14 COMPUTER AIDED ENGINEERING DESIGN
Δε
ε
θ
Δθ
ε
θ
2 2 Δθ θ Δθ θ Δθ
3
3
2
4
= + 43 = cos 334 + cos 44
∂
∂ ′
∂
∂ ′
rr′′
- sin – sin
cos cos
= =
- sin – sin
cos cos
3344
3344
4
4
1
2
3
4
3344
3344
rr
rr
rr
rr
′′
′′
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥⇒
⎡
⎣
⎢
⎤
⎦
⎥
′′
′′
⎡
⎣
⎢
θθ ⎤
θθ
Δθ
Δθ
Δε
Δε
Δθ
Δθ
θθ
θθ⎦⎦⎥
⎡
⎣
⎢
⎤
⎦
⎥
–1
1
2
Δε
Δε
(1.5)
This gives a recursive relationship
=
+
+
3
new
4
new
3
old
3
4
old
4
θ
θ
θΔθ
θΔθ
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥ (1.6)
The iteration is started with some estimated values of (, ).θθ 34 ′′ From Eq. (1.4) (ε 1 ,ε 2 ) is computed
and in the first step (Δε 1 ,Δε 2 ) is assigned as (ε 1 ,ε 2 ). Eq. (1.5) is solved to get (Δε 3 ,Δε 4 ) and (θ 3 ,θ 4 )
are updated using Eq. (1.6). Using the new values of (θ 3 ,θ 4 ), Eq. (1.4) is solved again to get (ε 1 ,ε 2 ).
This time (Δε 1 ,Δε 2 ) is computed as the difference between the current and previous values of (ε 1 ,ε 2 ).
Eqs. (1.4)-(1.6) are repeatedly solved until (Δε 1 ,Δε 2 ) and thus (Δθ 3 ,Δθ 4 ) are desirably small. For
givenθ 2 , therefore, positions of links 3 and 4(i.e. θ 3 and θ 4 ) are determined. For different values of
θ 2 , the procedure can be implemented to get the entire set of positions for the linkages 3 and 4.
Kinematic Coefficients, and Link Velocity and Acceleration
Consider Eq. (1.3) and note that θ 2 is the independent variable and (θ 3 ,θ 4 ) are dependent variables
(link lengths are constant). On differentiating Eq. (1.3) with respect to θ 2 on both sides
dX
d
rr
d
d
r
d
θ d
θθ
θ
θ
θ
θ
2 2233 θ
3
2
44
4
2
: – sin – sin – sin = 0
dY
d
rr
d
d
r
d
θ d
θθ
θ
θ
θ
θ
2 223 3 θ
3
2
44
4
2
: cos + cos + cos = 0
⇒
- sin – sin
cos cos
=
sin
- cos
33 44
3344
3
2
4
2
22
22
rr
rr
d
d
d
d
r
r
θθ
θθ
θ
θ
θ
θ
θ
θ
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎤
⎦
⎥
⇒ =
- sin – sin
cos cos
sin
- cos
= (say)
3
2
4
2
33 44
3344
–1
22
22
3
4
d
d
d
d
rr
rr
r
r
h
h
θ
θ
θ
θ
θθ
θθ
θ
θ
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥ (1.7)
h
d
d
h
d
(^3) d
3
2
4
4
2
= and =
θ
θ
θ
θ
are called the Kinematic Coefficients (KC) of the four bar mechanism with
respect to the driver crank. From Eq. (1.7), it can be observed that KC’s are functions of the link
lengths and instantaneous values of the angles. They are constants for a given position of the input
link. At any instant of time, to get the angular velocities ω 3 and ω 4 , and angular accelerations α 3
andα 4 of links 3 and 4, respectively
ω
θθ
θ
θ
ωω
θθ
θ
θ
3 33 ω
2
2
32 4
44
2
2
= = = ; = = = 42
d
dt
d
d
d
dt
h
d
dt
d
d
d
dt
h