Dynamic Stability of an Elastic Rotating System : Shell-Disc-Shaft

This paper presents a methodology and some results on the dynamic stability of an elastic rotating system consisting of oneand twodimensional members. These parts may contain different kinds of unsymmetries: either from massor stiffness imperfections or from anisotropic especially hydrodynamic bearings. The equations of motion are formulated using virtual work and an Finite Element approach. Special attention is paid to a kinematically consistent coupling of the elastic shell and disc. The eigenvalue extraction is based upon the method of Lanczos including a modal reduction and a correction process in order to ensure true diagonal system matrices. Some typical results for a shaft-disc-shell system with different bearings and imperfections are presented in detail.


INTRODUCTION
Rotating elastic systems including thin cylindrical shells, discs and shafts are widely used in mechani- cal, aeronautical and marine engineering.Modern de- sign of electrical Lmachines, for example, includes thin-walled bell-type rotors preferably made of com- posite material in multilayered mode.The elasticity of such shells may have a significant influence on the overall dynamic behavior of a system like that shown in Fig. 1.The mechanical and geometric properties of the other parts of the system, too, do effect the stabil- ity.From classical one-dimensional studies it is very well known, that imperfections and deviations from any symmetries can cause significant effects on the 33 dynamic stability.This paper presents results for a system with following members: Rotor shaft with small deviations compared with the ideal round shape of the cross-section.Bell-type thin shell made of homogeneous material, prestressed by the radial acceleration forces with lo- cal perturbations; for example caused by a discrete load system with mechanical properties of its own.Elastic disc connecting shaft and shell.This disc is prestressed, too, and may have mass and stiffness imperfections.Anisotropic, for example hydrodynamic bearings.
An overall analysis of such a coupled system has not been published in literature.Results for only parts of this system are especially given for flexible turbine rn FIGURE Elastic rotating system with hydrodynamic bearings and additional pair of discrete masses.
A global result of these papers is that the membran stresses in the rotating members have a significant influence on the stiffness of the system; in structural analysis this effect is called "geometric stiffness".The way to involve this effect into the analysis starts from a nonlinear representation of the strains e by means of the displacements u and slopes [.Rotating thin cylindrical shells of finite length with arbitrary boundary conditions including the geomet- ric stiffness are rarely treated in applied mechanics.
Plonski and Ruge [1990] presented some results for spinning shells with mass imperfections and different multilayered composites but with rigid disc and rigid bearings.There are some few papers dealing with finite perfect cylindrical shells by Suzuki et al.   1991], with ring-stiffened shells by Huang and Shu   [1992] and dealing with true three-dimensional solid and hollow cylinders; Seemann [1992].The situation for rotating rings is much easier and there are some profound studies by Clemens and Wauer [1985] and Wauer [1987] which confirm experimental results published by Endo et al. [1984].
Our attempt was to design a numerical tool which allows a true three-dimensional analysis of spinning coupled systems without imposing symmetry or iso- tropy conditions.We restrict ourselves to the station- ary situation with constant angular velocity fl and locally linear behavior of the fluid-film, which is non- linear in general.
Whatever basis we use for describing the state equations we will eventually get a system of linear- ized differential equations in the time domain with periodic coefficients like cos(2t), sin(2t).
Aq + BO + Cq-r, A M2, B 2D. M + D (D,t) + D s, (..):Bearing; (..)s: Structure; (..)L:Linear; (..)4s: Membrane Stiffening, r :radial mass-acceleration forces; Mi" Caused by inertia mass; Di" Damping; K" Stiffness.() Most of the degrees of freedom q in (1) describe the displacements and slopes of the continuous elastic members which are discritized by a Finite Element approximation.Consequently we decided to use a ro- tating basis A fixed to the undeformed constantly ro- tating configuration.Then only some few discrete displacements and slopes of the points where the bearings are fixed to the shaft are associated with periodic coefficients.

KINEMATICS, VIRTUAL WORK
The virtual work 8W of the internal elastic forces and the inertia forces is mainly described by the position vector 2p which depends on the displacement-vector fie of an arbitrary point P in Fig. 2 outside the mid- dle-surface of the shell. (2) The procedures for treating the disc and the shaft are rather similar.The vector fie consists of two typical parts: a translational and a rotational one.

p (3)
The rotational part in (3) results from linearizing the fundamental velocity equation da/dt ($ a.The displacements fi and slopes [ are measured with re- spect to the rotating basis A, where u and [ without vector-sign are the coordinates.
fi Au, 6 A6. (4) FIGURE 2 Notations, position-and displacement vector of a point P outside the middle-surface of shell.
The last, rotational, part in (5) is governed by the elastic slopes [, which are assumed to be indepen- dent of the displacements and to be very small: sin [3 [3, cos [3 1.This independent choice of slopes has several advantages: Simultaneous representation of translational and rotational inertia.
Simple continuity conditions with respect to the variational approach; C-continuity only.
Automatic treatment of shear elasticity.
The virtual displacement 8fi,, follows directly from (5), 8fir-A (u G3[). (7) The acceleration p is easy to describe if one uses the fundamental skew symmetric representation, 00 0 A-A-AD/" I-ADG'i , =G, 00-1 o o() for A with the local angular velocity o, which sim- plifies to o 12 a in our specific situation.
, , A (// + 2fGlti + 2G12/g AG ( (9) The virtual displacements from (7) and the accelera- tion from (9) are taken to describe the virtual work of the inertia forces.If the middle-surface with 0 is the geometric center in direction fi3 of the shell, that means f de 0, the rotational and translational inertia parts are decoupled; p is the mass density, the thickness of the shell.
8W y ,,',, pd{ &q d, (10) The virtual elastic work gU, here presented in detail for the cylindrical shell, consists of three parts U, Ua4, Us caused by bending moments rn, inplane-or membrane forces n and shear forces q.
U -B E K df; dq with m B E K, lfer u M=-DEe dfodl with n =DEe, lf T Us -G y dfo dq with q G T, The strains and curvatures e, /, K are functions of the displacements u and slopes [ and can be found in textbooks. .
-[2, [31   (14)   The nonlinear part in ( 13) must be incorporated into the analysis in order to describe the geometric stiff- ening.Simplified versions for this nonlinear part are known, for example given by Donnel and Marguerre.However, there are regions of distinct parameters, where results based on Donnel-Marguerre's approxi- mation differ too much from the better ones based on (13).Finally, the sum of the virtual works is taken to describe the state equations of the problem under consideration.
The column d in (15) depends on the accelerations (9) and strains (13), ( 14) which are functions of the slopes and displacements.

FINITE ELEMENT APPROACH
An independent treatment of displacements and slopes as introduced in section 2 allows for shear deformations and rotational inertia effects and has to ensure nothing but continuity with respect to u and [ along adjacent elements.This formulation and proce- dure was introduced by Reissner and Mindlin.
A further aspect with direct influence on the num- ber of degrees of freedom is the FEM-net .whichmust be able to describe the essential modes with respect to the dynamic behavior.The necessity to assign a rather fine net results from the wellknown effect in linear analysis that cylindrical shells tend to vibrate in first-order modes with several nodes in circumfer- ential direction.It has been found that a FEM-net with 40 elements in circumferential direction and 4 elements in direction of the shell-axis gives enough freedom to the shell.Adding the degrees of freedom for the disc and the shaft gives a total number of round about n 6000 DOFs incorporated in the col- umn q in (1).
Such an approximation is based upon an 8-node element with polynomials in both directions of the shell surface indicated by the coordinates { and Alternative interpolations use polynomials in direc- tion of the shell axis, for example a linear interpola- tion, and a kind of Fourier-expansion in circumferen- tial direction.+tral, O<-f;<-h, (16) t r cos p cos 2q)...sin q sin 2q)... ], p I/R, a r= [a 0 al a2...b b2... ].
Such an approach is normalized with respect to the overall-quantities of a total ring around the shell, not with respect to discrete nodes.Adding a discrete local mass imperfection in a special point at such a ring causes a distribution upon the whole element-mass- matrix.In other words, such a discrete mass (or stiff- ness) destroys the diagonal structure of the element matrices usually guaranteed by the orthogonality of the trigonometric functions in circumferential direc- tion.There is not doubt about the global flexibility of the classical two-dimensional polynomial approxima= tion.So we decided to adopt this approach through- out the whole analysis.The result of this procedure is the matrix equation (1) with matrices which are sub- structured in a very significant manner as is shown by P. RUGE and P. SENKER Senker [1993] in detail.The strategy to solve this equation consists of three main steps: 1. Treating the quasi-static situation with q qo, 0o 0 and taking into account constant parts )* of the matrices only.
(K + K + fD* a2M0)q0 1)2 R f. ( 18) The right hand column f represents the radial ac- celeration forces caused by the given rotation.The product azR is separated in order to make a com- parison with the corresponding part f/ZMoq o from the left side in (18).The nodal quantities in q are slopes and displacements caused by elasticity and of course their magnitude is less than one per thousand of the radius R of the shell; the mass- factors f and M do not alter this fact: rithms it has to be rewritten as first order  with unsymmetric matrices H and G in general.3. The time behavior of the variations q + is given by an exponential solution if the matrices involved are constant in time.Otherwise, if some of the matrices are periodic, an additional trigonometric summation must be incorporated as is well known from Hill's method.
+ q a exp(t) if matrices are constant.
(24) R f>> Mqo. ( The part fDB *, too, has been proved to be without numerical influence and finally the quasi static so- lution qo can be calculated from a simplified equation, with a constant coefficient matrix, which is not affected by the angular velocity . 2. The nodal quantities qo from (20) are supple- mented by linear time-dependent variations q+, q qo + q+(t), to study the local time behavior of the static solu- tion qo.Inserting (21) into the virtual work of the inertia and elastic forces and omitting higher than linear terms results in a linear homogeneous dif- ferential equation in the time domain with the geometric stiffness part K s which is a function of qo and appears automatically.
q+ [a 0 + a cos at +...+ b sin t +...] X exp(Xt) if matrices are a-periodic.( 25) Or, if one uses Floquet's approach, the Floquet transition matrix F for the first order system (23) must be calculated numerically by stepwise inte- gration along a period T 27r/), Pl FPo, Po p(t O),p p(t T). (26)   This finite difference equation is solved by p +Po.The eigenvalues qb from the eigenvalue prob- lem Fpo +Po indicate the stability behavior.
So far the steps which have to be done towards a time dependent solution q qo + q+(t) within a local region around the static situation qo.The essential part of this solution is hidden in the eigenvalues X or qb respectively.The local dynamic stability of the ro- tating system under consideration is guaranteed if and only if no real part g of any of the 2n eigenvalues in (24), ( 25) is greater than zero, M2q + + [2M1 + DB (t) + Ds]4 + + [KB(at + K + K s + aD1B (at)   aZM0]q + 0. ( In order to treat this equation by standard algo- ki i at-jtoi' j2= 1, Stability iff i -< 0, 1...2n; (27)   or if the modulus of no Floquet-eigenvalue + exceeds 1. (28) A typical problem arising from coupling disc and shell must be mentioned in order to treat the rotor system in a proper manner from the mechanical point of view.The FEM-representation consists of 3 nodal displacements but only 2 nodal slopes.With regard to the shell these slopes are [3, [32 along fi and I 2 as is shown in Fig. 3.For the disc these slopes are [32 and [33.
Shell: Shell 111 q-2a2, Disc" z)i-= 22 -t-3113 (29) The variational foundation of the virtual work representation demands for continuity with regard to dis- placements (there is no problem at all) and slopes.
The slopes 6j from the shell and [33 from the disc, however, miss their counterparts from the opposite structure.A mathematically consistent way of treating this situation was described by Kebari and Cassell  1991, 1992].They introduced an artificial rotation, 3 for the shell, is not satisfied a priori but is incorporated in a weak sense into the linear membran part of the virtual work, here shown for the shell only.
(32) Shell:, [33 (v,e u,n), (30) and 2 for the disc in a similar manner.By this the momentum equilibrium normal to the middle-surface 4 NUMERICAL ANALYSIS Keeping in mind what has to be done to analyse the homogeneous system Hp + Gp o in (23) of double order 2n 12000, either with constant or f-periodic .ateShell FIGURE 3 Different elastic slopes at the interface of disc and shell.4O P. RUGE and P. SENKER matrices, it becomes clear, that the order of the system must be reduced significantly.Even if there are no f-periodic parts in H and G these matrices still contain the angular velocity f as a constant parame- ter and it is necessary to solve the corresponding al- gebraic eigenvalue-problem for each distinct value fl separately.Nevertheless, if we restrict the analysis to the first, perhaps 100 eigen- values, out of the total number or not, this way is far from being competitive.An economic strategy .toget satisfying results starts from an auxiliary problem out of the original one ( 22), with symmetric matrices, even if the bearing stiffness is not symmetric in the true situation.A highly effi- cient method to calculate the lower spectrum of the symmetric problem (34) was described by Lanczos: The original pair M2, K of order n is transformed into a pair T, I of any order r less than n with a tridiagonal matrix T.
OriginalPair: oMzy Ky.35) is used to reduce the original complete homogeneous problem (22).It must be kept in mind, however, that the basis Z offers but an approximation, Y-[y...y] LZ, (37) for the basis Yr of the complete system.This can be seen from the products, (38) which are not purely diagonal as they should be.These errors have been proved to falsify the stability regions of the rotating system in a more or less sig- nificant manner.A simple correction, however, start- ing from a symmetric Cholesky decomposition of the error-matrix E,vi SS r allows the calculation of an improved modal basis Yr instead of Y by forward substitution: S'r Yy '.
(39) Indeed, this basis Yr gives true orthonormality with respect to M2, f.rM2f.I   (40)   and no more errors concerning regions of stability have been observed.

EXAMPLES AND CONCLUSION
Some typical results for a shaft-disc-shell system al- ready shown in Fig. are presented in detail.The angular velocity f is assumed to be constant and the dimensions and material data is choosen as follows: Geometry Shell: R 60 mm, 0.6 mm, R 300 mm.Disc:t 15mm, R 60mm.
Shaft: 11 220 mm, 12 30 mm, r 15 mm.Material E-2.06 l0 N/mm2, v 0.3, 9 7.8 10 -9 Ns2/mm4.The whole elastic system is modelled by n 6279 degrees of freedom.By modal reduction however, as was described in chapter 4, this amount was reduced to r 60 remaining DOF' s without influence concern- ing the stability behavior within the lower frequency range.The numerical effort in calculating the eigen- values X or qb was not essentially different comparing Hill' s and Floquet' s method.The results were found to be rather identical.Further informations and a lot of numerical details are presented by Senker [1993].
It is worth mentioning that there may be signifi- cant differences between the stability regions of a system either with rigid or with elastic shell.Fig. 4 shows these differences; first for 2-pads bearings in Fig. 4a, b and then for 4-pads bearings in Fig. 4c, d: The elasticity of the shell causes additional regions of instability and decreases the beginning of these regions.The method choosen to examine instability is Floquet's one.Obviously, different types of hydrodynamic bearings can cause rather different results.The same system, this time with elasticity for each of the three parts but with an oval shaft indicated by 1.8, Cq+C-10aN/m, gives results in Fig. 4e which are not too different from the situation in Fig. 4d for a round shaft.Finally a pair of discrete point-masses is added at the free end of the shell in a symmetrical manner.These two masses together are 3 percent of the total shell mass.Once more we found similar curves in Fig. 4f as in the previous situations but this time with an addi- tional heavily increasing region of instability which indicates global instability.These results are only some few out of a great amount which has been accomplished by the second author.To come to a conclusion, we can say that we have established a numerical tool suitable for dealing with the dynamic stability of rotating elastic systems including twodimensional members like shells and discs even with different kinds of unsymmetry.The eigenvalue extraction is based upon the method of Lanczos.It must be emphasized however, that this e) method applied to pairs of unsymmetric matrices is far from being established in an absolutely reliable manner.Consequently, further activities will concen- trate on this subject.

Acknowledgement
The results presented in this paper are based upon a research project which has been supported by the German Research Foundation--DFG.

FIGURE 4
FIGURE 4 Critical modulus I+I of Floquet eigenvalues versus angular velocity 2. a) 2-pads bearings with rigid shell, b) 2-pads bearings with elastic shell.(continued) FIGURE 4 (continued) c) 4-pads bearings with rigid shell, d) 4-pads bearings with elastic shell.(continued) FIGURE 4 ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy SystemsEconomic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.S SU UB BS SC CR RI IP PT TI IO ON N I IN NF FO OR RM MA AT TI IO ON N Volume 1 (2006), 4 issues per year Print ISSN: 1748-9237 Online ISSN: 1748-9245 Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM 3 member rates please email s su ub bs sc cr ri ip pt ti io on ns s@ @m ma an ne ey y. .cco o. .uuk k E ED DI IT TO OR RS S D Dr r F Fu uj ji io o A Ab be e NIMS, Japan D Dr r J Jo oh hn n H Ha al ld d, IPL-MPT, Technical University of Denmark, Denmark D Dr r R R V Vi is sw wa an na at th ha an n, EPRI, USA F Fo or r f fu ur rt th he er r i in nf fo or rm ma at ti io on n p pl le ea as se e c co on nt ta ac ct t: : Maney Publishing UK Tel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: subscriptions@maney.co.uk or Maney Publishing North America Tel (toll free): 866 297 5154 Fax: 617 354 6875 Email: maney@maneyusa.comFor further information or to subscribe online please visit w ww ww w. .mma an ne ey y. .cco o. .uuk k C CA AL LL L F FO OR R P PA AP PE ER RS S Contributions to the journal should be submitted online at http://ema.edmgr.comTo view the Notes for Contributors please visit: www.maney.co.uk/journals/notes/emaUpon publication in 2006, this journal will be available via the Ingenta Connect journals service.To view free sample content online visit: w ww ww w. .i in ng ge en nt ta ac co on nn ne ec ct t. .cco om m/ /c co on nt te en nt t/ /m ma an ne ey y