Free vibration of axially loaded, rotating Timoshenko shaft systems by the wave-train closure principle
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 the problem in the cases of kinematic discontinuity of change of cross-section and in the presence of a boundary. The characteristic function is calculated numerically in some region of \(\omega\) for different cases.
scientific article; zbMATH DE number 1341455