Asymptotic Locations of Eigenfrequencies of Euler–Bernoulli Beam with Nonhomogeneous Structural and Viscous Damping Coefficients
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A slender beam has two spatially nonhomogeneous damping terms. The first one acts opposite to the bending moment time derivative and is sometimes called structural damping, while the second acts opposite to the velocity and is called viscous damping. When these damping coefficients are constant, it is known that structural damping causes a strong attenuation rate that is frequency-proportional, whereas viscous damping causes a constant attenuation rate for all frequencies. In this paper, using the method of Birkhoff [Trans. Amer. Math. Soc., 9 (1908), pp. 219-231], [Trans. Amer. Math. Soc., 9 (1908), pp. 373-395], and Birkhoff and Langer [Proc. Amer. Acad. Arts Sci., (2) 58 (1923), pp. 51-128] explicit asymptotic expressions for the eigenfrequencies of the nonhomogeneous damping problem are derived. It is shown that the asymptotic patterns of the eigenspectrum remain similar to the coefficients case. The viscous damping effect is also shown to cause a constant shift to both the attenuation rates and the frequencies; thus it is overwhelmed by the structural damping effect. Because experimentally it has been observed that all eigenfrequencies of light beams essentially lie within the asymptotic regime, the asymptotic formulas derived herein should be useful in determining the pole assignment for feedback stabilization.
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