Techniques for Synthesizing FRFs From Analytical Models
Hasan G. Pasha, Randall J. Allemang and Allyn W. Phillips
University of Cincinnati – Structural Dynamics Research Lab (UC-SDRL), Cincinnati, Ohio, USA
Abstract. Modal correlation of test and analytical data is an important step in system identification and model
updating. The Frequency Response Assurance Criterion (FRAC) is one of the metrics that can be used to quantify
the strength of correlation between the test and analytical degrees of freedom (DOF). To calculate FRAC for test
and analytical data, frequency response functions (FRF) are required. Techniques to synthesize FRFs from finite
element models are discussed in this paper. Methods to represent damping in analytical models are also presented.
These techniques were applied to synthesize FRFs from a finite element model of a rectangular steel plate structure.
Comparing the synthesized FRFs with the measured FRFs for the rectangular plate structure aided in calibrating
the rectangular plate FE model. The techniques presented in this paper can be used to visually check if the test and
analytical data are well correlated and for calculating FRAC metric to quantify the strength of correlation.
Keywords. FRF synthesis, Damping in analytical models, Modal Correlation, FRAC, Model Calibration
Notation
Symbol
[·]∗
α
β
η
λ
ω
ψ
ζ
m
p
q
{x}
{ẋ}
{ẍ}
Apq
[C]
[CR ]
{f }
[H(ω)]
[K]
[Kj ]
[KR ]
[M ]
[Mi ]
[MR ]
[Nf ]
[Ni ]
[No ]
FRAC
FRF
Description
Complex conjugate [·]
Mass matrix multiplier for damping
Stiffness matrix multiplier for damping
Loss factor
System pole (rad/s)
Frequency (rad/s)
Mode shape coefficient
Damping ratio
Maximum mode number
Output DOF
Input DOF
Displacement vector
Velocity vector
Acceleration vector
Residue
Damping matrix of reduced model
Stiffness matrix of reduced model
Applied force
FRF Matrix ([ X
])
F
Stiffness matrix of full model
Stiffness matrix for material j
Stiffness matrix of reduced model
Mass matrix of full model
Mass matrix for material i
Mass matrix of reduced model
Number of frequency lines
Number of input degrees of freedom
Number of output degrees of freedom
Frequency Response Assurance Criterion
Frequency Response Function
1. Introduction
The finite element (FE) technique is used to represent complex structural systems. However, there is uncertainty
related to how well the FE model represents the true structure. Generally, modal testing is performed to correlate,
validate and calibrate the FE model. Such a FE model could then be used for any future studies instead of performing
resource intensive testing.
The order of the number of measurement DOFs is quite small compared that of a FE model. Model order reduction
of the FE model to measurement DOFs or expansion of the measurement DOFs to DOFs of the FE model should be
done before performing modal correlation. Modal correlation technique involves using various metrics to correlate
the analytical results with the experimental results. Generally, modes are paired and the relative error of modal
frequencies gives an estimate of the variation in modal frequencies. The Modal Assurance Criteria (MAC) and
Pseudo Orthogonality Check (POC) are metrics that could be used to check the linear independence of the modal
vectors.
These comparisons are based on the parameters derived from the FE model and parameters obtained by fitting a
model to the measured FRF data. A mechanism that could compare the raw measured FRF data with the inputoutput relationships derived from a FE model would give a better idea of the level of correlation, provide additional
insight to validate, and calibrate the FE model. The Frequency Response Assurance Criterion (FRAC) is a metric
that compares any two frequency response functions representing the same input-output relationship. This metric
can be used to check how well the analytical FRFs compare with the measured FRFs.
Techniques to synthesize a system FRF matrix from a FE model are discussed in the following section. Structural
examples of a rectangular steel plate are also presented to demonstrate the techniques discussed. Often damping is
ignored while developing a FE model. Many FE software packages now have a provision to prescribe damping using
standard parameters that correspond to established damping models. Implementation details on defining damping,
retrieving system matrices, and performing the system FRF computations are also discussed in this paper.
2. Synthesizing FRFs from Analytical Models
For a multi-degree of freedom dynamic system, the equations of motion are in the form shown in Eq. 1.
(1)
[M ] {ẍ} + [C] {ẋ} + [K] {x} = {f }
When the excitation is harmonic, the response of the system is expressed using the frequency response function
(FRF) matrix [H(ω)]. Typically, the FRF matrix is a No xNi xNf matrix. It is computed using Eq. 2.
(2)
−1
[H(ω)] = [K] − ω 2 [M ] + jω[C]
The FRFs can be synthesized from an analytical model in one the following ways described below.
2.1. Full space method. Once a model of the structure is developed in a FE software package, the mass, stiffness
and damping matrices of the full model can be retrieved. The FRF matrix for the system can be computed using
the expression shown in Eq. 2. The disadvantages of this approach are that since the order of the FE model is huge,
the system matrices are huge. Hence, computations take enormous amount of resources, both time and memory.
2.2. Synthesizing FRFs from reduced order models. In order to address the issue of the model order, the
full model could be reduced using condensation techniques. Both static and dynamic condensation techniques exist.
Static condensation techniques, such as the Guyan reduction, are simple, but a lot details are lost. As a result, the
reduced order model is not truly representative of the full model. Only the stiffness matrix of the system is considered
in static reduction, while mass and damping information are neglected. Dynamic condensation techniques consider
the mass and stiffness matrices in calculations, and therefore provide a better approximation of the full model.
Once the requirements of the analysis are determined, an appropriate condensation technique to perform model
order reduction of the full model can be selected and transformation of the full model’s mass, stiffness and damping
matrices to reduced space is performed. It is important to realize that the DOFs in the FE model that correspond
to the location of the measurement DOFs should be retained while performing the model order reduction. The FRF
matrix for the reduced model can be computed using the expression shown in Eq. 3.
(3)
[H(ω)] = [KR ] − ω 2 [MR ] + jω[CR ]
−1
2.3. Modal superposition method. Modal superposition method is a special form of model order reduction and
a quite popular technique in structural dynamics. The modal parameters, namely the modal frequencies and modal
coefficients, are used as generalized coordinates to perform the order reduction. The partial fraction model for FRF,
shown in Eq. 4, can be used for computing the FRF at response location p when the system is excited at reference
location q.
(4)
H(ω)pq =
m
X
A∗pqr
Apqr
+
jω − λr
jω − λ∗r
r=1
The residue Apqr and its complex conjugate A∗pqr can be computed using Eq. 5
(5)
Apqr = Qr ψpr ψqr
where Qr is the scaling constant for mode r, and ψpr and ψqr are the modal coefficients. These parameters can be
easily retrieved from a FE modal analysis project.
2.4. Implementation details. Frequency response function at multiple response locations due to unit force applied
at a particular reference location can be computed using FE software, such as ANSYS Workbench – Harmonic
Analysis module. However, in order to compute the FRF matrix for the whole system, this process can be inefficient
and slow. When the system FRF matrix is required, the system matrices could be retrieved from the FE model and
the FRF computation can be performed separately. The sections below provide information pertaining to retrieving
system matrices from ANSYS Workbench and computing the system FRF matrix using Matlab.
2.4.1. Retrieving system matrices with ANSYS Workbench. The ANSYS Workbench has submodules to perform
specific analyses. The Modal Analysis module, as the name suggests, can be used to perform modal analysis
of a structure. In order to retrieve the system matrices of a structure, a Modal Analysis project needs to be
setup. This requires defining the material model, building or importing the geometry, setting up and executing
the analysis. If the geometry is simple, it can be built using ANSYS Workbench Design Modeler module. However, if the geometry is complex it can be built using any design software and imported into ANSYS Workbench.
In order to retrieve the system matrices from the output database, the option to save the output database after
the analysis is complete should be set. System matrices are generally sparse. They can be retrieved as dense
or sparse matrices from ANSYS. A command object should be added to retrieve the system matrices at a specific memory location. Code snippets to retrieve the system matrices as sparse and dense matrices are shown below.
! Stiffness *DMAT,MatKD,D,IMPORT,FULL,file.full,STIFF *PRINT,MatKD,Kdense.matrix
! Mass *DMAT,MatMD,D,IMPORT,FULL,file.full,MASS *PRINT,MatMD,Mdense.matrix
! below commands create sparse matrix
*SMAT,MatKS,D,IMPORT,FULL,file.full,STIFF *PRINT,MatKS,Ksparse.matrix
For Mass *SMAT,MatMS,D,IMPORT,FULL,file.full,MASS *PRINT,MatMS,Msparse.matrix
2.4.2. Computing the FRF. Once the system matrices or modal parameters are retrieved, Matlab could be used to
compute the FRF matrix using the relations shown in previous sections. Open source codes to read sparse matrices
generated by ANSYS and converting them to Matlab format exist. The frequency resolution can be controlled using
the Matlab program to provide reasonable results in stipulated execution time.
3. Damping
In an ideal vibratory system, the energy changes its form back and forth between potential energy and kinetic energy.
However, since there are losses associated with the energy conversion in a practical system, the vibration eventually
dies out in the absence of any external excitation to the system. The energy loss is quantified by damping. There
are several mechanisms by which energy dissipation can take place, such as, friction, hysteresis, etc. It is difficult to
model damping based on microscopic phenomena. Simple mathematical models exist that can be used to quantify
damping, e.g. viscous damping model, structural damping model.
3.1. Modeling damping with ANSYS Workbench. The system damping matrix in ANSYS Workbench R14.5
is computed using Eq. 6 [1]. ANSYS has provision to prescribe damping by specifying various parameters.
(6)
[C] = α[M ] +
|
Nmb X
2
2
αim [Mi ] + β + ζ [K] +
βjm + ζj [Kj ] +
ω
ω
j=1
i=1
{z
} |
{z
}
Ne
X
N
ma
X
Mass Damping
Structural Damping
[Ck ]
Ng
X
[Gl ]
+
k=1
| {z }
Element Damping
l=1
| {z }
Gyroscopic Damping
+
Nv
X
1
[Cm ]
ω
m=1
|
{z
}
Viscoelastic Damping
The values for α, β and ζ can be set at the global or the material level. In a structure with multiple material layers,
for example when a coating material is used, the damping properties of the coating material can be different from
the base material. The terms highlighted in blue in Eq. 6 indicate that material specific damping can be specified.
In addition, a viscous damper can be modeled using the spring element connector and specifying the damping values,
which is referred as element damping in Eq. 6.
4. Frequency Response Assurance Criteria (FRAC)
Any two frequency response functions representing the same input-output relationship can be compared using a
metric known as the Frequency Response Assurance Criterion (FRAC). Once the FRFs are synthesized from FE
model, they can be compared with the measured FRFs and the FRAC can be computed to understand the strength of
correlation. For example, the FRAC for two FRFs, measured FRF Hpq (ω) and analytical FRF Ĥpq (ω), representing
the relation between output DOF p and input DOF q, can be computed using the expression shown in Eq. 7. The
frequency resolution of the analytical FRF data and measured FRF data should match.
(7)
FRACpq
2
P
ω2
∗
(ω)
ω=ω1 Hpq (ω)Ĥpq
= Pω2
Pω2
∗
∗
ω=ω1 Hpq (ω)Hpq (ω)
ω=ω1 Ĥpq (ω)Ĥpq (ω)
5. Experimental Example
A FE model of a rectangular steel plate structure of dimensions 0.86 m x 0.57 m x 0.0063 m (34”x22.5”x.25”) was
developed using ANSYS Workbench R14.5. A cold rolled steel rectangular plate structure was fabricated (E =
2.05x1011 Pa (2.9734x107 psi), ν = 0.29 and ρ = 7850 kg/m3 (0.2836 lb/in3 )), with 160 points marked on a
0.05 m x 0.05 m (2”x2”) grid. Each of these 160 points were impacted and FRFs were measured at 21 reference
locations using uniaxial accelerometers.
The system mass, damping and stiffness matrices were obtained from the FE model. A named selection of points
corresponding to the impact locations (which also included the sensor locations) was created. The displacement
for the named selection represents the modal coefficients. The analytical FRFs were generated using the full space
method from the analytical model.
5.1. FRF comparison. The FE model was calibrated to the match measured modal frequencies and modal vectors.
The FE model was validated by comparing analytical FRFs from the model with the measured FRFs. In addition,
the results obtained for two perturbed mass configurations with unconstrained boundaries were compared with the
predictions from the updated model to check its robustness. Fig. 1 shows a comparison of the driving-point and
cross-point FRFs for the rectangular plate.
5.2. Effect of structural damping. The structural damping does not affect the modal frequency. Its effect is only
to limit the response at resonance. The effect of structural damping on the plate was studied. The FRF for various
values of loss coefficient values are plotted in Fig. 2. It is evident that the modal frequencies do not change with
increasing η values, but the amplitude of the response at resonance is reduced.
6. Conclusions
• Modal correlation, updating, validation and calibration are important tasks in order to reduce uncertainty
in predictions from a FE model. Various metrics to perform modal correlation are available. The Frequency
Response Assurance Criterion (FRAC) aims to quantify the strength of correlation between measured and
synthesize FRFs. It is particularly advantageous to correlate FRFs as it works directly on raw input-output
relations.
(a) Driving-point FRFs
(b) Cross-point FRFs
Figure 1: Comparison of FRFs
Figure 2: Rectangular Plate – Effect of Structural Damping
• Techniques to synthesize frequency response functions from analytical models were discussed. FRFs can be
synthesized by full order model, reduced order model or by modal superposition. Full order method can be
resource intensive. The reduced order method or the modal superposition method provide reasonable results
while being less resource intensive.
• FE packages, such as ANSYS, have provisions to output system or element mass, stiffness and damping
matrices. ANSYS also has the capability to prescribe damping for a model by specifying parameters for
standard damping models. Implementation details to retrieve the system matrices and to compute the
system FRF matrix were discussed.
• A rectangular plate structure was used to demonstrate how the analytical FRFs could be compared with
measured FRFs. The effect of structural damping on the FRF of the steel plate structure was also presented.
References
[1] ANSYS, Inc., ANSYS Mechanical Linear and Nonlinear Dynamics, Lecture 3: Damping, ANSYS release 14.5, 2013