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Spectral finite element for vibration analysis of cracked viscoelastic Euler–Bernoulli beam subjected to moving load

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Abstract

In this article, a spectral finite element (SFE) model is presented for vibration analysis of a cracked viscoelastic beam subjected to moving loads. The dynamic shape functions are derived from the exact solution of the governing wave equations and are utilized for frequency-domain representation of a moving load. It is considered with either constant velocity or acceleration; then, the force vector for each spectral element is evaluated. The cracked beam is modeled as two segments connected by a massless rotational spring; thus, the beam dynamic stiffness matrix is extracted in frequency domain by considering compatibility conditions at the crack position. The effects of change in velocity and acceleration of moving load, crack parameters, and viscoelastic material properties on the dynamic response of the SFE beam model are investigated. The accuracy of SFE results is compared with that of finite elements. The results show the ascendency of the SFE model, as compared to FEM, for reducing the number of elements and computational effort, but increasing numerical accuracy.

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Abbreviations

SFEM:

Spectral finite element method

FEM:

Finite element method

DSM:

Dynamic stiffness matrix

FFT:

Fast Fourier transform

a :

Acceleration of moving load (m/s2)

b :

Width of beam (m)

\({C_{\theta}}\) :

crack depth-to-beam height ratio-dependent function for rotational spring

\({{\varvec{d}}({i\omega})}\) :

Spectral nodal displacement (\({4 \times 1}\)) (m, rad)

\({F({x, i\omega})}\) :

Spectral components of loading (N)

F 0 :

Amplitude of moving load (N)

f 1 :

Fundamental frequency of beam (Hz)

\({{\varvec{f}}_{c}({i\omega})}\) :

Equivalent spectral nodal force vector (\({4 \times 1}\)) (N, N m)

\({{\varvec{f}}_{d} (\omega)}\) :

External spectral nodal force vector (\({4\times}\)) (N, N m)

h :

Height of beam (m)

k :

Complex-valued wave number (rad/m)

L :

Length of beam (m)

\({l_{{\rm c}}}\) :

Crack location in beam (m)

\({l_{{\rm e}}}\) :

Length of spectral element (m)

M(x, t):

Bending moment (N m)

N :

Number of harmonic components of DFT

\({{\varvec{N}}_{{\rm VB}} ({x, i\omega})}\) :

Dynamic shape functions for transverse displacement (\({1 \times 4}\))

P(t):

Position of moving load in any spectral element as a general function (m)

Q(x, t):

Transverse shear force (N)

\({{\varvec{S}}_{{\rm VB}} ({i\omega})}\) :

Dynamic stiffness matrix

t :

Time (s)

V :

Velocity of moving load (m/s)

w(x, t):

Transverse displacement (m)

\({W_{n} ({x; i\omega_{n}})}\) :

Spectral components of transverse displacement (m)

x :

Location coordinate (m)

\({\delta (x)}\) :

Dirac delta function

\({\varepsilon}\) :

Depth of crack (m)

\({\lambda}\) :

Crack depth-to-beam height ratio (m/m)

\({\tau_{{\rm d}}}\) :

Viscoelastic coefficient

\({\omega}\) :

Sampling frequency (rad/s)

FEM:

finite element method

SFEM:

spectral finite element method

DSM:

Dynamic stiffness matrix

\({{\prime}}\) :

Partial differentiation with respect to x

n :

Number of harmonic component of sampling

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Authors and Affiliations

  1. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Vahid Sarvestan, Hamid Reza Mirdamadi, Mostafa Ghayour & Ali Mokhtari

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  1. Vahid Sarvestan
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  2. Hamid Reza Mirdamadi
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  3. Mostafa Ghayour
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  4. Ali Mokhtari
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Correspondence to Hamid Reza Mirdamadi.

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Sarvestan, V., Mirdamadi, H.R., Ghayour, M. et al. Spectral finite element for vibration analysis of cracked viscoelastic Euler–Bernoulli beam subjected to moving load. Acta Mech 226, 4259–4280 (2015). https://doi.org/10.1007/s00707-015-1491-3

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  • DOI: https://doi.org/10.1007/s00707-015-1491-3

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