Nomograms for Synthesizing Crank Rocker Mechanism with a Desired Optimum Range of Transmission Angle
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1 International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME Volume 3, Issue 3 (015 ISSN (Online Nomograms for Synthesizing Crank Rocker Mechanism with a Desired Optimum Range of Transmission Angle Dr. Khaled M. Khader Abstract The transmission angle is a significant criterion of mechanisms design. It's the smaller angle between the coupler link and the follower (output link of a planar mechanism. Transmission angle value which equal to 90 o provides the best characteristics to the mechanism motion, while this motion becomes very hard with values nearly 0 o or 180 o. As well as, the mechanism transmission motion may be impossible at 0 o or 180 o. Furthermore, the force-transmission characteristics of a crank rocker mechanism can be judged by concerning the transmission angle range through a whole crank rotation. Obviously, design nomograms which is presented in this paper can be consider as an effective fast tool helps the designers for direct selection of the suitable mechanism links ratios providing a definite synthesized transmission angle range. Keywords Transmission angle, Mechanism, Synthesis, Design techniques. T I. INTRODUCTION HE transmission angles of a planar crank rocker mechanism can be defined as the smaller angle between the coupler link and the follower rocker link as in [1]. It can be also defined as the smaller angle between the direction of absolute velocity vector of rocker link and direction of velocity difference vector of driving link at the connection joint as in []. It is well known that, the main objectives of mechanisms designers are low input torque variation, optimum transmission angle, smooth transmission, low periodic loads in bearing and fewer vibrations. Moreover, the transmission angle plays an important role for providing a good or bad mechanism motion's characteristics. Hence, a reasonable optimal transmission angle can treat the most problems in mechanisms as in [3]. The low fluctuation of torque can't be always guaranteed with a large value of transmission angle. Also, through a large error motion, very small or very large transmission angle outcomes undesirable noisy mechanism as mentioned in [4]. Unfortunately, optimizing the mechanism linkages with considering their force transmission characteristics simultaneously with synthesizing the transmission angle is a complicated problem. On the other hand, this optimizing problem can be considerably simplified if the transmission Khaled Khader is member of the teaching staff with the Department of Production Engineering & Mechanical Design, Faculty of Engineering, University of Menoufia, Shebin El-kom, Menoufia, Egypt. (corresponding author's phone: ; fax ; khkhm6@hotmail.com angle range is restricted to be bounded by a two specific limits as in [5]. Clearly, the maximum and minimum transmission angles are highly depending on the rocker angle. Moreover, the reasonable deviation of the transmission angle value from 90 o is significant for providing a good characteristics to mechanism motion and can guarantee a smooth motion without undesirable vibrations at high speed as in [3], [6]. On the other hand, the large transmission angle variation plays an effective role in reducing the effectiveness of force transmission where mechanisms having a wide deviation ranges of transmission angle from 90 o, reveal undesired operational characteristics at high speed like noise and jerk as in [], [7]. The mechanisms designers are always looking for designing mechanisms with a suitable linkages which have transmission angle range is as small as possible, especially in the existence of significant joint friction as in [1], [3]. Synthesis of four bar mechanism for polynomial function generation dealing with maximum and minimum transmission angles is presented in [8]. In addition to, an analytical synthesis of crank rocker mechanisms dealing with the motion between two small separated positions of minimum and maximum transmission angles is presented in [9]. Also, a shape optimization for path synthesis of crank rocker mechanisms dealing with optimal transmission angle is presented in [10]. Moreover, synthesis of planar four bar mechanism is presented in [11] for exact three coupler positions which can be attained with transmission angles less than 18 % of the optimum value of 90 o. As well as, [1] presented a new method of designing four bar function generators with a good effective transmission angle which can be considered an optimum angle, in a minimax sense, when their extreme values have variations are equally around 90 o. Solved example for analytical optimizing four bar transmission angle is presented in [13] through supposing the dimension of coupler link in addition to dimensions ranges of fixed, crank and rocker links. Furthermore, transmission angle range 30 o -150 o is recommended in [14]. As well as, some different transmission angle ranges are suggested like 35 o -145 o ; 40 o -140 o ; 45 o -135 o as in [6]. A few publications are dealing with selecting the transmission angle of crank rocker mechanisms using the graphical techniques. The minimum transmission angle can be selected from design charts depending on the value of angle 155
2 International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME Volume 3, Issue 3 (015 ISSN (Online between two dead-center positions of the rocker and the corresponding angle of the crank in addition to assuming three linkages dimensions as in [15], [16]. Hence, the design nomograms which is presented in this paper can be consider as a fast tool for directly selecting the suitable ratios of mechanism links lengths which give a definite synthesized transmission angle range. II. TRANSMISSION ANGLE The transmission angle (μ of a planar crank rocker mechanism (O 1 AB O is indicated in Fig. 1 as Fig. 1 Transmission Angle Where; fixed, crank, coupler and rocker links lengths are r 1, r, r 3 and r 4, respectively. Furthermore, the transmission angle ( μ of the crank rocker mechanism can be formulated as 1 r4 (r1 r r1r cos cos (1 r4 III. MINIMUM AND MAXIMUM TRANSMISSION ANGLE Maximum transmission angle (μ max and the minimum transmission angle (μ min of a planar crank rocker mechanism is shown in Fig. as 1 r4 (r1 - r min cos ( r4 1 r4 (r1 r max cos (3 r 4 IV. THEORETICAL FORMULATIONS OF DESIRED NOMOGRAMS Clearly, transmission angle can be considered as the optimal one when this angle has maximum and minimum values which have variations equally around 90 o as in [1]. Hence, the desired design nomograms (charts of optimal transmission angle are using the meaning of μ min =(π/-δ and μ max =(π/+δ, where δ can be assumed related to the designer requirements. Thus, the summation of μ min and μ max is equal to (π. Hence, cosines of μ min and μ max can be written as cos min cos max 0 (4 Also, the pervious equation can be rewritten as r4 (r1 - r r4 (r1 r 0 (5 r4 r4 As well as, the pervious equation can be simplified as r r4 r1 (6 Clearly, using linkages lengths ratios instead of using linkages lengths can simplify the designers work. Thus, the pervious equation can be rewritten as R R3 R4 1 (7 Where, ratios R, R 3 and R 4 can be written as R r / r1, R3 / r1 and R 4 r4 / r1 (8 The desired charts (nomogrames will depend on assuming only two values. The first assumed value is the desired rocker angle (ϕ I which occurs at the minimum transmission angle and the second one is the desired transmission angle range through supposing value of (δ. These two angles (ϕ I & δ in addition to coupler angle (ɵ I are indicated in Fig. 3 which shows the rocker angle (ϕ I at the minimum transmission angle μ min as Fig. Maximum and Minimum Transmission Angle Orientation Maximum value of transmission angle (μ max occurs at 180 o of crank angle while minimum value of transmission angle (μ min occurs at 0 o of crank angle as in [8]. Hence, μ min and μ max can be formulated as in [8], [17] in two equations as Fig. 3 Minimum Transmission Angle Geometry The triangle sine rule can be applied for the triangle ABO which is indicated in Fig. 3 as in [18]. Hence, the three fraction equalities can be shown as 156
3 International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME Volume 3, Issue 3 (015 ISSN (Online r1 r r4 (9 sin min sin ( I sin I The pervious equation can be rewritten using the linkages ratios as 1 R R3 R 4 (10 sin min sin ( I sin I Hence, the pervious equation can be written again as 1 R R3 R 4 (11 sin min sin I sin ( I min Where, sin (π - ϕ I = sin ϕ I and ɵ I = ϕ I - μ min. Thus, a new equation can be written form the pervious equation as sin R 3 ( I (1 R sin (1 min Also, the pervious equation can be rewritten as sin R3 A (1 R where A I (13 sin min As well as, a new equation can be written form Eq. (11 as sin ( min R 4 I (1 R (14 sin min Also, the pervious equation can be rewritten as sin ( I min R 4 B (1 R where B (15 sin min Both R 3 and R 4 which appeared in (7 can be substituted with another forms which appeared in (13 and (15 respectively. Equation (7 can be written in new form as R A (1 R B (1 R 1 (16 The pervious equation of R can rewritten as ( A B 1R ( A B R ( A B 1 0 (17 Thus; A B A B 1 R (18 A B 1 A B 1 It is well known that, the crank is shorter than the fixed link of the crank rocker mechanism, means that R less than one. Thus, the plus sign of square root in (18 can be rejected because of the value (A +B /(A +B -1 is greater than one. Hence, the pervious equation can be reduced to be; A B A B 1 R (19 A B 1 A B 1 Now, the value R can be calculated using the pervious equation. Also, the values R 3 and R 4 can be calculated using (13 and (15 respectively. These three values R, R 3 and R 4 are only depending on the two values (A & B which are only depending on the values of (ϕ I & μ min as sin I sin ( I min A and B (0 sin min sin min Furthermore, these three ratios R, R 3 and R 4 are only the functions of (ϕ I and (δ. Hence, design charts (nomograms can be drawn easily for synthesizing the linkages ratios (R, R 3 and R 4 of crank rocker mechanism through assuming the value of (δ which determine the minimum transmission angle (μ min and assuming the value of rocker angle (ϕ I which corresponds (μ min. It is well known that, the time ratio (T r is considered as an important criteria through designing the crank rocker mechanism. Also, satisfying the condition (μ min +μ max = π can make the time ratio is unity through designing the crank rocker mechanism as in [19]. The time ratio (T r can be formulated as follow; 1 Tr (1 360 ( 1 Where; 1 1 (R R 3 R 4 cos ( 1 (R R 3 and; 1 1 (R 3 R R 4 cos (3 (R 3 R Rocker angle (ϕ II corresponds (μ max can be formulated as; 1 (1 R R 4 R 3 cos (4 II (1 R R 4 Clearly, the rocker angle (ϕ, rocker angular velocity ( ϕ ' and rocker angular acceleration ( ϕ " are important factors of mechanism performance. Solving the equation of rocker angle (ϕ which is a function of (ψ, r 1, r, r 3 and r 4 is presented as two possible solutions in [0] which can be reformulated as F0 F1 cos F sin (5 Where; F 0 1 R R 4 R3 R cos (6 F1 R 4 (1- R cos F R R sin 4 Then, the sin (ϕ and cos (ϕ can be formulated as F 0F F 1 F1 F F0 sin F1 F F 1 F (7 F F1 F F0 F 0F1 cos F 1 F F 1 F Thus, the rocker angle (ϕ can be calculated through comparing the values and signs of sin (ϕ and cos (ϕ which are shown in the previous equation. As well as, rocker angular velocity ( ϕ ' can be found by derivation of (5. V. NOMOGRAMS (CHARTS FOR SYNTHESIZING OF CRANK ROCKER MECHANISM It is easy now to draw three nomograms (charts as an easy tools in order to help the mechanisms designers where they can directly select the suitable mechanism links ratios (R, R 3 and R 4 which give them a definite synthesized transmission angle 157
4 International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME Volume 3, Issue 3 (015 ISSN (Online range. These charts can be drawn through assuming different values of rocker angle (ϕ I and assuming different values of (δ which give corresponding different values of minimum transmission angles (μ min. Hence, substituting the assumed values (ϕ I & μ min in (13, (15 and (19 in order to calculate the linkages ratios R, R 3 and R 4. The relation between (R and the rocker angle (ϕ I which corresponds to the desired minimum transmission angle (μ min is shown in Fig. 4 through different curves of different values of (δ. Fig. 6 Relation between R 4 and rocker angle ϕ I which corresponds to minimum transmission angle μ min Fig. 4 Relation between R and the rocker angle ϕ I which corresponds to minimum transmission angle μ min Furthermore, a compact chart which is shown in Fig. 7 presents the relation between (R, R 3 & R 4 and the rocker angle (ϕ I which corresponds to desired minimum transmission angle (μ min through different curves of different values of (δ. These charts can be used through drawing a vertical line from the assumed rocker angle (ϕ I to intersect the curve of assumed value of (δ, hence drawing a horizontal line till the desired linkages ratios. As well as, the relation between (R 3 and the rocker angle (ϕ I which corresponds to the desired minimum transmission angle (μ min is shown in Fig. 5 through different curves of different values of (δ. Fig. 7 Relation between (R, R 3 & R 4 and rocker angle ϕ I of μ min Fig. 5 Relation between R 3 and rocker angle ϕ I which corresponds to minimum transmission angle μ min Also, the relation between (R 4 and the rocker angle (ϕ I which corresponds to the desired minimum transmission angle (μ min is shown in Fig. 6 through different curves of different values of (δ. VI. RESULTS AND DISCUSSIONS As example of using the pervious nomogrames for synthesizing linkages ratios (R, R 3, and R 4 of crank rocker mechanism or using (13, (15 and (19 for calculating the accurate values of linkages ratios. Assume a mechanism designer select the desired rocker angle (ϕ I =90 o which corresponds to desired minimum transmission angle (μ min = 60 o and he also select the desired value of (δ= 30 o. Hence, the ratios (R, R 3, and R 4 can be 158
5 International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME Volume 3, Issue 3 (015 ISSN (Online determined as; R =0.086, R 3 = and R 4 = By substituting the pervious values of (R, R 3, and R 4 in ( and (3, the calculated values of minimum and maximum transmission angles are; (μ min = 60 o and (μ max = 10 o which are the same as the assumed desired values. As well as, results of six different mechanism are listed in Table I. Table I Results of six different mechanism The transmission angle (μ and the rocker angle (ϕ can be calculated using (1, (6 and (7 for each crank angle (ψ. The relation between (ψ and both (μ & ϕ is shown in Fig. 8. As well as, relation between (ψ and rocker angular velocity ratio is shown in Fig. 9 where the rocker angular velocity ratio equals to the rocker angular velocity (ω r divided by crank angular velocity (ω c. The curve in Fig. 9 of rocker angular velocity ratio indicates a symmetric values of this ratio above and down zero axis where the mechanism time ratio is unity. Nb. ϕ I δ μ min R R 3 R Nb. ϕ II ϕ max μ max Time ratio ψ 1 ψ Also, by substituting the values of (R, R 3, and R 4 in (1, (, (3 and (4 for calculating the values of (ψ 1, ψ, T r and ϕ II. The calculated values are; ψ 1 =3.98 o, ψ =03.98 o, T r =1 and ϕ II =139.1 o. Fig. 9 Relation between (ψ and rocker angular velocity ratio VII. CONCLUSION The transmission angle is a significant criterion of mechanisms design where transmission angle value which equal to 90 o provides the best characteristics to the mechanism motion, while this motion becomes very hard with values nearly 0 o or 180 o. The nomograms which presented here are dealing with a good effective transmission angle which can be considered an optimum angle, where its extreme values have variations equally around 90 o. These nomograms (charts are only depending on the value of desired minimum transmission angle (μ min and the value of the desired rocker angle (ϕ I which corresponds to the desired minimum transmission angle. Obviously, the presented design nomograms can be considered as an effective fast tool which helps the designers for directly selection of suitable mechanism links ratios giving a definite synthesized transmission angle range. Fig. 8 Relation between (ψ and both (μ & ϕ REFERENCES [1] A. Hall, in: Kinematic and Linkage Design, Prentice-Hall, Englewood Cliffs, NJ,, 1961, pp. 41. [] R. Hartenberg and J. Denavit, in: Kinematic Synthesis of Linkages, McGraw-Hill, New York, 1964, pp [3] S. Balli and S. Chand, Transmission Angle in Mechanisms-Triangle in Mech., Mechanism and Machine Theory, vol. 37, no., pp , 00. [4] J. Kimberlla, in: Kinematic Analysis and Synthesis, McGraw-Hill, New York, 1991, pp [5] R. Soylu, Analytical Synthesis of Mechanisms Part-1, Mechanism and Machine Theory, vol. 8, no. 6, pp , [6] P. Eschenbach and D. Tesar, Link Length Bounds on the Four Bar Chain, Journal of Engineering for Industry Trans. ASME, vol. 93, no. 1, pp ,
6 International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME Volume 3, Issue 3 (015 ISSN (Online [7] D. Tao, in: Applied Linkage Synthesis, Addison-Wesley, Reading, MA, 1964, pp [8] T. Patal, Synthesis of Four Bar Mechanism for Polynomial Function Generation by Complex Algebra, in National Conference in Recent Trends in Engineering & Technology, B.V.M Engineering Collage, Nagar Gujarat INDIA, May 011, pp [9] P. Rao, Kinematic Synthesis of Variable Crank-rocker and Drag linkage planar type Five-Bar Mechanisms with Transmission Angle Control, International Journal of Engineering Research and Application, vol. 3, no. 1, pp , 013. [10] G. Marín, F. Alonso and M. Castillio, Shape Optimization for Path Synthesis of Crank-Rocker Mechanisms Using a Wavelet-Based Neural Network, Mechanism and Machine Theory, vol. 44, no. 6, pp , 009. [11] G. Hassaan, Synthesis of Planar Mechanisms, Part III: Four-Bar Mechanisms for Three Coupler-Positions Generation, Global Journal of Advanced Research, vol., no. 4, pp , 015. [1] K. Gupta, Design of Four-Bar Function Generators with Mini-Max Transmission Angle, Journal of Engineering for Industry Trans. ASME, vol. 99, no., pp , [13] S. Jian, Analysis and Optimization of the Transmission Angle of Crank Rocker Mechanism, in International Conference in Mechatronic, Electronic, Industrial and Control Engineering MEIC, Shenyang, China, April 015, pp [14] J. Buśkiewicz, A Specific Problem of Mechanism Synthesis, International Journal of Applied Mechanics and Engineering, vol. 19, no. 3, pp , 014. [15] P. Gensen, Classical and Modern Mechanisms for Engineers and Inventors, CRC Press, USA, 1991, pp [16] A. Mallik, A. Ghosh and G. Dittrich, Kinematic Analysis and Synthesis of Mechanism, CRC Press, USA, 1994, pp. 6. [17] S. Matekar, and G. Gogate, Optimum Synthesis of Path Generating Four-Bar Mechanism Using Differential Evaluation and Modified Error Function, Mechanism and Machine Theory, vol. 5, pp , 01. [18] N. Ncees, Fundamentals Of Engineering Supplied-Reference Handbook, Seventh Edition, the National Council of Examiners for Engineering and Surveying, USA, 005, pp. 5. [19] P. Roth and F. Ferudenstain, Synthesis of a Four Bar Link Path Generating Mechanisms with Optimum Transmission Characteristics, in Transaction of Seventh Conference of Mechanism, Purdue University, Lafayette, Indiana, USA, pp , 196. [0] K. Feurenstine, Approximation Synthesis of the Four-Bar Linkage, Transaction. ASME, vol. 77, no. 6, pp ,
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