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Nonlinear forced vibration analysis of postbuckled beams

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Abstract

A numerical solution methodology is proposed herein to investigate the nonlinear forced vibrations of Euler–Bernoulli beams with different boundary conditions around the buckled configurations. By introducing a set of differential and integral matrix operators, the nonlinear integro-differential equation that governs the buckling of beams is discretized and then solved using the pseudo-arc-length method. The discretized governing equation of free vibration around the buckled configurations is also solved as an eigenvalue problem after imposing the boundary conditions and some complicated matrix manipulations. To study forced and nonlinear vibrations that take place around a buckled configuration, a Galerkin-based numerical method is applied to reduce the partial integro-differential equation into a time-varying ordinary differential equation of Duffing type. The Duffing equation is then discretized using time differential matrix operators, which are defined based on the derivatives of a periodic base function. Finally, for any given magnitude of axial load, the pseudo -arc-length method is used to obtain the nonlinear frequencies of buckled beams. The effects of axial load on the free vibration, nonlinear, and forced vibrations of beams in both prebuckling and postbuckling domains for the lowest three vibration modes are analyzed. This study shows that the nonlinear response of beams subjected to periodic excitation is complex in the postbuckling domain. For example, the type of boundary conditions significantly affects the nonlinear response of the postbuckled beams.

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Abbreviations

N :

Number of grid points in space domain

N t :

Number of grid points in time domain

x i :

Chebyshev–Gauss–Lobatto grid points

\({{\bf D}_t^{(1)}}\) and \({{\bf D}_t^{(2)}}\) :

Time differentiation matrix operators

\({{\bf D}_x^{(r)}}\) :

Space differentiation matrix operator

\({\tilde{\bf S}_x}\) and S x :

Integral matrix operator

ℓ :

Length of beam

m :

Mass per unit length

A :

Cross-section area

E :

Young’s modulus

I :

Moment of inertia

\({\hat{w}}\) and w :

Transverse displacement and its dimensionless form

\({\hat{\mu}}\) and μ :

Damping coefficient and its dimensionless form

\({\hat{P}}\) and P :

Axial load and its dimensionless form

\({\hat{F}(\hat{x})}\) and F(x):

Transverse load and its dimensionless form

\({\hat{\Omega}}\) and Ω:

Excitation frequency and its dimensionless form

ω :

Natural frequency

q(t):

Vibration amplitude

v(x, t):

Dynamic part of field variable w(x, t)

ψ (x):

Static part of field variable w(x, t)

ϕ (x ):

Normalized vector of the linear vibration mode shape around the postbuckling configuration

T :

Period of response

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 41635-3756, Rasht, Iran

    M. Faghih Shojaei, R. Ansari, V. Mohammadi & H. Rouhi

Authors
  1. M. Faghih Shojaei
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  2. R. Ansari
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  3. V. Mohammadi
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  4. H. Rouhi
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Correspondence to R. Ansari.

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Faghih Shojaei, M., Ansari, R., Mohammadi, V. et al. Nonlinear forced vibration analysis of postbuckled beams. Arch Appl Mech 84, 421–440 (2014). https://doi.org/10.1007/s00419-013-0809-7

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