Explicit frequency equation and mode shapes of a cantilever beam coupled in bending and torsion

Exact explicit analytical expressions which give the natural frequencies an d mode shapes of a bending-torsion coupled beam with cantilever end conditi on are derived by rigorous application of the symbolic computing package RE DUCE. The expressions are surprisingly concise and very simple to use. By w ay of illustration in this paper, they are used to determine the natural fr equencies and mode shapes of a cantilever wing with substantial coupling be tween the bending and torsional modes of deformation. The results are compa red with exact published results to confirm the correctness and accuracy of the expressions. The derived expressions can be used to solve bench-mark f ree vibration problems as an aid in validating the finite element and other approximate methods. They are also intended for future applications in aer oelastic and/or optimization studies. Computer implementation and a compari son 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 alternat ive numerical method involving determinant evaluation and matrix manipulati on. (C) 1999 Academic Press.