9.6 The Hamilton Principles and Their Applications
555
Fig. 9.6 The simple
pendulum motion diagram
l
m
θ
u tt − a
2 u xx = 0
a
2
=
τ
ρ
This is the string vibration equation.
Example 9.6.3 As shown in Fig. 9.6, let the swing link length of the simple pendulum
be l, the mass of the pendulum bob is m. Neglecting the mass of the swing link, use
Hamilton’s principle to establish the motion equation of the simple pendulum.
Solution Choosing the included angle θ of the swing link and the plumb line as
the generalized coordinates, and choosing the level surface of suspension point O as
the zero position of potential energy, then the kinetic energy, potential energy and
Lagrange function of the pendulum are respectively
T =
1
2
m(l ˙
θ)
2
V = −mgl cos θ
L = T − V =
1
2
m(l ˙
θ)
2
+ mgl cos θ
According to the Hamilton’s principle, the variation of the functional
J =
1
2
t 1
t 0
[m(l ˙
θ)
2
+ 2mgl cos θ ]dt
is equal to zero, namely δ J = 0, the Euler equation can be obtained
−mgl sin θ − ml
2 ¨
θ = 0