Multi-Objective Optimization of Kinematic Characteristics of Geneva Mechanism

Mechanics, Materials Science & Engineering, March 2017

ISSN 2412-5954

Multi-Objective Optimization of Kinematic Characteristics of Geneva Mechanism Using High-Tech Optimization Methods8 Arash Mohammadzadeh Gonabadi 1,a, Mohammad Nouri Damghani 1,b 1

Department of mechanical engineering, Semnan University, Semnan, Iran

a

arash_mg@semnan.ac.ir

b

mnoori@semnan.ac.ir DOI 10.2412/mmse.26.65.331 provided by Seo4U.link

Keywords: Geneva wheel mechanism; dynamic modeling; kinematic characteristics; multi-objective optimization; GA; ICA

ABSTRACT. This research is aimed toward using variable input speed for triple-objective optimization of kinematic characteristics of a Geneva wheel mechanism. High-tech optimization methods including genetic algorithm (GA) and imperialist competitive algorithm (ICA) are used in this study. The objective functions in both methods are magnitude of the maximum output angular velocity, magnitude of the maximum output angular acceleration as well as magnitude of the maximum output angular jerk of driven wheel. The motion equations of Geneva mechanism are first extracted with their boundary conditions determined. Then, above objective functions are minimized using the considered algorithms.

Utilization of the input velocity may contribute to improve output kinematic characteristics of the mechanism.

Introduction. Developments achieved in the industries and widespread application of mechanisms throughout various industries have motivated researchers to rise their efforts in the course of ell as performance of automatic machines. Today, correspondingly, many undergoing researches are dedicated to analysis of mechanisms as well as their synthesis. In conventional methods, a constant input speed is assumed for any mechanism; so as an improvement in mechanism performance is impossible in many cases where researchers are forced to design a new mechanism with higher kinematic performance [5]. For instance, the researchers designed a new mechanism to enhance kinematic characteristics of a cam mechanism [10].

The idea to use variable input speed into mechanisms was first implemented in a cam mechanism whose input mechanism was, in fact, the output of a Whitworth quick-return mechanism [13]. These researches utilized variable input speeds to reduce the cam dimensions and consequently, its pressure angle. Other researchers showed, in both experimental and theoretical way, that considering a variable input speed, one can control and improve output characteristics of a mechanism [2, 4, 14].

Optimization of kinematic characteristics of a Geneva mechanism is addressed in this research.

Geneva mechanism is one of the most applied and simple mechanisms for generating periodic motions. Such mechanisms are widely used in many devices such as clocks, machine tools, printing and pressing machineries, packaging machineries as well as automotive machines. Although numerous kinds of tools are designed for the sake of periodic motion generation, but Geneva mechanisms are among the first choices for this purpose due to their simplicity, lower price, higher safety factor, long life and also smooth motion curves they produce [1, 7-8].

While simple Geneva mechanisms (with constant input speed) are popular, they suffer from a set of associated deficiencies. In Geneva mechanisms, output plots are not under control by the designer with angular acceleration of driven wheel having non-zero values at the beginning and the end of 8

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contact. This issue causes an angular jerk to be generated at the beginning and the end of contact.

Furthermore, the angular jerk has non-zero values along the contact. Such an angular jerk will result in shock, vibration, abrasion and also noise generation which may lead to serious damages to the production line as well as machineries at high speeds. In addition, the maximum angular acceleration over the contact between the pin and the slot is as high as it leads the mechanism to exhibit a poor performance in accurate applications. Many modification techniques are already proposed to adjust a Geneva mechanism including placing two Geneva wheels besides each other [7], a Geneva wheel with curved slots [9], a Geneva wheel with two input cranks [11] and a flywheel with a quad-rod mechanism [16].

Controlling input speed of the mechanism is one of the methods used to control its output characteristics. In recent years, variable input speed is widely used to improve kinematic as well as dynamic characteristics of many popular mechanisms [3, 6]. Although extensive studies have focused on Geneva mechanisms and designing different structures for them, little has been done on optimization of their characteristics via optimization procedures. Heidari et al. used a single-objective optimization via genetic algorithm (with acceleration as the objective function) to improve output characteristics of a Geneva wheel [15]. The researchers implemented a double-objective optimization method, namely NSGA II, as well as MOSO optimization method to improve output characteristics of a Geneva wheel [12] In this research, two triple-objective optimization methods, namely the genetic algorithm and imperialist competitive algorithm, are presented in order to achieve better design points within a fourslot Geneva mechanism. The objective functions included the maximum angular velocity, the maximum angular acceleration as well as the maximum angular. The optimization goal is to find polynomial coefficients of input angular displacement function (as a variable input) and to generate plots of suitable input angular velocities which cause an improvement in kinematic characteristics of the Geneva wheel.

Motion Equations for Four-Slot Geneva Mechanism The Geneva mechanism is the one used to transform continuous rotational motion into intermittent rotatory motion. Here a four-slot Geneva wheel is used that is an intermittent gear where the drive wheel has a pin that reaches into a slot of the driven wheel and thereby advances it by one step. The drive wheel also has a raised circular blocking disc that locks the driven wheel in a position between steps (see Figure 1).

Fig. 1. The four-slot Geneva mechanism.

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According to Figure 1 we have: (1) and

(2)

Combining Eqs. (1) and (2) we will get:

(3)

Taking derivative of the Eq. (3) one may obtain angular speed and acceleration of the driven wheel as follows:

(4)

and

(5)

In a four-slot Geneva wheel, the wheel makes 90 degrees rotation when a 360-degree revolution of the driving wheel is realized. Furthermore, the same distance is assumed from the contact point of the pin toward center points of the driving and the driven wheel (Geneva wheel) at the starting moment when the pin contacts the slot (the beginning of the contact). The starting moment of motion is considered to be the moment at which pin contacts the slot. Geneva wheel will complete one fourth of its motion by the time , when the driving wheel has rotated 90 degrees. Dimensionless motion equations are derived using the following dimensionless parameters:

(6) and

(7) also

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(8)

Time variable, , and angular variable, , vary in the ranges of and , respectively, so that dimensionless parameters of and vary within the ranges of and , respectively. Substituting the Eqs. (6), (7) and (8) into the Eqs. (4) and (5), angular velocity, angular acceleration and angular jerk relationships may be rewritten as follows:

(9)

and

(10)

where,

(11)

Based on the above equations, one may suggest that all kinematic characteristics (including output angular velocity, acceleration and jerk) depend on the input angular displacement, while output characteristics can be controlled by manipulating input angular displacement. Angular displacement, angular velocity, angular acceleration, and angular jerk are plotted in Figure 2 for a constant input angular speed of 1 rad/sec and constant coefficient of . As shown on this figure, the angular acceleration along with its derivative has non-zero values at the beginning and the end of the contact, while they exhibit high amplitude fluctuations along the contact. Such fluctuations are not appropriate for the mechanism and may cause damages and shocks to the mechanism as well as production line.

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4

Angular Velocity

3

2

1

0 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular

a) 20

Angular Acceleration

10

0

-10

-20 -0.5

-0.3

-0.1

0.1

0.3

0.5

0.3

0.5

Dimensionless Angular

b) 100

Angular Jerk

0

-100

-200 -0.5

-0.3

-0.1

0.1

Dimensionless Angular

c) Fig. 2. The plots of angular velocity (a), angular acceleration (b) and angular jerk (c) for the driven wheel (Geneva wheel).

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Optimization Methods This section deals with minimization of the magnitude of the maximum angular velocity, the magnitude of the maximum angular acceleration as well as the magnitude of the angular jerk (as the objective functions) of a Geneva wheel. A polynomial form is considered for the input angular displacement function (as the variable input). This function can be utilized to simply calculate input angular speed values. Input polynomial coefficients are considered as the design variables. In general, polynomial function and its associated boundary conditions are defined as follows:

(12)

Also (0) = 0.5 ,

(13)

(1) = 0.5

From the Eqs. (12) and (13), we have:

(14)

The value of angular speed of the driving wheel must always be positive; this is considered as a constraint through the problem, so as we have:

(15)

Here a 7 order polynomial is taken into account as the input function (input angular displacement function) for the mechanism. Considering the boundary conditions of the problem, 6 out of 8 coefficients of this polynomial are unknown which are to be taken as the design variables.

Genetic Algorithm This highly powerful algorithm has the capability of solving constrained and unconstrained, linear and nonlinear, single- and multi-variable as well as continuous- and discrete-variables problems [2128]. Optimization parameters depicted in Table 1 are used to solve the optimization problem via GA. Table 1. Parameters of the optimization problem, genetic algorithm (GA).

Number of population

Number of generation

Probability of mutation

Probability of crossover

90

400

0.015

0.7

Imperialist Competitive Algorithm MMSE Journal. Open Access www.mmse.xyz

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This is an extremely powerful algorithm to find the global optimum point for complex problems [1726]. To solve the problem via this algorithm, we used the optimization parameters shown in Table 2.

Table 2. Parameters of the optimization problem, imperialist competitive algorithm (ICA).

Initial countries

Early empire

90

8

The number of repetitions 0.5

2

0.02

1900

Results Once optimization was performed (using MATLAB), a set of points are obtained in terms of a convergence plots drawn in Figures 3 and 4. The optimum point obtained in this research is more improved compared to those given in previous researches [21-28].

Fig. 3. Globally optimized points (GA).

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Fig. 4. Globally optimized points (ICA) - (Red: best fitness & Black: mean fitness).

According to Figure 3, the optimum point has an angular jerk of 22.0464, an angular acceleration of 7.6855 and an angular velocity of 2.4421 in terms of magnitude. Furthermore, the best chromosome

(16) The optimum polynomial may be expressed as follows:

(17)

The curves of angular displacement and angular velocity of the driving wheel are plotted in Figure 5 for the optimum point (obtained via GA). As can be seen, the provided constraint in the Eq. (15) is met while the value of angular velocity of the driving wheel is positive all along the contact.

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0.5

Angular Displacement

0.3 0.1 -0.1 -0.3 -0.5 0

0.2

0.4

0.6

0.8

1

0.8

1

Dimensionless Time

a) 1.8

Angular Velocity

1.6

1.4 1.2 1

0.8 0.6 0

0.2

0.4

0.6

Dimensionless Time

b) Fig. 5. Curves of angular displacement (a) and angular velocity (b) of the input disk after optimization, GA method.

Figure 6 compares kinematic characteristics of the Geneva mechanism (including angular velocity, angular acceleration and angular jerk of the Geneva wheel) for constant input speed (i.e. before optimization) with those for variable input speed (i.e. after triple-objective optimization by GA).

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4

Angular Velocity

3

2

1

0 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular Constant Input Speed

Optimization (GA)

a) 15

Angular Acceleration

10 5 0 -5 -10

-15 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular Constant Input Speed

Optimization (GA)

b) 50

Angular Jerk

0 -50 -100 -150 -200 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular Constant Input Speed

Optimization (GA)

c) Fig. 6. Curves of angular velocity (a), angular acceleration (b) and angular jerk (c) of Geneva wheel before and after optimization, GA method.

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According to Figure 4, the optimum point obtained by imperialist competitive algorithm has an angular jerk of 21.5687, an angular acceleration of 7.8631 and an angular velocity of 2.4296 in terms of magnitude. Furthermore, the best chromosome with all 6 design variables has the following form

(18) The optimum polynomial may be expressed as follows:

(19)

The curves of angular displacement and angular velocity of the driving wheel are plotted in Figure 7 for the optimum point (obtained via ICA). As can be seen, the provided constraint in the Eq. (15) is met while the value of angular velocity of the driving wheel is positive all along the contact.

0.5

Angular Displacement

0.3 0.1

-0.1 -0.3 -0.5 0

0.2

0.4

0.6

0.8

1

Dimensionless Time

a) 1.8

Angular Velocity

1.6 1.4 1.2

1 0.8 0.6 0

0.2

0.4

0.6

0.8

1

Dimensionless Time

b) Fig. 7. Curves of angular displacement (a) and angular velocity (b) of the driving wheel after optimization, ICA method.

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Figure 8 compares kinematic characteristics of the Geneva mechanism (including angular velocity, angular acceleration and angular jerk) for constant input speed (i.e. before optimization) with those for variable input speed (i.e. after triple-objective optimization by ICA).

4

Angular Velocity

3

2

1

0 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular

Constant Input Speed

a) 15

Angular Acceleration

10 5 0 -5

-10 -15 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular Constant Input Speed

Optimization (ICA)

b) 50

Angular Jerk

0 -50 -100 -150 -200 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular Constant Input Speed

Optimization (ICA)

c) Fig. 8. Curves of angular velocity, angular acceleration and angular jerk of Geneva wheel before and after optimization, ICA method.

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Table 3 presents the results when considering a constant input angular velocity (before optimization) as well as optimized results from [1], [15] and [16] along with those optimized via two methods presented in this research.

Table 3. A comparison over the results obtained for constant input speed and those given by different optimization methods.

The maximum angular velocity 3.7922 2.5517 2.6134 2.5212

Constant input velocity of 1 rad/s Optimized input via [1] Double-objective optimization (NSGA II), [15] Triple-objective optimization, [16] Triple-objective optimization, present research via GA Triple-objective optimization, present research via ICA

The maximum angular acceleration

The maximum angular jerk

13.3370 6.96 6.8874 6.907

186.199 36.7 34.975 34.2064

2.4421

7.6855

22.0464

2.4296

7.8631

21.5687

Numerical Method Finally, the obtained results were modelled in MSC Visual Nastran Desktop 4D (see Figure. 9). It is one of the powerful dynamic software which can simulate dynamic motion [29-33]. The results had good agreement with each other as is observed on Figure 10.

Fig. 10. Curve of angular velocity of Geneva wheel after optimization, was modelled in MSC. Visual Nastran, ICA method.

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Angular Velocity

3

2

1

0 -0.5

-0.3

-0.1

0.1

0.3

0.5

Dimensionless Angular MSC. Visual Nastran

MATLAB

Fig. 11. A comparison of angular velocity of Geneva wheel after optimization, MATLAB and MSC. Visual Nastran Desktop 4D, ICA method.

Summary. According to Figures 6 and 7, the maximum values of angular velocity, angular acceleration and angular jerk were significantly reduced after optimization via either GA or ICA method. The values of magnitudes of the maximum angular velocity, the maximum angular acceleration as well as the maximum angular jerk were decreased by 35.6%, 42.38% and 88.16%, respectively, for GA method, while they are reduced by 35.9%, 41.04% and 88.42%, respectively, for ICA method. It is worth to note that the optimized values of angular velocity and jerk were slightly more satisfying via ICA rather than those obtained by GA. In addition, the results of this study in terms of maximum value of angular velocity as well as the maximum angular jerk are improved with respect to the results of [1], [15] and [16]. Such an improvement reduces the risk of shock and damages to the production line and machinery along the contact. It should be noted that selecting higher number of objective functions will let the designer to achieve superior kinematic characteristics when designing a Geneva mechanism with variable input speed. Therefore, for similar case study with non-linear equations such as Geneva and Four-bar mechanism ICA method is one of the significant methods for dynamic optimization.

References [1] Al-Sabeeh, A.K. 1993. Double-crank external Geneva mechanism, Journal of Mechanical Design, 115: 666-670.

[2] Bickford, J.H. 1972. Mechanism for intermittent motion, Industrial press INC, New York.

[3] Coelo, C.A. 2004. Handling multiple objective with particle swarm Ooptimization, IEEE Tranactions on Evolutionary Computation, 8: 256-279.

[4] Fenton, R.G. 1976. Geneva mechanism connected in series, Journal of Manufacturing Science and Engineering, 97: 603-608.

[5] Heidari, M. 2010. Production and modification of intermittent rotatory motion using slider, worm gear and cam mechanisms, MSc Thesis, Tehran University, Tehran, Iran.

[6] Heidari, M., Zahiri, M. and Zohoor, H. 2008. Optimization of Kinematic Characteristic of Geneva Mechanism by Genetic Algoritm, World Academy of Science: Engineering and Technology, 44: 387395.

[7] Hunt, H.K., Fink, N. and Nayar, J. 1960. Proceedings of the Institution of Mechanical Engineers (London), 174: 643-656.

MMSE Journal. Open Access www.mmse.xyz

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Mechanics, Materials Science & Engineering, March 2017

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[8] Lee, H.P. 1998. Design of a geneva mechanism with curved slots using parametric polynomial, Mechanism and Machine Theory, 33: 321-329.

[9] Lee, J.J. and Cho, C.C. 2002. Improving kinematic and structural performance of Geneva mechanism using the optimal cotrol method, Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science, 216: 761-774.

[10] Mills, J.K., Notash, L. and Fenton, R.G. 1993. Optimal design and sensitivity analysis of flexible cam mechanisms, Mechanism and Machine Theory, 28: 563-581.

[11] Mundo, D. and Yan, H.S. 2007. Kinematic optimization of ball-screw transmisson mechanisms, Mechanism and Machine Theory, 42: 34-47.

[12] PurMohammadi, A., Felezi, M.E. and NarimanZadeh, N. 2013. Optimization of kinematic characteristics of Geneva mechanism with constant input speed, 8th students Conference on Mechanical Engineering, Malayer University, Iran.

[13] Rothbart, H.A. 1956. Cams: design, dynamics and accuracy, Wiley, New York.

[14] Tesar, D. and Matthew, G.K. 1976. The dynamics synthesis, analysis and design of modeled cam systems, Lexington Book, Lexington.

[15] Yan, H.S. and Yan, G.J. 2009. Integrated control and mechanism design for the variable inputspeed servo fuor-bar linkage, Mechatronics, 19: 274-285.

[16] Benjamin Ivorra, Bijan Mohammadi, Angel Manuel Ramos ,A multi-layer line search method to improve the initialization of optimization algorithms, European Journal of Operational Research, Volume 247, Issue 3, 16 December 2015, Pages 711-720 [17] Weihong Zhang, Hu Liu, Tong Gao, Topology optimization of large-scale structures subjected to stationary random excitation: An efficient optimization procedure integrating pseudo excitation method and mode acceleration method, Computers & Structures, Volume 158, 1 October 2015, Pages 61-70 [18] Julie Coloigner, Laurent Albera, Amar Kachenoura, Fanny Noury, Lotfi Senhadji, Seminonnegative joint diagonalization by congruence and semi-nonnegative ICA, Signal Processing, Volume 105, December 2014, Pages 185-197 [19] Visa Koivunen, Traian Abrudan, Riemannian optimization in complex-valued ICA, Advances in Independent Component Analysis and Learning Machines, 2015, Pages 83-94 [20] Arash Mohammadzadeh, A.Ghoddoosian, M. NooriFlexible Rotors with Particle Swarm Opt Engineering - (Vol. 5 N. 3) - Papers 5 (3): 490-496.

[21] sandwich panels based on yielding and buckling criteria by using imperialist competitive -35 [in Persian].

[22] and Reviews in Applied Sciences - (Vol. 23 N. 1) - Papers 23 (1): 54-64.

[23] Domestic Hot Water System by the Use of Imperialist Competitive Algorithm with the Help of Exer 78, 2015 [24] Nader Mohammadi, Farahnaz Fallah Tafti, Ahmad Reza Arshi, Arash Mohammadzadeh, loading Shoes Using

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496, 2014 [25] Water Using a Neural Network Method in the Munic Industrial Engineering, Volume 5, Pages 73-85, 2012 [26] [Amir

Mohammadzadeh,

Nasrin

Mahdipour,

Arash

Mohammadzadeh,

Mohammad etwork

[27] Arash Mohammadzadeh, N. Etemadee. 2011. " Optimized Positioning of Structure Supports with PSO for Minimizing the Bending Moment." International Review of Mechanical Engineering (Vol. 5 N. 3) - Papers 5 (3): 422-425.

[28] Mohammad Nouri Damghani, Arash Mohammadzadeh Gonabadi (2016). Analytical and Numerical Study of Foam-Filled Corrugated Core Sandwich Panels under Low Velocity Impact.

Mechanics, Materials Science & Engineering, Vol 7. doi:10.2412/mmse.6.55.34 [29] Mohammad Nouri Damghani, Arash Mohammadzadeh Gonabadi (2016). Investigation of Energy Absorption in Aluminum Foam Sandwich Panels By Drop Hammer Test: Experimental Results. Mechanics, Materials Science & Engineering, Vol 7. doi:10.2412/mmse.6.953.525 [30] M Nouri Damghani, A Mohammadzadeh Gonabadi (2017). Numerical study of energy absorption in aluminum foam sandwich panel structures using drop hammer test. Journal of Sandwich Structures & Materials. First published date: January-11-2017. doi:10.1177/1099636216685315 [31] M.Noori-Damghani, H.Rahmani, Arash Mohammadzadeh, S.Shokri-Pour. 2011. "Comparison of Static and Dynamic Buckling Critical Force in the Homogeneous and Composite Columns (Pillars)." International Review of Mechanical Engineering - (Vol. 5 N. 7) - Papers 5 (7): 1208-1212.

[32] Mohammad Nouri Damghani, Arash Mohammadzadeh Gonabadi (2017). Numerical and Experimental Study of Energy Absorption in Aluminum Corrugated Core Sandwich Panels by Drop Hammer Test. Mechanics, Materials Science & Engineering, Vol 8. doi:10.2412/mmse.85.747.458 [33] tional Review of Mechanical Engineering - (Vol. 4 N. 7) - Papers 4 (7): 917-923.

Cite the paper Arash Mohammadzadeh Gonabadi, Mohammad Nouri Damghani (2017). Multi-Objective Optimization of Kinematic Characteristics of Geneva Mechanism Using High-Tech Optimization Methods. Mechanics, Materials Science & Engineering, Vol 8. doi:10.2412/mmse.26.65.331

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