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# lecture_21 (dragged) - MA 36600 LECTURE NOTES FRIDAY MARCH...

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MA 36600 LECTURE NOTES: FRIDAY, MARCH 6 Mechanical and Electrical Vibrations Hooke’s Law. Say that we have a mass m which is attached to a spring. Consider four forces on the mass m : Gravity: Newton’s Law of Gravity states that F g = m g . Restoring Force: Hooke’s Law states that F s = k ( L + u ) for some positive constant k . Damping: Say that the mass is in a viscous fluid, one which gives a type of damping. For example, the mass may be a ff ected by air resistance, or the mass may be in a liquid. Then the force of damping is proportional to the velocity i.e., F d = γ u for some positive constant γ . External: Say that the mass is acted upon by an outside mechanical force. We simply write this as F e = F ( t ) for some function. Newton’s Second Law of Motion states that F = m a . That is, m u + γ u + k u = F ( t ) . Hence the position u = u ( t ) satisfies a linear second order di ff erential equation. Note that in general, if we have an equation in the form a y + b y + c y = f ( t ) where a , b , and c are constants, we identify m
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