Desired Boundary Conditions Research Articles

Natural frequencies of membranes having mixed boundary conditions

A number of approximate methods have been given for determining the natural frequencies of membranes of arbitrary shape and uniform boundary conditions. In this paper, a method is presented whereby natural frequencies can be determined for membranes of arbitrary shapes with a portion of the boundary clamped (no displacement) and a portion free (vanishing slope). The method is developed through expanding the displacements in a series of functions, each of which satisfies the governing differential equation, but not the mixed boundary condition. A variational principle is used to find those expansion coefficients which give the best approximation to the desired boundary condition. The required variational principle is developed from Hamilton's principle through the same relaxation of the class of admissible functions used to develop Reissner's principle from the Theorem of Minimum Potential Energy. Results for a rectangular membrane, clamped on three edges, with the fourth edge partially clamped and partially free, are given, and are compared with results obtained for the same geometry through a collocation procedure and through a finite element analysis.

Fundamental-frequency approximation methods

The vibration of a sandwich plate is used, as an illustrative problem, to evaluate the effectiveness of some methods of obtaining one-term approximations for the lowest eigenvalue of coupled systems. It is shown that, in using a method of weighted residuals, the boundary conditions satisfied by both the trial and the weighting function play significant roles in the accuracy of the eigenvalue. Moreover, the desired boundary conditions for the weighting function may differ from those of the trial function, so that the Galerkin method, in which the trial function and weighting function coincide, may not be the best method to use. In addition, it is shown that the values of some physical parameters may play a critical role in the accuracy obtained with a given trial function. It is demonstrated that good approximations of the fundamental frequency are obtainable upon uncoupling a set of differential equations, but uncoupling may lead to non-self-adjoint boundary conditions, in which case the weighting function should satisfy boundary conditions which are adjoint to those of the trial function. Attention is also called to a method, due to Lanczos, which does not require a choice of trial or weighting functions, is insensitive to the form in which the problem is stated, appears to require less effort for a given degree of accuracy, and is applicable to non-self-adjoint as well as self-adjoint systems.

N-Burn Optimal Analytic Trajectories

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

P1 Blackness Theory for Infinite Cylinders

P1 blackness theory for infinite cylinders is developed in a manner analogous to the case of infinite slabs by defining effective diffusion theory cross sections in the “black” region which reproduce the desired boundary condition at the surface. Results of numerical analysis of a model problem are presented and the relative improvement of P1 blackness theory over normal P1 theory is compared with similar results for infinite slabs.

Experiments on Turbulent Heat Transfer in a Tube With Circumferentially Varying Thermal Boundary Conditions

An experimental investigation, supported by analysis, was performed to determine the heat transfer characteristics for turbulent flow in a circular tube with circumferentially varying wall temperature and wall heat flux. Air was the working fluid. The desired boundary conditions were achieved by electric heating within the wall of a tube whose thickness varied circumferentially. In this way, ratios of maximum-to-minimum wall heat flux as large as two were attained. Local heat transfer coefficients, deduced from the experimental data, display a circumferential variation that is substantially smaller than the heat flux variation. In general, lower heat transfer coefficients correspond to circumferential locations of greater heating, while higher coefficients correspond to locations of lesser heating. The predictions of prior analyses appear to overestimate the circumferential variation of the heat transfer coefficient. A specially designed probe was employed to measure the radial and circumferential temperature distributions within the flowing airstream. On the basis of these measurements, as well as from the heat transfer results, it is concluded that, in the neighborhood of the wall, the tangential turbulent diffusivity is greater than the radial turbulent diffusivity. The axial thermal development was found to be more rapid on the lesser-heated side of the tube than on the greater-heated side. Experimentally determined circumferential-average heat transfer coefficients agreed well with the predictions of analysis.