session #9 3d mechanisms: mobility, analysis, synthesis€¦ · mobility analysis • each body has...
TRANSCRIPT
![Page 1: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/1.jpg)
Session #93D Mechanisms:
Mobility, Analysis, Synthesis
![Page 2: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/2.jpg)
Concept Question
• How many DOF does this mechanism possess?
1)1 2)2 3)0 4) I don’t know
![Page 3: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/3.jpg)
Types of Kinematic Pairs
Source: Figure 4 in Suh, C. H. Computer Aided Design of Mechanisms Part B. Self-published book, 1992. Courtesy of Dr. C. H. Suh. Used with permission.
![Page 4: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/4.jpg)
Mobility Analysis
• Each body has 6 DOF• Different joints remove
different #s of DOF• R – 5• C – 4• P – 5• S – 3• H – 5• PL – 3
Source: Figure 4 in Suh, C. H. Computer Aided Design of Mechanisms Part B. Self-published book, 1992. Courtesy of Dr. C. H. Suh. Used with permission.
![Page 5: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/5.jpg)
Some 4 Bar Spatial Mechanisms
Source: Figure 7 (top) and Figure 8 (bottom) in Suh, C. H. Computer Aided Design of Mechanisms Part B. Self-published book, 1992. Courtesy of Dr. C. H. Suh. Used with permission.
![Page 6: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/6.jpg)
Challenge
• Compute the DOF of the generic RRRR mechanism
• Sketch an RRRR mechanism that has one DOF as a matter of fact
![Page 7: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/7.jpg)
Sarrus Mechanism
Source: Figure 16 in Suh, C. H. Computer Aided Design of Mechanisms Part B. Self-published book, 1992. Courtesy of Dr. C. H. Suh. Used with permission.
![Page 8: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/8.jpg)
Franke Mechanism
Source: Figure 17 in Suh, C. H. Computer Aided Design of Mechanisms Part B. Self-published book, 1992. Courtesy of Dr. C. H. Suh. Used with permission.
![Page 9: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/9.jpg)
Stewart Platform
7 Bodies12 S Joints
How many degrees of freedom (assuming the legs have a fixed length)?Where are the DOF?
Figure by MIT OCW.
Z
Y3.48
![Page 10: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/10.jpg)
Double Wishbone Suspension
Figure removed for copyright reasons.Exploded component view of AH Sprite/MG Midget Suspension and Steering.
![Page 11: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/11.jpg)
Challenge• Make a schematic of a double wishbone
suspension showing all rigid bodies (you may exclude the springs and dampers)– Upper wishbone– Lower wishbone– Kingpin– Knuckle– Wheel– Tie rod
• See if you can determine its mobility (how many DOF and where?)
![Page 12: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/12.jpg)
Formula SAE Suspension
http://web.mit.edu/fsae/concept/
Courtesy of MIT Motorsports. Used with permission.
![Page 13: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/13.jpg)
Challenge
• Make a schematic of just – the lower two arms– the upper two arms – the knuckle
• Is it equivalent to a double wishbone, kingpin, and a knuckle?
• Prepare an argument one way or other and be ready to “defend” it
![Page 14: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/14.jpg)
Multi-Link Suspension? Bodies? S Joints? R Joints? C Joints
How many DOF?Where are the DOF?
Figure removed for copyright reasons.Honda Accord suspension.
![Page 15: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/15.jpg)
McPherson Suspension
Car frame
Steering link
Shock absorberand spring
Lower control arm
Figure courtesy of “Casito.” Source: Wikipedia.
![Page 16: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/16.jpg)
HTMs in 3DRotation about the x axis Rotation about the y axis
Rx εx( )
1
0
0
0
0
cos εx( )sin εx( )
0
0
sin εx( )−
cos εx( )0
0
0
0
1
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟
⎠
≡ Ry εy( )
cos εy( )0
sin εy( )−
0
0
1
0
0
sin εy( )0
cos εy( )0
0
0
0
1
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟
⎠
≡
Rotation about the z axis Translation in all 3 directions
Rz εz( )
cos εz( )sin εz( )
0
0
sin εz( )−
cos εz( )0
0
0
0
1
0
0
0
0
1
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟
⎠
≡ T δx δy, δz,( )
1
0
0
0
0
1
0
0
0
0
1
0
δx
δy
δz
1
⎛⎜⎜⎜⎜⎜⎝
⎞
⎟⎟⎟
⎠
≡
![Page 17: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/17.jpg)
Simulating an R Joint
Ru u φ,( )
u1( )2 1 cos φ( )−( )⋅ cos φ( )+
u1 u2⋅ 1 cos φ( )−( )⋅ u3 sin φ( )⋅+
u1 u3⋅ 1 cos φ( )−( )⋅ u2 sin φ( )⋅−
0
u1 u2⋅ 1 cos φ( )−( )⋅ u3 sin φ( )⋅−
u2( )2 1 cos φ( )−( )⋅ cos φ( )+
u2 u3⋅ 1 cos φ( )−( )⋅ u1 sin φ( )⋅+
0
u1 u3⋅ 1 cos φ( )−( )⋅ u2 sin φ( )⋅+
u2 u3⋅ 1 cos φ( )−( )⋅ u1 sin φ( )⋅−
u3( )2 1 cos φ( )−( )⋅ cos φ( )+
0
0
0
0
1
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
≡
Rpu p u, φ,( ) T p1 p2, p3,( ) Ru u φ,( )⋅ T p1− p2−, p3−,( )⋅≡
![Page 18: Session #9 3D Mechanisms: Mobility, Analysis, Synthesis€¦ · Mobility Analysis • Each body has 6 DOF • Different joints remove different #s of DOF •R – 5 •C – 4 •P](https://reader034.vdocuments.us/reader034/viewer/2022052104/603f223ab108b802da6e7211/html5/thumbnails/18.jpg)
Next Steps
• Lab Friday 10 MAR• Next session Tuesday 14 MAR