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Structural Engineering and Mechanics Volume 22, Number 6, April20 2006 , pages 701-717 DOI: https://doi.org/10.12989/sem.2006.22.6.701 |
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On the natural frequencies and mode shapes of a multiple-step beam carrying a number of intermediate lumped masses and rotary inertias |
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Hsien-Yuan Lin and Ying-Chien Tsai
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| Abstract | ||
| In the existing reports regarding free transverse vibrations of the Euler-Bernoulli beams, most of them studied a uniform beam carrying various concentrated elements (such as point masses, rotary inertias, linear springs, rotational springs, spring-mass systems, ? etc.) or a stepped beam with one to three step changes in cross-sections but without any attachments. The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multiple-step Euler-Bernoulli beams carrying a number of lumped masses and rotary inertias. First, the coefficient matrices for an intermediate lumped mass (and rotary inertia), left-end support and right-end support of a multiple-step beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of lumped masses and rotary inertias on the dynamic characteristics of the multiple-step beam are also studied. | ||
| Key Words | ||
| multiple-step beam; lumped mass; rotary inertia; exact natural frequency; mode shape; integration constants. | ||
| Address | ||
| Hsien-Yuan Lin; Dept. of Mechanical and Electro Mechanical Engineering, Nat. Sun Yat-Sen University, 70, Lien-hai Road, Kaohsiung 804, Taiwan, ROC Ying-Chien Tsai; Dept. of Mechanical and Electro-Mechanical Engineering, Nat. Sun Yat-Sen University, 70, Lien-hai Road, Kaohsiung 804, Taiwan, ROC (Dept. of Mechanical Engineering, Cheng Shiu Univ., Kaohsiung 833, Taiwan, ROC) | ||