Experimental measurements on the actual hardware result in a physical
check of the accuracy of the mathematical model. If the model predicts the same behavior that is actually
measured, it is reasonable to extend the use of the model for simulation, thus reducing the
expense of building hardware and testing each different configuration. This type of modeling
plays a key role in the design and testing of aerospace vehicles and automobiles, to name only
two.
2) Modal analysis is also used to locate structural weak points. It provides added insight
into the most effective product design for avoiding failure. This often eliminates the tedious trial and error procedures that arise
from trying to apply inappropriate static analysis techniques to dynamic problems.
3) Modal analysis provides information that is essential in eliminating unwanted noise or
vibration. By understanding how a structure deforms at each of its resonant frequencies, judgments can be made as to what the
source of the disturbance is, what its propagation path is, and how it is radiated into the environment.
In recent years, the advent
of high performance, low cost minicomputers, and computing techniques
such as the fast Fourier transform have given birth to powerful
new "instruments" known as digital Fourier analyzers
(see Figure 1). The ability of these machines to quickly and accurately
provide the frequency spectrum of a timedomain signal has
opened a new era in structural dynamics testing. It is now relatively
simple to obtain fast, accurate, and complete measurements of
the dynamic behavior of mechanical structures, via transfer function
measurements and modal analysis.
Techniques have been developed
which now allow the modes of vibration of an elastic structure
to be identified from measured transfer function data,l2. Once
a set of transfer (frequency response) functions relating points
of interest on the structure have been measured and stored, they
may be operated on to obtain the modal parameters; i.e., the natural
frequency, damping factor, and characteristic mode shape for the
predominant modes of vibration of the structure. Most importantly,
the modal responses of many modes can be measured simultaneously
and complex mode shapes can be directly identified, permitting
one to avoid attempting to isolate the response of one mode at
a fume, i.e., the so called ''normal mode'' testing concept.
The purpose of this article
is to address the problem of making effective structural transfer
function measurements for modal analysis. First, the concept of
a transfer function will be explored. Simple examples of one and
two degree of freedom models will be used to explain the representation
of a mode in the Laplace domain. This representation is the key
to understanding the basis for extracting modal parameters from
measured data. Next, the digital computation of the transfer function
will be shown. In Part 11, the advantages and disadvantages of
various excitation types and a comparison of results will illustrate
the importance of choosing the proper type of excitation. In addition,
the solution for the problem of inadequate frequency resolution,
nonlinearities and distortion will be presented.
The Structural Dynamics Model
The use of digital Fourier
analyzers for identifying the modal properties of elastic structures
is based on accurately measuring structural transfer (frequency
response) functions. This measured data contains all of the information
necessary for obtaining the modal (Laplace) parameters which completely
define the structures' modes of vibration. Simple one and two
degree of freedom lumped models are effective tools for introducing
the concepts of a transfer function, the eplane representation
of a mode, and the corresponding modal parameters.
The idealized single degree
of freedom model of a simple vibrating system is shown in Figure
2. It consists of a spring, a damper, and a single mass which
is constrained to move along one axis only. If the system behaves
linearly and the mass is subjected to any arbitrary time varying
force, a corresponding time varying motion, which can be described
by a linear second order ordinary differential equation, will
result. As this motion takes place, forces are generated by the
spring and damper as shown in Figure 2.
The equation of motion of
the mass m is found by writing Newton's second law for the mass
(AFAR = ma ), where ma is a real inertial force,
where x(t)
and x(t)
denote the first and
second time derivatives of the displacement x(t).
Rewriting equation
(1) results in the more familiar form:
and m, c, and k are the mass,
damping constant, and spring constant, respectively. Equation
(2) merely balances the inertia force 
, the damping force 
, and the spring force (kx)
, against the externally applied
force,
.
The multiple degree of freedom
case follows the same general procedure. Again, applying Newton's
second law, one may write the equations of motion as:
and
It is often more convenient to write equations (3) and (4) in matrix form:
or equivalently, for the general ndegree of freedom system,
and the previously defined force, displacement, velocity, and acceleration terms are now
bdimensional vectors.
The mass, stiffness, and damping
matrices contain all of the necessary mass, stiffness,
and damping coefficients such that the equations of mohon
yield the correct time response when arbitrary input forces are
applied.
The timedomain behavior
of a complex dynamic system represented by equation (6) is very
useful information. However, in a great many cases, frequency
domain information turns out to be even more valuable. For example,
natural frequency is an important characteristic of a mechanical
system, and this can be more clearly identified by a frequency
domain representation of the data. The choice of domain is clearly
a function of what information is desired.
One of the most important
concepts used in digital signet processing is the ability to transform
data between the time and frequency domains via the Fast Fourier
Transform (FFT) and the Inverse FFT. The relationships between
the time, frequency, and Laplace domains are well defined and
greatly facilitate the process of implementing modal analysis
on a digital Fourier analyzer. Remember that the Fourier and Laplace
transforms are the mathematical tools that allow data to be transformed
from one independent variable to another (time, frequency or the
Laplace svariable). The discrete Fourier transform is a
mathematical tool which is easily implemented in a digital processor
for transforming hmedomain data to its equivalent frequency
domain form, and vice versa. It is important to note that no information
about a signal is either gained or lost as it is transformed from
one domain to another.
The transfer (or characteristic)
function is a good example of the versatility of presenting the
same information in
three different domains. In
the time domain, it is the unit impulse response, in the frequency
domain the frequency response function and in the Laplace or sdomain,
it is the transfer function. Most importantly, all are transforms
of each other.
Because we are concerned with
the identification of modal parameters from transfer function
data, it is convenient to return to the single degree of freedom
system and write equation (2) in its equivalent transfer function
form.
The Laplace Transform.Recall that a function
of time may be transformed into a function of the complex variable
s by:
The Laplace transform of the
equation of motion of a single degree of freedom system, as given
in equation (2), is
This transformed equation can be rewritten by combining the initial conditions with the
forcing function, to form a new F(s):
It should now be clear that we have transformed the original ordinary
differential equation into an algebraic equation where
s is a complex variable known as the Laplace operator. It is also
said that the problem is transformed from the time (real) domain
into the s (complex) domain, referring to the fact that
time is always a real variable, whereas the equivalent
information in the sdomain is described by complex
functions. One reason for the transformation is that the mathematics
are much easier in the sdomain. In addihon, it is generally
easier to visualize the parameters and behavior of damped linear
systems in the sdomain.
Solving for X(s) from equation
(9), we find
The denominator polynomial
is called the characteristic equation, since the roots of this
equation determine the character of the hme response. The roots
of this characteristic equation are also called the poles
or singularities
of the system.
The roots of the numerator polynomial are called the zeros
of the system.
Poles and zeros are critical frequencies. At the poles the function
x(s) becomes infinite; while at the zeros, the function becomes
zero. A transfer function of a dynamic system is defined as the
ratio of the output of the system to the input in the sdomain.
It is, by definition, a function of the complex variable s. If
a system has m inputs and n
resultant outputs,
then the system has m x n
transfer functions.
The transfer function which relates the displacement to the force
is referred to as the compliance transfer function and is expressed
mathematically as,
From equations (10) and (11), the compliance transfer function is,
Note that since s is complex, the transfer function has a real
and an imaginary part. The
Fourier transform is obtained by merely substituting jw for a.
This special case of the transfer function is called the
frequency response function In
other words, the Fourier transform is merely the Laplace transform
evaluated along the jw or frequency axis, of the complex Laplace
plane.
The analytical form of the
frequency response function is therefore found by letting s =jw
By making the following substitutions
in equation (13),
Cc = critical damping coefficient
we can write the classical form of the frequency response function so,
However, for our purposes,
we will continue to work in the sdomain. The above generalized
transfer function, equation (12), was developed in terms of compliance.
From an experimental viewpoint, other very useful forms of the
transfer function are often used and, in general, contain the
same information. Table I summarizes these different forms.
The sPlane. Since
s is a complex variable, we can represent all complex values of
s by points in a plane. Such a plane is referred to as the splane.
Any complex value of s may be located by plotting its real component
on one axis and its imaginary component on the other. Now,
the magnitude of any function, such as the compliance transfer
function, H(s),
can be plotted
as a surface above the plane of Figure 4. This requires a threedimensional
figure which can be difficult to sketch, but greatly facilitates
the understanding of the transfer function. By definition, s =
a + jw where a is the damping
coefficient and
w is the angular
frequency.
The inertance transfer function
of a simple two degree of freedom system is plotted as a function
of the s variable in Figure 5. The transfer function evaluated
along the frequency axis (s=jw) is the Fourier transform or the
system frequency response function. It is shown by the heavy line.
If we were to measure the frequency response function for this
system via experimental measurements using the Fourier transform,
we would obtain a complexvalued function of frequency. It
must be represented by its real (coincident) part and its imaginary
(quadrature) part; or
equivalently, by its magnitude and phase. These forms are shown in Figure 6.
In general, complex mechanical
systems contain many modes of vibration or "degrees of freedom."
Modern modal analysis techniques can be used to extract the modal
parameters of each mode without requiring each mode to be isolated
or excited by itself.
Modes of Vibration. The equations of motion of an n degree of freedom system can be written
as
Where, F(s)
= Laplace transform
of the applied force vector
X(s) = Laplace transform of
the resulting output vector
B(s) = Ms2 + Cs + K
s = Laplace operator
B(s) is referred to as the system matrix. The transfer matrix, H(s) is defined
as the inverse of the system matrix, hence it satisfies the equation.
X(s) = H(s) F(s) ( 16)
Each element of the transfer matrix is a transfer function.
From the general form of the
transfer function described in equation (16), H(s)
can always be written in partial fraction form as:
where n = number of degrees of freedom
pk = kth
root of the equation obtained by setting the determinant of the
matrix B(s) equal to zero
ak = residue matrix for the
km root.
As mentioned earlier, the
roots pk are
referred to as poles of the transfer function. These poles are
complex numbers and always occur in complex conjugate pairs, except
when the system is critically or supercritically damped. In the
latter cases, the poles are realvalued and lie along the
real (or damping) axis in the splane.
Each complex conjugate pair
of poles corresponds to a mode of vibration of the structure.
They are complex numbers written as
Where * denotes the conjugate,
a. is the modal damping coefficient, and `uk is the natural
frequency. These parameters are shown on the eplane
in Figure 8. An alternate set of coordinates for defining the
pole locations are the resonantirequency, given by
and the damping factor, or percent of critical damping, given by:
The transfer matrix completely
defines the dynamics of the system. In addition to the poles of
the system (which define the natural frequency and damping), the
residues from any row or column of H(s)
define the system
mode shapes for the various natural frequencies. In general,
a pole location, Pit, will be the same for all transfer functions
in the system because a mode of vibration is a global property
of an elastic structure. The values of the residues, however,
depend on the particular transfer function being measured. The values
of the residues determine the amplitude of the resonance in each
transfer function and, hence, the mode shape for the particular
resonance. From complex variable theory, we know that if we can
measure the frequency response function (via the Fourier transform)
then we know the exact form of the system (its transfer function)
in the splane, and hence we can find the four important
properties of any mode. Namely, its natural frequency, damping,
and magnitude and phase of its residue or amplitude.
While this is a somewhat trivial
task for a single degree of freedom system, it becomes increasingly
difficult for complex systems with many closely coupled modes.
However, considerable effort has been spent in recent years to
develop sophisticated algorithms for curvefitting to experimentally
measured frequency response functions.'2 This allows the modal
properties of each measured mode to be extracted in the presence
of other modes.
From a testing standpoint,
these new techniques offer important advantages. Writing equation
(16) in matrix form gives:
If only one mode is associated
with each pole, then it can be shown that the modal parameters
can be identified from any row
or column of the transfer function matrix [H],
except those corresponding
to components known as node points. In other words, it is impossible
to excite a mode by forcing it at one of its node points (a point
where no response is present). Therefore, only one
row or column
need be measured.
To measure one column on
the transfer matrix, an exciter would be attached to the structure
(point #1 to measure column #1; point #2 to measure column #2)
and responses would be measured at points #1 and
#2. Then the transfer
function would be formed by computing,
To measure a row of the transfer
matrix, the structure would be excited at point #1 and the response
measured at point #1. Next, the structure would be excited at
point #2 and the response again measured at point #1. This latter
case corresponds to having a stationary response transducer at
point #1, and using an instrumented hammer for applying impulsive
forcing functions. Both of these methods are referred to as single
point excitation techniques.
Complex Mode Shapes. Before
leaving the structural dynamic model, it is important to introduce
the idea of a complex mode shape. Without placing restrictions
on damping beyond the fact that the damping matrix be symmetric
and real valued, modal vectors can in general be complex valued.
When the mode vectors are real valued, they are the equivalent
of the mode shape. In
the case of complex modal vectors, the interpretation is slightly
different.
Recall that the transfer matrix
for a single mode can be written as:
where
ak = (n * n)
complex residue
matrix.
pk = pole location of mode k.
A single component of H(s)
is thus written
as
where
rk / 2j = complex residue of mode k.
Now, the inverse Laplace transform of the transfer function of equation (24) is the impulse
response of mode k; that
is, if only mode k was excited by a unit impulse, its time domain
response would be
where
A phase shift in the impulse
response is introduced by the phase angle Ok
of the complex
residue. For Ok=°, the
mode is said to be "normal" or real valued. It is this
phase delay in the impulse response that is represented by the
complex mode shape. Experimentally, a real or normal mode is characterized
by the fact that all points on the structure reach their maximum
or minimum deflection at the same time. In other words, all points
are either in phase or 180° out of phase. With a complex
mode, phases other than 0° and 180~ are possible. Thus, nodal
lines will be stationary for normal modes and nonstationary! or
"traveling" for complex modes. The impulse response
for a single degree of freedom system and for the two degree of
freedom system represented in Figure 5 are shown in Figure 9.
The digital Fourier Analyzer has proven to be an ideal for measuring
structural frequency response functions quickl! and accurately.
Since it provides a broadband frequency spectrum very quickly
(e.g., ~ 100 ms for 512 spectral lines when implemented in microcode),
it can be used for obtaining broadband response spectrums from
a structure which is excited by a broadband input signal. Furthermore,
if the input and response time signals are measured simultaneously,
Fourier transformed, and the transform of the response is divided
by the transform of the input, a transfer function between the
input and response points on the structure is measured. Because
the Fourier Analyzer contains a digital processor, it possesses
a high degree of flexibility in being able to postprocess
measured data in many ways.
It has been shown1,2 that the
modes of vibration of an elastic structure can be identified from
transfer function measurements by the application of digital parameter
identification techniques. HewlettPackard has implemented
these technicltles on the HP 5451B Fourier Analyzer. The system
uses a single point excitation technique. This approach, when
coupled with a broadband excitation allows all modes in the bandwidth
of the input energy to be excited simultaneously The modal frequencies,
damping coefficients, and residues (eigenvectors) are then extracted
from the measured broadband transfer functions via an analytical
curve fitting algorithm. This method thus permits an accurate
definition of modal parameters without exciting each mode individually.
Part II of this article will address the problem of making
transfer function measurements
The data shown in
Figure 10 was obtained
by using the HewlettPackard HP 5451B Fourier Analyzer to
measure the required set of frequency response functions from
a simple rectangular plate and identify the predominant modes
of vibration. Figure 10A shows a typical frequency response function
obtained front using an impulse testing technique on a flat aluminum
plate. Input force was measured with a load cell and the output
response was measured w ith an accelerometer. After 55 such functions
were measured and stored, the modal parameters vvere identifiecl
via a curve fitting algorithm. In addition, the Fourier Analyzer
provided an animated isometric display of each mode. the results
of which are shown in Figures 10B 10F.
The Transfer and Coherence Functions
The measurement of structural
transfer functions using digital Fourier analyzers has many important
advantages for the testing laboratory. However, it is imperative
that one have a firm understanding of the measurement process
in order to make effective measurements. For instance, digital
techniques require that all measurements be discrete and of finite
duration. Thus, in order to implement the Fourier transform digitally,
it must be changed to a finite form known as the Discrete Fourier
Transform (DFT). This means that all continuous time waveforms
which must be transformed must be sampled (measured) at discrete
intervals of time, uniformly separated by an interval At. It also
means that only a finite number of samples N can he taken and
stored. The record length T is the n
The effect of implementing
the DFT in a digital memory is that it no longer contains magnitude
and phase information at all frequencies as would he the case
for the continuous Fourier transform. Rather, it describes the
spectrum of the waveform at discrete frequencies and with finite
resolution
up to some maximum frequency, Fmax, which according to
Shannon's sampling theorem,
obeys
As a direct consequence of
equation (27), we can write the physical law which defines the
maximum frequency resolution obtainable for a sampled record of
length, T.
When dealing with real valuedtime
functions, there will be N points in the record. However, to completely
describe a given frequency, two values are required; the magnitude
and phase or, equivalently, the real part and the imaginary part.
Consequently, N points in the time domain can yield N/2 complex
quanhties in the frequency domain. With these important relationships
in mind, we can return to the problem of measuring transfer functions.
The general case for a system
transfer function measurement is shown below
The linear Fourier spectrum
is a complex valued function that
results from the Fourier transform of a time waveform. Thus, Sx
and Sx have a real (in phase or coincident) and imaginary (quadrature)
parts.
In general, the result of
a linear system on any time domain input signal, x(t), may be
determined from the convolution of the system impulse response,
h(t), with the input signal, x(t), to give the output, y(t).
This operation may be difficult
to visualize. However, a very simple relationship can be obtained
by applying the Fourier transform to the convolution integral.
The output spectrum, Sy, is the product of the input spectrum,
Sx, and the system transfer function, H(f).
In other words, the transfer
function of the system is defined as:
The simplest implementation
of a measurement scheme based on this technique is the use of
a sine wave for x(f). However, in many cases, this signal has
disadvantages compared to other more general types of signals.
The most general method is to measure the input and output time
waveforms in whatever form they may be, and to calculate H
using Sx,
Sx, and the Fourier
transform.
For the general measurement
case, the input x(t) is not sinusoidal and will often be chosen
to be random noise, especially since it has several advantages
when used as a stimulus for measuring structural transfer functions.
However, it is not generally useful to measure the linear spectrum
of this type of signal because it cannot be smoothed by averaging;
therefore we typically resort to the power spectrum.
The power spectrum of the
system input is defined and computed as:
where
Sx* = Complex conjugate of
Sx
where
where
Sy* = Complex conjugate of
Sy.
The cross power spectrum between
the input and the output is denoted by 
and defined as,
Returning to equation (31),
we can multiply the numerator and denominator by Sr*
This shows that the
transfer funccan be expressed as the ratio of the cross power
spectrum to the input auto power spectrum.
There are three important
reasons for defining the system transfer function in this way.
First, this technique measures magnitude and phase since the cross
power spectrum contains phase information. Second, this formulation
is not limited to sinusoids, but may in fact be used for any arbitrary
waveform that is Fourier transformable (as most physically realizable
time functions are). Finally, averaging can be applied to the
measurement. This alone is an important consideration because
of the large variance in the transfer function estimate when only
one measurement is used. So, in general,
where
denotes the ensemble
average of the cross power spectrum and 
or
represents the ensemble
average of the input auto power spectrum.
As an added note, the impulse
response h(t) of a linear system is merely the inverse transform
of the system transfer function,
Reducing Measurement Noise
The importance of averaging
becomes much more evident if the transfer function model shown
above is expanded to depict the "realworld" measurement
situation. One of the major characteristics of any modal testing
system is that extraneous noise from a variety of sources is always
measured along with the desired excitation and response signals.
This case for transfer function measurements is shown below.
Since we are interested in
identifying modal parameters from measured transfer functions,
the variance on the parameter estimates is reduced in proportion
to the amount of noise reduction in the measurements. The digital
Fourier analyzer has two inherent advantages over other types
of analyzers in reduction of measurement noise; namely, ensemble
averaging, and a second technique commonly referred to as post
data smoothing which may be applied after the measurements are
made.
Without repeating the mathematics
for the general model of a transfer function measurement in the
presence of noise, it is easy to show that the transfer function
is more accurately written as:
where the frequency dependence
notation has been dropped and,
This form assumes that the noise has a zero mean value and
is incoherent with the measured input signal. Now, as the
number of ensemble averages becomes larger, the noise
term 
becomes smaller and the ratio 
/ 
, more accurately estimates
the true transfer function. Figure 11 shows the effect of
averaging on a typical transfer function
measurement.
The Coherence Function
To determine the quality of
the transfer function, it is not sufficient to know only the relationship
between input and output. The question is whether the system output
is tocaused by the system input. Noise and/or nonlinear
effects can cause large outputs at various frequencies, thus introducing
errors in estimating the transfer function. The influence of noise
and/or nonlinearities, and thus the degree of noise contamination
in the transfer function is measured by calculating the coherence
function, denoted by 
where
The coherence function is easily calculated on a digital Fourier analyzer when transfer
functions are being measured. It is calculated as:
If the coherence is equal
to 1 at any specific frequency, the system is said to have perfect
causality at that frequency. In other words, the measured response
power is caused totally by the measured input power (or by sources
which are coherent with the measured input power). A coherence
value less than 1 at a given frequency indicates that the
measured response power is greater than that due to the measured
input because some extraneous noise is also contributing to the
output power.
When the coherence is zero, the output is caused totally by sources
other than the measured input. In general terms, the coherence
is a measure of the degree of noise contamination a measurement.
Thus, with more averaging, the estimate of coherence contains
less variance, therefore giving a better estimate of the noise
energy in a measured signal. This is illustrated in Figure 12.
Since the coherence function indicates the degree of causality in a transfer function it has
two very important uses:
1) It can be used qualitatively to determine how much averaging is required to reduce
measurement noise.
2) It can serve as a monitor on the quality of the transfer function measurements.
The transfer functions associated
with most mechanical systems are so complex in nature that it
is virtually impossible to judge their validity solely by inspection.
In one case familiar to the author, a spacecraft was being excited
with random noise in order to obtain structural transfer functions
for modal parameter identification The transfer and coherence
functions were monitored for each measurement. Then, between two
measurements the coherence function became noticeably different
from unity. After rechecking all instrumentation, it was
discovered that a random vibration test being conducted in a separate
part of the same building was providing incoherent excitation
via structural (building) coupling, even through a seismic isolation mass.
This extraneous source was increasing the variance on the measurement
but would probably not have been discovered without use of the
coherence function.
Summary
In Part 1, we have introduced
the structural dynamic model for elastic structures and the concept
of a mode of vibration in the Laplace domain. This means of representing
modes of vibration is very useful because we are interested in
identifying the modal parameters from measured frequency response
functions. Lastly, the procedure for calculating transfer and
coherence functions in a digital Fourier analyzer were discussed.
In Part II, we will discuss various techniques for accurately measuring structural transfer
functions. Because modal parameter identification algorithms work
on actual measured data, we are interested in making the best
measurements possible, thus increasing the accuracy of our parameter
estimates. Techniques for exciting structures with various forms
of excitation will be discussed. Also, we will discuss methods
for arbitrarily increasing the available frequency resolution
via band selectable Fourier analysis -the socalled zoom
transform.
References
I. Richardson, M, and Potter,
R., "Identification of the Modal
Properties of an Elastic Structure
from Measured Transfer
Function Data," 20th
I S.A., Albuquerque, N.M., May
1974.
2. Potter, R. and Richardson,
M, "Mass, Stiffness and Damping
Matrices from Measured Modal
Parameters " I.S.A. Conference
and Exhibit, New York City,
October
1974
3. Roth, P. R., "Effective
Measurements Using Digital Signal
Analysis," I.E.E.E.
Spectrum, pp 6270,
April 1971.
4. Fourier Analyzer Training
Manual, Application Note 1400
HewlettPackard
Company.
5. Potter, R. "A General
Theory of Modal Analysis for Linear
Systems," HewlettPackard
Company, 1975 (to be published).
6. Richardson, M., "Modal
Analysis using Digital Test Systems,"
Seminar on Understanding Digital
Control and Analysis in Vibration Test Systems, Shock
and Vibration Information Center publication, May 1975.
