Vibration of Plates with Constrained V-Notches or Cracks

Abstract

This paper reports the first known free vibration solutions for thin circular plates with V-notches having various edge conditions. The classical Ritz method is employed with 2 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 to account for the bending moment singularities at the sharp re-entrant corner of the V-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondimensional frequencies for circular plates having a free circumferential edge and various combinations edge conditions of the V-notch. Accurate (5 significant figures) frequencies are presented for clamped-free, clamped-hinged, and hinged-free notches for the spectra of notch angles, causing a re-entrant vertex corner of the radial edges. For very small notch angles, a clamped-free, clamped-hinged, or hinged-free radial crack ensues. One general observation is that, for the range of notch angles considered, there is a substantial increase in the first 6 frequencies as the notch depth increases. The frequency increase with increasing notch depth is more pronounced in the higher modes than the lower ones, and is quite substantial for segmental plates with notch angles equal to 180 deg. A large reduction in frequencies is also observed as the notch angle decreases at a constant notch depth. A new database of accurate frequencies and mode shapes for sectorial, semicircular and segmental plates is presented with which future solutions drawn from alternative numerical procedures and finite element and boundary element techniques may be compared. Normalized contours of the transverse vibratory displacement are shown for plates with various notch depths and notch angles of 5, 30, 60, 90, and 180 deg.