Nonlinear vibrations of rectangular laminated thin plates

Jongsik Woo; Sudhakar Nair

doi:10.2514/3.10898

Nonlinear vibrations of thin plates with variable thickness: Application to sound synthesis of cymbals

The Journal of the Acoustical Society of America ◽

10.1121/1.5091013 ◽

2019 ◽

Vol 145 (2) ◽

pp. 977-988

Author(s):

Quoc Bao Nguyen ◽

Cyril Touzé

Keyword(s):

Variable Thickness ◽

Nonlinear Vibrations ◽

Sound Synthesis ◽

Thin Plates

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

Journal of Applied Mechanics ◽

10.1115/1.1767167 ◽

2004 ◽

Vol 71 (4) ◽

pp. 551-559 ◽

Cited By ~ 18

Author(s):

Zhongping Bao ◽

Subrata Mukherjee ◽

Max Roman ◽

Nadine Aubry

Keyword(s):

Boundary Conditions ◽

Young’S Modulus ◽

Cross Sections ◽

Nonlinear Vibrations ◽

Young's Modulus ◽

Thin Plates ◽

Radius Of Gyration ◽

Initial Tension ◽

Various Boundary Conditions ◽

Analytical Expressions

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.

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