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Generalized function based solution to free vibration of Euler-Bernoulli beam with arbitrary number of concentrated masses
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DOI：10.1016/j.mechrescom.2026.104629.png)
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Abstract
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In the differential governing equation of the free vibration of an Euler-Bernoulli beam, the presence of multiple concentrated masses is modeled with the Dirac delta function. A new solution to the beam free vibration, which is based on the generalized functions and their distributional derivative rule, is provided. In the new solution, the shear force discontinuities due to the inertial forces cause by the concentrated masses are incorporated. As the result, the eigenvalue problem formulation with this new solution only needs to enforce the beam boundary conditions. In comparison with the determinant of a 4(N + 1) x 4(N + 1) matrix (N: the number of concentrated masses) formulated by the analytical method, the eigenvalue problem formulated by the new solution form is the determinant of a 4 x 4 matrix irrespective of the number of concentrated masses, which significantly reduces the computation. In comparison with the Laplace transform and the Galerkin method, the procedure of formulating an eigenvalue problem is significantly simplified with this new solution. Furthermore, compared with the piece-wise solution form of the analytical method, the one-piece form of this new solution is convenient for enforcing the beam boundary conditions and finding the eigenvectors: The new solution form is with a recursive relation, from which the concise and analytical forms of the characteristic equations are derived and the eigenvectors are readily obtained.
Keywords:
Generalized function
Beam
Concentrated mass
Eigenvalue problem
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