Zur Berechnung der Eigenfrequenzen von eingespannten Rechteckplatten bestimmen die Verfasser nach einer kiirzlich entwickelten Methode untere und nach dem R a y l e i g h -Ritz-Verfahren obere Schranken. Die Anwendbarkeit der neuen Methode wird klargestellt, und es werden numerische Ergebnisse fur eine Symmetrieklasse bei Rechteckplatten und fiir eine Symmetrie-Unterklasse bei puadratischen Platten angegeben. Die Schranken verbessern friiher veroffentlichte strenge Ergebnisse und bestatigen die Brauchbarkeit der Methode zur Berechnung unterer Schranken.The authors apply a recently developed method for lower bounds and the R a y l e i g h -R i t z procedure for upper bounds to the estimation of the frequencies of free vibration of clamped rectangular plates. The applicability of the lower bound procedure is demonstrated and calculations are presented for a symmetry class for rectangular plates and for a sub-symmetry class for square plates. T h e resulting bounds represent improvements over previously published rigorous results and demonstrate the utility of the lower bound method. B npennoxeHHoCi pa6ol.e aBTopbI onpenensm n n~ CO6CTBeHHblX YacToT a a~p e n n e n~b~x npHMoyronbaHx naamkinox Hnxme r p a m i w no noBoMy MeTony II sepxme no MeTony P e a e H -PHTqa. I I o x a a b~s a e~c~ npmvenHMocTb H O B O~O MeToRa, nam1 mcnemme p e a y n b~a~b~ n n~ macca cmmeTpm npHMoyronbabix njIacTHHoK II nonmacca CmmeTpmi KsanpaTabIx nnacTHaoK. I'paamm ynyqmamT paHee HaHHbie II nonTsep%namT n o n e a~o c~b MeTona n n~ swmcneam H I I x n H x r p a a w .
NomenclatureThe following symbols are not defined in the text: V a r 7 4 u, ul, ua Generic names for displacement functions Laplacian differential operator (= a2/i3z2 + a2/ay2) Biharmonic differential operator (= a4/ax4 + 2 a4/ayz ax2 + a4/ay4)
The authors use a recently developed procedure for lower bounds and the Rayleigh ‐ Ritz procedure for upper bounds to estimate frequencies of free vibration of thin rectangular cantilever plates. The applicability of the lower bound procedure is demonstrated, and calculations are given for a symmetry class of normal modes. The lower bounds seem to be the first rigorously obtained for this problem. The upper bounds are improvements over previously published rigorous results.
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