Free vibration of axially loaded, rotating Timoshenko shaft systems by the wave-train closure principle
type: Item
instance of: scholarly article
title: Free vibration of axially loaded, rotating Timoshenko shaft systems by the wave-train closure principle
zbMATH Open document ID: 1064.74586
DOI: 10.1016/S0020-7683(98)00172-3
author: Bongsu Kang
author: Chin An Tan
published in: International Journal of Solids and Structures
publication date: 1999-01-01T00:00:00Z
review text: The free vibration analysis of a rotating Timoshenko shaft system with multiple discontinuities (elastic supports, rotor masses, cross sectional changes) is presented. The equation governing the transverse and rotational motions of a homogeneous interval is given as follows: \[ \frac{\partial^4u} {\partial z^4} -(1+\alpha) \frac{\partial^4u} {\partial z^2 \partial t^2} +2i\beta\left(\frac{\partial^3u} {\partial z^2\partial t}- \frac{\partial^3u} {\partial t^3}\right)+ \alpha\frac {\partial^4u} {\partial t^4}- d\frac{\partial^2u} {\partial z^2}+ \alpha d\frac {\partial^2u} {\partial t^2}=0, \tag{1} \] where \(u\) is the displacement at position \(z\) and time \(t\), the constants \(\alpha\), \(\beta\), \(d\) are expressed through the parameters of the shaft, the rotation frequency and the axial force. The equation (1) admits a solution of the form \(u(x,t)=C e^{i(\gamma z+\omega t)}\), where \[ \gamma(\omega)=\pm \frac{1} {\sqrt 2} \left(A(\omega)\pm \sqrt{A^2(\omega)-4B(\omega)}\right)^{\frac 12} \] are the solutions of the corresponding biquadratic characteristic equation. The authors claim that the discriminant \(A^2-4B\) is semi-positive definite for most engineering applications. Besides this assumption (for real \(\omega\)) they restrict their attention to the case of \(A>0\) and \(B<0\). Then the solution (of frequency \(\omega\)) is \[ u(z,t)= \left(C^+_1e^{-i\Gamma_1z}+C^-_1e^{i\Gamma_1z}+ C^+_2e^{-\Gamma_2z}+C^-_2e^{\Gamma_2z}\right) e^{i\omega t}, \] where \(\Gamma_j\) are real. The positive (negative) travelling wave is considered \[ {\mathbb C}^+e^{i\omega t}=\binom {C_1^+e^{-i\Gamma_1z}} {C_2^+ e^{-\Gamma_2z}} e^{i\omega t} {\mathbb C}^- e^{i\omega t}= \binom {C_1^-e^{i\Gamma_1z}} {C_2^-e^{\Gamma_2z} e^{i\omega t}}. \] These waves are related by \( {\mathbb C}^{-}= {\mathbf rC}^+, \) where \({\mathbf r}\) is the \(2\times 2\) reflection matrix. The authors express the elements of this matrix (and those of the reflection matrix) through the parameters of…
reviewed by: V. Pivovarchik
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 74K10
zbMATH DE Number: 1341455
zbMATH Keywords: rotating Timoshenko shaft
zbMATH Keywords: reflection matrix
zbMATH Keywords: transmission matrix
zbMATH Keywords: cross-section change
zbMATH Keywords: vibration frequency
zbMATH Keywords: cutoff frequency
MaRDI profile type: Publication
Wikidata QID: Q127840598
Recommended article: Free vibration characteristics of rotating Timoshenko beams
Recommended article: Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia
Recommended article: Free vibration of an extensible rotating inclined Timoshenko beam
Recommended article: Free vibration analysis of thin rotating cylindrical shells using wave propagation approach
Recommended article: On the problem of free torsional vibration of variable diameter circular shafts by the boundary integral equation method
Recommended article: Free Vibration Analysis of Rotating Axially Functionally Graded Tapered Timoshenko Beams
Recommended article: Modal analysis of rotating shafts: A body-fixed axis formulation approach
Recommended article: Transverse free vibration of non-uniform rotating Timoshenko beams with elastically clamped boundary conditions
description: scientific article; zbMATH DE number 1341455