Formula Sheet for Exam 3 (Spring 2022)
Rotation matrix:
1
[𝑅𝑋,𝜙 ] = [0
0
0
𝑐𝑜𝑠 𝜙
𝑠𝑖𝑛 𝜙
0
𝑐𝑜𝑠 𝜃
− 𝑠𝑖𝑛 𝜙] , [𝑅𝑌,𝜃 ] = [ 0
𝑐𝑜𝑠 𝜙
− 𝑠𝑖𝑛 𝜃
𝑢𝑥2 𝑣(𝛿) + cos 𝛿
[𝑅𝛿,𝑢 ] = [𝑢𝑥 𝑢𝑦 𝑣(𝛿) + 𝑢𝑧 sin 𝛿
𝑢𝑥 𝑢𝑧 𝑣(𝛿) − 𝑢𝑦 sin 𝛿
𝑢𝑥 𝑢𝑦 𝑣(𝛿) − 𝑢𝑧 sin 𝛿
𝑢𝑦2 𝑣(𝛿) + cos 𝛿
𝑢𝑦 𝑢𝑧 𝑣(𝛿) + 𝑢𝑥 sin 𝛿
𝑐𝑜𝑠 𝜓
0 𝑠𝑖𝑛 𝜃
1
0 ] , [𝑅𝑍,𝜓 ] = [ 𝑠𝑖𝑛 𝜓
0 𝑐𝑜𝑠 𝜃
0
𝑢𝑥 𝑢𝑧 𝑣(𝛿) + 𝑢𝑦 sin 𝛿
𝑢𝑦 𝑢𝑧 𝑣(𝛿) − 𝑢𝑥 sin 𝛿 ] ,
𝑢𝑧2 𝑣(𝛿) + cos 𝛿
Statics of mechanism
Fao + Fa1 = 0
− 𝑠𝑖𝑛 𝜓
𝑐𝑜𝑠 𝜓
0
0
0]
1
𝑣(𝛿) = 1 − 𝑐𝑜𝑠 𝛿
Dynamics of mechanism
Fa 0 + Fa1 = m1aCG1
Tao + W1  Fa1 = 0
Ta 0 + R1 j  Fa 0 + R2  j  Fa1 = I1 j
Gears
Disk Cam Profile
Pitch circle
P=Pitch point
Driving
Driven
Equations for 3D kinematics:
 = 0 or locked
a
a
b1
a1
a0
 0
b1

a
b1

a1

a0
b1 =  R ,uˆa  ( b1 − a0 ) + a0
0 

b
b1
a1
a0
a =  R ,uˆa  ( a1 − a0 ) + a0
0 

b1
b =  R ,uˆa  ( b1 − a ) + a
uˆa =  R ,uˆa  uˆa1
0 
