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 $f_0$ 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 $f_0$ 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, $f_0\sim L^{-1/2}$, just like hanging chains. These two power 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 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.

Keywords:
natural frequency; elastic plate; power law

Graphical Abstract