Forced Vibration of Beams by Eigenmatrix Method

by Bennosuke Tanimoto, Prof. of Civ. Engrg.; Kanazawa Inst. of Tech., Nonoichi, Kanazawa, Japan; and Head of the Inst. for Developments at Operational Method (IDOM), Tobishima Construction Co., Inc., Kudan, Tokyo, Japan,
Shotaro Natsume, Assoc. Prof.; Dept. of Civ. Engrg., Shinshu Univ., Nagano, Japan,
Kiyoshi Ishikawa, Asst.; Dept. of Civ. Engrg., Shinshu Univ., Nagano, Japan,

Serial Information: Journal of the Structural Division, 1979, Vol. 105, Issue 12, Pg. 2725-2749

Document Type: Journal Paper


The eigenmatrix method and the operational displacement one are presented for the forced vibration problem of beams and frames. The differential equation for flexure is due to the classical Bernoulli-Euler's theory, and it can be extended to the excellent Timoshenko's one. The stress-strain relationship can also be extended from Hooke's law to the visco-elastic Voigt's law. Avoiding the prevailing lumped mass or finite element philosophy, the structures are treated as continuum solid body as they appear. The method consists of the three traditional branches in structural analysis: (1) Eigenvalue problem, (2)boundary-value problems, and (3)initial-value problem. The eigenmatrix method is in a sense the algebraic transfer matrix one, in which case the load term can be separated, and it is suited for one-dimensional structures. The operational displacement method is fitted for frames, in which case the complete stiffness matrix is tridiagonal.

Subject Headings: Vibration | Beams | Displacement (mechanics) | Frames | Matrix (mathematics) | Differential equations | Stress strain relations | Viscoelasticity

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