The Enhanced Frequency Response Function (eFRF):
Scaling and Other Issues
Allyn W. Phillips, PhD, Research Assistant Professor
Randall J. Allemang, PhD, Professor
University of Cincinnati
Structural Dynamics Research Laboratory (SDRL)
Cincinnati, OH 45221-0072 USA
ABSTRACT
The Enhanced Frequency Response Function has become an important element in utilizing the Complex
Mode Indicator Function (CMIF) as a modal parameter estimation method. Historically, the enhanced
frequency response function has been used to determine only the damping and the damped natural frequency.
In order to be able to utilize the enhanced frequency response function as a complete modal parameter
estimation method, the issue of scaling (modal mass or modal A) must be properly addressed. This paper
reviews and presents the development of the eFRF along with a solution to the scaling issues. A comparison
of properly scaled and improperly scaled results are presented for two example structural systems.
Nomenclature
N = Number of modes.
N i = Number of inputs.
N o = Number of outputs.
N e = Number of effective modes.
[H] = Frequency response function matrix.
[ ] H = Hermitian (conjugate transpose) of matrix.
⎡ Λ ⎦ = Eigenvalue matrix (diagonal).
[V ] = Eigenvector matrix matrix.
⎡ Σ ⎦ = Singular value matrix (diagonal).
[U ] = Left singular vector matrix (unitary).
[V ] = Right singular vector matrix (unitary).
{u } = Left singular vector (unitary).
{v } = Right singular vector (unitary).
Q r = Modal scaling for mode r.
ψ pr = Modal coefficient for DOF p, mode r.
{ψ }r = Modal vector for mode r.
[M ] = Reduced mass matrix.
M r = Modal mass for mode r.
1. Introduction
Modal parameter estimation has evolved and
developed extensively over the last twenty-five
years due to an increased use and availability of
numerical techniques involving least squares
estimation
and
eigenvalue/singular
value
decomposition methods [1-3]. Often times, the
development of modal parameter estimation
methods appear to be quite complicated and
difficult to understand. Much effort has gone into
simplifying the theoretical development of these
methods to focus on the common characteristics.
One common characteristic of most modal
parameter estimation methods is the two stage
development of the algorithm [4]. Traditional
time/frequency domain modal parameter estimation
algorithms estimate temporal information (modal
frequencies or poles) in the first stage and spatial
information (modal vectors and modal scaling) in
the second stage. The CMIF used in this paper is an
example of a spatial domain method based upon the
complex mode indicator function and the enhanced
frequency response function. Spatial domain
algorithms, like this one, estimate the spatial
information (modal vectors) in the first stage and
the temporal information (modal frequencies and
modal scaling) in the second stage. For this
technique to be effective, the issue of modal scaling
must be addressed. The following section presents
the theory and historical development of the
Enhanced Frequency Response Function and an
approximation based upon the use of singular value
decomposition scaled to permit estimation of
residue for the enhanced fundamental response.
{ψ }Ts [M]{ψ }s = M s
2. Enhanced Frequency
Response Functions (eFRF)
A virtual measurement, known as the enhanced
frequency response function (eFRF), is used to
identify the modal frequencies and scaling of a
single degree-of-freedom characteristic that is
associated with each peak in the CMIF [7-9]. The
eFRF is developed based upon the concept of
physical to modal coordinate transformation and is
used to manipulate frequency response functions so
as to enhance a particular mode of vibration. The
left singular vectors, associated with the peaks in
the CMIF, are used as an estimate of the modal filter
which accomplishes this.
2.1 eFRF - Theoretical Definition
Starting with:
H pq (ω ) =
2N
Σ
r=1
Q rψ prψ qr
jω − λ r
(1)
2N
Σ ψ pr
r=1
Q rψ qr
jω − λ r
(2)
The enhanced frequency response function (eFRF)
can now be defined:
eFRF r (ω ) =
eFRF s (ω ) = M s
Q sψ qs
jω − λ s
(8)
The last equation indicates that an estimate of the
reduced mass matrix is needed to develop the
enhanced frequency response function. In many
situations, when the mass distribution is adequately
represented by the measured degrees-of-freedom
(good spatial representation), the modal vector can
be used directly. Essentially, for the eFRF to yield
only one degree-of-freedom, this vector must
represent a modal filter for that mode [9]. In reality,
the goal of the eFRF is to allow a simple modal
parameter estimation algorithm to be used to
estimate modal frequency and scaling. If other
modes are still observable in the eFRF, these modes
can be handled with residuals in the modal
parameter estimation.
2.3 eFRF - CMIF/SVD Development
Redefining :
H pq (ω ) =
{ψ }Ts [M]{ψ }r = 0 (7)
Q rψ qr
jω − λ r
(3)
The eFRF r (ω ) is defined as the Enhanced
Frequency Response Function for mode r. The
eFRF has only to do with the input location and is
constant for a given column of the FRF matrix.
2.2 eFRF - Historical Development
The eFRF can be formulated from measured
frequency response function data in the following
manner:
In the technique utilized in the following examples,
the left singular vectors, associated with the
singular values at the peaks in the CMIF, are used
as the modal filter.
The frequency response function matrix is assumed
to have been corrected for transducer orientation.
Although this is not required for the computation of
damped natural frequency and damping, this will
simplify the correction for modal scaling. Starting
with a full FRF matrix the development of the eFRF
using singular vectors is:
eFRF r = {ur }T [H] {ur }
eFRF r = {ur }T
Σk
(9)
{ψ k } {ψ k }T
{u } (10)
M A k ( jω − λ k ) r
Assuming the spatial resolution
describes the mass distribution,
adequately
{ur }T {ψ k } << {ur }T {ψ r } ;k ≠ r
(11)
Then the
Q rψ qr
{H(ω )} = Σ {ψ }r
jω − λ r
r=1
(4)
eFRF s (ω ) = {ψ }Ts [M]{H(ω )}
(5)
2N
2N
eFRF s (ω ) = {ψ }Ts [M] Σ {ψ }r
r=1
From the orthogonality conditions:
Q rψ qr
jω − λ r
(6)
eFRF r ≈
{ur }T {ψ r } {ψ r }T {ur }
M Ar ( j ω − λ r )
(12)
Letting,
⎧ ⎫
(13)
⎨ur sc ⎬ = γ {ur } ; scaled to be dominantly real.
⎩ ⎭
T
T
⎧ ⎫ ⎧ ⎫⎧ ⎫
⎨ur sc ⎬ ⎨ur sc ⎬ ⎨ur sc ⎬
eFRF r ≈ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭
M Ar ( j ω − λ r )
⎧ ⎫
⎨ur sc ⎬
⎩ ⎭
(14)
possible that {v} may not be a strict subset of {u},
in this case the scale factor, SF, must be estimated
from the common subset of the input/output degrees
of freedom, (ie. the driving point degrees of
freedom):
However,
T
⎧ ⎫ ⎧ ⎫
⎨ur sc ⎬ ⎨ur sc ⎬ ≈ 1
⎩ ⎭ ⎩ ⎭
(15)
⎧
⎫
SF r = {v r (drvPt)}+ ⎨ur sc (drvPt)⎬
⎩
⎭
(17)
⎧ ⎫
⎨v r sc ⎬ = SF r {v r }
⎩ ⎭
(18)
Therefore,
eFRF r ≈
M Ar
1
( jω − λ r )
(16)
Unfortunately, the DOFs spanned by the input and
output are not typically the same. In fact, the inputs
may not even be a strict subset of the outputs. In
this case, additional scaling/correction is required in
order to utilize the measured frequency response
functions.
Assuming operation only on mode r, drop the
subscript r
⎧ {v sc } ⎫
{u sc }T {u sc } = {u sc }T ⎨
⎬
⎩{v unmeasured }⎭
(19)
{u sc }T {u sc } ≈ α {u sc }T {v sc }
(20)
{u sc }T {v sc } = {v sc }T {v sc }
(21)
However,
H
H
unmeasured
Taking norms,
{v sc } = SF {v}
(22)
||{v sc }||2 = |SF|
(23)
1 = ||{u sc }T {u sc }||2 ≈ α ||{v sc }||2
(24)
Implies that,
Figure 1. Frequency Response Space
The eFRF is typically used, together with single
degree-of-freedom modal parameter estimation
methods, to estimate the frequency and damping of
the associated modal frequency. In order for the
eFRF to also be used to estimate the modal scaling
(modal mass and/or modal A), the correct scaling
(correct magnitude and phase) of the eFRF must be
accounted for. Since the right and left singular
vectors in the singular value decomposition are
unitary and scaled consistently as a set, the arbitrary
scaling of the left singular vector must be accounted
for in the eFRF computation. For this case, where
the modal vector used in the eFRF is estimated
from the left singular vector associated with a peak
in the CMIF plot, the eFRF is scaled by utilizing the
values of the left and right singular vectors,
associated with the significant singular value, at the
driving point locations as follows. Note that it is
Therefore,
α ≈
1
|SF|2
(25)
Substituting,
⎛
⎞
{u sc }T {u sc } α {v sc }T {v sc }
⎝
⎠
eFRF ≈
M A ( jω − λ )
eFRF(ω ) = α {u sc }T [H(ω )]{v sc }
(26)
(27)
2.4 eFRF - Simplified Expression
The mechanism for scaling the eFRF can be
simplified for many cases. By observing that the
right singular vector v resulting from the SVD is
SF r = {v r (drvPt)}+ {ur (drvPt)}
(28)
SF r*
{u }T [H(ω )]{v r }
eFRF(ω ) =
|SF r |2 r
(29)
⎧ ⎫ SF r*
ur
⎨ur sc ⎬ =
⎩ ⎭ |SF r |
(30)
3. Issues
There are several issues that can affect the modal
scaling results. One important issue is the
transducer orientation conventions which generally
defines positive acceleration as from the transducer
base through the top and positive force as
compression. However, even if the convention is
satisfied, sign errors can still occur if any signal
conditioning causes signal inversion. Assuming
that the modal vector estimate has been scaled to be
dominantly real, this error will typically result in an
eFRF with the phase starting at 180° and falling to
0° resulting in a Modal A that is dominantly
negative imaginary (or negative modal mass).
This sign issue is often causes problems with
impedance
heads.
Since
by
convention
compression is positive force and base through top
positive acceleration, an impedance head generally
has an implied negative driving point characteristic
as shown in Figure 2 (positive force is in the
opposite direction from positive acceleration).
the charge amplifier. However, if similar
transducers and signal conditioning (same
manufacturer and type) are used for all
measurement channels, then it is expected that only
a single sign error might occur between the
accelerometers and load cells. This error will
results in the entire FRF matrix being multiplied by
-1.
4. Circular Plate Example
The data presented in Figures 3 through 5 is
representative of the seven reference circular plate
test object. Figure 3 shows the basic CMIF plot.
Several cross eigenvalue effects are noticeable (for
example, peaks in the 3rd and 4th CMIF curves
between 1250 and 1300 Hz.) Figure 4 has mode
tracking enabled. This provides a clearer indication
of the evolution of the shape as a function of
frequency. Finally, a more sharply defined CMIF
plot using only the imaginary part of the FRF is
shown in Figure 5.
Complex Mode Indicator Function
−5
10
−6
10
Magnitude
generally dominant real, the above expression can
be simplified and rewritten in terms of the original
SVD vector as:
−7
10
−8
10
−9
10
Impedance Head
+Force
+Acceleration
Figure 2. Conventional Orientation
Further problems occur when mixing transducer
types. When using charge mode transducers, often
a positive charge signal becomes inverted through
200
400
600
800
1000
1200
Frequency, Hz
1400
Figure 3. CMIF - Untracked
1600
1800
363.326 Hz, 0.00993 zeta, [Ma= 462.258,3479.863j]
Complex Mode Indicator Function
−5
Phase (deg)
10
180
0
−4
10
−6
10
−5
−6
−7
Magnitude
Magnitude
10
10
10
−7
10
−8
10
−8
10
−9
10
−9
10
200
400
600
800
1000
1200
Frequency, Hz
1400
1600
200
400
600
800
1800
1000
1200
Frequency (Hz)
1400
1600
1800
Figure 6. eFRF - Complex Vector
Figure 4. CMIF - Tracked
Phase (deg)
362.300 Hz, 0.00898 zeta, [Ma= −92.585,3372.070j]
Complex Mode Indicator Function
−5
180
0
−4
10
10
−5
10
−6
10
Magnitude
−6
−7
Magnitude
10
10
−7
10
−8
10
−8
10
−9
10
−9
10
200
200
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Figure 7. eFRF - Complex Vector
−10
10
400
400
600
800
1000
1200
Frequency, Hz
1400
1600
1800
180
0
−4
10
−5
10
−6
Magnitude
The properly scaled enhanced FRFs using complex
vectors (showing the repeated root), Figures 6 and
7, and using quadrature vectors (showing the first
360 Hz mode), Figure 8, are shown for the 360 Hz
peak. These plots show both the eFRF and the
estimated/synthesized SDOF fit for the region.
Note that most single degree-of-freedom modal
parameter estimation methods would work well on
this data.
Phase (deg)
363.424 Hz, 0.00970 zeta, [Ma= −29.575,3509.277j]
Figure 5. Imaginary CMIF - Tracked
10
−7
10
−8
10
−9
10
200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Figure 8. eFRF - Quadrature Vector
Utilizing only the first 360 Hz mode for
presentation, the following figures demonstrate the
characteristics and results of failure to properly
Phase (deg)
363.326 Hz, 0.00993 zeta, [Ma=−462.258,−3479.863j]
180
0
−4
10
−5
10
−6
Magnitude
scale the magnitude and phase of the eFRF. Figure
9 demonstrates the error associated with failing to
correct the eFRF magnitude. Figure 10 shows the
error resulting from not correcting the eFRF
magnitude and not correcting for the phasing
difference between u and v. Figure 11 shows the
attendant error resulting from incorrect transducer
orientation. Note that although the eFRF magnitude
and phase change dramatically, the estimated modal
frequency and damping are unaffected. This is
because the estimation of frequency and damping
utilizes relative information, whereas the estimate
of Modal A utilizes absolute information.
10
−7
10
−8
10
−9
10
200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Phase (deg)
363.326 Hz, 0.00993 zeta, [Ma=1342.297,10104.759j]
180
Figure 11. eFRF - Orientation Error
0
Figures 12 and 13 show the comparison of the
original FRF to the synthesis for the properly scaled
results, complex and quadrature vector respectively.
Figure 14 shows the error resulting from missing
the magnitude scaling. Figure 15 shows the error
resulting from incorrect magnitude and phase
scaling. Finally, figures 16 and 17 show the results
from incorrect transducer orientation.
−5
10
−6
Magnitude
10
−7
10
−8
10
200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Phase (deg)
Synthesis Overlay − Input 1y; Output 1y
−9
10
180
0
−5
10
Figure 9. eFRF - Missing Scaling
−6
10
180
−7
0
Magnitude
Phase (deg)
363.326 Hz, 0.00993 zeta, [Ma=−5763.321,−1602.437j]
−5
10
−6
Magnitude
10
10
−8
10
−9
10
−10
−7
10
−8
Figure 12. Synthesis Overlay - Complex Vector
10
10
−9
10
200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Figure 10. eFRF - Missing Scaling and Phasing
200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Synthesis Overlay − Input 1y; Output 1y
Phase (deg)
Phase (deg)
Synthesis Overlay − Input 1y; Output 1y
180
0
−5
10
−6
−6
10
−7
Magnitude
Magnitude
−7
10
−8
10
−9
10
−8
10
−9
10
10
−10
200
0
−5
10
10
10
180
−10
400
600
800
1000
1200
Frequency (Hz)
1400
1600
10
1800
Figure 13. Synthesis Overlay - Quadrature Vector
200
800
1000
1200
Frequency (Hz)
1400
1600
1800
−6
180
2
0
1
Imaginary
Phase (deg)
600
Figure 16. Synthesis Overlay - Orientation Error
Synthesis Overlay − Input 1y; Output 1y
−5
10
x 10
0
−1
−6
10
−2
200
−7
Magnitude
400
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
10
−6
1.5
x 10
−8
10
1
Real
0.5
−9
10
0
−0.5
−1
−10
10
200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Figure 14. Synthesis Overlay - Missing Scaling
−1.5
200
Figure 17. Synthesis Overlay - Orientation Error
Phase (deg)
Synthesis Overlay − Input 1y; Output 1y
5. Summary - Conclusions
180
0
−5
10
−6
10
Magnitude
−7
10
−8
10
−9
10
−10
10
200
400
600
800
1000
1200
Frequency (Hz)
1400
1600
1800
Figure 15. Synthesis Overlay - Missing Scaling
and Phasing
The Enhanced Frequency Response Function
(eFRF) combined with the modal vector estimates
produced by CMIF results in an effective, efficient
modal parameter estimation technique. It has been
used successfully on many different types of
structures and on data that was not amenable to
more sophisticated techniques[10]. However, as has
been shown in this paper, strict attention must be
paid to the magnitude and phase scaling of the
eFRF in order to extract reliable estimates for
Modal A (or modal mass). Failure to properly
account for the magnitude or phase results in
estimates that, while resonably accurate as to
frequency and damping, are improperly scaled (wrt.
Modal A) and therefore unsuitable for modeling or
prediction.
6. References
[1]
Strang, G., Linear Algebra and Its
Applications, Third Edition, Harcourt
Brace Jovanovich Publishers, San Diego,
1988, 505 pp.
[2]
Lawson, C.L., Hanson, R.J., Solving Least
Squares Problems, Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey, 1974, 340
pp.
[3]
Jolliffe, I.T., Principal Component
Analysis Springer-Verlag New York, Inc.,
1986, 271 pp.
[4]
Allemang, R.J., Brown, D.L., Fladung, W.,
"Modal Parameter Estimation: A Unified
Matrix Polynomial Approach", Proceedings,
International Modal Analysis Conference,
pp. 501-514, 1994.
[5]
Allemang, R.J., Brown, D.L., "Modal
Parameter Estimation" Experimental Modal
Analysis
and
Dynamic
Component
Synthesis, USAF Technical Report, Contract
No.
F33615-83-C-3218,
AFWALTR-87-3069, Vol. 3, 130 pp., 1987.
[6]
Shih, C.Y., Tsuei, Y.G., Allemang, R.J.,
Brown, D.L., "Complex Mode Indication
Function and Its Application to Spatial
Domain Parameter Estimation", Mechanical
System and Signal Processing, Vol. 2, No. 4,
pp. 367-377, 1988.
[7]
Fladung, W.D., Phillips, A.W., Brown,
D.L., "Specialized Parameter Estimation
Algorithms for Multiple Reference Testing",
Proceedings, International Modal Analysis
Conference, pp. 1078-1087, 1997.
[8] Allemang, R. J., "Investigation of Some
Multiple Input/Output Frequency Response
Function Experimental Modal Analysis
Techniques",
Doctoral
Dissertation,
University of Cincinnati, 1980, 358 pp.
[9] Shelley, S.J., "Investigation of Discrete
Modal Filters for Structural Dynamic
Applications",
Doctoral
Dissertation,
Department of Mechanical Engineering,
University of Cincinnati, 269 pp., 1991.
[10]
Phillips, A.W., Allemang, R.J., "The
Complex Mode Indicator Function (CMIF)
as a Parameter Estimation Method",
Proceedings, International Modal Analysis
Conference, pp. 705-710, 1998.