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Primary Resonance of Nonlinear Spinning Timoshenko Shaft Based on a Novel Third-order Approximation Model Derived from Geometrically Exact Nonlinear Model

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Abstract

In this paper, a set of generalized nonlinear equations of motion for spinning Timoshenko shafts is derived using the concept of a geometrically exact approach. In order to investigate the primary resonance of the shaft, the multiple scale method is applied to the discrete equations of motion. In this study, the effects of shear deformation, rotary inertia, gyroscopic terms, and linear damping were considered. To show the advantages of the Timoshenko models, a comparison is made between the results of Timoshenko and classical models. As a result, it can be seen that in the Timoshenko model, the amplitude of the vibration is directly related to the slenderness ratio of the shaft. Also, linear and nonlinear shear terms can affect the primary resonance of spinning shafts and their effects are more noticeable in higher vibrational modes.

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Abbreviations

A:

Cross-sectional area

Ai, ai :

Unit vector in X1X2X3 and x1x2x3 frame

C:

Transformation matrix

c:

Damping coefficient

D, \(\overline{D}\) :

Location of any point along the deformed reference line and cross-section

D, \(\overline{d}\) :

Location of any point along the undeformed reference line and cross-section

E:

Elasticity modulus

G:

Shear modulus

Gi :

Contravariant base vector tangent to Ai

gi, gi :

Covariant and contravariant base vector tangent to ai

I:

Moment of inertia

K:

Curvature of shaft after deformation

Ks :

Shear correction factor

L:

Length of shaft

n:

Mode number

T:

Kinetic energy

U:

Strain energy

u:

Axial displacement

v, w:

Transverse displacements

\(\Delta\) :

Identity matrix

\(\tilde{\omega }_{XO}\) :

Angular velocity of deformed frame relative to spinning frame

( ~):

Cross product operator

\(\vec{\tilde{\Omega }}\) :

Angular velocity of spinning frame relative to inertia frame

\(\xi_{1}, \, \xi_{2}\) :

Eccentricity of unbalanced shaft

\(\Gamma\) :

Green Lagrange strain tensor

\(\chi\) :

Deformation gradient tensor

\(\gamma, \,\kappa\) :

Generalized strains

\(\theta, \,\psi\) :

Bending rotation

\(\rho\) :

Mass density

\(\sigma\) :

Detuning parameters of spin

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Mofatteh Avenue, P.O. Box 15719-14911, Tehran, Iran

    H. Beiranvand & S. A. A. Hosseini

Authors
  1. H. Beiranvand
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  2. S. A. A. Hosseini
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Corresponding author

Correspondence to S. A. A. Hosseini.

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The authors declare no competing interests.

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Appendix 1

Appendix 1

Euler angles:

The Euler rotation matrix can be introduced using three Eulerian angles around body axes. We call these three angles,\(\psi \left( {x,t} \right),\theta \left( {x,t} \right),\beta \left( {x,t} \right)\) respectively.

figure a

Finally, the Euler rotation matrix can be obtained as:

$$\begin{gathered} C = \left[ {\begin{array}{*{20}c} {{\text{c}} \left( \theta \right){\text{c}} \left( \psi \right)} & {{\text{c}} \left( \theta \right){\text{s}} \left( \psi \right)} \\ { - {\text{s}} \left( \psi \right){\text{c}} \left( \beta \right) + {\text{s}} \left( \beta \right){\text{s}} \left( \theta \right){\text{c}} \left( \psi \right)} & {{\text{c}} \left( \beta \right){\text{c}} \left( \psi \right) + {\text{s}} \left( \beta \right){\text{s}} \left( \theta \right){\text{s}} \left( \psi \right)} \\ {{\text{s}} \left( \beta \right){\text{s}} \left( \psi \right) + {\text{c}} \left( \psi \right){\text{c}} \left( \beta \right){\text{s}} \left( \theta \right)} & { - {\text{c}} \left( \psi \right){\text{s}} \left( \beta \right) + {\text{c}} \left( \beta \right){\text{s}} \left( \theta \right){\text{s}} \left( \psi \right)} \\ \end{array} } \right. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. {\begin{array}{*{20}c} { - {\text{s}} \left( \theta \right)} \\ {{\text{s}} \left( \beta \right){\text{c}} \left( \theta \right)} \\ {{\text{c}} \left( \theta \right){\text{c}} \left( \beta \right)} \\ \end{array} } \right] \hfill \\ \end{gathered}$$

where \(c\left( * \right) = \cos (*)\,\,,\,\,\,s\left( * \right) = \sin \left( * \right)\).

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Beiranvand, H., Hosseini, S.A.A. Primary Resonance of Nonlinear Spinning Timoshenko Shaft Based on a Novel Third-order Approximation Model Derived from Geometrically Exact Nonlinear Model. Iran J Sci Technol Trans Mech Eng (2024). https://doi.org/10.1007/s40997-024-00760-3

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