Computer Aided Engineering Design

INTRODUCTION 13

unknown magnitude and known direction. The components of the vectors along x and y axes can be
expressed as:

X : r 2 cosθ 2 + r 3 cos θ 3 + r 4 cos θ 4 – r 1 cos θ 1 = 0

Y : r 2 sin θ 2 + r 3 sin θ 3 + r 4 sin θ 4 – r 1 sin θ 1 = 0 (1.2)

Here,θ 2 is the known crank angle and ωθαθ 2222 = ̇ ̇ ̇, = are also given. Since θ 1 = 0, Eq. (1.2) is
reduced to

X : r 2 cosθ 2 + r 3 cos θ 3 + r 4 cos θ 4 – r 1 = 0

Y: r 2 sinθ 2 + r 3 sin θ 3 + r 4 sin θ 4 = 0 (1.3)

Evaluating Link Positions
Eq. (1.3) is nonlinear if they are to be solved for θ 3 and θ 4 for given steps of θ 2. Newton’s method
converts the problem into an iterative algorithm suitable for computer implementation. Let the

estimated values be (, ).θθ 34 ′′ If the guess is not correct, Eqs. (1.3) will be different from zero, in

general. Let the errors be given by:

Xr : cos 22334411 θθθε + cos r ′′ + cos r – = r

Yr : sin 2233442 θθθε + sin r ′′ + sin r = (1.4)

For small changes (Δθ 3 ,Δθ 4 ) in the change in error (Δε 1 ,Δε 2 ) is given by the Taylor’s
series expansion up to the first order. That is

Δε

ε

θ

Δθ

ε

θ

1 1 Δθ θ Δθ θ Δθ

3

3

1

4

= + 4333444 = – sin – sin

∂

∂ ′

∂

∂ ′

rr′′

Figure 1.3 Schematic of a four-bar mechanism

B

4

K

3

A

2

O

1 r 1

r 3

r 2

r 4

x

θ 3

θ 4

θ 2

y