Part II: Analytical Dyadic Synthesis

Linkage Synthesis: Advanced Techniques

Topics:

1) ground-pivot specified methods (3 position).

2) Burmester theory for the 4 position problem and

3) 4 point synthesis of slider crank, fixed slider

4) 3 point synthesis of slider crank, rotating slider

5) 3 point synthesis of slider crank, rotating slider, ground-pivot specified.

5) basic optimization techniques for linkages and synthesis in commercial software.

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

Eq. 1.1

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

Eq. 1.2a-c

3. Expand the loop equation into 2 scalar equations:

Eq. 1.3

4. Eliminate unknown j using the square and add technique:

Eq. 1.4

5. Isolate angle z, first expand trig functions:

Eq. 1.5

6. Then simplify:

Eq. 1.6

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

Eq. 1.7

8. Rewrite the new linear equation:

Eq. 1.8

9. Construct the set of equations for three positions:

Eq. 1.9

10. Cast in matrix form:

Eq. 1.10

11. And solve:

Eq. 1.11

12. Now solve for dyad unknowns:

Eq. 1.12

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

Discuss Eq. 1.2:

  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. 1.2.

Discuss the solution procedure for Eq. 1.2

  1. Nonlinear in unknowns, and, no free-choice will make this linear directly.
  2. The angles j can be eliminated from each vector equation using the “square and add” procedure
  3. This results in 3 equations and 3 unknowns for the 3 precision position problem (explain)
  4. Since these equations are nonlinear in the unknowns, use a variable substitution to linearize in a new set of variables.
  5. Solve linear set of equations, solve unknowns

Four position problem: 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


Solution for four prescribed positions: Burmester

Procedure:

1. Start with the vector loop equation in standard form:

Eq. 2.1

2. Write this vector loop for positions 1-4:

Eq. 2.2 a-c

3. Rewrite as homogenous equations:

Eq. 2.3

4. Cast into matrix form:

Eq. 2.4

5. In this form, it is clear that either the vector is zero (which it cannot be due to the third term, 1) or that the equation cannot be solved with matrix inversion (i.e., it is invertible). This means that the determinant of the coefficient matrix must be zero:

Eq. 2.5

Or

Eq. 2.6

Where the ’s represent the determinants of the cofactors:

Eq. 2.7 a-d

The beta’s are the unknown terms, all components in the ’s are known and can be calculated (remember, they are in complex form)

Equation 2.6 looks like the typical loop closure equation for a fourbar. The standard way to treat this is to select values for 2 (the input), and calculate the corresponding values for 3, 4.

Eq. 2.8

Eq. 2.9 a-e

Expand into x, y (real, imaginary)

Eq. 2.10 a-b

Square and add:

Eq. 2.11

or

Eq, 2.12

Solving for 4:

Eq. 2.13

Eq. 2.14

2 solutions exist for 4. Now solve for 3:

Eq. 2.15

Now go back and solve for the dyad. Pick up with eqs. 2.2, choose two to solve for the two unknown (vectors) in the dyad.

Eq. 2.16

Eq. 2.17

Remember, two sets of Wl, Zl for each value of 2. Next, solve for the circle point and center point:

Eq. 2.18

These are called Burmester point pairs (BPP). Using these BPP’s and all values of 2 that result in a closed virtual linkage (eq. 2.6) result in two center point and coupler point curves (two curves corresponding to two set of solutions for 3, 4). As a reminder, the maximal values for 2 can be derived as:

Analytical techniques for linkages with Sliders:

Method #1: Four-point Synthesis of a slider crank (fixed-axis slider)

This figure represents the layout for the first case of four position synthesis of the slider crank. Here, W represents the crank and Z the coupler

1. Start by writing the dyad loop equation in the standard form.

3.1

This equation can be written for four positions and cast in matrix form as:

3.2

We can solve this equation for four positions using the same technique as in the previous problem classically referred to as Burmester synthesis. In this case, the set of input rotations, (’s) are generally prescribed along with the set of output displacements (’s) and ’s are the unknown connector angles to solve for. W and Z are the connector and crank dyad – these need to be determined. The slider orientation is fixed, known and given in S. Note that the change in notation, Z as the first vector and W the second in the dyad is to make this problem match the standard Burmester notation.

Method #2: Three-point Synthesis of a slider crank (moving-axis slider)

This figure represents the layout for the first case of four position synthesis of the slider crank. Here, W represents the crank, E the coupler and D the slider

1. Start by writing the dyad loop equation in the standard form.

, j=2, 3Eq. 4.1

where

Eq. 4.2

In this equation, the unknown parameters to determine are the dyads D and E. The q’s are also unknown, if these are treated as free choices, the system can be solved for dyad D, E as a system of linear equations.

Method #3: Three-point Synthesis of a slider crank (moving-axis slider, ground pivots specified)

This figure represents the layout for the first case of four position synthesis of the slider crank. Here, W represents the crank, E the coupler and D the slider

1. Start by writing the dyad loop equation with the ground pivot included.

, j=1, 2, 3Eq. 5.1

Isolate the unknown j (and combine knowns W and G):

Eq. 5.2

Eliminate the unknown j’s by expanding into Cartesian, square and add.

Eq. 5.3

The problem is now solved following the same procedure as in section 1, starting with eq. 1.4. The solution culminates in values for D, E and z to give the desired dyad.

Part II -1

ME 3610 Course Notes - Outline