Exact Theoretical Solution Research Articles

An Optical Boundary-Layer Probe

The applicabilh^ of the schlieren-photomultiplier technique to obtain quantitative density measurements in the laminar boundary layer induced by a traveling shock wave in a shock tube is investigated. Tests were conducted at a Mach number of 1.58 so that the data could be compared with the exact theoretical solution tabulated by Mirels. The data obtained are in good agreement with the theory if the distance of the light beam above the floor of the shock tube is adjusted to fit the theoretical curve; this would not be necessary if a larger shock tube were used. Values of the transition Reynolds number were also determined which are slightly less than those found by Martin using an interferometer. I t is shown that this technique is sensitive enough to detect changes in density that are only 0.006 per cent of atmospheric density.

Some Effects of Curvature on Frames

Induced radial loadings due to curvature can cause drastic changes in the stress distribution on the cross section of aircraft frames from that predicted by elementary bending theory. In the present paper, the exact theoretical solution for the stress distribution is reviewed for simple plate elements. An approximate solution for the stress distribution which is applicable to more general cross section shapes is then developed. Good agreement between the two theories is indicated for the basic model where they can be compared. An effective area concept is defined, and design charts showing its variation with the curvature parameter are presented. Experimental investigation of the stress distribution and ultimate crippling capacity of typical frame specimens is outlined. Excellent agreement was obtained between test results and the approximate theoretical solution.

Transverse Vibration of a Two-Span Beam Under Action of a Moving Constant Force

Abstract The problem relates to the transient vibration of a symmetrical, continuous, simply supported two-span beam which is traversed by a constant force moving with constant velocity. The beam is of slender proportions, flexure alone being considered. Damping is zero, and there is no mass associated with the moving force. Exact theoretical solutions for bending stress have been derived in general form. They consist of three infinite series, each related to one of three time eras as follows: (a) Where force is crossing first span; (b) is crossing second span; (c) has left the beam. Each term of a series is related to a natural mode of vibration. Quantitative theoretical studies show the variation in individual terms of the series, and also in summations of the first five terms, as the traversing velocity is varied. A mechanical model with electrical recording of stress was employed to obtain a more complete quantitative solution than was feasible analytically. The agreement between theory and experiment was reasonably good. Large magnifications of stress (of the order of 2.5) were found in the neighborhood of resonance with the fundamental mode.

Ultrasonic Radiation from Constrained Piston Sources and Rings

Quantitative knowledge of the ultrasonic field near a circular piezoelectric source radiating into a fluid is desirable for some uses (interferometry, absorption, and diffraction measurements). The exact theoretical solutions available for numerical calculations assume an ideal piston source, which is not necessarily correct in practice. Two studies of a non-ideal piston source are reported here: (1) If the piston is assumed to be partially constrained in its outer portions, the axial sound field can still be computed by the Rayleigh method. It does not differ markedly from the ideal case, in the region of greatest practical interest. Certain experiments which have been quoted as substantiating the ideal piston may therefore not be significant. (2) A circular ring source can be treated by King's method, in integrals of Bessel functions. Some non-ideal pistons are equivalent to assemblies of such rings. The narrow ring source has interesting properties: at moderate and large distances it produces a beam more highly directive than that from the circular source. The ring source also produces an axial intensity following the inverse square law, at short ranges, more closely than that of the circular source