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Nonlinear Static Analysis and Superharmonic Influence on Nonlinear Forced Vibration of Timoshenko Beams

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Advances in Materials, Mechanical and Industrial Engineering (INCOM 2018)

Part of the book series: Lecture Notes on Multidisciplinary Industrial Engineering ((LNMUINEN))

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Abstract

In the present work, static analysis and subsequently superharmonic influence on the nonlinear dynamic behavior of externally excited thick beams are investigated. Energy equations are derived considering Timoshenko beam theory. For the static analysis, classical Ritz method is followed. Nonlinear load–deflection response is obtained considering various geometric parameters such as length-to-depth ratio and load application points. For the vibration analysis, differential equations are obtained considering the Lagrange’s equation. Subsequently, harmonic balance method is employed for multi-DOF systems, which reduce the differential equations into nonlinear set of algebraic equation. These equations are tackled by enforcing an iterative scheme based on modified direct substitution method. Simple harmonic assumption although provides a very good prediction for small amplitude vibration problem. However, it is inadequate for the system having large amplitude vibration. It is shown that for accurate solution higher-order harmonics must be considered.

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Abbreviations

A j :

Temporal coordinates

B j :

Temporal coordinates

b :

Width of beam

C j :

Temporal coordinates

E :

Young’s modulus

F :

Amplitude of loading

h :

Depth of beam

K1, K2, K3, K4:

Stiffness parameters

k3, k4:

Stiffness parameter (dimensionless)

L :

Length of beam

M1, M2, M3:

Inertial parameters

m 3 :

Inertial parameter (dimensionless)

N :

Number of polynomial terms

T * :

Kinetic energy

T ** :

Kinetic energy (dimensionless)

U :

Longitudinal displacement

U * :

Potential energy

U ** :

Potential energy (dimensionless)

u :

Axial displacement (dimensionless)

W :

Transverse displacement

w :

Transverse displacement (dimensionless)

x f :

Load application point

ν :

Poisson’s ratio (dimensionless)

ξ :

Normalized axial coordinate

ξ f :

Xf/L

ρ :

Mass density

Ф:

Polynomial functions

Ψ:

Rotational displacement

ψ :

Normalized rotational displacement

Ω:

Frequency of excitation

ω :

Normalized frequency

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Authors and Affiliations

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

    Brajesh Panigrahi & Goutam Pohit

Authors
  1. Brajesh Panigrahi
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  2. Goutam Pohit
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Corresponding author

Correspondence to Brajesh Panigrahi .

Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, Jadavpur University, Kolkata, West Bengal, India

    Prasanta Sahoo

  2. Department of Mechanical Engineering, University of Aveiro , Aveiro, Portugal

    J. Paulo Davim

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Panigrahi, B., Pohit, G. (2019). Nonlinear Static Analysis and Superharmonic Influence on Nonlinear Forced Vibration of Timoshenko Beams. In: Sahoo, P., Davim, J. (eds) Advances in Materials, Mechanical and Industrial Engineering. INCOM 2018. Lecture Notes on Multidisciplinary Industrial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-96968-8_19

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  • DOI: https://doi.org/10.1007/978-3-319-96968-8_19

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  • Online ISBN: 978-3-319-96968-8

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