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=ー, 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ー out of phase. With a complex mode, phases other
than 0ー 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.