Showing papers in "AIAA Journal in 1965"

Reduction of stiffness and mass matrices

Abstract: Just as it is often necessary to reduce the size of the stiff­ness matrix in statical structural analysis, the simulta­neous reduction of the nondiagonal mass matrix for natural mode analysis may also be required. The basis for one such reduction technique may follow the procedure used in Ref. 1 for the stiffness matrix, namely, the elimination of coordinates at which no forces are applied.

Maximum likelihood estimates of linear dynamic systems

Abstract: This paper considers the problem of estimating the states of linear dynamic systems in the presence of additive Gaussian noise. Difference equations relating the estimates for the problems of filtering and smoothing are derived as well as a similar set of equations relating the covariance of the errors. The derivation is based on the method of maximum likelihood and depends primarily on the simple manipulation of the probability density functions. The solutions are in a form easily mechanized on a digital computer. A numerical example is included to show the advantage of smoothing in reducing the errors in estimation. In the Appendix the results for discrete systems are formally extended to continuous systems.

Dynamic Analysis of Structural Systems Using Component Modes

Abstract: A method is developed for analyzing complex structural systems that can be divided into interconnected components. Displacements of the separate components are expressed in generalized coordinates that are defined by displacement modes. These are generated in three categories: rigid-body, "constraint," and "normal" modes. Rigid-body modes are convenient where displacements are denned in inertial space for dynamic analysis. "Constraint" modes are included to treat redundancies in the interconnection system. "Normal" modes define displacements relative to the connections. Generalized mass, stiffness, and damping matrices are determined for each component, as are generalized forces. The requirement of system continuity gives rise to equations of displacement compatibility at the connections. These serve as equations of constraint among the component coordinates and are used to construct a transformation relating component coordinates to system coordinates. This transformation is used to derive system properties and forces from component properties and forces. System equations of motion are formulated and solved to determine system response. Component responses are found using the transformation. Connection forces are computed from the component equations. Each component can then be isolated and treated separately.

Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem

Abstract: A variational theorem is presented which is equivalent to the elastic field equations expressed in terms of the displacements and a function of the mean pressure. The variational theorem is particularly useful in the development of approximate structural analysis techniques for incompressible and nearly incompressible materials. An approximate analysis method derived by utilizing the Ritz technique in conjunction with the variational principle is illustrated. The results obtained from the application of this procedure to a sample problem are contrasted with those obtained by a similar technique that employs the theorem of minimum potential energy. The comparison illustrates that the results obtained from the analysis based upon the theorem of minimum potential energy becomes quite inaccurate for nearly incompressible materials whereas the solution technique based upon the new variational theorem yields accurate results.

Structural analysis of axisymmetric solids.

Abstract: N the aerospace industry, the stress analysis of complex axisymmetric structures of arbitrary shape subjected to thermal and mechanical loads is of considerable interest. Rocket nozzles and cases, solid-propellant grains, and spacecraft heat shields are practical examples of such structures. Although the governing differential equations for solids of revolution have been known for many years, closed form solutions have been obtained for only a limited number of structures; thus, the stress analyst must rely on experimental or numerical techniques. The finite difference method has been the most popular of the numerical techniques; however, for structures of composite materials and of arbitrary geometry, this procedure is difficult to apply. In the present investigation, the finite element idealization is used as the basic numerical procedure. This technique has been applied successfully in the stress analysis of many complex structures.1"3 In Ref. 4 impressive results were obtained in the analysis of axisymmetric shells approximated by a series of truncated cone elements. The approach, which is presented here, is similar in many respects to existing methods used in the analysis of two-dimensional stress problems.5"7 Recently, the finite element method was applied to the structural analysis of axisymmetric solids subjected to axisymmetric loads.8 In the present paper, the finite element method is used in the determination of stresses and displacements developed within elastic solids of revolution which are subjected to axisymmetric or nonaxisymmetr ic loads. Emphasis is placed on the application of the technique to complex aerospace structures.

Elastic stability of thin-walled cylindrical and conical shells under axial compression

Abstract: D E L I P R Results of an extensive experimental program on the stability of cylindrical and conical shells under axial compression are presented and discussed. The experimental data indicate that the buckling coefficient varies with radius-thickness ratio. A study of other data in the literature showed that most of the experimental results fell within or near the scatterband obtained in the present evaluation. A lower bound design curve is also contained in the paper.

Theoretical considerations of panel flutter at high supersonic Mach numbers.

Abstract: The general characteristics of panel flutter at high supersonic Mach numbers are examined theoretically. Linear plate theory and two-dimensional first-order aerodynamics are used. The paper attempts to clarify the important role of damping, the relationship between traveling and standing wave theories of panel flutter, and the effects of edge conditions. The solution procedures and general mathematical behavior may be of interest in other stability problems characterized by the appearance of complex eigenvalues.

Consistent matrix formulations for structural analysis using finite-element techniques.

Abstract: The discrete-element stiffness matrix technique for formulation of linear structural problems in engineering mechanics is examined to develop techniques that give exact or closely approximate solutions for static load-displacement, elastic stability, and dynamic response problems. An exact relationship is derived for determining the coordinate load matrix equivalent to a general distributed load function. Use of the load matrix in static load-displacement problems results in an exact solution for the coordinate displacements consistent with the theoretical basis used for constructing the stiffness matrix. Explicit expressions are derived for a finite-displacement matrix for beam and plate elements for use in formulating the general elastic-stability problem. The approach discussed provides closely approximate buckling loads that are upper bounds to the precise solution. The dynamic problem, including elastic-stability considerations, is formulated using a consistent mass matrix approach. This approach provides closely approximate natural frequencies that are upper bounds to the exact solution. The data required for stiffness, load, finite-displacement, and mass matrices for a system composed of Timoshenko beam elements with linearly varying properties are provided. An example cantilever stepped beam problem is solved to illustrate the techniques involved and the exact or closely approximate nature of the solutions obtained.

Some effects of finite particle volume on the dynamics of gas-particle mixtures

Abstract: In the analysis of two-phase flows in which a gas carries along a large number of small particles, the volume of the particles is customarily assumed to be negligible. This assumption often is well justified, but if either the mass fraction of the particles or the gas density is sufficiently high, the particle volume fraction may become significant. It is thus important to establish the conditions at which this parameter should be included in a flow analysis. The particles may be considered as incompressible by comparison with the gas, so that a finite particle volume fraction appears as an additional variable in the basic equations. Consequences of a finite particle volume are examined for isentropic changes of the mixture, for frozen and equilibrium speed of sound, and for both the frozen flow immediately behind a shock wave and the equilibrium flow that is established further downstream. The errors that would result from neglecting the particle volume range from insignificant to large. For example, even for gas-particle density ratios as low as 10 ~, the equilibrium flow velocity behind a shock front is quite sensitive to changes of the volume of the particles if they represent more than one-half of the mass of the mixture.

Liquid surface oscillations in longitudinally excited rigid cylindrical containers.

Abstract: Harmonic and superharmonic liquid surface motions observed for longitudinally excited rigid cylindrical tank

Dynamic stability of a flexible missile under constant and pulsating thrusts

Abstract: The stability of a flexible missile, idealized as a uniform free beam under an end thrust, is investigated. A simplified control system is incorporated to obtain directional stability. It is shown that, in the absence of the control system, the critical thrust magnitude is associated with coalescence of the two lowest bending frequencies. With the control system included, it is found that the critical thrust magnitude corresponds to a reduction of the lowest frequency to zero. The results are related to typical thrust levels found in large, present-day missiles. Also considered is the effect of periodically varying thrust perturbations on the stability of the vehicle. In this case, parametric instabilities are found to exist. It is found that, when the beam is very stiff longitudinally, instabilities are most likely to occur when the frequency of the thrust variation is in the vicinity of 1) twice any of the bending frequencies, 2) the sum of any two of the bending frequencies, or 3) the difference of any two of the bending frequencies. With low longitudinal compliance, significant instabilities are also found to occur when the frequency of the thrust variation is in the vicinity of one of the longitudinal natural frequencies.

Plastic buckling of axially compressed cylindrical shells

Abstract: The plastic buckling of cylindrical shells under axial compression is studied analytically and experimentally. Attention is given to both the nonlinear and nonconservative aspects of the stress-strain relation. The effect of unloading is investigated in an exact manner for a hinge model which is proposed and in both an exact and an approximate manner for a geometrically perfect shell. Thirty tests are reported on cylindrical shells of 2024-T4 aluminum with radius to thickness ratios of 120 through 10 and length to radius ratios of 0.20 to 7. Specimens were prepared with three different types of end conditions and were tested either through a ball and socket arrangement or flat ended between smooth bearing blocks. A simple (jy incremental theory gives results very close to J% deformation theory and does predict both buckling strength and the geometry of buckling for thick and moderately thick shells.

Interaction of a starting vortex as well as a vortex street with a traveling shock wave

Abstract: The passage of a shock front and the associated drift flow over a two-dimensional airfoil, at an angle of attack mounted in a shock tube, generated a columnar spiral starting vortex that interacted with the normal reflected shock. Interferometric investigations of this interaction extended considerably earlier schlieren observations by Hollingsworth and Richards. The generation of a progressive cylindrical acoustic wave front centered on the transmitted vortex and its alternate compression-rarefaction nature, as predicted by linearized analyses by Hollingsworth and Richards and later by Ribner, was experimentally verified and deviations from the predicted pressure amplitude distribution were discovered. A semiempirical relation for the wave-front pressure-amplitude distribution (expressed as a linear combination of quadrupole, dipole, and monopole acoustic sources) was found to represent closely the experimental results. The time variation of the core density of the transmitted vortex was deduced. The spiral vortex decayed essentially as a circular, incompressible viscous vortex until a neighboring high-density region appeared and the vortex decayed at a faster rate. The interaction of the reflected shock with the vortex street generated behind a circular cylinder, resulted in an abrupt change in vortex spacing and rapid dissolution of the vortex street. No well-defined acoustic wave fronts were evident behind the reflected shock front.

Sublimation of graphite at hypersonic speeds

Abstract: In this paper, a new theoretical model is presented for the sublimation of graphite at hypersonic flight speeds. The aerothermochemical interactions between dissociated air and graphite are treated by means of a nine-component model, including O, O2, N, N2, CO, CO2, C, C3, and CN. The mass-transfer rate, the heat-transfer rate, and the skin-friction coefficient are determined numerically and are then correlated by means of algebraic equations as a function of stagnation pressure, stagnation enthalpy, and wall temperature in the high-Reynoldsnumber stagnation-point flow regime.

Theory of laminar near wake of blunt bodies in hypersonic flow.

Abstract: 21 Feldman, S., "Hypersonic gasdynamic charts for equilibrium air," Avco Rept. (1957). 22 Nagamatsu, H. T., Workman, J. B., and Sheer, R. E., Jr., "Hypersonic nozzle expansion of air with atom recombination present," J. Aerospace Sci. 28, 833-837 (1961). 23 von Karman, T. and Tsien, H. S., "Boundary layer in compressible fluids," J. Aeronaut. Sci. 6, 227-232 (1938). 24 Kendall, J. M., Jr., "An experimental investigation of leading edge shock wave-boundary layer interaction at Mach 5.8," J. Aeronaut. Sci. 24,47-56 (1957). 25 Howarth, L., Modern Developments in Fluid Dynamics High Speed Flow (Oxford University Press, London, 1953), Vol. 1, p. 402. 26 von Karman, T., "Turbulence and skin friction," J. Aeronaut. Sci. 1,1-20 (1934). 27 Van Driest, E. R., "Turbulent boundary layer in compressible fluids," J. Aeronaut. Sci. 18,145-160 (1951). 28 Li, T. Y. and Nagamatsu, H. T., "Effects of density fluctuations on the turbulent skin friction of an insulated flat plate at high supersonic speeds," J. Aeronaut. Sci. 18, 696-697 (1951). 29 Persh, J., "An analytical investigation of turbulent boundary layer heat transfer coefficients for supersonic and hypersonic flow," Naval Ordnance Lab. Rept. 4099 (1955). 30 Lin, C. C., Turbulent Flows and Heat Transfer: High Speed Aerodynamics and Jet Propulsion (Princeton University Press, Princeton, N. J., 1959), Vol. V.

Axial buckling of pressurized imperfect cylindrical shells

Abstract: Buckling load of cylindrical shell with asymmetric and axisymmetric imperfections and calculation of upper bound to buckling load for shell with axisymmetric imperfections

Elastic stability of thin-walled cylindrical and conical shells under combined internal pressure and axial compression

Abstract: Nomenclature D = flexural stiff ness of shell wall Et*/[ 12(1 - */2)] E = Young's modulus of shell wall material L = axial length of cylinder or cone / = slant length of cone n = number of circumferential waves in cone buckled under external pressure n = number of circumferential waves in equivalent cylinder buckled under external pressure PO = critical axial compression load p = total axial load at buckling P = uniform internal or external hydrostatic pressure POT, PQ — critical external pressure pe = critical external pressure for equivalent cylinder R = cylinder radius RI = radius of small end of cone Rz = radius of large end of cone t = shell wall thickness a = semivertex angle of cone v = Poisson's ratio of wall material Pav = average radius of curvature of cone (Ri + Rz)/% coso;

Necessary conditions for singular extremals

Abstract: 10 Tisserand, F., Traite de Mechanique Celeste, Tome IV (Gauthier-Villars, Paris, 1896), Chap. XII. 11 Lagerstrom, P. A. and Kevorkian, J., "Numerical aspects of uniformly valid asymptotic approximations for a class of trajectories in the restricted three-body problem," AIAA Progress in Astronautics and Aeronautics: Celestial Mechanics and Astrodynamics, edited by V. G. Szebehely (Academic Press, New York, 1964), Vol. 14, pp. 3-34. 12 Martikan, F., Emery, L., Santora, F., Garceau, T., and Jazwinski, A., "Trajectories in the earth-moon system," Lunar Flight Handbook (Martin Co., Baltimore, Md., 1963), Chap. IV. 13 Egorov, V. A., "Certain problems on moon flight dynamics," The Russian Literature of Satellites, Parti (International Physical Index, Inc., New York, 1958), pp. 115-174. 14 Dallas, S. S., "Moon-to-earth trajectories," Jet Propulsion Lab. TR 32-412 (June 1,1963).

A new technique for the numerical analysis of nonequilibrium flows

Abstract: References 1 Kovasznay, L. S. G., "Turbulence in supersonic flow," J. Aeronaut. Sci. 20, 657 (1953). 2 Ribner, H. S., "Convection of a pattern of vorticity through a shock wave," NACA Rept. 1164 (1954). 3 Hollingsworth, M. A. and Richards, E. J., "A schlieren study of the interaction between a vortex and a shock wave in a shock tube," Aeronautical Research Council Rept. 17985, Fluid Motion Sub-Committee 2323 (1955). 4 Hollingsworth, M. A. and Richards, E. J., "On the sound generated by the interaction of a vortex and a shock wave," Aeronautical Research Council Rept. 18257, Fluid Motion SubCommittee 2371 (1956). 5 Ribner, H. S., "The sound generated by interaction of a single vortex with a shock wave," University of Toronto, Institute of Aerophysics Rept. 61 (1959). 6 Ruetnik, J. R. and Witmer, E. A., "Transient aerodynamics of two-dimensional airfoils," Part 2, Wright Air Development Center TR 54-368, AD151018 (1958). 7 Lamb, H., Hydrodynamics (Dover Publications, Inc., New York, 1945), 6th ed. Chap. VII, p. 232. 8 Abbott, I. H., von Doenhott, A. E., and Strivers, L. S., Jr., "Summary of airfoil data," NACA Rept. 824 (1945). 9 Lighthill, M. J., "Jet noise," AIAA J. 1, 1507-1517 (1963). 10 Powell, A., "Theory of vortex sound," J. Acoust. Soc. Am. 36,177-195(1963). 11 Curie, N., "The influence of solid boundaries upon aerodynamic sound," Proc. Roy. Soc. (London) A231, 505-514 (1955). 12 Lamb, H., Hydrodynamics (Dover Publications, Inc., New York, 1945) 6th ed. Chap XI, p. 592. 13 Roshko, A., "On the development of turbulent wakes from vortex streets," NACA TN 2913, pp. 42-43 (1953). 14 Weeks, T. M. and Dosanjh, D. S., "Interaction between an advancing shock wave and opposing jet flow," AIAA J. 1, 15271533 (1963). Birkhoff, G. and Zarantonello, E. H., "Jets, wakes, and cavities," Applied Mathematics and Mechanics No. 2 (Academic Press Inc., New York, 1957), Chap. XIII, pp. 282-283.

Regression rates of nonmetalized hybrid fuel systems

Abstract: A laboratory-scale slab burner has been used to characterize the regression rate of three different binder compounds with oxidizers varied from 100% fluorine to 100% oxygen. The oxidizer flow rate and pressure were varied from 0.014 to 0.17 lb/in.2-sec and 20 to 160 psia, respectively. In the regions of low flow rate, the regression rate was independent of pressure and increased as the 0.8 power of the specific total flow rate for each of the propellant formulations studied. At the higher flow rates, the regression rate was nearly independent of flow but increased markedly with pressure. Increasing the percent oxygen resulted in a reduction in the regression rate. The classical hybrid regression rate law was extended to include the effects of condensed-phase surface products and nonunity Prandtl number. Agreement between experimental and predicted regression rates was good in the low flow-rate regions where regression rates were independent of pressure. However, the classical turbulent heat-transfer model did not account for the observed pressure dependence of regression rate in the high flowrate regions. Rate-limiting chemical kinetic processes were postulated as the most likely cause of the observed pressure dependence.

Optimality of intermediate-thrust arcs of rocket trajectories

Abstract: A new necessary condition is derived for the optimality of intermediate-thrust arcs of rocket trajectories in vacuum. The new necessary condition, together with previously known necessary conditions, ensures the absolute optimality of "sufficiently short" intermediatethrust arcs. For inverse-square central fields, the new condition reduces to a requirement that the primer vector must point inward; its radial component must be negative. Intermediate-thrust arcs that satisfy the new condition cannot join to arcs of a finite maximum thrust, nor join directly (without impulses) to arcs of zero thrust, except in a very restricted set of special cases. They are, therefore, of theoretical rather than practical importance.

Comparison of measured and predicted flows through conical supersonic nozzles, with emphasis on the transonic region.

Abstract: Measured and predicted flows through conical supersonic nozzles compared, emphasizing transonic region

New directions in lifting surface theory

Abstract: speed of sound, dimensionless aspect ratio of planar lifting surface wing semichord, ft chordwise dimension of area element in supersonic and transonic theory local wing chord, ft reference chordlength, ft 2L/PxU*S = lift coefficient coefficient of pitching moment (p — ?>oo)/(pooM/2) = pressure coefficient depth of x-y plane below free liquid surface, ft or dimensionless depth below free surface, referred to I U/(gl) = Froude number gravitational constant, ft/sec height of wing midspan above ground plane, ft; also downward bending or heaving displacement, ft reference value of bending displacement at wingtip ( —1) /2 = complex unit wk/U or tal/U = reduced frequency kbi/b = reduced frequency based on areaelement size kernel function of subsonic integral equation reference length, ft upward normal force or force per unit span acting on lifting surface chordwise and spanwise coordinates of "receiving" area element, dimensionless flight Mach number static pressure distance between points, dimensionless {(x £) + (1 M)[(j/ T?) + (z 2o(i?))]) /2 = elliptical distance curvilinear coordinates measured spanwise along and normal to surface, dimensionless E S/STIP = normalized s projected cylindrical lifting surface area, ft or dimensionless; also spanwise distance on cantilever wing, measured in semispans. time coordinate, sec fluid velocity components, dimensionless flight speed, f ps velocity component normal to S, dimensionless function describing wake surface rectangular Cartesian coordinates, dimensionless, see Fig. 1 height of S above x-y plane, dimensionless displacement of wing above its projected surface, dimensionless angle of attack measured from zero lift : (M — 1) = cotangent of Mach angle discontinuous jump in a quantity, passing from lower to upper surface of wing

Behavior of fully stressed design of structures and its relationshipto minimum-weight design.

Abstract: The iterative method of analysis and fully stressed redesign does not always converge to a minimum-weight design. Using the Kuhn-Tucker optimality condition of nonlinear programing, the necessary condition for the equivalence of the two methods of design is found and a computationally practical method for verification of the optimality of fully stressed design is obtained. Where the fully stressed design is not optimum a method is suggested for determination of the optimum which reduces the dimensionality of the problem by dividing the design variables into free and fully stressed ones. Then, the optimization is decentralized into an optimal search on free design variables and fully stressed design of the rest of the variables.

Application of matrix displacement method to linear elastic analysisof shells of revolution.

Abstract: The paper describes the application of the matrix displacement method to the linear elastic analysis of shells of revolution. The shell is idealized as a series of conical frusta, joined at nodal circles. In the present paper the use of the idealization to handle asymmetric deformations is demonstrated. Also shown is how approximations to the mass and stiffness matrices may be consistently derived; these may then be used to solve both static and dynamic problems. The effect on the results of static analyses of improving the approximation to the stiffness matrix, using the procedure suggested by Pian, is demonstrated and discussed. Results are presented of static analyses of several shell configurations, and these are compared with solutions by other methods. It is found that the method provides accurate solutions for shells of revolution under both axisymmetric and asymmetric loadings by using a reasonable number of elements.