NATURAL FREQUENCY OF A HEAVY FLEXIBLE PLATE: POWER LAW EVOLUTION AS A FUNCTION OF LENGTH

Abstract

THIS THEORETICAL AND EXPERIMENTAL WORK DEALS WITH THE SCALING LAWS FOLLOWED BY THE NATURAL FREQUENCY F₀OF A HANGING, HEAVY AND FLEXIBLE PLATE AS A FUNCTION OF ITS LENGTH L. WHEN THE PLATE LENGTH L IS SMALL ENOUGH, IT BEHAVES AS AN ELASTIC PLATE WHOSE WEIGHT CAN BE NEGLECTED: IT IS WELL KNOWN THAT F₀ EVOLVES AS A FUNCTION OF L^-2. NEVERTHELESS, WHEN THE PLATE LENGTH IS INCREASED, THE MASS HAS TO BE TAKEN INTO ACCOUNT, AND THE PREVIOUS EVOLUTION IS NOT VALID ANYMORE. IN THE CASE OF LONG ELASTIC PLATES, F₀ ~ L^-1/2,JUST LIKE HANGING CHAINS. THESE TWO KINDS OF SCALING LAWS DEPEND ON THE RATIO L/L_C, WHERE L_C IS A CRITICAL LENGTH THAT WRITES AS A FUNCTION OF THE PLATE MASS AND THE FLEXURAL RIGIDITY. AFTER THE THEORY IS DEVELOPED AND THE PLATE MOTION EQUATION IS SOLVED USING A GALERKIN EXPANSION, WE FIND THE THEORETICAL EVOLUTION OF THE NATURAL FREQUENCIES AS A FUNCTION OF LENGTH. EXPERIMENTS WERE PERFORMED WITH THREE DISTINCT MATERIALS AND THE NATURAL FREQUENCY WAS MEASURED FOR DIFFERENT LENGTH VALUES. OUR DATA POINTS FIT THE ABOVE-MENTIONED LIMIT CASES AND THE INTERMEDIATE CASE WAS CALCULATED THANKS TO OUR GALERKIN EXPANSION.