Table Of ContentIntroduction
The 27 ht Leeds-Lyon Symposium on Tribology was held at the Institut National des Sciences
Appliqu6es de Lyon from Tuesday 5 ht to Friday 8 ht September 2000. The central theme was"
"Tribology Research: From Model Experiment to Industrial Problem: A Century of Efforts in
Mechanics, Materials Science and Physico-Chemistry". To celebrate the Year 2000 the
organisers tried to achieve two different goals. First, to bring together contributions from the
three major scientific fields that make up the Tribological community: Mechanics, Physico-
Chemistry and Materials Science. The second goal was to attract contributions ranging from
complex industrial problems to model experiments.
To fulfil the first aim, eminent researchers in their field were asked to organise a session on a
topic they thought important. Important because past developments had greatly changed the
field or future efforts were necessary in that field. The organisers would like to thank
Professor K.L. Johnson, Dr. J. Greenwood, Professor C. Kajdas and Professor D. Kuhlmann-
Wilsdorf for providing the framework of the three main sessions, and thereby contributing to
the success of this year's symposium.
The symposium was opened on Tuesday afternoon with two keynote addresses by Professors
J. Frane and W.A. Goddard. Professor Frane, winner of the 1999 Gold Medal in Tribology
gave a detailed technical lecture on the dynamic behaviour of an elastic shaft supported by
hydrodynamic beatings. Professor Goddard gave a global overview of first principles multi-
scale modelling of physicochemical aspects of tribology. The organisers were very pleased
that Professor D. Dowson was present to chair this session.
The Symposium Review board had examined many abstracts of which some 90 were
accepted for presentation. In view of the large number of interesting proposed papers, it was
decided to organise parallel sessions. Each morning and afternoon was opened by a single
session focussing on this years' theme, followed by four parallel sessions. Up to five papers
per session were presented, and due to the animated discussions, more than one chairman had
difficulties adhering to the allocated time.
The reviewing process, which has proved efficient and stimulating, was extended and
formalised. The full manuscript was reviewed anonymously and depending on the comments,
minor or major corrections were required. The reviewing process has also allowed us to
improve the quality of the written English. The organisers would like to thank the great
majority of the authors for supplying their manuscript before the conference, the reviewers for
their quick and expert work, and finally the authors for quickly implementing these
comments. A complete list of reviewers appears at the end of the Introduction.
The Year 2000 symposium banquet was held in the "Abbaye de Collonges" where the
conference delegates were welcomed by the world renowned French Chef Paul Bocuse. The
guests greatly enjoyed the cuisine, the atmosphere created by his professional staff and the
entertainment provided by the magnificent organ. The participants toasted Professor J-M
vi
Georges on the occasion of his retirement. The organisers would like to thank Professor
Georges for his continued support of the Leeds-Lyon Symposium.
The traditional cultural event took place on Thursday evening, in the historic "Chapelle de la
Trinit6". The delegates attended a unique presentation of medieval music performed by the
seven members of the Lyonnais ensemble "Musica Nova", led by one of the four sopranos
Anne Quentin. In this symbolic year, the concert traced vocal and instrumental musical
history, dating from the first centuries with pure Gregorian music to J.S. Bach.
The Friday barbecue party was organised as usual by the laboratory staff. The Symposium
tour on Saturday visited the Burgundy region and in particular Beaune with its world famous
15 ht century H6tel-Dieu. The tour included a visit to a traditional cooper and wine tasting in
the cellar of the famous house Bouchard-Ain6 et ills.
In order to honour one of founders of the Leeds-Lyon Symposium on Tribology, the LMC has
decided to inaugurate the Maurice Godet Award. This prize, to be awarded at every Lyon
conference, will be given to the best paper and presentation by a young researcher. This prize
commemorates the work of Maurice Godet and the importance he attached to attracting young
and promising researchers to the field of Tribology. This year's prize was awarded
unanimously to Miss A. Brown for the paper entitled: "Spring-supported thrust beatings used
in hydroelectric generators: Finite element analysis of the pad deformation", co-authored by
Professor J.B. Medley and Dr. J.H. Ferguson. The prize was awarded by Professor L.
Rozeanu, a long time contributor and supporter of the Leeds-Lyon Symposium.
The organisers would like to thank all the members of the Laboratoire de M6canique des
Contacts for participating in the organisation and thereby contributing to the success of the
vii
Leeds-Lyon Symposium. In particular they would like to thank Professor L. Flamand for
arranging the financial aspects and Mrs A.-M. Colin for handling the entire administration.
The organisers gratefully acknowledge the financial support received from the following
companies (cid:12)9
INSA Direction Villeurbanne, France
PSA PEUGEOT-CITROEN Paris, France
Technocentre RENAULT Guyancourt, France
RHODIA CHIMIE Courbevoie, France
SHELL Thornton, UK
SKF-ERC Nieuwegein, The Netherlands
SNR Roulements Annecy, France
TOTAL Paris, France
This support allowed us to offer students reduced fees, and we were very pleased to see the
large number of students actively contributing to the success of the conference.
The Leeds-Lyon Symposia have now covered a wide range of topics, as shown in the
following list. The essential aim is to select each year a topic of current interest to tribologists
and to contribute to the further advance of knowledge in this field.
1 Cavitation and Related Phenomena in Lubrication Leeds 1974
2 Super Laminar Flow in Beatings Lyon 1975
3 The Wear of Non-Metallic Materials Leeds 1976
4 Surface Roughness Effects in Lubrication Lyon 1977
5 Elastohydrodynamics and Related Topics Leeds 1978
6 Thermal Effects in Tribology Lyon 1979
7 Friction and Traction Leeds 1980
8 The Running-in Process in Tribology Lyon 1981
9 Tribology of Reciprocating Engines Leeds 1982
01 Numerical and Experimental Methods in Tribology Lyon 1983
11 Mixed Lubrication and Lubricated Wear Leeds 1984
21 Mechanisms and Surface Distress Lyon 1985
31 Fluid Film Lubrication - Osborne Reynolds Centenary Leeds 1986
14 Interface Dynamics Lyon 1987
51 The Tribological Design of Machine Elements Leeds 1988
61 Mechanics of Coatings Lyon 1989
71 Vehicle Tribology Leeds 1990
81 Wear Particles : From the Cradle to the Grave Lyon 1991
91 Thin Films in Tribology Leeds 1992
20 Dissipative Processes in Tribology. Lyon 1993
12 Lubricants and Lubrication Leeds 1994
22 The Third Body Concept: Interpretation of Tribological
Phenomena Lyon 1995
iiiv
23 Elastohydrodynamic : Fundamentals and Applications in
Lubrication and Traction Leeds 1996
24 Tribology of Energy Conservation London 1997
25 Lubrication at the Frontier : The Role of the Interface and
Surface Layers in the Thin Film and Boundary Regime Lyon 1998
26 Thinning Films and Tribological Interfaces Leeds 1999
27 Tribology Research : From Model Experiment to
Industrial Problem Lyon 2000
We look forward to the 28 ht Leeds-Lyon Symposium which, exceptionally, will be held in
Vienna from Monday 3 dr to Friday 7 ht September 2001 under the title (cid:12)9 "Boundary and Mixed
Lubrication : Science and Application".
Ton Lubrecht G6rard Dalmaz
List of reviewers
Adams G. Dubourg M.-C. Kajdas C. Newall J.P.
Baillet L. Dudragne G. Kapsa P. Olver A.
Bair S. Dwyer-Joyce R. Kennedy F. Poll G.
Bec S. Ehret P. Kernizan C. Priest M.
Belin M. Evans H. Krupka I. Richetti P.
Bou-Sa'fd B. Flamand L. Kuhlmann-Wilsdorf D. Sainsot P.
Brendl6 M. Fletcher D. Le Mogne T. Salant R.
Briscoe B. Fouvry S. Loubet J.-L. Seabra J.
Cann P. Frane J. Lubrecht T. Sidoroff F.
Chandrasekar S. Gardos M. Martin G. Soom A.
Chang L. Georges J.-M. Martin J.M. Sugimura J.
Ciulli E. Girodin D. Mazuyer D. Taylor C.
Colin F. Hartl M. Medley J. Torrance A.
Coulon S. Iordanoff I. Mess6 S. Venner C.
Dalmaz G. Jabbarzadeh A. Meurisse M.-H. Vergne P.
Descartes S. Johnson K. Morales Espejel G. Ville F.
Dowson D. Jolkin A. N61ias D. Vincent L.
Tribology Research: From Model Experiment to Industrial Problem
G. Dalmaz et al. (Editors)
(cid:14)9 2001 Elsevier Science B.V. All rights reserved.
DYNAMIC BEHAVIOUR OF ELASTIC SHAFT SUPPORTED BY HYDRODYNAMIC BEARINGS
Jean FRENE and Olivier BONNEAU
Laboratoire de M6canique des Solides-UMR CNRS 6610, Universit6 de Poitiers, SP2MI, Bd Pierre et Marie
Curie, T616port 2, BP 30179- 86960 Futuroscope - Chasseneuil Cedex
High speed rotors present a lot of stability problems especially when the speed of rotation passes through a
critical speed. The non-linear dynamic behaviour of the fluid bearings has, in that case, an important effect. This
study presents several models of bearing dynamic behaviour taking into account the flexibility of the shaft:
Squeeze film damper (SFD) behaviour is described. This element gives a lot of damping but its behaviour is
totally non-linear. The coupling between the axial thrust bearing behaviour and the bending vibrations of the
shaft is also studied.
1. INTRODUCTION linear models) and rotor dynamic models. The
coupling between fluid element and the shaft is done
In the turbomachinery field the designer has to by a non linear approach: the simulation of the non
model the shaft with high accuracy and has to take linear dynamic behaviour of the flexible rotor is a
into account all the shaft surroundings. A lot of time step by step approach done in the shaft modal
disciplines are concerned by this study: structural basis.
mechanics, acoustics, fluid mechanics, heat transfer, In the second part of the paper some applications
lubrication,... The difficulty, but also the challenge of this dynamic model are detailed:
of the rotor study, comes from this diversity of - Results obtained for a flexible shaft mounted in a
subjects and from the choice of different models. squeeze film damper (SFD) have been chosen; this
Until a few years ago, each discipline was example is very interesting because of the high non
developing its own models: linear behaviour of the fluid element. Numerical and
- very detailed shaft (finite element models) experimental results are presented and the
mounted in a simple support (with linear stiffness comparison shows the good accuracy of the
and damping coefficients) for structure mechanics numerical model.
approach; - The non-linear coupling between axial motion and
- very sophisticated bearing models (non-linear) the bending vibrations of the shaft is also presented.
but with simple shafts (rigid) in lubrication This coupled behaviour is due to the thrust bearing
applications. effects and to the shaft flexibility.
This earlier work gives useful results for a first
level of approach but these are inadequate when the
mechanism works in severe conditions (near the 2. FLUID ELEMENTS (FLUID BEARINGS,
critical speed for example). In such conditions it is ANNULAR SEALS)
necessary to analyze the coupled behaviour. The
work presented here 1 to 5 is a synthesis of 2.1. Presentation
several studies done with coupled models. These Shafts are generally supported by two kinds of
models take into account shaft flexibility in fluid elements: ball bearings and (or) fluid bearings. The
surroundings. Non linear shaft support behaviour ball bearings give near-linear behaviour and this
will be specially emphasized. technology is not studied in this work.
In the first part of the paper one describes Fluid bearings have a complex behaviour;
bearing dynamic models (both linear and non- linearisation can be carried out but the behaviour is
sometimes quite non linear. Three kinds of fluid In the case of low viscosity fluids, for a large
bearings are studied here: the traditional bearing, clearance or when rotation speed is high the flow
the squeeze film damper and the axial thrust could become turbulent. Then two turbulent
bearing. viscosity coefficients are introduced. They are a
Annular seals may also have an important effect function of turbulent parameters: local Reynolds
on shaft behaviour 6 to 10. The problems induced number, axial and circumferential pressure gradient
by these components are related to turbulent flow and roughness. Some turbulent flow models have
and inertia effect in the fluid which are due to the been applied 6 to 9 to calculate turbulent viscosity
low fluid viscosity (in liquid hydrogen turbopump coefficients.
for example).
.......-----
2.2. Fluid film modelling "O I
The characterization of fluid elements requires
solution of the Navier-Stokes equations. This
solution gives the values of the pressure field in the
film. Two cases appear:
- The film thickness is small compared to the
other dimensions, thus this condition allows us to
simplify the problem. An elliptic differential
equation: the Reynolds equation is obtained.
-It is impossible to neglect one dimension in
relation to the others so it is necessary to solve the
complete Navier-Stokes 3D systems. This approach
is necessary for example in a labyrinth seal with
very deep grooves. This aspect will not be
described in this study. Arghir 10 developed
numerical models adapted to this sort of seal
geometry.
2.2.1. Lubrication equations
We presem in this part the model adopted for
Figure .1 Film characteristics.
journal bearing; the axial thrust bearing solution is
similar 11 and we do not present it here. To solve the Reynolds equation, different
The assumption of a thin film thickness allows boundary conditions should be applied:
us to write the Reynolds equation as follows: - Feed boundary conditions (feed pressure in holes,
external pressure...)
1 'E 3h - Flow conditions, which characterize rupture in the
h a
film.:
R2 8~ * a/ kxSO* -~ u, kz ~ = -Sommerfeld conditions: no rupture of the film (2
n film).
(cid:12)9 (cid:12)9 O) 6;tl
= Xcos 0 +Ysin 0 +--~ -Gtimbel solution: rupture when numerical
200*
pressure is negative (n film).
-Reynolds conditions: the pressure and its
Where: R and L are respectively the radius and the
derivative are nil when rupture of the film occurs.
length of the bearing, h is the film thickness, P is the
-Film reformation conditions: the pressure and its
pressure field, oc is the angular shaft velocity, t~ is
derivative are nil when rupture of the film occurs and
the dynamic viscosity of the fluid, 0* and z are
the flow through the inactive part of the film is
respectively the circumferential and axial
taking into account for the film reformation.
parameter. All the different parameters are shown in
Then the Reynolds equation is integrated using
Figure .1
one of the following methods:
- Short bearing theory: the circumferential of the fluid load, Ax and Ay are small displacements
pressure variation effect is neglected. The fluid film near the static equilibrium and ~:A and 'SA are small
force is formulated analytically and computing time
velocity variations.
is small.
- Finite length bearing theory. In spite of its
yF (x o + Ax, oY + Ay, ,~/A r;A = yF (Xo, Yo,0,0) +
accuracy this solution implies heavy numerical
calculations which are not suitable for the
+ Ax/CTy/+ Ay/--~)+ A:~(~ A~,(---~)
application discussed here.
,t oJ o t )o o
- Unidimensional theory. This solution, inspired
from Rhode and Li 12, is based on an axial ~Cxa-I- - Ay-~--)-I- ....
parabolic pressure assumption. A variational t~
calculus implies a new equation of Reynolds where 0
only one space variable 0 appears. The gain in time
is important compared to the finite length model.
Then a first order development is carried out:
These methods have been tested and their
accuracies depend on the problem geometry.
Integration of the pressure film gives fluid film + Ax, 0Y + Ay, Ai, ,,~A Aii, ),~A - F x (0)~
oX(y?
forces on the rotor. + Ax, 0Y + Ay, ,~:A ,,~A Aii, A~,)- F(0)J
Fx= 2Yf 2~ ~~ P(0, z)cos 0Rd0dz ;ff( } = a- ji };AAI - jib }~AI ~:A - jim }~AI ~A
=yF ~
2V~ ~t P(0, z)sin 0Rd0dz
where fx and fy are the additional loads created by
displacement and velocities, aij ' bij and mij are the
stiffness, damping and added mass coefficients. The
2.2.2. Dynamic behaviour
last coefficients (added mass) exist only for inertia
The fluid element influence on rotor dynamics is
flow.
characterized by the hydrodynamic forces generated
Two numerical approaches may be used to
by the pressure field. These forces are non-linear
calculate these coefficients:
forces of the position and velocity of the shaft
- a perturbation method 31
center in the bearing. In general the shaft is
- a numerical differentiation 41 to 16.
submitted to a combination of several forces:
These dynamic coefficients may be used to
a static load (weight, belt tension,...)
determine the linear stability of a rigid shaft
-
a dynamic load (unbalance, shaft vibration)
supported by two identical bearings or to estimate
T- he static load implies an equilibrium position
the dynamic response of a shaft (to obtain critical
(stable or unstable) and the dynamic load creates an
speeds for example). However this model is false
orbital trajectory of the shaft. Depending on the
when the amplitude is large or when the equilibrium
conditions, two approaches can be carried out: a
is unstable, it is necessary then to obtain a non linear
linear approach and a non-linear one.
solution.
2.2.3. Linear theory
2.2.4. Non linear model
The aim of this approach is to linearize the bearing
This approach is a numerical study of the
behaviour around an equilibrium state. This theory
equations of motion which are integrated by a step
is based on a small displacement hypothesis. Thus,
by step method: at each step, the Reynolds equation
close to a static equilibrium state, a first order
is solved to evaluate the film forces, then the
development may be done. In these expressions F x
fundamental principle of mechanics is integrated to
and Fy are the components in the (.2, ~) reference obtain speeds and positions of the next step.
3. ROTOR MODELLING obtain an acceptable precision. A change of variable
is done in the following form: {8}= g{q} where
Two models could be considered, first the shaft
r is the matrix obtained with modal vectors. {gi },
is considered rigid and symmetrical (in geometry
and }q{ is the modal vector of displacement. A new
and in loading). It could be compared to a mass
system is written where the bearing effect appears in
mounted in a fluid bearing. Then the equations of
the modal stiffness matrix.
motion are simple (two degrees of freedom). This
model is valid when the speed of rotation is far from
}~{m + }lc{c + }q{'k = }f{
a critical speed. This simple model is useful for a
first approach but in this study a more advanced
model is presented which concerns an elastic shaft. with :
The rotor is modeled with typical beam finite t/v c= t/v CIr
m :
elements including gyroscopic effects 1. The
differential system, with }8{ the node displacement
k' = tzg Kk }f{ = ~t }F{
vector is the following:
To do a non linear calculus the linear bearing
effect
gniraebK should be subtracted from modal
stiffness k' and should be introduced as a non linear
Where M:is the mass matrix, C is the gyroscopic effect. Then
matrix, K is the stiffness matrix }bnuF{ are the
m{~} + c{ft} + k{q} = {f} + }lnF{
unbalance forces, ~rF{ } are the non linear bearing
with k = 'k l- /I t gniraebk s I~/
forces and ~gF{ / are the gravity forces. The damping
does not appear explicitly but it exists in the ~rF{ }
{Fnl} is the modal non linear force obtained, at each
terms as a non-linear fluid damping.
This system has 4(n+ )1 degrees of freedom (n is time step (see 4-1), by a modal basis change of the
the number of nodes). The iterative non-linear real non linear force. These real non-linear forces are
calculation will be very expensive in computer time. calculated in the physical coordinates.
To reduce the degrees of freedom a modal approach
is used 17-18.
Two approaches are available to obtain modal 4. FLUID STRUCTURE COUPLING
base:
-to take a free motion shaft but then it is The coupling between fluid element and the shaft
important to conserve a lot of modes. is done by a non-linear approach. The simulation of
-to take a judicious equivalent bearing stiffness. the non-linear dynamic behaviour of the flexible
This stiffness is not the real one (because an rotor is a step by step approach done in the modal
uncentralised squeeze film damper has no linear basis. The flow chart is as follows:
stiffness). It is a symmetrical stiffness and its value a) Beginning with initial values of modal
leads to a representative modal base. Then this positions and velocities.
stiffness may be subtracted from the modal stiffness b) Calculus of external modal forces.
matrix. c) Calculus of physical displacements and
Then the undamped fundamental principle of velocities in bearings (by modal basis change).
mechanics on the rotor could be written: d) Non linear bearings forces calculus (in the real
basis).
r{M } -q (Kr -q sgniraebK }8{) = 0 e) Computation of all forces in the modal basis
(by modal basis change).
f) Modal acceleration computation.
The solution has the following form:
g) Time integration (variable step Euler method).
8{-=}8{ 0 }ert and the first 6 modes are calculated.
h) Shaft speed is incremented.
Lacroix 17 has shown that 6 modes are enough to i) The process begins again in b.
4.1. Flexible shaft on a Squeeze Film Damper Bearing span: 0.8 rn,
4.1.1. Squeeze Film Damper (SFD) Rotor diameter 0.06 rn,
Results obtained for a flexible shaft mounted in Bearing length: 0.015m,
a squeeze film damper (SFD) have been chosen, Diameter: 0.09 rn,
this example is very interesting because of the high Radial clearance 0.05 mm.
non-linear behaviour of this kind of fluid element.
The basic idea for this kind of bearing is to 4.1.3. Results
support a ball bearing in a fluid bearing (figure 2). A test rig was developed by Kassai at the
The rotation is ensured by the ball bearing and the National Institute of Applied Sciences 1 & [ 19]. The
oil film is squeezed between two non rotating rings shaft displacements are measured by eddy current
(the external ring of the ball bearing is jammed in proximity probes. Numerical and experimental
rotation). Two technologies for SFD are possible. A results are obtained for a linear rotation speed
first kind of assembly consists in mounting, in variation from 8500 rpm to 13500 rpm during 22 .s
parallel with the SFD centering springs [19-20]. Numerical results obtained by short bearing theory
The second technology is without centering springs and experimental results are shown in figure 4a and
then some stiffness problems appear: the creation of 4b (oil temperature = 25~ ~t=0.05 Pa.s) and in
a significant pressure field (to support a static load) figure 5a and 5b (oil temperature = 60~ ~t=0.0135
being dependent on a threshold of minimum Pa.s). These figures show the displacement
perturbations. The results presented here are amplitude of the shaft in the middle of the rotor
obtained only with this second technology. (where the dynamic amplitude is maximum) versus
\ ~ o ,
rotational speed.
edutilpmA )mar(
4.O t
0.3
2.0
1.0
I
I
I
Figure 2 - Scheme of a Squeeze Film Damper. o I ' I ' I ' I ' I '
0009 00001 00011 00021 00031 of speed noitator )mpr(
4.1.2. Shaft
Figure 4a. Experimental results, t~ = 0.05 Pa.s.
The shaft is a rotating flexible shaft supported at
one end by a ball bearing and at the other end by an
active squeeze film damper without a centralizing
spring (figure 3).
Figure 3. Shaft representation.
Results are obtained for a linear rotation speed
variation from 7000 to 15000 rpm. The shaft and
bearing characteristics are the following: Figure 4b. Numerical results, t~ = 0.05 Pa.s.
edutilpmA (mm) The same conclusion has been obtained
0.4u concerning the effect of the radial clearance (a high
radial clearance is better closed to its critical speed
and a low radial clearance is necessary for speeds
different from the critical speed 3.
5. ACTIVE SQUEEZE FILM DAMPER
An active squeeze film damper can be chosen to
0 present an optimal value of clearance or viscosity.
I I I I I
9000 10000 11000 00021 13000 speed of rotation (rpm)
Figure 5a. Experimental results, g = 0.0135 Pa.s. 5.1. Variable clearance
To perform the control, the active squeeze film
edutilpmA (mm)
damper has been modeled. The idea is to regulate the
0Au
radial clearance by a parameter x corresponding to
the position of a conical squeeze film damper in its
housing (figure 6).
0.2
I ' I ' I ' I ' I '
9000 10000 11000 12000 13000 speed of rotation (rpm)
Figure 5 b. Numerical results, t~ = 0.0135 Pa.s.
The comparison between experimental and
numerical results is good. There is a critical speed
around 11800 rpm for la=0.05 Pa.s and around
11500 rpm for ~t=0.0135 Pa.s. The experimental
and numerical amplitudes are very similar. Figure 6. Active squeeze film damper.
Comparison between figure 4a (respectively 4b) Some interesting work has been published 22-
and 5a (respectively 5b) shows that the decrease of 23 on conical squeeze film dampers. The dynamic
viscosity (which could be due to an increase in oil study is carried out with a simple shaft
temperature) gives an increase of damping. In fact (symmetrical, centered, with one stiffness) and with
low viscosity leads to more displacement in the a synchronous circular whirl.
bearing and consequently higher energy dissipation In our work the shaft is modelled with these 6 first
may occur. modes and the trajectory can be of any shape. The
The conclusion of these first results is the numerical results are obtained with the same shaft as
following: the influence of the squeeze film damper previously. The idea is to monitor the radial
is very important and its effect is very different clearance by the rotational speed, the clearance
depending on the rotor speed. When the rotational evolution is a linear evolution from 0.05 mm to 0.1
speed is close to a critical speed, the squeeze film mlrl.
damper must dissipate a lot of energy, consequently Figure 7 presents the amplitude of the middle of
the viscosity must be small. Conversely a large the rotor and figure 8 gives the displacement in the
viscosity is better for speeds different to the critical squeeze film damper for 0.05 nma, 0.1 mm and for a
speed (in that case the amplitude in the SFD is linear evolution. It is remarkable (figure 7) that the
decreasing). shaft amplitude is filtered around the critical speed.
