Hence the spring oscillates with a smaller and

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. Hence the spring oscillates with a smaller and smaller amplitude. The motion decays to zero as time goes. Overdamping If c is large, i.e. c 2 > 4 mk , Then we still have real roots r = ` c ˚ q c 2 ` 4 mk 2 m Example 3 Assume that the object with mass 2 kg, k = 128 adn c = 40 . Find the position of the mass at any time t . Solution: 2 d 2 x dt 2 + 40 dx dt + 128 x = 0 d 2 x dt 2 + 20 dx dt + 64 x = 0 r 2 + 20 r + 64 = 0 ( r + 4)( r + 16) = 0 r 1 = ` 4 and r = ` 16 We get the general solutions x ( t ) = c 1 e ` 4 x + c x e ` 16 x . When t goes to infinity, e ` 5 t goes to 0 . The motion decays to zero as time goes. But here the oscillations do not occur. 14.3 Forced Vibrations. Suppose that, in addition to the restoring force and the damping force, the motion of the spring is affected by an external force F ( t ) depending on the time. The Newton’s Second Law implies that m d 2 x dt 2 = ` kx ` c dx dt + F ( t ) or m d 2 x dt 2 + kx + c dx dt = F ( t ) which is a non-homogeneous equation. A commonly occurring type of the external force is a periodic function. For example, F ( t ) = F 0 cos ( ! 0 t ) : 3
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We know how to solve these equations by the undetermined coefficients. Most of cases the particular solution looks like A cos ( ! 0 t ) + B sin ( ! 0 t ) . As we have seen before, sometimes, the frequency ! 0 coincides with the natural frequency ! as in the general solution of the homogeneous equation. Then we should times x to get a particular solution. In particular, this means we may have larger amplitude. This phenomenon is called resonance.
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