Page 221 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Lagrange’s Equation Chap. 7
208

The excess coordinates exceeding the number of degrees of freedom of the
system are called superfluous coordinates, and constraint equations equal in num­
ber to the superfluous coordinates are necessary for their elimination. Constraints
are called holonomie if the excess coordinates can be eliminated through equa­
tions of constraint. We will deal only with holonomie systems in this text.
Examine now the problem of defining the position of the double pendulum of

Fig. 7.1-2. The double pendulum has only 2 DOF and the angles 6 ^ and 62
completely define the position of and m2. Thus, and 62 are generalized
coordinates, i.e., 6 ^ = and 82 = Q2-

The position of and m 2 can also be expressed in rectangular coordinates
X, y. However they are related by the constraint equations
If = x'i + yj
l¡ = (X2 - X ^ + (y 2 - y i f
and hence are not independent. We can express the rectangular coordinates x¿, y¿
in terms of the generalized coordinates and 62
C = /j sin . ^2 = li sin 6 ^ + ¡2 sin 62

j
= /i cos y2 ^ + ^2 ^2
and these can also be considered as constraint equations.
To determine the kinetic energy, the squares of the velocity can be written in
terms of the generalized coordinates:
^2
(MO

u j = i | + 3)2 = [ / i ^ i + I282 cos (02 “ ^1)] T [¡2^2 {^2 ~ ^1)]

The kinetic energy
-r 1 2 . 1 2
T = 2^\^t + i^2^2
is then a function of both q = 6 and q = 9:
T = T[q^,q2,...,q\,q2,---) (7.1-2)