N.S. Bardell

Free vibration of a thin cylindrical shell with periodic circumferential stiffeners

Abstract The theory is developed for obtaining the propagation constants of a thin uniform cylindrical shell, periodically stiffened by uniform circular frames of general cross-section. The free wave motion is analyzed and the stop and pass bands of free wave motion in the structure are located. Hysteretic damping is included. The natural frequencies of two stiffened finite cylindrical shells are deduced. The relative effects of the frame cross section and pitch on the free vibration characteristics of the whole structure are discussed.

Free vibration analysis of a flat plate using the hierarchical finite element method

Abstract The hierarchical finite element method is used to determined the natural frequencies and modes of a flat, rectangular plate. Ten different boundary conditions—including free edges and point supports—are considered in this paper. Extensive results are presented for each case (including the variation of frequency with the aspect ratio and the Poisson ratio), and these are shown to be in very good agreement with the work of other investigators. This confirms both the applicability and accuracy of solution of the HFEM to problem of this type.

Free vibration of a thin cylindrical shell with discrete axial stiffeners

Wave propagation around a cylindrical shell which is reinforced at regular intervals by flexible stiffeners parallel to the shell generator is considered. The shell itself is restricted to a section between two circumferential frames on to which the shell is simply supported. The structure effectively constitutes a one-dimensional periodic system and is analyzed as such. Four degrees of freedom are allowed between each periodic element and equations are set up for the four pairs of propagation constants which characterize the possible wave motions. Symmetric or asymmetric stiffener sections may be accommodated in the analysis together with structural damping. Computed propagation constants are presented for two different stiffener cross-sections, each pitched at two different intervals around the shell. Natural frequencies are calculated for one of these.

Free vibration analysis of coplanar sandwich panels

Abstract A comprehensive vibration study of simple three-layer sandwich plates, based on the h - p version of the finite element method, is presented. The methodology incorporates a new set of trigonometric functions to provide the element p -enrichment—these functions exhibit good convergence characteristics, and enable the medium frequency regime to be explored at minimum computational expense. Elements may be joined together to model more general coplanar assemblies, and the trade-off between h -division and p -enrichment is discussed. Excellent agreement has been found with the work of other investigators, and new results are presented for (i) a completely free, symmetric section, rectangular sandwich panel whose core thickness is varied as a function of the overall plate thickness whilst the mass per unit area is maintained constant, and (ii) a cantilevered, T-planform, asymmetric section, sandwich plate. The results from this latter case are compared with those forthcoming from a proprietary finite element package; outstanding agreement is obtained, and a reduction of over 30% in the total number of degrees of freedom is demonstrated.

Free vibration of an orthogonally stiffened flat plate

Abstract A flat plate, reinforced by a regular orthogonal array of uniform beams, is analyzed by using techniques developed for studying wave propagation in two-dimensional periodic structures. A “plane-wave” type of motion is considered which may be characterized by different propagation phase constants in the x- and y-directions. The hierarchical finite element method is used to set up the governing equations of free wave motion, and these are then solved as an eigenvalue problem for the frequencies at which particular waves will propagate. Plots of phase constant surfaces vs. frequency are presented for a number of different plate-beam configurations. Excellent agreement is found between some of these and the results of earlier investigators. When the plate is supported by flexible beams in both directions, wave propagation is found to commerce at zero frequency. At higher frequencies alternating (and overlapping) attenuation and propagation bands occur. The nature and explanations of these are discussed. Wave speed surfaces vs. frequency are also presented and these give insight into the critical coincidence frequencies of the plate under acoustic excitation.

Free vibration of an orthogonally stiffened cylindrical shell, part I: Discrete line simple supports

Abstract The hierarchical finite element method is used to establish the stiffness and mass matrices of a cylindrically curved rectangular panel. Some natural frequencies and modes of two such panels, each with different boundary conditions, are then determined. Excellent agreement if found between this work and that of other investigators. These stiffness and mass matrices are then combined with periodic structure theory to analyze an orthogonally stiffened cylindrical shell. This analysis is formulated for a “plane wave” type of motion which is characterized by different propagation constants in the axial and circumferential directions. The governing equations of free vibration are then solved as a matrix eigenvalue problem for the frequencies at which particular waves will propagate. Results are presented in the form of phase-constant surface stacks, and clearly show the qualitative effects of varying the major shell parameters b a , R a and R h .

Free and forced vibration analysis of thin, laminated, cylindrically curved panels

A comprehensive vibration study of thin, laminated, cylindrically curved shell panels (based on the shell theory of Love with a modification by Arnold and Warburton) is conducted by using the h-p version of the finite-element method (FEM). Polynomially enriched stiffness and mass matrices are derived from classical shell theory using Symbolic Computing, and then stored in algebraic form for a single, generic element. A number of such elements may then be combined to form the global stiffness and mass matrices for a more general co-axial and/or co-circumferential assembly. Any of the classical edge conditions, or point corner supports, may be accommodated in the analysis; forcing may be applied through one or more point forces acting normal to the shell surface. Excellent agreement has been found with the work of other investigators, and some new results are presented for a multiply supported curved panel made from the aluminium-glass-fibre hybrid GLARE. The h-p method is shown, by example, to offer an efficient means of conducting typical repetitive sensitivity analyses, such as varying the fibre orientation and the stacking sequences of a given panel.

Free vibration of an orthogonally stiffened cylindrical shell, part II: Discrete general stiffeners

Abstract A thin cylindrical shell is considered, stiffened axially by equi-pitched, identical stringers and circumferentially by equi-pitched, identical frames. Generality of strigner and frame section is allowed. The structure is analyzed as a two-dimensional periodic structure by using wave propagation techniques in conjunction with the hierarchical finite element method. Results are presented in the form of phase-constant surfaces plotted against frequency. It is shown that free wave-motion can propagate in the infinite structure from zero frequency. A small frequency band has been identified in which predominantly flexural waves cannot propagate. Some experiments, which have been performed on a one-quarter scale fuselage model, confirm the main findings of the theoretical analysis.

Free vibration analysis of thin coplanar rectangular plate assemblies — Part I: theory, and initial results for specially orthotropic plates

Abstract This paper communicates a new h - p finite element methodology for studying the free vibration of generally orthotropic coplanar plate assemblies. In Part I, hierarchically enriched stiffness and mass matrices of a generally orthotropic rectangular plate element are derived using symbolic computing. These may then be combined, via a special connectivity matrix, to form the global stiffness and mass matrices of a more general coplanar plate structure. A variety of different boundary conditions may be accommodated in the model by specifying whether an element edge is simply supported, clamped, free, or whether an element corner is point supported. The natural frequencies, and the associated normal modes, are then sought from the resulting matrix-eigenvalue problem. Detailed comparisons with the work of other investigators confirm the validity of the methodology for specially orthotropic plates; the validation for generally orthotropic plates, and plate assemblies, is addressed in Part II.

FREE-VIBRATION ANALYSIS OF THIN RECTANGULAR LAMINATED PLATE ASSEMBLIES USING THE H-P VERSION OF THE FINITE-ELEMENT METHOD

A vibration study of thin, laminated plate assemblies is conducted by using the h-p version of the finite element method (FEM). Polynomially-enriched stiffness and mass matrices are derived from classical plate theory using Symbolic Computing, and then stored in algebraic form for a single, generic element. A number of such elements may then be combined to form the global stiffness and mass matrices for a more general planar assembly. Any of the classical edge conditions, or point corner supports, may be accommodated in the analysis, and the natural frequencies are sought from a standard matrix-eigenvalue problem. Excellent agreement has been found with the work of other investigators. The h-p method is shown, by example, to offer considerable savings in computational effort when compared with the standard h-version of the FEM. One further development of the method is presented which illustrates how it might form the basis of a condition monitoring measuring technique based on natural frequency shifts arising from non-propagating, through-the-thickness, crack damage.