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Unformatted text preview: 3.4: Mechanical Vibrations Mass  Spring  Dashpot x Equilibrium Point m c Setup: Attach a mass, m , to a spring and a dashpot. x H t L is the distance from equilibrium. If x H t L > 0, then the spring is stretched. If x H t L < 0, then the spring is compressed. A d a s h p o t is a piston that always opposes motion. Assume that the force of opposition is proportional to velocity with constant of proportionality c > 0. Hence, we have F R =  c v =  c dx dt . If c = 0, then the motion is u n d a m p e d . If c „ 0, then the motion is d a m p e d . Hooks Law : The restorative force, F S , is proportional to the distance stretched and opposes motion. So, for k > 0, F S =  k x . External Force F E = F H t L . If F E = F H t L = 0, then the motion is f r e e . If F E = F H t L „ 0, then the motion if f o r c e d . Newton's Law : F T = m a = m d 2 x dt 2 . Total Forces on Mass : F T = F S + F R + F E . Hence, we need to solve m d 2 x dt 2 + c dx dt + k x = F E = F H t L . Free Undamped Motion: F H t L = 0, c = 0. Then, we need to solve the basic second order, linear, homogeneous differential equation with constant coefficients: m d 2 x dt 2 + k x = d 2 x dt 2 + k m x = d 2 x dt 2 +Ω 2 x = 0, where we define Ω = k m . Thus, we have a characteristic equation of r 2 +Ω 2 = r = –Ω i . Thus, our solution can be written as a linear combination of sines and cosines, x H t L = A cos H Ω t L + B sin H Ω t L = C B A C cos H Ω t L + B C sin H Ω t LF , where C = A 2 + B 2 = C @ cos H Α L cos H Ω t L + sin...
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 Fall '11
 KeithEmmert
 Exponential Function, Mass, Cos, Dashpot

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