1. Week 6 2. Solving ODEs using Fourier series (forced oscillations)
Consider a damped and forced oscillator:

2. The oscillator is governed by Newton’s Second Law:
(1)
Here x(t) is the is the deviation from the equilibrium position, t is the time variable, Fe(t) is an external force,
directed ‘backwards’
where k is the spring modulus and c is the damping coefficient.

3. ω is called the natural frequency of the oscillator.
Thus, Eq.(1) is or
where μ = c/m and ω2 = k/m, and r = Fe/m.
Comments:
ω is called the natural frequency of the oscillator.
Keep in mind that Fe – and hence, r – may be functions of t.

4. Example 1:
Find the general solution of
(2)
where
Eq. (2) is a linear non-homogeneous ODE  we can find its general solution as the sum of the general solution of the corresponding homogeneous ODE + a particular solution of the full ODE.
However, since we are going to represent r(t) by its Fourier series, we’ll have to find a particular solution for each term of the series.

5. The plan:
Find the solution xh of the homogeneous equation.
Expand r(t) in a Fourier series
Find a particular solution xp,n of the non-homogeneous equation for each term of the series.
The general solution is x = xh + xp,1 + xp,
Soln:
Step 1:

6. Step 2: Find the Fourier coefficients of r(t):
r(t) is odd  a0 = an = 0.
Find bn:
hence

7. Step 3: Use the Method of Undetermined Coefficients to find a particular solution of the non-homogeneous equation for each term of the Fourier series of r(t), e.g.
n = 1:
n = 2:

8. n = n: Step 4: The general solution of Eq. (2) is
This solution is valid only if
(3)

9. If condition (3) does not hold, the solution isn’t periodic and, thus, cannot be expressed in terms of a Fourier series.
Physically, condition (3) eliminates resonance between the external force and the oscillator’s natural frequency.
What can happen if resonance occurs is shown in the following videos:
Friction causes resonant oscillations to saturate.
Thus, if we ‘return’ the friction term in Eq. (2), the amplitude of resonant oscillations will initially grow, but eventually stabilise.