Synthesis of Planer Eight Bar Mechanism for Function and Path Generation

Transcription

1 GRD Journals- Global Research and Development Journal for Engineering Volume 1 Issue 12 November 216 ISSN: Synthesis of Planer Eight Bar Mechanism for Function and Path Generation Ramanagouda C. Biradar Assistant Professor Department of Mechanical Engineering WIT, Solapur Deepak M. Kalai Assistant Professor Department of Mechanical Engineering DKTE, Ichalakaranji Rahul B. Patil Assistant Professor Department of Mechanical Engineering WIT, Solapur Kashinath I. Swami Assistant Professor Department of Mechanical Engineering WIT, Solapur Sureshbabu P. Mankani Assistant Professor Department of Mechanical Engineering WIT, Solapur Abstract Linkages have an advantage of easily modifying their output by suitable adjustments compared to cams and geared mechanisms. But at the same time it is usually difficult to change the dimensions of mechanisms every time to do different tasks. Also we cannot synthesize exactly the necessary mechanism because of limitation of maximum precision points. This problem is addressed by using adjustable Crank-Rocker linkages that are capable of generating multiple paths with a simple adjustment of the length of the rocker guider. Little work has been done in the area of synthesis of adjustable four-bar and six bar linkages for function and continuous path generation, especially of adjustable crank-rocker linkages. The present work is an attempt to synthesize adjustable eight bar mechanism to overcome this difficulty. The link lengths are computed using Cheybychev s spacing and Fruedenstein s equations. A procedure is developed to compute the values of design variable which satisfies the input output relationship. Analytical and programmable method is developed in Matlab to generate desired function and path. Keywords- Planar eight bar mechanism, Adjustable link, Fruedensteins method, Chebychiv spacing I. INTRODUCTION The important and complementary area called kinematics synthesis, where mechanism is created to meet certain motion specification is touched upon only by a consideration of the simple aspect of planar linkage synthesis. Linkages include garage door mechanisms, car wiper mechanisms, gear shift mechanisms. They are a very important part of mechanical engineering which is given very little attention. Maximum application of mechanisms is found in mechanical systems. Although there are many phases in the design process which can be approached in a well ordered, scientific manner, the overall process is by its very nature as much an art as a science. It calls for imagination, intuition, creativity, judgment, and experience. Types of Synthesis Problems. Function Generation It is defined as the correlation of an input motion with an output motion in a mechanism. A function generator is conceptually a black box that delivers some predictable output in response to a known input. Example~ of function generation includes the automobile accelerator, the control stick in an aircraft, and the piston in an engine mechanism. Path Generation defined as the control of a point in the plane such that it follows some prescribed path. This is typically achieved with at least four bars, wherein a point in the coupler traces the desired path. No attempt is made in path generation to control the orientation of the link that contains the point of interest. However, it is common for the timing of the arrival of the point at particular locations along the path to be defined. This case is called path generation with prescribed timing and is analogous to function generation in that a particular output function is specified. Motion Generation (Rigid-body Guidance) defines locations through which the tracer point passes and the corresponding orientation of the coupler link at those rocations. Examples of motion generation include the power lift, gate on a truck, the lift mechanism on a dumpster truck, and the windshield wipers on an automobile.[1] Most engineering design practice involves a combination of synthesis and analysis. However, one cannot analyze anything until it is synthesized into existence.[3] The optimal synthesis model is set based on the positional structural error of the All rights reserved by 64

2 slider guider introduced, which can effectively reflect overall difference between the desired and generated paths avoid the difficulty of selecting corresponding comparison points on the two paths and can be calculated conveniently.[4] A. Crank Rocker Mechanism Fig. 1: The crank and rocker four bar mechanism. The four bar linkage illustrate in fig1 is called crank-and-rocker mechanism. Thus, link2, which is the crank, can rotate through a full circle, but the rocker, link4, can oscillate. In general, we shall follow the commonly accepted practice of designated the frame or fixed link as a link1.link3 in fig1.13 is called the coupler or the connecting rod.with the four-bar linkage, the position problem generally consists of finding the directions of the coupler link and the output link. Or rocker, when the link dimensions and the input crank position are known. II. SYNTHESIS OF ADJUSTABLE MECHANISM Fig. 2: Adjustable Mechanism Adjustable mechanisms allow design of machines which can perform several duties without costly alterations. The adjustments are performed by changing the distance between two adjacent joints. Although adjustable mechanisms find many applications, studies to develop synthesis procedures for these mechanisms are not as numerous as those on other devices. Although adjustable linkage mechanisms are multidegree freedom systems, the application of theories of multidegree freedom systems to adjustable mechanisms is sparse. A technique to synthesize a four bar adjustable function generator operating in three phases to produce two specified functions is presented here and is based on eight bar linkage theory. A maximum of three precision points for each function have been selected in the illustrative example. The adjustment is attained by changing one link length of the four bar mechanism in first phase. All rights reserved by 65

3 III. METHODOLOGY The synthesis of 8-Bar mechanism involves different methodology and different equations. This work started with Chebychev spacing method which gives precision points. Using Precision points in Fruedenstein s equations link lengths are determined. Here 8-bar mechanism is divided into three phases of four bar mechanism, phase-1 is four bar mechanism (L1 L2 L3 L4), phase- 2 one more four bar linkage (L1 L4 L5 L6) and phase-3 is crank-rocker mechanism (L1 L6 L7 L8). Fig. 3: Eight bar mechanism with rocker as output IV. RESULTS AND DISCUSSION A. Function Generation Considering a function y sin(x), x 9, y 1, and then derive b response to this function. Using Cheybychev s spacing method select three positions of the mechanism by using input/output angles and their ranges to decide one value of the three design variables. Using first position values input/output angles are calculated which are necessary to find the value of design variation variable in b fallowing ranges. Initial angle 5, 3 6, 9, Final angle f =11, f =12 and Range, The link lengths are determined by using Fruedenstein s equation which is as follows. In the third phase of the four bar linkage the input angle swings with 9. Hence response of Design variation variable ( b) with input angle ( ) for phase-3 is as fallows. The design variation variable b is given by b b a 2 B 2sin Where, 6 K1 ( 1 K2) cos 6 28 tan 2 K3 1 tan 1 K2 6 K1 cos acK 3 c d 8 B tan 2 (1) (2) (3) (4) All rights reserved by 66

4 Table 1: Sample readings of design variation variable ( B) corresponding to input angle ( ) SL.NO. INPUT ANGLE ( ) DESIGN VARIABLE( B) Fig. 4: the B response corresponding to input angle (θ16) B. Calculation of Precision Points According to Cheybychev s spacing, X j = (5) Where, n = 3 for three position Ss = starting displacement of slider Sf = final displacement of slider For j=1,2,3 in equation (5) three precision points are calculated then using these precision point corresponding angles θ1,θ2,θ3 are determined using fallowing equation. Since it is given that the entire problem is divided into three segments, each represented by unique function. For the problem fallows the equation (6) For 12 For 24 ( ) (7) the problem fallows the equation (8) the problem fallows the equation ( ) (9) The three positions of slider corresponding to three positions of input angles are calculated. All rights reserved by 67

5 Table 2: Input and output angles for phase-iii of four bar mechanism SL.NO. INPUT ANGLE OUTPUT ANGLE Synthesis of Planer Eight Bar Mechanism for Function and Path Generation C. Path Generation The path or coupler curve of the eight bar mechanism is calculated by analytical equations and plotted using MATLAB software. The coupler curves are plotted for the important cases considering exact length, max length and min length of the coupler. The equations are as follows. In four bar mechanism, the phase-1 variables are r1= (L 1 ) = (d) and r2= (L 2 ) = (a) and r3= (L 3 ) = (b) and r4= (L 4 ) = (c). From above figure λ, β, f, γ are given by For finding λ, λ = β + γ + α Where α =, 6, 12, For finding β, f = ( For finding γ, from figure. { } { } Co-ordination of coupler point D is given by Xc = r 2 cos ϕ + e cos λ Yc = r 2 sin ϕ + e sin λ Where Xc and Yc are the co-ordinations of coupler point 1/ 2 Fig. 5: Four-bar mechanism for path generation { { } } { } (1) All rights reserved by 68

6 Fig. 6: Coupler curve for different angles Fig. 7: Coupler curve with EXACT length mm Fig. 8: Coupler curve with MAX length mm All rights reserved by 69

7 Fig. 9: Coupler curve with MIN length mm D. Motion Generation The motion or coupler curve of the eight bar mechanism is calculated by analytical equations and plotted using MATLAB software. The coupler curves are plotted for the important cases considering exact length, max length and min length of the coupler. Fig. 1: Four bar linkage for motion Fig. 4. Shows four bar linkage in one general position with a coupler point located at its first precision P 1. Let r 1, r 2, r 3, and r 4 are the ground, crank, coupler, and follower links respectively. The positions P2 and P3 are precision positions of coupler curve. These positions are achieved by rotating the crank and input rocker link, through ϕ 2 and ϕ 3 which by using the synthesis procedure. Consider a global co-ordinate system X and Y at the first precision point P 1. Let, define the other two desired positions P 2 and P3 with respect to global co-ordinate system. The position difference vectors P 21 =P 1 -P 2, P 31 =P 1 -P 3 and P 41 =P 1 -P 4 have an angles δ 2, δ 3 and δ 4 respectively. The above four position difference vectors define the displacements of the output motion point P from P 1 to P 2, P 1 to P 3 and P 1 to P 4 respectively as shown in figure. In the first case consider left half dyad for 3 positions to solve for r 21 and Z 1. And the same is extended for four and five positions to solve r 21 and Z 1. First vector loop equation for motion from P 1 and P 2 is given by, r 22 + Z 2 P 21 Z 1 r 21 =, (11) Second loop for motion from P 1 to P 3 is given by, All rights reserved by 7

8 r 23 +Z 3 P 31 Z 1 r 21 =, (12) Substituting complex number equivalents for vectors then r 2 e j(θ 2 + ϕ 2 ) + ze j(ϕ + α 2 ) P 21 e jδ 2 ze jϕ r 2 e jθ 2 = (13) r 2 e j(θ 2 + ϕ 3 ) + ze j(ϕ + α 3 ) P 31 e jδ 3 ze jϕ r 2 e jθ 2 = (14) After expanding and simplifying r 2 e jθ 2(e jϕ 2 1)+ze jϕ (e jα 2 1)=P 21 e jδ 2 (15) r 2 e jθ 2 (e jϕ 3 1) + ze jϕ (e jα 3 1) = P 21 e jδ 3 (16) Resultant r2 of vectors r2, r22 and r23 is the same in all three positions, since it represents rigid link. The same can be said for the resultant z of vectors Z1, Z2 and Z3. Equations (15) and (16) are vector equations \each contain two scalar equations. By considering Euler s identity separate real and imaginary terms. Real part: r 2 cos θ 2 (cosϕ 2 1) r 2 sinθ 2 sinϕ 2 + z cosϕ(cosα 2 1) z sinϕ sinα 2 = P 21 cosδ 2 (17) r 2 cos θ 2 (cosϕ 3 1) r 2 sinθ 2 sinϕ 3 + z cosϕ(cosα 3 1) z sinϕ sinα 3 = P 21 cosδ 3 (18) Imaginary part: r 2 sin θ 2 (cosϕ 2 1) r 2 cosθ 2 sinϕ 2 + z sinϕ(cosα 2 1) z cosϕ sinα 2 = P 21 sinδ 2 (19) r 2 sin θ 2 (cosϕ 3 1) r 2 cosθ 2 sinϕ 3 + z sinϕ(cosα 3 1) z cosϕ sinα 3 = P 21 sinδ 3 (2) For simplification purpose X and Y components of two unknowns r2 and Z are substituted in Equation (17) and (18). X and Y components of r2 and Z are r 21 x = r 2 cosθ 2 ; r 21 y = r 2 sinθ 2 ; Z 1 x = Zcosϕ; Z 1 y = Zsinϕ; r 21x (cosϕ 2 1) r 21y sinϕ 2 + Z 1x (cosα 2 1) Z 1y sinα 2 = P 21 cosδ 2, (21) A=cosϕ2-1 B=sinϕ2 C=cos2 1 D=sinα2 E=cos3 1 F=sinϕ3 G = cosα3 1 H= sinα3 I = cosϕ4 1 J = sinϕ4 K= cosα4 1 L= sinα4 M=cosϕ5 1 N= sinα4 O = cosα2 1 R = sinα5 A1= cosϕ2 1 B1=sinϕ2 C1= cosα21 D1=sinα2 E1=cosϕ3 1 F1= sinϕ3 G1=cos3 1 H1=sinα3 I1= osϕ4 1 J1=sin4 K1=cosα4 1 L1=sinα4, M1=cosϕ5 1 N1=sinϕ5 O1=cosα5 1 R1=sinα5 P2=P21 cosδ2 P3=31cosδ3 P4=P41cosδ4 P6=P21sinδ2, P7=P31sinδ3 P8=P41sinδ4 P9=P51sinδ5 P5=P51cosδ5 r 21x (cosϕ 2 1) r 21y sinϕ 3 + Z 1x (cosα 3 1) Z 1y sinα 3 = P 31 cosδ 3, (22) r 21y (cosϕ 2 1) + r 21x sinϕ 2 + Z 1y (cosα 2 1) + Z 1x sinα 2 = P 21 cosδ 2, (23) r 21y (cosϕ 3 1) + r 21x sinϕ 3 + Z 1y (cosα 3 1) + Z 1x sinα 3 = P 31 cosδ 2, (24) The equations (21) and (24) are solved for r 21x, r 21y, Z 1x and Z 1y.By using appropriate assumed values of ϕ 2 and ϕ 3 (angular rotations of link 2), for the left dyad. The same steps are to be used the right dyad r 4 S. For which the vector loop equations are, r 42 + S 2 P 21 S 1 r 41 =, r 43 + S 3 P 31 S 1 r 31 =, Next solving the procedure for real part and imaginary part is same. Here we should consider the r 4 and S for right dyad By setting co-efficient of all equations which contain the assumed and defined terms equal to some constants. For three coupler positions: The following matrix is to be solved to determine L21x, L21y, Z1x, and Z1y. For left Dyad The above matrix is to be solved to determine L41x, L41y, S1x, and S1y. ( r, ) variables, so it could finish any planar synthesis by controlling the variable r For right Dyad. All rights reserved by 71

9 Fig. 11: Motion generation with Exact length mm Fig. 12: Motion generation with MAX length mm Fig. 13: Motion generation with MIN length mm All rights reserved by 72

10 V. CONCLUSION Here the output of the eight bar mechanism is taken as rocker link. The function, motion and path for phase-i and phase-2 are generated by considering four bar and six bar mechanism separately then by considering eight bar mechanism function and path are generated to satisfy input and output relations. In the present work, an analytical and programmed method is developed in MATLAB to synthesize planar eight bar linkage mechanism for function and path generation by adjusting the link length. From the results, it is clear that the rate of variation of adjustable link is reasonable and is a function of link length and input output relationship. This can be controlled well by changing link length and input output relationship. It provides the engineer wider design space to choose design variables with precise time. REFERENCES [1] Mekonnen Gebreslasie and Alem Bazezew: Synthesis, Analysis And Simulation Of A Four-Bar Mechanism Using Matlab Programming Journal of EAEA, Vol. 18,21. [2] Arthur G. Erdman, George N. Sander, Mechanism Design: Analysis and synthesis, volume-1, and volume-2 Prentice Hall of India Private Limited, New Delhi, (1988). [3] Joseph Edward Shigley, John Joseph Uicker, JR., Theory of machines and mechanisms, second edition, McGraw-Hill, New York, (1995) [4] Robert L. Norton, Design of Machinery, An Introduction to the Synthesis and Analysis of Mechanisms and Machines, third edition, Tata McGraw-Hill Publishing Company limited, New Delhi. [5] K.J.Waldron and G. L.Kinzel, MATLAB PROGRAMS for Textbook Kinematics, Dynamics, and Design of Machinery. [6] Rudra Pratap: Getting started with MAT LAB 7, OXFORD University press, (28). [7] Y.Kirani Singh, B.B.Chaudhuri: MATLAB Programming: PHI learning pvt Ltd, Delhi, (29). [8] H.Zhou, Edmund H.M.Cheung, Analysis and optimal synthesis of adjustable linkages for path generation Mechatronics 12 (22) [9] Qiong Shen, Yahia M. Al-Smadi, Peter J. Martin, Kevin Russell, Raj S. Sodhi, An extension of mechanism design optimization for motion generation Mechanism and Machine Theory 44 (29) All rights reserved by 73