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Unformatted text preview: 22 Springs: Part II (Forced Vibrations) Let us look, again, at those mass/spring systems discussed in chapter 17. Remember, in such a system we have a spring with one end attached to an immobile wall and the other end attached to some object that can move back and forth under the influences of the spring and whatever friction may be in the system. Now that we have methods for dealing with nonhomogeneous differential equations (in particular, the method of educated guess), we can expand our investigations to mass/spring systems that are under the influence of outside forces such as gravity or of someone pushing and pulling the object. Of course, the limitations of the method of guess will limit the forces we can consider. Still, these forces happen to be particularly relevant to mass/spring systems, and our analysis will lead to some very interesting results — results that can be extremely useful not just when considering springs, but also when considering other systems in which things vibrate or oscillate. 22.1 The Mass/Spring System In chapter 17, we derived m d 2 y dt 2 + γ dy dt + κ y = F . to model the mass/spring system. In this differential equation: 1. y = y ( t ) is the position (in meters) at time t (in seconds) of the object attached to the spring. As before, the Y –axis is positioned so that (a) y = 0 is the location of the object when the spring is at its natural length. (This is the “equilibrium point” of the object, at least when F = 0 .) (b) y > 0 when the spring is stretched. (c) y < 0 when the spring is compressed. In chapter 17 we visualized the spring as laying horizontally as in figure 22.1a, but that was just to keep us from thinking about the effect of gravity on this mass/spring system. Now, we can allow the spring (and Y –axis) to be either horizontal or vertical or even at some other angle. All that is important is that the motion of the object only be along the Y –axis. (Do note, however, that if the spring is hanging vertically, as in figure 22.1c, then the Y –axis is actually pointing downward .) 437 438 Springs: Part II (Forced Vibrations) (a) (b) (c) y ( t ) y ( t ) y ( t ) Y Y Y m m m Natural length of the spring Figure 22.1: Three equivalent mass/spring systems with slightly different orientations. 2. m is the mass (in kilograms) of the object attached to the spring. 3. κ is the spring constant, a positive quantity describing the “stiffness” of the spring (with “stiffer” springs having larger values for κ ). 4. γ is the damping constant, a nonnegative quantity describing how much friction is in the system resisting the motion (with γ = 0 corresponding to an ideal system with no friction whatsoever)....
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 Summer '09
 Differential Equations, Equations, Cos, Angular frequency, y0

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