Interaction of Parametric and Forced Vibrations in High Speed Rotor-Bearing-Systems BERNHARDT WEYH

In this paper the interaction of periodic parameter excitation with forced excitations of in general non commensurable basic frequencies is under consideration. The method of enlarged systems serves as a toolbox, where the central idea of the method and the structure of the enlarged system are in the same way derived as the complete homogeneous and the manifold of inhomogeneous solutions are discussed. The application to a realisation of a high speed twin-disc rotor-bearing-system of a textile spinning turbine illustrates the resonance effects that haven’t been characterized before.


MOTIVATION AND INTRODUCTION
The development of machines emphasizes single functional units or modules.Priority is given to projects which increase machine performance and productivity.These developments do not always comply with the rules of traditional mechanical engi- neering.New effects and phenomena that haven't been analysed before must be considered.However this can be supported and accelerated by modern so- phisticated modelling techniques and computer simu- lations.
A part of the development of the heart of a rotor spinning machine--the high speed rotor bearing of the spinbox-turbine, see Hanzel [1985], installed in the latest machine generation, see Socha et al.  1991 ],--will be considered here.Vibration phenom-81 ena occurring at very high speeds (80000-130000 rpm) called for modification of the twin-disc bearing.Weck et al. [1984] have supposed that the elastic in- homogeneities of the twin-disc coating and the driv- ing belt have effects on rotor vibration.A simple plain model will lead to a set of inhomogeneous differential equations with periodic parameter excitation and periodic forcing, first mentioned by Weyh [1993], where the basic frequencies are quite different and normally non commensurable.
Vibrations of linear parameter excited systems are mainly analysed by simulation techniques (initial value problems).A detailed survey of these methods for the stability problem including literature specifi- cations are given in Friedemann et al. [1977] and Weyh et al. [1991].The quality of the numerical re- sults is essentially dependant on the eigen-values, the B. WEYH parameter excitation frequency and, if present, the forced excitation frequencies.If the eigen-values are situated far from each other, the integration-time in- creases in the same way as the integration interval increases if parameter and/or excitation frequencies are different.In general long-time simulations are necessary to analyse the steady state vibrations, be- cause it must be guaranteed that the eigen-vibrations have been faded away.A wide numerical study has to be done normally, to find out general information about solution manifold, phenomena and parameter- dependance, see for example Diekhans [1981] and Laschet [1988].
These time-consuming problems can be avoided when replacing the time-variant basic system by a time-invariant description of higher order with the same or in the case of approximation nearly the same quality of solution.This idea can be realized by a simple substitution of the system states.
The method is quite similar to Hill and Bolotin (Bolotin [1964]) for the homogeneous part, because the enlarged system matrices are equivalent to the generalized Hill matrices introduced in Weyh [1989,  1990], where the approximation error and the conver- gence problems are widely discussed.Here the new idea serves for interpretation and generalization of the method.It enables us to find general resonance conditions of the interaction of parameter and forced excited vibrations with commensurable or non com- mensurable basic frequencies.
M(t)y + P(t) + Q(t)y h(t) (2) where dots denote derivatives with respect to time t.
The Tp 2rr/)-periodic, in general real, f f-sys- tem matrices will be represented by finite complex Fourier-matrix-series with j= --1), to simplify the following procedures.h(t) stands for the Tf 2wkq-periodic f 1-forcing- vector.

Enlarged System
The complex substitutions yp(t) y(t)dpat,)p -P(1)P (p -P(1)P stands for all values p of the integer interval [-P, P]) transform the basic system (2) un- der consideration of the complex series of system matrices (3) into 2P + coupled matrix differential equations of basic order 2f, see Weyh [1993] and Vieh- Hweger [1993].With the help of the hyper-coordi- nate-vector TRANSFORMATION

AND SYSTEM-ENLARGEMENT Basic System
With the help of the f 1-dimensional generalized coordinate-vector Y (Y, Y2 yf)T (1) a general structure of mathematical models of peri- odic parameter excited, inhomogeneous linear systems can be described by r ..yr_ r r rr Y (Y-v,. 1, Y0' Yl Ye) (5) the derived enlarged system can be represented in a compact linear hyper-matrix form M + P + Q y + ge(t, , :, y) h(t) (6) of the order N 2f(2P + 1) and is exactly equivalent to the basic system (2).The periodic parameter ex- cited, ordinary differential system (6) consists on the left side of a constant and a time-variant part.The structure of the constant part can simply be seen from the N N-dimensional coefficient-matrices in the sub-matrix representations given for an example in HIGH SPEED ROTOR-BEARING SYSTEMS 83 Table I.The hyper-matrices M, P, Q can be con- structed algorithmically by means of the system co- efficient matrices M, P, Q of the basic system (2)   given in (3).The hyper-matrices are closely con- nected with the Hill matrix approximations by Bolo- tin [1964] and are exactly the same as given in the generalized Hill matrix approximation by Weyh  [1989, 1990] if using real instead of complex Fouri- er-series representations in (3) and (4).The time- varying part of (6) consists of Tp-periodic coefficient- matrices of higher harmonic ordersmand this is the speciality of the enlarged systemmonly higher than the substitution order P. The f 1-dimensional sub- vectors hp(t) h(t)da', yp -P(1)P ( 7) oi the hyper-vector of inhomogeneity on the right side of ( 6) h(t) (hTp(t),.. h r (t), h(t) hr(t),, hr(t)) r (8)   can be interpreted as amplitude modulations of the forcing function of the basic system (2) on the har- monics of the parameter excitation as carrier frequency.

Approximate System
By neglecting the higher harmonic parameter excita- tion g,(o), equation ( 6) reduces to a time-invariant matrix equation of the order N M + P + Qy h(t) (9) A detailed discussion of the approximation errors and the convergence of homogeneous solutions can be found in the same way in Weyh [1989]    of the Coefficient-Matrices M, P, Q of the Enlarged System with K 3, P 2.

THEORETICAL CONCLUSIONS
Instead of considering parameter excited differential equations (6) of order 2f, constant systems (9) of or- der N can be solved.The complete solution of the approximate system can be calculatedmand this is the essential advantagemby the well known algorithms of constant systems.In particular the follow- ing conclusions can be derived:

Stability Problem
The eigen-value problem of the approximate system (MX 2 + P + Q)a 0 (10) leads to (1)N eigen-values 1 i,] --JOOi,y, )i -f(1)f :/: 0, yj -P(1)P ( 11) It can be shown (for a detailed derivation see Naab et al. [1988], Neumann et al. [1994]) that the eigen- values coincide with the 2f characteristic exponents i + yto of the basic system (2) except for multiple of the parameter excitation frequency f and approxima- tion errors Aid in the form -[-J O') -at-jD.
i,] "q-J c i] -}-Aid (12)   From the afore-mentioned connection it follows, that the stability conditions of the periodic system (2), normally analysed with the help of the characteristic multiplier method (Weyh et al. [1991]), are directly recognizable by the real parts of the eigen-values of (10) (see Mtiller et al. [1976]) here in the chosen approximation.The system (2) is: asymptotically stable, if all eigen-values of (10) have negative real parts unstable, if at least one eigen-value of (10) has a positive real part ReX > 0, )for one Only for critical eigen-values )t of (10) with ReX 0, two different cases must be distinguished.
Let v stand for the multiplicity of the eigen-value with a vanishing real part and d N rank(--2 + PX + Q) stand for the defect of the accompanying characteristic matrix.If ReX -< 0 with 1 (1)N and ReX 0, then the system (2) is: limit stable for d F otherwise unstable for dm < F m.
Limit Stability Conditions and Parametric

Resonances
It follows from ( 12), that all real parts id of the eigen-values of the approximate system (9) exist in a 2P + times multiplicity, as in the critical case.If oid, oi, are two imaginary parts of one critical eigenvalue multiplicity of the approximate system, then the connection with the imaginary parts of the eigen- values of the basic system OOi OO -jO, OOi,k O0 -Jr-kl, i f(1)f4= O,)j,k P(1)P holds if the approximation errors Aid Ai,k are small.
From the real structure of the technical starting prob- lem it follows, that a complex eigen-value is followed by its conjugate 0 -to_ i.A critical pair of eigen- values of one critical eigen-value multiplicity of the approximate system can then become multiple in the form Rehi < O, yl (1)N ---, J1TI k j (13)   Taking into account fOl,k as an imaginary part of a second critical eigen-value multiplicity, the equality o) j 0.)_i,k leads to the combination resonance condition of the sum type m + co mQ, jm k-j ( 14) while using the conjugate of m, the last condition stands for the difference type.

Homogeneous Solutions
The substitution (4) shows, that the required homo- geneous solution of the basic system (2) where the scalars c stand for the integration constants considering the initial conditions while the vectors a(0 t) stand for the centre parts of the eigen-vectors of the enlarged system

Inhomogeneous Solutions
Inhomogeneous solutions as a result of standard forc- ing (constant, harmonic, periodic, quasiperiodic of the basic system can be calculated easily.The har- monic excitation including a constant part h(t) ; hqe TM (17) q=-I serves as an example for the construction of an inho- mogeneous solution and leads to an excitation vector of the approximated system hq, (...0T, 0T, 0T, hq, 0T,   In case of non commensurable basic frequencies , f (18) is quasiperiodic, otherwise 2-rrv/f-periodic (v  integer).The inhomogeneous solution of the approx- is given by sums of products of amplitude:vectors with the harmonic excitation functions.The unknown amplitude-vectors in (20) result from bqp Gqp(q'q + pf) hqp by using the frequency response matrices Gqp(qrl + p12) [-(q + pl))ZM + )(q-q + pf) The corresponding solution of the basic system is then given by the f 1-dimensional center part of (2O).

Resonance Conditions and Interactions
The special representation of the frequency response matrix Gqp(q'q + pf) allows the elementary interpretation of the resonance phenomena and the interaction of parametric with forced excitation.

APPLICATION TO OPEN-END ROTOR-BEARING-SYSTEMS Description of System
The picture in Figure shows the lower part of a high speed Open-End rotor-bearing with a spinning rotor, carrying the Open-End spinning turbine at its front end, Socha et al. 1991 ].The bearing-system consists of two rigid coupled twin discs mounted in two bear- ing shells which are fixed in a bearing house.The drive of the spinning rotor follows on from a trans- mission belt crossing the rotor, schematically sketched in Figure 2. (Up to 120 spinning rotors are driven by the same belt from a central drive in one spinning machine.)The necessary tangential belt wrap is produced by the elastic supported billy-roller which presses the rotor between the elastic coating of the twin disc pairs for the radial bearing.Due to a minimal deviation from parallelism of the twin disc axis a permanent thrust load acts on the rotor and squeezes its back end against a ball step bearing for where the abbreviations of elements of the damping matrix are given with the expressions p (dt 1%-at2)c0s2 %-d21 c0s2 (28) P2 (1 2)sin g cos + d21 sin /cos / P2-(l + 2)sin28 + d21 sin2 / and, by exchanging d for c formally in the expres- sions of p/, the elements of the stiffness matrix q/ are introduced.
The parameter excitation of the system results from the elastic inhomogeneities of the twin disc coatings and can be represented by periodically vary- ing damping and stiffness coefficients

Structure of Model Equation
For first predictions a simple mechanical, three-mass model with four degrees of freedom, on the base of Figure 2, can be used.The generalized 4-dimensional coordinate-vector, introduced by (1), y (X, Y, $2, $3) T ( 26) describes in this model the horizontal X and the ver- tical Y displacement of the spinning rotor axis.$2 stands for the movement of the acting part of the transmission belt in the direction of contact with the billy-roller, while in the same direction $3 represents the motion of the axis of the billy-roller.
The constant parts of the system matrices of (3) are defined by the real symmetric matrices M 0 diag(m + m., m + m., m2, m3) P?I /9?2 -d21 cos "y 0 P0 P22 with the basic frequency 1) of the twin discs and the small number K 3 of harmonics as given in the example in Table I.Considering this arrangement, the time-variant parts of the system matrices are given by the complex symmetric coefficient-matrices Pk Pl P2 0 0 P2 0 0 00 Lsym 0 ql q2 0 0 lk q2 0 0 (30) 00 sym 0 B. WEYH of the trigonometric matrix-series (3), with the abbre- viations of the complex elements sk sk -+-J(lOt; 4ott2)] sin 3 cos 3 (3) k and qij by exchanging ot for [3 and d for c respec- tively as mentioned before.The minus or plus sign of (31) stands for positive or negative vatues of k 4 0.
Outer system excitation results from the centrifu- gal force of the imbalance F, m,e'q (32) dominantly caused by the spinning process.The com- plex vector of excitation forces (17) consists of a con- stant q 0 and a harmonic q -1, part (33) h 0 (0, 0, -F 2, -F3) T h (-Fu,J-Fu, O,O) T as a result of the initial tensions F 2, F 3 and the uni- form rotation q xlt of the spinning turbine.It is assumed that the system drive realizes motions with- out slip and that the spinning turbine rotates at a con- stant angular velocity q q.Therefore the rotational frequency of the twin disc fl rlr/R ( 34) is directly connected with the rotor frequency.The relation r/R will later decide on periodic (commensurable frequencies) or quasiperiodic (non commensu- rable frequencies) inhomogeneous solutions.
The system parameters, see  rotor/turbine imbalance active part of transmission belt billy-roller twin disc coating (mean value) twin disc coating (varying cos-coeff.)twin disc coating (varying cos-coeff.)twin disc coating (varying sin-coeff.)twin disc coating (varying sin-coeff.)belt coating (normal) billy-roller damper twin disc coating (mean value) twin disc coating (varying cos-coeff.)twin disc coating (varying cos-coeff.)twin disc coating (varying sin-coeff.)twin disc coating (varying sin-coeff.)belt coating (normal direction) billy-roller spring spinning turbine frcquenoy FIGURE 3 Visualization of resonance conditions in the parameter space: twin-disc frequency f against spinning turbine frequency "q.
sential background information about the modelling process.

Results
Resonance points For quick predictions and a first orientation in the parameter space of interest--para- metric frequency via forced frequency--the reso- nance conditions ( 13), ( 14), ( 25) are sketched in Fig- ure 3 for the spinning system presented before in a simple approximation j (1) 1, p (1) 1. Parametric resonance lines are given by the horizontal lines, forced resonance lines by the vertical lines, main parametric resonance lines by some of the hor- izontal lines, while the skew lines stand for the inter- action of forced with parametric resonances.The in- tersections of the described resonance lines with the thick frequency line "q fR/r, see (34), stand for the points where resonances or large amplitudes can oc- cur.In case of higher approximations further reso- nances radiate out from the eigen-frequency-points marked on the q-axis.Normally the importance of the resonances decreases with increasing resonance order m and approximation order P.This effect de- pend on the convergence of harmonic amplitudes of the parameter excitation (28), (29).
Response spectra The solutions of equation ( 21) with ( 22) are calculated with a constant step rate Axl and exemplary the vertical vibrations of the spinning rotor (Y-direction) are plotted point-by-point against the excitation frequency "q in the following figures.The amplitude and phase curve (of inhomogeneous FIGURE 5 Amplitude curve and phase shift curve of undamped system against turbine frequency (first order approximation).FIGURE 6 Amplitude curve and phase shift curve of damped system against turbine frequency (first order approximation).
solutions) are shown in Figure 4 against spinning tur- bine frequency q.Only four isolated forced reso- nances in the case of homogeneous elasticity of the twin disc coating are shown.These two plots for the undamped system without parameter excitation are used as references.In Figure 5 two neighbouring interaction resonances can be detected beside each main forced resonance in the undamped case, while in the damped case, Figure 6, only little information about these interactions are recognizable.In the last four plots hull curve representations of the centre part of the amplitude  3) for "q 10 000 [l/s].It should be known that in Figures 7-10 the phase shift of damping and stiffness inhomogeneities of the twin-disc has been changed from 180 to 0 by changing the parameter intensity coefficients of the second twin-disc pair to q [* 100 l/s] FIGURE 8 Hull curve of damped system against turbine fre- quency (first order approximation).FIGURE 10 Hull curve of damped system against turbine fre- quency (10th order approximation).
0.05 Parallel to the afore-mentioned effects further resonances appear and can be characterized with the help of a completed Figure 3.
Stability information Complementary to this an ap- proximate stability analysis, for example discussed in Neumann et al. 1994], separates stable from unstable parameter regions.Here unstable solutions are situ- ated in parameter regions with high amplitudes, see Figure 3, or it follows from high damping that the most unstable regions dissapear.

CONCLUSIONS
In this paper the results of the method of enlarged systems for the calculation of periodic parameter ex- cited problems including the interaction with forced excitations have been presented.The central idea and FIGURE 9 Hull curve of undamped system against turbine fre- quency (10th order approximation).
the structure of the enlarged system have been analy- sed.In general the enlargement of the system equa- tions allows: the approximate calculation of the characteristic exponents, including a stability definition and a new interpretation of the well known conditions of parameter and combination resonances, the approximate calculation of the complete homo- geneous solution, the approximate calculation of the inhomogeneous solution and a complete overview of resonance phenomena re- sulting from interaction of parametric and forced excitation with in general non commensurable ba- sic frequencies.
Exceptionally the method is based only on the procedures of constant systems the engineer is familiar with.
General homogeneous and inhomogeneous solu- tions have been derived and applied to a realization of the high speed twin-disc rotor-bearing-system of a textile spinning turbine.The new effects of interac- tion of forced and parametric excitation on the rotor vibration have been discussed.adjungate of a matrix argument c integration constant det [.] determinant of a matrix argument f number of degrees of freedom i, j, k, l, m, p, q indeces time Yi coordinates X, Y, $2, $3 -)" derivative with respect to time ) imaginary unit a () 1-th eigen-vector of enlarged system bqp amplitude-vector of quasiperiodic component bss hull curve amplitude-vector of quasiperiodic component gp(.) higher harmonic time-variant part of enlarged system h(t) forced excitation-vector of basic system hq coefficient-vector of periodic forced excitation of basic system h(t) hyper-excitation-vector hp(t) sub-vector of inhomogeneity of ex- citation of enlarged system hqp coefficient-vector of quasiperiodic excitation of enlarged system y coordinate-vector yp(t) substitution-vector N order of enlarged system P substitution (approximation) order TU period of forced excitation T_e period of parametric excitation Gqp(.) frequency response matrix M(t), P(t), Q(t) time-variant system matrices of ba- sic system M, P, Q Fourier-coefficient matrices of system matrices-expansions M, P, Q constant hyper matrices of enlarged system 8i real part of.eigen-value of basic system aid.
real part of eigen-value of enlarged system xl excitation frequency oo ij eigen-value of enlarged system imaginary part of eigen-value of ba- sic system imaginary part of eigen-value of en- larged system eigen-value approximation error parametric frequency For additional information see also Table II.

FIGURE 2
FIGUREPicture of lower part of Open-End spinning box in the latest machine generation.

FIGURE 4
FIGURE 4 Amplitude curve and phase shift curve of constant undamped system against turbine frequency.

FIGURE 7
FIGURE 7 Hull curve of undamped system against turbine fre- quency (first order approximation).

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TABLE Example of the
Table II, point to es-