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

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

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3 Graphical Representation Displacement, Velocity and Acceleration

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4 Energy Conservation Equilibrium Position No dissipation T + V = constant Potential Energy At Equilibrium k δ st +mg = 0,

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

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6 Energy Conservation (contd) If V = 0 at the equilibrium position,

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

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8 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,

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9 The Phugoid (contd) Vertical momentum equation Energy conservation T = D (to first order) Equations of motion

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10 The Phugoid (contd) h and v satisfy a Harmonic Oscillator Equa- tion Natural frequency and Period Light aircraft v ft/s τ 20s Solution

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11 The Phugoid (contd) Integrate v equation

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