Fundamental concepts 11
interest. Hence, here we shall concentrate on the kinematic analysis
methods that provide the essential information.
2.2.1 The vector method with complex numbers
The simplest and most effective method for the analysis of planar mechanisms is the vector method involving complex numbers, which was first
introduced by Bloch (Beggs, 1966). In a planar mechanism shown in Figure
2.6(a), let [x, y] be a fixed reference frame with its origin at one end of link
1, Figure 2.6(b). Link 1 can be represented by a vector p 1 ,
,
where p 1 is the length of p 1 . This can also be written in complex polar
form with real and imaginary parts corresponding to the Cartesian
coordinates
,
in which j is the standard imaginary unit. The subsequent links, e.g. links 2
and 3, can be written as
and
.
In a mechanism where one or more closed chains can be identified, around
any closed chain with n links there must be
.
(2.5)
Eq. (2.5) is an important vector equation called the closure equation.
(a)
(b)
Figure 2.6 (a) A planar linkage and (b) its vector representation.