Lecture D31 : Linear Harmonic Oscillator Spring-Mass System Spring Force F = ?kx, k > 0 Newton’s Second Law m¨x+kx = 0 (Define) Natural frequency (and period) ωn = radicalBigg k m parenleftbigg τ = 2piω n parenrightbigg Equation of a linear harmonic oscillator ¨x+ω2nx = 0 1 Solution General solution x(t) = Acosωnt+Bsinωnt or, x(t) = C sin(ωnt+φ) Initial conditions x(0) = x0 ˙x(0) = ˙x0 Solution, x(t) = x0cosωnt+ ˙x0ω n sinωnt or, x(t) = radicalBig x20 +(˙x0/ωn)2sin(ωnt+tan?1(x0ωn˙x 0 )) 2 Graphical Representation Displacement, Velocity and Acceleration 3 Energy Conservation Equilibrium Position No dissipation T +V = constant Potential Energy V = 12k(x+δst)2 ? 12kδ2st ?mgx At Equilibrium ?kδst +mg = 0, V = 12kx2 4 Energy Conservation (cont’d) Kinetic Energy 1 2m˙x 2 Conservation of energy d dt(T +V) = m˙x¨x+kx˙x = 0 Governing equation m¨x+kx = 0 Above represents a very general way of de- riving equations of motion (Lagrangian Me- chanics) 5 Energy Conservation (cont’d) If V = 0 at the equilibrium position, V = 0 T = Tmax for x = 0 V = Vmax T = 0 for x = xmax → Tmax = Vmax 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 (h0, v0) h = h0 +h′ v = v0 +v′ ? L = W (≡ mg) for v = v0, but L ～ v2, L mg = v2 v20 ≈ v20 v20 (1+2 v′ v0 +...) 8 The Phugoid (cont’d) ? Vertical momentum equation m¨h = L?mg ¨h = g(1+2v′ v0 ?1) ? Energy conservation T = D mgh0 + 12mv20 = mgh+ 12mv2 (to first order) → gh′ +v0v′ = 0 ? Equations of motion ¨h′ +2 g2 v20 h ′ = 0 ¨v′ +2 g 2 v20 v ′ = 0 9 The Phugoid (cont’d) h′ and v′ satisfy a Harmonic Oscillator Equa- tion Natural frequency and Period ωn = √2g v0 τ = √2piv0 g Light aircraft v0 ～ 150ft/s → τ ～ 20s Solution h′ = Asin parenleftBigg√2g v0 t parenrightBigg v′ = ? gv 0 Asin parenleftBigg√2g v0 t parenrightBigg 10 The Phugoid (cont’d) Integrate v′ equation x′ = A√2cos parenleftBigg√2g v0 t parenrightBigg x = x0 +x′ 11