Natural frequency of a heavy flexible plate: power law evolution as a function of length

Abstract This theoretical and experimental work deals with the power law evolution followed by the natural frequency f0 of 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 power laws depend on the ratio L/Lc, where Lc 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 systematically measured for a wide length interval. Our data points fit the above-mentioned limit cases and the intermediate case was calculated thanks to our Galerkin expansion.