Using Hamilton’s principle, exact equations of motion for non-linear planar and spatial Euler–Bernoulli beams are derived. In the existing non-linear Euler–Bernoulli beam formulations, some elastic terms are dropped by differentiation from the incomplete Green–Lagrange strain tensor followed by negligible elastic deformations of cross-sectional frame. On the other hand, in this article, the exact strain field concerning considerable elastic deformations of cross-sectional frame is used as a source in differentiations. As a result, the achieved closed-form equations are exact and more accurate than formerly reported equations in the literature. Moreover, the applicable dynamic model of inextensional beams which is fully accurate, yet simple has been shown. The planar and inextensional dynamic models have been compared with the existing dynamic models in the literature, and the proposed dynamic models demonstrate significant improvements in the numerical results. Finally, experiments on the carbon fibre rod...

https://journals.sagepub.com/doi/pdf/10.1177/0954406211420206

Exact formulations of non-linear planar and spatial Euler-Bernoulli beams with finite strains

On coupled longitudinal and lateral vibrations of elastic rods

Using the fully-enhanced variation of elastic potential energy the secondary extra elastic terms have been revealed in the dynamic model of a nonlinear 3D Euler-Bernoulli beam undergoing negligible elastic orientation. They have been sensed neither in the nonlinear 3D Euler-Bernoulli beam nor in the enhanced nonlinear 3D Euler-Bernoulli beam. They complete the issue of the new elastic terms of the enhanced nonlinear 3D Euler-Bernoulli beam. The whole of the extra elastic terms have been shown in the motion equations and the boundary conditions of a fully-enhanced nonlinear 3D Euler-Bernoulli beam undergoing negligible elastic orientation.

Modified Nonlinear 3D Euler Bernoulli Beam Theory

Past researchers suggested that “static instabilities” exist at certain rotational speeds of whirling rods. This paper shows these instabilities are an artefact of the material constitutive laws that are being used well outside their range of applicability. An alternative approach is developed where strains due to rotation are separated from the superimposed vibration. This enables the generally predicted lowering of longitudinal natural frequencies with rotational speed shown to be simply a result of the bulk changes in the geometry of whirling rods. Steady-state equations of whirling rods are formulated in Lagrangian coordinates. Due to the nonlinear nature of the governing equations, an original numerical method is applied to solve the problem. Numerical results are compared with analytical results obtained from the linearized uniaxial model. There is close agreement between these two models at low angular velocities. However, at high angular velocities, discrepancies between them arise, confirming tha...

Longitudinal vibration frequencies of steadily whirling rods

http://scientiairanica.sharif.edu/article_3242_2c06dddb327de578add90a5edfbdab59.pdf

Dynamic Model of a Mobile Robot with Long Spatially Flexible Links

Bunding 45 2130 Eighth Street, Suite 1, Wright-Patterson AFB, OH 45433-7765 In this paper, a model for the deformation of a rotating prismatic rod-like body is developed and analyzed. The novel feature of the model is its incorporation of the Poisson effect. As a result, it provides realistic solutions for the deformed states of steadily rotating rods. The model presented in this paper is also simplified to a nonlinear version of a classical model for rotating rods, Numerical continuation is used to solve the boundary value problems associated with these models.

J. S. Turcotte

Papers

On the Steady Motions of a Rotating Elastic Rod