THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

20Vibration and Vibration Control20.1 IntroductionNoise often results from vibration. Many sources of vibration exist and they includeimpact processes, such as blasting, pile driving, hammering, and die stamping; machinerysuch as motors, engines, fans, blowers, and pumps; turbulence in fluidssystems; and transportation vehicles. Attenuation of vibration generally cuts downon the noise level and in many cases lengthens the service life of the machinery itself.Damping, correction of imbalances, and configuration of flow paths constitutethe principal measures of cutting down on the deleterious effects of vibration.20.2 Modeling Vibration SystemsWe commence with a basic one-degree-of-freedom system of Figure 20.1, whichconsists of a mass, a spring, and a damper (also referred to as a dashpot). Thesystem has only one degree of freedom because it is constrained to move onlyin the x-direction. Summing up all the forces acting on the mass and applyingNewton’s second law, we obtainwherem d2 xdt + C dx + kx = f (t) (20.1)2 dtm = masst = timeC = coefficient of dampingk = linear elastic constantDamping occurs from energy dissipation due to hysteresis, sliding friction, fluidviscosity, and other causes. The damping force may be proportional to the velocity(as stated in the above equation) or it may even be proportional to some other powerof velocity. Sliding friction is often represented as a constant force in a direction585