Download 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
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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
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 deriving equations of motion (Lagrangian Mechanics)
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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 Equation
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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