Concept of Coupled Displacement Field for Large Amplitude Free Vibrations of Shear Flexible Beams

Abstract

Continuum solutions for solving the large amplitude free vibration problem of shear flexible beams using the energy method involves assuming suitable admissible functions for the lateral displacement and the total rotation. Use of even, single-term admissible functions leads to two coupled nonlinear temporal differential equations in terms of the lateral displacement and the total rotation, the solution of which is rather involved. This situation can be effectively tackled if one uses the concept of a coupled displacement field wherein the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation of the shear flexible beam. This approach leads to only one undetermined coefficient, in the case of single-term admissible functions, which can easily be used in the principle of conservation of total energy, neglecting damping, to solve the problem. Finally, one gets a nonlinear ordinary differential equation of the Duffing type which can be solved using any available standard method. The effectiveness of the concept discussed above is brought out through the solution of the large amplitude free vibrations, in terms of the fundamental frequency, of uniform shear flexible beams, with axially immovable ends, using single-term admissible functions.

Large amplitude free vibrations of simply supported moderately thick rectangular plates using coupled displacement field method

“…By applying harmonic balance method [2] and simplifying, the expression for frequency ratio can be written as:…”

“…However much attention was not received in the continuum correlation of the Finite element analysis combined with the CDF in the past years except in the formulation presented by Zhou [1] where the two independent variables (transverse displacement and rotation) are integrated using coupling equation. The large amplitude free vibration behavior of uniform Timoshenko beams using CDF method was successfully demonstrated [2]. In this study, using CDF method the large amplitude free vibration behavior of uniform shear deformable moderately thick rectangular plates was studied and exact closed form solutions were obtained.…”

Theoretical analysis and a parametric study on forced vibration of large statically deformed curved beam with rigid links and moving support

“…Some of such reported models include closed-form solution as eigenvalue problem, 18 elliptic integral-based approach, 19 homotopy analysis method, 20 harmonic balance method, 21 Ritz method, 22 Galerkin method, 23 etc. For thick beams having a low slenderness ratio, rotary inertia and shear deformation effects influence the vibration characteristics significantly which are analyzed in the framework of Timoshenko beam theory 24,25 and several higher-order shear deformation theories. 26 For regular uniform thick beams, some of the reported analysis methods are general differential quadrature method, 27 variational differential quadrature method, 28 etc.…”

A weak form quadrature element method for nonlinear free vibrations of Timoshenko beams

“…Adopting a space averaging approach, Foda (1999) studied the nonlinear vibrations of simply supported Timoshenko beams using the multiple scales method. Thereafter, various novel methods and theories have been applied to analyze the nonlinear vibrations of Timoshenko beams, such as the differential quadrature method (Zhong and Guo, 2003;Guo et al, 2004;Liao and Zhong, 2008), the spline-based differential quadrature method (Zhong and Liao, 2007), the coupled displacement field method (Rao et al, 2006), the boundary element method (Sapountzakis and Dourakopoulos, 2009a, b), the modified couple stress theory (Asghari et al, 2010), the harmonic balance method (Gunda et al, 2010), the homotopy analysis method (Kargarnovin and Jafari-Talookolaei, 2010), the homotopy perturbation method (Mojahedi et al, 2011), and the strain gradient theory (Asghari et al, 2012).…”