## 68 Vibrating beams tubes and disks

(a) Isotropic solids

Anything vibrating at its natural frequency without significant damping can be reduced to the simple problem of a mass, m, attached to a spring of stiffness, K. The lowest natural frequency of such a system is

Specific cases require specific values for m and K. They can often be estimated with sufficient accuracy to be useful in approximate modeling. Higher natural vibration frequencies are simple multiples of the lowest.

The first box in Figure 6.7 gives the lowest natural frequencies of the flex-ural modes of uniform beams with various end-constraints. As an example, the first can be estimated by assuming that the effective mass of the vibrating beam is one quarter of its real mass, so that where m0 is the mass per unit length of the beam (i.e. m is half the total mass of the beam) and K is the bending stiffness (given by F/S from Section 6.3); the estimate differs from the exact value by 2%. Vibrations of a tube have a similar form, using I and m0 for the tube. Circumferential vibrations can be found approximately by 'unwrapping' the tube and treating it as a vibrating plate, simply supported at two of its four edges.

The second box gives the lowest natural frequencies for flat circular disks with simply supported and clamped edges. Disks with curved faces are stiffer and have higher natural frequencies.

(b) Metal foams: scaling laws for frequency

Both longitudinal and flexural vibration frequencies are proportional to *JE/p, where E is Young's modulus and p is the density, provided the dimensions of m mGi ~4~

Vibrating beams, tubes and disks Ci 3.52

 p n 22.4 U y Beams,tubes f= Natural frequency (s-1) mo = pA = Mass/Length (kg/m) p = Density (kg/m3) A = Section area (m2) I = See Figure 6.1 f= Natural frequency (s-1) mo = pA = Mass/Length (kg/m) p = Density (kg/m3) A = Section area (m2) I = See Figure 6.1 9.87 2.68 1.44 2.94 Disks Disks m1 = p t = Mass/Area (kg/m2) t = Thickness (m) R = Radius (m) v = Poisson's ratio m1 = p t = Mass/Area (kg/m2) t = Thickness (m) R = Radius (m) v = Poisson's ratio Figure 6.7 Vibrating beams, tubes and disks the sample are fixed. The moduli of foams scale as (p/ps)2, and the mass as (p/ps). Thus the natural vibration frequencies of a sample of fixed dimensions scale as f/fs = (p/ps)1/2 - the lower the density of the foam, the lower its natural vibration frequencies. By contrast, the natural vibration frequencies of panels of the same stiffness (but of different thickness) scale as f/fs D (p/psy1/6 - the lower the density, the higher the frequency. And for panels of equal mass (but of different thickness) the frequencies scale as f/fs D (p/psy1/2 - the lower the density, the higher the frequency.
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