Analytic Approach to
Mechanism Design
http://www.engr.colostate.edu/me/program/courses/ME324/notes/PositionAnalysis.ppt
ME 324
Fall 2000
1
Position synthesis
Chapter 4 Analytic Position Analysis
Imaginary
Axis
• A vector can be
represented by a
complex number
• Real part is x-axis
jR sin q
• Imaginary part is yaxis
• Useful when we
begin to take
derivatives
2
Point A
RA
q
R cos q
Real
Axis
Position synthesis
Derivatives, Vector Rotations in the
Complex Plane
Imaginary
• Taking a derivative of
B
a complex number will
RB = j R
result in multiplication
by j
C
A
• Each multiplication by
RA
RC = j2 R = -R
Real
j rotates a vector 90°
CCW in the complex
plane
R = j3 R = - j R D
D
3
Position synthesis
Labeling of Links and
Link 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
4
Link 3, length b
B
Coupler
Link 2,
length a
Link 4,
length c
A
Link 1, length d
Pivot 02
Ground Link
Pivot 04
Position synthesis
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
5
3
q3
A
2
q2
B
4
q4
1
q1 = 0°
Position synthesis
More on Complex Notation
• Polar form:
re jq
• Cartesian form:
r cosq + j r sinq
• Euler identity:
±e jq = cosq ± j sinq
• Differentiation:
jq
6
de
jq
= je
dq
Position synthesis
The Vector Loop Technique
• Vector loop equation:
R2 + R3 - R4 - R1 = 0
• Alternative notation:
R3 b
RAO2 + RBA - RBO4 - RO4O2 = 0
q3
a A
nomenclature - tip then tail
R2
q2 d
• Complex notation:
R1
aejq2 + bejq3 - cejq4 - dejq1 = 0 O
2
• Substitute Euler equation:
a (cos q2+j sinq2) + b (cos q3+j sinq3)
- c (cos q4+j sinq4) - d (cos q1+j sinq1) = 0
7
B
R4
q4
c
O4
Position synthesis
Vector Loop Technique - continued
• Separate into real and imaginary parts:
Real:
a 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
8
Position synthesis
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
9
Position synthesis
Vector Loop Summary
•
•
•
•
•
•
•
10
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
Position synthesis
Example:
Analytic Position Analysis
• Input position q2 given
• Solve for q3 & q4
b=2.14
q 3 =?°
a=1.6
q 2 =51.3° d=3.5
11
c=2.06
q 4 =?°
Position synthesis
Example:
Vector Loop Equation
R2 + R3 - R4 - R1 = 0
aejq2 + bejq3 - cejq4 - dejq1 = 0
1.6ej51.3Þ + 2.14ejq3 - 2.06ejq4 - 3.5ej0° =
R3
0
R2
b=2.14
q 3 =?°
c=2.06
a=1.6
q 2 =51.3° d=3.5
12
R4
R1
q 4 =?°
Position synthesis
Example:
Analytic Position Analysis
aejq2 + bejq3 - cejq4 - dejq1 = 0
a(cosq2+jsinq2) + b(cosq3+jsinq3) - c(cosq4+jsinq4)
- d(cosq1+jsinq1)=0
Real part:
a cos q2 + b cos q3 - c cos q4 - d = 0
b=2.14
1.6 cos 51.3 + 2.14 cos q3
q 3 =?°
- 2.06 cos q4 - 3.5 = 0
a=1.6
Imaginary part:
c=2.06
a sin q2 + b sin q3 - c sin q4 = 0
q 2 =51.3° d=3.5
1.6 sin 51.3 + 2.14 sin q3
- 2.06 sin q4 = 0
13
Position synthesis
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
b=2.14
Solve simultaneously to yield
q 3=21Þ
2 solutions.
a=1.6
Open solution:
c=2.06
q3 = 21Þ, q4 = 104°
q 4=104Þ
q 2=51.3Þ d=3.5
14
Position synthesis
Review - Law of Cosines
A +B -C
cosq =
2AB
2
2
2
A2 + B2 - C 2 
q = arccos

2AB


q
A
B
C
15
Position synthesis
Transmission Angles
• Transmission angle is the angle
between the output angle and the
coupler
m2
180- m2
• Absolute value of the acute angle
• Measure of quality of force
acute
transmission
m1
• Ideally, as close to 90° 180- m1
as possible
16
Position synthesis
Extreme Transmission Angles Grashof Crank Rocker
• For a Grashof fourbar,
extreme values occur
when crank is collinear
with ground
m2
b
c
a
d
m1
For the extended position shown:
m1=arccos [ (b2+(a+d) 2 - c2)/2b (a+d) ]
m2=180° - arccos [ (b2+c2 - (a+d)2 )/2b c ]
17
Position synthesis
Extreme Transmission Angles Grashof Crank Rocker
For the overlapped case shown:
m2
b
m1=__________________________
a
c
m1 d
m2=__________________________
18
Position synthesis
Extreme Transmission Angles Grashof Double Rocker
• Remember: coupler
makes a full revolution
with respect to
rockers
• Transmission angle
varies from 0° to 90°
19
Position synthesis
Extreme Transmission Angles Non-Grashof Linkage
• Transmission angle is zero m b m =0
1
2
degrees in toggle position:
a
output rocker & coupler
d
• Other transmission angle given
as:
m2=__________________________
• Similar analysis for other toggle
position
20
c
Position synthesis
Calculation of Toggle Angles
• The input angle, q2 , for the first
toggle position given as:
a
b
q2
q2=__________________________
• Similar analysis for the other toggle
position
21
c
d
Position synthesis