Influence Coefficient Matrix Research Articles

An Improved Mach-Box Approach for Supersonic Oscillatory Pressures

An analytical refinement of the Mach-box approach has been developed which provides greatly improved supersonic oscillatory pressure distributions with almost no increase in computer time over that required by the original formulation. The refinement consists, in part, of applying physically justifiable weighting factors to the terms of the aerodynamic influence coefficient matrix. Unlike approaches that employ curve smoothing to eliminate the erratic undulations in computed pressures obtained by the basic method, the present analysis produces well-behaved pressure distributions while retaining essential flow discontinuities. Furthermore, the present method is more cost-effective than approaches relying on refinements in the aerodynamic grid to obtain comparable improvements in the calculated pressures.

Stability of Shells Attached to Elastic Core

The effect upon the buckling strength of a soft elastic core attached to a shell of revolution subjected to axisymmetric loads is investigated. The axisymmetric stress problem is solved by using the finite element method to solve the body of revolution problem with an axis of material symmetry subjected to Fourier expandable thermal, body force, and surface traction loading. The governing equations are derived for a triangular toroidal continuum element attached to shell elements. The elastic core influence coefficient matrix of the core is derived by applying the unit line load at the interface nodal point of the core. The inversion of the influence coefficient matrix yields an equivalent stiffness matrix of the core which is then combined with the shell stiffness matrix. Results from this investigation are in good agreement with available analytical and experimental results for cylindrical shells. New results are obtained for conical and spherical shells.

Eigenvalue properties of structural mean-axis systems

The authors review some of the properties of the pseudoinverse and oblique pseudoinverse of a linear transformation T from one finite-dimensional inner-product space into another, and then use these properties and a theorem of Milne (1968), which states that the oblique pseudoinverse can be expressed in terms of a weak generalized inverse and two projection operators, in order to compute a mean-axis influence coefficient matrix for the dynamic analysis of an elastic body. Some eigenvalue invariance properties of the mean-axis structural dynamics equations are demonstrated, and on the simple example of a uniform beam, it is shown that the finite frequencies and mode shapes of the mean axis structural system are identical to the nonzero frequencies and mode shapes of the free structure.

Analysis of moderately thick flanged reactor vessels

Abstract A computer method is presented for the analysis of moderately thick flanged shells of revolution such as are used for reactor pressure vessels. The shell may be subjected to symmetrical or unsymmetrical loads and a thermal environment. The method employs a finite element discretization for modelling the flange portions, and a theory appropriate to moderately thick shells for the remainder of the pressure vessel. The governing differential equations for the shell portions are put in the Goldberg-Bogdanoff first-order form and integrated numerically using a scheme such as a Runge-Kutta process. The finite element stiffness matrix for a flange region is used to form a superelement influence coefficient matrix, permitting the flange region to be treated as a giant step in the numerical integration process.

Calculation of contact stresses at contacts between conical bodies

A method for calculating the elastic-stress distribution in two axi-symmetrical conical bodies caused by frictionless contact between them is described. Finite-difference methods were used to solve the governing differential equations. Equilibrium and compatibility over the contacting surfaces was achieved by using the finite-difference solutions to derive a matrix of influence coefficients. By means of an iterative technique, using this matrix, the portion of each surface in the contact zone and the stresses on it were found. With equilibrium and compatibility achieved, the stresses throughout the body of each component were calculated. A selection of results for one case is given.

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

On the Bending of Spherical Shells

The bending behavior of spherical shells is reduced to the solution of a second-order differential equation with complex coefficients. Influence coefficients and functions of truncated spherical shells are evaluated and presented in curve form for the analysis of structures composed of spherical segments. These coefficients are determined by an exact and an asymptotic solution of the governing equation. The asymptotic solution is valid over the entire range of the independent variable when the ratio of shell radius to thickness is greater than 100. An uncoupling criterion is given by which it is possible to establish whether the effect of loads applied at one edge of the shell is transmitted to the other edge. This criterion is determined by the cross-product terms in the influence coefficient matrix.

Solution of Subsonic Nonplanar Lifting Surface Problems by Means of High-Speed Digital Computers

The method proposed in this paper is based on an approximate solution of the integral equation which represents the potential flow about a finite wing, with no restrictions beyond those necessary for linearization. After assuming the usual series representation of the wing surface vorticity distribution, the solution is achieved by approximating portions of the kernels of the transformed integral equation by single and double Fourier series and performing termwise integrations analytically. This is followed by the routine inversion of the aerod3mamic influence coefficient matrix, after satisfying appropriate boundary conditions at selected control points. In this procedure the number of control points used is limited only by the storage capacity of the computer. Control points may be located so as to cover the entire wing surface, with due regard to certain physical requirements, permitting the accurate representation of complicated mean surface shapes. An evaluation of the proposed method is included. Comparisons with other theoretical methods and electrical analogy tank results are used to substantiate the accurac} of the proposed method when applied to plane wings. A final evaluation involves a comparison of calculated surface pressure distribution with wind-tunnel measurements on a swept, tapered wing with a cambered and twisted mean surface. The agreement evidenced in the latter comparison has the same order of overall accuracy as similar comparisons on plane wing planforms. In either case, the results given by the proposed method are within the accuracy requirements for most aircraft design studies.

Aerodynamic Influence Coefficients for Delta Wings in Steady Supersonic Flow

A routine numerical method of solving integral equations arising in steady supersonic flow past flexible delta wings, with supersonic and subsonic leading edges, is presented. The wing is subdivided into diamond shaped panels, by a network of equi‐spaced Mach lines, and the effect of placing unit downwash at each panel in turn is calculated. The resulting influence function, which has been tabulated, is then formed into the required matrix of influence coefficients. Within the limitations of linearized theory the results are valid for all Mach numbers and mode shapes, including chordwise bending. Although applied here to delta wings the method is equally suited to wings of arbitrary plan form. An aeroelastic efficiency calculation, for a sonic leading edge 45 deg. delta wing of representative torsional stiffness, is included for comparison with strip theory.

Aeroelastic Problems of Low Aspect Ratio Wings

A method is presented for calculating influence coefficients in multispar wings of any plan form. Shear deflexion, chordwise bending and taper are taken into account. All the bending material is concentrated at the rib and spar booms and the skin is assumed to carry only shear. The method is particularly useful in the early stages of design as it is rapid and gives a good internal load distribution. The choice of the method, the assumptions made, and the idealization of the structure are discussed. The structure is reduced to a system of beams and torque boxes, the latter connected to the beams by vertical shear at their corner nodes. The internal forces are first expressed as functions of the displacements at the nodes by means of stiffness coefficients. Next the equations of equilibrium, at each node, are established. Finally the conditions of compatibility are brought in. The solution of this system of equations gives the matrix of influence coefficients. Wherever vertical loads only are applied it is possible to solve for the moment equations of straight beams separately from the remainder of the structure, hence from the stiffness matrix. For convenience, these straight beams are organized into sub‐matrices whose moment equations are ‘reduced’. The specific end conditions also allow further reductions. The internal loads can be obtained by substituting the influence coefficients back into the sub‐matrices and by considering the equilibrium of the beams. The angular displacements of the nodes may also be calculated in a similar manner by using the ‘reduced’ rows from the sub‐matrices.