ROTATING DISC VIBRATION ANALYSIS
SCANNING LDV
A.B. Stanbridge,
Mechanical
Engineering
M. Martarelli
WITH A CIRCULAR-
& D.J. Ewins
Imperial College
Department, Exhibition Road, London SW7 26X, UK
ABSTRACT
frequencies and the location of nodal lines is indeterminate,
depending, usually, on the position of applied excitation.
The vibration of circular discs, both stationary and rotating,
has been studied using a circular-scanning
laser Doppler
vibrometer (LDV).
Natural modes of such discs usually
involve circumferential mode shapes with an integer number
of waves around the circumference.
These may combine to
give operating deflection shapes (ODSs) which are standing
or iravelling waves, or a combination of both. Examples of
rotating discs with contacting and non-contacting excitation
are included.
In a case of parametric excitation (squeal) of
a pin-on-disc rig, vibration involved a standing wave fixed in
space.
An n-nodal diameter mode shape may also be visualised as
being the summation of two travelling waves of equal
amplitudes, travelling circumferentially in opposite directions,
each with a wavelength of 2nln radians, and a wavespeed of
wD/n radls, WD being the vibration frequency in radianslsec,
[I].
Under some circumstances,
particularly with rotating
discs, both modes of the pair may be excited, giving an
operating deflection shape (ODS) in which the forward- and
backward-travelling
waves have different amplitudes and
resulting in an ODS which is part travelling, and part
standing wave.
NOMENCLATURE
A circular-scanning
laser Doppler vibrometer (LDV) can, in
principle, 121, provide complete information about out-ofplane vibration operating deflection shapes (ODSs) of such
disc-like components, and this paper investigates the nature
of these in practical rotating and non-rotating situations.
vibration frequency in disc coordinates
vibration frequency in fixed coordinates
circumferential wave number, nodal diameters
scan frequencies
Qe, Qr
disc rotational speed
s
f, Ffa, Rb radii
e
angle
t
time
WI
w
n
.\‘\\\1/ /’
,’.’,‘I’.
j ‘.-.
GB
1 INTRODUCTION
Natural mode shapes of quasi-axisymmetric
discs can
generally be defined by circumferential
Fourier series in
which just a few terms, usually only one, have a significant
These are often described as being ‘nmagnitude, [I].
diameter
modes’
implying
modes
with
a cosne
circumferential mode-shape and hence n equi-spaced nodal
diameters.
Figure 1 An n = 3 nodal diameter mode showing
and anti-nodes.
nodes
The technique has been applied to a thin steel disc, where
vibration could be induced either with an electro-magnet or
mechanically, via a jockey wheel, in a stationary or a rotating
disc.
Reference is also made to an application to the
problems of disc-brake squeal, using a pin-on-disc rig. In
this case, vibration was a limit-cycle instability, without any
discrete frequency input excitation.
In fact the natural modes occur in spatially-orthogonal
pairs,
in which the nodal radii in one mode of the pair become antinodes in the other, and vice versa (Figure 1). In a perfectly
axi-symmetric disc the two modes will have identical natural
464
2 PRACTICAL
CONSTRAINTS
4 EXCITATION
Commercial LDVs can easily be used in a continuous-scan
role, provided they incorporate x, y beam deflection mirrors
which can be controlled externally.
If sine and cosine wave
inputs are applied to the two mirrors, the measurement point
can be made to scan continuously around a circle.
In this
way, every point around the scan circle is covered, at a rate
equal to the difference between the scan speed and the disc
rotational speed, if moving.
Provided that there is a steady
state vibration, the LDV response is periodic, and it will have
a line spectrum which can be obtained directly by an FFT.
In most cases phase spectrum data, which are needed in
other contexts and which complicate the signal processing,
are unnecessary here.
OF ROTATING
DISCS
If the disc is rotating at an angular speed, S, the LDV scan
speed, relative to the disc, Qo, will be (Q, -s) , negative
speeds indicating rotation in the opposite direction to S.
In some circumstances
it is possible to apply sinusoidal
vibration to a rotating disc from a fixed point in space.
If w
is the excitation frequency in the non-rotating frame of
reference, n-diameter ODSs are excited, in rotating, disc,
coordinates, at frequencies
(m--,S)
and (m+nS);
i.e.
a,
= (0 k &)
, [3]. In principle, an LDV scanning at Q0 will,
under these circumstances,
have a response with four
frequency components; at (m- nS) ? n(Q, - S) and at
(w + nS) f n(Q B - S)
LDVs are susceptible
to speckle
noise, an optical
phenomenon that is inescapable
because the instrument
relies on scattered light returned from an optically-rough
surface.
Speckle noise produces, effectively, very short
drop-out spikes, irregularly spaced in time, which equate in
the frequency
domain to a broad-band
random noise
spectrum which limits the accuracy and resolution of the
LDV measurements.
The severity of the speckle noise
increases as the scan speed relative to the surface is
increased.
So far as the present applications
are
concerned therefore, circular scanning is more suited to
fairly low rotational speed applications.
That is to say, at:
ClJ-d2,
(1)
o+ n(S2, - 2s)
(2)
CO+&,
(3)
and w - n(Q2, - 2s)
(4)
At resonant conditions, either frequencies (1) and (2), or (3)
and (4), are to be expected to predominate in the LDV
output signal.
An LDV measures the velocity of the point addressed, in the
direction of the incident measurement beam.
As normally
applied, the LDV is set up on the axis of rotation of the disc,
so that it actually measures the vibration resolved onto a line
which is not exactly perpendicular to the surface. The angle
involved is usually no more than about IO”, and the
correction is therefore quite small.
Of more importance, if
measurements are to be made on a rotating disc, is that the
scan axis must be aligned with the rotation axis, otherwise
the LDV detects a proportion of the rotational velocity, [4],
apparent in the LDV output signal as a component at the
LDV scan frequency.
This interfering signal is present in
most of the illustrations included here.
Disc peripheral
speeds can be very high compared with the vibration
velocities to be measured and this interference can be
overwhelming unless the alignment is very precise. For this
reason also, the method is unsuitable for high-speed rotors.
3 SCANS ON NON-ROTATING
AND RESPONSE
DISCS
It should be noted, incidentally, that when w = 0, the disc
experiences
vibration
at a frequency
nS.
With a
symmetrical disc, if nS coincides with a natural frequency, a
backward-travelling
wave will be set up in the disc, if there
are steady forces in the non-rotating frame of reference.
The result is a standing wave in the non-rotating frame of
reference -the classic ‘critical speed’ condition.
4.1
Test Measurements
on a Symmetrical
Disc
Discs and disc-like components can be excited in operating
machines from non-contacting sources due to, for example,
A simple test rig
magnetic fields or aerodynamic forces.
has been used to check these phenomena: a 450 mm
diameter, 1.5 mm thick circular disc could be rotated by an
electric motor, and excited by an electromagnet
in close
proximity to its surface. An LDV was set up to scan the disc
in a circle, near its circumference, at 10 Hz.
Under non-rotating conditions, excitation at 366 Hz, for
example, forced a resonance with, as evidenced by the LDV
spectrum (Figure 2), a 6 nodal diameter ODS.
Fairly obviously, if an LDV measurement beam is scanned at
a rotational frequency, Q,, around a (non-rotating)
disc
vibrating in an n-diameter mode, its output spectrum will
contain two components: one due to the forward-travelling
wave (relative to the scan) and one due to the backwardtravelling wave.
These will appear, respectively,
at
frequencies (o+, - nQ,) and (w, + &,).
An FFT of the
One may note from Figure 2 that the forward and backward
ODS components were, in this case, not excited with exactly
equal amplitudes, so that this was not a pure standing wave
vibration.
A g-nodal diameter ODS could also be excited under
rotating conditions.
The example shown in Figure 3
involved rotation at 2.44 Hz (146 rpm), with excitation at
351.7 Hz.
The ODS was, in this case, predominantly a
backward-travelling
wave.
LDV
signal
will
therefore
give
an
unambiguous
measurement of the magnitude of these component waves,
of the frequency of vibration, wo, and of n.
465
0 1w x.3 300 &W MO 6W
Figure 2 Non-contact
excitation of a symmetrical
nonrotating disc. Circular-scan
(10 Hz) LDV spectrum,
g-nodal dia. ODS.
Excitation in disc coordinates was at W+ nS = 366.3Hz.
The frequency separation of the travelling-wave
peaks in
Figure 3 was at frz(Q,, - ,Q,) = +45.4 Hz, rather than
+60Hz for a non-rotating
situation, as in Figure 2.
Figure 4 Rotating
Figure 3 Non-contact
excitation of a rotating
symmetrical
disc, Disc speed 2.44 Hz, Excitation at
351.7 Hz, LDV spectrum, scan at 10 Hz, B-nodal dia. ODS
Disc, excited
space
from a fixed point in
%lz
Figure 5 Contact excitation
of a symmetrical
nonrotating disc. Excitation at 369 Hz, Circular (10 Hz)
LDV scan spectrum, B-nodal dia. ODS.
Figure 4 shows the same rig and disc, arranged so that
sinusoidal vibration could be applied directly with a normal
shaker, by means of a jockey wheel mounted in ballbearings, attached to the exciter drive.
Figure 5 shows the
spectrum of an LDV signal obtained with the disc not
rotating, with excitation at an apparent resonant frequency,
369 Hz, and using a circular LDV scan at IO Hz. Sideband
peaks are seen at 310 Hz and 412 Hz, which indicates an
ODS with (412-310)/(2x1 0) = 6 nodal diameters.
Note that
the fixed-point contact appears to have enforced a singlemode ODS in this non-rotating situation; i.e. equal sideband
amplitudes, indicating a standing wave, stationary ODS
wave but on its own it gives no information about the
This
vibration frequency
or the spatial wavelength.
information
can, however,
be derived
by a repeat
The easiest
measurement at a different LDV scan speed.
approach is to reverse the scan direction, so that Q, is
negative.
In this particular case, Figure 6(b), the vibration
peak then appeared at 290 Hz. These two frequencies are
(w + n&J and (w - r&Q, and they correspond to an n = 6
backward-travelling
wave.
The vibration frequency, in disc
coordinates was o + n,S = 366.6 Hz (compared with 369 Hz
Some low-frequency vibration
at the stationary condition).
is apparent in Figure 6; this was due to the interaction
between the imperfectly-flat
disc and the shaker pressed
Movement of the shaker, at multiples of
against it.
rotational speed, as it rotated, produced corresponding
inertia force excitation.
A similar experiment was conducted with the disc rotating at
2.44 Hz. Under these conditions, resonance was observed
with excitation at 352 Hz, Figure 6(a) displaying the LDV
output spectrum.
There is a single response peak at 412
Hz. The single peak indicates the presence of a travelling
466
Figure 7, for example, shows an ODS measured with a IO
Hz circular scan, with non-contacting magnetic excitation at
363.6 Hz and the disc rotating at 2.5 Hz. The n = 6 ODS
pattern is more distorted than before, with n = 0, 2 and 4
components, but only a single mode is excited, forward- and
backward-travelling
components being, within experimental
uncertainty, equal.
Similar effects were observed with contacting excitation, via
the jockey wheel - the split modes were, again, excited
independently, depending on the excitation frequency.
i
xc
em
HZ
6(a)
1
~I
---me
1
09r
Figure 8 Contact excitation of a rotating asymmetrical
disc, Disc speed 1.2 Hz, Excitation at 360 Hz, 10 Hz LDV
scan spectrum, combination
of two 6-nodal dia modes.
6(b)
Figure 6 Contact excitation of a rotating symmetrical
disc, Disc speed 2.44 Hz, Excitation at 352 Hz,
B-nodal dia. ODS, IO Hz LDV scan spectra,
(a) forward, (b) backward.
4.2
Test Measurements
on an Asymmetric
Figure 8 illustrates a somewhat special example: it shows a
circular-scan ODS signal spectrum with a disc rotational
speed of 1.2 Hz (72 rpm) and excitation at an apparent
resonance peak at 360 Hz.
In this case, excitation was
produced, in the disc frame of reference, at 352.8 and 367.2
Hz, both of which frequencies happened to coincide with the
two split, 6-nodal diameter natural frequencies.
With a IO
Hz circular scan, response in these gave the four prominent
peaks shown in Figure 8. This, exceptionally, is an example
where all four response components, (I), (2) (3) and (4)
above, are present simultaneously.
The forward- and
backward-travelling
waves are not completely balanced, i.e.
both response modes are predominantly
standing waves,
but with a travelling wave component, rotating against the
scan direction.
Disc
The disc was later mistuned by attaching two 18 gm weights
on a diameter, near its periphery, to split the orthogonal
modes.
Basically, in this situation, a sinusoidally-excited
ODS, at a resonance peak, usually involved only one of
these modes-a
standing wave in disc coordinates.
4.3
Rig
In the pin-on-disc rig already referred to, an inclined pin is
pressed against a rotating disc.
If the pin arrangement is
suitably adjusted, the disc will exhibit limit-cycle parametric
vibration, ‘squeal’, even at very low speeds of rotation. The
arrangement
is simpler than for a vehicle brake disc
assembly, there being, for instance, point contact rather than
area contact where the pin and disc are pressed together.
The phenomenon
is essentially non-linear, being frictioninduced, but the disc itself generally vibrates in a sinusoidal
vibration with an n-diameter ODS.
Figure 7 Non-contact
Circular-scan
Pin-On-Disc
excitation of an asymmetrical
rotating disc
(IO Hz) LDV, 6-nodal dia. ODS.
467
The disc concerned was made from aluminium, 16” (406
mm) diameter and 1” (25.4mm) thick. Figure 9 shows the
LDV squeal signal spectrum produced by pressing the pin
against the (uniform) disc, rotating at 0.0317 Hz (1.9 rpm),
with a circular LDV scan at 10 Hz. Response peaks at 2270
and 2330 indicated excitation of a 3-diameter, standingwave ODS at 2300 Hz
The disc was next made asymmetric by attaching a weight
near the periphery, so as to split the two 3-diameter mode
frequencies.
The squeal could be induced nearly as easily
as before, but the squeal frequency could be heard to
alternate between the two natural frequencies, depending on
the orientation of the disc relative to the pin, Figure 11. The
ODS response pattern, as might be expected, was not fixed
in space but moved somewhat irregularly with the disc,
depending on the position of the excitation pin.
5 CIRCULAR
AREA SCANS
As described above, a circular scan is achieved by directing
the LDV measurement point around a circle with a radius rat
a uniform rate, so that 0 = Q,,t
The radius, r, can also be
If Rr > Q0
varied, for example so that r = R, + R, cosQrt
the scanning circle will then spiral in and out to cover an
annular area, Figure 12(a); if Q, > Q,,. then a ‘daisy’ scan is
produced, Figure 12(b), in which the LDV scans to and fro
along a radius which sweeps at a lower frequency around
the circular area.
IH;
Figure 9 Circular scan LDV spectrum.
Pin-on-disc Rig. Squeal with a symmetric disc
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jJIi./,
.?,il
;
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.&
Il#ii
j‘k’(//(I
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Figure 12 Circular
Mirror sw-~~-11oc.3 1mo
ow
Area Scans
The xand ymirror drive signals required to achieve this take
Figure 10 Circular scan LDV signal and mirror drive
signal.
Pin-on-disc Rig. Squeal with a symmetric disc
the form:
x = (R, + R, cosQ,f)cosSl,t
and y=(R,, +R,cosQ,t)sin!2,$.
The LDV (time) signal modulation was seen to be fixed in
time relative to the mirror drive signals, Figure IO, with an
anti-node at the pin contact position.
The disc was
therefore rotating slowly through a standing wave, which
was fixed in the non-rotating frame of reference.
035
03’
025
Mistuning
y;;i@
u
[HIGH1 C-------I
Hz
P
Figure 13 LDV signal
spectrum
area scan.
for a ‘daisy’
circular
HIGHAn example of a daisy scan LDV spectrum for the disc in the
pin-on-disc rig, with the pin retracted and with direct
excitation at the relevant 3-diameter mode, with Rr = 20 Hz
and Q, = 1.1 Hz, is shown in Figure 13. It resembles that
LOW
Figure 11 Pin-on-Disc Rig. Mistuned Disc.
Pin position, P, for high and low squeal frequencies.
468
for a radial line scan, but each sideband is split, the spacing
between each pair being +nslO. Straight-line scan theory,
see [2], may be used to determine
the radial ODS,
In this
modulated by ncos8 in the circumferential direction.
particular case there are only two sideband pairs on either
side of the central pair, so that the radial ODS distribution
derived is simply a quadratic expression in r. The result of
applying this process to the sideband data in Figure 13 is the
3D area ODS presented in Figure 14. Determination of the
radial mode shape requires
that signs (or complex
component amplitudes) for the sideband components be
This in turn generally means that the vibration
established.
frequency, o, and the scan speed relative to the disc
surface, must be known precisely.
Circular area scanning is
therefore not so easily applied to rotating discs because
these frequencies may be inaccessible.
Figure 14 3-nodal diameter area ODS derived
in Figure 13.
from data
6 CONCLUSIONS
ACKNOWLEDGEMENTS
Axisymmetric
and near-axisymmetric
discs have natural
mode shapes in spatially-orthogonal
pairs, at identical or
These may be excited alone or
close natural frequencies.
in any combination,
depending on whether the disc is
stationary or rotating, on the nature of the excitation and on
A circular-scanning
the proximity of the natural frequencies.
LDV is extremely useful in giving an immediate diagnosis
and quantification of such vibration, provided there is a line
of sight.
(Speckle noise may also be a problem with highspeed wheels.)
The scanning LDV response measurement technique was
developed under the auspices of of BRITE (VALSE) Project
BE97-4126, and the authors are grateful to the EU and the
other partners in the project for their support.
We would
particularly
like to acknowledge
the involvement
of the
FV/FLP group at Bosch, who have encouraged our interest
in disc vibration, and who provided the pin-on-disc rig.
REFERENCES
In the laboratory tests described here, with single-point,
resonant, sinusoidal contact excitation of axi-symmetric
discs, standing wave vibration was always excited in a nonrotating disc, and pure travelling wave ODSs when rotating.
With non-contacting excitation there was, in each case, a
mixture of standing and travelling wave ODS.
[I]
Medigholi, H., Robb, D.A. and Ewins, D. J.,
Simulation of Vibration in a Disc Rotating Past a Static
force, IMAC 10, pp 802 - 809. 1992.
[2] Stanbridge, A. B. and Ewins, D. J., Modal Testing
Using a Scanning LDV, Mechanical Systems and Signal
Processing, Vol. 13(2), pp 255 - 270, 1999.
A pure travelling wave appears as a single-frequency
vibration; its nature can easily be determined by a second
measurement with a reversed scan direction.
[3] Wildheim, S. J., Excitation of Rotationally
Periodic Shuctures, ASME Journal of Applied
Mechanics, Vol. 46, pp 878 - 882, 1979.
With the pin-on-disc rig, provided the disc was uniform, the
‘squeal’ induced was a standing wave in space, through
which the disc rotated, with an anti-node at the pin which
produced the excitation.
[4] Castellini P. and Paone N., Developmenf of the
Tracking Laser Vibrometer: Performance and
Uncertainty Analysis, Review of Scientific Instruments,
American Institute of Physics, (to be published).
Mistuning will separate each pair of natural frequencies and,
with sinusoidal contact-excitation,
at resonance, only one
mode, a standing-wave
in the disc’s frame of reference,
With a
could be excited, whether rotating or non-rotating.
mistuned disc in the pin-on-disc rig, the squeal mode (and
frequency) switched to and fro, depending on the position of
the radial nodes relative to the excitation source.
469