Abstract

Accuracy of Modelling and Identification of Malfunctions in Rotor Systems: Experimental Results

Nicolò Bachschmid

Paolo Pennacchi

Ezio Tanzi

Andrea Vania

Politecnico di Milano Dipartimento di Meccanica Pizza L. da Vinci, 32. I-20133. Milano. Italy

nicolo.bachschmid@polimi.it, paolo.pennacchi@mecc.polimi.it

Introduction

The aim of a E.C. funded research project named MODIAROT (Model based Diagnosis of Rotor Systems in Power Plants) is to evaluate the capability, efficiency and reliability of malfunction identification techniques. Those are mainly based on the models of the system and on a comparison of calculated results with experimental results. One of the techniques which can be used is the least square fitting approach in the frequency domain. In a first step of this project, the available models are tested for reliability and limits of validity through a benchmark, and some refinements of the models are proposed. The second step involves experimental results obtained on several different test rigs for the validation of the models. The third step relates to the modelling of different malfunctions and to the experimental validation on the different test rigs of these malfunction models. The last step is dedicated to the identification of these malfunctions from shaft vibration measurements.

In the following some results obtained on the test rig of the Politecnico di Milano are shown: the results are related to the validation of the models of the system (rotor, bearings, supporting structure), of the models of some malfunctions (e.g. unbalance, rotor bow, coupling misalignment) and of the least square fitting identification procedures.

Description of the Test Rig

The test rig, shown in Figure 1, is composed of two rigidly coupled rotors supported on four elliptical shaped oil film bearings, designed to have three critical speeds within the operating speed range of 0-6000 rpm, driven by a variable speed electric motor. The rotor system is mounted on a flexible steel foundation, with several natural frequencies in the operating speed range. Two proximity probes in each bearing measure the relative shaft displacements, or the journal orbits; two accelerometers on each bearing housing measure its vibrations, and two force sensors on each bearing housing measure the forces which are transmitted to the foundation. The forces which are transmitted from the shaft to the bearing can be calculated by adding the inertia forces of each bearing housing to the force measured by the force sensors.

The absolute vibration of the shaft is calculated by adding to the relative displacement measured by the proximitors, the absolute bearing housing displacement, which is obtained integrating twice the acceleration measured by the accelerometers.

Validation of the System Models

The system is composed by the rotors, the oil film bearings and the supporting structure. Each one of these is considered separately and validated.

Validation of the Rotor Model

The rotors have been modelled by means of beam elements; different mass and stiffness diameters are used for modelling the rotors and special rules have been applied to the sections in which a sharp increase in diameter occurs (Genta, 1995). Several approaches that use beam finite elements exist in literature in order to solve the problem of behaviour of rotors that present sharp diameter changes (see f.i. Rao, 1991 and Bigret, 1980). These methods provide for inserting, between the two parts of diameter D₁ and D₂, an intermediate element of diameter D_m, which is arithmetic mean of the previous ones.

The length of the intermediate finite beam element l is obtained according to eq. (1), which significance is shown graphically in Figure 2:

The angle a has been chosen equal to 37°. In this way, the calculated and experimental eigenfrequencies of the free-free shafts resulted in very good agreement. Therefore two different diameters are introduced in the schematisation of the rotor, shown in Figure 3:

The damping matrix DR is given by the following equation:

The structural damping of the rotor, represented by the matrix D_R^struc, is equal to:

The numerical values of coefficients a and b are considered later on. Different values have been used for the model of the shaft alone and for the complete model of the test rig.

A first check of the model has been carried out by measuring the free-free natural frequencies of each rotor (suspended on ropes) and comparing these values with the calculated ones. Table 1 summarizes this comparison:

The agreement between the frequencies can be considered very good. A further check of the accuracy and the validity of the rotor model can be done by comparing the measured shaft vibrations with the calculated ones. The latter are obtained by applying to the rotor, in correspondence of the bearings, the forces transmitted through the oil film (which are measured by the force transducers and corrected by the bearing housing inertia forces) and in correspondence to some other point the exciting force (such as unbalance e.g.).

Since it resulted that several exciting causes were coexisting on the rotor system (original unknown unbalance distribution, rotor bow, residual coupling misalignment, mechanical run-out of some shaft journal a.s.o.), the vector differences of the vibrations related to two different run down transients must be considered (and linearity must be assumed). The first set of measures contains all the original unknown exciting causes, while the second one the same exciting forces plus the concentrated known unbalance, e.g. applied to one of the existing balancing planes.

The model of the rotor with coupled shafts will be validated by imposing an unbalance of 3.6 104 kgm at -90° on the balancing plane number 7 on the long shaft (see Figure 3) and by comparing the simulated journal absolute displacements with the measured vibrations. The model of the rotor alone is used, and the bearing oil film is substituted by its reaction forces, which have been measured indirectly by the load cells. The experimental data (relative journal-bearing displacements, forces and accelerations of the bearings) have been obtained with a procedure called ‘tracking filter’, i. e. the acquisition of amplitude and phase of the different harmonic components by means of the spectrum analyser, making the rotor follow a very slow run-up. The rotation speed, used for the validation, ranges between 500 and 2700 rpm.

The measured quantities need to be processed before they can be used in the simulation. First, referring to one of the bearings, a rotation of the reference system has to be done in order to have the relative journals-bearings displacements expressed in horizontal and vertical directions. Then, by calling xⁿ_rel the vector of the n^th harmonic component of the relative journals-bearing displacements in horizontal and vertical direction (which can be assumed to be 2p/W periodic, where W is the rotating speed) and aⁿ_b the vector of the measured n^th harmonic component of the bearing accelerations in the same directions, the nth harmonic component of the journal absolute displacements can be obtained as:

By calling f ⁿ_b the vector of the n^th harmonic component of the measured forces at the bearing, the n^th harmonic component of the oil film forces acting on the journal can be obtained as:

where m = 9 kg is the mass of the bearing housing. The n^th harmonic components in horizontal and vertical directions of the absolute displacements of all the rotor journals and of the oil film forces, calculated as described, can be respectively collected in the vectors Xⁿ_e,bal and Fⁿ_e,bal for the situation of balanced rotor and in the vectors Xⁿ_e,unb and Fⁿ_e,unb for the situation of unbalanced rotor.

The effect on the test rig behaviour of adding an unbalance mass to the balanced situation can be obtained as the difference of the above defined complex vectors, by obtaining the vectors:

Since the unbalance gives a 1^st harmonic excitation, the major changes arise in the 1^st harmonic (i.e. the 1 ´ revolution harmonic component) of the above quantities. Therefore, by starting from the 1^st harmonic of the oil film forces DF_e and applying it on the journal nodes of the model of the rotor, the ‘analytic displacements’ can be calculated. In fact, by calling M_R, D_R and K_R the mass, internal damping and stiffness matrices respectively of the shaft and DX_athe vector of the displacements of all nodes (four complex components in each node) due to unbalance only, the following force balance equation holds in the frequency domain, referring to the 1^st harmonic force and vibration components:

where DF_e are the difference of measured oil film forces in unbalanced and balanced situation and DF_unb are the known exciting forces acting on the shaft due to the unbalance.

The analytical displacements can be obtained with the inversion of rotor elastodynamic matrix:

However, all these data handling may magnify the effect of measuring errors, which may not be negligible, considering also that the bearing housing accelerations are rather small in wide frequency ranges. The errors in the A/D conversion are further magnified by the double integration required to produce absolute vibrational displacements from accelerometer; these errors could invalidate the results in the lower frequency range. The effect of these errors are shown e.g. in Error! Reference source not found., which shows the amplitude and the phase of the l ´ component of horizontal and vertical vibration of the shaft in correspondence of bearing 3, as a function of rotational speed (Bode diagrams). The solid line represents the experimental values, measured by the proximity probes, converted in the fixed (vertical-horizontal) reference frame and added to the absolute vibrations (displacements) of the bearing housing that are obtained by integrating twice the acceleration. The thin line represents the calculated vibrations obtained by applying the measured forces and the unbalance to the free-free rotor.

The agreement, between the experimental and the calculated case, ranges from good to acceptable. The scattering of the calculated results is probably due to the following reason: the rotor is a free-free beam and the known unbalance force and the heavily ‘processed’ oil film forces are applied to it. Some small measuring errors, introduced also by the tracking filter analysis, are possibly magnified by the described data processing method. Therefore, the system of forces applied to the free-free rotor is not ‘balanced’ and causes vibrations, which are higher than the experimental ones. In order to reduce the scattering of the results, the coefficients a and b used to calculate D_R^struc (see eq. (3)) are fixed respectively equal to 5 and 0.001, in order to have a high structural damping of the rotor.

Bearing in mind that, by comparing calculated results in a benchmark of computer codes, unexpected high discrepancies have been found between results obtained with different codes but the same model, as reported by Bachschmid et al. (1997a), the agreement obtained here between calculated and experimental results can be considered almost good.

Validation of the Oil Film Bearing Model

As shown in a previous analysis on the accuracy of calculated results obtained with different oil film models and codes (see f.i. Lund, 1987, Jecmenica et al., 1998, Bachschmid et al., 1997a), in general a rather good agreement of the different results has been found. Therefore, also a good agreement of calculated results with experimental results was expected. Insufficient oil flow rate, as has been ascertained later on, and unexpected thermal effects have instead deviated the results from calculated values: the agreement in the statical positions is very poor, while a better agreement has been found in dynamical conditions.

The geometry of the four lemon shaped bearings used on the test rig are summarised in Table 2. The bearing numbering starts at the left hand side of Figure 1.

The stiffness and damping coefficients of the oil film have been calculated as a function of the rotor speed in the range from 500 to 6500 rpm. The characteristics of the oil were:

The diagrams of the stiffness and damping coefficients are reported in Figure 5 to Figure 10. The damping cross terms are not reported since they result very small.

The statical bearing loads, which correspond to the values measured by the load cells, are reported in Table 3.

The reliability of the stiffness and damping coefficients used to schematise the oil film of bearings is now analysed. The dynamic oil film forces, obtained from measured quantities, are compared with the analytically calculated ones for this purpose. The ‘experimental forces’ are the measured and corrected forces in a balanced situation of the rotor, whereas the ‘analytic forces’ are obtained from the measured relative journal-bearing displacements in the vertical and horizontal direction, for a particular speed and bearing in the following way:

In eq. (10), F_R^x and F_R^y are the ‘analytical’ oil film forces, x_rel are the complex 1^st harmonic components of the relative displacement in the frequency domain, k and r are the oil film stiffness and damping coefficients.

The 1^st harmonic component only of the relative displacements is considered. In Figure 11 and Figure 12, the comparison between analytic and experimental oil film forces is shown for bearing 2 and 3, in the range from 500 to 3000 rpm.

From the Figure 11 and Figure 12, a rather good agreement is found in general between analytical and experimental forces, except the higher frequency range. Similar results are obtained also for the bearings 1 and 4.

Validation of the Supporting Structure Model

The supporting structure has been analysed by modal analysis (see f.i. Freeswell et al., 1996, Smart et al., 1998), before installing on it the rotors, and its identified modal model, with only few degrees of freedom, has been introduced in the complete model of the system, according to a procedure described in Cheli et al. (1992).

The modal parameters (generalised mass, stiffness and damping) relative to the different eigenmodes of the foundation have been obtained by means of modal identification techniques starting from the experimental transfer functions of the supporting structure, defined by means of measurements carried out on the test rig after having removed the shafts. In fact, the FRF has been obtained applying a sinusoidal force on the bearing housing of the 3^rd bearing in horizontal direction with frequency W by means of an electrodynamic exciter. The force frequency has been changed slowly in the range from 10 Hz to 100 Hz (slow sine sweep) and, at the same time, the bearing housing accelerations in horizontal and vertical direction (8 d.o.f.) have been measured. In this way, the minimum number of d.o.f. of the foundation are considered, i.e. the translational d.o.f. in the connections between shafts and foundation, in order to simulate the behaviour of whole system. The modal parameters of foundation are reported in Table 4. In the range of frequency 10-100 Hz, nine eigenmodes have been defined.

In order to validate the modal parameter estimates, the experimental frequency response functions, due to a single input excitation, have been compared with the regenerated data obtained by substituting the results of the modal analysis into the following expression:

In eq. (11) the term hij represents the frequency response function evaluated at the i d.o.f., due to an excitation applied at the j d.o.f. The terms y_riand y_rjare the components of the r-th normal modes associated to the degrees of freedom i and j. The regenerated data obtained with eq. (11) have been compared to the respective experimental data (FRF) measured at the four supports of the test rig. A sample of the results obtained by the comparison is reported in Figure 13. The good curve fitting shows that the accuracy of the identified modal parameters was good. Similar results are obtained also for the other bearings.

The validation of the supporting structure model is performed by comparing the measured bearing housing displacements due to the unbalances of the rotor, with those calculated by applying the measured forces to the model of the supporting structure.

The comparison is made in the full speed range from 600 to 5700 rpm, using only the l ´ components of forces and displacements.

The ‘analytic displacement’ are calculated applying the measured forces in correspondence of the bearing to the modal model of foundation through eq. (12) (see f.i. Hoshiya et al.(1984)):

(12)

where:

The comparison between experimental results and calculated results is shown in Figure 14 and Figure 15 for bearings 3 and 4. The fitting is rather good for rotor speeds below 3000 rpm. Above this rotating speed the system non-linearities become more effective and the fitting is not so good as before. Similar results are obtained for bearings 1 and 2 too.

Validation of the Complete Model (Rotor + Bearings + Foundation)

After the validation of the models of the single parts, in this section the reliability of the complete model of PdM test rig will be checked. The experimental data for the validation have been obtained by imposing an unbalance of 3.6 10-4 kgm at –90° on the balancing plane number 2 on the short shaft (see Figure 3). In a similar way to the validation of the rotor, the validation of the complete model consists in the comparison between analytic absolute journal displacements obtained applying the forcing system due to the unbalance to the model of the test rig, with the experimental ones defined by the difference between the measured journal vibration in unbalanced and balanced conditions. Since the unbalance mass gives a forcing system with frequency equal to the speed of the rotor W , the 1st harmonic component of measured quantities is only considered, under the hypothesis of linear system. The vector DX_an represents the displacements of all nodes (4 complex components in each node) in frequency domain; referring to the 1^st harmonic force components the following force balance equation holds:

where DF_unb are the known exciting forces due to unbalance. So we can obtain easily:

The internal damping of the rotor (a and b coefficients) has been completely neglected in these calculations, because in this case the scattering of the results, which was previously described, does not occur, since the measured forces are not used for the simulation. The comparison between analytic and experimental displacement is shown in the Figure 16 and Figure 17 for bearings 2 and 3. Similar results are obtained for bearings 1 and 4.

The agreement between calculated and measured results seems to be rather good. Critical speeds and supporting structure resonances are generally well reproduced in amplitude and phase. Therefore, the complete model of the test rig can be considered validated.

The Fault Models

The models of unbalance and rotor bow are simply represented respectively by a concentrated rotating force, proportional to the square of the rotating speed, and by two rotating equal and opposite constant moments, which force the rotor to deflect in similar way to the measured deflection (see Figure 18).

The rigid coupling angular and radial misalignment may also be represented by a system of forces and moments applied to the coupling flanges, which force the rotor to deflect and bow (see Figure 19).

Application of Identification Procedures

The least square fitting procedure for fault identification, described in Bachschmid (1997b), has been applied to different experimental results obtained on the test rig to which different faults have been applied.

Unbalance Identification

In this paragraph the identification methodology of the faults, in particular of the unbalance, will be validated through experimental data referred to the unbalance on the long shaft. The results of the identification depend obviously on the speed set used for the least square fitting procedure. The speed values used in the unbalance identification are in the range from 600 to 2700 rpm with a step of 300 rpm, as shown in Figure 20, where the amplitude of the experimental absolute journal 1 ´ displacements are presented (the asterisks show the speeds which have been used for the identification procedure).

These were obtained by the tracking filter analysis of run down transient and by vector differences of unbalanced rotor results minus those of the balanced rotor which is the reference situation.

Using this speed set, the relative identification residual can be calculated for each possible node of application of the unbalance. The most probable location of the unbalance is where the residual reaches its minimum; in its correspondence, amplitude and phase of the fault is determined as shown in Figure 21: these values and the position are rather close to the real ones.

Bow Identification

The reference situation is not available in this case, i.e. the experimental data of the rotor without bow were not measured, since the rotor has a permanent bow.

For this reason the experimental data obtained after balancing are used and the effect of the known balancing masses are subtracted: the experimental data which are due to the bow alone are obtained in this way, assuming the original unbalance negligible. The effect of the balancing masses is obviously calculated with the model of the system.

The speeds used are from 600 to 2700 rpm with a step of 300 rpm, the same of the unbalance identification, as shown in Figure 22 where also the experimental 1 ´ rev. displacements are represented.

On the contrary of the unbalance case, the experimental absolute journal displacements used for the comparison have also an "analytical" component, being the algebraic sum of the measured vibrations in the reference situation and the vibrations due to the balancing masses, calculated with the complete model.

These data are compared with the analytical displacements due to a force system which represents the bow according to Figure 18.

In this case, the application nodes of the bow force system have been previously fixed in correspondence of the bearings. This force system, which models the bow, can be validated comparing the bow induced on the rotor by the force system and the bow measured on the shafts at certain sections.

The bow has been evaluated by rotating slowly the rotor on its bearings, by measuring the displacements at the sections shown in Figure 23, and by extracting the first harmonic component from these data.

For the comparison, the analytical bow has been calculated with the rotor model fixed on vertical and horizontal springs with a standard stiffness of 10⁶ N / m in correspondence of the journals. The results are also shown in Figure 23, and the agreement between calculated and measured statical bow seems to be rather good.

Since the experimental data in the reference situation are available only for the speeds between 500 and 2700 rpm, the comparison of the dynamical behaviour will be limited in this range. A very good agreement in amplitude and phase between calculated and experimental journal vibrations for the bow force system was achieved as shown in Figure 24 and Figure 25 for bearings 1 and 3. Similar results are obtained for the others bearings.

Rigid Coupling Misalignment Identification

An angular misalignment can be introduced in a coupling by inserting a tapered spacer with the same diameter of the flanges and with an angle between the flat faces (see Figure 26).

In the test rig the angle between the two flat faces was 0.545 mrad at -120°. A cylindrical spacer with parallel faces has been used to generate the pre-fault reference situation.

The static deflection induced by the coupling misalignment has been measured through comparators, obtaining amplitude and phase of the rotor eccentricity at some sections; in this case, the deflection is given by the difference between post and pre-fault situation. The comparison between the experimental and the calculated deflection is shown in Figure 27.

The experimental data obtained as usual as vector differences of the 1 ´ rev. components obtained by tracking filter analysis of the run-down transient of misaligned rotor minus that one of the reference rotor (with the cylindrical spacer) and the speeds used for the identification procedure are shown in Figure 28.

The results of the identification method are shown in Figure 29, where a sharp minimum occurs in correspondence of the coupling. The identified amount of angular misalignment is 4.5 mrad at - 112°, which is close to the real value.

The comparison of experimental absolute journal vibrations and the analytical ones is shown in Figure 30 and Figure 31 for bearings 2 and 3. Similar results have been obtained for the others bearings too. The results of the comparison show a good fitting, except for the peaks of the first critical speed of the short shaft in vertical direction (2300 rpm), where the calculated results do not reproduce in a good way the experimental ones. Also the second critical speed of the long shaft in horizontal direction (2900 rpm) is not well-reproduced by the analytical simulation.

Capability of Distinguishing Different Faults

Finally, the capability of distinguishing the faults which cause similar effects is analysed with the data used for the rigid coupling misalignment of the rotor. The considered faults are:

The identification speed sets are the same used in Figure 28 and the obtained results are reported in Figure 32. The best results are obtained with the coupling misalignment fault; with only angular misalignment the best localisation is obtained, whereas with the combined angular and radial misalignment the residual has a lower value, but the localisation is weaker. The residuals obtained with other faults are always higher. Therefore the identification procedure indicates the coupling misalignment fault as the most probable candidate.

Conclusions

System models and model based identification procedures have been validated by means of experimental data obtained on a 4 oil film bearing 2 rotors test rig. Although the models are not always reliable and accurate enough to reproduce truly the experimental behaviour, they can be successfully used for model based malfunction identification procedures in the frame of an automatic diagnostic system for turbogenerators in power plants.

Acknowledgments

The financial support of Birite-Euram Contract BRPR-CT95-0022 is gratefully acknowledged.

Presented at DINAME 99 – 8th International Conference on Dynamics Problems in Mechanics, 4-8 January 1999, Rio de Janeiro. RJ. Brazil. Technical Editor: Hans Ingo Weber.

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