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**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

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Solution General solution or, Initial conditions Solution, or,

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**Graphical Representation**

Displacement, Velocity and Acceleration

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**Energy Conservation No dissipation T + V = constant Potential Energy**

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

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**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)

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**Energy Conservation (cont’d)**

If V = 0 at the equilibrium position,

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**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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**The Phugoid Idealized situation**

Small perturbations (h′, v′) about steady level flight (h0, v0) L = W (≡ mg) for v = v0, but L ∼ v2,

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**The Phugoid (cont’d) Vertical momentum equation**

Energy conservation T = D (to first order) Equations of motion

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**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

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The Phugoid (cont’d) Integrate v′ equation

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