Lecture D 31 Linear Harmonic Oscillator SpringMass System

Lecture D 31 : 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 1

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

Graphical Representation Displacement, Velocity and Acceleration 3

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

Energy Conservation (cont’d) Kinetic Energy Conservation of energy Governing equation Above represents a very general way of deriving equations of motion (Lagrangian Mechanics) 5

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

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

The Phugoid Idealized situation • Small perturbations (h′, v′) about steady level flight (h 0, v 0) • L = W (≡ mg) for v = v 0, but L ∼ v 2, 8

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

The Phugoid (cont’d) h′ and v′ satisfy a Harmonic Oscillator Equation Natural frequency and Period L i g h t a i r c r a f t v 0 ∼ 1 50 f t / s → τ ∼ 2 0 s Solution 10

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