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damping - Chapter 3 SDOF Systems Free Damped Response 3.1...

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Chapter 3 SDOF Systems: Free Damped Response 3.1 Introduction W hen we looked at a mass-spring system we noticed that the predicted response was a sinusoidal oscillation that lasted forever. This is of course patently untrue in that all real systems would stop oscillating after a while. This difference between reality and our mathematical model is due to the absence of any kind of dissipative mechanism in our model. Dissipation of energy can occur due to various friction (or damping) forms. There are three main types that we will briefly discuss here. The first is the Coulomb damping (dry friction) that is caused whenever two solid bodies slide past each other. Without getting into a phenomenological description of why and how these forces come about, we restrict ourselves to an empirical description that the damping force would look as shown in Fig. 3.1. Note that the x -axis is velocity. The force is clearly a nonlinear function of velocity, and as such quite undesirable. The second type of mechanism is structural (or internal) damping that comes about because of -5 0 5 -1.5 -1 -0.5 0 0.5 1 1.5 Velocity Force Figure 3.1: Coulomb damping 37
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38 CHAPTER 3. SDOF SYSTEMS: FREE DAMPED RESPONSE -5 0 5 -600 -400 -200 0 200 400 600 800 Velocity Force --Slope=c Figure 3.2: Viscous damping cyclic stresses in materials. If you take a paper clip and bend it back and forth a few times, you will notice that it gets warm; this increase in temperature is coming from the internal friction that causes dissipation of the energy that we put into the system by bending it. This is a common occurrence in all kinds of rotating machinery and needs to be included in more complicated models than we will encounter in this book. Also, some materials like rubber have a high degree of internal friction and should always be modeled with it. The third type of friction that we will be primarily concerned about comes from interaction with fluids and arises due to viscous friction. This can be unintentional such as the frictional drag felt in an automobile (or aircraft, of course) as it moves through the medium of air. Or, it can be intentionally introduced such as lubricating oil to separate two solid objects that have relative motion (a door hinge is an obvious example). In this case, rather than eliminating friction, we are striving to replace Coulomb friction with fluid friction. Fluid-film bearings (including air bearings such as in computer hard drives) and automotive shock absorbers are some examples. In the following we will restrict ourselves to fluid friction. Anytime we expect some amount of fluid friction, we will assume that it can be accurately modeled using a piston-cylinder mechanism or a dashpot. The force-velocity characteristic is assumed to be linear as shown in the figure, Fig. 3.2.
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damping - Chapter 3 SDOF Systems Free Damped Response 3.1...

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