Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
CA 02348320 2001-05-18
MODAL ANALYSIS METHOD AND APPARATUS THEREFOR
Field of the invention
The present invention relates to modal analysis of a structure for
determining dynamic vibratory characteristics thereof, and more particularly
to
modal analysis method and apparatus using acoustical excitation to impart
vibration to the structure under test.
Brief descriation of the prior art
Modal analysis techniques have been recently applied to many vibratory
testing applications, and particularly in Environmental Stress Screening (ESS)
tests such as those performed in printed circuit boards (PCB's) manufacturing
as
part of quality control procedures.
According to conventional ESS procedures for testing PCB's,
determination of the vibration spectrum required for testing a particular PCB
is
usually an empirical matter. Induced fatigue and precipitation of latent
defects are
generally not estimated considering the actual stress within the circuit, but
are
rather empirically estimated from the vibration level as measured. Types of
defects that are precipitated with a stimulation using random vibrations are
mainly related to poor solders, component or substrate defects, connector
problems, poor securing of cables and components, and structural problems.
Methods of determining the spectrum of a vibrating excitation typically range
from
the study of vibrating behavior with comparison of the global response to
predetermined optimum vibration levels, to the use of spectrums previously
employed with success for other similar products. An intermediary method
consists of introducing typical defects in a product and then increasing the
vibration level until these defects repetitively precipitate, which method
requires
to apply long-continued vibrating stimulation, typically of about 10 minutes
or
more. In order to improve efficiency over these known methods, a structural
model characterizing the vibration response of a product can be build prior to
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determine the spectrum of vibrating stimulation likely to produce the target
frequency response profile. For this purpose, modal analysis techniques are
used, such as those described in the applicants' papers " Modal analysis of
electronic circuit using acoustical sources ", 4t" Annual IEEE Accelerated
Stress
Testing, 1998, and "Experimental modal analysis using acoustical sources ", 1
Tn
Canadian Congress on Applied Mechanics, 1999. Modal analysis essentially
consists in establishing a theoretical model in terms of vibration parameters
including resonance frequencies and damping factor associated with main
modes of vibration. Then, values of these vibration parameters are determined
experimentally using either a mechanical or acoustical source of vibration,
such
as disclosed in the inventor's prior International PCT application no. WO
01/01103 to the applicants as published on Jan. 4, 2001, along with
conventional
vibration measuring instrumentation. From the obtained vibration parameters
values, vibrating stimulation levels required to comply with ESS testing
requirements can be predicted as well as optimal vibration spectrums. Acoustic
excitation is a very attractive, non-contact approach for excitation of
flexible
structures. Unfortunately, an acoustical source does not produce a localized
force on the structure under test, and therefore a plurality of vibration
transducers
(accelerometers) directly mounted on the article under test have been required
heretofore, such as taught in the above-cited publications from the
applicants. A
complex set-up of transducers and cables must be realized to perform modal
analysis of a specific structure to be tested, implying time-consuming
calibration
procedures.
Summary of the invention
It is therefore an object of the present invention to provide an acoustic-
based modal analysis method and apparatus for determining dynamic vibration
characteristics of a structure, which minimizes the number of output vibration
transducers required.
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According to the above object, from a broad aspect of the present
invention, there is provided a modal analysis method for determining dynamic
vibration characteristics of a structure under acoustic excitation. The method
comprises a first step of providing nt complementary sets of correlated input
acoustic pressure-related data in the frequency domain representing m
complementary acoustic excitation signals, all sets of data being provided
according to n spatially distributed locations associated with the structure
with
m>_n , each set including reference input acoustic pressure-related data
provided
according to a reference one of the locations. The method further comprises
steps of providing m corresponding complementary sets of output vibration data
in the frequency domain in response to the acoustic excitation at a reference
point on the excited structure corresponding to the reference location and
providing m corresponding complementary sets of n input transfer functions
characterizing the correlation between each set of input acoustic pressure-
related data and the reference input acoustic pressure-related data. The
method
further comprises steps of obtaining n structural transfer functions
characterizing
each set of input acoustic pressure-related data from relations between the m
sets of n input transfer functions and the m sets of output vibration response
data and deriving from the structural transfer functions the dynamic vibratory
characteristics of the acoustically excited structure.
According to a further broad aspect of the invention, there is provided a
modal analysis method for acoustically determining dynamic vibration
characteristics of a structure, the method comprising steps of: a) generating
an
acoustic excitation signal toward n spatially distributed locations associated
with
the structure while the latter is held to allow vibration thereof, one of said
locations being a reference location; b) sensing the acoustic excitation
signal at
the locations to produce a corresponding set of n correlated input acoustic
pressure-related electrical signals, one of the electrical signals being a
reference
signal associated with the reference location; c) converting the set of n
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correlated input acoustic pressure-related electrical signals into a set of
correlated input acoustic pressure-related data in the frequency domain
including reference data associated with the reference signal; d) sensing
induced output vibration in response to the acoustic excitation at a reference
point on the excited structure corresponding to the reference location to
produce an output vibration response electrical signal; e) converting the
output vibration response electrical signal into a set of output vibration
response data in the frequency domain; f) providing a set of n input transfer
functions characterizing the correlation between the input acoustic pressure-
related data and the reference data; g) performing said steps a) to f) for m-1
complementary acoustic excitation signals with m>_n , to produce m-1
complementary sets of input acoustic pressure-related data and to produce
m-1 complementary sets of output response vibration data; h) obtaining n
structural transfer functions characterizing each set of input acoustic
pressure-related data from relations between the m sets of n input transfer
functions and the m sets of output vibration response data; and i) deriving
from the structural transfer functions the dynamic vibratory characteristics
of
the structure.
According to a still further broad aspect of the invention, there is
provided modal analysis apparatus for determining dynamic vibration
characteristics of a structure. The apparatus comprises acoustical source
means capable of generating m complementary sets of correlated
acoustic excitation signals toward n spatially distributed locations
associated with the structure, one of said locations being a reference
location, and a structure holder provided with attachment means for holding
the structure while allowing thereof to vibrate under the acoustic excitation
signals. The apparatus further comprises acoustic sensor means
responsive to the acoustic excitation signal at the locations to produce m
complementary sets of n correlated input acoustic pressure-related electrical
signals, one of the electrical signals being a reference signal associated
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4A
with said reference location, and Fourier transform means for converting the
sets
of correlated input acoustic pressure-related electrical signals into sets of
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correlated input acoustic pressure-related data in the frequency domain
including
reference data associated with the reference signal. The apparatus further
comprises vibration sensing means responsive to induced output vibration in
response to the acoustic excitation at a reference point on the excited
structure
5 corresponding to the reference location to produce m complementary output
vibration electrical signals and Fourier transform means for converting the
output
vibration electrical signals into m sets of output vibration data in the
frequency
domain. The apparatus further comprises data processor means responsive to
the sets of correlated input acoustic pressure-related data and to the sets of
output vibration data for providing n input transfer functions characterizing
the
correlation between each set of acoustic pressure-related data and the
reference
data, for obtaining n structural transfer functions characterizing each set of
input
acoustic pressure-related electrical data from relations between the m sets of
n
input transfer functions and the m sets of output vibration response data, and
for
deriving from the structural transfer functions the dynamic vibratory
characteristics of the structure.
Brief description of the drawinas
Preferred embodiments of a modal analysis method and apparatus
according to the invention will now be described in view of the accompanying
drawings in which:
Fig. 1 is a schematic view of a preferred embodiment of a modal analysis
apparatus according to the present invention;
Fig. 2 is a perspective view of a PCB holder and loudspeaker provided on
a preferred embodiment of the apparatus according to the invention;
Fig. 3 is a block diagram of the Multiple-Inputs/Single Output model on
which is based the principle set forth by the invention;
Fig. 4 is a flow chart representing the simulation process performed to
verify performance of the method according to the invention;
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Fig. 5 is a graph showing an example of structural transfer function HAY for
i=2 that has been identified by the method and apparatus of the present
invention;
Fig. 6 is a graph showing an example of amplitude values for a first mode
that has been identified by the method and apparatus of the present invention.
Detailed description of the areferred embodiments
In the present specification, the apparatus and method according the
present invention will be described in view of a particular application
dealing with
PCB's as tested structures. However, it is to be understood that the
application
scope of the present invention is by no means limited to PCB's or like
flexible
structures, but to any other structure for which dynamic vibration
characteristics
has to be determined.
Referring to Fig. 1, the modal analysis apparatus according to the
invention as generally designated at 10 comprises a structure holder 12 having
a
main frame 13 provided with attachment means in the form of a fixture 14
having
adjustable clamps 16 for securing a PCB 18 at a peripheral portion thereof to
allow vibration under acoustic excitation, as will be explained later in
detail. The
fixture 14 is preferably of a similar design as the fixture described in the
above-
cited published international PCT application no. WO 01 /01103 to the
applicants.
As shown in Fig.2, the fixture 14, which is designed to receive a single PCB
18 in
the example shown, comprises a generally rectangular outer frame 15 provided
with a recessed planar portion 17 defining a central opening 19 to be aligned
with
a locating reference pattern 24 printed on a mat 23 or directly on the floor,
by
positioning the legs 21 of main frame 13 accordingly. The clamps 16 are
mounted on fixture planar portion 17, which clamps having mounting blocks 27
that can be locked in a predetermined position along the corresponding sides
of
the frame 15 by set screws 29 extending through corresponding bores provided
on the sides of frame 15, and through corresponding threaded bores provided on
blocks 27. Alternatively, the sides of frame 15 may be provided with elongated
i;
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4 1
slots (not shown) to allow position adjustment for the blocks 27. Each clamp
16
includes a spring-biased clamping member 31 cooperating through pivot 33 with
a base member 35 having a pair of lateral flanges 37 being rigidly secured to
the
corresponding block 27 with screws 39. To the forward end of each clamping
member 16 is secured a mounting spacer 42 fixed in a position parallel to a
corresponding PCB edge with a set screw 44 vertically extending through the
forward end of clamping member 31. Each mounting block 27 is provided with a
rib (not shown) having an end that is vertically aligned with the mounting
spacer
42 when the clamp is in a lock position, defining a tight space for receiving
and
maintaining the PCB edge adjacent portion. Facing ends of mounting spacers 42
and corresponding ribs are aligned with rubber pads 45 to ensure that the PCB
edge surface is not damaged by the clamps 16 when the latter are brought in a
lock position. The fixture 14 is designed to allow the mounting of a
sufficient
number of clamps 16 located on the periphery of the PCS to allow the latter to
vibrate according to some vibration modes characterizing the structure, as
will be
explained later in more detail.
Turning back to Fig. 1, as part of an acoustical source and disposed under
fixture 14 is an acoustical transducer or loudspeaker 20 to be located at a
stable
position with respect to the locating reference pattern 24, which allows
positioning of the loudspeaker 20 at selected specific locations with
reference to
the central opening 19 of fixture 14, as will be explained later in more
detail. The
acoustical source further includes driver means operatively coupled to
loudspeaker 20, in the form of an audio amplifier 25 responsive to an input
signal, such as a white noise, generated by a signal generator 26. The
apparatus further comprises a set of acoustical sensors in the form of a
plurality
of microphones 28; , with i=l,n , which are disposed at spatially distributed
locations associated with the structure, generally according to a two-
dimensional
configuration. The value for n and the appropriate configuration for the
microphones are dictated by the particular modal analysis to be performed. A
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g
selected one of microphones 28; identified as 28k is considered as a reference
microphone disposed at a reference location, as will be explained later in
more
detail. Alternatively, a lesser number of microphones can be used by
performing
successive tests with the same microphones relocated at different positions. A
vibration sensor in the form of an accelerometer 38 is disposed at a reference
point on the structure or PCB 18, which reference point is spatially
associated
with the location of reference microphone 28k being vertically aligned with
accelerometer 38. The microphones 28, to 28" are coupled to corresponding n
inputs provided on a conventional conditioning amplifier 32, which also
receives
at input 41 the output vibration signals coming from accelerometer 38. It is
to be
understood that separate conditioning instrumentation for the microphone
signals
and accelerometer signal can also be provided, as well known in the art. The
conditioned outputs of signal conditioning amplifier 32 are fed to
corresponding n
inputs of a Fourier transform converter which is preferably a Fast Fourier
Transform analyzer generating converted data in the frequency domain toward a
data processor device such as computer 40 for further processing.
The principles on which is based the present invention will now be
explained in detail. The non-contact modal analysis technique according to the
invention is based on a particular Multiple Inputs Single Output model (MISO)
using correlated acoustical excitation signals. The mode shapes and modal
parameters of the structure are given by the identification of the Frequency
Response Functions (FRF) obtained by acoustic pressures measurements of the
excitation in the near field of the structure at a predetermined number of
locations
in accordance with the considered number of degrees of freedom, and by a
single acceleration measurement of the structure response. Then, dynamic
vibration characteristics of a structure under test, including natural
frequencies,
mode shapes and damping factors, can be determined using conventional
derivation techniques.
i
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Referring to Fig. 3, a MISO system generally designated at 47 is defined
by the application of several input forces F;(w) with i=1,2,...,k,...,n ,
which forces
F;(w) are in the form of acoustic excitation signals as sensed by microphones
28;
at n spatially distributed locations associated with the structure, which
5 microphones generate correlated input acoustic pressure-related electrical
signals, and by the measurement of a single vibration response Y(a), as sensed
by accelerometer 38 at a reference point on the excited structure. When a set
of
perfectly coherent external acoustic forces F;(w) is applied to the structure
we
can define the input transfer function between a force I and a force j as
being
HF;Fj(w). This relation is expressed as follows by choosing a force Fk(~) as a
reference force associated with a reference location on the structure:
S~~=H~~xS~~
wherin .SFkFi is the cross-spectrum between the reference force Fk(~) and a
force
i, and SFkFk is the auto-spectrum of the reference force Fk(r~). The n input
15 transfer functions HFk~;(~) characterize the correlation between the input
acoustic
pressure-related signal and the reference signal. These input transfer
functions
HFkFn~) depend on the characteristics of the acoustic excitation and vary for
each
acoustic load case a according to the amplitude and phase relations between
the
forces. The dynamic mechanical system is characterized by a series of
structural
transfer function H;Y(w) which are specific to the structure and depend on the
modal parameters. The cross-spectrum between reference force Fk(w) and
response Y(w) is expressed as follows:
,fir ~r S~F
m
We express the total response Y(~) according to Fk(~) in the following way:
Y=~Hrr H~~Fk
I ~. ~ i
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The measurements of each input force F;(w) with i=1,2,...,k,...,n and of the
vibration response Y(tv), expressed as frequency domain data through Fourier
analysis, lead to only one equation with n unknowns which are the structural
transfer functions H;y(w) of the structure, in terms of the n input transfer
functions
5 HFkFi(~) characterizing the correlation between the input acoustic pressure-
related data and the reference data. It is thus necessary to increase the
number
of equations available to m ~ n to be able to derive all H;~.(u~). By exciting
the
structure with complementary load cases with a = a...m and m ~ n and by
measuring each set of forces F;(w) and response Y(a), it is possible to
express
10 the system of relations in matrix form as follows:
Hn~ H~~ a a Her Y Fka
H~r~....HFkFr a
a
H~~ b b b ...H~FHZr Y F~b
H~r~ H~~ . b b
H~~ a ...HF1FH3r Y ~ a
H~~ a a = a~ lFk
Hn~ a
H~i~~m~I~~~m~l~x~~m~ ...H~F~m)H"r Y~myF'x(m)
wherein HF,~;(a) are the input transfer functions between Fk(a) and F; (a) for
the
load case a with a = a,...m.
15 In other words, once a first set of input acoustic pressure-related data
Fk(a) for a first load case a=a is provided with its corresponding set of
output
response vibration data, the same type of data is obtained in a same manner as
explained above for m-1 complementary acoustic excitation signals ,
corresponding to m-1 further load cases with min , to produce m-I
20 complementary sets of input acoustic pressure-related data and m-1
complementary sets of output response vibration data. In practice, a
particular
load case will be associated to a specific position of the loudspeaker 20 with
respect to the reference pattern shown in Fig. 2. It is to be understood that
any
other means to provide a plurality of load cases, such as using a plurality of
25 spatially distributed loudspeakers, are contemplated in practicing the
present
invention. Then, the n structural transfer functions H;y(r~) characterizing
each set
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11
of input acoustic pressure-related data can be obtained from a system of
relations between the m sets of n input transfer functions and the m sets of
output vibration response data. The system of relations can be easily solved
by
any appropriate technique such as inversing or pseudo-inversing techniques if
m
> n to obtain:
{H~r~,~t~=~H~F(a~~,~m~~F~Y(a~k(a~~~~~.ra
where FRF is the vector (m x 1) containing the FRF between the force Fk(a)
with
a=a,..m, and the acoustic response of the system Y(a). More specifically, the
above system can be solved to identify the n structural transfer function Hey
where m = n H ~~.r a = Z ~,~, ~FRF ~m
where m ) n H ~.~ o = P~~;, ~ Z~'"~", ~ FRF ~m
wherein:
R~> = Z~',~m~ Zc"~n>
Once the H;Y are obtained for each frequency, the n first mode shapes of the
system with associated natural frequencies and damping factors can be derived
using any usual techniques, such a peak amplitude method, as described by
D.J . Ewins, « Modal Testing : Theory and practice », Research Studies Press,
1984. There exist a number of modal analysis methods which, although different
in their detail, all share the same basic assumption: namely, that in the
vicinity of
a resonance the total response is dominated by the contribution of the mode
whose natural frequency is closest. The methods vary as to whether they
assume that all the response is attributed to that single mode or whether the
other modes' contributions are represented by a simple approximation. The
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simplest of these methods is one which has been used for a long time and which
is sometimes referred to as the peal-amplitude or peak-picking method. This is
a
method which works adequately for structures whose FRF exhibit well-separated
modes which are not so lightly-damped that accurate measurements at
30 resonance are difficult to obtain but which, on the other hand, are not so
heavily
damped that the response at a resonance is strongly influenced by more than
one mode. Although this appears to limit the applicability of the method, it
should
be noted that in the more difficult cases, such an approach can be useful in
obtaining initial estimates to the parameters required, thereby speeding up
the
35 more general curve-fitting procedures described later. The method is
applied as
follows:
(i) first, individual resonance peaks are detected on the FRF plot,
corresponding to mathematical expression (2) above, and the frequency of
maximum response taken as the natural frequency of that mode ~x ;
40 (ii) second, the maximum value of the FRF is noted I&I and the frequency
bandwidth 0~ of the function for a response level of ~~2 is determined. The
two points thus identified as rr~ and r.~ are the half-power points;
(iii) The hysteretic damping loss factor of the mode in question can now be
estimated from the following formulae:
45 ~''-~C~2-~f'2~ =0 /Ctk
which damping loss factor r~. is related to the damping factor by a factor 2
as
shown in equation (3) below;
(iv) Last, we may consider ~al as an estimate for the modal constant of the
mode being analyzed, corresponding to or mode shape as expressed by
50 equation (10) above, by assuming that the total response in this resonant
region
is attributed to a single term in the general FRF series.
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12A
The method according to the invention has been proven through numerical
simulation performed on a plate with simply supported boundary conditions
using
a number of load cases m=n=7, i.e. considering seven (7) acoustic excitation
locations p1 to p~ associated with the structure as shown in the graph of Fig.
6,
and according to a simulation process illustrated on Fig. 4. Input forces
F;(~) with
i=1,2,...,k,...,n are calculated at step 50 according to a point source
radiation
model as well known in the art, to provide the m complementary sets of
correlated input acoustic pressure-related data in the frequency domain
representing m complementary acoustic excitation signals. At step 52, The
first
natural frequencies ~",m and mode shapes are obtained with an analytical plate
model defined from, plate thickness, late length, plate width, Young's modulus
material density of the plate, Poisson's coefficient, and parameters related
to the material properties of the plate. The damping factor can be calculated
as
follows:
2~
wherein 0~ is the frequency bandwidth corresponding to half of natural
frequency amplitude. Then, at following step 54, a usual modal superposition
algorithm is applied to derive the theoretical structural transfer functions f-
l~;Y(~)
as well as the vibration response Y(~) of the structure.
~ i. ~ ~ i
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13
At following step 56, the MISO model with coherent excitations is used to
determine the mode shapes and the structural transfer functions H;~.(w) of the
plate according to the model of the invention. As shown in Fig. 5 wherein both
theoretical and modeled structural transfer functions i-1~2v(~),H2~~(~) are
plotted for
the 0 - 400 Hz frequency range, it can be seen that both curves mutually
correspond in amplitude and frequency in the area of the natural or resonance
frequencies. Fig. 6 shows a mode shape comparison for the first resonance
frequency of the plate, wherein amplitude values associated with the seven (7)
excitation locations pi to pl were used along with boundary conditions to
interpolate amplitude values associated with p8 and p9 . Then, a validation of
the
MISO modeling is performed at step 58, wherein a Mode Assurance Criteria
(MAC) analysis between the theoretical and modeled mode shapes, as described
in D.J . Ewins, « Modal Testing : Theory and practice » Research Studies
Press,
19&4, yields to MAC=1 fo all seven (7) identified mode shapes, thus indicating
a
perfect mode shape ident~cation using the model according to the invention.