Research Papers

Exact Vibration Solution for Exponentially Tapered Cantilever With Tip Mass

[+] Author and Article Information
C.Y. Wang

 Department of Mathematics, Michigan State University, East Lansing, Michigancywang@math.msu.edu

C. M. Wang

 Engineering Science Programme and Department of Civil and Environmental Engineering, National University of Singapore, Kent Ridge, Singapore 119260, Singaporeceewcm@nus.edu.sg

J. Vib. Acoust 134(4), 041012 (May 29, 2012) (4 pages) doi:10.1115/1.4005835 History: Received May 05, 2011; Revised November 15, 2011; Published May 29, 2012; Online May 29, 2012

This technical note is concerned with the free vibration problem of a cantilever beam with constant thickness and exponentially decaying width. Existing analytical results for such a vibration beam problem are found to be incomplete because lower frequencies could not be obtained. Presented herein is the exact characteristic equation for generating the complete vibration frequencies for the considered vibrating beam problem. Also the note treated for the first time such a tapered cantilever beam with a tip mass. The exact solutions (frequencies and mode shapes) are important to engineers designing such tapered beams and the results serve as benchmarks for assessing the validity, convergence and accuracy of numerical methods and solutions.

Copyright © 2012 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Exponential cantilever with constant thickness

Grahic Jump Location
Figure 2

The decrease of (exaggerated) width of the exponential beam. From top: c = 0.1, 0.5, 1, 2.

Grahic Jump Location
Figure 3

(a) Mode shapes for the exponential cantilever with clamped base and no tip mass. c=1,ν=0,γ=∞. (b) Mode shapes for the exponential cantilever with clamped base and tip mass. c=1,ν=10,γ=∞.



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