My title - Departamento de Matemática da Universidade do Minho

7 OSCILLATIONS AND CYCLES 52
Dumped oscillations.
Adding friction to an harmonic oscillator we get
¨q = −2α ˙q − ω 2 q ,
where α > 0 is some friction coefficient. The guess q(t) = e −αt y(t) gives ÿ = δy where the
“discriminant” is δ = ω 2 − α. Find the general solution, draw pictures and discuss the cases
α 2 < ω 2 (under-critical damping), α 2 = ω 2 (critical damping), and α 2 > ω 2 (overcritical damping).
Show that the energy
decreases with time outside equilibrium points.
E(q, ˙q) = 1 2 ˙q2 + 1 2 ω2 q 2
Underdamped, critical and overdamped oscillations (phase portrait and time series).
7.2 Mathematical pendulum and Jacobi’s elliptic integrals
Mathematical pendulum.
The Newton equation
I ¨θ = −mgl sin θ
models the motion of an idealized pendulum (meaning a point mass attached to a wire of negligible
weight, under a constant gravitational force) with mass m and length l, where I = ml 2 is the
moment of inertia, g is the gravitational acceleration (near the Earth’s surface), and θ is the angle
of the wire with the origin θ = 0 located at the stable equilibrium point. Observe that in the limit
of small oscillations we could safely replace sin θ ≃ θ and we are back to the harmonic oscillator.
The energy
E = 1 2 ˙θ 2 − mgl cos θ
is a constant of the motion. We can define the resonant frequency ω = √ mgl/I and write the
equation as
¨θ = −ω 2 sin θ
To simplify thinks, let’s take ω = 1. Solving form ˙q 2 the we see that the motion with energy E is
given implicitly by the “elliptic integral”
∫
dθ
t = √
2(E − cos(θ))