Vibration Analysis of a Finite Lightweight Locally Resonant Beam Suspended with Periodic Force-Moment-Type Resonators inside Using an Exact Wave-Based Approach

Abstract

This paper employs and develops the exact wave-based vibration analysis approach to investigate the propagation properties of a designed finite lightweight locally resonant (LR) beam with two-degree-of-freedom (2-DOF) force-moment-type resonators attached periodically inside. By deriving the propagation, reflection, and transmission matrices of the structural discontinuities, the vibration of the LR beam can be described as structural waves. By assembling wave relations into the beam, the approach shows high efficiency because the forced vibration problem of the lightweight LR structure is turned to be the solution to a related set of matrix equations. The accuracy of the developed approach is validated with two examples carried out using the finite element method. In addition, the influence of the main parameters of the LR beam is studied and we found that the increase in the mass of the resonator and the stiffness of the spring are more sensitive in broadening the width and increasing the center frequency of the band gap of the designed lightweight LR beam. The proposed structure and analysis approach in this paper may provide an exact and efficient means for the design and analysis of structures in which damping and lightweight properties are required, such as space-arm and the framework of antennas in the field of aerospace.

Vibration Analysis of Locally Resonant Beams with L-Joint Using an Exact Wave-Based Vibration Approach

“…With the existence of discontinuities along a uniform beam, the propagation situation of the exact wave is expressed by Equations (7)–(9) with a single frequency, assuming the distance from point A to B in a beam is as represented in Figure 1 [ 37 ]. The relations of propagation matrix and wave vectors at A and B are defined as: in which Where , , , and , respectively, signify the wave coefficients of forward propagation and opposite propagation at points A and B .…”

“…At the point x = 0, the waves and are formed by the external forces and moment as depicted in Figure 2 [ 37 ], where transverse forces, longitudinal forces, bending moments are defined as F , Q , and M , and the equilibrium and continuity equations are: where the amplitude vector of the excited wave is given: …”

“…As states in Ref. [ 37 ], the production of applied force by the resonators is considered, where the host beam receives injecting waves. Combining Equations (7)–(9), (20) and (21), and uniting the expressions of transverse force in Equation (45) with Equations (22)–(24), the relations of vibration waves at point A can be obtained as: where the coefficient matrices in Equation (35) are: …”

“…So far, researchers have proposed the exact wave-based vibration approach for analyzing beams and better simplifying the vibration analysis of such structures [ 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 ]. Based on the influence of discontinues and planar frame structure, especially on bending and longitudinal vibration, the further interpretation concerning the raised approach is shown by Mei [ 29 , 30 ], which greatly simplifies analysis of the vibration properties of planar frames.…”

“…[ 33 ], which is dramatically beneficial to solving vibration problems in combined distributed and discrete systems. Additionally, the analysis of a finite Timoshenko beam carrying force-moment type resonators [ 34 ], separated force-moment type resonators [ 35 ], and two-degree-of-freedom force type resonators [ 36 ] are further studied, where designing and modeling are better simplified by applying the wave-based vibration approach expanded to the band-gap’s analysis, as well as the analysis of lightweight LR beams investigated by Lv et al [ 37 ]. The above studies illustrated the high efficiency of the wave-based vibration approach and made the contribution for further studies.…”