This paper describes a new nonlinear finite element formulation of a spatial beam element. In the large deformation of a beam, dominant motion is due to the rigid body rotation and the strains are always remaining small. Therefore, using the co-rotational element coordinates, strain-displacement equation is assumed to be linear in these element coordinates and the geometric nonlinearity is considered in the transformation between the element coordinates and the global coordinates. Thus, only the usual linear stiffness matrix is first formulated in the element coordinates, and then the geometric (nonlinear) stiffness matrix is derived from the variation of the transformation matrix.
In Part 1, the authors presented a new nonlinear finite element formulation of a spatial beam element, and obtained an unsymmetric tangent stiffness matrix. In this paper, firstly we solve two examples of plane and spatial deformations and show that the use of artificially symmetrised tangent stiffness matrix does not decrease the accuracy of the solution. Then, three numerical examples of spatial post-buckling problems are presented, indicating the accuracy and the efficiency of the proposed method.
An example of hypersonic turbulent flow is numerically studied using Navier-Stokes equations, where the LES analysis includes the complicated effects of compressibility and chemical reactions of air. In a 2-dimensional finite-length channel, the counter-direction flows are introduced in the upper and lower portions, essentially generating a supersonic turbulent shear layer: The interactions are quite complicated due to the presence of shock waves, expansion waves, contact discontinuities, vortices, and endothermic dissociation reactions. It is found out that the flow patterns are changed by the aspect ratio of the channel and the incident Mach number. The resulting turbulent flows are entirely nonsteady and yet approximately periodical. The flowfields are not significantly altered by introducing an LES model; the phenomenon is primarily controlled by wave interactions.
A boundary element method program for unsteady supersonic flow around arbitrary configurations has been developed. The fundamental formulation is based on Morino's method, the application of which to unsteady supersonic flow had been very limited before the present work. Linear distribution of doublet is adopted to ensure the continuity of the strength on panel edges. The exponential terms in the integrands are approximated by using Maclaurin expansion. Flow around oscillating wings is simulated and the results are shown to be consistent with previously known numerical results or with wind tunnel tests. The effects of the angle of attack and the wing thickness, both of which could not be treated by previous methods, are evaluated and it is shown that the effect of the latter is not negligible to assess the unsteady aerodynamic force on wings correctly.
Demand for aviation has been increasing, particularly in major airports. It is desirable to handle more traffic in the terminal areas of these airports. However, its increase results in that of Air Traffic Control (ATC) operations and complicated traffic handling. A limit of ATC handling capability depends upon the air traffic controller's workload. Therefore, we conducted a series of terminal ATC simulation experiments, focusing on arriving traffic. This paper presents the results of the simulations. The traffic handling capability is discussed based upon the aircraft trajectory data, and then the number of aircraft handled is analyzed in relation to the ATC instructions, which is a measure of the controller's workload.