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Transverse vibrations of arbitrary non-uniform beams

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Abstract

Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases.

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Abbreviations

u :

displacement

E :

Young’s modulus

I :

inertialmoment

ρ :

density

A :

area

p :

exciting force

x :

coordinate

f 1(x):

dimensionless bending stiffness

f 2 (x):

dimensionless linear density

L :

length of beam

β :

dimensionless frequency

ω :

cycle frequency

D :

Wronski determinant

c :

integral constant

C :

constant

a :

scale factor

i:

imaginary unit

t :

time

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Authors and Affiliations

  1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, P. R. China

    Shu-qi Guo  (郭树起)

  2. School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, 050043, P. R. China

    Shao-pu Yang  (杨绍普)

Authors
  1. Shu-qi Guo  (郭树起)
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  2. Shao-pu Yang  (杨绍普)
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Guo, Sq., Yang, Sp. Transverse vibrations of arbitrary non-uniform beams. Appl. Math. Mech.-Engl. Ed. 35, 607–620 (2014). https://doi.org/10.1007/s10483-014-1816-7

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  • DOI: https://doi.org/10.1007/s10483-014-1816-7

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