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Free vibration analysis of a rotating double-tapered beam using the transfer matrix method

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Abstract

In this study, a transfer matrix method is developed to analyze the effect of the centrifugal force on the natural frequencies of a rotating double-tapered beam, whose cross-sections have linearly reduced height and width. The root of the differential equation is determined based on the Bernoulli-Euler theory and the effects of the hub radius, taper ratio, and rotation speed, by applying the Frobenius method. In particular, the effect of the centrifugal forces is analyzed based on the relationship between the bending and additional strain energies generated by centrifugal stiffening. The shape function of the displacements depends on the variations in the taper ratio and rotation speed. In addition, the strain energies are varied. Various examples are presented to demonstrate the influences of the strain energies. Furthermore, the effects of the centrifugal force on the natural frequencies of the rotating double-tapered beam are investigated by analyzing the variation in the predicted strain energy.

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Abbreviations

w(x, t):

Bending displacement

ζ (= cx/L):

Nondimensional variable

U :

Strain energy

T :

Kinetic energy

EI(x):

Bending stiffness varying by tapering

EI(x):

Bending stiffness of a uniform beam

EI(x):

Mass per unit length varying by tapering

m 0 :

Mass per unit length when c = 0

ω :

Angular frequency

c :

Taper ratio

a i+1 :

Coefficients of the Frobenius series

F(x):

Centrifugal force

A:

Matrix for arbitrary constants

C:

Matrix for the state quantity at the start point

Q:

Matrix for the state quantity at the end point

T:

Transfer matrix

Z0, Z1 :

Matrix for the state vectors

f (ζ, k):

Function of the Frobenius series

r H :

Hub radius

Φ:

Slope of the deformation curve

M(x,t):

Bending moment

V(x,t):

Shear force

S B :

Bending strain energy

S F :

Additional strain energy by the centrifugal force

C B :

Contribution rate of the bending strain energy

C F :

Contribution rate of the additional strain energy by the centrifugal force

U max :

Maximum strain energy

Ω:

Rotational angular speed

\(\overline {\rm{\Omega }} \) :

Nondimensional rotation speed

\(\overline \omega \) :

Nondimensional natural frequency

T max :

Maximum kinetic energy

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Acknowledgments

The authors gratefully acknowledge the financial support for this research from the National Research Foundation of Korea (Grant number NRF-2018R1D1A1B07047019).

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Authors and Affiliations

  1. Department of Mechanical System Engineering, Kyonggi University, 154-42, Gwanggyosan-ro, Yeongtong-gu, Gyonggi-do, Suwon-si, 16227, Korea

    Jung Woo Lee & Jung Youn Lee

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  1. Jung Woo Lee
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Correspondence to Jung Woo Lee.

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Recommended by Editor No-cheol Park

Jungwoo Lee received B.S., M.S., and Ph.D. degrees from Kyonggi University in 2002, 2004, and 2017, respectively. He is currently an Assistant Professor at the Department of Mechanical System Engineering of Kyonggi University. His research interests are in structural vibration, composite structures, continuum mechanics, and the transfer matrix method.

Jung Youn Lee is a Professor of Dept. of Mechanical System Engineering at Kyonggi University, where he has been since 1996. He received a B.S., an M.S. and his Ph.D. from Hanyang University in 1979, 1989 and 1992, respectively. His research interests are in system identification, structural modification, inverse problem, modal analysis and sensitivity analysis of vibration.

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Lee, J.W., Lee, J.Y. Free vibration analysis of a rotating double-tapered beam using the transfer matrix method. J Mech Sci Technol 34, 2731–2744 (2020). https://doi.org/10.1007/s12206-020-0605-6

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  • DOI: https://doi.org/10.1007/s12206-020-0605-6

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