14 COMPUTER AIDED ENGINEERING DESIGN
Δεε
θ
Δθε
θ
2 2 Δθ θ Δθ θ Δθ
3
32
4= + 43 = cos 334 + cos 44∂
∂ ′∂
∂ ′
rr′′- sin – sin
cos cos
= =- sin – sin
cos cos
3344
33444
41
23
43344
3344rr
rrrr
rr′′
′′⎡
⎣⎢⎤
⎦⎥⎡
⎣⎢⎤
⎦⎥⎡
⎣⎢⎤
⎦⎥⇒⎡
⎣⎢⎤
⎦⎥′′
′′⎡
⎣⎢θθ ⎤
θθΔθ
ΔθΔε
ΔεΔθ
Δθθθ
θθ⎦⎦⎥⎡
⎣⎢⎤
⎦⎥–1
1
2Δε
Δε
(1.5)
This gives a recursive relationship
=+
+3new4new3old
3
4old
4θ
θθΔθ
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 θ3
2
44
4
2: – sin – sin – sin = 0dY
d
rr
d
d
r
d
θ d
θθ
θ
θ
θ
θ
2 223 3 θ3
2
44
4
2: cos + cos + cos = 0⇒- sin – sin
cos cos
=sin- cos
33 44
33443
2
4
222
22rr
rrd
d
d
dr
rθθ
θθθ
θ
θ
θθ
θ⎡
⎣
⎢⎤
⎦
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡
⎣
⎢⎤
⎦
⎥⇒ =- sin – sin
cos cos
sin- cos
= (say)3
2
4
233 44
3344–1
22
223
4d
d
d
drr
rrr
rh
hθ
θ
θ
θθθ
θθθ
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