Instytut Podstawowych Problemów Techniki

Streszczenie:

This study develops a model of the dynamics of the extended 2D Gao beam and simulates it. Here, the static model studied by Dyniewicz, Shillor and Bajer (Meccanica, 2024), is modified by incorporating inertial terms to account for dynamic behavior. The beam model expands the 1D Gao beam, which can oscillate around a buckled position, and the Timoshenko beam, which factors in shear effects in the beam’s cross sections. The resulting model consists of two highly nonlinear wave equations, alongside specified initial and boundary conditions. A finite element method (FEM) algorithm is created and executed to analyze the system’s vibrations induced by a periodically oscillating longitudinal compressive force. The simulation results are discussed, highlighting the ways the initial conditions influence the solutions, which are graphically illustrated through phase portraits. From an engineering viewpoint, this thick Gao beam model is notable for its relative simplicity. Similarly to the Timoshenko beam model, it includes shear effects, yielding a wave-like equation of motion. Considerations of the shear are essential for accurately analyzing thicker beams, as traditional models that overlook them may fail to capture the true system behaviors. Consequently, this extended Gao model offers more realistic outcomes in dynamic scenarios.

Słowa kluczowe:

Extended Gao beam, Dynamic oscillations, Vibrations about buckled state, Simulations

Afiliacje autorów:

Streszczenie:

This work derives and simulates a two-dimensional extension of the nonlinear Gao beam, by adding the cross-sectional shear variable, similarly to the extension of the usual Bernoulli-Euler beam into the Timoshenko beam. The model allows for oscillatory motion about a buckled state, as well as adds vertical shear of the cross sections, thus reflecting better nonlinear thick beams. The static model is derived from the principle of virtual elastic energy, and is in the form of a highly nonlinear coupled system for the beams transverse vibrations and the motion of the cross sections. The model allows for general distributive transversal and longitudinal loads and a compressive horizontal load acting on its edges.

The model is simulated numerically, using the dynamic version for better insight into the steady solutions. The terms that distinguish our numerical solutions from the solutions of the Gao beam, described in the literature, are highlighted. The numerical scheme and its characteristic finite element matrices allow us to obtain simulation results that demonstrate the type of vibrations of our extended and modified beam, and also the differences between these solutions and those of the Gao beam model.

Słowa kluczowe:

Gao beam, nonlinear vibration

Afiliacje autorów:

Streszczenie:

This paper models, analyzes and simulates the vibrations of a nonlinear Gao beam that is subjected to a moving mass or a massless point-force. Such problems arise naturally in transportation systems such as trains or trams. The dynamics of the system as the mass or the force move on the beam are investigated numerically in the cases when the vibrations are about a buckled state, and in the cases when the mass is positive or vanishes. The simulations are compared to those of the Euler–Bernoulli linear beam and the differences are highlighted. It is seen that the linear beam may be used only when the loads are small, while the Gao beam allows for moderate loads. The simulations are based on a time-marching finite elements algorithm for the model that has been developed and implemented. The results of representative and interesting computer simulations are depicted. The existence of weak solutions of the model is established using a variational formulation of the problem and results about variational set-inclusions.

Słowa kluczowe:

dynamic vibrations, buckling of a Gao beam, moving point-load

Afiliacje autorów:

Streszczenie:

A model for the dynamics of a Gao elastic or viscoelastic nonlinear beam that is subject to a horizontally moving vertical point-force is modeled and computationally studied. In particular, the behavior and vibrations of the beam as the mass is moving on it is investigated. Such problems arise naturally in transportation systems with rails. A time-marching finite element numerical algorithm for the problem is developed and implemented. Results of representative simulations are depicted and compared to the behavior of a linear Euler beam with a moving mass.

Słowa kluczowe:

Gao beam, moving inertial point load, Finite Element simulations, beam vibration

Afiliacje autorów: