ABSTRACT
This formula tells us the "shape" into which the rod must initially be deformed to give the motion of Eq. (10-26) when released. That is, up to this point we have been rather vague about the initial deformation of the rod, merely calling itf(x). We see now that to get a certain natural frequency to exist alone, not just "any old" shape f(x) may be used. Figure 10-4 is a plot ofEq. (10-27). Note that C simply determines the scale or magnitude of the oscillation of any plane in the rod; if we double C, for example, all amplitudes are doubled. At any chosen location, such as x 1, the rod vibrates longitudinally according to
JTXt JT (E 6 u(x1, t) = Ccos L cos L yf;t = A 1 coswt (10-28)
a simple harmonic oscillation. The frequencies w given by (1 0-24) are called the natural frequencies of the rod;
note that there are an infinite number of them. This is characteristic of distributedparameter vibration models and also of the real systems which they represent; they really do have an infinite number of natural frequencies. Just as in lumped-parameter models, if an external driving force acts at a frequency equal to a natural frequency the undamped system builds up an infinite motion. The "dynamic deflection curve" of Fig. 10-4 is called the mode shape. Each natural frequency has its own characteristic mode shape; for n = 2 we have w = (2n: 1 L)-JETP and the mode shape of Fig. 10-5. In order to produce vibrations at a single natural frequency only, it is necessary to initially deform the rod into precisely the cosinusoidal shapes such as Fig. I 0-4 or 10-5. What if the initial deformation were some arbitrary shape? We would then find in general that all the natural frequencies would be excited in varying degrees depending on the shape of the particular deformation imposed. The motion is then a superposition of the various mode shapes, each at a different amplitude. The same Fourier series used in Chapter 9 is also used in these sorts of problems
x=L Figure 10-4 Mode shape for first natural frequency.
