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Dynamic Analysis of Bi-directional Functionally Graded Beam with Geometric Nonlinearity

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Abstract

Purpose

The objective of the present work is to carry out simulation studies for prediction of nonlinear dynamic characteristics of functionally graded uniform beams.

Methods

The material properties are considered to be varying simultaneously along thickness and length direction. Euler–Bernoulli beam theory is used to generate the constitutive relations and two distinct methodologies are employed, viz., whole domain method and finite element method. Geometric nonlinearity is incorporated in the methodology using Von Karman’s strain–displacement relations.

Results

The results are presented as backbone curve and mode shape in free vibration and frequency–response curve and operational deflection shape in forced vibration. The effect of material gradation on dynamic behaviour is investigated using different values of material gradient parameter. The nonlinear system is seen to display hard spring behaviour. A comparative study between WDM and FEM also shows the correctness of present study.

Conclusion

In free vibration, a similar trend has been found in all cases that natural frequency increases with response amplitude. This behaviour is due to the stretching effect associated with large deflection resulting in additional stiffening of the system. Increase in vibration frequencies with response amplitude is formally known as hardening type nonlinearity. The same is also observed in forced vibration analysis where the frequency–response curve bends towards right.

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Data availability

Both original data used and generated in the research, that supports the results and analyses are included in the article.

Abbreviations

b :

Width of the beam

d i :

Displacement coefficients

E 0 :

Elastic modulus of the beam material at (x=0)

e :

Error limit

{f}:

Force vector

h :

Thickness of the beam

[K]:

Stiffness matrix

L :

Length of the beam

l :

Length of the beam element

[M]:

Mass matrix

M x :

Bending stress couple

N x :

Direct stress resultant

N :

Shape function

nu, nw :

Number of constituent functions for u and w

ng :

Number of Gauss points

p :

External load

\(\overline{p}\) :

Excitation force

{q}:

Displacement field

T :

Kinetic energy of the system

t :

Time coordinate

u, w :

Displacement field in x and z-axis

U :

Strain energy stored in the system

V :

Potential energy of the external forces

δ :

Variational operator

ρ 0 :

Density of the plate material at (x=0)

ω :

Non-dimensional frequency parameter

α, ϕ :

Set of orthogonal functions for u and w

Ε :

Normal strain

θ :

Normal rotation

κ x, κ z :

Material gradient parameters

ξ :

Normalized axial coordinate

π :

Potential energy

λ :

Relaxation parameter

σ :

Normal stress

υ :

Poisson’s ratio

Ω :

Excitation frequency

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Authors and Affiliations

  1. Department of Mechanical Engineering, School of Engineering, University of Petroleum and Energy Studies, Dehradun, 248007, Uttarakhand, India

    S. Kumar

  2. Department of Mechanical Engineering, National Institute of Technology, Rourkela, 769008, Orissa, India

    H. Roy

  3. Department of Mechanical Engineering, Jadavpur University, Kolkata, 700032, India

    A. Mitra

  4. Department of Mechanical Engineering, Indian Institute of Technology, Delhi, 110016, India

    K. Ganguly

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  1. S. Kumar
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Kumar, S., Roy, H., Mitra, A. et al. Dynamic Analysis of Bi-directional Functionally Graded Beam with Geometric Nonlinearity. J. Vib. Eng. Technol. 12, 3051–3067 (2024). https://doi.org/10.1007/s42417-023-01032-1

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  • DOI: https://doi.org/10.1007/s42417-023-01032-1

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