Chapter13.1 - Simple harmonic motion Equation of motion:...

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Unformatted text preview: Simple harmonic motion Equation of motion: Trial solution: Note: x is real, but B and can be complex! Derivatives: Equation for : General solution: Velocity: t Be x = = kx dt x d m 2 2 2 2 2 = + x dt x d x e B dt x d x e B dt dx t t 2 2 2 2 ; = = = = ~ dt d = = + 2 2 2 2 x x ) ( ) ( t i t i t i t i De Ce i De i Ce i dt dx t v = = = i = 2 , 1 t i t i De Ce t x + = ) ( Simple harmonic motion Use initial conditions and the fact that x and are real: Solution: C is a complex number: Final solution: ) Re( 2 )* ( * t i t i t i t i t i Ce Ce Ce e C Ce x = + = + = ) cos( + = t A x + + + = = = = + )) sin( ) Re(cos( ) Re( ) Re( 2 ) ( t i t A e A e e A x e A C t i i t i i * Re Re ) ( ) ( Im Im ) ( C D D C D C D C i v D C D C D C x = = = = + + = Damped harmonic motion Include non-conservative force, e.g. drag force: b is damping coefficient Equation of motion in 1D: Damping parameter: Units of frequency Natural frequency: Damped oscillator equation: 2 2 2 2 = + + = + + = x m k dt dx m b dt x d kx dt dx b dt x d m bv kx ma v F b = m b = 2 2 2 = + + x dt dx dt x d m k = Damped harmonic motion Trial solution: Derivatives: Substitute: Three cases are possible: 2 2 2 = + + x dt dx dt x d t Be x = x e B dt x d x e B dt dx t t 2 2 2 2 ; = = = = = + + = + + 2 2 2 2 x x x 2 2 2 , 1 ) 2 ( 2 = damped) y (criticall 2 2 ). 3 ed) (underdamp 2 ) 2 ( ; 2 ). 2 d) (overdampe 2 ) 2 ( ; 2 ). 1 2 , 1 2 2 2 , 1 2 2 = = = = <...
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Chapter13.1 - Simple harmonic motion Equation of motion:...

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