Search results for "Nonlinear beam"
showing 5 items of 5 documents
Efficient formulation of a two-noded geometrically exact curved beam element
2021
The article extends the formulation of a 2D geometrically exact beam element proposed by Jirasek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier-Bernoulli hypothesis, with a possible extension to shear-flexible beams. Numerical procedures …
BEAM ELEMENT UNDER FINITE ROTATIONS
2021
The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law. The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease comput…
Implicit analytic solutions for a nonlinear fractional partial differential beam equation
2020
Abstract Analytic solutions in implicit form are derived for a nonlinear partial differential equation (PDE) with fractional derivative elements, which can model the dynamics of a deterministically excited Euler-Bernoulli beam resting on a viscoelastic foundation. Specifically, the initial-boundary value problem for the corresponding PDE is reduced to an initial value problem for a nonlinear ordinary differential equation in a Hilbert space. Next, by employing the cosine and sine families of operators, a variation of parameters representation of the solution map is introduced. Due to the presence of a nonlinear term, a local fixed point theorem is employed to prove the local existence and u…
Spatiotemporal Nonlinear Beam Shaping
2016
The reshaping of multimode waves in optical fibers is a process where the spatial and spectral degrees of freedom are inherently coupled. Our experiments demonstrate that pumping a graded-index multimode fiber with sub-ns pulses from a microchip Nd:YAG laser leads to supercontinuum generation with a uniform bell-shaped spatial beam profile.
Efficient finite difference formulation of a geometrically nonlinear beam element
2021
The article is focused on a two-dimensional geometrically nonlinear formulation of a Bernoulli beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first-order differential equations. These equations are then discretized by finite differences and the boundary value problem is converted into an initial value problem using a technique inspired by the shooting method. Accuracy of the numerical approximation is conveniently increased by refining the integration scheme on the element lev…