Vibrations of circular plates with clamped v-notches or rigidly constrained radial cracks

This study offers the first known free vibration data for thin circular plates with clamped V-notches. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include: (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained; and (2) corner functions which account for the bending moment singularities at the sharp corner of the V-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of non-dimensional frequencies for circular plates with clamped notches. Numerical results are obtained for plates having their circular edges completely free. Accurate (five significant figure) frequencies are presented for a wide spectrum of notch angles (0°, 5°, 10°, 30°, 60° and 90°) and depths. For very small notch angles, a rigidly constrained radial crack ensues. Some general findings are that, for the spectrum of notch angles examined, the first six frequencies increase as the notch depth increases, more so in the higher modes than the lower ones. The frequency increase with increasing notch depth is quite substantial for the semi-circular plates, and for segmented plates with sector angles less than 180°. For a constant notch depth, it is found that there is a substantial reduction in the first six frequencies as notch angle decreases. Normalized contours of the transverse vibratory displacement are shown for plates having 90° and 5° notches of various depths ranging from deep to very shallow. The first known frequencies and mode shapes for sectorial, semi-circular and segmented plates with clamped radial edges are also presented as special cases of the title problem.