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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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Vibrations and Waves Chapter 11. Most object oscillate (vibrate) because solids are elastic and they will vibrate when given an impulse Tuning forks,

Vibrations and Waves Chapter 11. Most object oscillate (vibrate) because solids are elastic and they will vibrate when given an impulse Tuning forks,

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