Transcript Linkage analysis - Stanford University
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Analytic Approach to Mechanism Design http://www.engr.colostate.edu/me/program/courses/ME324/notes/PositionAnalysis.ppt
ME 324 Fall 2000
Position synthesis
Chapter 4 Analytic Position Analysis Imaginary Axis
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• 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 jR sin q R A q R cos q Point A Real Axis
Position synthesis
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Derivatives, Vector Rotations in the Complex Plane • Taking a derivative of a complex number will result in multiplication by j B C • Each multiplication by j rotates a vector 90 °
R C
=
j
2 R = -R CCW in the complex plane
R D
=
j
3 R = -
j
R
R B
=
j
R D Imaginary
R A
A Real
Position synthesis
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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
Link 3, length b B Coupler Link 2, length a A Pivot 02 Link 1, length d Ground Link Link 4, length c Pivot 04 Position synthesis
Angle Measurement Convention
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• All angles measured from angle of the ground link • Define q 1 = 0 ° • One DOF, so can describe all angles in terms of one input, usually q 2
2 3
q 3
A
q 2
1
q 1 = 0 °
B 4
q 4
Position synthesis
More on Complex Notation
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• Polar form:
re j
q • Cartesian form:
r
cos q +
j r
sin q • Euler identity: ±
e j
q = cos q ±
j
sin q • Differentiation:
j
q
de d
q =
je j
q
Position synthesis
The Vector Loop Technique • Vector loop equation:
R 2
+
R 3
-
R 4
-
R 1
= 0
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• Alternative notation:
R AO2
+
R BA
-
R BO4
-
R O4O2
= 0
R 3
nomenclature - tip then tail • Complex notation: ae j q 2 + be j q 3 - ce j q 4 - de j q 1 = 0
R 2
O
a A
q 2
d
• Substitute Euler equation: 2 a (cos q 2 +j sin q 2 ) + b (cos q 3 +j sin q 3 ) - c (cos q 4 +j sin q 4 ) - d (cos q 1 +j sin q 1 ) = 0 q 3
b R 1 B c R 4
O 4 q 4
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Vector Loop Technique - continued • Separate into real and imaginary parts: Real: a cos q 2 a cos q 2 since q 1 + b cos q 3 + b cos q 3 = 0, cos q 1 - c cos q 4 - c cos q 4 = 1 - d cos q 1 - d = 0 , = 0
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Imaginary: ja sin q 2 + jb sin q 3 a sin q 2 since q 1 + b sin q = 0, sin 3 q 1 - jc sin = 0 q - c sin q 4 4 - jd sin = 0 , q 1 = 0
Position synthesis
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Vector Loop Technique continued a cos q 2 a sin q 2 + b cos + b sin q 3 q 3 - c cos - c sin q 4 q 4 - d = 0 = 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
Position synthesis
Vector Loop Summary
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• 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
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Example: Analytic Position Analysis • Input position q 2 • Solve for q 3 & q 4 given b=2.14
q 3 =?
° a=1.6
c=2.06
q 2 =51.3
° d=3.5
q 4 =?
°
Position synthesis
Example: Vector Loop Equation
12 R 2
+
R 3
ae j q 2 + be j
R
q 3
4
-
R 1
- ce j q 4 1.6e
j51.3Þ + 2.14e
j = 0 q 3 - de j q 1 = 0 - 2.06e
j q 4 0 - 3.5
e j 0 ° = b=2.14
q 3 =?
°
R 3 R 2
c=2.06
a=1.6
R 4
q 2 =51.3
° d=3.5
q 4 =?
°
R 1 Position synthesis
Example: Analytic Position Analysis ae j q 2 + be j q 3 - ce j q 4 - de j q 1 = 0 a(cos q 2 +jsin q 2 ) + b(cos q 3 +jsin q 3 ) - c(cos q 4 +jsin q 4 ) - d(cos q 1 +jsin q 1 )=0 Real part: a cos q 2 + b cos q 3 - c cos q 4 1.6 cos 51.3
+ 2.14 cos q 3 - 2.06 cos q 4 - 3.5 = 0 - d = 0 a=1.6
Imaginary part: a sin q 2 + b sin q 3 - c sin q 4
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1.6 sin 51.3
+ 2.14 sin q 3 - 2.06 sin q 4 = 0 = 0 q 2 =51.3
° b=2.14
q 3 =?
° d=3.5
c=2.06
Position synthesis
Solution: Open Linkage 2 equations from real & imaginary equations 1.6 cos 51.3
1.6 sin 51.3
+ 2.14 cos q 3 + 2.14 sin q 3 - 2.06 cos - 2.06 sin q 4 q 4 - 3.5 = 0 = 0 2 unknowns: q 3 & q 4 b=2.14
Solve simultaneously to yield 2 solutions.
q 3 =21Þ a=1.6
Open solution:
c=2.06
q
3 = 21Þ,
q
4 = 104 °
q 2 =51.3Þ d=3.5
q 4 =104Þ
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Review - Law of Cosines cos q =
A
2 +
B
2 -
C
2 2
AB
q = arccos
A
2 +
B
2 2
AB
-
C
2 A q B C
Position synthesis
Transmission Angles
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• 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° 180 m 1 as possible m 1 acute 180 m 2 m 2
Position synthesis
Extreme Transmission Angles Grashof Crank Rocker
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• For a Grashof fourbar, extreme values occur when crank is collinear with ground b m 2 a m 1 For the extended position shown: m 1 =arccos [ (b 2 +(a+d) 2 - c 2 )/2b (a+d) ] m 2 =180 ° - arccos [ (b 2 +c 2 - (a+d) 2 )/2b c ] d c
Position synthesis
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Extreme Transmission Angles Grashof Crank Rocker For the overlapped case shown: m 1 =__________________________ m 2 =__________________________ a b m 1 d m 2 c
Position synthesis
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Extreme Transmission Angles Grashof Double Rocker • Remember: coupler makes a full revolution with respect to rockers • Transmission angle varies from 0 ° to 90°
Position synthesis
Extreme Transmission Angles Non-Grashof Linkage
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• Transmission angle is zero degrees in toggle position: output rocker & coupler a m 2 b • Other transmission angle given as: m 2 =__________________________ • Similar analysis for other toggle position m 1 =0 d c
Position synthesis
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Calculation of Toggle Angles • The input angle, q 2 , for the first toggle position given as: b a q 2 q 2 =__________________________ • Similar analysis for the other toggle position d c
Position synthesis