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Lecture 8 Notes

Lecture 8 Notes - Today Forced vibrations Newtons 2nd Law...

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Today • Forced vibrations • Newton’s 2nd Law with external forcing. • Forced mass-spring system without damping away from resonance. • Forced mass-spring system without damping at resonance. • Forced mass-spring system with damping. • Review questions.
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Forced vibrations (3.8) • Newton’s 2nd Law: ma = kx γ v + F ( t )
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Forced vibrations (3.8) • Newton’s 2nd Law: ma = kx γ v + F ( t ) spring force
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Forced vibrations (3.8) • Newton’s 2nd Law: ma = kx γ v + F ( t ) spring force drag force
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Forced vibrations (3.8) • Newton’s 2nd Law: ma = kx γ v + F ( t ) spring force drag force applied/external force
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Forced vibrations (3.8) • Newton’s 2nd Law: ma = kx γ v + F ( t ) mx + γ x + kx = F ( t ) spring force drag force applied/external force
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Forced vibrations (3.8) • Newton’s 2nd Law: ma = kx γ v + F ( t ) mx + γ x + kx = F ( t ) spring force drag force applied/external force • Forced vibrations - nonhomogeneous linear equation with constant coefficients.
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Forced vibrations (3.8) • Newton’s 2nd Law: ma = kx γ v + F ( t ) mx + γ x + kx = F ( t ) spring force drag force applied/external force • Forced vibrations - nonhomogeneous linear equation with constant coefficients. • Building during earthquake, tuning fork near instrument, car over washboard road, polar bond under EM stimulus (classical, not quantum).
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Forced vibrations (3.8) • Without damping ( ). γ = 0
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t )
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t ) forcing frequency
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t ) mx + kx = 0 forcing frequency
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t ) mx + kx = 0 x h ( t ) = C 1 cos( ω 0 t ) + C 2 sin( ω 0 t ) forcing frequency
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t ) mx + kx = 0 x h ( t ) = C 1 cos( ω 0 t ) + C 2 sin( ω 0 t ) ω 0 = ? forcing frequency
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t ) mx + kx = 0 x h ( t ) = C 1 cos( ω 0 t ) + C 2 sin( ω 0 t ) ω 0 = k m forcing frequency
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t ) mx + kx = 0 x h ( t ) = C 1 cos( ω 0 t ) + C 2 sin( ω 0 t ) ω 0 = k m natural frequency forcing frequency
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Forced vibrations (3.8) • Without damping ( ). γ = 0 mx + kx = F 0 cos( ω t ) mx + kx = 0 x h ( t ) = C 1 cos( ω 0 t ) + C 2 sin( ω 0 t ) ω 0 = k m • Case 1: ω = ω 0 natural frequency forcing frequency
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Forced vibrations (3.8) • Without damping ( ).
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