Synthesis of Mechanism

Synthesis of Mechanism
Theory of Machine
Introduction of synthesis
Types of synthesis
Synthesis of a four bar chain
Freudenstein’s equation for four bar mechanism
Precision point for function generator
(Chebychev spacing method)
Bloch method

1. Theory Of Machine Lecture 1-2-3 New chapter: Synthesis of Mechanisms Subtopics: Introduction of synthesis, Types of synthesis, Synthesis of a four bar chain, Freudenstein’s equation for four bar mechanism, Precision point for function generator ( Chebychev spacing method), Bloch method .

2. INTRODUCTION: Kinematic synthesis determines the size and configuration of mechanisms Once this is defined, then material selection, stress analysis and dynamic analysis are used to shape the components to ensure performance and reliability.

3. Types of synthesis: Kinematic synthesis for a mechanical system is described as having three general phases , known as  1. Type synthesis, 2. Number synthesis and 3. Dimensional synthesis

4. Type synthesis:   It is the process of determining possible mechanism structures to perform a given task or combination of tasks without regard to the dimensions of the components. Choosing the type of mechanism to perform the required function

5. Example of type synthesis: To transmit power Selection of any one drive out of a rope drive, a belt-pulley transmission or a gear train Given a task to be produced by a mechanism, find the type that will best perform it, e.g., a linkage, a cam mechanism, a gear train, or a combination of these.

6. Number synthesis : The  number of links and the number of joints needed to produce the required motion are calculated.

7. Dimensional synthesis : The proportions or lengths of the links, or angles, necessary to satisfy the required motion characteristics are found out. The main objective here is to find the dimensions defining the geometry of the various links and joints of the kinematic chain Given a task to be produced by a mechanism, find its relevant geometric parameters.

8. Dimensioning involves two phases : Functional dimensioning includes the determination of the fundamental dimensions of the machine parts, prior to the shaping of all its parts. It is the functional dimensioning where kinematic synthesis plays a major role . 2. Mechanical dimensioning pertains to the dimensioning of the machine elements for stress, strength, heat capacity, and dynamic-response requirements

9. Synthesis is the procedure by which identification of the specific mechanism ( Type synthesis ), and appropriate dimensions of the linkages are identified ( Dimensional synthesis ) to get desired motion.

10. Kinematic Analysis vs. Kinematic Synthesis The fundamental problems in mechanism kinematics can be broadly classified into: (a) Analysis : For Given linkage find the motion of its links, for a prescribed motion of its input joint(s). (b) Synthesis: For Given a task find the linkage that best performs the task.

11. The task at hand can be one of three, in this context: (a) Function generation: the motion of the output joint(s) is prescribed as a function of the motion of the input joint(s); (b) Motion generation (rigid-body guidance): the motion of the output link(s) is prescribed in terms of the motion of the input link(s) or joint(s); (c) Path generation: the path traced by a point on a floating link—a link not anchored to the mechanism frame—is prescribed as a curve, possibly timed with the motion of the input joint(s).

12. Two types of dimensional synthesis: 1. Exact synthesis : Number of linkage parameters available is sufficient to produce exactly the prescribed motion . Problem leads to—linear or, most frequently, nonlinear—equation solving . 2. Approximate synthesis : Number of linkage parameters available is not sufficient to produce exactly the prescribed motion . Optimum dimensions are sought that approximate the prescribed motion with the minimum error . Problem leads to mathematical programming (optimization)

13. In-line slider crank mechanism Approximate synthesis Design problem: Determine the appropriate lengths L2 and L3 of the crank and coupler respectively to achieve the desired stroke

14. From the kinematic diagram in the figure we conclude that the center of the crank rotation is on the constrained path of the slider and that can take on any value since it does not affect the stroke, however a shorter connecting arm yields greater velocities and accelerations for the slider, therefore its length should be as large as possible

15. Mechanism to move a link between two positions Some applications require a link to move between two fixed positions, when these positions are specified the design problem is termed two point synthesis, it can be achieved using a single pivot or by using the coupler in a four bar mechanism.

16. Two point synthesis using a pivot Design problem: Identify the location of the pivot and angle of rotation given the initial and final locations of two points on the link

17. a. Geometrical Synthesis: 1. Draw the lines that connect the initial and final locations of each point 2. Construct the perpendicular bisectors s to these lines 3. The intersection is the location of the pivot

18. 4. The angle between the lines connecting the pivot and the end locations of one of the points is the angle of rotation.

19. Two point synthesis using a four bar mechanism Design problem: Identify the location of the pivots and the lengths of all four links Geometrical Synthesis:

20. 1. Draw the lines that connect the initial and final locations of each point 2. Construct the perpendicular bisectors s to these lines 3. The pivots can be placed anywhere on the perpendicular bisectors

21. 4. The lengths can then be measured graphically Note that longer pivoting lengths will rotate at smaller angles, this produces larger transmission angles and reduces the forces required to drive the linkage.

22. Mechanism to move a link between three positions Some applications require a link to move between three fixed positions, when these positions are specified the design problem is termed three point synthesis and can be achieved using the coupler in a four bar mechanism. Design problem: Identify the location of the pivots and the lengths of all four links

23. Geometrical Synthesis: 1. Draw the lines that connect the initial and final locations of each point 2. Construct the perpendicular bisectors to these lines 3. The intersections of two of the perpendicular bisectors are the pivot locations.

24. 4. The lengths can then be measured graphically

25. The  synthesis , or design, of  four - bar  mechanisms is important when aiming to produce a desired output motion for a specific input motion

26. By kinematic synthesis we mean the design or creation of a mechanism to attain specific motion characteristics. In this sense, Synthesis is the very essence of design because it represents the creation of new hardware to meet particular requirements of motion: displacement; velocity; acceleration; individually or in combination.

27. Classifications of Synthesis Problem The problems in synthesis can be placed in one of the following three categories : Function generation Path generation Body guidance These are discussed as follows : Function generation In designing a mechanism, the frequent requirement is that the output link should either rotate, oscillate or reciprocate according to a specified function of time or function of the motion of input link. This is known as function generation.

28. A simple example is that of designing a four bar mechanism to generate the function y = f (x). In this case, x represents the motion of the input link and the mechanism is to be designed so that the motion of the output link approximates the function y. Note : The common mechanism used for function generation is that of a cam and a follower in which the angular displacement of the follower is specified as a function of the angle of rotation of the cam. The synthesis problem is to find the shape of the cam surface for the given follower displacements.

29. 2. Path generation In a path generation, the mechanism is required to guide a point (called a tracer point or coupler point) along a path having a prescribed shape. The common requirements are that a portion of the path be a circular arc, elliptical or a straight line. Examples of such devices include James Watt's straight line linkage 3. Motion Generation Body guidance In body guidance, both the position of a point within a moving body and the angular displacement of the body are specified. The problem may be a simple translation or a combination of translation and rotation.

30. Output motion is a set of positions of a line defined as x, y, z successive locations. Guiding a rigid body through a finite set of positions and orientations . A good example is a landing gear mechanism which must retract and extend the wheels, having down and up locked poses with specific intermediate poses for collision avoidance,

31. Function Generation using Freudenstein's Equation A presentation of using Freudenstein's Equation in the synthesis of four-bar linkages that will mechanically generate functions like and almost anything else we can think of.

32. Question: Discuss Synthesis of Four Bar Mechanism Freudenstein's Equation

33. Consider a four bar mechanism as shown in Fig. The synthesis of a four bar mechanism, when input and output angles are specified, is discussed below :

34. Once the values of k1, k2 and k3 are known, then the link lengths a, b, c and d are determined by using equation (ii). In actual practice, either the value of a or d is assumed to be unity to get the proportionate values of other links .

35. Example: Design a four bar mechanism to co-ordinate the input and output angles as follows : Input angles = 15°, 30° and 45° ; Output angles = 30°, 40° and 55°.

36. …..( i ) …..(ii) …..(iii)

37. Assuming the length of one of the links, say a as one unit, we get the length of the other links as follows : We know that K1 = d / a or d = K1 * a = 0.905 units

38. Question: Discuss Precision Point Selection using Chebyshev Spacing The structural error in a function generator is simply the error between the mathematical function and the actual mechanism, usually expressed as a percentage. A good choice of precision points will help reduce the structural error. One good choice for the three precision points is using Chebyshev Spacing, which is simply a kind of equal spacing around a circle, then projection onto the horizontal bisector of the circle With Freudenstein's Equation we are limited to three precision points. We also have the bounds of the interval on x as given.

39. Chebychev spacing  Precision Points for Function Generation In designing a mechanism to generate a particular function, it is usually impossible to accurately produce the function at more than a few points. The points at which the generated and desired functions agree are known as precision points or accuracy points These points must be located so as to minimize the error generated between these points. The best spacing of the precision points, for the first trial, is called Chebychev spacing.

40. According to Freudenstein and Sandor , the Chebychev spacing for n points in the range Xs ≤ X≤ Xf ( i.e. when X varies between Xs and Xf ) is given by where Xj = Precision points Δ X = Range in X= Xf − Xs, and j = 1, 2, ... n The subscripts s and f indicate start and finish positions respectively.

41. The precision or accuracy points may be easily obtained by using the graphical method as discussed below--- Chebychev spacing . 1. Draw a circle of diameter equal to the range Δ X= Xf − Xs . 2. Inscribe a regular polygon having the number of sides equal to twice the number of precision points required, i.e. for three precision points, draw a regular hexagon inside the circle, as shown in Fig. 3. Draw perpendiculars from each corner which intersect the diagonal of a circle at precision points X1, X2, X3.

42. Freudenstein's Equation The development begins with the loop closure equation for a four-bar linkage, as shown in Figure below:

43. Loop Closure Equation The loop closure equation simply sums the position vectors around the complete four-bar linkage, and in vector form is given by Each link has length r and is at angle , hence the complex form may be written as …..1 ….2

44. We can expand (2) using Euler's identity, then separate Real and Imag terms. Before doing that, notice that the angle of link 1 is Finally, for conciseness, use the short form With these substitutions, equation (2) yields the two equations ….3 ….4

45. Since the input of our mechanism will be link 2, and the output will be link 4, …5

46. Where …6 …7 …8 …9

47. Let the minimum and maximum values of independent variable x be called Xi and Xf . Our three precision points X1; X2; X3 will between Xi and Xf ; the sequence will be [ Xi X1 X2 X3 Xf ] Chebyshev solution for N points; expressed for 3 points using notation it is

48. Summery Introduction of synthesis, Types of synthesis, Synthesis of a four bar chain, Freudenstein’s equation for four bar mechanism, Precision point for function generator ( Chebychev spacing method),