F.W. Williams

Free vibration of composite beams - An exact method using symbolic computation

An exact dynamic stiffness matrix method has been developed to predict the free vibration characteristics of composite beams (or simple structures assembled from them) for which the bending and torsional displacements are (materially) coupled. To achieve this, an explicit expression is presented for each of the elements of the dynamic stiffness matrix of a bending-torsion coupled composite beam. This was made possible by performing symbolic computing with the help of the package Reduce. Programming the stiffness expressions in Fortran on a SUN SPARC station indicates about 75% savings in computer time when compared with the matrix inversion method normally adopted in the absence of such expressions. The derived dynamic stiffness matrix is then used in conjunction with the Wittrick-Williams algorithm to compute the natural frequencies and mode shapes of composite beams with substantial coupling between bending and torsional displacements. The results obtained from the present theory are compared with those available in the literature and discussed.

Coupled bending-torsional dynamic stiffness matrix for timoshenko beam elements

Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix elements of a uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. Application of the developed theory in the context of wings, blades and grillages is discussed with particular reference to an established algorithm. Programming the derived stiffness expressions on a VAX computer indicates about 87% savings in computer time when compared with the matrix inversion method normally adopted in the absence of such expressions. The correctness of the stiffness expressions is numerically checked up to machine accuracy against the corresponding stiffnesses from the inversion method. The stiffnesses are also checked up to nine figure accuracy against those obtained from a comparable approximate method.

Optimum design using VICONOPT, a buckling and strength constraint program for prismatic assemblies of anisotropic plates

A computer program for obtaining the optimum (least mass) dimensions of the kind of prismatic assemblies of laminated, composite plates which occur in advanced aerospace construction is described. Rigorous buckling analysis (derived from exact member theory) and a tailored design procedure are used to produce designs which satisfy buckling and material strength constraints and configurational requirements. Analysis is two to three orders of magnitude quicker than FEM, keeps track of all the governing modes of failure and is efficiently adapted to give sensitivities and to maintain feasibility. Tailoring encourages convergence in fewer sizing cycles than competing programs and permits start designs which are a long way from feasible and/or optimum. Comparisons with its predecessor, PASCO, show that the program is more likely to produce an optimum, will do so more quickly in some cases, and remains accurate for a wider range of problems.

Coupled bending-torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element

Abstract Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to machine accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.

Physical interpretation of the symplectic orthogonality of the eigensolutions of a hamiltonian or symplectic matrix

Abstract The physical interpretation of the adjoint symplectic orthogonality between the eigenvectors of a Hamiltonian matrix, or of a symplectic matrix, is shown in this note to be that it corresponds to the well-known Betti reciprocal theorem.

On the free vibration analysis of spinning structures by using discrete or distributed mass models

Abstract The analysis of the natural frequencies and modes of vibration of a spinning structure is in general complicated by the presence of gyroscopic, or Coriolis, forces, leading to a complex Hermitian dynamic stiffness (or impedance) matrix. It is shown that if the analysis is performed by using a discrete model with N degrees of freedom, the leading principal minors of the N th order dynamic stiffness matrix exhibit a type of Sturm sequence property. This leads to a theorem which can be used for the systematic calculation of the natural frequencies of either a discrete system which is assembled from sub-structures, or an assembly of distributed mass members. In both of these cases the order n of the matrix is less than the number of degrees of freedom, and its determinant possesses poles as well as zeros. The theorem is identical with a corresponding one for non-spinning structures previously derived by the authors. The application of the theorem to a spinning two-dimensional frame, with distributed mass members, is discussed in some detail.

Extension of the Wittrick-Williams Algorithm to Mixed Variable Systems

A precise integration algorithm has recently been proposed by Zhong (1994) for dynamic stiffness matrix computations, but he did not give a corresponding eigenvalue count method. The Wittrick-Williams algorithm gives an eigenvalue count method for pure displacement formulations, but the precise integration method uses a mixed variable formulation. Therefore the Wittrick-Williams method is extended in this paper to give the eigenvalue count needed by the precise integration method and by other methods involving mixed variable formulations. A simple Timoshenko beam example is included.

Flexural vibration of axially loaded beams with linear or parabolic taper

Abstract Curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections. Previous curves have only included axial forces in a very limited number of cases, whereas here axial forces within the range of greatest practical importance can always be allowed for. The present curves cover 11 combinations of end conditions, three types of taper, all taper ratios between 0·2 and 1, and all axial forces between − P c and 0·6 P c where P c is the critical buckling load for pure compression and can be found from curves provided (a list of principal nomenclature is given in Appendix 2). The cross sections covered include thin circular ones of constant thickness and thin-walled cross sections consisting of constant thickness flat plates which all have their breadths tapering in the same way. Truncated wedges of varying depth and constant width are also covered. The taper is linear between the ends, or linear between a maximum value at the centre and equal values at the ends, or parabolic with a maximum value at the centre. Simple examples show how the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1%. The curves also illustrate how natural frequencies and buckling loads of thin-walled members are altered by changing the amount and type of a members taper while keeping its wall thickness and mass constant. Finally, it is shown that, for the ranges covered by the curves, frequencies which separate the first five natural frequencies can be found a priori . The theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy. Exact Bernoulli-Euler dynamic stiffnesses, which allowed for axial force effects, were used for these uniform components. In Appendix 1 the effects of using exact Timoshenko stiffnesses instead of the Bernoulli-Euler ones are illustrated.

Review of exact buckling and frequency calculations with optional multi-level substructuring

Abstract This review covers the many applications of the Wittrick-Williams algorithm, which ensures that no critical buckling loads, or natural frequencies of undamped free vibration, are missed even when using the ‘exact’ member equations obtained by solving the appropriate differential equations. The review includes: plane and space frames; prismatic assemblies of isotropic or anisotropic plates, including in-plane plate shear loads; exact multi-level substructuring; design; damping; efficient solution of rotationally or linearly repetitive structures; use of Lagrangian multipliers; programmable pocket calculator methods; program listings for small computers and; references to large computer programs.

Exact dynamic stiffnesses for an axially loaded uniform Timoshenko member embedded in an elastic medium

This paper gives exact dynamic stiffness coefficients for an axially loaded Timoshenko member embedded in an elastic medium and shows how to use them in a general theory for finding the natural frequencies of plane or space frames. The member is considered to have a uniform distribution of mass and the stiffnesses optionally allow for the separate or combined effects of axial load, rotatory inertia and shear deflection. Axial, torsional and flexural responses are assumed to be uncoupled, as happens for doubly symmetric cross-sections. The frames covered include those with beams on elastic foundations and/or supported by piles or other members embedded in an elastic medium. Buckling problems can be solved by setting the frequency parameter to zero.