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Lecture D31 : Linear Harmonic Oscillator

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. Solution. General solution. or,. Initial conditions.

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Lecture D31 : Linear Harmonic Oscillator

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  1. 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

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

  3. Graphical Representation Displacement, Velocity and Acceleration

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

  5. Energy Conservation (cont’d) Kinetic Energy Conservation of energy Governing equation Above represents a verygeneral way of de-riving equations of motion (Lagrangian Me-chanics)

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

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

  8. The Phugoid Idealized situation • Small perturbations (h′,v′) about steady level flight (h0, v0) • L = W(≡ mg) for v = v0, but L ∼ v2,

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

  10. 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

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

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