Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V...
Transcript of Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V...
![Page 1: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/1.jpg)
• Time derivatives of the loop-closure expressions allow the analysis of velocities & accelerations, i.e.:
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
VELOCITY CLOSURE
ACCELERATION CLOSURE
![Page 2: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/2.jpg)
• Review of time derivatives of displacement
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 3: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/3.jpg)
• Example: Velocity analysis of the offset slider-crank
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
2
3
1
0
Displacement closure:
![Page 4: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/4.jpg)
• Taking the time derivative:
• Rearranging and substituting:
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 5: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/5.jpg)
• Taking the time derivative:
• Rearranging and substituting:
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 6: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/6.jpg)
• Taking the time derivative:
• Rearranging and substituting:
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 7: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/7.jpg)
• Taking the time derivative:
• Rearranging and substituting:
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 8: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/8.jpg)
• Splitting into real and imaginary eqns
• The solution for is obtained by solving the imaginary equation as:
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
REAL
IMAGINARY
![Page 9: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/9.jpg)
• Substituting this solution back into the real equation gives the other unknown:
Velocity & Acceleration Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
REAL EQUATION
![Page 10: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/10.jpg)
• The velocity of one point can be expressed as the velocity of another point, plus the relative velocity of the two points
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 11: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/11.jpg)
• Example
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
![Page 12: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/12.jpg)
• Example
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
VB
Absolute velocity of point B
![Page 13: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/13.jpg)
• Example
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
VA
Absolute velocity of point A
![Page 14: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/14.jpg)
• Example
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
VB/A
Relative velocity of point B w.r.t. point A
![Page 15: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/15.jpg)
• Example
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
VA
VB/A
VB
![Page 16: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/16.jpg)
• Note that:
– The direction and magnitude of VA is a known function of the input angular velocity, ω2
– The mechanism’s joints define the direction of many of the remaining relative and absolute velocities
– This information can be manipulated to find the velocity (direction and magnitude) of points not on the input link
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 17: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/17.jpg)
• There are 4 distinct cases where relative velocity analysis is applied (though only 3 are non-trivial)
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
Same Point Different Points
SameLink
Different Links
Case 1TRIVIAL CASE
Case 2DIFFERENCE
MOTION
Case 3RELATIVE MOTION
Case 4DIFFERENCE & RELATIVE
MOTION
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• Case 2: Different points on the same link
– Want to find velocity of point B w.r.t. point A (VB|A)
– Take the derivative of the rel. position vector (RB|A)
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
VA
VB/A
VB
![Page 19: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/19.jpg)
• Case 2: Different points on the same link
– Examining this result gives simple method for calculation:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
VA
VB/A
VB
Always = 0 for a rigid link(no length change)
Equivalent to rotation through 90° in the sense (CW or CCW) of ωB|A (i.e. ω3)
So, for a rigid link :
VB|A = (rB|A)(ω3), ┴ RB|A
![Page 20: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/20.jpg)
• Case 2: Different points on the same link
– Recalling that VB = VA + VB|A we can set up a system of equations to solve for VB:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
RA
RB/A
RB
A
B
O4O2
θ2
θ3
θ4
ω2
VA
VB/A
VB
Tricky to approach analytically, but graphical methods can be used, and can be much more intuitive
![Page 21: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/21.jpg)
• Case 2 Example (Supp Ex V1)
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
![Page 22: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/22.jpg)
• Solve using a graphical method
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
![Page 23: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/23.jpg)
• Find VB by relative velocity analysis:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
0V
1 mm = 5 mm/s
![Page 24: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/24.jpg)
• VA: direction, magnitude both known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
0V
1 mm = 5 mm/s
![Page 25: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/25.jpg)
• VA: direction, magnitude both known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
1 mm = 5 mm/sC
B
AO2 O4
3 4
2
ω2
![Page 26: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/26.jpg)
0V
VA
1 mm = 5 mm/s
A
• VA: direction, magnitude both known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
![Page 27: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/27.jpg)
0V
VA
1 mm = 5 mm/s
A
• VB|A: direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
![Page 28: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/28.jpg)
• VB|A: direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
0V
VA
1 mm = 5 mm/s
A
![Page 29: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/29.jpg)
• VB|A: direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
0V
VA
dir(VB|A )
1 mm = 5 mm/s
A
![Page 30: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/30.jpg)
• VB: direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
VA
dir(VB|A )
1 mm = 5 mm/s
A
C
B
AO2 O4
3 4
2
ω2
![Page 31: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/31.jpg)
• VB: direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
0V
VA
dir(VB|A )
1 mm = 5 mm/s
A
![Page 32: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/32.jpg)
• VB: direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
A
![Page 33: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/33.jpg)
• Solution is obtained by intersection & measurement
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
A
C
B
AO2 O4
3 4
2
ω2
![Page 34: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/34.jpg)
• Solution is obtained by intersection & measurement
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
A
BVB
C
B
AO2 O4
3 4
2
ω2
![Page 35: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/35.jpg)
• Solution is obtained by intersection & measurement
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
A
BVB
X
C
B
AO2 O4
3 4
2
ω2
![Page 36: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/36.jpg)
• Noticing that , we measure:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
A
BVB
![Page 37: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/37.jpg)
• Noticing that , we measure:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
VB|A
A
BVB
X
![Page 38: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/38.jpg)
• Noticing that , we measure:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
VB|A
A
BVB
X
![Page 39: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/39.jpg)
• And compute:
• Where the direction of rotation is inferred from the direction of
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
VB|A
A
BVB
X X
![Page 40: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/40.jpg)
• Similarly:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
VB|A
A
BVB
X X X
![Page 41: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/41.jpg)
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
VB|A
A
BVB
• Now, VC can be found in a variety of ways:
– Intersect relative velocity directions w.r.t. A & B
– Compute directly, e.g. VC = VA+(ω3 X AC)
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X X
![Page 42: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/42.jpg)
• Using the first method, note that:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
X X X
C
B
AO2 O4
3 4
2
ω2
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
VB|A
A
BVB
![Page 43: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/43.jpg)
• Using the first method, note that:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X
and ,
X
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
VB|A
A
BVB
![Page 44: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/44.jpg)
• Using the first method, note that:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X
and ,
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
dir(VC|A)
VB|A
A
BVB
X
![Page 45: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/45.jpg)
• Using the first method, note that:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X
and ,
X
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
dir(VC|A)
VB|A
A
BVB
![Page 46: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/46.jpg)
• Using the first method, note that:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X
and ,
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
dir(VC|A)
VB|A
dir(V
C|B )
A
BVB
X
![Page 47: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/47.jpg)
• Intersection gives the solution for VC
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X X
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
dir(VC|A)
VB|A
dir(V
C|B )
A
BVB
![Page 48: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/48.jpg)
• Intersection gives the solution for VC
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
dir(VC|A)
VB|A
VC
dir(V
C|B )
A C
BVB
X
![Page 49: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/49.jpg)
• Intersection gives the solution for VC
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
X X
0V
VA
dir(VB)
dir(VB|A )
1 mm = 5 mm/s
dir(VC|A)
VB|A
VC
dir(V
C|B )
A C
BVB
X X
![Page 50: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/50.jpg)
• Another Case 2 Example (Supp Ex V2)
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
4
![Page 51: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/51.jpg)
• Solve using the same graphical method:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
1 mm = 10 mm/s
![Page 52: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/52.jpg)
• Find VB by case 2 analysis:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
1 mm = 10 mm/s
![Page 53: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/53.jpg)
• VA : magnitude, direction both known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
1 mm = 10 mm/s
![Page 54: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/54.jpg)
• VA : magnitude, direction both known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
1 mm = 10 mm/s
![Page 55: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/55.jpg)
• VA : magnitude, direction both known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
AV
A
1 mm = 10 mm/s
![Page 56: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/56.jpg)
• VB|A : direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
AV
A
1 mm = 10 mm/s
![Page 57: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/57.jpg)
• VB|A : direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
AV
A
1 mm = 10 mm/s
![Page 58: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/58.jpg)
• VB|A : direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
dir
(VB
|A)
AV
A
1 mm = 10 mm/s
![Page 59: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/59.jpg)
• VB : direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0Vdir
(VB
|A)
AV
A
1 mm = 10 mm/s
C
B
A
O23
2
ω2
4
![Page 60: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/60.jpg)
• VB : direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0Vdir
(VB
|A)
AV
A
dir(VB)
1 mm = 10 mm/s
C
B
A
O23
2
ω2
4
![Page 61: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/61.jpg)
• Intersection & measurement give:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
dir
(VB
|A)
AV
A
dir(VB) VBV
B|A
B
1 mm = 10 mm/s
![Page 62: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/62.jpg)
• Intersection & measurement give:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
dir
(VB
|A)
AV
A
dir(VB) VBV
B|A
B
1 mm = 10 mm/s
X
![Page 63: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/63.jpg)
• And ω3 is found from:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
4
X0V
dir
(VB
|A)
AV
A
dir(VB) VBV
B|A
B
1 mm = 10 mm/s
X
![Page 64: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/64.jpg)
• Solve for VC by intersection:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
dir
(VB
|A)
AV
A
dir(VB) VBV
B|A
B
1 mm = 10 mm/s
X X
![Page 65: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/65.jpg)
• Solve for VC by intersection:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
dir
(VB
|A)
AV
A
dir(VB) VBV
B|A
B
dir(V
C|A )
1 mm = 10 mm/s
X X
![Page 66: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/66.jpg)
• Solve for VC by intersection:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
dir
(VB
|A)
AV
A
dir(VB) VBV
B|A
dir(VC|B
)
B
dir(V
C|A )
1 mm = 10 mm/s
X X
![Page 67: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/67.jpg)
• Measuring then gives:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
A
O23
2
ω2
40V
dir
(VB
|A)
AV
A
dir(VB) VBV
B|A
dir(VC|B
)
B
C VC
dir(V
C|A )
1 mm = 10 mm/s
X XX
![Page 68: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/68.jpg)
• Case 3: Coincident points on different links
– Occurs for slides & pistons, cams & followers:
• Two points on different links momentarily occupy the same point in the plane
• Each has a different absolute velocity, therefore a relative velocity exists
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 69: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/69.jpg)
• Case 3: Coincident points on different links
– Calculate the slide (relative) velocity, VB3|B4
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
B2, B3, B4
42
3
• So far, we have taken the derivative of the relative position vector, RB3|B4
• But how can we express this vector for two coincident points?
• Intuitively, we can imagine displacing the slide by some small distance along the slide, then drawing RB3|B4
• Taking the limit as the displacement approaches zero, we can see that RB3|B4
has zero length, and is directed along the tangent to the slide (link 4) at point B
![Page 70: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/70.jpg)
• Case 3: Coincident points on different links
– Calculate the slide (relative) velocity, VB3|B4
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
θslide
B2, B3, B4
42
3
• So:
• Taking the derivative:
![Page 71: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/71.jpg)
• Case 3: Coincident points on different links
– Calculate the slide (relative) velocity, VB3|B4
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
θslide
B2, B3, B4
42
3
• Simplifying gives the final expression:
• Note that this deceptively simple expression hides the potentially difficult task of finding the slide tangent angle
• In the following examples, straight slides are used to avoid this hassle (tangent angle = link angle for a straight slide)
![Page 72: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/72.jpg)
• Case 3 Example (Supp Ex V3)
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
O2
2
ω2
4
O4
3
![Page 73: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/73.jpg)
C
B
O2
2
ω2
4
O4
3
• Solve graphically
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
1 mm = 10 mm/s
![Page 74: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/74.jpg)
C
B
O2
2
ω2
4
O4
3
• Use case 3 analysis to find VB4:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
1 mm = 10 mm/s
![Page 75: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/75.jpg)
C
B
O2
2
ω2
4
O4
3
0V
1 mm = 10 mm/sC
B
O2
2
ω2
4
O4
3
• VB2: direction, magnitude both known:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
B2
VB2
1 mm = 10 mm/s
![Page 76: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/76.jpg)
0V
B2
VB2
1 mm = 10 mm/s
0V
B2
VB2
dir(V
B2|B
4 )
1 mm = 10 mm/sC
B
O2
2
ω2
4
O4
3
• VB2|B4: only direction is known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
O2
2
ω2
4
O4
3
![Page 77: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/77.jpg)
C
B
O2
2
ω2
4
O4
3
C
B
O2
2
ω2
4
O4
3
0V
B2
VB2
dir(V
B2|B
4 )
1 mm = 10 mm/s
0V
dir(VB4)
B2
VB2
dir(V
B2|B
4 )
1 mm = 10 mm/s
• VB4: direction known, magnitude unknown
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
![Page 78: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/78.jpg)
• Obtain VB4 by intersection
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
O2
2
ω2
4
O4
3
0V
dir(VB4)
B2
VB2
dir(V
B2|B
4 )
VB4
B4
VB
2|B
4
1 mm = 10 mm/s
X
![Page 79: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/79.jpg)
• ω4 follows immediately from VB4:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
O2
2
ω2
4
O4
3
0V
dir(VB4)
B2
VB2
dir(V
B2|B
4 )
VB4
B4
VB
2|B
4
1 mm = 10 mm/s
X X
![Page 80: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/80.jpg)
• VC follows immediately from ω4, or by:
•
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
O2
2
ω2
4
O4
3
0V
dir(VB4)
B2
VB2
dir(V
B2|B
4 )
VB4
B4
VB
2|B
4
1 mm = 10 mm/s
X X
![Page 81: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/81.jpg)
• VC follows immediately from ω4, or by:
• ,
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
O2
2
ω2
4
O4
3
X X0V
dir(VB4)
B2
VB2
dir(V
B2|B
4 )
VB4
B4
VB
2|B
4
dir(VC|B4)
1 mm = 10 mm/s
![Page 82: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/82.jpg)
• VC follows immediately from ω4, or by:
• , ,
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
O2
2
ω2
4
O4
3
X X0V
dir(VB4)
B2
VB2
dir(V
B2|B
4 )
VB4
B4
VB
2|B
4
dir
(VC)
dir(VC|B4)
1 mm = 10 mm/s
![Page 83: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/83.jpg)
C
B
O2
2
ω2
4
O4
3
• Measuring:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
0V
dir(VB4)
B2
VB2
dir(V
B2|B
4 )
VB4
B4
VB
2|B
4
dir
(VC)
dir(VC|B4)
C
VC
1 mm = 10 mm/s
X X X
![Page 84: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/84.jpg)
• Another Case 3 Example (Supp Ex V4)
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
![Page 85: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/85.jpg)
• Note: The 4-Bar was solved in Ex V1
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
![Page 86: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/86.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• VC is found by scaling arguments:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
VA
VB
VB|A
A
B
0V
1 mm = 5 mm/s
![Page 87: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/87.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• VC is found by scaling arguments:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
VA
VB
VB|A
A
B
VC|A
0V
1 mm = 5 mm/s
![Page 88: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/88.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• Measuring:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
VA
VB
VB|A
A
B
VC|A
0V
C
VC
1 mm = 5 mm/s
X
![Page 89: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/89.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• VD5 is found by Case 3 analysis:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
X
VA
VB
VB|A
A
B
VC|A
0V
C
VC
1 mm = 5 mm/s
![Page 90: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/90.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• VD5 : only the direction is known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
X
VA
VB
VB|A
A
B
VC|A
dir(VD5 )
0V
C
VC
1 mm = 5 mm/s
![Page 91: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/91.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• VD5|C : only the direction is known
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
X
VA
VB
VB|A
A
B
VC|A
dir(VD5 )
0V
dir(VD5|C
)
C
VC
1 mm = 5 mm/s
![Page 92: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/92.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• Measuring:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
X
VA
VB
VB|A
A
B
VD5
VC|A
dir(VD5 )
0V
dir(VD5|C
)
VD5|C
C
D5
VC
1 mm = 5 mm/s
X
![Page 93: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/93.jpg)
C
B
AO2 O4
3 4
2
ω2
D5, D6
5
6
• ω5 is found immediately from:
Relative Velocity Analysis
MECH 335 Lecture Notes © R.Podhorodeski, 2009
VA
VB
VB|A
A
B
VD5
VC|A
dir(VD5 )
0V
dir(VD5|C
)
VD5|C
C
D5
VC
1 mm = 5 mm/s
X X X
![Page 94: Velocity & Acceleration Analysis - ariel.ac.il · •Note that: –The direction and magnitude of V A is a known function of the input angular velocity, ω 2 –The mechanism’s](https://reader034.fdocuments.us/reader034/viewer/2022042209/5ead390cbc8dc47aad5088c9/html5/thumbnails/94.jpg)
END OF LECTURE PACK 3