-This paper presents the results obtained analysing the dynamic behaviour and natural vibration modes of the components of a turbine-compressor unit, connected by toothed coupling. The study was undertaken with the aim of improving the dynamic behaviour of the system, identifying some critical characteristics, mainly linked to problems of misalignment which lead to elevated vibration levels in some bearings and to the rupture of the lubrication film. An accurate 3D parametric virtual model of the system was used, integrating FEM and Multibody calculation programmes to perform a dynamic analysis of the various components considered as deformable bodies. Information regarding some characteristics significant for the dynamics of the system was obtained in experimental trials, allowing the validation of the numerical models. In particular, the contact of the teeth of the turbine and compressor hubs with the teeth of the coupling bell were simulated, as well as the hydrodynamic effect of the lubrication in the shaft bearings. The analysis of the results highlighted the need to use models with deformable elements, and allowed the determination of the limiting conditions of misalignment. Index Terms-3D Flexible modeling, Computational dynamic analysis, misalignmen, bearings, lubrication film.
Management of a modern petrochemical plant, where the
sound functioning of the machines installed guarantees the
reliability and continuity of production, must include
continuous monitoring by means of dedicated systems. Checking
for vibration is essential in verifying the correct functioning
of mechanical systems and will evidence possible anomalies
at their incipient stage [
The simplified Reynolds equations proposed by Ocvirk and
Dubois [
The P2005A turbo-compressor in the ethylene production plant of Enichem Priolo is used to compress the mixture of hydrocarbon gases emitted at the head of the quench column. The machine, constructed by Nuovo Pignone, consists of three main parts: the steam driven turbine, the centrifuge compressor which compresses the gas, and the toothed coupling which transmits the torque from the turbine to the compressor. The axial turbine, delivers a maximum power of 20835 kW at 4190 rpm. The shaft rests on two radial hydrodynamic bearings of the Michell type (tilting pad bearings) and a hydrodynamic axial thrust bearing of the Kingsbury type. The steam and lubrication oil seals are secured by a series of rings cut on the rotor and mounted on the stator with a labyrinth system. The six-stages centrifuge compressor of horizontal open case type, with three intake flanges and one delivery flange positioned in the lower half-casing, again rests on two segmented hydrodynamic bearings and a Kingsbury-type thrust bearing (Tab. I).
The Maag toothed coupling, Zud8 type, in AISI 8740 steel, is 976.3 mm in length with a maximum diameter of 383 mm. The two bells, internally toothed, rest on the hubs and are connected together by a sleeve collar; two metal Orings, pressure mounted in circumference slots cut into the teeth of the bell, allow the bell-cylinder system an axial slip of 6 mm. The hubs, toothed externally with 88 teeth, are keyed on the turbine and compressor shafts; two keys and two threaded locking rings prevent, respectively, tangential and axial movement. The smaller longitudinal dimensions of the hub teeth compared to those of the bells, combined with radial and tangential gap, allow the shafts to become misaligned. A gas balancing system, which uses a disk keyed to the shaft, makes it possible to limit the axial thrust and the vibrations. To prevent excessive vibrations from damaging the seals, during operation a control system verifies that radial movement at the bearings never exceeds the allowed tollerances (0:24 0:28 mm for the compressor bearings and 0:3 0:358 mm for the turbine bearings).
The numerical model was developed using: the ADAMs
calculation programme to construct the multibody model of
the turbo-compressor; and the MSC/NASTRAN calculation
programme which, through modal analysis of the components,
was used to generate transfer files simulating the flexible
behaviour of the parts in the multibody code [
The transfer file contains the stiffness and damping matrices
of dimensions (6Nx6N), where N is the number of points used
in modelling the flexible parts; five for the turbine, two for
the coupling and twelve for the compressor. A characteristic
aspect of this modelling is the use of kinetic reference systems,
KRF (Kinematic Reference Frame), integral with each rigid
part making up the discretized flexible body. In this approach,
each substructure of the FEM is represented by a superelement
characterised by the above stiffness and damping matrices[
The construction of the finite element model required a simplification of these geometries, eliminating some features irrelevant to the dynamic behaviour. The number of hexahedral elements and nodes are reported in Table II.
For each shaft, three-component forces were applied at the
mid-line of each bearing: the two radial components x and z
(Fig. 2) reproduce the hydrodynamic reaction of the bearing
on the journal, while the y component, using a bistop function,
simulates the thrust bearing [
Integrating the differential equations of Reynolds, in accordance with the Ocvirk and Dubois approximation for short bearings, the characteristics of the lubrication fluid in the bearings were obtained (eqq. 1-7) which determine, together with the static component of the load, the elastohydrodynamic reaction of the bearing on the pivot (eq. 8).
0 !b rb b3 "0 c3 (1 sin2 0 "2)2 + 0 + 3 "0 cos 0 sin 0 + 2(1 + "20) cos2 0
4(1 "20) 25 (1 "02)3 0 !b rb b3 " c3 (1 + 2"20) sin2 4(1 kXY = kXZ = kY Z = kZZ = rXX = # (6) # (7) (8) rXZ = rZX = 0 rb b3 "0 c3 4 sin2 (1 "20)02 + + 3 "0 cos 0 sin 0 2(1
Three-component vector forces were used to model the forces exchanged between the teeth of the hubs and those of the bells. Using IMPACT and BISTOP functions, these vector forces limit the axial, radial and tangential movements of the hubs inside the bells.
The axial movement of the sleeve collar-bells system ( 3 mm) is limited by the z components of the forces of contact between the teeth simulating the behaviour of the two
O-rings pressure mounted inside the bells. The x component of the vector force is applied to the tooth flank in the tangential direction, while the y component is applied in the radial direction (Fig. 4 and Fig. 5).
VI. MODEL VALIDATION AND ANALYSIS OF RESULTS
The model, was validated on the basis of experimental
measurements conducted at the Enichem Priolo plant using
inductive proximity transducers, which measure the radial and
axial movements of the shafts in their bearings. The
experimental measurements of the monitoring system were filtered
and only those frequencies of major interest appear in the
spectra, i.e. frequencies up to the value at the speed of rotation
and at multiples of this speed [
The numerical modal analysis was performed using the Lanczos method of constants.
Since the frequency corresponding to maximum speed (4000 rpm) is 66 Hz, the possibility of torsional resonance with the two shafts and the coupling is improbable and, therefore, the operating anomalies are considered to be the result of shaft misalignment. Thus, as well as an analysis of the dynamic behaviour of the system under normal operating conditions, a misalignment between the shafts was simulated with values of
Fig. 6. Frequency spectrum of the movements in the internal turbine bearing. 40 and 90, the maximum values admitted by the manufacturer at speed and in transient, respectively.
Fig. 7 shows the displacement calculated during six revolutions at a speed of 3920 rpm at the two sensors mounted at 45 with respect to the vertical, in the turbine bearing located near the coupling.
The displacements, calculated under normal operating conditions, have amplitudes of less than 1.5 mils (0.0381 mm) and in the real system can, therefore, be considered background noise produced by the surface roughness of the pivot. The coincidence of the values measured by two sensors indicates that the pivot rotates with an almost constant eccentricity. The values of the displacements for rigid and deformable elements are compared in Fig. 8.
The need to use a model with deformable elements appears evident. Although this model presents considerable difficulty in construction and longer calculation times, it yields results which are comparable to those measured experimentally. In the model with rigid elements, instead, the displacement values are incompatible with the normal operation of the machine, above all considering the fact that they were calculated with zero misalignment.
Fig. 9 shows a comparison of the displacements and relative spectra of the turbine bearing at the sensor at +45 for misalignments of zero, 40 and 90. A misalignment of 40 leads to displacements of about 3 mils (0.0762 mm), the limiting value beyond which the critical operation alarm is activated. A misalignment of 90. produces displacements of almost 6 mils (0.1524 mm) which result in the machine blocking, given that this exceeds the maximum allowed radial play of 0.13 mm. The frequency diagrams evidence that increasing the misalignment results in a decrease in the amplitude of the vibrations at 3920 rpm (65.3 Hz) and conversely, a considerable increase at double the rotation speed. Comparing these data with those of Fig. 4, it can be deduced that during the experimental measurements the machine was operating with a misalignment of less than 40. The misalignments, therefore,
Fig. 10. Comparison of displacements in the internal and external bearings. provoke vibrations at frequencies which are multiples of the frequency at operating speed.
Finally, Fig. 10 shows a comparison of the displacements calculated in the two bearings (internal and external with respect to the coupling) of the compressor and turbine, when a misalignment of 40 is simulated. In the turbine, the values of the displacements are substantially coincident and show a fairly regular trend due to the fact that jibbing between the turbine itself and the coupling was provoked to simulate the misalignment. In the compressor, the internal bearing shows a trend similar to that of the turbine, while the external bearing shows a rather irregular trend as a result of being dragged by the coupling and thus oscillating around the equilibrium position.
The possibility of constructing a reliable numerical model with flexible bodies able to simulate the dynamic behaviour of the turbo-compressor in a realistic manner also made it possible to obtain complete information regarding the stress and strain states of the various machine components while operating. In particular is was possible to analyse the impulsive interactions between the teeth of the hubs and those of the bells Fig. 11. jibbing).
Strains in the coupling and bearings (misalignment of 90 with when jibbing occurs, evaluating the stress and strain states. The simulations of misalignment with jibbing between the axis of the turbine and that of the coupling reproduce the most severe operating conditions; the bells present elevated strains (Fig. 9) accompanied by a displacement of the centres of rotation of the pivots in their bearings. The stress peaks are concentrated on the external surface of the bells and at the toothing, reaching stresses close to the elastic limit (0.21 GPa).
Figure 9 shows the strain distributions in the zones of the bells and in the bearings where the stresses reach maximum values, remaining, however, below the elastic limit. From the analysis of the dynamic behaviour of the turbo-compressor, it emerged that in misalignments of up to 90, the strains in the bells and bearings in any case allow the machine to operate; the limits of vibration anticipated for the pivots are not exceeded and the lubrication film in the bearings is not ruptured. During the simulations of jibbing, it was possible to distinguish two phases; in the first phase a considerable amount of energy is absorbed by the bells, followed by a phase of settling in which the shafts find a new position of dynamic equilibrium on the bearings and the strain energy of the bells decreases.
The dynamic optimization of a turbo-compressor, simulated through a complete model with flexible bodies which reproduced all the interactions between the parts of the system was performed. The validity of the model was verified through the comparison of data obtained by calculation with that measured experimentally provided by ENICHEM. The models usefulness is demonstrated by making it possible to study typical situations of malfunction of the real system, such as the jibbing of the toothed coupling, the most frequent cause of alarms signalled by the vibration monitoring system. The study allowed the simulation of the functioning of the rotors at operating speed, performing an investigation into the maximum permitted misalignment between the shafts and the coupling. It was possible to verify the entity of the displacements of the bearing pivots, and analyse the strains and the stress states in the toothed bells of the turbine-compressor coupling.
Through the exact determination of the natural frequencies of the three main components of the system, it was possible to affirm that the critical operating characteristics are not linked to phenomena of resonance produced by applied forces originating in the bearings.
This study was made possible by the helpfulness of Enichem Priolo. The authors wish to thank Engineer Antonio Rosolia, who supplied the data and material necessary for the development of this research.