Numerical and Experimental Investigations of Coupling Effects in Anisotropic Elastic Rotors

It is known that in elastic disc-shaft systems in particular, the one-nodal-diameter mode of the discs can be highly coupled with the bending modes of the shaft. Consequently, when the system rotates, the elastic modes of the flexible discs are coupled with the gyroscopic modes of the flexible shaft equipped with rigid discs. In the paper this coupling effect is investigated numerically and experimentally.


INTRODUCTION
Traditionally, the analysis of the dynamic behavior of rotating disc-shaft systems is performed using two different approaches.The rotordynamic ap- proach is focused on shaft behavior and assumes that the effect of wheel flexibility is negligible.On the other hand, the bladed-disc approach assumes that the shaft is rigid and analyses the behavior of the different wheels separately.These approaches are usually efficient and accurate enough.However, there is an increasing number of cases where coupling effects between bladed-disks and shafts cannot be ignored.Possible couplings have been pointed out in early studies by Dopkin and Shoup 264 H. IRRETIER et al.Khader (1984), Khader et al. (1990, 1991), Wu and  Flowers (1992).Particular features such as inertial coupling, mistuning influence and forced response behavior have been examined.Even recently, an analytical procedure has been developed by Chun  and Lee (1996) in order to study the influence of rotation, blade stagger and blade pretwist on coupled behaviors.
These analytical approaches, while very useful, are based on simplified formulations and thus cannot be applied efficiently to real structures with complex geometries.On the contrary, direct and precise finite element modeling of the whole coupled system would be attractive, but cannot be performed due to the unrealistic computer re- sources needed.Consequently, intermediate model- ing techniques have to be proposed and assessed.
A first type of intermediate model consists in analysing the shaft with rigid disc and the bladed assemblies separately, and then solving the coupled problem after a modal reduction based on the calculated mode shapes (Zhang et al., 1994).This model should give quite accurate results when the link between the shaft and the disc is rigid,, but it is believed to be too restrictive for general applica- tions.Dealing with the whole coupled assembly, direct modeling is possible using an axisymmetrical description of the rotor (G6radin and Kill, 1984;  Stephenson and Rouch, 1993).The resulting model should be effective in certain cases but it is penalised by the axisymmetrical hypothesis and by the modeling difficulties associated.Finally, it appears that the most interesting models are those based on the cyclic symmetrical properties of the structure (G6radin and Kill, 1986; Hohlrieder et al., 1994;  1997; Jacquet-Richardet et al., 1996).However, these models remain heavy and, in this case, special procedures should be developed in order to main- tain the computer requirements at an acceptable level.The formulation presented here is based on a global analysis of rotating assemblies modeled using a finite element technique.The undamped non-rotating system is first analysed using the wave propagation method associated with a component mode reduction.Then, the whole flexible system, submitted to centrifugal and gyroscopic effects, is analysed after a modal reduction.
In parallel to the analytical and numerical work dealing with rotating flexible bladed-disc-shaft assemblies, efforts are made to extend experimental modal analysis techniques on rotating structures (Irretier and Reuter, 1995;1997).Generally, this   requires that the modal identification algorithms are based on time-variant systems, to account for anisotropic effects in the bladed-discs or shafts of the rotating assembly (Reuter, 1997; Irretier, 1998).In addition to the numerical procedure and results described in the paper, corresponding ex- perimental results are presented which are based on this extended modal testing and identification technique.

THEORETICAL BACKGROUND
The method presented here has been developed in order to deal with turbomachinery wheels with com- plex shapes and geometry.The motion equations, expressed in a body fixed coordinate system, of a flexible bladed-disc mounted on a flexible shaft rotating at a given uniform angular velocity, can be expressed as: [KE + KG({,:5}s KS]{,:5} {FC(f2)}, (1) [M]{('}d if-[CM + CR]{(}d where [M] is the mass matrix, [CM] the mechanical damping matrix, [CR] the gyroscopic matrix, [KE]  the elastic stiffness matrix, [KG] the geometric stiffness matrix, [KS] the supplementary stiffness matrix and {FC} the nodal centrifugal force vector.
6}s is the static equilibrium position of the structure under centrifugal loading and {6}d is the small amplitude dynamic displacement around the static position.For a given rotation speed f, the solution of the non-linear system (1) using a Newton- Raphson procedure gives the static displacement vector.The stiffness matrix is then known and the dynamic problem can be solved.When dealing with bladed-disc--shaft assemblies, the size of the problem can be reduced if their rotationally periodic characteristics are taken into account.Such structures are constituted with N identical jointed sectors.According to the wave propagation theory in periodic media, Thomas   (1979), the dynamic displacement vector of the different sectors p is related to the corresponding quantities of a reference sector by the following phase relations: {6}-{6} cos(p-1)/3n + {(5,} sin(p- where {6} is the vibrational displacement of sector p, {(5 c}, {6,} are generalized quantities associated with the basic sector and/3n 2rcn/N is the phase difference between the displacement of two adjacent sectors.N is the total number of sectors and n, Fourier order, takes the discrete values: Applying the wave propagation relations (3), the dynamic problem (2) is divided into small size systems associated with each of the possible phase angles [Mn] where {Sn} {5c, 5"s) t.When dealing with isolated bladed-discs, damping and gyroscopic effects are usually neglected and (5) becomes a Hermitian system which is solved only once, for an operat- ing speed.When dealing with bladed-disc-shaft assemblies, gyroscopic effects can no longer be neglected and the variations of frequencies with respect to the rotation speed are relatively large.Consequently, it becomes necessary to solve a different complex and non-Hermitian eigenvalue problem for each value of the rotation speed in the operating range.In this case, using only the tradi- tional reductions (Craig and Bampton, 1968) leads to a lengthy procedure which is still not compu- tationally efficient.In the following, we shall show that the analysis cost of bladed-disc-shaft assembl- ies can be considerably reduced, without appreciable loss of accuracy, provided the rotating mode shapes are written as a linear combination of the associated non-rotating mode shapes.

REDUCED DYNAMIC PROBLEM
Let us first consider the undamped system at rest (f 0).Thus (5) reduces to: The solution of ( 6) is performed after a condensa- tion based on the efficient Craig and Bampton (1968) substructuring method.This solution gives the frequencies and mode shapes of the non- rotating system (Jacquet-Richardet et al., 1996).
All the mode shapes calculated are grouped into modal matrices [], used for the reduction of the rotating system (5).Assuming that: (7) system (5) becomes: [m]{6,n} + [cn]{dln} + [k]{%} {0} (8)   with: The solution of ( 6) and ( 8) is performed for all the possible values of the phase parameter/3 2rm/N given by (4).The frequencies of the rotating system are given directly.The corresponding mode shapes are obtained after applying the two successive transformations, ( 7) and (3), to the calculated eigenvectors.The modes of the shaft can occur only with n 0 or n (n 0 torsion and longi- tudinal modes, n bending modes).The modes of bladed-discs are always classified using an analogy with axisymmetric modes, which are mainly characterised by nodal lines lying along the diameters of the structure and having a constant angular spacing.They are either zero (n =0), one (n 1), two (n 2) or more (n > 2) nodal diameter bending or torsion modes.Inertial disc-shaft coupling effects are important for modes associated with n 0 and n 1, because these modes are the only ones where a shaft motion can occur.The disc modes with zero-nodal-diameter, which are char- acterised by a resultant axial force, interact with the longitudinal shaft deformations.The modes with one-nodal-diameter, which exert a net pitching moment and a shearing force, interact with the shaft bending modes.
In order to be able to identify possible resonance points, the frequencies are calculated for the whole operating range of the structure and the results classically reported on a Campbell diagram, which illustrates the dependence of the natural frequencies on the rotor speed f.When using the proposed analysis method, the non-rotating mode shapes of the assembly are calculated only once.Then, the following steps are involved for each rotation speed considered: the static problem is solved; the dynamic problem (5) is reduced according to (7); the reduced system (8) is solved.
Compared to traditional procedures, the computer cost saving is considerable and allows precise analyses of complex industrial structures using efficient workstations.

APPLICATION
For the verification of the numerical model and analysis and to study coupling effects of an elastic disc on a flexible shaft under rotation, a test stand was designed as shown in Fig. (Reuter, 1997).It consists of a thin circular steel disc (outer radius 300mm, thickness 3mm) fixed in an overhung position on an elastic circular steel shaft (length 175 mm, diameter 15 mm).The supported shaft prolongation (length 101.5 mm, diameter 28 mm) in two ball bearings can be considered to be rigid in comparison to the flexibility of the overhung shaft.Two different types of disc tuning states were investigated: (I) Completely tuned and (II) mis- tuned by two diametral masses (each 0.047 kg at a radius of 280 mm as visible in Fig. 1).For systems (I) and (II) numerical calculations were carried out at INSA Lyon by the technique described above while for system (II) additional experimental investigations were carried out at the University of Kassel.For these tests, a step sine excitation by a magnetic exciter with an integrated force trans- ducer was used (Fig. 2) and the response was measured by an eddy current pick-up, both without contact with the rotating disc.Both are visible in Fig. 1, too, and the schematic representation of the test stand is given in Fig. 3 (Irretier and Reuter, 1995).From the excitation and response signals, frequency response functions were determined for the rotating system based on a theory for time- variant systems and, by curve fitting algorithms in the frequency domain.The modal parameters were identified with respect to the stationary Dame (index s, Fig. 3).other reference frames, e.g., the rotating frame (Reuter, 1997).
Considering the numerical model, a portion of 54 of the disk-shaft assembly is meshed using 108 isoparametric brick finite elements with 20 nodes.
The resulting mesh, presented in Fig. 4 (dark zone) comprises about 550 finite element nodes and, consequently, about 1650 degrees of freedom.The matrices associated with the disk elements are evaluated using a reduced Gaussian integration to avoid numerical shear locking.Damping effects are neglected.The natural frequencies associated with the first modes calculated are conveyed onto the Campbell diagram presented in Fig. 5.At rest, these modes can be classified as follows.The first mode is mainly a shaft first bending mode, influenced by the disk mode with one-nodal-diameter.The second mode is a disk mode without any nodal diameter or nodal circle (umbrella mode).The third mode is a stationary pick-up nodal diameter stationary magnetic exciter   shaft bending mode strongly coupled with the one- nodal-diameter disk mode.Finally, the fourth mode is a disk mode with two nodal diameters.
The progression of the disk modes with rotation is only induced by centrifugal effects (stress stiffening and spin softening).As predicted, no coupling occurs between these modes and shaft bending.When rotating, each shaft mode splits into back- ward and forward branches, due to gyroscopic effects.When the speed increases, there is a clear coupling between the first and second shaft bending modes.

Influence of Disk and Shaft Flexibility
The influence of disk flexibility on the overall behavior is illustrated by comparison of the results presented in Fig. 5, with the results obtained using a rotordynamic approach (rigid disk).The compar- ison, which naturally involves only shaft modes, is presented in Fig. 6.When examining Fig. 6, it appears clearly that the influence of disk flexibility is considerable here.At rest, the first frequency given by the rotordynamic approach is overesti- mated by a factor of 31% and the second one is overestimated by a factor of 87%.This last result illustrates the coupled nature of the second shaft bending mode with the one-nodal-diameter disk mode.Considering the evolution of frequencies with rotation, the forward branches (lower branches in the rotating frame) are much more influenced by disk flexibility effects than the back- ward branches.The influence of shaft flexibility is illustrated in Fig. 7, where the flexible disc-shaft assembly results (solid lines) are compared with results associated with a rigid shaft model (dotted lines).
The rigid shaft results were obtained by considering the disk isolated and clamped at its inner diameter.
Figure 7 confirms that the zero-and two-nodal- diameter disk modes, identified previously, are not influenced by shaft bending.On the contrary, the highly coupled nature of shaft bending and one-nodal-diameter disk bending is here also high- lighted.For the flexible assembly, the one-nodal- diameter mode has been identified as a component of the first and third modes: at rest fl =2 14.2 Hz and f3=39.4Hz.The pure one-nodal-diameter disk mode is just between these two frequencies: f--21.6Hz.When the speed increases, it can be noticed that the backward branch of the first mode and the forward branch of the third mode of the flexible assembly, evolve toward a pure one-nodal- diameter disk mode.

Influence of Mistuning
The influence of mistuning has been tested by adding two diametrically opposed point masses of m=0.047kg each, on the disk, at a radius R 280 mm.Due to these two masses the assembly is no longer axisymmetric and, consequently, the choice of the basic cyclic sector is no longer free but should take up half of the structure.The resulting mesh, based on the same element size as previously, comprises 360 finite elements and 2200 nodes.In this case, the computational effort needed is obviously much more considerable.
Frequency splitting can be noticed due to mistuning.As the mistuning introduced in the system is relatively low, Table I shows that the splitting induced is not very pronounced.For comparison, the test results are also presented in Table I (with the exception of mode 3 which was not investigated in the tests).The experimental frequen- cies for the first shaft bending mode (n= 1) are almost 6% higher than the numerical results.The main reason for this discrepancy is the slightly longer overhung shaft considered in the numerical calculations (180 mm instead of 175 mm).For the elastic disc mode frequencies with n 0 and n 2, the difference between the test and numerical results is much less, being around 1%.The splitting tendency due to mistuning is very well predicted for the n and n 2 diameter mode as can be seen in Table I.
The Campbell diagram for the mistuned assembly is presented in Fig. 8 for the modes under consideration.Besides the curves representing the numerical results, test results are also shown by included points.Again, the agreement between both results is excellent with slightly higher devia- tions for the first shaft bending mode, as already indicated earlier for zero rotation.

CONCLUSION
A numerical technique was presented for the calcula- tion of natural frequencies and mode shapes of coupled elastic shaft-disc systems under rotation.
It is based on a finite element modal approach combined with the wave propagation method for cyclic symmetric structures and a component mode reduction technique, both to reduce the required numerical effort considerably.Confirmed by related experimental results for a rotating mistuned disc on a flexible overhung shaft, the proposed numerical method proves to be a powerful technique for the frequency analysis of these types of rotating struc- tures.The results show that the one-nodal-diameter modes of the disc are highly coupled with the shaft bending modes for the test example considered.For these coupled modes, the influence ofrotation on the natural frequencies becomes apparent by both the centrifugal stiffening as well as the gyroscopic effect.
As expected, the umbrella mode and the two-nodal- diameter mode of the disc which are not coupled with the shaft flexibility are only influenced by the centrifugal stiffening effect.

FIGURE
FIGUREElastic disc on a flexible shaft.

FIGURE 3
FIGURE 3 Experimental modal analysis of rotating structures.

FIGURE 4
FIGURE 4 Finite element mesh of the assembly.

FIGURE 7
FIGURE 7 Influence of shaft flexibility.

E
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