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A simple noniterative method for recovering a space‐dependent load on the Euler‐Bernoulli beam equation
Author(s) -
Liu CheinShan,
Jhao WunSin,
Chang ChihWen
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5226
Subject(s) - eigenfunction , mathematics , mathematical analysis , cantilever , robustness (evolution) , beam (structure) , bernoulli's principle , eigenvalues and eigenvectors , physics , optics , thermodynamics , biochemistry , chemistry , quantum mechanics , engineering , gene , aerospace engineering
When adjoint eigenfunctions are adopted as the test functions in Green's second identity for the Euler‐Bernoulli beam equation, we can develop a quite simple noniterative numerical algorithm to recover an unknown space‐dependent external force H ( x ) exerted on the beam. The spatial parts of the adjoint eigenfunctions are used as the bases to expand the unknown function H ( x ), where we view the two end values of H ( x ) as two unknown coefficients for the simply supported and hinged‐clamped beams, and the left end value of H ( x ) as an unknown coefficient for the cantilevered beam. We can derive closed‐form solutions of the expansion coefficients, and thus closed‐form series solutions of H ( x ). Consequently, we have a noniterative method to recover the unknown force H ( x ) supplemented by the noisy final time displacement data. Numerical examples demonstrate the accuracy, efficiency, and robustness of the novel methods in the recovery of unknown forces on the simply supported, cantilevered, and hinged‐clamped beams.