Analytic%20Approach%20to%20Mechanism%20Design

1
Analytic Approach toMechanism Design
http//www.engr.colostate.edu/me/program/courses/M
E324/notes/PositionAnalysis.ppt

• ME 324
• Fall 2000

2
Chapter 4 -Analytic Position Analysis
ImaginaryAxis

• A vector can be represented by a complex number
• Real part is x-axis
• Imaginary part is y-axis
• Useful when we begin to take derivatives

Point A
jR sin q
RA
q
RealAxis
R cos q
3
Derivatives, Vector Rotations in the Complex Plane
Imaginary

• Taking a derivative of a complex number will
  result in multiplication by j
• Each multiplication by j rotates a vector 90 CCW
  in the complex plane

B
RB j R
A
C
RA
RC j2 R -R
Real
D
RD j3 R - j R
4
Labeling of Links andLink Lengths

• Link labeling starts with ground link
• Labeling of link lengths starts with link
  adjacent to ground link
• Makes no sense - just go with it

Link 3, length b
B
Link 4, length c
Coupler
A
Link 2, length a
Link 1, length d
Ground Link
Pivot 02
Pivot 04
5
Angle Measurement Convention

• All angles measured from angle of the ground link
• Define q1 0
• One DOF, so can describe all angles in terms of
  one input, usually q2

3
B
q3
4
A
q4
2
q2
1
q1 0
6
More on Complex Notation

• Polar form re jq
• Cartesian form r cosq j r sinq
• Euler identitye jq cosq j sinq
• Differentiation

j
q
de
j
q

je
d
q
7
The Vector Loop Technique

• Vector loop equationR2 R3 - R4 - R1 0
• Alternative notation RAO2 RBA - RBO4 - RO4O2
  0nomenclature - tip then tail
• Complex notationaejq2 bejq3 - cejq4 - dejq1
  0
• Substitute Euler equationa (cos q2j sinq2) b
  (cos q3j sinq3)- c (cos q4j sinq4) - d (cos
  q1j sinq1) 0

R3
b
B
R4
q3
A
a
q4
c
R2
q2
d
R1
O2
O4
8
Vector Loop Technique - continued

• Separate into real and imaginary parts
• Reala cos q2 b cos q3 - c cos q4 - d cos q1
  0
• a cos q2 b cos q3 - c cos q4 - d 0, since
  q1 0, cos q1 1
• Imaginary ja sin q2 jb sin q3 - jc sin q4 -
  jd sin q1 0
• a sin q2 b sin q3 - c sin q4 0, since q1
  0, sin q1 0

9
Vector Loop Technique -continued

• a cos q2 b cos q3 - c cos q4 - d 0
• a sin q2 b sin q3 - c sin q4 0
• a,b,c,d are known
• One of the three angles is given
• 2 unknown angles remain
• 2 equations given above
• Solve simultaneously for remaining angles

10
Vector Loop - Summary

• Draw and label vector loop for mechanism
• Write vector equations
• Substitute Euler identity
• Separate into real and imaginary
• 2 equations, 2 unknown angles
• Solve for 2 unknown angles
• Note there will be two solutions since mechanism
  can be open or crossed

11
ExampleAnalytic Position Analysis

• Input position q2 given
• Solve for q3 q4

b2.14
q
?
3
a1.6
c2.06
q
?
q
51.3
d3.5
4
2
12
ExampleVector Loop Equation

• R2 R3 - R4 - R1 0
• aejq2 bejq3 - cejq4 - dejq1 0
• 1.6ej51.3Þ 2.14ejq3 - 2.06ejq4 - 3.5ej0 0

R3
b2.14
R4
q
?
3
R2
c2.06
a1.6
q
?
q
51.3
d3.5
4
2
R1
13
ExampleAnalytic Position Analysis

• aejq2 bejq3 - cejq4 - dejq1 0
• a(cosq2jsinq2) b(cosq3jsinq3) -
  c(cosq4jsinq4) - d(cosq1jsinq1)0
• Real part
• a cos q2 b cos q3 - c cos q4 - d 0
• 1.6 cos 51.3 2.14 cos q3 - 2.06 cos q4 - 3.5
  0
• Imaginary part
• a sin q2 b sin q3 - c sin q4 0
• 1.6 sin 51.3 2.14 sin q3 - 2.06 sin q4 0

14
Solution Open Linkage

• 2 equations from real imaginary equations
• 1.6 cos 51.3 2.14 cos q3 - 2.06 cos q4 - 3.5
  0
• 1.6 sin 51.3 2.14 sin q3 - 2.06 sin q4 0
• 2 unknowns q3 q4
• Solve simultaneously to yield2 solutions.
• Open solution
• q3 21Þ, q4 104

15
Review - Law of Cosines
2
2
2
A

B
-
C
cos
q

2
AB
q
A
é
ù
2
2
2
A

B
-
C
q

arccos
ê
ú
B
2
AB
ë
û
C
16
Transmission Angles

• Transmission angle is the angle between the
  output angle and the coupler
• Absolute value of the acute angle
• Measure of quality of force transmission
• Ideally, as close to 90 as possible

m2
180??- m2
acute
m1
180?- m1
17
Extreme Transmission Angles - Grashof Crank
Rocker

• For a Grashof fourbar, extreme values occur
  when crank is collinear with ground
• For the extended position shown
• m1arccos (b2(ad) 2 - c2)/2b (ad)
• m2180 - arccos (b2c2 - (ad)2 )/2b c

m2
c
b
a
d
m1
18
Extreme Transmission Angles - Grashof Crank
Rocker

• For the overlapped case shown
• m1__________________________
• m2__________________________

m2
c
b
m1
a
d
19
Extreme Transmission Angles - Grashof Double
Rocker

• Remember coupler makes a full revolution with
  respect to rockers
• Transmission angle varies from 0 to 90

20
Extreme Transmission Angles - Non-Grashof Linkage
b

• Transmission angle is zero degrees in toggle
  position output rocker coupler
• Other transmission angle given as
• m2__________________________
• Similar analysis for other toggle position

m10?
m2
a
c
d
21
Calculation of Toggle Angles

• The input angle, q2 , for the first toggle
  position given as
• q2__________________________
• Similar analysis for the other toggle position

b
a
c
q2
d