Identiﬁcation of the dynamic force and moment characteristics of annular gaps using linear independent rotor whirling

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Identiﬁcation of the dynamic force and

moment characteristics of annular gaps

using linear independent rotor whirling

motions

Maximilian M. G. Kuhr*

Chair of Fluid Systems

Technische Universität Darmstadt

maximilian.kuhr@fst.tu-darmstadt.de

Abstract

Nowadays, most studies on the dynamic properties of annular gaps focus only on the force char-

acteristics due to translational motions, while the tilt and moment coefﬁcients are less well studied.

Therefore, there is hardly any reliable experimental data for the additional coefﬁcients that can be used

for validation purpose. To improve this, a test rig ﬁrst presented by [

] is used to experimentally

determine the dynamic force and moment characteristics of three annuli of different lengths. By using

active magnetic bearings, the rotor is excited with user-deﬁned frequencies and the rotor position and the

forces and moments induced by the ﬂow ﬁeld in the annulus are measured. To obtain accurate and reliable

experimental data, extensive preliminary studies are carried out to determine the known characteristics

of the test rig rotor and the added mass and inertia imposed by the test rig. Subsequently, an elaborate

uncertainty quantiﬁcation is carried out to quantify the measurement uncertainties. The experimental

results, i.e. the 48 rotordynamic coefﬁcients, are compared to a calculation method by Kuhr

[16]

, Kuhr et al.

[18]

. It is shown that the presented experimental data agree well with the calculation method, especially

for the additional rotordynamic tilt and moment coefﬁcients. Furthermore, it is shown that the annulus

length signiﬁcantly inﬂuences the coefﬁcients of the ﬁrst sub-matrix. A dependence of the additional

coefﬁcients on the length is recognisable, but less pronounced. Even for the shortest investigated annulus,

i.e.

L:=˜

L/˜

1, the stiffness coefﬁcients due to the forces from the angular motion of the rotor are of

the same order of magnitude as the stiffness coefﬁcients due to the forces from the translational motion.

This supports recent results by Kuhr

[16]

, Kuhr et al.

[18]

, indicating that the additional coefﬁcients

become relevant much earlier than assumed throughout the literature, cf. Childs [3].

I. Introduction

Reliable and accurate experiments provide the foundation for the validation of any calculation

model. By comparing the model to real components under well controlled conditions, conclusions

can be drawn regarding the accuracy and reliability of the model itself. Regarding the dynamic

force and moment characteristic of annular gaps, most experimental studies focus only on the

forces due to translational motions, while the tilt and moment coefﬁcients are less well studied.

Therefore, there is hardly any reliable experimental data for the additional coefﬁcients that can

be used for validation purpose. This is of particular importance since the additional tilt and

*Corresponding author

arXiv:2210.09935v1 [physics.flu-dyn] 18 Oct 2022

Force and moment characteristics of annular gaps

moment characteristics can have a signiﬁcant impact on the rotordynamic behaviour of modern

turbomachinery, i.e the eigenvalues and eigenfrequencies, cf. Gasch et al.

[8]

, Feng and Jiang

[7]

, Kim and Palazzolo

[15]

. Recent results by Kuhr

[16]

, Kuhr et al.

[18]

for example show that

the additional tilt and moment characteristics are becoming relevant much earlier than assumed

throughout the literature. However, due to the limited experimental data, the results, which are

solely based on simulation results, are hardly validated.

In modern turbomachinery, two essential narrow annular gaps exist, applying ﬂuid-ﬁlm forces

and moments on the rotating shaft [

]. First, journal bearings which are either oil or media

lubricated and second, annular seals or damper seals. The usage of low viscous ﬂuids like water

or cryogenic liquids for lubrication purpose, often leads to an operation at high Reynolds numbers,

resulting in turbulent ﬂow conditions and signiﬁcant inertia effects [

]. However, this

paper neither focus on annular seals or journal bearings, but concentrate on a generic geometry

valid for both machine elements, cf. ﬁgure 1.

In general, the dynamic characteristics of annular gaps are modelled by using classical me-

chanical elements, namely springs, dampers and masses. Accordingly, the system is characterised

by using stiffness

, damping

and inertia

coefﬁcients. Throughout the paper, the tilde

characterises dimensional variables. The generalised equation of motion, including forces and

moments of the annular gap ﬂow yields

−











=





KXX ˜

KXY ˜

KXα˜

KXβ

KYX ˜

KYY ˜

KYα˜

KYβ

KαX˜

KαY˜

Kαα ˜

Kαβ

KβX˜

KβY˜

Kβα ˜

Kββ













αX

βY







+





CXX ˜

CXY ˜

CXα˜

CXβ

CYX ˜

CYY ˜

CYα˜

CYβ

CαX˜

CαY˜

Cαα ˜

Cαβ

CβX˜

CβY˜

Cβα ˜

Cββ













αX

βY







+





MXX ˜

MXY ˜

MXα˜

MXβ

MYX ˜

MYY ˜

MYα˜

MYβ

MαX˜

MαY˜

Mαα ˜

Mαβ

MβX˜

MβY˜

Mβα ˜

Mββ













αX

βY







(1a)

−











=˜

KI˜

KII

KIII ˜

KIV





αX

βY







+˜

CI˜

CII

CIII ˜

CIV





αX

βY







+˜

MI˜

MII

MIII ˜

MIV





αX

βY







. (1b)

Within the equation,

and

are the induced forces and moments of the annulus

acting on the rotor. The translational motion, velocity and acceleration of the rotor is given

and

, whereas

αX

and

βY

denotes the angular motion, velocity

and acceleration of the rotor around the

and

axis. The describing coefﬁcients depend on

three different characteristics of the annulus: (i) the geometry of the annular gap, i.e. the gap

length

, the shaft radius

, the mean gap clearance

and the gap function

h=˜

h(ϕ

with

the circumferential and axial coordinate

; (ii) the operating conditions, i.e. the eccentric and

angular position of the shaft

αX

βY

, the distance of the centre of rotation

from the annulus

entrance, the mean axial velocity

, the angular velocity of the shaft

Ω

and the pre-swirl velocity

at the annulus inlet

Cϕ|z=0

; (iii) the lubricant characteristics, i.e. the dynamic viscosity

and the

ﬂuid density

. On dimensional ground, the dimensionless rotordynamic coefﬁcients are only

a function of 8 dimensionless measures: (i) the dimensionless annulus length

L:=˜

L/˜

, (ii) the

relative eccentricity

ε:=˜

e/˜

, (iii, iv + v) the normalised angular displacements

α:=˜

LαX/˜

and

β:=˜

LβY/˜

around the dimensionless fulcrum

zT:=˜

zT/˜

, (vi) the modiﬁed Reynolds number in

Force and moment characteristics of annular gaps





 







 



 





 











   

























 













 







Figure 1: Schematic drawing of an eccentric operated generic annular gap with axial ﬂow and pre-swirl.

circumferential direction

ψRe∗

ϕ:= (˜

h/˜

R) ( ˜

Ω˜

R˜

h/˜

ν)nf

, (vii) the ﬂow number

φ:=˜

Cz/(˜

Ω˜

, (viii)

the dimensionless pre-swirl

Cϕ|z=0:=˜

Cϕ|z=0/(˜

Ω˜

, cf. [

]. Here,

ψ:=˜

h/˜

is the gap clearance

and

is an empirical constant describing an arbitrary line within the double logarithmic Moody

diagram

Kij,Cij,Mij =fL,ε,α,β,zT,Re∗

ϕ,φ,Cϕ|z=0. (2)

The dimensionless rotordynamic coefﬁcients are separately deﬁned for the forces and moments

acting on the rotor due to translational and angular motions

KI:=2˜

h˜

Kij

$˜

Ω2˜

R3˜

L,KII :=2˜

h˜

KII

$˜

Ω2˜

R3˜

L2,KIII :=2˜

h˜

KIII

$˜

Ω2˜

R3˜

L2,KIV :=2˜

h˜

KIV

$˜

Ω2˜

R3˜

L3. (3)

Here, the ﬁrst two stiffness coefﬁcients represent dynamic force coefﬁcients due to translational

and angular motion, whereas the later ones represent the moment coefﬁcients due to translational

and angular motion. The indices represent the corresponding sub-matrices of equation 1. The

damping and inertia terms are deﬁned accordingly

CI:=2˜

h˜

$˜

Ω˜

R3˜

L,CII :=2˜

h˜

CII

$˜

Ω˜

R3˜

L2,CIII :=2˜

h˜

CIII

$˜

Ω˜

R3˜

L2,CIV :=2˜

h˜

CIV

$˜

Ω˜

R3˜

L3, (4a)

MI:=2˜

h˜

$˜

R3˜

L,MII :=2˜

h˜

MII

$˜

R3˜

L2,MIII :=2˜

h˜

MIII

$˜

R3˜

L2,MIV :=2˜

h˜

MIV

$˜

R3˜

L3. (4b)

As stated before, the vast majority of experiments focus solely on the dynamic characteristics

due to translational motion, i.e. sub-matrix I, of either journal bearings or annular seals, cf.

[

]. Additionally, an adequate uncertainty quantiﬁcation regarding the systematic and

statistic uncertainty of the measurement signal is often missing when presenting the experimental

data. For a detailed list of the theoretical and experimental investigations, see the review articles

by Sternlicht and Rieger [36], Lund [19], Pinkus [28], Tiwari et al. [39,40], Zakharov [41].

Force and moment characteristics of annular gaps

i. Experimental identiﬁcation of all 48 coefﬁcients

In the following, we focus only on the experimental identiﬁcation of all 48 rotordynamic coefﬁcients,

i.e. the dynamic characteristics including the tilt and moment coefﬁcients. One of the ﬁrst

experimental identiﬁcation of the all coefﬁcients of an annular seal was presented by Kanemori

and Iwatsubo

[13

14]

. Here a test rig was presented, consisting of two rotors embedded in each

other. By bearing and driving both rotors independently of each other, small translational and

angular motions inducing frequency-dependent forces and moments on the rotor are imposed

around the static equilibrium position By measuring both the motion of the rotor and the induced

forces and moments on the rotor, the rotordynamic coefﬁcients are identiﬁed using parameter

identiﬁcation methods, cf. [

]. Essentially, the inﬂuence of the axial pressure difference

∆˜

as well as the angular frequency of the rotor

Ω

and the pre-swirl

Cϕ|z=0

and the combination of

cylindrical and conical whirls is investigated.

In contrast to the two embedded rotors used by Kanemori and Iwatsubo, Neumer

[25]

and

Matros et al. [22] use active magnetic bearings (AMBs) to position and excite the rotor. Here, the

main advantage is that the bearings act as actuators and sensors at the same time. By measuring

the electric current of the electromagnets as well as the rotor position, the force of the bearing is

determined based on the force-current relation. This three-dimensional characteristic can either be

determined by an extensive calibration procedure or by calculation of the partial derivation of the

ﬁeld energy stored in the volume of the air gap with respect to the air gap itself, cf. Maslen and

Schweitzer

[21]

. However, the force-current relation of the bearing is often linearised resulting in a

signiﬁcantly high uncertainty of the measured forces. Furthermore, it is only valid at the main

coordinate axes of the magnetic bearing. Within their studies, Neumer and Matros et al. focus

mainly on the inﬂuence of the angular frequency of the rotor

Ω

as well as the inﬂuence of the

axial pressure difference

∆˜

on the rotordynamic coefﬁcients. Here, in addition to the stiffness,

damping and inertia coefﬁcients from the induced forces due to translational motion, only stiffness

and damping coefﬁcients from the induced moments on the rotor due to translational motion are

given.

The aforementioned references are the exclusive bases for all 48 rotordynamic coefﬁcients.

Accordingly, there is a need for further reliable experimental investigations of the dynamic

properties, especially regarding the additional rotordynamic tilt and moment coefﬁcients and their

dependencies according to equation 2. For this purpose, a test rig originally presented by Kuhr

et al.

[17]

is extended in the following sections for the experimental determination of the dynamic

force and moment characteristic using four linear independent whirling motions.

II. Annular gap flow test rig

[

] present a test rig to experimentally investigate the static force characteristics of generic

annuli, cf. ﬁgure 2. The test rig mainly consists of two active magnetic bearings supporting

the rotor and to act as an inherent displacement, excitation and force measurement system. In

contrast to the test rigs used by Neumer and Matros et al., the force is obtained by measuring

the magnetic ﬂux density

in the air gap between the rotor and the active magnetic bearing.

Therefore, each pole of each electromagnet is equipped with a hall sensor. The force applied per

pole is proportional to the magnetic ﬂux density

FH∝˜

. The test rig is designed to investigate

annuli with relative lengths in a range of 0.2

≤L≤

1.8 and relative clearances 10

−3≤ψ≤

−2

Furthermore, it is capable of applying pressure differences across the annulus up to 13

bar

resulting in ﬂow numbers up to

φ≤

5. For a detailed description of the test rig, its calibration

and the measurement uncertainty of the used equipment, refer to [

]. In contrast to the

Force and moment characteristics of annular gaps

AXIAL BEARING

MOTOR

GAP MODULE

MAGNETIC BEARING

150

FOUNDATION

Figure 2:

The annular gap test rig at the Chair of Fluid Systems at the Technische Universität Darmstadt, cf. Kuhr

[16], Kuhr et al. [17].

formerly presented solely static force results, the test rig is extended to experimentally determine

the dynamic force and moment characteristics of the investigated annuli.

i. Identiﬁcation using four linear independent whirling motions

Since the rotordynamic force and moment characteristics cannot be measured directly, they must

be determined by using parameter identiﬁcation methods, i.e. linear and quadratic regressions.

The method used here is similar to that of [

] and requires the excitation of the rotor at user-deﬁned

frequencies and amplitudes. This is made possible by controlling the electric currents within the

active magnetic bearings utilizing a feed-forward compensation (FFC) embedded in the controller.

Subsequently, the induced dynamic forces and moments acting on the rotor

and

well as the motion of the rotor in the four degrees of freedom,

and

αX

βY

are measured. The

measurement data acquisition of all sensors is carried out simultaneously based on a multifunction

IO module with a sampling rate of 6000 Hz per channel over a duration of ﬁve seconds.

To identify the rotordynamic coefﬁcients, the generalised equation of motion 1is ﬁrst trans-

ferred into the frequency domain, yielding the complex equation of motion

−











=





KXX ˜

KXY ˜

KXα˜

KXβ

KXY ˜

KYY ˜

KYα˜

KYβ

KαX˜

KαY˜

Kαα ˜

Kαβ

KβX˜

KβY˜

Kβα ˜

Kββ













DαX

DβY







. (5)

Here,

Kij

are the complex stiffness coefﬁcients depending on the precessional frequency

F=F(˜

and

M=F(˜

are the Fourier transformations of the induces forces and moments

on the rotor, whereas

Di=F(˜

are the Fourier transforms of the translational and angular

excitation amplitudes. The real part of the complex stiffness

<˜

Kij

contains the stiffness and

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IdenticationofthedynamicforceandmomentcharacteristicsofannulargapsusinglinearindependentrotorwhirlingmotionsMaximilianM.G.Kuhr*ChairofFluidSystemsTechnischeUniversitätDarmstadtmaximilian.kuhr@fst.tu-darmstadt.deAbstractNowadays,moststudiesonthedynamicpropertiesofannulargapsfocusonlyontheforcechar-a...

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Identiﬁcation of the dynamic force and moment characteristics of annular gaps using linear independent rotor whirling

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