     Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory has been widely used in many engineering problems. In the Euler-Bernoulli beam theory, a kinematic hypothesis is made such that plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation. With this assumption, the shear deformation of the beam is neglected. As observed by researchers, however the shear deformation of a beam component at high frequencies becomes noticeable. As such, the Euler-Bernoulli hypothesis may not be suitable for mid-frequency (medium-frequency) vibration analysis. In that sense, the Euler-Bernoulli beam theory, which is valid at low frequencies, may overestimate the stiffness of a beam structure if its dynamic response at relatively high frequencies is investigated. It is well known that both bending deformation and shear deformation are described in a Timoshenko beam model.

      Besides beam models, accurate and efficient analysis tools are also important to the outcome of mid-frequency analysis. Conventional low-frequency analysis tools, such as finite element analysis (FEA), do not work well for mid-frequency vibration problems because they require a huge number of degrees of freedom and consequently costly computation, and are sensitive to material properties and boundary conditions. On the other hand, high-frequency analysis tools, such as the statistical energy analysis (SEA) and its variations, are unsuitable for mid-frequency problems because they describe the vibrational behaviors of a multi-body structure in a global manner and cannot provide detailed local information about displacements and internal forces. According to the literature, a computationally efficient and accurate tool is in demand for mid-frequency vibration problems

   This paper presents an analytical method for mid-frequency vibration analysis of beam structures modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The main contribution of this work is the utility of the augmented DTFM in mid-frequency analysis of Timoshenko beam structures for the first time. In the development, the governing equations and boundary conditions of beam components are systematically cast in an s-domain spatial state-space formulation, without discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be determined in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at high frequencies, can be directly obtained from the space-state formulation, which can be extremely difficult, if not impossible, with energy-based methods.

    The mid-frequency analysis by the augmented DTFM is validated with the FEA in numerical simulation, where the efficiency and accuracy of the proposed method is shown. In the meantime, the Timoshenko beam model and the Euler-Bernoulli beam model are compared, which reveals the difference of the two beam models in mid-frequency vibration analysis. Particularly, the obvious effects of shear deformation on the vibrational behaviors of a beam structure at medium and high frequencies indicate the necessity of consideration of shear deformation in vibration analysis of beam structures in a mid- or high-frequency region.

 


Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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