Upper and Lower Bounds of Frequencies for Cantilever Bars of Variable Cross Section

Invoking the boundary conditions $(0) = = LMr),$(=>) = 0, where <t>o(r) = 4>{t, 0), we determine A and B and rewrite (17) in the form $(??) = e [ft -\ q-1 j" J -j g-V» J The integral equation (20) Substituting where Example We consider the half space described by (13a, 6) with p = 1, corresponding to the wave-speed distribution Ci(l + £), and the surface loading oo(0 = (r,e-" r . Setting k = (1 + J) 2 and m = 1 in (66) and (9a, 6), we obtain Substituting or, in terms of the original variables, (24) (25) ( 26) The ratio of the second term to the first in Upper and Lower Bounds of 3 Numbers in brackets designate References at end of Note. Manuscript received by ASME Applied Mechanics Division, March 7, 1966. 8 = 1.0, where 5 is the ratio of the diameter at the free end to the diameter at the fixed end. The percentage deviation in the tables is calculated according to the formula: (upper bound-lower bound) Percent deviation = X (100) average value where the "average value" is the average of the upper bound and the lower bound. Calculated deviations are rounded off to the nearest tenth. The following notation is used in the text