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

  • Brajesh PanigrahiEmail author
  • Goutam Pohit
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)

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.

Keywords

Nonlinear static behavior Large amplitude vibration Harmonic balance method Modified direct substitution technique 

Nomenclature

Aj

Temporal coordinates

Bj

Temporal coordinates

b

Width of beam

Cj

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

m3

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)

xf

Load application point

Greek Symbols

ν

Poisson’s ratio (dimensionless)

ξ

Normalized axial coordinate

ξf

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia

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