Virtual Measurements in
Experimental Structural Analysis
Randall J. Allemang, PhD
Structural Dynamics Research Lab
University of Cincinnati
Cincinnati, Ohio, USA 45221-0072
COBEM97
UC
Virtual Measurements - Introduction
‹ What
are virtual Measurements?
‹ Virtual
measurements are any measurements that are
not directly derived from physical sensors
‹ Virtual measurements are developed from physical
sensors via weighting and/or linear combinations
(linear transformations)
‹ Virtual measurements are frequently used to reduce
large physical data sets or to enhance characteristics
within a physical data set
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Physical Measurements
‹ Physical
measurements are generated as a
function of placement of a physical sensor in a
static/dynamic environment
‹ Physical measurements are difficult to interpret
when multiple inputs/sources are generating the
physical response of the sensor
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Virtual Measurements - Concept
‹ Virtual
measurements utilize a linear
transformation of physical sensors in order to
preserve/enhance/clarify information in the
original physical measurements
‹ The linear transformation is chosen depending
upon the desired result, data reduction or data
enhancement
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Virtual Measurements - Purpose
‹ Data
Reduction - Reduce a large amount of
redundant data to a manageable set (common in
modal parameter estimation)
‹ Data Enhancement - Optimize the characteristic
in the data relative to a specific mode of the
structural system (modal filter)
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Virtual Measurements - History
‹ Virtual
measurements have been commonly
formulated to understand the underlying nature of
multiple input, multiple output (MIMO) data
acquisition/analysis situations
‹ Common examples are partial coherence, virtual
coherence, principal force analysis
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Virtual Measurements - History
‹ Principal
Force Example
LM GFF
MM ......
=
MM ...
MNGFF
11
GFF
Ni 1
...
GFF22
...
GFF33
...
...
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...
GFF1 Ni
...
...
...
...
... GFFNi N i
OP
PP
PP
PQ
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Virtual Measurements - History
‹ Principal
‹ Plot
Force Example
of eigenvalues
GFF = V Λ V
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Virtual Measurements - History
‹ Auto
Power Spectrum of Forces
H-Frame
Example
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Virtual Measurements - History
‹ Principal
Force Analysis
H-Frame
Example
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Virtual Measurements
‹ Linear
Transformation:
‹ General
Case - Reduce information in data set,
commonly based upon eigenvalue decomposition (ED)
or singular value decomposition (SVD).
‹ Special Case - Preserve information in data set relative
to one or more modes of vibration, commonly based
upon analytical or experimental modal vectors.
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Virtual Measurements
‹ General
Linear Transformations
‹ Frequency
Response Function Application
H' = T H
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Virtual Measurements
‹ General
Linear Transformations - FRF
‹ Eigenvalue
H (ω )
Decomposition - Output DOF Space
N o × Ni N s
T
Ne × No
H (ω )
H
Ni N s × N o
= V
No × No
l ql q l q n s
= v1 v2 ... vk ... v N e
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Λ
No × No
V
H
No × No
T
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Virtual Measurements
‹ General
Linear Transformations - FRF
‹ Eigenvalue
H (ω )
Ni × N o N s
T
Decomposition - Input DOF Space
H (ω )
N e × Ni
H
N o N s × Ni
= V
Ni × Ni
Λ
l ql q l q n s
= v1 v2 ... vk ... v N e
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Ni × Ni
V
H
N i × Ni
T
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Virtual Measurements
‹ General
Linear Transformations - FRF
‹ Singular
H
Value Decomposition - Output DOF Space
N o × Ni N s
T
Ne × No
= U
No × No
Σ
No × No
V
l ql q l q n s
= u1 u2 ... uk ... u N e
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H
N o × Ni N s
T
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Virtual Measurements
‹ General
Linear Transformations - FRF
‹ Singular
H
Value Decomposition - Input DOF Space
Ni × N o N s
T
N e × Ni
= U
Ni × Ni
Σ
Ni × Ni
V
l ql q l q n s
= u1 u2 ... uk ... u N e
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H
Ni × N o N s
T
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Typical Physical Measurements (280)
Automotive Example
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Typical Virtual Measurements (20)
Automotive Example
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Virtual Measurements
‹ Specialized
Linear Transformations
‹ Reciprocal
Modal Vector - Modal Filter Concept
‹ Enhance characteristics particular to each mode of
vibration
‹ Requires an estimate of each mode of vibration
‹ Generates a modal coordinate
‹ Valid with time or frequency domain data
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Virtual Measurements
‹ Specialized
Linear Transformations
‹ Reciprocal
Modal Vector - Modal Filter Concept
l q lxq
1, r = s
R
lφ q lψ q = ST0, r ≠ s
ηr = φ r
T
T
r
s
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Virtual Measurements
‹ Specialized
‹ Reciprocal
Linear Transformations
Modal Vector - Method 1
Φ
Φ
Ψ = I
T
T
= Ψ
−1
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Virtual Measurements
‹ Specialized
Linear Transformations
‹ Reciprocal
Modal Vector - Method 1
Ψ
T
Φ
T
M Ψ = I
≈ Ψ
T
M
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Virtual Measurements
‹ Specialized
‹ Reciprocal
l
q
H (ω )
Linear Transformations
Modal Vector - Method 2
N
p
=∑
r =1
lφ q lH (ω )q
T
r
p
Qrψ
l
q
( jω − λ )
ψr +
pr
r
=
Qrψ
pr
( jω − λ r )
ψ
*
Qr
*
pr
*
r
m
r
( jω − λ )
ψ *r
ψ
l q mψ r ( jω − λ )
+ φr
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*
r
*
Qr
*
pr
*
r
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Virtual Measurements
‹ Modal
Filter Estimates
‹ Reciprocal
Modal Vector - Method 1
‹ Reciprocal Modal Vector - Method 2
‹ Analytical Modal Vectors and Mass Matrix
‹ Experimental Modal Vector Estimate
‹ SDOF - MDOF Parameter Estimation
‹ Complex Mode Indicator Function (CMIF)
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Virtual Measurements
‹ Complex
‹ Plot
H
Mode Indicator Function
of singular values
N o × Ni
= U
N o × Ni
Σ
Ni × Ni
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V
H
Ni × Ni
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Virtual Measurements
‹ Complex
Mode Indicator Function
Circular Plate
Example
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Virtual Measurements
‹ Complex
Mode Indicator Function
Circular Plate
Example
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Virtual Measurements
‹ Complex
Mode Indicator Function
‹ Maxima
in the primary and successive CMIF plots
indicate the location of a modal frequency
‹ The singular vector associated with each maxima is a
good estimate of the modal vector
‹ This estimate of the modal vector is used as a modal
filter to isolate one modal coordinate in the enhanced
frequency response function (eFRF)
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Virtual Measurements
‹ Enhanced
Frequency Response Function
2N
H pq (ω ) = ∑
r =1
2N
Qrψ
ψ qr
( jω − λ r )
H pq (ω ) = ∑ ψ pr
r =1
pr
Qrψ qr
( jω − λ r )
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Virtual Measurements
‹ Enhanced
Frequency Response Function
eFRFr (ω ) =
Qrψ qr
( jω − λ r )
Qsψ qs
l q lH (ω )q = lφ q ∑ lψ q ( jω − λ )
eFRFr (ω ) = φ r
T
T
2N
r
s
s =1
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Virtual Measurements
‹ Enhanced
Frequency Response Function
Circular Plate
Example
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Virtual Measurements
‹ Enhanced
Frequency Response Function
Circular Plate
Example
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Virtual Measurements
‹ Modal
Filter, Time Domain - Example
‹ Space
Truss Application
‹ Six (6) Input Actuators
‹ Thirty-Seven (37) Response Sensors
‹ Eight (8) Modal Filters
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Virtual Measurements
‹ Modal
Filter, Time Domain - Example
‹ Physical
Measurements
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Virtual Measurements
‹ Modal
Filter, Time Domain - Example
‹ Physical
Measurements
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Virtual Measurements
‹ Modal
Filter, Time Domain - Example
‹ Virtual
Measurements
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Virtual Measurements
‹ Modal
Filter, Time Domain - Example
‹ Virtual
Measurements
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Virtual Measurements
‹ Kalman
Filtered Order Tracking - Example
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Virtual Measurements
‹ Kalman
Filtered Order Tracking - Example
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Summary/Conclusions
‹ Virtual
measurements are powerful tools which
can be useful in understanding experimental
structure analysis data
‹ Virtual measurements depend upon data set
arrays with sufficient spatial information and
linear, time invariant characteristics
‹ Virtual measurements yield reasonable solutions
to practical problems
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Future Applications
‹ Since
virtual measurements can be generated
quickly as long as the linear transformations are
known a priori, there are a number of interesting
real time applications
‹ Flight
flutter parameter estimation
‹ Multi-axis sensors (load, acceleration)
‹ Rigid and flexible body control
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