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 𝑓𝑓 0 of a hanging, heavy and flexible plate as a function of its length 𝐿𝐿. When the plate length 𝐿𝐿 is small enough, it behaves as an elastic plate whose weight can be neglected: it is well known that 𝑓𝑓 0 evolves as a function of 𝐿𝐿 −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, 𝑓𝑓 0 ∼ 𝐿𝐿 −1/2 , just like hanging chains. These two power laws depend on the ratio 𝐿𝐿/𝐿𝐿 𝑐𝑐 , where 𝐿𝐿 𝑐𝑐 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.

Stochastic Higher-order Finite Element Model for the Free Vibration of a Continuous Beam resting on Elastic Support with Uncertain Elastic Modulus

“…In this section, a continuous beam on 5 elastic supports, as shown in Figure 1, is investigated. The geometrical and material parameters of the beam are: length L=10m, height h=0.6m, mean of Young's modulus E=30×10 5 MPa, coefficient of variation of a random field of elastic modulus σ=0.1, and mass density ρ=2400kg/m 3 . To valid the proposed approach, it is compared with Monte Carlo simulation.…”

“…Beams and frames are used in many engineering applications, such as civil engineering [1][2][3], bridge engineering, and aerospace engineering. The importance of dynamic problems has been the subject of many research studies [4][5][6][7][8][9]. In civil engineering, the structures resting on foundations are very important [10][11][12][13][14][15].…”