Deviations from Poisson statistics in the spectra of free rectangular thin plates

The counterintuitive fact that wave chaos appears in the bending spectrum of free rectangular thin plates is presented. After extensive numerical simulations, varying the ratio between the length of its sides, it is shown that (i) frequency levels belonging to different symmetry classes cross each other and (ii) for levels within the same symmetry sector, only avoided crossings appear. The consequence of anticrossings is studied by calculating the distribution of the ratio of consecutive level spacings for each symmetry class. The resulting ratio distributions disagree with the expected Poissonian result. They are then compared with some well-known transition distributions between Poisson and the Gaussian orthogonal random matrix ensemble. It is found that the distribution of the ratio of consecutive level spacings agrees with the prediction of the Rosenzweig-Porter model. Also, the normal-mode vibration amplitudes are found experimentally on aluminum plates, before and after an avoided crossing for symmetrical-symmetrical, symmetrical-antisymmetrical, and antisymmetrical-symmetrical classes. The measured modes show an excellent agreement with our numerical predictions. The expected Poissonian distribution is recovered for the simply supported rectangular plate.

Figure 1

(a) Rectangular plate of length $L$, width $C$ and thickness $h$; it has flexural rigidity $D$ and density $\rho$. (b) Eighth bending mode at frequency $f_8=308.28$ Hz, showing its absolute vertical displacement for a simulated, free vibrating rectangular plate, where $L=800$ mm, $C=0.355$ m, $h=0.00635$ m, $E=69$ GPa, $\nu =0.33$ and $\rho =2700$ kg/${\mathrm{m}}^3$.

Figure 2

Reflected bending waves on (a) a simply supported boundary and (b) a free boundary.

Figure 3

Bending spectrum of the rectangular plate with (a) free boundary conditions and (b) with simply supported boundary conditions. The $SS$ symmetry class (see the text) is drawn as thicker magenta lines, $SA$ as thick green lines, $AS$ thin blue, and $AA$ modes as thinner red lines.

Figure 4

Profile histograms for free (orange) and simply supported (black) plate, for the ratio of consecutive level spacings within the interval $r\in [0,2]$, compared with Poisson (cyan), GOE (blue) and semi-Poisson (thick red) distributions. (a) $SS$ symmetry, (b) $SA$ symmetry, (c) $AS$ symmetry, and (d) $AA$ symmetry class.

Figure 5

Average histogram profile for all symmetry classes of the free plate (orange) compared with the best fit of RMT transition models: Brody-Atas (dotted blue), Izrailev (green) and Rosenzweig-Porter (thick magenta); also the average profile histogram distribution for the simply supported plate is shown (black).

Figure 6

Left: the frequency spectrum as a function of the length $L$ for the SA symmetry pointing a sequence of bending wave amplitudes numerically calculated, through an avoided crossing, for the free rectangular plate. The colors follow the same scale as in Fig. 1, indicating the nodes by green and maxima by red. Right: comparison between measured modes (darker) and their corresponding simulated ones (brighter). Modes A and C correspond to a 500-mm length plate, while modes B and D to an 800-mm length plate.

Figure 7

Evolution through an avoided crossing for the AS symmetry class. The figure is ordered as in Fig. 6. Mode A is for a plate with $L=400$ mm and mode C for $L=500$ mm; modes B and D correspond to $L=800$ mm.

Figure 8

Evolution through an avoided crossing for the SS symmetry class. The figure is ordered as in Fig. 6. Modes A and C correspond to a plate with $L=500$ mm; modes B and D are for $L=800$ mm.