Dynamic response characteristics of multiplicative faults in a misaligned-bowed rotor-train system integrated with active magnetic bearings

Abstract

The present study deals with the modeling and response analysis of two coupled rotor systems that are subjected to the coupling angular misalignment accompanied by the shaft bow and integrated with an active magnetic bearing (AMB). The innovative aspect of the work lies in modeling a rotor-train system in the contemporaneous presence of the residual bow and the misalignment in the shaft, which has multiplicative effect. To study the responses, a fundamental model of the rotor-AMB system has been derived. The angular misalignment introduces an additional coupling stiffness (ACS) and changes the intact stiffness. An excitation function has been chosen to model the misalignment, which gives both odd and even frequency components in the full spectrum. In the misaligned system, the presence of bow generates a multiplicative load, which also affects the multi-harmonic responses and makes it more challenging to interpret. The AMB has been used to control vibrations caused by multiplicative faults, and its adoption represents a unique approach to reduce vibrations. The present control model modulates the supply current to the AMB via PID (proportional-integral-derivative) controller. The study numerically simulates the steady-state responses and analyses the harmonic components by examining the time-domain responses, orbit, and displacement spectra. Furthermore, the study is extended to examine the responses acquired from an experimental test rig constructed in the laboratory. The angularly misaligned and residual bowed shaft responses from the test rig were captured with the help of proximity probes. Finally, the proposed model is validated by comparing the pattern of the orbit plots and full spectra obtained from the numerical simulation and experiment and are found to be consistent.

Appendices

Appendix A: Deflection expression for the simply supported beam

Deflection of a simply supported beam with an offset load (Gere, [39]) at any given cross section can be written as;

$$ \begin{gathered} y = \frac{Pbx}{{6lEI}}\left( {l^{2} - x^{2} - b^{2} } \right){\text{ for }}0 < x < a \hfill \\ y = \frac{Pb}{{6lEI}}\left[ {\frac{l}{b}\left( {x - a} \right)^{3} + \left( {l^{2} - b^{2} } \right)x - x^{3} } \right]{\text{ for a}} < x < l \hfill \\ \end{gathered} $$

(A.1)

Now, the slopes at the bearing supports on the left and right sides of the beam can be expressed as:

$$ \varphi_{{c_{1} }} = \frac{{Pb\left( {l^{2} - b^{2} } \right)}}{6lEI}{;}\;\;\;\;\varphi_{{c_{1} }} = \frac{{Pa\left( {l^{2} - a^{2} } \right)}}{6lEI} $$

(A.2)

Herein, P represents the length of the shaft, l denotes the length of the shaft, E is the young’s modulus of the material, I refers to the polar mass moment of inertia, a and b represent the distances between the left and right support, while x stands for the distance of as specific cross section from the left support, respectively.

Appendix B: Routh–Hurwitz stability criteria

This criterion utilizes the characteristic equation of the system for the determination of the stable condition, according to Routh–Hurwitz criterion, if all the roots of the characteristic equation are in left half plane of the s-plane, then system is stable (Tiwari [23]).

If the characteristic equation of the system is

$$ a_{o} + a_{1} s + a_{2} s^{2} + a_{3} s^{3} + \ldots \ldots + a_{n} s^{n} = 0,a_{n} > 0 \, $$

(B.1)

If the following condition is fulfilled, then stable is stable;

$$ D_{1} = a_{n - 1} > 0{ ; }D_{2} = \left| {\begin{array}{*{20}c} {a_{n - 1} } & {a_{n} } \\ {a_{n - 3} } & {a_{n - 2} } \\ \end{array} } \right| > 0 $$

(B.2)

$$ D_{3} = \left| {\begin{array}{*{20}c} {a_{n - 1} } & {a_{n} } & \ldots & \ldots & 0 \\ {a_{n - 3} } & {a_{n - 2} } & \ldots & \ldots & \vdots \\ \vdots & \vdots & \ddots & \ldots & \vdots \\ \vdots & \vdots & \ldots & \ddots & \vdots \\ 0 & 0 & \ldots & \ldots & {a_{1} } \\ \end{array} } \right| > 0,D_{n} = a_{o} D_{n - 1} > 0 $$

(B.3)

To ensure the rotor-AMB system’s stability during its entire operation, the PID controller is tuned using the Routh–Hurwitz stability criteria, the stability conditions can be stated as follows:

$$ m > 0{;}\;\;\; \, K_{D} > 0{;}\;\;\; \, K_{I} > 0{;}\;\;\; \, K_{P} = \frac{1}{{k_{i} k_{x} k_{sn} }}\left( {\frac{{mK_{I} }}{{K_{D} }} + k_{s} } \right) $$

(B.4)

Appendix C: Transfer function of the proposed rotor-AMB system for Nyquist stability criteria

The closed-loop transfer function for the proposed rotor-AMB system, as shown in block diagram shown in Fig. 6, is given by

$$ G\left( s \right) = \frac{{G_{CMA} \left( s \right)}}{{1 + G_{CMA} \left( s \right)G_{sn} \left( s \right)}} $$

(C.1)

With $G_{CMA} \left( s \right) = G_{C} G_{M} G_{A}$, where, $G_{C}$ be the transfer function of the controller,$G_{M}$ be the transfer function of magnetic bearing,$G_{A}$ be the transfer function of amplifier gain $k_{x}$, $G_{sn}$ be the transfer function of the overall sensor correction gain $k_{sn}$(Tiwari [23]).

$$ G_{B} \left( s \right) = \frac{{k_{p} s + k_{I} + k_{D} s^{2} \, }}{s} $$

(C.2)

$$ G_{M} = \left( {\frac{{k_{{i_{2} }} }}{{\left( {m_{2} s^{2} + \left( {c_{2xx} + c_{h} } \right)s + \left( {k_{2xx} + k_{{s_{2} }} } \right) - {\text{j}}\omega c_{h} } \right)}}} \right){;}\;\;\; \, G_{M} = k_{x} {;}\;\;\;G_{sn} = k_{sn} $$

(C.3)

System’s overall transfer function is given by:

$$ G\left( s \right) = \frac{{\frac{{\left( {k_{P} s + k_{I} + k_{D} s^{2} } \right)k_{{i_{2} }} k_{s} }}{{\left( {m_{2} s^{3} + \left( {c_{2xx} + c_{h} } \right)s^{2} + \left( {k_{2xx} + k_{{s_{2} }} - {\text{j}}\omega c_{h} } \right)s} \right)}}}}{{\frac{{\left( {m_{2} s^{3} + \left( {c_{2xx} + c_{h} } \right)s^{2} + \left( {k_{2xx} + k_{{s_{2} }} - {\text{j}}\omega c_{h} } \right)s} \right) + \left( {k_{p} s + k_{I} + k_{D} s^{2} } \right)k_{{i_{2} }} k_{s} k_{sn} }}{{\left( {m_{2} s^{3} + \left( {c_{2xx} + c_{h} } \right)s^{2} + \left( {k_{2xx} + k_{{s_{2} }} - {\text{j}}\omega c_{h} } \right)s} \right)}}}} $$

(C.4)

$$ G\left( s \right) = \frac{{\left( {k_{D} s^{2} + k_{P} s + k_{I} } \right)k_{{i_{2} }} k_{x} }}{{m_{2} s^{3} + \left( {c_{2xx} + c_{h} + k_{D} k_{{i_{2} }} k_{x} k_{sn} } \right)s^{2} + \left( {k_{2xx} + k_{{s_{2} }} - {\text{j}}\omega c_{h} + k_{P} k_{{i_{2} }} k_{x} k_{sn} } \right)s + k_{I} k_{{i_{2} }} k_{x} k_{sn} }} $$

(C.5)

The denominator of Equation represents the characteristic equation of the proposed rotor model and can be expressed as;

$$ m_{2} s^{3} + \left( {\left( {c_{2xx} + c_{h} } \right) + \left( {k_{D} k_{{i_{2} }} k_{x} k_{sn} } \right)} \right)s^{2} + \left( {k_{2xx} + k_{{s_{2} }} - {\text{j}}\omega c_{h} + \left( {k_{p} k_{{i_{2} }} k_{x} k_{sn} } \right)} \right)s + k_{I} k_{{i_{2} }} k_{x} k_{sn} $$

(C.6)