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Research Papers

# Eigenfrequency Computation of Beam/Plate Carrying Concentrated Mass/Spring

[+] Author and Article Information
Yin Zhang

State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics,Â Chinese Academy of Sciences, Beijing 100190, Peopleâ€™s Republic of Chinazhangyin@lnm.imech.ac.cn

J. Vib. Acoust 133(2), 021006 (Mar 03, 2011) (10 pages) doi:10.1115/1.4002121 History: Received October 19, 2009; Revised May 26, 2010; Published March 03, 2011; Online March 03, 2011

## Abstract

With the adsorption of analyte on the resonator mass sensor, the system eigenfrequencies will shift due to the changes of inertial mass and structural rigidity. How to model those changes and formulate the eigenfrequency computation is very important to the mass sensor application, which results in different accuracies and requires different amounts of computation. Different methods on the eigenfrequency computation of a beam and a plate carrying arbitrary number of concentrated mass/spring are presented and compared. The advantages and disadvantages of these methods are analyzed and discussed. A new method called finite mode transform method (FMTM) is shown to have good convergence and require much less computation for a beam carrying concentrated mass/spring. Because the previous finite sine transform method (FSTM) has only been applied to compute the eigenfrequency of the plate with four edges simply supported carrying a single concentrated mass, here a generalized FSTM is also presented for the case of the same plate carrying arbitrary number of concentrated mass and spring. When the total number of concentrated mass and spring is small, FMTM and FSTM are demonstrated to be very efficient.

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## Figures

Figure 1

The schematic diagram of a beam carrying a concentrated mass and spring. The concentrated mass is located at x=u1 and the concentrated spring is at x=v1.

Figure 2

The convergence study on FMTM computation on the fundamental frequency of the beam with a concentrated mass as the mode number increases. Two cases of Î±1=20 and Î±1=100 are presented and the concentrated mass is located at ÎĽ1=0.1.

Figure 3

The first and second mode shapes of the beam with a concentrated mass when Î±1 is taken as 0, 1, and 20, respectively. The concentrated mass is also located at ÎĽ1=0.1.

Figure 4

The schematic diagram of the distribution of the concentrated masses and springs in the plate

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