Cantilever End Conditions Research Articles

Explicit Modal Analysis of an Axially Loaded Timoshenko Beam With Bending-Torsion Coupling

Exact analytical expressions for the natural frequencies and mode shapes of a uniform bending-torsion coupled Timoshenko beam are presented. The beam is taken to be axially loaded, and for which cantilever end conditions apply. A symbolic computing package (REDUCE) has been used, which simplifies the analysis greatly. Results are presented for an illustrative example, which confirms the accuracy of the method, and provides a convenient benchmark for the validation of the finite element or other alternative approximate methods. The approach offers the prospect of aeroelastic development, and is computationally efficient, thus holding out the promise of eventual optimization. [S0021-8936(00)01002-3]

EXPLICIT FREQUENCY EQUATION AND MODE SHAPES OF A CANTILEVER BEAM COUPLED IN BENDING AND TORSION

Exact explicit analytical expressions which give the natural frequencies and mode shapes of a bending–torsion coupled beam with cantilever end condition are derived by rigorous application of the symbolic computing package REDUCE. The expressions are surprisingly concise and very simple to use. By way of illustration in this paper, they are used to determine the natural frequencies and mode shapes of a cantilever wing with substantial coupling between the bending and torsional modes of deformation. The results are compared with exact published results to confirm the correctness and accuracy of the expressions. The derived expressions can be used to solve bench-mark free vibration problems as an aid in validating the finite element and other approximate methods. They are also intended for future applications in aeroelastic and/or optimization studies. Computer implementation and a comparison of solution times show that there is more than a four-fold advantage in c.p.u. time when using the explicit expressions as opposed to the alternative numerical method involving determinant evaluation and matrix manipulation.

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

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.