OutofPlane Free Vibration Analysis of a Horizontally Circular Curved Beam Carrying Arbitrary Sets of Concentrated Elements
by JongShyong Wu, (corresponding author), (Professor, Dept. of Systems and Naval Mechatronic Engineering, National ChengKung Univ., Tainan, Taiwan 70101, Republic of China Email: jswu@mail.ncku.edu.tw) and YungChuan Chen, (Graduate student, Dept. of Systems and Naval Mechatronic Engineering, National ChengKung Univ., Tainan, Taiwan 70101, Republic of China.)
Journal of Structural Engineering, Vol. 137, No. 2, February 2011, pp. 220241, (doi: http://dx.doi.org/10.1061/(ASCE)ST.1943541X.0000290)
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Document type: 
Journal Paper 
Abstract: 
For convenience, a horizontally circular curved beam without any attachments is called a "bare" curved beam and the one carrying any attachments is called a "loaded" curved beam, in this paper. For the outofplane free vibrations of bare curved beams, one can find some exact solutions from the existing literature, but this is not true for those of the loaded curved beams. One of the main reasons for the last situation is due to the difficulty of solving a complexvariable eigenvalue equation. It is well known that the halfinterval method is one of the simplest techniques for searching the roots of an eigenvalue equation. However, it suffers difficulty if the eigenvalue equation is a determinant form (H(ω)=0) with some (or all) of its coefficients [H_{i,j}(ω)] being the complex numbers, because it is difficult to find a trial root (ω^{t}) so that both the real part H_{R} and imaginary part H_{I} of the associated determinant value H(ω^{t}) are equal to zero simultaneously (i.e., H_{R}=H_{I}=0). Furthermore, the magnitude of the determinant value is greater than or equal to zero (i.e., overline H = the squareroot of H_{R}^{2}+H_{R}^{2} ≥ 0). To overcome the last difficulty, this paper presents a technique to replace all complex coefficients of the eigenvalue equation by the real ones, so that the conventional halfinterval method may be easily applied to determining the "exact" solution for the natural frequencies and mode shapes of outofplane free vibrations of a uniform curved EulerBernoulli beam carrying arbitrary sets of concentrated elements in various boundary conditions, where each set of concentrated elements includes a lumped mass, a linear spring, a bending spring and a twisting (torsional) spring. To confirm the reliability of the presented theory and the developed computer program, most of the exact solutions for natural frequencies and mode shapes obtained from the presented approach are compared with the "approximate" ones obtained from the conventional finiteelement method and good agreements are achieved. 
Author Keywords: 
 Bare curved beam 
 Loaded curved beam 
 Concentrated elements 
 Natural frequency 
 Mode shape 
 Exact solution 
