TopicISynthesisPartI Detail

ME 3610 Course Notes - Outline Part II -1

Part I: Mechanism Synthesis Mechanism synthesis in kinematics consists of formalized techniques used in design of mechanisms. We call these techniques dimensional synthesis to determine the critical dimensions of the mechanism. The following table names some of the more common techniques: Mechanism class Name Application Comments Ref.

Linkages Planar Graphical 2 position

Driving dyad Quick return

N. pg 93

Graphical Body guidance

3 position N. pg 99

Graphical Path generation

3 positions, w/ or w/o prescribed timing

S&E, Ch. 2

Graphical Function generation

3 positions S&E, Ch. 2

Freudensteins equation technique

Analytical technique, initially for function generation

Initial application of solutions at precision positions - Loop Closure equations

N. pg 176

Loop closure equation technique


Dyadic synthesis Analytical, dyads Forms equations for up to n positions

N. Ch. 5

Chebyshev spacing Theory for error limiting at precision positions

M&R

Cognate linkages Roberts-Chevyshev theorem -Three different linkages generate the same coupler point curve

N. pg 123

Hrones & Nelson atlas of coupler curves

Path generation N. pg 113

Burmester theory 4 position body guidance

S&E, Ch. 3

Five position with Sylvesters dyalitic eliminant

Analytical 5 position body guidance

S&E, Ch. 3

Order synthesis Synthesis applied to infinitely separated positions (to match velocity and acc. specifications

S&E, Ch. 3

Linkages Spatial Dyadic synthesis equations

Up to seven positions can be met with SS dyad (compare to 5 RR)

M&R, S&E


Cams Graphical M&R Conjugate geometry course notes Gears References: N. - Norton , R. L., 2004, Design of Machinery: An Introduction to the Synthesis and Analysis of

Mechanisms and Machines, 3rd Ed., McGraw-Hill. S&E - Sandor, G.N. and Erdman, A. Advanced Mechanism Design: Analysis and Synthesis, Vol.

II, Prentice Hall. M&R - Mabie, H. H., and C. F. Reinholtz, 1987, Mechanisms and Dynamics of Machinery,

Fourth Edition, Wiley.


Linkage Synthesis: This section will review some of the most common and techniques for synthesizing linkages. These section will cover the following topics: 1) review basic graphical and analytical synthesis techniques. 2) center/circle point curves for choosing free choices (3 position), 3) ground-pivot specified methods (3 position). 4) Burmester theory for the 4 position problem and 5) basic optimization techniques for linkages and synthesis in commercial software. Review basics of linkage synthesis

1. Definitions:

a. Synthesis: To create a mechanism given desired task

b. Analysis: To determine the motion characteristics (task) given a mechanism.

c. Grashof mechanism

d. Toggle position

e. Types of sixbars


2. Forms of synthesis: a. Type synthesis:

Choosing the type of mechanism best suited to the task a. Ex: Gear trains, linkages, cams, actuation methods, and # of links/joints the

mechanism should have. b. Degrees of freedom.

b. Dimensional synthesis:

Determine the significant dimensions of the mechanism

c. Classical Synthesis problems: Motion generation Path generation


Function generation d. Defects that may occur: Branch defect Grashof defect Order defect


2. Graphical Synthesis Techniques: 2 positions Toggle positions Equal forward/reverse drive times: Locate the driving dyad ground pivot along the chord line Quick return-type mechanisms Driving dyad ground pivot not located along the chord line


3. Graphical Synthesis Techniques: 3 positions a. Introduction: b. why three positions? The graphical approach for linkage synthesis is based on the geometric construction of a circle from three points (i.e., finding the center of a circle defined by three positions). Interestingly, four positions (and even five) can define a unique circle. However, simple geometric construction techniques for these cases do not exist, therefore three positions are the problems we solve. c. How many positions total could be solved? Five, (three link-lengths and two off-sets). This can be seen in considering d. What are Precision Positions Positions which will be met exactly (precisely) during linkage motion e. How are Precision Positions selected? Precision positions should be selected to best represent the overall desired motion. If some exact points are required (for example a pick-up or drop-off point), then these can be used as precision


positions. Note that no other desired position (other than precision positions) will be necessarily met by linkage motion. One formal method for choosing precision points is Chebyshev spacing Four different three-position techniques will be discussed: motion generation, path generation, function generation, and motion generation with prescribed ground pivots.


B.1. Motion Generation: Motion generation is the workhorse of the linkage synthesis problems. In motion generation, the position and orientation of a body are to be guided (hence the other name, body guidance). The procedure proceeds as follows: 1) Specify 3 positions of the body (the precision positions) 2) Choose 2 moving pivots on the body (coupler or circle points), A& B. Locate A1, A2, A3 and B1, B2, B3. 3) Find the center of points Ai and Bi: 1. create lines A1A2, A2A3, B1B2, and B2B3 2. Draw perpendicular bisectors 3 Find the intersection of these bisectors to give OA and OB. 4) Construct the linkage, check for defects 5) iterate as necessary, choosing new coupler points Notes:

1. there are 4 infinities of solutions corresponding to choosing the circle points, A & B. 2. Choosing the 3 body positions to represent the task presents an iteration 3. A & B do not need to be on the body. 4. Once OA and OB are chosen, check for defects.


B.2. Path Generation: Path generation is a subset of motion generation (only body positions, not orientations are specific). Two approaches are used to solve a path generation problem.

One approach would define an additional set of angles as prescribed input timing and then proceed in a manner somewhat like motion generation.

A second approach is to look at coupler curves: Curves defined by points on a coupler link (non-grounded link in a four-bar). An infinite number of coupler curves exist for one four-bar, and there are infinite possibilities of fourbars (the coupler curve in general is a 6th order curve).

The Hrones and Nelson Atlas of fourbar coupler curves can be used to choose curves, or select a suitable software program.


B.3. Function Generation: The function generation problem relates creates a functional relationship between the rotations of the input and output links of a fourbar.


B. 4. Motion Generation with Specified Ground Pivots: Given a motion generation task and two ground pivots specified, create a 4-bar linkage Process uses inversion: to consider the motion of a device with different links considered

as the reference or ground


Procedure: 1. Specify 3 precision positions of the body 2. Choose 2 ground pivots, OA and OB. Now, using inversion, locate OA2, OA3 and OB2, OB3. i.e., consider the body fixed and the pivots moving around the body.

2.1 First, measure the position of OA relative to the body in the second position, and then draw OA (call it OA2) relative to the body in the first position using these measurement. Do the same to locate OA3 (measure relative to the body in the third position, draw it relative to the body in the first position).

2.2 Repeat for OB 3. Find the center of points OA and OB in the usual manner. 4. Draw the linkage in the first position


Locate three precision positions and grd pivots Locate OA2


Locate OA3 Locate Coupler point A


Repeat for coupler points B Draw in the linkage


4. Analytical Synthesis Techniques: Analytical synthesis techniques lend themselves to computer solution can automate the synthesis process and present much better tools for linkage synthesis. (The trade-off is that the techniques are somewhat less-intuitive initially to a beininning mechanism designer).

The analytical techniques began with Freudenstien, who essentially solved the geometric synthesis equations in an analytical fashion.

The techniques we will use are called dyadic synthesis and developed out of Sandors work (extended by Erdman).

Analytical Dyadic Synthesis of Linkage or Dyadic Linkage Synthesis: The key idea behind dyadic linkage synthesis is to consider a linkage as composed of a set of dyads. Each dyad must perform the motion desired of the linkage. Therefore, the synthesis process can be reduced to synthesizing the motion of a set of dyads independently and then combining them to create entire linkage. Dyad: Two-link pair Consider for example a four-bar:


Notation 1. Point P on the coupler traces the output position while gives the orientation of coupler

(and body) 2. W and Z are vectors representing the dyad in position 1. 3. W rotated by angle j is given by We^(i*j) 4. Zl and Zr must have the same rotation () 5. j gives left-hand input timing 6. j gives right-hand input timing


Procedure: (From here on, we will consider the body-guidance problem, with body position and orientation given)

1. Represent the four-bar as 2 coupled dyads 2. Synthesize one dyad at a time 3. Move one dyad from the first precision position to the next 4. Write a vector loop equation to represent the unknown dyad vector at known positions (In

the standard form solution, each loop equation will include the dyad in the first and jth position)

5. For each single loop equation, there are 5 u.k. parameters (Wl, Zl, j) 6. Make appropriate free choices 7. Solve the system equations for the unknowns 8. This results in 1 dyad that satisfies the precisions points. Solve for second to complete

the four bar (with the requirement that the coupler rotation is consistent)


1. Write a vector-loop equation: njee llj

il

il

jj ==++ 2,01 WZPPZW Or: ( ) ( ) ( )111 PPZW =+ jilil jj ee This is called the Standard-form equation 2. For three positions (n=3), there are 2 vector equations for the left dyad: ( ) ( ) ( ) 21211 22 ==+ PPZW ilil ee ( ) ( ) ( ) 31311 33 ==+ PPZW ilil ee Note the number of unknowns in the above equations: Knowns: P1, P2, P3, 2, 3Unknowns: Wl, Zl, 2, 3 6 Number of equations: 4


3. For four positions (n=4), there are 3 vector equations for the left dyad: (above 2 plus): ( ) ( ) ( )1411 44 PPZW =+ ilil ee

Note the number of unknowns in the above equations: Knowns: P1, P2, P3, 2, 3Unknowns: Wl, Zl, 2, 3, 4 7 Number of equations: 6 4. This process can be repeated. Look at all possibilities in the following table:


Table I: Number of positions Vs. number of solutions for the Std Form Equation on a Body Guidance Problem

# of positions

(j)

# of scalar

equations

# of Scalar unknowns

# of Free

Choices

# of Solutions

Solution Technique

2 2 5 (W,Z, b2)

3 O(infinity)^3 So Easy!

3 4 6 (W,Z, b2, b3)

2 O(infinity)^2 Straight forward (Linear

equations in general)

4 6 7 (W,Z, b2, b3)

1 O(infinity) Medium-difficulty

(Burmester) 5 8 8 (W,Z,

b2, b3, b4)0 Finite Hard

Analytically (but not

impossible)


Free Choices, a few more comments: 1. Proper selection of free choices leads to a set of linear equations in the unknowns. 2. Consider other sets of free choices, discuss their merits and disadvantages.


Solving the Standard Form Equation for 3 positions:

1. Recall the two loop closure equations for 1 dyad: ( ) ( ) ( ) 21211 22 ==+ PPZW ilil ee Eq. 2a ( ) ( ) ( ) 31311 33 ==+ PPZW ilil ee Eq. 2b 2. Make free choices such that only W and Z are unknown. The equations are known linear and can be solved as:

( )( 12

13

PPdZcWPPbZaW

=+ )=+

ll

ll Eq. 3

where: ( ) ( )( ) ( 1,1 ,1,1 33

22

====

ii

ii

eeee

dcba )

3. Cast in matrix form:

Eq. 4 rhsZW

A =

l

l


where ( )( )

=

=

13

12,PPPP

rhsdcba

A

Note that matrix A and vector b are complex. How would you expand (Eq. 4) such that A and b are not complex? 4. Now solve for the unknown dyad vectors, Wl and Zl

Eq. 5 rhsAZW 1=

l

l

5. Methods to do this (matrix inverse) Cramers Rule Gauss-Jordan Elimination Matlab


Using Matlab to solve the Std. Form Equation:

Assigning complex vectors: >>a=exp(beta2*i)-1 >>b=exp(alpha2*i)-1 Creating matrix and vector >>A=[a,b;c,d] >>rhs=[(P2x-P1x)+i*(P2y-P1y); (P3x-P1x)+i*(P3y-P1y)]; Invert and multiply >>x=inv(A)*b Extract results >>W=x(1) >>Wx=real(W);Wy=imag(W)


Complete the Fourbar: Now solve for the right hand dyad For a body-guidance problem, s are the same, the s are the free choices What are free choices for a path-generation problem? Reconstruct the four-bar using the two dyads Check for defects, performance, etc.


Analytical Dyadic Synthesis: Thought-Provoking Questions 1) Generate the standard form equation, W(eibj - 1) + Z(eiaj - 1) = dj for a body guidance problem (draw a figure). List the knowns and unknowns. Given three positions, describe a closed-form solution technique. 2) Create a table for the body guidance problem that demonstrates the maximum number of positions that can be solved with a four-bar, and list the unknowns, free-choices, and solutions for all smaller positions. 3) Given a path generation problem w/o prescribed input timing (the only givens are the Pj's, determine the maximum number of positions that can be synthesized with a four-bar linkage. Support/prove your result. 4) Derive the standard form equation for a function generation problem. List the knowns and unknowns. Also, list the number of free-choices to solve for three positions. 5) Create a table for the function generation problem that demonstrates the number of positions possible along with knowns, unknowns, free choices, and number of solutions. 6) For a body guidance problem, given the three positions and thus two free-choices, list all possible combinations of two free choices. Discuss the merits of these various choices. 7) Show how to set up the equations to solve three position body-guidance problem if the ground pivots are to be made as free-choices. 8) Given a function generation problem, determine the maximum number of precision pairs that can be synthesized with a four-bar linkage.

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TopicISynthesisPartI Detail