Nonlinear Vibrations of Beams, Strings, Plates, and Membranes Without Initial Tension

The subject of this paper is nonlinear vibrations of beams, strings (defined as beams with very thin uniform cross sections), plates and membranes (defined as very thin plates) without initial tension. Such problems are of great current interest in minute structures with some dimensions in the range of nanometers (nm) to micrometers (μm). A general discussion of these problems is followed by finite element method (FEM) analyses of beams and square plates with different boundary conditions. It is shown that the common practice of neglecting the bending stiffness of strings and membranes, while permissible in the presence of significant initial tension, is not appropriate in the case of nonlinear vibrations of such objects, with no initial tension, and with moderately large amplitude (of the order of the diameter of a string or the thickness of a plate). Approximate, but accurate analytical expressions are presented in this paper for the ratio of the nonlinear to the linear natural fundamental frequency of beams and plates, as functions of the ratio of amplitude to radius of gyration for beams, or the ratio of amplitude to thickness for square plates, for various boundary conditions. These expressions are independent of system parameters—the Young’s modulus, density, length, and radius of gyration for beams; the Young’s modulus, density, length of side, and thickness for square plates. (The plate formula exhibits explicit dependence on the Poisson’s ratio.) It is expected that these results will prove to be useful for the design of macro as well as micro and nano structures.