Abstract
This paper derives an analytical model of a circular beam with a T-shaped cross section for use in the high-frequency range, defined here as approximately 1 to 50 kHz. The T-shaped cross section is composed of an outer web and an inner flange. The web in-plane motion is modeled with two-dimensional elasticity equations of motion, and the left portion and right portion of the flange are modeled separately with Timoshenko shell equations. The differential equations are solved with unknown wave propagation coefficients multiplied by Bessel and exponential spatial domain functions. These are inserted into constraint and equilibrium equations at the intersection of the web and flange and into boundary conditions at the edges of the system. Two separate cases are formulated: structural axisymmetric motion and structural non-axisymmetric motion and these results are added together for the total solution. The axisymmetric case produces 14 linear algebraic equations and the non-axisymmetric case produces 24 linear algebraic equations. These are solved to yield the wave propagation coefficients, and this gives a corresponding solution to the displacement field in the radial and tangential directions. The dynamics of the longitudinal direction are discussed but are not solved in this paper. An example problem is formulated and compared to solutions from fully elastic finite element modeling. It is shown that the accurate frequency range of this new model compares very favorably to finite element analysis up to 47 kHz. This new analytical model is about four magnitudes faster in computation time than the corresponding finite element models.
Highlights
The work derived is a direct extension of a previous effort [1,2], that modeled the dynamics of a circular T-beam
In the new model derived here, the Donnell shell formulation is replaced with a Timoshenko-type shell formulation [4], which includes shear deformation and rotary inertia terms, resulting in a higher-frequency range of analysis
Any location of the beam can be chosen for displacement output, the web’s outer surface was investigated here because this location is pertinent to the analysis of reinforced cylindrical shells
Summary
The work derived is a direct extension of a previous effort [1,2], that modeled the dynamics of a circular T-beam. This work develops an analytical model of a circular T-shaped beam and extends the frequency range of this system compared to previous modeling efforts It is primarily intended for use in models that have reinforced cylindrical shells that need improved accuracy at higher frequencies. Bernoulli and Euler [5] derived the first accurate beam model in Cartesian coordinates and this model was extended to cylindrical coordinates This theory uses the assumption that all sections rotate orthogonal to the neutral axis of the beam. Tran-Van-Nhieu [20] researched the problem with an emphasis on Bloch–Floquet wave scattering These papers [16,17,18,19,20] have generally modeled the ribs as having lumped parameter behavior. Modeling the beam in this manner allows the web and flange equations of motion to incorporate higher-order dynamic effects and this makes the overall model much more accurate at higher frequencies
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