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Unformatted text preview: Lecture 8 We studied the phenomenon of resonance in mass-spring-damper sys- tem in the 6th lecture. The key idea was that large amplitude of oscillation can be induced if the external forcing frequency is close to the natural frequency of forcing The force in question can indeed come from an external source, but it could also come induced by the dynam- ics of the system. In this lecture, we will study several examples of forcing, the first one related to bad noises you all heard coming from washing machines at times. If the load inside the drum of a washing machine is not in a perfectly symmetric balance, noise can occur, ind- ucating that there are lateral oscillations of the drum occuring. Thats not good but is not easily avoidable in design. Lets figure out why. Rotating balance: Consider a mass-spring system of mass M , stifness k and damping coefficient c with an attached mass m rotating at a distance l from the center of mass M , at constant angular velocity , as shown in figure 2. The equation of motion for the mass M reads M x + c x + kx = F ( t ) , (1) where F ( t ) is the force in the vertical direction imposed on M by the rotation of m . The acceleration of m is obtained as a combination of the transferred acceleration of M and centripetal acceleration: a m = a M- l 2 e r where a M , a m are accelerations of mass M and m , respectively (note: there is no angular acceleration since is constant), and e r is the unit vector in the direction of the rod of length l to which the mass m is attached. In the vertical direction, with unit vector j a mx j = x j- l 2 sin( t ) j Newtons law for the mass m in vertical direction reads...
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