Page 25 - Fall 2006

Fig. 3. A flexural, or bending wave propagating through a plate or beam (ampli- tudes highly exaggerated). As with pure shear, the wave propagates along the plate or beam axis, while deforming the structure transversely. Unlike pure shear, how- ever, a bending wave causes the plate or beam cross sections to rotate about the neutral axis.
and shells, like the example shown in Fig. 3. Bending waves deform a structure transversely, so that they excite acoustic waves in neighboring fluids (we will learn about this phe- nomenon in part 2 of this tutorial). Although longitudinal and shear wave behavior is simple—similar to that of acoustic waves in air or water—bending waves are far more compli- cated. In particular, the speed of a bending wave depends not only on the elastic moduli and density of the structural mate- rial it travels through, but also on the geometric properties of the beam or plate cross section. Also, bending wave speeds are dispersive, with the curious property of depending on their frequency of oscillation.
I will not derive the bending wave equations for beams and plates in this article, but will show them, along with their corresponding wave speeds. The wave equation and wave speed for flexure in thin7 beams are
, and (9a)
. (9b)
Whereas the wave equations for longitudinal and shear (and acoustic) waves are second order, the bending wave equation has a fourth order variation with space. Also, note that the wave speed does not appear explicitly in the flexural wave equation, and that the wave speed depends on frequency, as we learned earlier.
Although the thin beam bending wave equation is more complicated than those for pure longitudinal and shear waves, it is still fairly simple. However, when flexural wavelengths become short with respect to the beam thickness, other terms become important—such as resistance to shear deformation and the rotary mass inertia. Unfortunately, including these effects compli- cates the wave equation and sound speed considerably, leading to the thick beam8 wave equation and wave speed:
(10a) , and
. (10b)
Two new components appear in the thick beam bending wave equation: a fourth order dependence of motion on both
time and space, and a fourth order dependence on time. Some new combinations of factors also appear: KAG is the shear factor that is the product of area, shear modulus, and the correction fac- tor K, which is the fraction of the beam cross section which sup- ports shear; and I/A, which represents the rotary inertia.
When shear resistance and rotary inertia are negligible (which is the case for waves with long wavelengths with respect to thickness), the wave equation and wave speed reduce to the simpler forms shown earlier for thin, or Bernoulli-Euler beams. Since long wavelengths imply low frequencies, thin beam theo- ry is sometimes called a low frequency limit of the general, thick beam theory. For very high frequencies, the shear resist- ance terms become dominant, so that the flexural wave equa- tion simplifies to the shear wave equation, and the bending wave speed approaches the shear wave speed:
. (11)
The only difference between a shear wave in a beam and one in an infinite structural material is the shear correction factor K.
Although flexural wave theories for plates are derived in different ways than those for beams, the general thick plate9 wave equation and wave speed are essentially the same as those for beams, but for a wave propagating in two dimensions:
, and
(12a)
. (12b)
D is a combination of terms called the flexural rigidity:
. (13)
As with beams, the low frequency (thin plate) limits of the thick plate equations are simpler, but still dispersive:
, where
.
(14a)
(14b)
And, at high frequencies flexural waves in plates approach pure shear waves, where
. (15)
For homogenous isotropic plates, the shear correction factor is 5/6 (for beams, K depends on the geometry of the cross section). The wave speeds of thin and thick plates, along with lon- gitudinal and shear wave speeds in a 10 cm thick steel plate are shown in Fig. 4. The low and high frequency limits of the gen- eral thick plate wave speed are evident in the plot. So, longitu- dinal waves are faster than shear waves, which are faster than bending waves. Also, waves in stiffer materials are faster than
Structural Acoustics Tutorial 23