Exact solution for longitudinal vibration of elastically constrained biperiodic rod

Author Type

Faculty

Co-Author Type 1

Faculty

Co-Author Type 2

Outside Researcher

Co-Author Type 3

Outside Researcher

College

Engineering and Computer Science

Department

Ocean and Mechanical Engineering

Document Type

Article

Publication/Event/Conference Title

ZAMM Zeitschrift Fur Angewandte Mathematik Und Mechanik

Publication Status

Version of Record

Abstract

This article is focused on the exact determination of the eigenfrequencies of an elastically constrained biperiodic elastic rod. The rod is composed of biperiodic cells of stepped continuous rod elements. The axial displacement solution in each cell can be expressed from the resolution of a second-order differential equation. After expressing the continuity conditions between each cell (displacement and normal force continuity), it is possible to relate the solution of each cell concerning its neighbors. The differential eigenvalue problem of the continuous biperiodic rod is converted into a linear difference eigenvalue problem associated with the coefficients of the expressed solution in each cell. A transcendental equation for the natural frequency of the elastically restrained continuous biperiodic rod is obtained from the resolution of the discrete linear difference eigenvalue problem. This transcendental equation is valid regardless of the number N of biperiodic cells, with N more prominent than 2. This general expression is validated by the natural frequency values obtained using a direct method for a few cells. This transcendental equation generalizes the results available in the literature for a free-free biperiodic rod or a fixed-fixed biperiodic rod. The eigenfrequencies of the biperiodic rod are compared to those of a homogenized rod, also in the presence of elastic restraints. The validity of the finite homogenized rod theory is confirmed when the number of cells is sufficient, even in the presence of elastic restraints.

DOI

10.1002/zamm.70300

Publication Date

12-1-2025

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