Showing papers in "Journal of the Aeronautical Sciences in 1948"

Finite Deflections of Sandwich Plates

Abstract: This paper gives the basic differential equations for finite transverse deflections of sandwich plates under the following assumptions. The plate consists of a core layer and of two face layers of such construction that the face-parallel stresses in the core and the variation of the face stresses over the thickness of the face layers are negligible. The resultant equations permit the analysis of the effect of transverse shear stress deformation and transverse normal stress deformation in the core on the overall behavior of the plate. I t is shown that, in general, the effect of the transverse normal stresses in the core is negligibly small compared with the effect of the transverse shear stresses. The equations, when simplified by the omission of the transverse normal stress terms, are brought into a form suitable for the solution of rectangular-plate problems. It is further shown that the range of deflections for which the linear ("small deflection") theory is adequate decreases in accordance with a simple explicit formula as the core is made softer relative to the faces. The finite deflection equations are used to obtain plate-buckling equations that include the effect of transverse shear stress deformation on buckling loads. A typical buckling problem is solved and discussed. • (1) I N T R O D U C T I O N I THIS NOTE we derive a system of differential equations for the small finite deflections of sandwich plates. This system of equations is a generalization of the now well-known results for homogeneous plates, where the problem can be reduced to two simultaneous equations for the transverse deflection w of the middle surface and for an Airy stress function F. We consider a sandwich plate consisting of a core layer of thickness h — t and two face layers of thickness t each. We assume tha t / is small compared with h and tha t the values of the elastic constants Ef, Gf for the face layers are large compared with the values of the elastic constants ECf Gc for the core layer. We further assume tha t the products tEf, tGf are large compared with the values of hEC) and hGc. On the basis of t h e assumption tha t t

The Boundary Layer of Yawed Cylinders

Abstract: Starting with the Navier-Stokes equations of motion for viscous fluids, Prandtl 's argument regarding the orders of magnitudes of quantities in a laminary boundary layer is repeated for the special case of a yawed cylinder of infinite length. I t is shown that for this case the chordwise flow is given by the same equations as for unyawed (plane) flow about the same cylinder and that the spanwise boundary-layer flow can be calculated by integration of a linear second-order differential equation. This result leads to useful conclusions—e.g., that the location of laminar separation of the chordwise flow is independent of yaw. As an example, the flow over a flat plate, at zero incidence, having a sweptback leading edge, is treated. I t is found that the boundary-layer flow is always in the direction of the external stream and has the Blasius velocity profile—i.e., it is unaffected by sweepback of the leading edge. As a second example, a certain class of cylinders suggested by Prandtl is considered in the yawed condition. The flow about these in the plane case is known, the spanwise boundary-layer flow is calculated approximately, and some characteristic features are presented graphically.

Spectrum of Locally Isotropic Turbulence

Abstract: The concept of locally isotropic turbulence is used in Kolmogoroff's sense. Kolmogoroff's second similarity hypothesis is not used, but other hypotheses are suggested for the similarity of energy transfer in the energy spectrum of turbulence. The spectrum thus obtained is identical with the —5/3 power law in the nonviscous range but falls off rapidly in the viscous high-frequency end. The correlation coefficients agree with Kolmogoroff's predictions for both small r and for large r and give explicit results in the intermediate range. The only free constant in the present theory is R0, a certain Reynolds Number that corresponds to the smallest eddies. Extensive comparison with available energy spectrum and correlation measurements have been made.

Problem Types in the Theory of Perfectly Plastic Materials

Abstract: Typical problems of the theory of perfectly plastic materials are discussed for the example given by the structure shown in Fig. 1. A simple graphical representation is developed which facilitates the study of the states of stress and residual stress produced by slowly varying the load. For monotonically increasing load, three domains of mechanical behavior are denned: elastic deformation, contained plastic deformation, and unrestricted plastic flow. In the domain of contained plastic deformation the instantaneous stresses in the structure depend only on the instantaneous load and can be found from the minimum principle of Haar and von Karman. However, if unloading is permitted, the stresses depend on the complete history of loading, and the principle of Haar and von Karman must be replaced by a minimum principle recently developed by H. J. Greenberg. Cases in which the precise loading program is known beforehand are not often encountered in engineering. As a rule, only the extremes are known between which the load will vary. Under certain conditions, the structure will then "shake down" to a state of residual stress such that all further variations of the load between the given extremes are supported in a purely elastic manner. This "state of residual stress" is independent of the precise program of loading. Conditions for the existence of such a state of residual stress are discussed, and a minimum principle is conjectured from which this state may be determined. Finally, structural stability in the plastic range is discussed. I t is pointed out that the customary formulation of the stability problem for conservative systems (elastic range) is not adequate for nonconservative systems (plastic range). ANY METHOD of theoretical or experimental stress analysis is based on a stress-strain law. Hooke's law is generally accepted as an adequate basis for stress analysis in the elastic range. In the plastic range, however, no stress-strain law has found equally general acceptance. The selection of a stress-strain law adequate for the treatment of a specific problem is therefore important. This selection is facilitated by a classification of the problems encountered in the theory of plasticity. In the following, such a classification of problems is attempted for plastic materials that do not exhibit strain-hardening (perfectly plastic materials). To simplify the mathematical work as much as possible, Received February 25, 1948. * This paper was presented at the Symposium on Plasticity held at Brown University on February 9-10, 1948, under the joint sponsorship of the Office of Naval Research and the Bureau of Ships (Contract N7onr-358). t The author is indebted to F. R. Shanley for valuable suggestions concerning the discussion of structural stability in the plastic range. t Professor of Applied Mechanics. the various problem types will be discussed for the example given by the system shown in Fig. 1. The two-force members OA, OB, OC join the point 0 to the fixed points A, B} C. The system is symmetric with respect to the vertical OB, and the load P is acting along this axis of symmetry. The subscripts 1, 2, 3 will be used to refer to the bars OA, OB, OC, respectively, and the forces acting in these bars will be denoted by Si, S2, S3. On account of the symmetry of the system, Si — S3. Consider, first, the mechanical behavior of the system in the elastic range. The elastic strain energy of the system is given by an expression of the form

Variational Method in the Theory of Compressible Fluid

Abstract: The variational principle in hydrodynamics was first studied by Hargreaves, who has shown that, when the hydrodynamic equations are satisfied, the integrand in the variational principle is a linear function of the pressure. For steady, irrotational flows, Bateman ,3 has shown that the integrand is the pressure only, which can be written as a function of the velocity potential 4>. A direct method to obtain approximate solution in the case of subsonic flow of a compressible fluid can therefore be formulated following the Rayleigh-Ritz method instead of solving the original nonlinear differential equations. In applying this method to the case of a compressible fluid passing an arbitrary airfoil, however, there are two difficulties: first, since the fluid region is infinite, Bateman's integral is not directly applicable; second, it is not easy to assume the potential function satisfying the boundary conditions. The nonlinear problem of compressible flow passing an arbitrary airfoil has been formulated in this paper. I t is proved that, by conformal transformation, the velocity potential can be set up to satisfy the boundary conditions in the general case, and, hence, the Rayleigh-Ritz method can be applied. I t is also shown that, by using the Theory of Residues in the integration and a modified Crout method in solving the resulting simultaneous equations, the computational labor can be mate' daily reduced. Since the variational principle is valid for subsonic, sonic, and supersonic flows, the possibility of applying it to the study of mixed flow is indicated. A numerical example of the nonlinear problem of the compressible flow around a circular cylinder is carried out, and the results compare favorably with those found by the existing methods.

The Flow of a Perfect Fluid Through an Axial Turbomachine with Prescribed Blade Loading

Abstract: The theory of the three-dimensional flow through an axial turbomachine, associated with variation of circulation along the blade length, is described as an extension of the classical theory of finite wings and is simplified to a problem in axially symmetric rotational fluid motion by considering an infinite number of blades in each row. The problem is linearized by considering the vorticity generated by the blades to be transported by the mean velocity. The linearized problem leads to well-known partial differential equations and is solved for the radial, tangential, and axial velocity components induced by a single row of stationary or rotating blades with finite chord and prescribed loading. The particular case for which the blade chord approaches zero and the tangential velocity changes discontinuously is associated with the theory of the Prandtl lifting line for finite wings. Because the problem has been linearized, the solution for a multistage turbomachine follows by superposition of appropriate single rows. Analytical expressions and graphical values for the velocity components are given for a single stationary or rotating blade row of given loading with a hub/ t ip ratio of 0.6 and a blade aspect ratio of 2. The corresponding discontinuous approximation is compared with the more nearly exact solution and is shown to constitute a useful approximation to the solution for a finite blade chord when the discontinuity is located appropriately. An exponential approximation for the velocity components, deduced from the analysis, allows rapid estimation of the rate at which the equilibrium velocity profiles develop ahead of, and behind, a blade row and, using the superposition principle, provides a simple means of approximating the velocity distribution in a multistage turbomachine and of discussing mutual interference of blade rows.

Heat transfer to bodies traveling at high speed in the upper atmosphere

Abstract: A general method has been developed, using the methods of kinetic theory, whereby the surface temperatures of bodies can be calculated for steady flight at any speed in a rarefied gas. The particular solution was made for a flat plate; however, the calculations can be easily extended to bodies of arbitrary shape.

Theoretical Pressure Distributions for a Thin Airfoil Oscillating in Incompressible Flow

Abstract: Formulas are derived for the theoretical pressures acting on a thin airfoil and aileron combination oscillating with harmonic motion of small amplitude in incompressible flow. Calculations of pressure distribution are made at several representative values of the frequency parameter 1/k for airfoil translation, for airfoil rotation about the quarter-chord, and for hinged-surface flapping motions. The tabulated data thus obtained are plotted. INTRODUCTION T FORCES AND MOMENTS for the harmonically oscillating airfoil have previously been •derived by Theodorsen without the pressure formulas having been explicitly stated. A number of German pape r s 2 6 have been written covering various phases of the subject. Of these, Dietze has developed pressure formulas (for translation, pitch about the leading edge, and control surface flapping) which, when referred to the quarter-chord axis, can be shown to be in agreement with those developed in this paper. In these German investigations, as in Theodorsen's, the primary objective was to obtain the lifts and moments; therefore, computations were not made to show typical pressure distributions. • In a recent investigation to determine the magnitude and phase of aileron hinge moments due to an aileron oscillating harmonically in a transonic airstream, pressure-measuring units were installed in the wing and aileron upper and lower surfaces to measure the oscillating pressures. The pressure pickups were designed specifically for the conditions existent in that test and were, therefore, not satisfactory for measuring the oscillating pressures a t the lower dynamic pressures of normal flight speeds. Formulas were derived and computations for theoretical oscillating pressures were made for the range of parameters expected in tha t test in order t ha t comparisons with the experimental pressures could be made. The derivations and the tables of computed pressures for the incompressible flow are presented below in anticipation of the development of . pressure-measuring techniques to check the theoretical flutter coefficients in the usual velocity range.

Inelastic Column Theories and an Analysis of Experimental Observations

Abstract: The inelastic column buckling theories as usually presented are found to be rather confusing. Many experimental results still cannot be explained satisfactorily by the theories as they stand. I t appears to be pertinent, therefore, to define the inelastic buckling problem once more from a more rigorous mathematical point of view and to give a more rigorous mathematical treatment of the problem. The effect of assuming a constant tangent modulus on the buckling load in the tangent modulus formula is discussed, and many anomalies of column behavior in the inelastic region are explained. In making such a study, it is important to emphasize the difference between an ideal column and an actual column and the difference between the buckling load and the ultimate load. Southwell's method for analyzing column tests, which was originally proposed for the case of elastic buckling, is now shown to be valid for inelastic buckling. I t is also shown that , in applying the method, instead of analyzing load and deflection measurements, simultaneous load and strain readings can be used. Thus, it is easier to measure the strain more accurately.

Stability of Boundary Layers and of Flow in Entrance Section of a Channel

Abstract: Lin's approximate formula for the minimum critical Reynolds Number is used to study the entrance flow into a channel, the boundary layer with pressure gradient, and the boundary layer with suction. I t is found that the results thus obtained compare favorably with much more elaborate calculations wherever they are available. In the case of the entrance flow into a channel, the new result is found—that the entrance section is comparatively more stable than the fully developed parabolic velocity distribution.

Surface-Pressure Gradient and Shock-Front Curvature at the Edge of a Plane Ogive with Attached Shock Front

Abstract: The flow of air at high Mach Numbers past a sufficiently sharp plane ogive is characterized by the presence of a curved shock front attached to the edge of the ogive. This paper presents and carries out a simple method for computing the exact surfacepressure gradient and shock-front curvature at the edge of the ogive. . A comparison is made between the exact edge-pressure gradient and that given by two approximate methods of computing the pressure distribution along an ogive. The numerical results, presented graphically, cover all ogives with attached shock fronts for Mach Numbers up to 13.

Plastic Flow Characteristics of Aluminum-Alloy Plate

Abstract: Commercial hot-rolled P/Vin. 24ST aluminum plate possesses a, marked degree of crystallographic anisotropy. Tensile tests a t various orientations showed tha t the yield strength is only slightly dependent upon the crystallographic anisotropy. Ratios of the transverse strains to the longitudinal strain varied from — 0.30 to —0.57, while for an isotropic material the ratio would b e 0 . 5 0 .

Low-Speed Flutter and Its Physical Interpretation

Abstract: The phenomenon of low-speed flutter of an oscillating airfoil is investigated. The problem resolves itself into the study of two phenomena: tha t of zero air-speed classic flutter and tha t of an oscillating airfoil with no vortex shed. I t is seen tha t both cases involve oscillation of the airfoil about a nodal line located three-quarters of the chord aft of the leading edge. The study is based on the fundamental theory developed by Theodorsen. The relationship between the parameters required for zerospeed flutter for the general case is developed, and the physical interpretation of this phenomenon is discussed. For the latter case, the solution of the equations of motion, based on noncirculatory flow, and the equations resulting from the Kut ta condition result in a relationship between the inertia and elastic parameters for which the airfoil will flutter (i.e., maintain oscillations a t constant amplitude) for all air speeds. The physical interpretation and the mechanism of this phenomenon are discussed. Practical implications, such as the importance of nodal-line location and the effect of the physical parameters of the airfoil on nodal line location, are discussed. The possibility of a new approach to the flutter problem is suggested—that of employing the ground vibration modes of the airplane to study the existence of a low flutter speed and as a design guide to raise the flutter speed. , Conclusions are also drawn regarding the influence of aspect ratio and the relative importance of theoretical aerodynamic coefficients against values of these quantities measured in the wind tunnel.

On The Kinematics of Turbulence

Abstract: A short review of different correlation coefficients and spectra is presented. Some new relations between these characteristics are given for homogeneous and isotropic turbulence. The "form of a turbulent particle" has been given from the point of view of correlation. The study of numerous experimental correlation curves shows that , where their second moments are represented as functions of their first moments, the resulting points lie near a straight line. The correlation curves may be represented by functions of the general form

A Numerical-Graphical Method of Charactersitics for Axially Symmetric Isentropic Flow

Abstract: A graphical-numerical method of characteristics, based on the Tollmien method, is developed for axially symmetric isentropic flow problems Complete rules of procedure are presented for the application of the method Similar methods have been given by Ferrari and Ferri As an illustration, the flow around a conical-nosed body is calculated In this problem a procedure is given for approximating the curved shock wave, assuming constant entropy behind the shock wave

The Theory of the Impact Tube at Low Pressures

Abstract: A theoretical analysis has been made for an impact tube of the relation between free-stream Mach Number and the impact and free-stream pressures and densities for extremely low pressures. I t is shown that the results differ appreciably from the corresponding continuum relations.

The Disturbed Flapping Motion of Helicopter Rotor Blades

Abstract: The article deals with the flapping motion of a helicopter blade when disturbed by external forces (gust, variation in cyclic pitch). The investigation is based on a rectangular and untwisted blade with the flapping hinge on the rotor axis. The disturbance gives rise to nonperiodic oscillations, which are described by a linear differential equation with periodic coefficients. In solving this equation, a criterion is obtained which permits rapid determination of the conditions of stability. Numerical evaluation shows that the disturbed motion is stable, though at a certain critical advance ratio partial destabilization sets in which increases with increasing flying speed. The influence of coupling between flapping angle and blade pitch is also considered. Practically, the disturbed motion is finished after one revolution of the rotor.