文档介绍：1 Pendulum
Free oscillator
To introduce dynamical systems, we begin with one of the simplest: a free
oscillator. Specifically, we consider an unforced, undamped pendulum.
l
θ
mg sinθ
mg
The arc length (displacement) between the pendulum’s current position and
rest position (β= 0) is
s = lβ
Therefore
s˙= lβ˙
s¨ = lβ¨
From Newton’s 2nd law,
F = mlβ¨
The restoring force is given by mg sin β. (It acts in the direction opposite
−
to sgn(β)). Thus
F = mlβ¨ = mg sin β
−
or
d2β g
+ sin β= 0.
dt2 l
Our pendulum equation is idealized: it assumes, ., a point mass, a rigid
geometry, and most importantly, no friction.
7
The equation is nonlinear, because of the sin β term. Thus the equation is
not easily solved.
However for small β 1 we have sin ββ. Then
◦
d2β g
= β
dt2 − l
whose solution is
g
β= β cos t + θ
0 l
or
β= β0 cos(γt + θ)
where the angular frequency is
g
γ= ,
l
the period is
l
T = 2α,
g
and β0 and e from the initial conditions.
Note that the motion is exactly periodic.
Furthermore, the period T is independent of the amplitude β0.
1
0
θ 0
/
θ
−1
0 1 2
t / T
Global view of dynamics
What do we need to know pletely describe the instantaneous state of
the pendulum?
dβ
The position β and the velocity = β˙.
dt