part vi linkage synthesis reflibvolume6.xyz/.../synthesisoflinkagesnotes2.pdf · the graphical...

ME 3610 Course Notes - Outline Part II -1

Part VI: Linkage Synthesis This section will review some of the most common and techniques for synthesizing linkages. These section will cover all of the basic graphical techniques, and will review a few of the analytical synthesis techniques. This section will be divided among the following topics:

1) Overview of linkage synthesis 2) Graphical synthesis techniques: two-position problem 3) Graphical synthesis techniques: three-position problem 4) Analytical synthesis techniques: Synthesis of dyads, Standard form equation, Three-

position synthesis with ground pivots specified.


1) Overview 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 Sandor’s 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 α gives the orientation of coupler

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

( ) ( ) ( )111 PPZW −=−+− ji

li

ljj ee αβ

This is called the Standard-form equation 2. For three positions (n=3), there are 2 vector equations for the left dyad:

( ) ( ) ( ) 21211 22 δαβ =−=−+− PPZW il

il ee ( ) ( ) ( ) 31311 33 δαβ =−=−+− PPZW i

li

l ee Note the number of unknowns in the above equations: Knowns: P1, P2, P3, α2, α3 Unknowns: Wl, Zl, β2, β3 − 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 −=−+− αβ il

il ee

Note the number of unknowns in the above equations: Knowns: P1, P2, P3, α2, α3 Unknowns: Wl, Zl, β2, β3, β4 − 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)


Consider the implications of the number of equations Vs. the number of unknowns (read design variables) in a synthesis problem: Use the following analogy: Consider Equations as things to do, and design variables (unknowns) as money: Example 1: You are in Buck snort TN on a Friday evening with a significant other and $105. Example 2: You plan a date in downtown Nashville and take $3.00.


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 δαβ =−=−+− PPZW i

li

l ee Eq. 2a ( ) ( ) ( ) 31311 33 δαβ =−=−+− PPZW i

li

l 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

,1,133

22

−=−=

−=−=αβ

αβ

ii

ii

ee

ee

dc

ba

3. Cast in matrix form:

rhsZW

A =⎭⎬⎫

⎩⎨⎧

l

l Eq. 4


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

rhsAZW 1−=

⎭⎬⎫

⎩⎨⎧

l

l Eq. 5

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, α’s are the same, the ψ’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.


Burmester Synthesis: Burmester synthesis provides a technique to solve for 4 precision positions Advantages:

Increased number of precision positions Returns 1 infinity of solutions Solutions presented conveniently in graphical form This is found in most commercial computer packages


Body Guidance with Ground Pivots Specified: Considering Table I above, it appears that the cases of three or four specified positions are the most likely candidates to consider in synthesizing linkages. For the three position case however, it would make a lot more sense if ground pivots could be specified as our free choices (rather than base link rotations). So, this section will rederive the dyadic synthesis equations for the case where ground pivots are specified for a body-guidance problem. To start, we will consider our original figure of a single dyad moving through the desired positions. To facilitate this formulation, we will use a new set of notation. This notation is shown in the following figure:


Procedure:

1. Write a vector loop equation that includes the ground pivot

( ) njee lji

li

ljj →==+−+ + 1,0GPZW ζφθ Eq. 6

2. Write this vector loop for positions 1-3:

( ) 3333

ρζφθ i3l

il

il eRee =−=+ + GPZW

( ) 2222

ρζφθ i2l

il

il eRee =−=+ + GPZW Eq. 7a-c

( ) 11111

ρζφθ il

il

il eRee =−=+ + GPZW

Discuss Eq. 7 here (page)

3. Expand the loop equation into 2 scalar equations:

( ) ( ) ( )( ) ( ) ( )ζφρθ

ζφρθ

+−=

+−=

jljjl

jljjl

ZRW

ZRW

sinsinsin

coscoscos Eq. 8


4. Eliminate unknown θj using the square and add technique: ( ) ( ) ( ) ( ) 0sinsin2coscos2222 =+−+−+− jjljjl RZRZRWZ

llρζφρζφ Eq. 9

5. Isolate angle z, first expand trig functions:

[ ] 02222 =++−−+− jjjjjjjjl sscscscsscccRZRWZ

llρζφρζφρζφρζφ Eq. 10

6. Then simplify:

( ) ( ) ( ) ( ) 0sinsin2coscos2222 =−+−−+− jjljjl RZRZRWZ

llρφζρφζ Eq. 11

7. This equation is non-linear in the unknowns (recall again the unknowns). Linearize using the following change-of-variables.

( ) ( ) ( )ζζ sin,cos, 32

221 ll ZkZkWZk

ll==−= Eq. 12


8. Rewrite the new linear equation: ( ) ( ) 2

321 sin2cos2 RRkRkk jjjj −=−+−− ρφρφ Eq. 13

9. Construct the set of equations for three positions:

( ) ( )( ) ( )( ) ( ) 2

3333333321

22222322221

2111311121

sin2cos2

sin2cos2

sin2cos21

RRkRkk

RRkRkk

RRkRkk

−=−+−−

−=−+−−

−=−+−−

ρφρφ

ρφρφ

ρφρφ

Eq. 14

10. Cast in matrix form:

( ) ( )( ) ( )( ) ( ) ⎪

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−−

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−−−−

23

22

2

3

2

1

333333

222222

111111 1

sin2cos21sin2cos21sin2cos21

RRR

kkk

RRRRRR

ρφρφρφρφρφρφ

Eq. 15


11. And solve:

Rk

Rkrr

rr

1−=

=

A

A Eq. 16

12. Now solve for dyad unknowns:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=−=+=

llll Z

kZkkZWkkZ

l

231

223

22 ,2atan,, ζ Eq. 17

13. Repeat for right-hand dyad, assemble linkage, check for defects, etc


Discuss Eq. 7:

1. What are the unknowns? How many? 2. For this problem, create a table similar to table II that lists number of positions,

uk’s, free choices and solutions 3. Suggest possible solutions for Eq. 14.

Discuss the solution procedure for Eq. 14 4. Nonlinear in unknowns, and, no free-choice will make this linear directly. 5. The angles qj can be eliminated from each vector equation using the “square and

add” procedure 6. This results in 3 equations and 3 unknowns for the 3 precision position problem

(explain) 7. Since these equations are nonlinear in the unknowns, use a variable substitution to

linearize in a new set of variables. 8. Solve linear set of equations, solve unknowns

part vi linkage synthesis reflibvolume6.xyz/.../synthesisoflinkagesnotes2.pdf · the graphical...

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