Abstract
In this paper, an analytical modeling approach for the flexural vibration analysis of the nonuniform double-beam system is proposed via an improved Fourier series method, in which both types of translational and rotational springs are introduced to account for the mechanical coupling on the interface as well as boundary restraints. Energy formulation is employed for the dynamic description of the coupling system. With the aim to treat the varying thickness across the beam in a unified pattern, the relevant variables are all expanded into Fourier series. Supplementary terms with the smoothed characteristics are introduced to the standard Fourier series for the construction of displacement admissible function for each beam. In conjunction with the Rayleigh–Ritz procedure, the transverse modal characteristics of nonuniform double-beam system can be obtained by solving a standard eigenvalue problem. Instead of solving the certain value of nonideal boundary conditions, the continuous spring stiffnesses of the boundary conditions are considered, and the rotational restrains are introduced in the coupling beam interface. Numerical results are then presented to demonstrate the reliability of the current model and study the influence of various parameters, such as taper ratio, boundary, and coupling strength on the free vibration characteristics, with the emphasis put on the rotational restraining coefficients on the beam interface. This work can provide an efficient modeling framework for the vibration characteristics study of the complex double-beam system, especially with arbitrary varying thickness and coupling stiffness.
Highlights
An analytical modeling approach for the flexural vibration analysis of the nonuniform double-beam system is proposed via an improved Fourier series method, in which both types of translational and rotational springs are introduced to account for the mechanical coupling on the interface as well as boundary restraints
With the aim to treat the varying thickness across the beam in a unified pattern, the relevant variables are all expanded into Fourier series
Numerical results are presented to demonstrate the reliability of the current model and study the influence of various parameters, such as taper ratio, boundary, and coupling strength on the free vibration characteristics, with the emphasis put on the rotational restraining coefficients on the beam interface. is work can provide an efficient modeling framework for the vibration characteristics study of the complex double-beam system, especially with arbitrary varying thickness and coupling stiffness
Summary
Where w(x) is the transverse vibration displacement eld function, KT0 and KR0 are respectively the sti ness coe cients for the translational and rotational springs at the end x 0, and similar meaning can be deduced for the right end of x L. e subscript i means that this variable is associated with the ith beam member. Substituting the admissible function Equation (9) into the elastically connected double-beam system Lagrangian Equations (1)–(4), minimizing it with respect to all the unknown Fourier series coefficients and truncating the Fourier series into finite number n N, one will obtain the system characteristic equation in the matrix form:. Where K and M are the stiffness and mass matrices for the elastically connected double-beam system, respectively, and A is the unknown Fourier series coefficient vector By solving such standard eigenvalue problem, all the modal parameters can be obtained
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