Structural Influence Coefficients Research Articles

Flow-structural interaction inside a solid rocket motor during ignition transient

In this article, a numerical procedure for the analysis of the flow-structural interaction inside a solid rocket motor (SRM) is developed from existing codes. This computer code simultaneously models the developing flowfield and the associated propellant grain deformation during the ignition transient period of SRM operation. It was created by coupling together an ignition-transient flow code, which provides a detailed picture of the time-dependent flowfield and flame spreading inside the motor, with a set of structural influence coefficients, which enable us to calculate the grain deformation that results from a given surface pressure distribution. Also included in the numerical package is a grid-generation code, which generates a grid mesh for the internal-flow passages, based on the deformed shape of the propellant grain. This grid mesh then is used in the ignition transient flow code. As these three component calculations march forward in time, they interact continuously; in this way the flowfield and grain shape are updated continuously. The computer analysis was validated by and used to study the static test failure of the Titan solid rocket motor upgrade (SRMU) that occurred on April 1, 1991 at Edwards Air Force Base. It has since been used to predict the stability of other motors.

Equations of motion of a quasisteady flight vehicle utilizing restrained static aeroelastic characteristics

Mean axes provide the usual reference in maneuvering and dynamic response analyses of flexible vehicles. Attached or structural axes have also been used because the flexibility characteristics of the structure are only determined for a restrained structure. If the structural axes are employed, a relationship is required for the orientations of the structural axes relative to the mean axes, since the equations of motion determine the orientations of the mean axes. If this relationship is not considered, as it has not been in a number of publications, the solution to the equations of motion of the structural axes is not invariant with the choice of support configuration used in the calculation of the structural flexibility influence coefficients and is, therefore, incorrect. This relationship and the correct equations of motion for the structural axes are presented, and the correctness of the formulation is demonstrated numerically in studies of longitudinal maneuvering of an example forward-swept-wing airplane at constant forward velocity. Although the paper assumes constant airspeed in the basic developments, it concludes with discussions of aeroelastic speed derivatives and aeroelastic effects on drag. The paper assumes quasisteady equilibrium of the structure with its applied loads throughout.

A Computational Scheme for Structural Influence Coefficients of Certain Planar Wings

The objective of this work has been to reduce the amount of numerical computation work involved in obtaining structural influence coefficients, especially for special-purpose computer programs involving wings that have elastic axes that can be approximated by a series of straight line noncollinear segments. The need for repetitive runs, probably due to different mass arrangements, enhances the desirability of the technique. A general matrix of influence coefficients which allows the user to handle the TV-segmented case is derived and presented. The decreased number of integral evaluations is summarized for a three-segmente d case as an illustration [only 12 integrals compared to 27 (straight); only 12 integrals compared to 42 (noncollinear)]. For collinear segments of practical wings, the order of required integral evaluations is shown to differ by N2 and by more for noncollinear cases. Therefore, a significant reduction in computation effort is claimed.

Reply by Author to Rebuttal by W. P. Rodden

Rodden, W. P., A Method for Deriving Structural Influence Coefficients from Ground Vibration Tests, AIAA Journal, Vol. 5, May 1967, pp. 991-1000. Natke, H. G., private correspondence with W. P. Rodden, July 11,1966. Flanelly, W. G., and Herman, A., The State of the Art of System Identification of Aerospace System Identification of Vibrating Structures, Reprints of Papers presented at ASME Winter Meeting, 1972, pp. 121-131. Falkner, V. M., The Calculation of Aerodynamic Loading on Surfaces of any Shape, R&M 1910, British Aeronautical Research Council, 1943. Watkins, C. E., Woolston, D. S., and Cunningham, H. J., A Systematic Kernel Function Procedure for Determining Aerodynamic Forces on Oscillating or Steady Finite Wings at Subsonic Speeds, NASAR-48, 1959. Guyan, R. J., Reduction of Stiffness and Mass Matrices, AIAA Journal, Vol. 3, Feb. 1965, p. 380.

Aeroelastic Loads Predictions using Finite Element Aerodynamics

Aerodynamic and structural influence coefficients are utilized to determine the load distributions, deflections, and trim parameters for a vehicle in quasi-static aeroelastic equilibrium. A matrix formulation is used to solve the various quasi-static aeroelastic problems. Nonlinearities in the aeroelastic trim equations are accounted for by an iteration of the classical closed form solution. Aerodynamic and structural idealizations are related by a surface spline transformation. Solutions are developed for symmetric, antisymmetric, and asymmetric load conditions on symmetric vehicles of general geometric shapes which may include both lifting surfaces and lifting bodies. INTRODUCTION With the advent of large order matrix solutions for the analysis of complex structures a need has arisen for a complementary approach to the external loads problem. Finite element structural analysis techniques demand that the external loads be distributed over the structure at discrete points. Therefore, shear, moment, and torque distributions along a psuedoelastic axis are no longer sufficient to define the external load distributions required by the stress analysis. A general finite element approach to the problem of the determination of aeroelastic loads on a flexible vehicle flying in a state of quasi-static equilibrium is presented here. The vector point loads available from this solution are directly applicable to matrix structural analyses. Structural and aerodynamic influence coefficients obtained from finite element idealizations of the aircraft are utilized as a basis for the method. The technique is primarily an extension of the method first suggested by ~ e d m a n ( l ) * and later generalized by odde en.(^) This work extends the efforts of the above-mentioned authors by * Numbers in parentheses designate References at end of paper.

Dihedral effect of a flexible wing

The dihedral effect of a flexible wing is considered as a critical aeroelastic problem because of the importance of dihedral in determining lateral-directional dynamic stability and the large changes in dihedral that result from symmetrical wing bending during longitudinal maneuvers that approach limit load factor. Some recently declassified experimental data of the rolling moment caused by sideslip measured at low speed on an aeroelastic model of the Douglas XA3D-1 airplane are presented. A review of some early simplified methods of analysis is made, and the correlation of one of the methods with the experimental results is shown. Finally, the problem of estimating wing dihedral effect is reformulated in terms of structural and aerodynamic influence coefficients in order to utilize recent advances in lifting surface theory. The experimental data and other examples cited indicate that the aeroelastic change in dihedral effect at limit load factor can equal the value measured on rigid models.

Reply by Authors to W. P. Rodden, J. A. McGrew, and T. P. Kalman

2 Hitch, H. P. Y., Modern Methods of Investigating Flutter and Vibration, Journal of the Royal Aeronautical Society, Vol. 68, No. 642, June 1964, pp. 357-373. 3 Rauscher, M., Station Function and Air Density Variations in Flutter Analysis, Journal of the Aeronautical Sciences, Vol. 16, No. 6, June 1949, pp. 345-353. 4 Schmitt, A. F., A Least-Squares Matrix Interpolation of Flexibility Influence Coefficients, Journal of the Aeronautical Sciences, Vol. 23, No. 10, Oct. 1956, p. 980. 5 Rodden, W. P., Further Remarks on Matrix Interpolation of Flexibility Influence Coefficients, Journal of the Aerospace Sciences, Vol. 26, No. 11, Nov. 1959, pp. 760-761. 6 Rodden, W. P. and Revell, J. D., Influence Coefficients from Slender Wing Theory, Rept. NA-57-1437, Sept. 1957, North American Aviation Inc., Los Angeles, Calif. 7 Rodden, W. P. and Revell, J. D:, Status of Aerodynamic Influence Coefficients, Rept. TDR-930(2230-09)TN-2, Nov. 22, 1961, Aerospace Corp., El Segundo, Calif. 8 Andrew, L. V., et al., Unsteady Aerodynamics for Advanced Configurations, FDL-TDR-64-152, Pt. I, II, and IV, May 1965, Air Force Flight Dynamics Lab., Wright-Patterson Air Force Base, Ohio. 9 Kordes, E. E., Kruszewski, E. T., and Weidman, D. J., Experimental Influence Coefficients and Vibration Modes of a Built-Up 45° Delta-Wing Specimen, TN 3999, May 1957, NACA. 10 Rodden, W. P., A Method for Deriving Structural Influence Coefficients from Ground Vibration Tests, AIAA Journal, Vol. 5, No. 5, May 1967, pp. 991-1000. 11 McGrew, J. A., Orthogonalization of Measured Modes and Calculation of Influence Coefficients, AIAA Journal, Vol. 7, No. 4, April 1969, pp. 774-776. Reply by Authors to W. P. Rodden, J. A. McGrew, and T. P. Kalman

Comment on "A Direct Matrix Method for the Divergence Problem"

T is curiously refreshing to be reminded from time to time of the struggle that accompanies the advancing state-of-theart of the aeronautical sciences. Lest we take our present position too much for granted, Ref. 1 has just appeared from out of the past to recall the period of the 1950's during which the field of static aeroelasticity made the transition from the slide-rule to the high-speed digital computer. The following historical recollections are offered as a more complete review of that dynamic period of progress. Equation (4) of Ref. (1) was almost implicit in Eq. (8-55) of Bisplinghoff, Ashley, and Halfman.2 The definition of static and oscillatory aerodynamic influence coefficients (AIC's) in Ref. 3 [Eq. (6) of Ref. 1 appears as Eq. (8) in Ref. 3] provided for a general formulation of the various aeroelastic problems which was outlined in the introduction to Ref. 4. A general purpose computer program was developed in Ref. 5 for the static aeroelastic analysis of the problems of rigid and flexible load distributions, divergence, estimation of rigid and flexible static and dynamic stability derivatives, and the correction of wind-tunnel data measured on flexible models. Both structural influence coefficients and AIC's were used and Eq. (4) of Ref. 1 is found as Eq. (23) in Ref. 5. The example problem used to demonstrate the computer program of Ref. 5 was also the jet transport wing of Bisplinghoff, Ashley, and Halfman.2 However, the AIC's were based on the subsonic lifting surface theory (at a Mach number of zero) of Runyan and Woolston.6'7 The first two divergence dynamic pressures were calculated to illustrate the divergence option, and were#i = 3786 and#2 = 26,951 psf. The fundamental divergence dynamic pressure corresponds to the velocity Vt = 1784 fps at sea level, with which the value in Ref. 1 using lifting line theory agrees well. Later calculations of the divergence characteristics of the jet transport wing were performed using incompressible strip theory in connection with a method for transient flutter analysis.8 The five divergence dynamic pressures were found to be q, = 2397, q2 = 9510, q3 = 23,555, q* = 38,208, and q$ = 69,914 psf. The fundamental divergence velocity at sea level is Ki = 1420 fps, as was mentioned in Ref. 8. If the aspect ratio correction used in Refs. 1 and 2 is made, the divergence velocity at sea level becomes Vi = 1950 fps and agrees with the result in Ref. 2, p. 440. The concern in Ref. 1 over the null eigenvalues from bending is unwarranted. The five nonzero eigenvalues of the example wing (using strip theory) were found easily by the power method because of their wide separation. The sixth through tenth eigenvalues were not zero (infinite divergence speeds) because of round-off errors in the eigen-matrix deflation required in the power method; however, the sixth eigenvalue was four orders of magnitude lower than the fifth and thereby gave some clue to what should have been expected.

A method for deriving structural influence coefficients from ground vibration tests.

A method is presented for finding the structural influence coefficients of an arbitrarily restrained system from the experimental results of an effectively free-free ground vibration test. A theoretical example is presented to illustrate the general procedure and to demonstrate the manner of convergence of the method. A discussion of nonorthogonality of measured modes is given, and a practical application is made to a delta wing specimen for which influence coefficients, modes, and frequencies have been measured, and the degree of correlation is discussed. Finally, a method for finding a complex influence coefficient matrix that includes structural damping is suggested for use in oscillatory aeroelastic analyses. Nomenclature

Aerodynamic Influence Coefficients from Strip Theory

T N ANY AEROELASTIC ANALYSIS by collocation methods, both structural and aerodynamic influence coefficients must be known. We consider here the derivation of the aerodynamic influence coefficients (AICs) from the simplest of the aerodynamic theories—that of strip theory in which the flow along any section of the wing can be considered as two-dimensional. The type of AICs necessary in an aeroelastic analysis must relate the controlpoint forces, F (rather than pressures) to the deflections, h, or streamwise slopes. In the present treatment, we prefer the deflections rather than streamwise slopes as the generalized coordinates because the deflections have a more general meaning on a cambered surface and deflection influence coefficients are more readily obtained from a structural analysis than slope influence coefficients. Accordingly, we define a steady matrix of AICs, [CHS], by