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Potential of Stator Vibration Response for the Diagnosis of Rub in a Rotor System

M. A. Mokhtar1*, A. K. Darpe2, Kshitij Gupta2

1Darbhanga College of Engineering, Darbhanga Mabbi, Lal Sahpur, Darbhanga – 846005, Bihar

2Mechanical Engineering Department, Indian Institute of Technology Delhi Hauz Khas, New Delhi – 110016, India

*Corresponding author: drasjadmokhtar@gmail.com Abstract

The rotating machines are more susceptible to failures due extreme set of working environment to meet the requirements of today’s competitive world hence should be investigated in details. Previous rotor-stator rub model studied only the dynamics of the rotor ignoring the fact that the stator is structural connectivity and con influence the dynamics of rotor. The present work studies A coupled rotor-stator model is used for the investigation where the rotor vibrations are transmitted to the stator through the bearing support in no rub situations. Therefore, the stator in the present model is dynamic and offers more realistic contact constraints as compared with earlier models for rub investigations. The stator has its own vibratory motion which influences the nature of interaction due to instantaneous clearance between the rotor and the stator. The cepstrum plots with respect to the data point indicate the presence of sub-harmonic frequency in the response characteristics that compliments the proposal that the stator vibration response data can be used as a potential source for the diagnosis of rub. Keywords: Rotor-stator coupled model, Stator vibration, Cepstrum, Rubbing, Lagrange Multiplier Method

1. Introduction

A rotating machine consists of rotor surrounded by non-rotating structural components known as stator.

The rotor is supported on bearings and the bearing housing is structurally connected to the stator. Fault diagnosis is an important aspect to avoid untimely failures on these machines during operation. The efficiency of these machines can be improved by closing the gap between the rotor and the stator. However, the reduced clearance increases the chances of rotor coming into contact with the stators at critical speeds.

Recent researches, while simulating the rotor-stator contact interaction, have modelled rotor system that allows incorporation of dynamic behaviour of stator through mathematical formulation. It includes the stator as a part of the rotor system and ensures dynamic stator contact boundary. In the simulation model, the stator is coupled through contact condition during rotor-stator interaction and independent when there is no contact.

ISME Journal of Mechanics and Design, Special Issue

Vol. 4, No. 2, 2020 pp. 18-26

(ISSN 2582-5690)

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In actual, the stator is structurally connected to the rotor through the support structure or bearing housing and the rotor vibrational energy is transferred to the stator even without direct contact. The present work explores this aspect of a rotating machine where investigations on the interaction between rotor and stator coupled system are carried out.

Past literatures have extensively studied on vibrational characteristics of rotor under rotor-stator contact. Rub diagnostic features such as sub-harmonics and super-harmonics are explored through experimental and theoretical work. [1,2]. Presence of 2nd and 3rd harmonics of the synchronous frequency is related with rub and further advised to look into the entire spectrum for more accurate interpretations. In a wind mill imbalance problem, it is observed that the super-harmonics upto order-9 and sub harmonics upto order-32 is present in the signal [3]. Literature reviews [4–6] on the rub phenomenon suggests to develop mathematical models more relevant with actual turbo-machineries and identify parameters that affects the dynamic characteristics of it [5]. With the advanced computational proficiencies, it is feasible to develop rub models that are more interactive and give reliable solutions.

The directional nature of rub and the resonance shift phenomenon are explored in order to distinguish rub from various other rotor faults such as misalignments, cracks etc. [7]. Assessment of rotor vibration signal became essential for design of efficient rotating machines and therefore turns out to be an integral part of the research and development. Various signal processing techniques such as Hilbert-Huang transform (HHT), full spectrum and empirical mode decomposition (EMD) are incorporated in the investigations of rub phenomenon [7–9].

Researchers have started looking for modelling rotor-stator interactions in such a manner that simulates the actual situation more closely. One such method adopted is the computational contact mechanics where the contact interaction between rotor and stator is captured in details. The advantage of this technique is that it allows the stator to interact dynamically with the rotor during contact. Investigation of rotor stator contact is carried out using Lagrange multiplier method and augmented Lagrange method [10–13].

These approaches make way to investigate the vibrational response of stator during its interaction with the rotor and open up the possibilities of rub diagnosis. Initially, most of the rub investigation models consider stator and rotor as two separate parts in the system although both are structurally connected in real cases. Childs [14] in his study on HPTP (high pressure fuel turbopump) has incorporated this fact while deriving the governing equations of motion for turbo pumps and studied the instability in the system. In real rotor problems, the sensor placements for vibration measurements on the rotor have always been challenging and cost adding exercise while placements of measurement sensors on housing or stators are very easy. It would be a great contribution in the rub analysis, if the stator vibration data can be used to extract features indicative of the rub phenomenon.

The present study is an extension of previous work [11] where a finite element based coupled rotor-stator model is created where the stator is structurally connected with the rotor through bearing housing. The rotor vibration transmits to stator through these structural connections or through direct contact during rub. The stator interacts with the rotor with its own vibratory motion that decides the nature of interaction. The earlier reported work [11] establishes a strong correlation between vibration responses of the rotor and the stator during rub. The present work supplements to the conclusions drawn [11] and hence can be further suggested to be used for extracting rub specific features.

2. Rotor-stator coupled system modelling

The rotor-bearing-stator (RBS) system, already explained in the authors’ previous work [11] which comprises of a shaft with a disc located at the centre and is supported between two bearings at the end. The

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shaft is flexible and the stator structure is connected to the bearing housing as shown in Fig. 1. The geometric and material properties of the rotor-bearing-stator system are shown in the table 1 below.

Table 1: Geometric and material properties of the rotor-bearing-stator system

Parameter Value

Length of the rotor between two support bearings 80 cm

Diameter of the rotor shaft 2 cm

Length of the stator beam along rotor length 80 cm

Area of cross section for stator beam 19.63 mm2

Density (rotor shaft, stator beam) 7800 kg-m-3

Elastic Modulus (rotor shaft, stator beam) 210 GPa

Shear Modulus (rotor shaft, stator beam) 79.6 GPa

Stiffness of the bearings at both the end 1×107 N/m2

Stiffness of the bearing housing 1×109 N/m2

The friction coefficient between stator and rotor 0.3

The rotor shaft is discretized into 14 finite beam elements. Each element consists of 3-translational and 3-

rotational degrees of freedom at each node. The stiffness of the bearing housing is considered to be high which is modeled as spring elements connected between bearing and the stator structure at the other end. The stator beam is discretised with 8 beam elements and connected with the bearing housing shown in Fig. 1b. The beam element for stator, at each node, consists of 2-translational (y, z) and 2-rotational (θy, θz) degrees of freedom. The governing equations of motion for the rotor-bearing-stator assembly is given by

(1)

The mass, damping and stiffness of the complete rotor-bearing-stator (RBS) system are represented by matrices M, C and K respectively. The gyroscopic effect (ωG) is included using gyroscopic matrix G, where ω is the rotational speed. Generalized force and displacement are represented by f and u vectors. Damping matrix D is evaluated using the concept of proportional damping (Rayleigh damping) with modal damping ratio assumed to be 0.007 and 0.05. The clearance ratio, cg = g/max(X(ω)) is defined as ratio of gap between rotor and stator (g) and the maximum allowable rotor deflection (X(ω)) in absence of the stator. The clearance ratio is a non-dimensional quantity and should be less than 1 for the rotor-stator interaction.

Fig. 1. The coupled rotor-bearing-stator (RBS) model: (a) Schematic diagram, (b) Finite Element Representation.

(a)

(b)

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2.1 Rotor-stator contact model The system natural frequencies evaluated on the coupled RBS system shown in Fig. 1 are enlisted in Table

2. The mode shapes corresponding to each natural frequency are shown in Fig. 2. The bending mode frequency of 32.64 Hz indicates predominant rotor flexure while the bending mode frequency of 106 Hz exhibits predominantly stator defection.

In the present study, the stator stiffness is defined with respect to the rotor stiffness and the influence of stator stiffness on the dynamics is observed. A parameter stiffness ratio, β is defined as follows.

s

ref

stator stiffness k

reference stiffness k (2)

The present simulations use the values of β as 1.0, 1.5 and 2.0 however the rest of the system parameters are not varied. Eigen value analysis used in the present work is discussed in detail in the previous work [11]. The mode shapes are redrawn as shown in Figure 2. The present study uses the rotor stator contact model which is based on Lagrange multiplier method explained in [15,16] and implemented in [10-13].

Table 2: Eigen values of the coupled rotor bearing stator (RBS) model for β = 1.0 [11]

Mode no. Frequency Comments

1st & 2nd 32.64 Hz Bending mode of rotor in horizontal and vertical plane

3rd & 4th 106.15 Hz Bending mode of stator in horizontal and vertical plane

5th 128.05 Hz rotor torsional

6th & 7th 198.84 Hz Rotor bending mode

8th & 9th 258.19 Hz Rotor bending mode

10th & 11th 382.8 Hz Stator bending mode

Fig. 2. Mode shapes of the coupled rotor-bearing-stator system (a) in y-direction, (b) in z-direction, (c) 3D view of superimposed mode shapes

Nodal position Displacement, z-direction

Displacement, y-direction

(a) (b)

(c)

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Using the concepts of variational mechanics, principle of virtual work and the Lagrange multiplier method [16], the governing equations of motion are obtained that is expressed as

(3)

where, GN is the normal contact constraint matrix. The normal-tangential-contact constraint matrix is represented by GNT (= GN + µGT). The nodal displacement, velocity and acceleration vectors are represented by u, u and u respectively. The radius of the disc is rd. Equation 3 is the governing equations of motion under contact constraint for the RBS system that will be solved by explicit numerical integration scheme. The solution procedure adopted is central difference scheme with forward increment Lagrange multiplier method [15]. The algorithm flow chart is shown in Fig. 3 that describe the procedure adopted for enforcement of contact constraint and solution for the rotor response.

Fig. 3. Algorithm for the response evaluation during rotor-stator contact.

3. Results and Discussions

The rotor stator contact phenomenon is investigated with the clearance ratio, cg = 0.7 of the rotor system

explained in Sec. 2. The speed ratio is varied from 0.5 to 7.0 with a uniform increment of 0.1. The value of cg = 0.7 indicates that the gap maintained between the rotor and stator is 70% of the maximum allowable rotor deflection in absence of the stator. The compliant stator, during its contact with the rotor provides additional support and hence additional stiffness to the rotor which increases its effective bending natural frequency generally referred as pseudo critical speed. At higher rotor spin speed of r = 2.9, 4.5 and 6.7 the sub synchronous

components also generate pseudo-resonance in the coupled rotor-bearing-stator system as

shown in Fig 4.

TNT N

0N - 0; 0; 0 1,.....i i

d N N Nr g for i q

Mu+Cu+Ku+G λ = f

G u+X

11 1 1 1

*

1 1

Tn n n nN N NT N

Tn n

c NT N

G G g u

u G

M

= M

1 1*

n ncu u u

Yes

No

1 1 12 3 4

X, X and X

t > specified

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Fig. 4. Spectrum cascade of rotor vibration response in vertical direction, r = 0.5 to 6.9.

Fig. 5 Spectrum cascade for vibration response of stator in vertical direction under rub.

The vibrations measured from casing/stator housing are not considered to be a direct measurement, although it could be a preferred choice for sensor placement. On the contrary, the present work, in continuation with the previously reported work [11], explores the potential of stator vibration response for the diagnosis of rub using a signal processing techniques. Stator vibration characteristics during rub are obtained for rotor speed ratio increasing from r = 0 to 7.0. Figure 5 depicts the spectrum cascade for vibration response of stator in vertical direction under rub. Comparing the rotor spectrum cascade (Fig. 4) with the stator vibration obtained during the coast-up operation (Fig. 5), it is observed that the stator vibration exhibits harmonics and sub-harmonics of rotational frequency similar to the rotor spectrum cascade shown in Fig 4. The pseudo resonance frequency is also observed in both rotor and stator vibratory response. The frequencies appearing in the stator spectrum has dominating presence near the stator natural frequency (106.2Hz) whereas in the rotor spectrum the larger peaks are observed close to the rotor’s pseudo resonance frequency (49Hz). Therefore, it is suggested to use the stator housing vibrational response features in the diagnosis of rub. The measurement of stator housing vibration is much easier as compared to it with the measurements from rotors directly.

In the past studies on rotor-stator contact the stator model is considered as a stationary element that applies reaction force on the rotor equal to the product of the stator stiffness and the theoretical compression value of the spring element. The present work allows the stator to vibrate even without rotor contact due to transmission of rotor vibrations through the bearing support structure. Hence the stator interacts with the rotor during rub through its own vibratory motion. The gap between rotor and stator thus depends on the instantaneous position of both the rotor and the stator and hence the influence the interaction between them. The present work thus captures the overall contact dynamics more appropriately and realistically

Frequency, Hz

Speed ratio, r 1X

1/2X

1/3X

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3.1 Cepstrum Analysis

In continuation to the previous work the rotor-stator contact model [11], the addition of cepstrum analysis

on stator vibration response further establishes and build confidence in the idea that the stator vibration can be usefull in diagnosing and anlysisng the phenomenon of rub. The stator vibration response using cepstrum analysis, explained here in the present work, indicates its importance in the rub diagnosis. A cepstrum procedure is an important application in machine diagnostics to identify the modulating frequencies. It is evaluated by taking the inverse Fourier transform of the logarithm of the estimated spectrum of the signal. Sometimes, it is the power spectrum of a logarithmic power spectrum. In the present analysis, the cepstrum is defined as

1[log( ( ( ))]xC f f x t (4)

where f and f -1 represent Fourier and inverse Fourier transform and x(t) is the time domain data for which the cepstrum procedure is applied. The cepstrum obtained for the cases with β = 1 and β = 1.5 for specified speed ratio is shown in Figure 6 that indicates the rotational frequencies and periodicity of the response characteristics in terms of data points Nr. The gap, in terms of data points, is represented by the parameter Nr and is defined as

(Hz)

(Hz)

Sampling rateNr

Rotational speed (5)

Speed ratio r = 1.5 in both the cases correspond to first order periodic response and their cepstrum clearly depicts the rotational frequencies (Figure 6a, b). For β = 1 and r = 3.3, the cepstrum analysis shown in Figure 6c indicates the 1∕2X frequency as the operating speed lies in the range of period-2 responses that is r = 2.8 to 4.1 (see Figure 8b in [11]). Period-2 response for β = 1.5, r = 2.7 and period-3 response for r = 4.5, β = 1 and r = 4.7, β = 1.5 is observed in the cesptrum analysis plot shown in Figure 6d-f respectively.

Fig. 6 Cepstrum plots for stiffness ratio, β = 1; (a) r = 1.5, (c) r = 3.3, (e) r = 4.5, and β = 1.5; (b) r = 1.5, (d) r =

2.7, (f) r = 4.7.

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The cepstrum plots with respect to the data point Nr indicate the presence of sub-harmonic frequency in the response characteristics that compliments the proposal that the stator vibration response data can be used as a potential source for the diagnosis of rub. The correlation of stator response with rotor response is consistently appearing in the bifurcation plots, spectrum cascade, cross-correlation coefficient plots [11] and cepstrum analysis (present work).

The main emphasis of the present study is divided in three aspects. (a) The structural connection between the stator and the rotor through bearing housing. This is practically fine even if the stator model is not realistic in its shape. (b) The effect or influence of a dynamically modeled stator where the clearance between the rotor and the stator depends on both rotor and stator vibratory responses. (c) A third aspect may be the fact that the present analysis is focused more on establishing the correlation between rotor and stator vibration response.

There is no doubt about it that a cylindrical shell model for stator will be more realistic however, for a single point contact or partial rub cases, the present rub model can find its place in the research contributions related to rub diagnosis. The beam model is relevant enough as long as the study is focused on single point rubbing which is a highly investigated case in rotordynamics. The present study opens up many research directions such as modeling of more realistic stator, having its structural connection, diagnosis through stator vibration measurements, precise investigations about the complicated response characteristics, etc.

4. Conclusions

The present work, in continuation to the author’s previous work [11], uses Lagrange Multiplier Method and explores the potentials of stator vibration response data for the identification of rub. The measurement of stator vibration response is very easy as compared with the complexity of measuring rotor vibration response in real machines. In order to ensure a realistic rotor-stator model, structural connectivity between rotor and stator is ensured. The cepstrum analysis on stator vibration data show coupling with rotor vibration data with the presence of rotational frequency and the order of periodicity in the response due to rub interaction. In authors’ earlier reported work correlation between rotor and stator vibrational features were established. The results discussed in this work further establish that the vibration signals captured from stator useful enough for rub identification. Hence, the present investigation proposes to include the stator vibration response data for the identification of rotor-stator contact.

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