Experimental Evaluation of a Modal Parameter Based Foundation Identification Procedure

Abstract

One approach to model foundations of existing turbomachinery installations uses motion measurements at bearing supports and at select points on the foundation to identify the modal parameters of an equivalent foundation. This paper describes an experimental evaluation of this approach. Discussed are the experimental rig, its commissioning, the procedure for obtaining the required measurements and preliminary identification results. The proposed approach could identify reasonably accurately foundation natural frequencies, but identification of damping ratios, modal masses and mode shapes was significantly influenced by input data errors. The resulting equivalent foundation could predict only approximately the frequency response of the rig. Further investigations are needed to improve these predictions prior to field applications.

1 Introduction

Correct modelling of a rotor bearing foundation system (RBFS) is an invaluable asset for the balancing and efficient running of turbomachinery. However, modelling of the system foundation is still problematic, particularly for existing installations [1–4]. Here we propose to use motion measurements of the rotor and foundation with the rotor in situ to identify the relevant modal parameters for an equivalent foundation, i.e. a foundation which, when substituted for the actual foundation, reproduces the vibration behaviour of the RBFS over the operating speed range of interest.

In earlier work, an identification approach was developed which successfully identified, via numerical experiments, the modal parameters of an equivalent foundation for an RBFS consisting of an unbalanced rotor supported by two hydrodynamic bearing pedestals fixed to a flexibly supported flexible foundation block [3]. However, Ref. [3] did not fully consider signal noise, measuring instrument limitations, nor the practical problems associated with the determination of the excitation forces. Hence, experimental evaluation of the modal parameter identification approach is in order.

Actually, the correct determination of the excitation forces is a major problem and two approaches have been suggested for this. The first approach uses the measured motions of the rotor journals in the hydrodynamic bearings to determine, via Reynolds equation, the pressure distributions in the fluid films. Integration of these pressures then yields the required transmitted forces. This approach does not need to know the rotor unbalance, nor the rotor model but is too dependent on a sufficiently accurate knowledge of the bearing clearances, lubricant viscosities and cavitation behaviours to provide appropriate excitation force data [5]. The second approach presumes a sufficiently accurate rotor model to bypasses the need to use Reynolds equation. However, this approach requires motion measurements at the same speeds for two rotor unbalance conditions to ensure a known unbalance. This approach was used to successfully identify the equivalent foundation of a simple pedestal RBFS [5]. It is the approach adopted in this paper which is concerned with the experimental evaluation of a modal parameter identification approach when applied to an RBFS wherein an unbalanced rotor is supported via two bearings on a flexibly mounted foundation block, allowing for ‘cross-talk’ between the bearings. Such an RBFS is significantly more complicated than the simple pedestal block evaluated in Ref. [5].

2 The Identification Equations

Assuming the equivalent foundation to be an n degree of freedom (DOF) system, a harmonic excitation frequency Ω, and periodic response with fundamental frequency Ω, one can show that the identification equations are given by $(k = 1, \ldots ,n)$:

$$ \left( { -\Omega ^{2} + i\Omega \zeta_{k} + \lambda_{k} } \right)\mathop \sum \limits_{j = 1}^{n} a_{jk} X_{j} - \mathop \sum \limits_{j = 1}^{n} \varPhi_{jk} F_{j} /m_{k} = 0 $$

(1)

Knowledge of $F_{j}$ and $X_{j}$ at a sufficient number of frequencies $\Omega$ suffices to identify the elements of $\varvec{\zeta}$, $\varvec{m},\varvec{\lambda}$ and $\varvec{\varPhi}$, which parameters define the desired equivalent system. The response is then given by:

$$ \tilde{X} =\varvec{\varPhi}[\varvec{m}\left( { -\Omega ^{2} \varvec{I} + 2i\Omega \varvec{\omega \xi } +\varvec{\lambda}} \right)]^{ - 1}\varvec{\varPhi}^{T} \tilde{F} $$

(2)

The derivation of the above equations and the numerical procedure to solve for the modal parameters are given in Ref. [3].

3 Experimental Procedure

3.1 Experimental Rig

Figure 1 shows the experimental rig. It consists of a three disc rotor, the motor end of which is driven via a flexible coupling by a motor with variable speed up to 10,000 rpm and is supported by two single row ball bearings while the other end is supported by a plain journal bearing. The disks provide balancing planes, with each disc having 16 threaded holes 22.5° apart, allowing for insertion of balance screws. Both bearings are mounted in aluminium pedestals which are bolted to an aluminium block which in turn is supported by two pairs of 12 mm × 5 mm rectangular steel bars (the pair on the left being 225 mm long while the pair on the right being 205 mm long). These bars are fastened to the underside of the aluminium block at one end and to a steel plate (which itself is clamped to a heavy steel table) at the other end, thereby providing flexible supports to the aluminium block. Two steel weights, each of mass 0.69 kg, are bolted onto the right end corners of the aluminium block in order to provide foundation natural frequencies which were estimated to fall within the planned operating speed range. Thus, the to be identified foundation comprised the bearing pedestals including all attachments thereto, the outer races of the ball bearings, the aluminium bock with the added steel weights and the support bars. Full details of the rotor, the bearings and housings, and the aluminium block are given in Ref. [5].

Fig. 1

Rotor-bearing-foundation rig

3.2 Instrumentation

Figure 2 shows the locations and measurement directions of the five displacement transducers or proximitors P1 … P5. P1 and P2 were mounted in the transverse vertical and horizontal directions as close as possible to the ball bearings while P3 and P4 were inserted into the journal bearing housing, again in the transverse vertical and horizontal directions. P5 was positioned in the plane of a notch on the coupling to provide a phase reference signal.

Fig. 2

Locations of the proximitors on the bearing housings

Figure 3 shows the locations and measurement directions of the six accelerometers A1 … A6. A1 was attached to the aluminium block in the longitudinal direction (LX). A2 and A3 were attached to the ball bearing housing in the transverse vertical (LY) and horizontal directions (LZ) while A4 and A5 were attached to the journal bearing housing, being in the same transverse plane as P3 and P4 but rotated 45° due to space limitations. A2–A5 served to measure the absolute motion of the foundation and, in conjunction with P1–P4, served to measure the absolute displacements of the rotor at the rotor/foundation connection points. Additionally, A6 was attached to the aluminium block approximately one third of the way between the bearing connection points, in the vertical (MY) direction. Thus, one should be able to identify an up to 6 DOF equivalent foundation. Two thermocouples were installed in the journal bearing to monitor inlet and outlet oil temperatures. A photodiode tachometer facing the striped outer edge of the rotor disk nearest to the motor provided 12 pulses per revolution.

Fig. 3

Accelerometer locations and measurement directions

Figure 4 shows the instrumentation setup. After passage through appropriate amplifiers, all signals were digitised by a 16 channel PC-30D data acquisition board and stored in a computer for subsequent processing using in house data processing software. P3 and P4 were also connected to an oscilloscope to monitor the journal orbit; and the photodiode tachometer was also connected to a counter to monitor rotor speed.

Fig. 4

Instrument flow chart

3.3 Determination of Foundation Natural Frequencies and Mode Shapes

Before the rotor was added to the rig, the natural frequencies of the foundation were determined by hammer tests. Six natural frequencies were found in the frequency range from 0 to 512 Hz with resolution of 0.125 Hz, viz. 25.500, 31.500, 40.625, 66.625, 229.000 and 481.000 Hz. None of the accelerometers was able to find all six. There was no longitudinal vibration frequency below 512 Hz. These natural frequencies formed the yardstick frequencies of the yet to be identified foundation.

A finite element model (FEM) of the foundation, was established by ANSYS 13.0 and used to estimate the first six undamped natural frequencies and corresponding mode shapes of the foundation. Figure 5 summarises the results. The aluminium block exhibits minimal flexure till the sixth mode. As expected, the agreement between the hammer test natural frequencies and FEA natural frequencies is only approximate owing to the limited number of elements which could be accommodated; but the results are felt to be close enough to provide a qualitative yardstick for the mode shapes.

Fig. 5

Natural frequencies and mode shapes of the foundation using FEM

3.4 Input Data

Since the residual rotor unbalance is unknown, identification theory requires measurements to be taken at identical speeds from two rotor rundowns with known unbalance added to the rotor prior to the second rundown. Measurements at ‘identical’ speeds were actually taken for three rundowns and Table 1 shows the respective added unbalances used for these rundowns. Known unbalances were added by inserting short steel set screws into the indicated discs into one or more of the threaded holes which were available at 30.5 mm radial distance from the rotor axis. The screw masses were measured via an electronic balance to an accuracy of 0.0001 g. Measurement data (accelerometer, proximitor and tachometer signals) for these three different unbalance distributions was taken over the speed range from 73 to 20 Hz (72.76–19.94 Hz) with step sizes no larger than 2 Hz. Below 20 Hz, the signals were too small and above 72 Hz, the harmonic orbit amplitude in the journal bearing exceeded half the radial clearance. The rundowns with added unbalance distribution $\varvec{U}_{\bf{2}}$ gave better signal differences compared to the rundown with $\varvec{U}_{\bf{1}}$, so the differences (at selected ‘identical’ speeds) between the measured accelerations and displacements obtained from rundowns with $\varvec{U}_{\bf{2}} \varvec{ }$ and $\varvec{U}_{\bf{0}} \varvec{ }$ were used for identification purposes. There were 67 ‘identical’ or matching speeds where the speed differences in the two rundowns were less than 0.2 %, as calculated from the FFT of the P5 signal.

Table 1 Added unbalance distributions for the three rundowns

Figure 6 shows the resulting magnitude differences obtained from accelerometers A1–A6 while Fig. 7 shows the resulting magnitude of the transmitted force differences, which were calculated from the measured absolute displacement differences of the rotor at the bearings, and the assumed known dynamic model of the rotor [6]. From hereafter, for the sake of brevity, the displacement differences and the force differences will simply be referred to as the displacements and the forces.

Fig. 6

Magnitude of displacement differences (rundowns with $\varvec{U}_{{{\mathbf{2}}\text{ }}}$ and $\varvec{U}_{{\mathbf{0}}}$)

Fig. 7

Magnitude of transmitted force differences (rundowns with $\varvec{U}_{{{\mathbf{2}}\varvec{ }}}$ and $\varvec{U}_{{\mathbf{0}}}$)

It can be seen the longitudinal displacement of the foundation (curve LX) is relatively small. The FEM analysis indicated that the longitudinal natural frequency was well above 512 Hz. Hence, accelerometer A1 signals were not used in the identification. Also, since the sixth natural frequency $\omega_{6}$ of 430.75 Hz was well above 72.75 Hz, the upper bound of the measurement frequency range, $\omega_{6}$ would have negligible effect on the measurement data, and hence could be ignored, leaving one to require the equivalent foundation to have 5 DOF. Note that there is minimal difference in the transverse horizontal displacements at the bearing connection points (curve LZ) and (curve RZ), even though it is these displacements which would reflect the presence of the natural frequency indicated by the $\omega_{5}$ mode in Fig. 5.

Hammer tests on the stationary rotor detected a natural frequency around 65 Hz. The actual natural frequencies of the rotor are more difficult to find, being influenced by the rotor support stiffness and gyroscopic effects; and one would expect two natural frequencies in the vicinity of 65 Hz. This is clearly indicated in Fig. 7 which shows peaks in the transmitted forces around 61.5 and 64 Hz. These large transmitted forces are significantly inaccurate as the in house software used to calculate these forces assumed a rotor with zero rotor damping. Hence, acceptable input measurement data was restricted to speeds at which the transmitted forces were calculated to be less than 10 N. This meant that the data for speeds in the range 61.68–67.74 Hz were discarded, leaving data for 60 speeds available for identification purposes.

3.5 Results and Discussion

The identification procedure outlined in Ref. [3] was applied to identify a 5 DOF equivalent foundation. When the data for all 60 speeds was used, it was only possible to identify the four natural frequencies within the operating speed range. To identify the fifth natural frequency (known from the hammer tests to be around 231 Hz), the input data set was restricted to speeds above 58 Hz (using only 10 speeds), whereupon a fifth natural frequency could be identified. It did not matter whether damping was assumed or not. In both cases, the fifth natural frequency was grossly inaccurate, but the other four were more accurately identified than when one assumed a 4 DOF. The identified parameters for zero damping are given in Table 2. The identified natural frequencies when damping was included were not the same but of similar accuracy. The identified damping ratios were generally small. Though the natural frequencies within the operating speed range are in reasonable agreement, the mode shapes, modal masses and damping ratios are known to be often significantly in error as vastly different values can be obtained when different data sets are used or when different weighting is applied to the data. The problem is compounded by the very small differences in the horizontal displacements at the bearing connection points.

Table 2 Identified modal parameters

Figure 8 compares the predicted frequency responses with the actual measured responses, using Eq. (2). Figure 8a, b assume zero damping in the identified foundation but differ significantly because in Fig. 8a all five modes are used to obtain the response whereas in Fig. 8b the fifth mode is not included as it is grossly inaccurate. The agreement between actual and predicted responses is fair in Fig. 8b whereas it is unacceptable in Fig. 8a.

Fig. 8

Measured (EXP) and identified (ID) frequency responses on the foundation. a 5 modes. Undamped. b 4 modes. Undamped. c 4 modes. Damped

It should be noted that the response predictions naturally exclude predictions at speeds in the near vicinity of the identified natural frequencies since, with assumed zero damping, Eq. (2) predicts infinite response at those frequencies. Figure 8c is the same as Fig. 8b except damping is assumed. As expected, with such low values of damping ratios, there is little difference between the response predictions in Fig. 8b, c. Inclusion of damping is of little value for this particular foundation since its damping is low compared to the overall damping in a system which includes hydrodynamic bearings; yet its inclusion adds to the solution complexity and round off error problems.

The discrepancy between predicted and actual responses could be due to a variety of reasons, including inaccuracy of the calculated excitation forces, excessive round off and measurement error and the need for a more accurate damping model for the foundation. Investigations to minimise errors due to these effects are continuing.

4 Conclusions

The resulting equivalent foundation could predict approximately the frequency response of an experimental rig; but further investigations are required to improve these predictions prior to field applications.

The natural frequencies of the rig foundation were reasonably well identified but the identification of the damping ratios, modal masses, and mode shapes was significantly influenced by input data errors, round off errors and ever present signal noise.