Oliver G. McGee

Exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges

Abstract The first known exact analytical solutions are derived for the free vibrations of thick (Mindlin) sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge. The general solutions to the Mindlin differential equations of motion contain noninteger order ordinary and modified Bessel functions of the first and second kinds, and six arbitrary constants of integration. By exercising a careful limiting process, three regularity conditions at the vertex of the radial edges are invoked to yield three equations of constraint among the six constants for sector angles exceeding 180° (re-entrant corners). Three additional linearly independent equations among the six constants are obtained by satisfying the three boundary conditions along the circular edge. Frequency determinant equations are derived for Mindlin sectorial plates with circular boundaries which are clamped, simply supported, or free. Nondimensional frequency parameters are presented for over a wide range of salient and re-entrant sector angles (30° ⩽ α ⩽ 360°), and thickness-to-radius ratios of 0.1, 0.2 and 0.4. Frequency results obtained for Mindlin sectorial plates are compared to those determined for classically thin sectorial plates, and the results are found to be considerably different than those derived from thin plate theory, particularly for the fundamental frequencies of plates having sector angles slightly greater than 180° when the circular boundary is free. The frequencies for 360° sectorial plates (i.e. circular plates having a hinged crack) are compared with those for complete circular ones.

Vibrations of cantilevered skewed trapezoidal and triangular plates with corner stress singularities

Abstract In a recent companion paper, the efficacy of admissible corner functions used in conjunction with mathematically complete polynomials has been demonstrated in an exhaustive amount of Ritz convergence tables apropos to cantilevered skew parallelogram plates. The present work extends the method to include skewed plates having trapezoidal and triangular planforms, and is the first known vibrational study encompassing the effect of bending stress singularities at the reentrant corner of such plates. Extensive and accurate nondimensional frequency tables and graphical charts are presented for a series of trapezoidal plates showing the effect of aspect ratio, chord ratio and skew angle. The importance of using admissible corner functions in skew trapezoidal plate vibrations is found to increase as the skew angle increases and as the aspect ratio and chord ratio decreases. Some theoretical and experimental data heretofore published for delta and skewed triangular cantilevered plates are compared with results obtained using the present method. No previous theoretical results for skewed trapezoidal cantilevered plates are known to exist.

Closed-form effects of warping and pretwist on the torsional vibration of thin-walled open-profile cantilevers

Abstract Theoretical natural frequencies are determined for the first three modes of torsional vibration of pretwisted cantilevers with thin-walled open profiles. The equation of motion and boundary conditions are derived from a multifilament model of a pretwisted thin-walled bar undergoing warp deformation and a moderately large twist about the elastic axis. Closed-form torsional frequencies and mode shapes are derived from the linear terms of the resulting nonlinear equations. As the length-to-leg ratio ( L/b ) and leg-to-thickness ratio ( b/t ) of typical open-profile cantilevers are varied, the individual and collective effects of warping and pretwist on the torsional frequencies are qualitatively defined. The refined derivation and closed-form solutions presented here reveal significant destiffening of the torsional frequencies due to warping-pretwist coupling in cantilevers having pretwisted unsymmetrical open-profiles. These coupling effects, which have been ignored in thin-walled bar vibration theories heretofore, are brought out by the Wagner effect and warping shear stresses acting on the profile.

Frequency-constrained optimization of space frames having general cross-sectional relationships∗

Abstract An optimality criteria (OC) method is presented for weight optimization of space frames having general cross-sectional relationships. The space frames support a sizeable amount of non-structural mass, while multiple natural frequency constraints, and minimum and maximum gauge restrictions are imposed on their design. The iterative design method involves alternately satisfying the constraints (scaling) and applying the Kuhn-Tucker (optimality) condition (resizing). The primary sizing variables (cross-sectional areas), and indirectly the secondary ones (two principal moments of inertia and a torsional constant), are uniformly scaled to the constraint surfaces using a nonlinear closed-form formulation. No exact scaling formulation for this class of problem has been proposed and tested in the optimization literature hitherto. The closed-form scaling procedure is united with an adaptable design strategy in which linear extrapolates of past-scaled design vectors are coupled with automatically-tuned OC ...