An analytical model for the determination of resonance frequencies of perforated beams

Abstract

In this paper, we develop closed expressions for the equivalent bending and shear stiffness of beams with regular square perforations, and apply them to the problem of determining the resonance frequencies of slender, regularly perforated clamped–clamped beams, which are of interest in the development of MEMS resonant devices. We prove that, depending on the perforation size, the Euler–Bernoulli equation or the more complex shear equation needs to be used to obtain accurate values for these frequencies. Extensive finite element method simulations are used to validate the proposed model over the full practical range of possible hole sizes. An experimental verification of the model is also presented.

Analytical modeling for the determination of nonlocal resonance frequencies of perforated nanobeams subjected to temperature-induced loads

“…In this section, the dynamics behavior of perforated nanobeam will be analyzed by using the Eringen's nonlocal elasticity theory and the Timoshenko beam theory containing the equivalent characteristic parameters for bending and shear stiffness developed by Luschi and Pieri [46]. To this purpose, we consider a nanobeam of length L, width band thickness h, with periodic square holes network of spatial period s P and size of hole d h .…”

“…Unlike the Euler-Bernoulli beam theory which takes in consideration the bending effect, Timoshenko [3] proposed a beam theory which adds the shear effect as well as the rotation effect to the bending effect [4,46]. By using the Timoshenko beam theory, the standard equations for the dynamic vibration of a perforated nanobeam subjected to temperature-induced loads can be given by two coupled differential equations expressed in terms of the flexural deflection wand the rotation angle ψ of the cross-section.…”

“…Luschi and Pieri [47] determined the analytical expressions for the equivalent parameters ρA eq , ρI eq , EI eq and GA eq as functions of the number of holes N and the filling ratio α¼(1Àβ).The analytical expressions of ρA eq , ρI eq , EI eq and GA eq have been determined by using Eqs. (6)-(9) respectively [46]: 1. Geometry shape of perforated nanobeamstructure with periodic square holesnetwork.…”

“…For many practical cases, the complete solution of (2) is not required. It can be shown [4] that for typical materials and crosssections, the rotation effect term ρI eq is 3-6 times smaller than the shear deformation term (EI eq .ρA eq /kAG eq ) [46], so that the right part ρI eq .∂ 2 ψ/∂t 2 of the second Timoshenko beam Eq. (2) can be neglected thus only bending and shear effects are considered.…”

“…In the last few years, the study of the effect of holes is concerned with the viscous losses around plates for the squeeze damping [43,44], the anchor losses in bulk plate resonators [45] and the resonance frequencies change for perforated beams [46]. This latter study may be considered as aclearly and very significant contribution in which Luschi and Pieri have developed analytical expressions for the equivalent bending stiffness and shear stiffness for perforated beam structures with periodic square holes network, and they have determined the resonance frequencies expression.…”