T.J. McDaniel

The analysis of curved multi-span structures

Abstract Analytical techniques, based on transfer matrix methods, are presented for the analysis of the forced vibrations of cylindrically curved multi-span structures. Specifically, analysis is presented of shell segments and closed shells stiffened by discrete stringers in the axial direction, as an idealization of skin-stringer fuselage structure. A procedure whereby the transfer matrix for a shell section can be derived directly from the matrix differential equation is discussed, and an improved transfer matrix for the open section stringer is obtained. Specific approaches to formulating the problem of a shell segment with arbitrary stringer spacing are compared with complementary methods used to analyze a closed shell with periodic stringer spacing. The possibility of representing the structure by a closed circular beam with multiple supports is also discussed. Numerical difficulties inherent in the different approaches are considered. Calculations of normal modes and natural frequencies of the undamped structure, as well as the forced response of damped structures, are illustrated with numerical examples.

A combined finite element-transfer matrix structural analysis method

A combined finite element-transfer matrix (FETM) method is developed to study the dynamics of orthogonally stiffened structures. Finite element procedures are used in FETM to formulate transfer matrices for structural components which are not one-dimensional. The resulting transfer matrices are used to reduce the large number of unknowns occurring in a standard finite element analysis by obtaining transfer matrix relationships over large units of the structure. A reduction in computer storage and computation time is obtained since the dimension of the final matrix equation to be solved is considerably reduced. The accuracy of the FETM results are verified by comparison with an exact solution to the limiting modes and frequencies of a periodically stiffened structure of infinite length. A reduction of computer time was demonstrated for the FETM by a comparison with the SAP-4 finite element program. The efficiency of the FETM method was further demonstrated by computing the frequency response of a two row-five span orthogonally stiffened plate structure using ordinary transfer matrix procedures.

Dynamics of rotationally periodic large space structures

Abstract The dynamic analysis of large area space structures is restricted by cost and utter size of the computer analysis. An efficient analysis must take advantage of construction periodicity inherent in many structures. In the analysis described here a finite element transfer matrix method is employed to eliminate internal degrees of freedom from the basic unit of a rotationally periodic space structure. Eigenfunctions of the resulting periodic unit transfer matrix are used to obtain frequency responses of the complete structure without increasing the analysis variables. Interpolation procedures are developed which significantly reduce the required computations, the dimension of the transfer matrix, and the number of eigenvalues/eigenvectors extractions required in a given frequency range. A substantial saving in the resulting computer analysis is obtained. For a 100 m parabolic dish structure with 3642 structural members contained in 44 units, the periodic analysis involves less than 5% of the variables of a finite element analysis. The frequency response functions obtained are used to compare added coatings, gyroscopic, and tuned damping devices for reducing the responses of large area structures. Transient response can be obtained by an inverse Fourier transform of the frequency response. The analysis also provides modal information for the periodic structure.

Dynamics of Circular Periodic Structures

The frequency response matrix is determined for a periodically supported, periodically damped, closed circular beam structure which is an approximate model of a skin-stringer aircraft fuselage structure. The analytical technique which is developed depends on the periodic nature of the structure to simplify the analysis. The analysis is a complementary approach to the transfer matrix method for determining frequency response. One purpose of the analysis is to circumvent the numerical and/or computer storage difficulties commonly found in computing the frequency response for typical aircraft fuselage structure from a transfer matrix analysis. Numerical difficulty is avoided by using the properties of the transfer matrix for one periodic unit and a consequence of the Cayley-Hamilton theorem to obtain an analytical solution for the frequency response matrix. The second purpose of the analysis is to investigate the spatial decay of response from a point or region of the structure which is being excited. This spatial decay isolates the response to a region of structure near the excitation and reduces the over-all dynamic stress level of the structure for a general excitation. Three damping devices that utilize a viscoelastic link to produce spatial decay are evaluated. Several numerical examples are shown.

Dynamics of cylindrical shells with variable curvature

Abstract The transfer matrix method is extended to the analysis of non-circular cylindrical panels. The exact solution for the transfer matrix of a panel with exponential curvature is obtained by solving exactly the variable coefficient differential equations of motion of the shell using a Laplace transform—difference equation technique. The results are compared with respect to accuracy and computer time with various approximate methods of computing the transfer matrix for the same panel. Natural frequencies and mode shapes for typical non-circular panels are computed and compared with a constant curvature panel to show the effects of variable curvature.

Dynamics of bi-periodic structures

Abstract Many structures considered for space applications are bi-periodic in their construction. Bi-periodicity means that two types of structural subassemblies, alternating in one or more directions, make up the structure. To gain insight into the dynamics of bi-periodic space structures a variety of one and two dimensional bi-periodic structures are considered. Results indicate that bands in which natural frequencies lie for periodic structures are further subdivided as a consequence of the bi-periodicity. Analytical solutions for the modes and frequencies of finite length one dimensional bi-periodic structures are obtained for general boundary conditions. A transmission method is developed to simplify the application of boundary conditions. It is found that some modes occur at frequencies which are outside the frequency bands predicted for bi-periodic structures. Two dimensional bi-periodic crossed beam grillage and truss structures are considered in this study. For cases where a separation of variables solution is possible the two dimensional structures exhibit similar properties to the one dimensional bi-periodic structures. Analytical solutions for the one and two dimensional bi-periodic structures considered above lead to a compact solution form similar to that of periodic structures analysis.

Solution bounds for varying geometry beams

Abstract The theory of differential and integral inequalities is applied to obtain upper and lower bounds to the transfer matrix for beams with varying geometry. Various techniques of generating and refining these bounds are investigated. Numerical results indicate that these bounds can be refined to produce numerical agreement of the upper and the lower bound to a given number of significant digits. Proceeding from bounds on the transfer matrix elements a theory is developed for determining upper and lower bounds on the natural frequencies and mode shapes and on the solution state vector for static loading of such beams. This procedure is then extended to the analysis of multispan beams with varying geometry. Numerical results are presented for various configurations.

DYNAMICS OF NON-CIRCULAR STIFFENED CYLINDRICAL SHELLS

Abstract The objective of this study is to assess the effect of non-circularity, i.e. spatially varying curvature, on the dynamics of stringer stiffened cylindrical shell structures. The transfer matrix method is used to study the effect of varying curvature on the normal modes and the frequency response of such structures. To obtain these results the transfer matrix for a varying curvature shell in conjunction with the transfer matrix techniques for stiffened circular structures are applied. A previously developed exact solution for the panel transfer matrix, where the curvature varies exponentially, is used in computing the numerical results. Responses of increasing curvature and decreasing curvature structures are compared with those of the circular structure.