Journal of Vibration and Acoustics 125 (1) 129-131 (2003)
PROPER ORTHOGONAL MODES OF A BEAM SENSED WITH STRAIN GAGES
M. S. Riaz
Department of Mechanical Engineering
Michigan State University
2555 Engineering Building
East Lansing, MI 48824
Telephone: 517-355-2980, 517-353-1750 (fax)
riazmuha@egr.msu.edu
1.0 INTRODUCTION
Proper orthogonal decomposition (POD) is a useful
experimental tool in dynamical systems, for example in
dimensionality studies [1,2] and reduced order modeling [3-6].
Application of POD in structural vibrations often involves
sensed displacements, x1, x2, …, xM, at M locations on the
structure. These displacements are sampled N times at a fixed
sampling rate to form displacement arrays xj = [xj(t1),…,xj(tN)]T,
j = 1,…, M. The means are often subtracted. An NxM
ensemble matrix X = [x1,…, xM] is then built. The MxM
correlation matrix is R = XTX/N. Since R is real and
symmetric, its eigenvectors form an orthonormal basis. The
eigenvectors are the proper orthogonal modes (POMs) and the
eigenvalues are the proper orthogonal values (POVs).
The POMs in certain nonlinear structures have resembled
the normal modes of the linearized system [1,7,8]. The POMs
may converge to linear normal modes in multi-modal free
responses of symmetric lightly damped lumped-mass linear
systems, but only if the mass matrix has the form mI, which can
be achieved by a coordinate transformation if the mass
distribution is known [9,10] and equivalently [11]. Analysis
and simulations show that this relationship is approximately true
in distributed parameter systems if the mass and the sensor
locations are uniformly distributed [12].
In this note, this application of POD for modal analysis is
tested on a cantilevered beam sensed with strain gages. The
objective is to verify the applicability of POD for modal
analysis in a simple experiment, and to bring forth issues that
arise from strain measurements. This parallels experimental
work with accelerometers [13].
2.0 SYSTEM DESCRIPTION
An experiment was made to emulate a cantilevered EulerBernoulli beam. The 0.394× 0.012× 0.00079 m3 beam of mild
steel had one end fixed in a steel clamp. The beam had Young’s
modulus E = 128 ×109 N/m2 and a density  = 7488 kg/m3.
Theoretical modal frequencies for this beam were
computed based on the known solution [14] and were compared
B. F. Feeny
Department of Mechanical Engineering
Michigan State University
2555 Engineering Building
East Lansing, MI 48824
Telephone: 517-353-9451, 517-353-1750 (fax)
feeny@me.msu.edu
to experimental modal frequencies obtained from the fast
Fourier transform of an impulse response. The six lowest
theoretical frequencies were 4.52, 28.36, 79.38, 155.6, 257.2,
and 384.2 Hz, and the experimental modal frequencies were
4.5, 27.2, 75.5, 147, 243, and 365 Hz.
The modal damping ratios were estimated as 0.0161, 0.016,
0.009, 0.0047, 0.0031, and 0.0022. These are rough estimates
due to coarse resolution on the FFT analyzer, and the
assumption of an ideal impulse input. Conventional modal
analysis [14] would involve measuring frequency response
functions between various input and output locations. Our beam
was so floppy that we were unable to generate a meaningful
signal on the impulse hammer. (This points to an advantage of
the POD method, for which sensed inputs are not generally
needed.)
The displacements at various points were measured with
the help of twelve strain gages in six half wheat-stone bridges
arranged for bending. The strain gages were mounted on the
beam at locations from the clamped end specified as x1=
0.00527m,
x2=0.0543m,
x3=0.1033m,
x4=0.1523m,
x5=0.2013m, x6=0.2503m. The gages were biased towards the
clamped end to improve strain sensitivity. The locations were
“optimized” by performing numerical simulations and iterating
the gage locations [15].
The displacements were estimated from the strain
measurements by using assumed-mode basis functions,
according to the following development.
By approximating
y( x, t )   i ( x)ui (t )
Ms
i 1
where Ms is the number of strain gage locations, and the i(x)
form a basis satisfying the geometric conditions. Then we can
write y = u, where yk = y(xk, t) are the elements of the Mdvector y, kj = j (xk) are the elements of Md Ms matrix , xk
are the desired displacement locations, and uj(t) are the
elements of Ms modal displacement vector u. Md is the number
of desired displacement measurements on the beam.
The axial strain ε(x) on the surface of a symmetric beam in
bending is related to the transverse displacement y(x) by
1
 xx ( x, t )  c
or
locations. The pseudo sensors are redundant sensors derived by
interpolating the measurements through the usage of the basis
functions, which in this case were the linear normal modes of
the cantilevered beam model. The motive for using the pseudo
sensors was to improve the resolution associated with the
rectangular rule integration that effectively underlies the
relationship between orthogonality of linear normal modal
functions, orthogonality between discrete POMs and the
uniformly discretized modal vector [12]. However, as we used
only first six strain measurements and six LNMs, the system
measurement only contains six independent displacements. The
pseudo sensors apparently performed an equivalent task to
Gramm-Schmidt orthonomalization used in converting discrete
POMs to continuous orthogonal functions [15]. Either approach
depends on the interpolating functions and how well they depict
the physics of the problem.
Figure 2 shows an example when 21 pseudo sensors were
used. The use of pseudo sensors provides some improvement,
and also helps in visualizing the highly undulatory modes.
 y
 cu
x 2
2
 c  u
where c is half the width of the beam, i = (xi) are the elements
of Ms strain vector , and ij = 2j(x)/x2|x=xi are the elements
of MsMs matrix .
Strains were measured at Ms locations on the beam, and
displacements were desired at Md locations. As such, the
MsMs matrix  is constructed based on the Ms actual straingage positions and the Ms basis functions, and the MdM matrix
 is built based on the Ms basis functions evaluated at Md
desired displacement locations.
Assuming  is invertible (as it is made up of evaluations of
linearly independent basis functions), we obtain y in terms of ,
such that
y
1

c
3.3 On the Choice of Basis Functions
The LNMs are typically unknown, and not available as a
basis for a strain-to-modal-coordinate conversion. As such, we
made example computations using various bases. Figure 3
shows the results for basis functions of the form i(x)=1cos(ix), where i=(2i+1)/2L, i=0,1,…,5.
These basis
functions are not orthogonal, but are linearly independent, and
meet the geometric boundary conditions and the zero-strain
condition at x=L. The POMs are not as good as the those with
a basis of LNMs, but are distinct from the basis functions.
Not shown are results for basis functions of the form i(x)
= sin(ix), where i = (4i+1)/4L, i=0,1,…,5. This basis only
satisfies the displacement boundary condition at x=0. It
imposes a false constraint of zero slope at x=L. The resulting
POMs nearly (not exactly) accommodated the other geometric
boundary condition, as responses obey the boundary conditions
[16]. But the POMs exhibited the erroneous zero slope at x=L.
Otherwise, the POMs were qualitatively representative of the
LNMs.
We also used orthogonal polynomials of degrees two
through seven as basis functions [17]. The results are not
shown, but the POMs were distorted from the true LNMs
toward the free end of the beam. This is probably because the
high-degree polynomials, evaluated at several values of x, are
not well conditioned for the computation.
3.0 RESULTS
The beam was excited with an impulse, and the resulting
multi-modal free vibration was monitored. The six strain
histories were simultaneously sampled and recorded in 0.2
second windows which were then pasted together in time. The
data was taken for 10 seconds with a sampling rate of 800
samples per second in all the tests, which was enough time to
capture the modal characteristics of the cantilever beam.
3.1 Theoretical Modes as a Basis
The first set of tests involved basis functions i(x)
consisting of the linear normal modes of the cantilevered beam
model. This may seem pointless, as the linear normal modes
are ultimately to be estimated by the POD. But the linear
normal modes provide the best basis for the strain-todisplacement conversion, resulting in the best displacement
estimates, and thus yield an illustrative example for the method.
Figure 1 shows the POMs, with Md = 6, compared with LNMs.
Continuous lines represent the LNMs and circles show the
POMs. The POVs corresponding to this plot were 2.88,
0.00374, 1.19e-5, 1.19e-6, 2.22e-8, and 1.53e-9.
Tests were done at various sampling rates and impulse
locations, and results were consistent for the lower three modes.
In all the cases we saw the maximum energy with the first
mode. When the input was applied near a nodal point, the
corresponding mode was excited less.
4.0 CONCLUSION
POD was applied to a beam sensed with strain gages.
Basis functions were used to convert strains to displacements.
Results were best when the basis functions were trivially chosen
as the LNMs. When the basis functions differed from the
LNMs, the POMs were still a reasonable approximation of the
3.2 Use of Redundant Sensors
We generated a large number of “pseudo sensors” on the
beam by evaluating the continuous functions at Md > 6
2
Dynamics, The Richard Rand 50th Anniversary Volume, pp.
159-143, Edited by A. Guran.
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angewandte Mathematik und Mechanik 77 (S1) S83-S84.
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Interpretation of Proper Orthogonal Modes in Vibrations,”
Journal of Sound and Vibration 211(4), 607-616.
[10] Kerschen, G. and Golinval, J. C., 2001, “Physical
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Value Decomposition,” Journal of Sound and Vibration 249(5)
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Hedgecock C. E., 1993, “The Time Correlation Method for
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Technical Conference, Sacramento, on CD-ROM.
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Orthogonal Decomposition for the Modal Analysis of
Homogeneous Structures,” Journal of Vibration and Control 8
19-40.
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Englewood Cliffs. New Jersey.
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Nonlinear Dynamics 22(4) 317-333.
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Orthogonal Decomposition (POD) of a Class of Vibroimpact
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[17] Riaz, M. S., and Feeny, B.F., 1999, “Proper Orthogonal
Decomposition of an Experimental Cantilever Beam,
Proceedings of the ASME Design Engineering Technical
Conferences, Las Vegas, September 10-13, on CD-ROM.
LNMs, and were distinct from the basis itself, even though the
basis was used to interpolate redundant sensors. The choice of
the basis functions is thought to influence the POD by its
influence on the quality of the displacement estimations.
Acknowledgments
This research was supported by the National Science
Foundation (CMS-9624347 and CMS-0099603).
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Figure 2: Proper orthogonal modes with the impulse applied
between 4th and 5th strain gage and using 21 pseudo sensors.
Circles: POMs. Lines: LNMs.
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Figure 1: Proper orthogonal modes with the impulse applied
between 4th and 5th strain gage from the clamp. Circles depict
the POMs, and lines show the theoretical LNMs.
4
Figure 3: Proper orthogonal modes using a basis made of
cosine functions. Circles: POMs. Lines: LNMs.
5