A FOURTH ORDER ENERGY STABLE FINITE DIFFERENCE SCHEME FOR A TIMOSHENKO BEAM EQUATIONS WITH LOCALLY DISTRIBUTED FEEDBACK

This work deals with the numerical solution of a control problem governed by the Timoshenko beam equations with locally distributed feedback. We apply a fourth-order Compact Finite Difference (CFD) approximation for the discretizing spatial derivatives and a Forward second order method for the resulting linear system of ordinary differential equations. Using the energy method, we derive energy relation for the continuous model, and design numerical scheme that preserve a discrete analogue of the energy relation. Numerical results show that the CFD approximation of fourth order give an efficient method for solving the Timoshenko beam equations.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

Numerical investigation on effect of internal flow on vortex-induced vibration dynamics of a full-scale mining riser

The dynamics of a full-scale pipe conveying fluid inside is investigated based on the finite element method (FEM). During the numerical simulation, the Euler–Bernoulli beam equations are used to model the motion of the full-scale pipe while the effect of internal flow is considered. And the semi-empirical time-domain model is applied to simulate the external hydrodynamic forces exerted on the pipe. Then the typical vortex-induced vibration (VIV) characteristics of the full-scale pipe considering both internal and external flows are analyzed. The results show that with the increase of the internal flow velocity, the natural frequencies of the full-scale pipe decrease and the in-line (IL) and cross-flow (CF) dominating modes are increased. Furthermore, the dominating frequencies in both IL and CF directions are not notably changed. And the IL and CF root-mean-square (RMS) values of amplitudes fluctuate at around the stable values due to the stable external hydrodynamic forces. It should be noticed that the IL and CF RMS strain values of the full-scale pipe are increased, especially for high external and internal flow velocities. The maximal RMS strain values in both IL and CF directions appear next to the pipe top, which could have an influence on the motion of the ship on the sea surface.

Aeroelastic Stability Analysis of Electric Aircraft Wings with Distributed Electric Propulsors

In this paper, the effect of distributed electric propulsion on the aeroelastic stability of an electric aircraft wing was investigated. All the electric propulsors, which are of different properties, are attached to the wing of the aircraft in different positions. The wing structural dynamics was modelled by using geometrically exact beam equations, while the aerodynamic loads were simulated by using an unsteady aerodynamic theory. The electric propulsors were modelled by using a concentrated mass attached to the wing, and the motor’s thrust and angular momentum were taken into account. The thrust of each propulsor was modelled as a follower force acting exactly at the centre of gravity of the propulsor. The nonlinear aeroelastic governing equations were discretised using a time–space scheme, and the obtained results were verified against available results and very good agreement was observed. Two case studies were considered throughout the paper, resembling two flight conditions of the electric aircraft. The numerical results show that the tip propulsor thrust, mass, and angular momentum had the most impact on the aeroelastic stability of the wing. In addition, it was observed that the high-lift motors had a minimal effect on the aeroelastic stability of the wing.

Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations

In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.