Vibrations of circular plates having v-notches or sharp radial cracks

This paper provides the first known free vibration data for circular plates having V-notches. A V-notch has bending moment singularities at its sharp corner due to the transverse vibratory motion. Atheoretical analysis is undertaken using two sets of admissible displacement functions, (1) algebraic-trigonometric polynomials and (2) corner functions. These function sets are used with the Ritz method. The first set guarantees convergence to the exact frequencies as sufficient terms are taken. The second set represents the corner singularities exactly, and accelerates convergence greatly. Numerical results are given for non-dimensional frequencies of completely free circular plates having various notch angles and depths. As the notch angle becomes very small, a sharp radial crack ensues. Convergence studies demonstrate the necessity of adding corner functions to achieve accuratefrequencies. Extensive, accurate (five significant figure) frequencies are presented for the spectrum of notch angles (0°, 1°,5°, 10°, 30°, 60° and 90°) and depths. The effect of the Poisson ratio on the frequencies in the case of shallow notches is also investigated. Sharp notches are found to reduce each ofthe first six frequencies from those of a complete circular plate, whereas large notch angles can increase some of the frequencies. Nodal patterns are shown for plates having 5° notches. The first known frequencies for completely free sectorial, semi-circular and segmented plates are also given as special cases.