1. Lecture D31 : Linear Harmonic Oscillator
Spring-Mass System
Spring Force F = −kx, k > 0
Newton’s Second Law
(Define) Natural frequency (and period)
Equation of a linear harmonic oscillator

2. Solution
General solution
or,
Initial conditions
Solution,
or,

3. Graphical Representation
Displacement, Velocity and Acceleration

4. Energy Conservation No dissipation T + V = constant Potential Energy
Equilibrium
Position
No dissipation
T + V = constant
Potential Energy
At Equilibrium −kδst +mg = 0,

5. Energy Conservation (cont’d)
Kinetic Energy
Conservation of energy
Governing equation
Above represents a very general way of de-riving equations of motion (Lagrangian Me-chanics)

6. Energy Conservation (cont’d)
If V = 0 at the equilibrium position,

7. Examples Spring-mass systems Rotating machinery
Pendulums (small amplitude)
Oscillating bodies (small amplitude)
Aircraft motion (Phugoid)
Waves (String, Surface, Volume, etc.)
Circuits
. . .

8. The Phugoid Idealized situation
Small perturbations (h′, v′) about steady level flight (h0, v0)
L = W (≡ mg) for v = v0, but L ∼ v2,

9. The Phugoid (cont’d) Vertical momentum equation
Energy conservation T = D
(to first order)
Equations of motion

10. The Phugoid (cont’d) h′ and v′ satisfy a Harmonic Oscillator Equa-
tion
Natural frequency and Period
Light aircraft v0 ∼ 150 ft/s → τ ∼ 20s
Solution

11. The Phugoid (cont’d)
Integrate v′ equation