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3.9we - 3.9 FORCED VIBRATIONS KIAM HEONG KWA 1 Forced...

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3.9: FORCED VIBRATIONS KIAM HEONG KWA 1. Forced Vibrations with Damping Continuing our investigation of the spring-mass system with damp- ing, we now suppose that a periodic external force F ( t ) = F 0 cos ωt is applied to the mass. Here F 0 and ω are numerically positive constants representing the amplitude and the frequency, respectively, of the force. The equation of motion is now (1.1) mu 00 + γu 0 + ku = F 0 cos ωt, where m , γ , and k are the mass, the damping coefficient, and the spring constant of the spring-mass system. Thus the motion is described by (1.2) u ( t ) = u c ( t ) + U ( t ) , where u c ( t ) is the motion of the mass in the absence of the external force and is the solution of the homogeneous equation (1.3) mu 00 + γu 0 + ku = 0 , while (1.4) U ( t ) = F 0 ( k - 2 ) 2 + γ 2 ω 2 ( k - 2 ) cos ωt + γω sin ωt is a particular solution of the full nonhomogeneous equation (1.1). In other words, u c ( t ) is a damped free vibration which dies out after sufficiently long time: lim t →∞ u c ( t ) = 0. Hence it is called the transient solution of the motion (1.2). Physically, the transient solution is the response of the system to the initial conditions. With the presence of damping, the energy put into the system by the initial conditions is dissipated with increasing time and thus the transient solution dies out eventually. In contrast, U ( t ) is the response of the system to the presence of the external force F ( t ) . It persists indefinitely as long as the external force F ( t ) is present and oscillates with he same frequency as F ( t ). It is thus called the steady-state solution or the forced response of the motion (1.2).
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