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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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