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Rotordynamic stability is crucial for high performance centrifugal compressors. In this paper, the weighted instrumental variable (WIV) based system identification method for rotating machinery stability is investigated based on a sine sweep forward excitation with an electromagnetic actuator. The traditional multiple input multiple output (MIMO) frequency response function (FRF) is transformed into a directional frequency response function (dFRF). The rational polynomial method (RPM) combined with WIV is developed to identify the rotor’s first forward mode parameters. This new approach is called the COMDYN method. Experimental work using the COMDYN method is carried out under different rotating speeds, oil inlet temperatures, and pressure conditions. Two sets of bearings with preloads 0.1 and 0.3 are investigated. A numerical rotor-bearing model is also built. The numerical results correlate reasonably well with the experimental results. The investigation results indicate that the new method satisfies the desired features of rotating machine stability identification. Furthermore, the system log decrement was improved somewhat with the increase of oil inlet temperature. The increase of oil supply pressure affects the rotor-bearing system stability very slightly. The results of this paper provide new and useful insights for potentially avoiding instability faults in centrifugal compressors.

Assuring the rotordynamic stability is of crucial importance for high performance centrifugal compressors. Improved rotordynamic stability assurance will be welcomed by both end users and centrifugal compressor manufacturers. The stability estimation method developed in this work combines complex dFRF, RPM, and WIV and is called the COMDYN method. It contributes to meeting the objective of better stability identification. Over the years, rotordynamicists have successfully followed the dual approach of excitation source reduction and vibration absorption through increased effective damping [

Large numbers of papers have focused on the rotordynamic stability, by both numerical and experimental work, on the component and system levels. On the component level, Childs [

On a system level, Baumann [

The instrumental variable (IV) method, with the reputation of unbiasedness as an estimator, was first introduced into the economic parameters estimation. Later, Kendall and Stuart [

In this paper, the traditional MIMO FRF matrix is transformed into the dFRF matrix from the real number field to the complex field using a transformation matrix. This eliminates or depresses the influence of forward and backward modal superposition. The theoretical basis for this complex dFRF method is provided in Lee [

In the experimental tests, two sets of bearings with preloads 0.1 and 0.3 are tested for their damping effect, as measured by the system log decrement, on the rotor-bearing system. This work also compares rotordynamic stability under different bearing loading configurations (LBP/LOP) both experimentally and theoretically. A numerical model was built to simulate the log decrement of the rotor-bearing system. The numerical results correlate with the experimental results reasonably well. Also, the effect of rotating speed as well as bearing lubrication oil pressure and temperature on the stability is investigated by both experimental and numerical methods. The result indicates that as the rotating speed increases, the log decrement decreases, approximately linearly, as the rotating speed becomes larger than the first natural frequency. It also shows that the system log decrement increases slightly as the supply temperature of the bearing lubricating oil increases. Although the effect of oil inlet pressure also has this tendency, the degree is very slight again. The LOP configuration provides more damping to the rotor system and the oil inlet temperature affects the log decrement in this configuration somewhat more than the LBP configuration. The results of this paper provide some new insights for potentially avoiding instability faults in centrifugal compressors.

Figure

Multidegrees of freedom of rotor-bearing system.

The number of measurement locations for most rotor systems is limited, which results in a limited number of rotor responses that can be used to characterize the system. Equation (

Usually, the PEM for estimating the FRF is first used to give the error function, including the parameters to be identified. The PEM is then used to obtain the minimum conditions of the 2-norm of error vector, which is the familiar OLS techniques. However, the OLS is a biased estimator. This paper applies an unbiased estimator, the WIV method, to estimate the parameters, which yields a more accurate estimate, even in high noise circumstances.

One way to obtain the error function involves using the RPM to establish the FRF model. For the FRF in (

After establishing the model, the error function can be written as

Substituting (

Here the footnote

Because the parameter vector

Traditionally, the parameters in (

To obtain an unbiased estimation, the IV matrix

Next premultiply (

Since the term

The main problem is how to choose the IV matrix. Fritzen [

Then, the objective function of the 2-norm error vector becomes

Obviously, the OLS estimation for (

Next, the WIV for parameter estimation becomes

Adapting the same process described in Fritzen [

Usually, the frequency point with the greatest amplitude in the amplitude spectrum has the highest SNR. Thus, weighting function values can be defined as

To investigate the stability of a flexible rotor supported by a set of two tilting pad bearings, a test rig was built some years ago in the ROMAC lab [

Photo of the rotor assembly.

Figure

Photo of bearing assembly.

As Figure

Current signal of the magnetic bearing exciter.

Figure

Waterfall plot of the test rig.

Waterfall for speed increase process (5-pad, LOP, preload 0.1)

Forward sine sweep at 10,000 rpm (5-pad, LOP, preload 0.1)

Forward sine sweep at 10,000 rpm (5-pad, LOP, preload 0.3)

Backward sine sweep at 9,000 rpm (5-pad, LOP, preload 0.3)

Forward sine sweep at 9,000 rpm (5-pad, LOP, preload 0.3)

Figure

Example of frequency response function (FRF) under sine sweep excitation (5-pad, LOP, preload 0.3, 10,000 rpm).

To predict the damped natural frequency and log decrement of the test rig, two analysis packages developed by the ROMAC Lab were used. The bearing software code MAXBRG [

FEM model of the stability test rig rotor.

First forward whirl eigenvalue and mode shape.

Theoretically, the log decrement of the first forward procession mode will decrease with the increase of the rotating speed. This tendency is mainly due to the dynamic performance of the bearing. With the increase of speed, the damping coming from the oil film decreases while the stiffness usually increases. The log decrement of the rotor-bearing system at different rotating speeds from 7,000 to 10,000 rpm was tested, with the results that are shown in Figure

Log decrement of the rotor at different rotating speed for LOP configurations (comparison between theory and experiment, preload 0.1).

Log decrement of the test rig at different speed

Differences between simulated and expermental result

Under practical situations, maintaining the oil inlet temperature constant in the whole performance test process is difficult, as Pettinato et al. [

Log decrement of rotor-bearing system at different oil inlet temperature (preload 0.1).

Log decrement of the test rig at different oil inlet temperature

Derivation between simulated and expermental results

The oil inlet pressure is potentially another factor affecting the system stability. The paper presents tests of the system log decrement at four different points. The inlet pressures were 1.0 bar, 1.25 bar, 1.5 bar, and 1.75 bar. Both LBP and LOP load orientation tests were conducted, respectively, with a bearing preload of 0.1. From the results shown in Figure

Log decrement of rotor-bearing system versus inlet pressure at different oil inlet temperatures (preload 0.1).

Figures

Log decrement of rotor-bearing system at different oil inlet temperature and pressure (preload 0.1).

LOP

LBP

This paper investigates a new method of determining the damping properties of rotors on fluid film bearings using a frequency domain forward sine sweep excitation, the complex dFRF combined with WIV—the COMDYN method. The complex frequency domain approach provides modal parameters with complex zeros and poles. The approach also determines a clear distinction between forward and backward modal responses including a new method of decomposing the response into forward and backward modal responses. It should be noted that the MOBAR method can also handle both forward and backward modal responses. However, the new COMDYN method differs from the MOBAR method in that it does not use the blocking technique to measure system log decrement during transient vibration decay as well as utilizing both input and output signals. The approach eliminates the influence of forward and backward excitation modal overlap. The new method can potentially be used to determine bearing stiffness and damping coefficients. It was shown in this work that pure sinusoidal excitation gives very good estimations of the system damping which is different from the theoretical literature development using pseudorandom excitation. No test results similar to this new complex dFRF method using the magnetic exciter on a flexible rotor and sinusoidal excitation have been published.

Further, the WIV approach was developed and implemented experimentally as an essential component of this research. Also, the work studied two key relationships between the system log decrement with the increase of rotating speed and oil inlet temperatures by experimental and numerical methods, based on two sets of tilting pad bearing with preloads 0.1 and 0.3. In the experimental research work, the COMDYN method is used to identify the log decrement of the rotor-bearing system at different rotating speeds, oil inlet temperatures, and inlet pressures with both LOP and LBP load orientations, respectively. Under the experimental conditions carried out in this paper, it can be concluded that the system log decrement decreases with the increase of rotating speed almost linearly after the operating speed passes through the first critical speed, similar to the results in [

Active magnetic bearings

Autoregressive moving average

Computational fluid dynamics

Directional frequency response function

Degrees of freedom

Fast Fourier transform

Frequency response function

Load between pads

Load on pads

Multiple input multiple output

Multiple output backward autoregression

Original equipment manufacturer

Ordinary least square

Rational polynomial method

Signal to noise ratio

Thermoelastic-hydrodynamic

Weighted instrumental variable.

Numerator in rational polynomials

Backward precession

Denominator in rational polynomials

Damping matrix

Complex number

Error vector

External forces applying on the rotor

Forward precession

Gyroscopic matrix

Frequency response function matrix

Measured frequency response function matrix

_{ mn}:

Frequency response in the

Stiffness matrix

Weighting value matrix

Weighting function value

Mass matrix

_{ N}:

Number of support points

Laplace frequency point

Parameter vector

Vector

Instrumental variable matrix

Rotor displacement, velocity, and acceleration

Rotor horizontal displacement degree of freedom; modal displacement

Rotor vertical displacement degree of freedom; adjoint matrix.

Rotor rotation about

Rotor rotation about

Eigenvalue

Angular frequency.

Bearing terms, finite element model

Cross-coupled stiffness terms, finite element model; complex number symbol

Lumped mass terms, finite element model

Shaft terms, finite element model.

The authors declare that they have no conflict of interests regarding the publication of this paper.

The work described in this paper was supported financially by the Natural Science Foundation of China (51135001) and “973” Program (2012CB026000). These supports are gratefully acknowledged.