chapter 5 v2008 v1 color noprint

Copyright 2008 Department of Mechanical Engineering - University of Massachusetts Lowell Chapter 5 - 1

Mechanical Engineering - 22.321 Design of Machinery

Chapter

ANALYTICAL LINKAGE SYNTHESIS



5.0 Introduction

Fundamentals of Position Analysis Established

Can now Analytically Synthesize LinkagesUse an Algebraic Procedure



5.1 Types of Kinematic Synthesis

Function Generation the correlation of an input function w/ an output function in a mechanism

Path Generation the control of a point in the plane so that it follows some prescribed path

Motion Generation the control of a line in the plane so that it assumes some sequential set of prescribed positions



5.2 Two-Position Synthesis for Rocker OutputExample 3-1 showed a simple graphical technique for the design of a non-quick-return Grashof fourbar linkage to drive a rocker through an angle

- Rocker excursion should not exceed 180o- Best to keep excursion to under 120o for min

transmission angle of 30o



Consider doing an analytical determination of link lengths for the driver dyad- First choose a suitable location on Link 4 to

attach Link 3, e.g. Point B



5.3 Precision Points

Precision Positions or PointsThe prescribed points or positions for successive locations of the output link (coupler or rocker) in the plane



Number of equations limits the number of precision points that can be synthesized

For example, Fourbar linkage can synthesize up to:

5 precision points for motion or pathgeneration (i.e. coupler output)

7 points for function generation(i.e. rocker output)



- 2 or 3 points set of linear equations- 4 or more points

set of nonlinear equations need to use a computer program,

e.g. MathCAD or Matlab

Caveat: May still have intermediate toggle points



5.4 Two-Position Motion Generation by Analytical Synthesis

Consider a fourbar linkage in one position1. With a coupler point located at a first

precision point P12. A second precision point P23. Rocker input is to go through an angle 2



4. Angle of the coupler at each precision position is defined by the angles of the position vectors z1 and z2

5. The angle corresponds to 3 in Position 1

6. The angle 2 denotes a change in orientation in going from the first to second precision position.

Note: Figure 5-1 is schematic and dimensions are unknown at this early point in the problem



Problem StatementDesign a fourbar linkage which will move a line on its coupler link s.t. 1. A point P on that line will first be at P1 and later

at P2 and 2. Rotate the line through 2 between those two

precision positionsFind: Lengths and angles of the four links The coupler link dimensions A1 P1 and B1 P1 as

given in Fig. 5-1



5.5 Comparison of Analytical and Graphical Two-Position Synthesis



5.6 Simultaneous Equation SolutionAnalytical synthesis leads to set of simultaneous equations.

2-Position Synthesis 2 Equations 3-Position Synthesis 4 EquationsSeveral solution approaches:

Spreadsheet (e.g. Excel) MathCAD or Matlab Calculator Longhand by Gaussian elimination method



5.7 Three-Position Motion Generation by Analytical Synthesis Same basic approach as defining two

dyads at each end of the fourbar linkage A pair of dyads for each precision point

Problem StatementDesign a fourbar linkage which will:1. Rotate a line on the coupler through 2 and 3

to precision points P1 and P2, respectively2. Point P is on the coupler lineFind Lengths and angles of the four links and coupler dimensions A1P1 and B1P1



Solution Process Locate Global reference frame at P1

(for convenience) As was done for the two-position synthesis,

first solve for the LHS and then repeat forthe RHS

Can write two vector-loop equations foreach side



Summarizing: Infinite number of solutions Designer must check to see if linkage is feasible



5.8 Comparison of Analytical and Graphical Three-Position Synthesis



5.9 Synthesis for a Specified Fixed Pivot Location

Common problem as the available locations for fixed pivots in most machines are limited

Our 4 free choices will be the x and y coordinates of the two fixed pivots

Approach Nonlinear equations containing transcendental functions of unknown angles.

As before: The solution process must be applied for the LHS and then the RHS



5.10 Center-Point and Circle-Point Circles

If we could find the loci of all possible solutions to the three-position synthesis problem.

We would then have an overview of the potential locations of the ends of the vectors



Consider holding one of the free choices, say 2, at an arbitrary value, then solve Equations 5.25 and 5.26 for 3 = 02. A circle will be generated

Circle is locus of points for roots of for the chosen 2

O2 coincides with the root ofHence, called a Center-Point Circle



Next consider the vector

Hold 2 constant at some arbitrary angle Let 3 = 02

A circle of all possible locations of the root of for the given 2 chosen.As is fixed to the tip of which in turn describes a circle about pivot O2.

This locus is called the Circle-Point Circle



The x,y components of vectors and are defined by Equations 5.25 and 5.26

(5.25)

(5.26)

Negating the x,y components of will give the coordinates of points on the circle-point circle



The x,y components of N=-Z-W define the points on the O2 center-point circle for any assumed value of 2 as 3 is iterated thru 02

Vector W is calculated using angles 2 and 3 and vector Z using angles 2 as 3

The process is repeated for the RHS Note: There are an infinity of solutions because one angle is being

chosen arbitrarily A computer program can help in making reasonable choices



Fig. 5-9 shows the circle-point and center-point circles for the Chebychev straight-line linkage for choices

Two larger circles are the center-point circles for the potential locations of the fixed pivots O2 and O4

The two smaller circles define the loci of the possible moving pivot locations



5.11 Four- and Five-Position Analytical Synthesis

Can use same basic technique as was used for 2- and 3-position analytical synthesis

Let angles be designed k, k and kwhere k=2nand where:

k denotes the precision positionn represents the number of

precision positions for which we are solving



5.12 Analytical Synthesis of a Path Generator with Prescribed Timing

Can use same approach for path generation w/ prescribed timing

Here we specify angles 2 and 3 of the input rocker to define timing

Thus, the free choices are now 2 and 3 Use the same Equations 5.25, 5.26, 5.30 and 5.31 Can be extended to as many as 5 precision points



5.13 Analytical Synthesis of a Fourbar Function Generator

Can still use a similar approach as was used for the synthesis of path generation

In this case, we do not care about the motion of the coupler

Coupler only exists to couple input link to output link

Coupler is simply a line from A to P P is simply a coupler point which happens to

coincide with the pin joint P has a simple arc motion



Function generator uses Link 2 as the input link and takes output from Link 4.

Relationship between the angles of Links 2 and 4 is the function generated.

The function is then:

NOT a continuous function. Holds only for the n discrete points specified.

Linkage must satisfy

Once again, write vector loop equations.



5.14 Other Linkage Synthesis Methods Many other techniques for the synthesis of

linkages have been discovered in recent years Most are involved and mathematically cumbersome Only a few admit a closed-form solution Most require an iterative numerical solution

Table 5-5 summarizes some of the existing fourbar linkage synthesis methods divided into three types



1. PrecisionA method that attempts to find a solution which will pass exactly through the desired precision points but may deviate from the desired path between these points



Limited to matching a number of points equal to the number of independently adjustable parameters that define the mechanism e.g.

Fourbar linkage 9 parameters. For up to five precision points, equations

can be solved for without iteration. For 6 to 9 points an iterative is required to

solve the equation set.



The 9 parameters are:- 4 link lengths- 2 coordinates of the coupler point

with respect to the coupler link- 3 parameters which define the

location and orientation of the fixed length in the global reference frame



2. EquationMethods that solve the tri-circular, trinodalsextic coupler curve to find a linkage that will generate an entire coupler curve that closely approximates a set of desired points on the curve.

3. OptimizedAn iterative optimization procedure that attempts to minimize an objective function that can be defined in many ways.

Allow larger numbers of points to be specified then do the precision methods.

Limited only by computer time and round-off error. Convergence depends on good initial choices.

chapter 5 v2008 v1 color noprint

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