Page 114 - Mechatronic Systems Modelling and Simulation with HDLs

6.2 MULTIBODY MECHANICS 103

Two mass points in the plane are rigidly joined together by a mass-free rod.
Their position is determined by two pairs of cartesian coordinates (x 1 ,y 1 ) and (x 2 ,
y 2 ). The condition induced by the rod can be described by the following equation
2
2
2
(x 2 − x 1 ) + (y 2 − y 1 ) − k = 0 (6.6)
We therefore have four cartesian coordinates and a bond equation, thus a total
of three degrees of freedom. In principle, however, the configuration of the two
particles can be described by the following generalised coordinates:

q 1 = x coordinate of the mid-point of the rod
q 2 = y coordinate of the mid-point of the rod
q 3 = angle φ of the rod.

The fourth coordinate q 4 would be the length of the rod, which is, however, con-
stant. So the associated bond equation
q 4 = k (6.7)

is trivial and can be disregarded. There thus remain three coordinates without a
further bond equation, i.e. three degrees of freedom. Only by this formulation in
generalised coordinates can we thus omit the consideration of constraint equations
in holonomous systems. In the following we will consider exclusively holonomous
systems.
It is often worthwhile going over to the generalised coordinates, which — as is
the case for pure cartesian coordinates — represent the configuration of the system,
i.e. the position of all particles. Coordinate transformations permit the conversion
between generalised and cartesian coordinates:
x 1 = x 1 (q 1 , q 2 ,... , q n , t)

x 2 = x 2 (q 1 , q 2 ,... , q n , t)
(6.8)
.
.
.
x 3N = x 3N (q 1 , q 2 ,..., q n , t)
The transition to generalised coordinates requires that forces acting upon the system
from outside are also present in generalised form. Whereas the forces in cartesian
coordinates can be simply split up into their x, y and z components, things are
more complicated in this case. For example, forces acting upon angular coordinates
become moments. The conversion rule for the generalised force Q i is naturally also
based upon the coordinate transformation x j and looks like this:
3N
∂x j

Q i = F j (i = 1, 2,..., n) (6.9)
∂q i
j=1