MECH 314 Rolling without Slipping 7-Bar Mechanism Velocity ... - CIM

2
⎡
⎣
0
0
ωBD
⎤
⎡
⎦ × ⎣
rBDx
rBDy
0
⎤
⎡
⎦ + ⎣
0
0
ωDF
⎤
⎡
⎦ × ⎣
rDF x
rDF y
0
⎤
⎡
⎦ + ⎣
0
0
ωEF G
⎤
⎡
⎦ × ⎣
rF Gx
rF Gy
0
⎤
⎡
⎦ − ⎣
0
0
ωGH
⎤
⎡
⎦ × ⎣
rHGx
rHGy
0
⎤
⎡
⎦ = ⎣
These lead to a detached coefficient form of four simultaneous linear equations in five homogeneous variables
⎡
−ωACrACy
⎢ ωACrACx ⎢
⎣ −ωBDrBDy
ωBDrBDx
−rCEy
rCEx
0
0
0
0
−rDF y
rDF x
−rEGy
rEGx
−rF Gy
rF Gx
rHGy
−rHGx
rHGy
−rHGx
⎡
⎤ ∆
⎢ ΩCE ⎥ ⎢
⎥ ⎢ ΩDF ⎦ ⎢
⎣ ΩEF G
⎤
⎥
⎦ =
⎡ ⎤
0
⎢ 0 ⎥
⎢ 0 ⎥
⎣ 0 ⎦
0
such that
where
ΩGH
ωCE = ΩCE
∆ , ωDF = ΩDF
∆ , ωEF G = ΩEF G
∆ , ωGH = ΩGH
∆
∆ =
ΩCE = −
ΩDF =
ΩEF G = −
ΩGH =
−rCEy 0 −rEGy rHGy
rCEx 0 rEGx −rHGx
0 −rDF y −rF Gy rHGy
0 rDF x rF Gx −rHGx
−ωACrACy 0 −rEGy rHGy
ωACrACx 0 rEGx −rHGx
−ωBDrBDy −rDF y −rF Gy rHGy
ωBDrBDx rDF x rF Gx −rHGx
−ωACrACy −rCEy −rEGy rHGy
ωACrACx rCEx rEGx −rHGx
−ωBDrBDy 0 −rF Gy rHGy
ωBDrBDx 0 rF Gx −rHGx
−ωACrACy −rCEy 0 rHGy
ωACrACx rCEx 0 −rHGx
−ωBDrBDy 0 −rDF y rHGy
ωBDrBDx 0 rDF x −rHGx
−ωACrACy −rCEy 0 −rEGy
ωACrACx rCEx 0 rEGx
−ωBDrBDy 0 −rDF y −rF Gy
ωBDrBDx 0 rDF x rF Gx
The link vectors are obtained by subtracting the coordinates of the first point subscript from the second, e.g.,
differences of point position vectors such as rAC = c − a.
⎡ ⎤
1.334
⎡
4.623
⎤ ⎡ ⎤
5.587
rAC = ⎣ 5.850 ⎦ , rCE = ⎣ 1.904 ⎦ , rEG = ⎣ 4.218 ⎦
⎡ ⎤
3.354
0
⎡ ⎤
−0.951
0
⎡ ⎤
−0.859
0
⎡
−4.456
⎤
rBD = ⎣ 2.179 ⎦ , rDF = ⎣ 2.845 ⎦ , rF G = ⎣ 6.948 ⎦ , rHG = ⎣ −2.028 ⎦
0
0
0
0
3 Results and a Graphical Check
Substituting the numerical values for the parameters in the detached coefficient matrix yields
⎡
⎢
⎣
−17.55
4.002
9.8055
−1.904
4.623
0
0
0
−2.845
−4.218
5.587
−6.948
⎤
−2.028
4.456 ⎥
−2.028 ⎦
−15.093 0 −0.951 −0.859 4.456
0
0
0
⎤
⎦