# Model Validation Using Coordinate Distance with Performance Sensitivity

Authors: Jiann-Shiun Lew
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

This paper presents an innovative approach to model validation for a structure with significant parameter variations. Model uncertainty of the structural dynamics is quantified with the use of a singular value decomposition technique to extract the principal components of parameter change, and an interval model is generated to represent the system with parameter uncertainty. The coordinate vector, corresponding to the identified principal directions, of the validation system is computed. The coordinate distance between the validation system and the identified interval model is used as a metric for model validation. A beam structure with an attached subsystem, which has significant parameter uncertainty, is used to demonstrate the proposed approach.


---

## Body

## 1. Introduction

Model
validation of structural dynamics is of great interest to both government and
industry [1]. Recently, a model validation workshop [2, 3] was organized by
Sandia National Laboratories to address the problem of certification of structures
under various forms of uncertainty. Following their formulation, an integrated
system consisting of a beam structure and an attached subsystem, shown in Figure 1, is the test structure used for study. In this model the physical elements
of the attached three degrees of freedom subsystem are the only ones exhibiting
significant parameter variations, all other parameters are known. The
substructure, along with its nonlinear connection, is considered for
calibration, and data are provided as a basis for the calibration of the
substructure model [2].A beam structure with an attached subsystem.In
the process of certifying structures for use in harsh dynamic environments, it
is often required that not only the main structure be capable of withstanding
the loads but also all the attached substructures. To ensure survivability of
all the substructures, Sandia in [2] has chosen a performance metric in terms
of the maximum acceleration magnitude of mass 3, top of the substructure, under
a shock force at position x8. For this study, the uncertain
parameters are the identified modal parameters (frequency, damping, and mode
shape) of subsystem, 15 parameters total.This
paper presents a model  validation methodology based on an interval modeling
technique for the structural  dynamics problem proposed by Sandia [2]. A
singular value decomposition technique [4] is applied to extract the principal
components of parameter change, where the sensitivity of performance is
included in the SVD process. From this process, an interval model is generated
and each interval corresponds to one identified bounded uncertainty parameter
with its associated principal direction. This interval modeling technique can
precisely quantify the uncertainty of a system with significant parameter
uncertainty [4]. The coordinate vector, corresponding to the identified principal
directions, of the validation system can be computed. The coordinate distance
between the validation system and the identified interval model is used as the
metric for model validation [5].


## 2. Model validation

In the model validation process, first an interval
modeling technique, given in the appendix, is applied for uncertainty
quantification.  The data used for model
uncertainty quantification are based on the identified modal parameters from 60
virtual experiments [2], generated from 20 identical systems selected from a
virtual pool and three levels of random excitation applied at mass 2. The modal
parameter vector of the subsystem is defined as
(2.1)p=[ω1ω2ω3ξ1ξ2ξ3ϕ11ϕ21ϕ31ϕ12⋯ϕ33]T,
where ωi is the ith
natural frequency, ξi is
the ith damping ratio, and ϕji is the jth component of the ith
mode shape. The interval modeling technique in the appendix is applied to generate
an interval model as
(2.2)P={p∣p=p0+∑j=115αjqj,αj∈[αj−,αj+]},
where p0 is the nominal parameter vector, and αj is the jth
identified bounded uncertainty parameter corresponding to the basis vector qj.
The coordinate vector of any validation system with parameter vector pv can be computed as
(2.3)βv=U−1Δpv,
with
(2.4)Δpv=pv−p0,U=[q1⋯q15],
where U is the basis matrix. The coordinate
distance between a validation system and the interval model is defined as
(2.5)dv=min{[βv−β(p)]T[βv−β(p)],p∈P},
where β(p) is the coordinate
vector of the subsystem with parameter vector p. This distance represents a metric of performance deviation
between a validation system and the identified interval model since the
weighting of performance sensitivity is included in SVD process [4, 5].


## 3. Discussion of results

There
are 60 sets of identified modal parameters used for model validation [2], generated
from 20 identical systems selected from a virtual pool with three levels of
shock input at mass 1. Figure 2 shows three modal parameters of 60
calibration systems and 60 validation systems as functions of the first
uncertainty parameter α1.
Variations in the natural frequencies are significant, around 100%, and
increase linearly as the first uncertainty parameter α1 increases. Natural frequencies of calibration systems and validation systems
share same variation characteristics. Damping and mode shape coefficients of
validation systems show bias from those of calibration systems. For example,
the mean value of ξ3 of the validation systems is around 30% lower
than that of the calibration systems when α1 is 0.1. For the second mode shape coefficient of the first mode in Figure
2, the mean value for the validation systems is always around 5% lower than
that of the calibration systems. Figure 3 shows the uncertainty parameters α2–α4 and
the identified interval bounds as functions of the first uncertainty parameter α1. The third interval length,
normalized to the first interval length (i.e., α1=1), drops to less than 10% (see Figure 3) of the
first interval length [4]. The model uncertainty is dominated by the first
uncertainty parameter α1. Natural frequency variations are
the dominant uncertainty corresponding to variations in α1.
In contrast to frequency variations, damping and mode shape variations behave
more like random variables, and they correspond to secondary uncertainties [4].
All α2 and α3 of validation systems are
inside the bounds or close to the boundary of the identified interval model.
All α4 of
validation systems are outside the bounds of the interval model, and this bias
is mainly contributed from the bias of mode shape and damping. Figure 4 shows
the coordinate distance of 60 validation systems from interval model. The
distance is mainly due to the bias of α4.Modal parameters of subsystems:
(a) natural frequency (rad/sec) of 1st mode, (b) damping ratio of 3rd mode, (c)
2nd mode shape coefficient of 1st mode. (circles) 60 calibration systems;
(asterisks)  60 validation systems.
(a)(b)(c)Coordinates and parameter bounds of uncertainty parameters: (a) 2nd uncertainty
parameter, (b) 3rd uncertainty parameter, (c) 4th uncertainty parameter. (circles) 60 calibration systems;
(asterisks) 60 validation
systems; — parameter bounds
of interval model.
(a)(b)(c)Coordinate distance of 60
validation systems from interval model: (circles) distance from interval model;
(asterisks)  distance contributed fromα4 bias.Figure5 shows the performance sensitivity to the identified uncertainty parameters αi. The sensitivity of
performance to the jth uncertainty parameters αj of the ith chosen
subsystem pi is defined as
(3.1)sijα=1a(pi)|∂a(pi)∂αj|,i=1,…,ns,
where a(pi) is the maximum
acceleration magnitude of the integrated system with subsystem parameter vector pi, and ns is
the number of parameter vectors. This sensitivity represents a percentage
change. The average sensitivity corresponding to the jth uncertainty
parameters αj is
defined as
(3.2)sjα=1ns∑i=1ns|sijα|.
Figure 5 shows that this
sensitivity is between 21% and 69%, corresponding to the original maximum
acceleration magnitude, and the sensitivity to α4 is 39% of
the maximum acceleration. Coordinate distance of all the validation systems is
between 0.03 and 0.07. This means that the maximum acceleration deviation between
the validation system and a system in interval model is insignificant (around
1% to 3%), based on the sensitivity in Figure 5. All the validation systems
are acceptable, based on the coordinate distance corresponding to performance
index of maximum acceleration.Sensitivity of performance to identified uncertainty parametersαj.Figure6 shows the maximum acceleration of the integrated systems with the
identified interval model, 60 calibration systems, and 60 validation systems
when an impulse force is applied at x8 position. The results
show that the identified interval model well represents and covers 60
calibration systems. The maximum acceleration of all the validation systems is
inside the envelope or close to the boundary of the interval model. As
expected, the validation systems are acceptable, based on the coordinate
distance results shown in Figure 4. This coordinate distance represents a
metric of the maximum acceleration deviation (percentage difference) between a
validation system and the identified interval model.Maximum acceleration with impulse
input: (circles)  interval
system; + 60 calibration systems; (asterisks) 60 validation systems.


## 4. Concluding remarks

This paper
presents a novel  approach for model validation of a system with an attached
subsystem that is exhibiting significant parameter uncertainty. An interval
modeling technique is applied for uncertainty quantification with the
performance sensitivity weighting in SVD process. The coordinate distance,
between the validation system and the identified interval model, is defined as
a metric for model validation. This distance represents a metric of the
possible performance deviation of the validation system from a system in
interval model. The results show that all the validation systems provided by
Sandia are acceptable, based on this distance metric. This demonstrates an
efficient tool for model validation, based on the interval model analysis. The
proposed technique in this paper can be extended to probability framework.



---

*Source: 298146-2008-09-08.xml*