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.
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