Page 27 - Fall 2006

plot the mode shapes for high m, and to compute the reso- nance frequencies of beams with free and simply supported boundary conditions. Notice how as m increases, π2(2m- 1)2/(4a2) (the wavenumber term for free beam modes) approaches m2π2/a2 (the wavenumber term for simply sup- ported beam modes). In Fig. 7, the mode shapes for a simply supported and free beam are shown for large m and km, and are nearly identical away from the boundaries.
Modes of plates
Mode shapes in flat plates look like those in beams, but are two-dimensional. We consider again simply supported boundary conditions (the analyst’s best friend) at the edges of a thin rectangular plate, where the transverse displacement field of a given mode shape of order (m,n) is:
, (19) and the corresponding resonance frequencies of the modes are . (20)
Now, our rigidity/mass term is D/ρh, rather than EI/ρA for beams, and our wavenumber is now a two-dimensional wavevector,
(21)
Fig. 6. First four mode shapes of a free beam. The dashed lines indicate the vibra- tion nodes, or locations of zero deformation.
Figure 6 shows sample mode shapes of a straight free beam. Here, dashed lines have been placed at the modal node lines (loca- tions of zero deformation). Whereas counting antinodes can determine mode order for beams with simple supports at their ends, counting nodes determines mode order for free beams.
The only difference between the resonance frequencies for beams with free (or other) boundary conditions and those for simply supported boundary conditions is the wavenum- ber term km. Fortunately, as the mode order m increases and the wavelengths become small with respect to the structural dimensions, the near field deformations around a structure’s edges influence the mode shapes and their resonance fre- quencies less. An exercise to confirm this phenomenon is to
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