Dynamic equations of motion for inextensible beams and plates

• Published in: Archive of applied mechanics (1991) Vol. 92; no. 6; pp. 1929 - 1952
• Main Authors: Deliyianni, Maria, McHugh, Kevin, Webster, Justin T., Dowell, Earl
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2022; Springer Nature B.V
• Subjects: Cantilever beams; Cantilever plates; Classical Mechanics; Engineering; Equations of motion; Lagrange multiplier; Mathematical models; Nonlinear differential equations; One dimensional models; Original; Partial differential equations; Rectangular plates; Stiffness; Theoretical and Applied Mechanics; 74K20; Inextensibility; Cantilever; Nonlinear elasticity; Energy harvesting; 74B20; 35L77; quasilinear PDE; 93A30
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-022-02157-7

The large deflections of cantilevered beams and rectangular plates are modeled and discussed. Traditional nonlinear elastic models (e.g., von Karman’s) employ elastic restoring forces based on the effect of stretching on bending, and these are less applicable to cantilevers. Recent experimental work indicates that elastic cantilevers are subject to nonlinear inertial and stiffness effects. We review a recently established (quasilinear and nonlocal) cantilevered beam model, and consider some extensions to two spatial dimensions, namely inextensible plates. Our principal configuration is that of a thin, isotropic, homogeneous rectangular plate, clamped on the one edge and free on the remaining three. We proceed through the geometric and elastic modeling to obtain equations of motion via Hamilton’s principle for the appropriately specified energies. We then enforce effective inextensibility constraints through Lagrange multipliers. Multiple plate analogs of the established 1D model are obtained, based on assumptions. In total, we present three distinct nonlinear partial differential equation models and, additionally, describe a class of “higher-order” models. Each model has particular advantages and drawbacks for both mathematical and engineering analyses. We conclude with a discussion of the various models, as well as some analytical problems.