. Due to motion transmission line from crank link to follower link, joint clearance between crank and coupler connection has an important role on path generation of point P. If the crank pivot (A0) is taken as the reference point, the mass center positions for moving links are given as follows: 2 c 3 2 3 xG2 cos h2 4 5 ¼ A0 G 2 4 5; ð7Þ y cG2 sin h2 2 4 2 4
xcG3 y cG3 xcG4 y cG4
3
2
5 ¼ L2 4
cos h2
3
2
5 þ r2 4
sin h2 3
2
5 ¼ L2 4
cos h2 sin h2
cos c2
2
3
5 þ A 0 G3 4
sin c2 3
2
5 þ r2 4
cos c2 sin c2
3
2
5 þ L3 4
cosðhc3 þ dÞ sinðhc3
cos hc3 sin hc3
3
3 5;
ð8Þ
þ dÞ 2
5 þ BG4 4
cos hc4 sin hc4
3 5:
ð9Þ
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Transmission angles for the model mechanism without and with clearance are defined as follows [10]: 2 L þ L22 L23 L24 2L1 L2 cos h2 l ¼ arccos 1 ; ð10Þ 2L3 L4 ! L21 þ A0 A0 2 L23 L24 2L1 A0 A0 cosðh2 wÞ c l ¼ arccos ; ð11Þ 2L3 L4 where A0 A0 and w are given, respectively A0 A0 ¼ ½L22 þ r22 þ 2L2 r2 cosðh2 c2 Þ1=2 ; " # 02 2 2 1 L2 þ A0 A r2 w ¼ cos : 2L2 A0 A0
ð12Þ ð13Þ
If the mechanism links are assumed to be rigid, direction variable of clearance vector can be derived by using Lagrange’s equation in the following form [6,7,11]: d oT oT oU oDF þ þ ¼ 0; ð14Þ dt o_c2 oc2 oc2 o_c2 where T, U and DF denote the kinetic energy, the potential energy and the dissipation function, respectively. The above mentioned terms are given as follows: T ¼ U¼
4 4 1X 1X 2 2 2 I i ðh_ ci Þ þ mi ½ð_xcGi Þ þ ð_y cGi Þ ; 2 i¼2 2 i¼2 4 X
mi gy cGi ;
ð15Þ ð16Þ
i¼2
DF ¼
4 1X 1 C hi ðh_ ci Þ2 þ C c2 c_ 22 ; 2 i¼2 2
ð17Þ
where i designates the link number. If the above terms are substituted into Eq. (14), Lagrange’s equation can be established in the following form: 4 X oxc oy c oy c ohc ohc ð18Þ hci i þ mi €xcGi Gi þ €y cGi Gi þ gmi Gi þ C hi h_ ci i þ C c2 c_ 2 ¼ 0: I i€ oc2 oc2 oc2 oc2 oc2 i¼2 4. Optimization procedure for path generation in mechanism with clearance Two objective functions (OF) are defined, firstly, to determine the direction of the joint clearance and secondly, to minimize the error between the desired and actual positions of point P. As seen from Eq. (18), the motion equation of mechanism has nonlinear character. Genetic algorithm approach is implemented to solve this equation, and so to determine the direction of joint clearance as a function of position variable of input link. This equation is considered as the first objective function, and it is expressed in the following form: ! 4 c c X oxcGi oy cGi oy cGi c ohi c c c ohi € _ þ mi €xGi þ €y Gi þ gmi þ C hi hi þ C c2 X ; I i hi Minimize F 1 ðX Þ ¼ f oX oX oX oX oX i¼2 Subject to
hj ðX Þ ¼ 0; xl 6 x j 6 x u ;
ð19Þ
xj 2 X ; where hj(x) are the equality constraints which are determined depending on parametric relations between joint clearance direction and input variable. X is a vector comprising the design variables corresponding to joint
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clearance direction, that is, c2 ; c_ 2 and €c2 xl and xu are the lower and upper bounds of the design variables, respectively. In path generation problems, the coupler point has to track a given path with minimum error. Objective function is a measure of the error between the desired and actual paths, and is usually expressed as the sum of the squares of the error at each point of the path [5]. For minimizing the error between desired and actual positions of point P, the second objective function is given as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N h i u1 X 2 2 ð20Þ ðP dxk P axk Þ þ ðP dyk P ayk Þ Minimize F 2 ðxÞ ¼ t N k¼1 Subject to
gm ðxÞ 6 0; xl 6 x m 6 x u ;
where P dxk ; P dyk are x- and y-coordinate values of the kth point in the desired path and P axk ; P ayk are the x- and ycoordinate values of the kth point in the actual path. The inequality constraints (gm(x)) consist of Grashof’s rule, that is, crank-rocker condition. xm denote the independent design variables, and they consist of link lengths (Li) and structural angle (b). Design variables for the second objective function are given in the following form: T
x ¼ bL1 L2 L3 L4 bA0 P c :
ð21Þ
5. Results
180
600
150
500
120
400
90
300
60
200
30
100
0
0
-30
0
60
120
180
240
300
-100 360
Input Position [Degree] (
):
γ 2 [Degree], (
):
γ2
[Rad/sec]
Fig. 3. Direction of joint clearance with respect to position of input link.
Radian/second
Degree
In the present paper, a theoretical model is used for the case of single clearance in a four-bar mechanism with rigid links, and 600 rpm is considered as running speed. ADAMS [12], a mechanical system simulation software, is used to model and simulate the mechanism, and also to verify the analytical results. The determining of the direction variable of the joint clearance and the optimum link parameters ðL1 ; L2 ; L3 ; L4 ; b; A0 P Þ in the model mechanism with joint clearance are realized using genetic algorithm on MATLAB software. Therefore,
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the variation of joint clearance direction with respect to position variable of input link, which is obtained from the first objective function using genetic algorithm, is outlined in Fig. 3. If the convergence rate of the energy terms in Eq. (18) to zero is considered as a measure of performance criteria for Genetic algorithm to solve the nonlinear motion equation, that is, to determine the joint clearance direction, the convergence error in this equation by using the obtained solution set is given in Fig. 4. As shown, maximum and average errors are 1.5 105 and 4.3907 107, respectively. It can be said that these results are acceptable in determining appropriate design parameters. Two case studies are implemented for observing the influences of joint clearance on the path generation and transmission angle. In the first study, joint clearance value is taken to be equal to 1 mm, and in the second one, this value is considered as 2 mm. In order to minimize the path error, the second objective function for the genetic algorithm application is developed. As a result, the original and optimized values of design parameters are given in Table 2. If the kinematic equations (Eqs. (5) and (6)) are solved for two different clearance values, the obtained path configurations for desired (without clearance), actual (with clearance) and optimized (optimized with clearance) mechanisms are outlined in Figs. 5 and 6, respectively. As shown in these figures, the path error reaches the largest value when the mechanism goes into dead-center positions. The path errors obtained for two different joint clearance values in the original and optimized mechanisms are given in Fig. 7. As shown, these errors for original mechanism are bigger than that of the optimized mechanism. When error values for x-
1.6 1.4
1.2
Error (x10-5)
1
0.8 0.6
0.4
0.2 0
0
60
120
180
240
300
360
Input Position [Degree] Fig. 4. Performance measure of GA for Lagrange equation.
Table 2 Optimized design variables for different joint clearance values Design variables
L1 (mm) L2 (mm) L3 (mm) L4 (mm) b (degree) A0 P ðmmÞ
Original values
400 100 360 240 20.3319 305.07
Optimum values with joint clearance r2 = 1.0 mm
r2 = 2.0 mm
399.91846 100.01917 358.89966 239,98779 20.37210 304.24561
399.90625 100.03125 357.82812 239.60156 20.48950 303.41797
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229
302
Right Dead-center position
300
Y-coordinate [mm]
280 286 266
280
260
240
196
220
Left Dead-center position
200 182 88
180 80
100
120
140
160
180
108
200
220
240
260
280
X-coordinate [mm] (
) : Desired Path (
) : Actual Path
(
) : Optimized Path
Fig. 5. Desired, actual and optimized path configurations for the joint clearance of 1 mm.
320
302
300
Right Dead-center position
Y-coordinate [mm]
280
260
286 266
280
240
196
220
Left Dead-center position
200 182
180 80
100
120
140
160
180
88
108
200
220
240
260
280
X-coordinate [mm] (
) : Desired Path (
) : Actual Path
(
) : Optimized Path
Fig. 6. Desired, actual and optimized path configurations for the joint clearance of 2 mm.
and y-directions of each mechanism are evaluated separately, it is clearly seen that path errors for the joint clearance of 1 mm decrease by 78.46% and 75.72% in x- and y-directions, respectively. Also, in the case of 2 mm clearance, these errors decrease by 78.45% and 75.69% in each directions, respectively. The transmission angles under two different clearance conditions during the whole motion cycle of the mechanism are shown in Figs. 8 and 9, respectively. As seen from related figures, the deviation of the trans-
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1 mm joint clearance (X-direction)
3 1.5 0 -1.5 -3 0
60
120
180
240
300
360
300
360
300
360
300
360
1 mm joint clearance (Y-direction) 3 1.5
Path error [mm]
0 -1.5 -3
0
60
120
180
240
2 mm joint clearance (X-direction)
3 1.5 0 -1.5 -3 0
60
120
180
240
2 mm joint clearance (Y-direction)
3 1.5 0 -1.5 -3 0
60
120
180
240
Input Position [Degree] ( (
) : Path error between desired and actual mechanisms ) : Path error between desired and optimized mechanisms
Fig. 7. Path errors for two joint clearance values.
mission angle from the desired value is larger when the joint clearance value increases. Also, it can be clearly seen that the biggest deviations of transmission angle from the desired value occurred at the extreme positions. The difference between desired and actual transmission angles, and also the deviation of the desired transmission angle relative to the optimized ones, as shown in Fig. 10, can be assigned as a measure of performance for the second optimization. As seen form Fig. 10, transmission angle errors between desired and optimized mechanisms are smaller than those of the other. These errors approximately decrease by 55.56% and 42.77% for 1 and 2 mm joint clearances, respectively.
6. Conclusion In this study, assuming that the presence of joint clearance between crank and coupler links, the effect of this factor on mechanism path generation and transmission quality is investigated. Joint clearance is considered as a massless virtual link and mathematical expression of its motion is derived by using Lagrange’s equation. An objective function is defined, and parametric relations between input variable and joint clearance direction are considered as the constraints for solving nonlinear differential equation by using genetic algorithm approach. So, the direction of joint clearance relative to input variable is obtained. Also, to minimize the deviations in the path generation and transmission angle arising from joint clearance, that is, for obtaining
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115 110
Transmission Angle [Degree]
105 100 95 90 112
85 80 75 70 102 130
65
230
60 55
0
60
120
180
240
300
360
Input Position [Degree] (
) : Desired µ
(
) : Actual µ
(
) : Optimized µ
Fig. 8. Transmission angle variation for the joint clearance of 1 mm.
115 110
Transmission Angle [Degree]
105 100 95 90
112
85 80 75 70 102 130
65
230
60 55
0
60
120
180
240
300
360
Input Position [Degree] (
) : Desired µ
(
) : Actual µ
(
) : Optimized µ
Fig. 9. Transmission angle variation for the joint clearance of 2 mm.
the desired values with minimum error, a second objective function is defined and optimization of the link parameters in mechanism is implemented. Crank-rocker situation corresponding to Grashof’s rule is considered as a geometric constraint to obtain the best approximation for the desired values.
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1 mm joint clearance
µ error [Degree]
1.5 1 0.5 0 -0.5 -1 -1.5
0
60
120
300
360
300
360
2 mm joint clearance
1.5
µ error [Degree]
180 240 Input Position [Degree]
1 0.5 0 -0.5 -1 -1.5
0
60 ( (
120
180 240 Input Position [Degree]
) : µ error between desired and actual mechanisms ) : µ error between desired and optimized mechanisms
Fig. 10. Transmission angle errors for two joint clearances.
As seen from the results, if the values of equivalent joint clearances are bigger, the mechanism is subjected to higher deviations from the desired characteristics. After the optimization process, the position errors of the point P for the joint clearance of 1 mm decrease by 78.46% and 75.72% in x- and y-directions, respectively. In the case of 2 mm joint clearance, these errors decrease by 78.45% and 75.69% in each directions, respectively. Also, transmission angle errors approximately decrease by 55.56% and 42.77% for two cases of joint clearance. These results can be seen as a measure of success of optimization strategy, constraints and inequalities. The proposed approach is a versatile method to carry out the optimum synthesis of planar mechanisms, and it can be easily adapted to different mechanisms having joints with clearance. Acknowledgements This work is a part of the research project FBA-07-06. The authors wish to express their thanks for financial support being provided by the Scientific Research Project Fund of Erciyes University, in carrying out this study. Appendix A Mechanism dynamic parameters used in Lagrange’s equation are given in Table A.1. Front and side views of the examined mechanism’s links are given in Fig. A.1, respectively. Partial differentiations of mass center positions for moving links and links’ position variables (hc3 and hc4 ) with respect to direction of joint clearance are given, respectively, 2 c 3 2 c 3 oxGi oxGi " # 6 oc 7 6 oh2 7 _xcGi 6 27 7 ¼ x2 6 ðA:1Þ 7; 4 oy c 5 þ c_ 2 6 4 oy cGi 5 y_ cGi Gi oh2 oc2
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Table A.1 Dynamic parameters for four-bar mechanism Parameters
Descriptions
Values
I2 I3 I4 m2 m3 m4 g x2 C hi C c2
Moment of inertia of the crank Moment of inertia of the coupler Moment of inertia of the follower Mass of the crank Mass of the coupler Mass of the follower Acceleration of gravity Angular velocity of input link Damping coefficient [11] Damping coefficient [11]
5.12 104 kg m2 8.85 103 kg m2 1.58 103 kg m2 0.121 kg 1.048 kg 0.071 kg 9.81 m/s2 62.83 rad/s 0.2 106 kg m s/rad 0.2 106 kg m s/rad
Fig. A.1. Two different views of investigated four-bar mechanism links.
2
"
€xcGi €y cGi
#
3
3 2 2 c 3 o xGi o2 xcGi 2 oxc 3 6 Gi 2 7 6 oh oc 7 6 oc2 7 7 6 oh oc2 6 2 27 6 2 7 2 7 ¼ x22 6 7 þ €c2 4 oy c 5 þ c_ 22 6 2 c 7; 6 2 c 7 þ 2x2 c_ 2 6 2 c 5 4 4 o yG 5 Gi o y o y 4 Gi 5 Gi i oc2 2 oh2 oc2 oc22 oh2 o2 xcGi
2
ðA:2Þ
ohc ohc h_ cz ¼ x2 z þ c_ 2 z ðz ¼ 3; 4Þ; oh2 oc2
ðA:3Þ
o2 hc o2 hcz ohc o2 hc € hcz ¼ x22 2z þ 2x2 c_ 2 þ €c2 z þ c_ 22 2z ðz ¼ 3; 4Þ: oh2 oc2 oc2 oc2 oh2
ðA:4Þ
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