Developable Modes in Vibrated Thin Plates

We investigate the normal modes of a developable cone singularity as observed in a circular sheet supported by a rigid circular frame and pushed at its center. When the center of the sheet is in addition submitted to a sinusoidal forcing, two types of bending modes, named here rolling and tilt modes, are parametrically excited. The rolling mode is an angular oscillation of the concave sector of the developable cone structure. If the amplitude of vibration is high enough, the rolling mode amplitude increases dramatically giving rise to both a continuous rotation of the concave sector and a material angular displacement of the sheet, similar to that produced by a moving wrinkle in a carpet.

Figure 1

Sketch of experimental setup. A rigid frame of radius $R=56\mathrm{mm}$ holds a circular sheet, of radius $R_s=73\mathrm{mm}$, which vertical static deflection $d$ is adjusted by a microdisplacement stage. An oscillatory vertical displacement of amplitude $d_F$ and frequency $f$ is superimposed. A schematic of the sheet shape (developable cone structure) is shown on the right. Shaded region indicates sector of the sheet which is not in contact with the cylindrical hoop (in white), while dashed lines delimit the concave sector.

Figure 2

Phase diagram as a function of forcing frequency $f$ and forcing amplitude $d_F$ for static center displacement $d=4\mathrm{mm}$. VIB stands for vibration of the sheet, ROT for rotation of the sheet, and APR for aperiodic rotation. The triangles pointing up correspond to thresholds measured for increasing amplitudes $d_F$ whereas all other symbols were obtained for decreasing amplitudes. Inset: rolling (RM) and tilt modes (TM).

Figure 3

Minima of the vibration (VIB) and rotation (ROT) tongues of Fig. 2, as a function of the center static displacement $d$. (a) Frequencies $f_0$ (ROT) and $f_1$ (VIB), and frequency of decaying free oscillations, ${\nu }_1$. (b) Thresholds in amplitude of forcing $d_F^0$ (ROT) and rotation $d_F^1$ (VIB).

Figure 4

Counterclockwise rotation at velocity $f/2$ in the ROT tongue. Center displacement, $d=4\mathrm{mm}$, forcing amplitude $d_F=3\mathrm{mm}$, and frequency $f=11\mathrm{Hz}$. The reflection of a fluorescent ring is seen; it is approximately circular in the convex sector of the $d$ cone and inverted in the concave part (axis marked by an arrow). The rotation velocity $\varphi (t)$ and the amplitude of the $d$ cone ${ϵ}_s(t)$ vary along a period of rotation. (a) $t=0$. Concave sector pointing to 6:30, high amplitude ${ϵ}_s$, slow rotation. (b) $t=16\mathrm{ms}$, medium ${ϵ}_s$, fast rotation. (c) $t=56\mathrm{ms}$, high ${ϵ}_s$, slow rotation. (d) $t=74\mathrm{ms}$, medium ${ϵ}_s$, fast rotation. (e) $t=92\mathrm{ms}$, small ${ϵ}_s$, very fast rotation; note that the shape is slightly asymmetric at fast rotation. (f) $t=106\mathrm{ms}$, medium ${ϵ}_s$, slow rotation.

Figure 5

Frequency of material rotation $f_r$ normalized with frequency of forcing $f$ and $d$ cone strength $ϵ=d/R$, vs amplitude of forcing $d_F$ over center displacement $d$. $d=3\mathrm{mm}$ (○), $d=4\mathrm{mm}$ (□), and $d=5.9\mathrm{mm}$ (⋄). Fit to equation $f_r=p/4f{ϵ}^2[1-4/\pi (d_F/d)+(d_F/d{)}^2/2]$, yielding $p=1.86$.