Dual flexible rotor system with active magnetic bearings for unbalance and coupling misalignment faults analysis

Abstract

Rotating machines are the backbone of the present industrial world. Early fault detection and conditioning of these machines are primary concern of the researchers associated in this field. There are various faults (assembly error, coupling misalignment, looseness, imbalance, rotor crack, etc.) that cause malfunction of rotating machinery. Imbalance is one of the oldest problem and still challenging to perfectly balance the rotor. Imbalance leads to another inherent fault, i.e., coupling misalignment, especially in dual rotor or rotor train system. Imbalance and misalignment cause excessive vibration in the system that tends to shatter failure of the critical components of rotating machinery. In this article, active magnetic bearings (AMBs) are utilized to suppress the excessive vibration generated due to imbalance and misalignment. To regulate the controlling current of AMB a proportional integral derivative (PID) feedback controller is employed. A quantification technique is suggested to evaluate the tuned AMB characteristics along with imbalance and coupling misalignment dynamic parameters. A finite element method (FEM) modelling with high-frequency reduction scheme is utilized to acquire reduced system equations of motion. There are two advantages of employing condensation scheme, first, it reduces the number of sensors required and second, only linear (practically measurable) degrees of freedom are present in equations of motion derived. A SIMULINK™ code is prepared to solve a reduced linear differential equation. The time series feedback signals (current and displacement) obtained are transformed into a frequency series utilizing Fast Fourier Transformation (FFT) and utilized in developed algorithm. To establish the accuracy and effectiveness of the methodology, the estimated parameters are evaluated under two different frequency bands against measurement and modelling error (5% variation in mass of the disc and bearing characteristic parameters).

Appendix A

The peak (fr+1, 1 and −fr) acquired from figure 9, are utilized to develop the algorithm as,

$$ \varvec{A}(\omega_{\varvec{i}} )\chi = \varvec{B}(\omega_{\varvec{i}} ) $$

(A.1)

with,

$$ \varvec{A}(\omega_{\varvec{i}} ) = [\varvec{A}_{\varvec{U}} (\omega_{\varvec{i}} ) { }\varvec{A}_{{\varvec{AMB}}} (\omega_{\varvec{i}} ) { }\varvec{A}_{\varvec{C}} (\omega_{\varvec{i}} )] $$

(A.2)

Regression matrix due to residual unbalance $\varvec{A}_{\varvec{U}} (\omega_{\varvec{i}} ) { }$

$$ \varvec{A}_{\varvec{U}} (\omega_{\varvec{i}} ) { = }\left[ {\begin{array}{*{20}c} { - \varvec{m}^{{\varvec{d}_{1} }} \omega^{2} \lambda_{{\varvec{U}_{1} }} } & 0 & { - \varvec{m}^{{\varvec{d}_{2} }} \omega^{2} \lambda_{{\varvec{U}_{2} }} } & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & { - \varvec{m}^{{\varvec{d}_{1} }} \omega^{2} \lambda_{{\varvec{U}_{1} }} } & 0 & { - \varvec{m}^{{\varvec{d}_{2} }} \omega^{2} \lambda_{{\varvec{U}_{2} }} } \\ \vdots & \vdots & \vdots & \vdots \\ \end{array} } \right] $$

Regression matrix due to active magnetic bearing $A_{AMB} (\omega_{i} ) { }$

$$ \varvec{A}_{{\varvec{AMB}}} (\omega_{\varvec{i}} ) { = }\left[ {\begin{array}{*{20}c} {\varvec{A}_{{\varvec{AMB}_{\varvec{x}} }} } & {\varvec{A}_{{\varvec{AMB}_{\varvec{y}} }} } \\ \end{array} } \right] $$

(A.3)

$$ \begin{aligned} A_{{AMB_{x} }} \, & { = }\left[ {\begin{array}{*{20}c} {re(\bar{X}_{i}^\circ \lambda_{{AMB_{x1} }} )} & {re(\bar{I}_{i} \lambda_{{AMB_{x1} }} )} & {re(\bar{X}_{i}^\circ \lambda_{{AMB_{x2} }} )} & {re(\bar{X}_{i} \lambda_{{AMB_{x2} }} )} \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots \\ \end{array} } \right] \\ A_{{AMB_{y} }} \, & { = }\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots \\ {im(\bar{X}_{i}^\circ \lambda_{{AMB_{y1} }} )} & {im(\bar{I}_{i} \lambda_{{AMB_{y1} }} )} & {im(\bar{X}_{i}^\circ \lambda_{{AMB_{y2} }} )} & {im(\bar{X}_{i} \lambda_{{AMB_{y2} }} )} \\ \vdots & \vdots & \vdots & \vdots \\ \end{array} } \right] \\ \end{aligned} $$

Regression matrix due to coupling stiffness and damping $A_{C} (\omega_{i} ) { }$

$$ A_{C} (\omega_{i} ) { } = [\begin{array}{*{20}c} {A_{{C_{k} }} (\omega_{i} )} & {A_{{C_{c} }} (\omega_{i} )} \\ \end{array} ] $$

(A.4)

$$ \begin{aligned} A_{{c_{x} }} & { = }\left[ {\begin{array}{*{20}c} {re(\bar{X}_{i} \lambda_{xx}^{{c_{k} }} )} & {re(\bar{X}_{i} \lambda_{xy}^{{c_{k} }} )} & {re(\bar{X}_{i} \lambda_{yx}^{{c_{k} }} )} & {re(\bar{X}_{i} \lambda_{yy}^{{c_{k} }} )} & {re(\bar{X}_{i} \lambda_{{\varphi_{x} \varphi_{x} }}^{{c_{k} }} )} & {re(\bar{X}_{i} \lambda_{{\varphi_{y} \varphi_{y} }}^{{c_{k} }} )} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {im(\bar{X}_{i} \lambda_{xx}^{{c_{k} }} )} & {im(\bar{X}_{i} \lambda_{xx}^{{c_{k} }} )} & {im(\bar{X}_{i} \lambda_{yx}^{{c_{k} }} )} & {im(\bar{X}_{i} \lambda_{yy}^{{c_{k} }} )} & {im(\bar{X}_{i} \lambda_{{\varphi_{x} \varphi_{x} }}^{{c_{k} }} )} & {im(\bar{X}_{i} \lambda_{{\varphi_{y} \varphi_{y} }}^{{c_{k} }} )} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \end{array} } \right] \\ A_{{C_{c} }} & = \left[ {\begin{array}{*{20}c} {im\left( {\bar{X}_{i} \lambda_{xx}^{{c_{c} }} } \right)} & {im\left( {\bar{X}_{i} \lambda_{xy}^{{c_{c} }} } \right)} & {im\left( {\bar{X}_{i} \lambda_{yx}^{{c_{c} }} } \right)} & {im\left( {\bar{X}_{i} \lambda_{yx}^{{c_{c} }} } \right)} \\ \vdots & \vdots & \vdots & \vdots \\ {re\left( {\bar{X}_{i} \lambda_{xx}^{c} } \right)} & {re\left( {\bar{X}_{i} \lambda_{xy}^{{c_{c} }} } \right)} & {re\left( {\bar{X}_{i} \lambda_{yx}^{{c_{c} }} } \right)} & {re\left( {\bar{X}_{i} \lambda_{yy}^{{c_{c} }} } \right)} \\ \vdots & \vdots & \vdots & \vdots \\ \end{array} } \right] \\ \end{aligned} $$

$$ B(\omega_{i} ) = \left[ {\begin{array}{*{20}c} {\omega^{2} M^{H(D + S)} re\left( {\bar{X}_{i} } \right) - K^{H(B + S)} re\left( {\bar{\eta }_{i} } \right) + \omega C^{H(B + S)} im\left( {\bar{\eta }_{i} } \right) + \omega^{2} G^{H(D)} im\left( {\bar{\eta }_{i} } \right)} \\ \vdots \\ {} \\ {\omega^{2} M^{H(D + S)} im\left( {\bar{X}_{i} } \right) - K^{H(B + S)} im\left( {\bar{\eta }_{i} } \right) + \omega C^{H(B + S)} re\left( {\bar{\eta }_{i} } \right) + \omega^{2} G^{H(D)} re\left( {\bar{\eta }_{i} } \right)} \\ \vdots \\ {} \\ \end{array} } \right] $$

(A.5)

Here $\bar{X}_{i}$ and $\bar{I}_{i}$ (i = fr+1, 0, −fr) are the displacement and current signals in frequency domain. $\lambda_{{U_{i} }}$, $\lambda_{{AMB_{xi} }} , \, \lambda_{{AMB_{yi} }} , \, {}^{o}\lambda_{{AMB_{xi} }} , \, {}^{o}\lambda_{{AMB_{yi} }}$(i = 1, 2) and $\lambda_{xx}^{{C_{k} }} , \, \lambda_{xy}^{{C_{k} }} , \, \lambda_{yx}^{{C_{k} }} , \, \lambda_{yy}^{{C_{k} }} , \, \lambda_{{\varphi_{x} \varphi_{x} }}^{{C_{k} }} , \, \lambda_{{\varphi_{y} \varphi_{y} }}^{{C_{k} }} ,\lambda_{xx}^{{C_{c} }} , \, \lambda_{xy}^{{C_{c} }} , \, \lambda_{yx}^{{C_{c} }} , \, \lambda_{yy}^{{C_{c} }}$ are the flag location corresponding to unbalance, AMB (displacement and current) and coupling stiffness and damping, respectively.