Kinematics Roadmap

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Kinematics Roadmap
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Linkage Synthesis and Analysis

• Analysis is used to determine a mechanisms
  performance when its desired dimensions are known
• Synthesis is used to design a mechanism when the
  desired performance is known
• Synthesis usually involves multiple analyses

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Complex Number Method
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Complex Number Review
A vector is rotated by multiplying by ei(rotation
angle)
bj
r2
W
Wj
q2
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Dyad or Standard Form - First Dyad
From the vector loop shown W Z dj - Weibj -
Zeiaj 0 W(eibj - 1) Z(eiaj - 1) dj
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Dyad or Standard Form - Second Dyad
From the vector loop shown U S dj - Ueigj -
Seiaj 0 U (eigj - 1) S (eiaj - 1) dj
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Complex Number Method Cont.
V
U
W
G
V Z - S
G W V - U
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Free Choices

• Two and three position synthesis problems result
  in more unknowns than equations. Some of the
  unknowns are selected as free choices
• free choices unknowns - equations
• Choose free choices that make the problem linear
• If the resulting mechanism is undesirable, choose
  new values for free choices

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Three Position Synthesis
For first dyad of the mechanism W(eib2 - 1)
Z(eia2 - 1) d2 W(eib3 - 1) Z(eia3 - 1)
d3 For second dyad of the mechanism U(eig2 - 1)
S(eia2 - 1) d2 U(eig3 - 1) S(eia3 - 1)
d3
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Three Position Synthesis
From the first dyad of the mechanism there are 12
variables and 4 equations therefore, 8
variables must be specified or chosen as free
choices. Function generation Motion
generation Path generation Path generation with
prescribed timing
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Three Position Synthesis Cont.
The dyad equations can be expressed in matrix
form (Axb)
For the first dyad
For the second dyad
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Cramers Rule
Given
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Three Position Synthesis Cont.
Use Cramers rule to solve for the unknowns
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Three Position Synthesis
Design a mechanism for path generation with
prescribed timing with the following specified
path
P2
d2 -10 5i d3 -15 - 2i b2 47.67o b3
92.10o
d2
P
P3
d3
This is 6 of 8 variables to be specified. To make
the equations linear, the other angles are chosen
as free choices
a2 -20o g2 15o a3 -10o g3
44.21o
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Three Position Synthesis
From Cramers rule
W 10ei60 Z 13.17e-i144.6 U 16.98ei88.4 S
22.5e-i124.3
V
S
U
P2
From the above vectors, the others can be found
d2
W
Z
G 7.00ei22.2 V 11.12ei79.9
G
P
P3
d3
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Nonlinear Synthesis

• The complex number synthesis equations for three
  position synthesis and good choices for free
  variables leads to a linear solution. Other cases
  may be solved that require the solution of
  nonlinear equations
• Four precision points
• Five precision points
• Angles as unknowns rather than free choices

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Multiloop Mechanisms

• Synthesis for multiloop mechanisms (such as six
  bars) may be solved by extending the above
  methods to include more loops. This increases
  the number of equations and unknowns, but is not
  a difficult extension

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Optimization

• Definition - Mathematical approach used to find
  the best values for parameters to obtain an
  optimal design.

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Optimization Applications in Mechanisms

• Minimize structural error
• Include velocity, acceleration, dynamics,
  mechanical advantage, etc.
• Ground pivot locations
• Help select among infinite number of values for
  free choices

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Structural Error

• In some applications the structural error
  incurred in precision point synthesis is
  unacceptable. The error may be minimized by using
  optimization methods. Some possible methods
  include the following
• Use precision point synthesis combined with an
  optimization method to search among the infinite
  possible solutions to find the optimal solution.
• Use optimization methods with displacement
  equations to minimize structural error without
  precision points.
• Optimization may be as simple as iterating, or it
  may be using state-of-the-art optimization
  programs.

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Optimization-Definitions

• Objective function - the function to be minimized
  (or maximized)
• Design variables - the variables that are changed
  to find the optimal solution
• Constraints - limits on how the design variables
  may be changed
• Example - Synthesize a linkage with reduced
  structural error to replace the given mechanism

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Optimization Techniques

• Closed-form (calculus)
• Iterative Search
• Numerical Methods (SQP, genetic algorithms,
  simulated annealing, etc.)

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Nonlinear Equations Example
Objective function Design variables q3,
q4 Minimize objective function. When
sufficiently close to zero, the values of the
design variables are the solutions of the
equations
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Optimization - Example

• Find the value of x such that f is minimized
  where f 10 x2 - 40x 2 and 0 x 100
• Identify the constraints, design variables, and
  objective function.

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Mechanism Example
Find a crank-rocker mechanism that has a coupler
curve that travels through a straight line for
100 degrees of crank rotation. Objective
function minimize sum of squares of
difference Constraints r2 lt r1 r2 lt r3
r2 lt r4 s l lt p q Design variables r1,
r3, r4, a3, b3 (set r2 1)
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Other Synthesis Methods

• Overlay method
• Circle-point and center point circles
• Ground pivot specifications
• Freudensteins equation
• Loop closure
• Coupler curves (Hrones and Nelson)
• Burmester theory