Adjustment of the boundary conditions in a finite element model
of a plate suspended with springs by an inverse method
E. Euler and Prof. Dr. H.Sol
Vrije Universiteit Brussel
Pleinlaan 2
1050 Brussel
Belgium
ABSTRACT
The aim of this study is to determine the material properties of composite structures in function of
temperature. Identification of material properties using vibration-based mixed numerical experimental
techniques [MNETs] often takes a freely suspended rectangular plate as test configuration. However, in a
remote-controlled furnace, contactless acoustic excitations are used and the vibration amplitude is
registered contactless with a laser beam or camera. In this configuration, a plate with a free-free suspension
is difficult to position in a stable way. An alternative in such cases is the suspension of the specimen using
steel helical extension springs. The mass, stiffness and damping of the suspension springs influences the
structural vibration. Material identification by using vibration-based mixed numerical experimental techniques
[MNETs] can only be successfully applied if the mathematical model – which is solved using the finite
element method – is appropriate. The mathematical model is appropriate if it represents mass, stiffness and
damping properties of the physical test setup in sufficient detail. Consequently, relevant properties of the
suspension system must be determined in order to start the material identification procedure. This paper
describes an inverse method to identify the mass, stiffness and damping properties of the suspension
system. This information is used to adapt the boundary conditions in the mathematical model.
The presented method can be described as follows. First the material identification procedure is applied to a
plate specimen at room temperature with free – free boundary condition. A mathematical model of the plate
specimen is developed. The material properties in this model are the above identified material properties.
The mathematical model is complemented with local mass and stiffness elements to model the suspension
system. An initial value for axial spring stiffness is provided by the manufacturer of the helical extension
springs. An initial mass value can be derived by reasoning. At this stage, a sensitivity analysis is carried out
to verify if rigid body modes of the system can be used to identify the spring stiffness and/or mass influence.
If not, helical extension springs with other properties must be used to suspend the physical plate specimen.
If correct helical springs are chosen, the translational rigid body frequency of the suspended system can be
used to identify the axial spring stiffness. The rotational rigid body modes of the suspended system are used
to identify the exact location of the spring elements in the mathematical model.
Next the relevant resonant frequencies of the suspended plate specimen are experimentally measured.
These experimentally obtained values are compared with the frequencies of the mathematical model. The
spring masses are modified in such a way as the difference between experimental and numerical
frequencies becomes minimal. At this stage a mathematical model is obtained which reflect the vibratory
behavior of the suspended plate system. It is our intent to apply the same approach to determine damping
properties of the suspension system as well. Finally, the mathematical model can be used to start an
identification procedure to determine the material properties of the plate specimen in function of
temperature.
Introduction
Traditionally, elastic material properties are measured with quasi static tests such as tensile tests or four
point bending tests. Alternatively, elastic material parameters can also be determined by using vibrationbased methods. This approach is founded on the fundamental relation that exists between the elastic
material properties of a structure and its vibratory behaviour. The first identification methods based on this
principle used analytical formulas to describe the vibratory behaviour of the specimens. In 1937 Förster [1]
used the Euler beam theory to link the elastic modulus to the resonant frequency of the fundamental flexural
mode of a beam-shaped test specimen. In 1945 Pickett [2] used Goens’ [3] approximate solution of the
Timoshenko beam equations [4] which resulted in a more accurate relation between the elastic modulus and
the resonant frequency. In 1961, Spinner and Teft [5] extended Pickett’s work to torsional frequencies,
enabling the identification of the shear modulus. Their work formed the base of the ASTM resonant beam
test procedure [6], which standardised material testing based on analytical vibration models.
The standardised resonant-based techniques are restricted to homogeneous isotropic materials. The limiting
factor in extending the resonant-based techniques to more complex materials is the use of analytical models
to describe the vibratory behaviour of the test specimens. In 1986, Sol [7] showed that it is possible to
replace the analytical formulas by special purpose finite element models. Unfortunately, the finite element
formulation cannot be reversed into a formulation that provides the material properties from the resonant
frequencies of the test samples. The material identification problem has to be solved in an inverse way:
starting from a set of trial values, the unknown material parameters have to be tuned in such a way that the
finite element model reproduces the measured resonant frequencies. Techniques based on this approach
are commonly referred to as Mixed Numerical-Experimental Techniques or MNETs [8].
Material identification by using vibration-based MNETs can be successfully applied if the mathematical
model – which is solved using the finite element method – is appropriate. The mathematical model is
appropriate if it represents mass and stiffness properties of the physical test specimen in sufficient detail.
Moreover, mass and stiffness properties of the suspension system of the test specimen must also be taken
into account and modelled correctly in the mathematical model.
Therefore, one mainly uses freely suspended test specimens. This is accomplished by using very thin and
light elastic bands. In this configuration, one assumes that mass and stiffness properties of the suspension
are negligible small with respect to the mass and stiffness properties of the physical structure. Consequently,
it is assumed that this manner of suspension has no impact on the vibratory behaviour of the test specimen.
However, in a remote-controlled furnace, contactless acoustic excitations are used and the vibration
amplitude is registered contactless with a laser beam or camera. In this configuration, a plate with a free-free
suspension is difficult to position in a stable way. An alternative in such cases is the suspension of the
specimen using steel helical extension springs. In this configuration it becomes possible to determine the
orthotropic material properties in function of temperature.
A suspension system making use of helical extension springs is proposed in this paper. Relevant properties
of the suspension system must be determined in order to start the material identification procedure. This
paper describes an inverse method to identify the mass and stiffness properties of the suspension system.
This information is used to adapt the boundary conditions in the mathematical model.
Material identification using vibration-based MNETs: the method
The method proposed by Sol [7] identifies the four in-plane engineering constants of an orthotropic material,
i.e. E1, E2, G12 and ν12, by using a vibration-based MNET. The method makes use of the fundamental
bending modes of two beam-shaped specimens, and the first three resonant frequencies of a plate-shaped
specimen.
Two beams with perpendicular longitudinal axis are cut from the test plate [Fig 1]. The fundamental flexural
mode of the two beams will yield an initial value for E1 and E2. The test plate is modified in such a way that
the ratio of the length and width of the plate becomes:
⎛L⎞
⎜ ⎟=
⎝B⎠
4
E1
E2
(1)
A plate with this aspect ratio has two coupled first-order flexural modes and, consequently, a maximal
Poisson’s ratio sensitivity. The first three mode shapes of a Poisson plate are respectively a torsion mode, a
saddle mode and a breathing mode. The frequency corresponding with the torsion mode has a good
sensitivity with respect to G12. The frequencies corresponding with the saddle and breathing mode are
sensitive with respect to E1, E2 and ν12. A good sensitivity of the frequency versus elastic parameter is a
prerequisite for a successful identification.
If one is not allowed to cut beam specimens from the plate, the identification of the material properties can
still be carried out. In this case, it is necessary to measure the first five resonant frequencies of the plate
structure. The first three mode shapes are still a torsion mode, a saddle mode and a breathing mode. Extra
information is contained in the two supplemental higher-order combined torsion-bending mode shapes of the
plate.
No matter which of the two options are chosen, a mathematical model of the plate is created. In the physical
model all mass and stiffness related properties are known except for the orthotropic material properties. All
known properties are implemented as such into the mathematical model. This mathematical model is solved
with finite elements for numerical resonant frequencies using a set of trial values for the unknown elastic
parameters.
The numerical and experimental frequencies are compared, and the trial values of the unknown elastic
parameters are corrected in order to minimise the difference between the two frequency sets. The improved
material properties are inserted into the FE-model and a new iteration cycle is started. Once the numerical
and experimental frequencies match, the procedure is aborted, and the desired material properties can be
found in the database of the FE-model.
Material identification using vibration-based MNETs: proposed suspension system
The method proposed by Sol [7] is based on the measurement of frequencies corresponding with certain
well defined mode shapes. The method can be successfully applied on two conditions. The mathematical
model of the experiment must be appropriate. The mathematical model is appropriate if it represents mass
and stiffness properties of the physical test specimen in sufficient detail including the effect of the
suspension system. Secondly, the suspension system allows the specimen to vibrate in certain well defined
mode shapes.
Until now, the first requirement is accomplished by suspending the specimen by very thin and light elastic
bands. In this configuration, one assumes that mass and stiffness properties of the suspension are negligible
small with respect to the mass and stiffness properties of the physical structure. Consequently, the
suspension system is not modelled in the mathematical model. The second requirement is accomplished by
hanging the plate at the nodal lines of the certain well defined mode shapes [Fig 2, Fig 3 and Fig 4].
Drawbacks of the current suspension system are that the threats are not high temperature resistant, the
plate is not positioned in a stable manner and during the test the plate is suspended in different points
depending on which mode shape one wants to measure. To investigate the temperature influence on the
orthotropic material properties of composite materials, the current system is not optimal.
The suspension system proposed consists out of four helical extension springs made out of steel. Figure 6
shows the suspended plate configuration. The springs are connected at the corner points of the plate. This is
done by drilling a small hole in the corner points of the plate. During the test the suspension points are not
modified. To use this system with success two things must be investigated. The mass and stiffness influence
of the springs on the plate vibratory behaviour. Investigate if the suspension system will still allow the plate
to vibrate in the relevant mode shapes.
Material identification using vibration-based MNETs: proposed method
First the material identification procedure is applied to a plate specimen at room temperature with free – free
boundary conditions. The first five resonant frequencies of the plate are measured. The torsion mode, the
saddle mode, the breathing mode and two combined torsion – bending modes. The plate is suspended with
elastic bands in the nodal lines of the relevant mode shapes. Minimal four holes must be drilled into the
plate in order to do so. In the physical model all mass and stiffness related properties are known except for
the orthotropic material properties. All known properties are implemented as such into the mathematical
model. The procedure yields the orthotropic material properties at room temperature i.e. E1, E2, G12
and ν12. These material properties are used further in the model.
Next the mathematical model is complemented with local mass and stiffness elements to model the
suspension system. An initial value for the axial spring stiffness can be calculated using the formula [9]:
k axial
=
Gd 4
64nR 3
(2)
with G de shear modulus, d the diameter of the coils cross section and n the number of coils. Often the axial
spring stiffness is also provided by the spring manufacturer. An equivalent initial mass meq value can be
derived for a spring of length L. The kinetic energy in pure planar rotation about a fixed point O of the spring
can be written as
T=
1
I 0ω 2
2
(3)
In which I0 is the mass moment of inertia about O and ω is the magnitude of the angular vector ω. The
kinetic energy about the fixed point O should be the same for the real spring versus the model with an
equivalent mass placed at a distance L. This can be written as,
1 2
ω ∫ r 2 dm
2 spring
1
meq L2ω 2
2
=
(4)
For a uniform spring dm = (ms/L)dr with ms the mass of the spring and L the length of the spring. Substituted
in the above equation gives
L
1 2 ms 2
ω
r dr
L ∫0
2
=
1
meq L2ω 2
2
(5)
1 2 ms 2
ω
L
2
3
=
1
meq L2ω 2
2
(6)
One can conclude that the equivalent mass meq for a spring is equal to
m eq
=
ms
3
(7)
At this stage, a sensitivity analysis is carried out. The parameters in the sensitivity analysis are the spring
axial stiffness k axial , the spring equivalent mass meq and the material properties of the specimen i.e. E1, E2,
G12 and ν12. The responses in the sensitivity analysis are the three first resonant frequencies of the system.
If the first modes are rigid body modes of the system then the sensitivity of the material properties versus
resonant frequencies is zero. Rigid body frequencies are sensitive to a change in spring stiffness. Rigid body
frequencies are insensitive with respect to the equivalent spring mass meq since the equivalent mass is very
low compared with the participating rigid body mass of the plate. The translational rigid body frequency of
the suspended system is used to identify the axial spring stiffness. The rotational rigid body modes of the
suspended system are used to identify the exact location of the spring elements in the mathematical model.
If the first modes are not rigid body modes, helical extension springs with other properties must be used to
suspend the physical plate specimen.
In order to determine the equivalent mass of the spring, a second sensitivity analysis is carried out. The
parameters in this second analysis are the spring axial stiffness
k axial
and the spring equivalent mass meq.
The responses in the sensitivity analysis are the five resonant frequencies of the system which generate
strain energy in the plate. In general one can state that the sensitivity of meq is high and the sensitivity of
k axial
is low with respect to the frequencies. Consequently, these five frequencies are used to determine
meq. The relevant resonant frequencies of the suspended plate specimen are experimentally measured.
These experimental values are compared with the frequencies of the mathematical model. The spring
masses are modified in such a way as the difference between experimental and numerical frequencies
becomes minimal.
At this stage a mathematical model is obtained which reflect the vibratory behaviour of the suspended plate
system. Finally, the mathematical model can be used to start an identification procedure to determine the
material properties of the plate specimen in function of temperature.
Material identification of a steel plate: illustration
Properties and dimensions steel plate:
0.12m x 0.12m x 0.00148m
M = 1672.5e-4 kg
Volume = [0.12 x 0.12 x 0.00148] – 4 x [0.00148 x pi x 0.00159²] – 4 x [0.00148 x pi x 0.00095²] = 212.5e-7
m³
ρ = 7870.6 kg/m³
Removed mass due to drilling a corner hole on the plate
mcorner-hole: 92.5e-6 kg.
Removed mass due to drilling a midside hole on the plate
mmidside-hole: 33.0e-6 kg.
Properties and dimensions helical spring:
Length unloaded spring = 0.05 m
Total mass = 37.6e-4 kg
Equivalent mass meq = 125.3e-5 kg
Material identification: plate with free-free boundary conditions
Convergence check numerical model:
Dimensions plate model: 0.12m x 0.12m x 0.00148m
Ex: 210GPa
Ey: 200GPa
Gxy: 70GPa
Νxy: 0.25
ρ: 7800 kg/m³
Number of
elements
Torsion
frequency
[Hz]
Saddle
frequency
[Hz]
Breathing
frequency
[Hz]
Torsion - bending Y
frequency
[Hz]
Torsion - bending X
frequency
[Hz]
10 x 10
20 x 20
30 x 30
40 x 40
50 x 50
60 x 60
100 x 100
316.4
320.4
321.1
321.3
321.3
321.3
321.3
487.1
499.2
501.5
502.3
502.7
502.9
503.2
582.5
594.5
596.8
597.6
598
598.2
598.5
810.2
835.3
840.1
841.7
842.4
842.7
842.9
818
843.4
848.3
849.9
850.6
850.9
851.2
Since calculation time is not an issue, a mesh size of 60 x 60 elements is retained.
Experimental results:
The plate is excited with a hammer impact. The free response [point velocity] of the plate is measured with a
Polytec OVF 303 laser sensor head. The data is captured and analyzed by an LMS.Pimento system with a
frequency resolution of 0.1 Hz. Free-free boundary conditions are assumed since the plate is suspended
with light elastic bands in the nodal points of the respective mode.
1
2
3
Mean
Torsion
[Hz]
Saddle
[Hz]
Breathing
[Hz]
Combined
Torsion-bendingX
[Hz]
Combined
Torsion-bendingY
[Hz]
376.2
376.2
376.1
376.2
526.9
526.9
526.9
526.9
626.7
626.5
626.5
626.6
944.9
944.8
944.8
944.8
937.6
937.5
937.5
937.5
Numerical model:
The material X-axis is in the direction of the global Y-axis. The material Y-axis is in the direction of the global
X-axis. Initial material values are used to start the updating process.
Ex = 210GPa
Ey = 200GPa
Gxy = 70GPa
Nuxy = 0.25
The first five frequencies based on the above initial material values, are
Torsion = 322.1 Hz
Saddle = 501.6 Hz
Breathing = 599.8 Hz
Torsion-BendingY = 845.5 Hz
Torsion-BendingX = 853.7 Hz
Mathematical model is shown in figure 7
mcorner-hole [92.5e-6 kg] is indicated in green color.
mmidside-hole [33.0e-6 kg] is indicated in red color.
Resonant frequencies are shown in figure 8 – figure 12
Material identification:
Reference responses: experimental measured resonant frequencies 376.2 Hz; 526.9 Hz; 626.6 Hz; 937.5
Hz; 944.8 Hz.
Parameters: Ex, Ey, Gxy and Nuxy.
Ex
Ey
Gxy
Nuxy
Torsion
Saddle
Breathing
Torsion-BendingY
Torsion-BendingX
Initial values
[N/m²]
Updated values
[N/m²]
Difference
(%)
2.10E+11
2.00E+11
7.00E+10
2.50E-01
2.29E+11
2.19E+11
9.58E+10
2.42E-01
9.21E+00
9.73E+00
3.69E+01
-3.28E+00
Fea using Updated
Values
[Hz]
Reference
Response Test
[Hz]
Difference
[%]
3.74E+02
5.26E+02
6.25E+02
9.42E+02
9.49E+02
3.76E+02
5.27E+02
6.27E+02
9.38E+02
9.45E+02
-6.20E-01
-1.84E-01
-1.87E-01
5.06E-01
4.96E-01
Material identification: suspended plate
An initial value for the axial spring stiffness = 40 N/m [80N/m in numerical model, two springs in parallel]
Equivalent mass meq for one axial spring = 125.3e-5 kg [250.6e-5 kg in numerical model, two springs
attached in real model]
1
2
3
Mean
RBTRANSZ [Hz]
8.6
8.6
8.6
8.6
RBROTX [Hz]
14.9
14.9
14.9
14.9
RBROTY [Hz]
13.6
13.6
13.6
13.6
Sensitivity analysis:
Reference responses = first three resonant frequencies suspended system
Parameter = spring axial stiffness, equivalent spring mass, Ex, Ey, Gxy and Nuxy. Three first frequencies
are rigid body modes.
Sensitivity matrix is shown in figure 13.
Translational rigid frequency is used to determine the axial spring stiffness. Model updating results are
shown in table
kaxial
RBTRANSZ
Initial Value
[N/m]
Updated Value
[N/m]
Difference
[%]
80
129.6
+6.2e+01
Fea using Updated
Value
[Hz]
Reference Response
Test
[Hz]
Difference
[%]
8.6
8.6
0
The rotational rigid body modes of the suspended system are used to identify the exact location of the spring
elements in the mathematical model. Figure 14 shows the mathematical model at this point.
Sensitivity analysis:
Reference responses = five first frequencies which contains strain energy
Parameter = spring axial stiffness, equivalent spring mass
Sensitivity matrix is shown in figure 15.
Determination meq using Torsion frequency:
Mequivalent
Torsion
Initial Value
[kg]
Updated Value
[kg]
Difference
[%]
25.06e-4
2.04e-4
-9.19e+1
Fea using Updated
Value
[Hz]
Reference Response
Test
[Hz]
Difference
[%]
365.4
365.4
0
Determination meq using Saddle frequency:
Mequivalent
Saddle
Initial Value
[kg]
Updated Value
[kg]
Difference
[%]
2.04e-4
1.36e-4
-3.31e+1
Fea using Updated
Value
[Hz]
Reference Response
Test
[Hz]
Difference
[%]
525.1
527.2
-4.0e-1
Determination meq using Breathing frequency:
Mequivalent
Saddle
Initial Value
[kg]
Updated Value
[kg]
Difference
[%]
1.36e-4
4.57e-5
-6.6e+1
Fea using Updated
Value
[Hz]
Reference Response
Test
[Hz]
Difference
[%]
617.2
617.2
0
Determination meq using Torsion-BendingY frequency:
Mequivalent
Saddle
Initial Value
[kg]
Updated Value
[kg]
Difference
[%]
4.57e-5
5.73e-5
+2.53e+1
Fea using Updated
Value
[Hz]
Reference Response
Test
[Hz]
Difference
[%]
927.7
927.7
0
Determination meq using Torsion-BendingX frequency:
Mequivalent
Saddle
Initial Value
[kg]
Updated Value
[kg]
Difference
[%]
5.73e-5
1.31e-4
+1.28e+2
Fea using Updated
Value
[Hz]
Reference Response
Test
[Hz]
Difference
[%]
933.2
933.2
0
References
[1] F. Förster. Ein neues Messverfahren zur Bestimmung des Elastizitäts-modulus und der Dämpfung. Z.
Metallkd, Vol. 29:109-115, 1937.
[2] G. Pickett. Equations for computing elastic constants from flexural and torsional resonant frequencies of
vibrating prisms and cylinders. Proceedings ASTM, Vol. 45:846-865, 1945.
[3] E. Goens. Über die Bestimmung des Elastizitätsmodulus von Stäben mit Hilfe von
Biegungsschwingungen. Annalen der Physik, Band 11:649-678, 1931.
th
[4] S. Timoshenko. Vibration problems in engineering. Wiley, New York, USA, 4 edition, 1974.
[5] S.Spinner and W.E.Teft. A method for determining mechanical resonance frequencies and for calculating
elastic moduli from these frequencies. Proceedings ASTM, Vol. 61:1209-1221, 1961.
[6] American Society for Testing and Materials. Standard C 1259-01, Standard Test Method for Dynamic
Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced Ceramics by Impulse Excitation of
Vibration, April 2001.
[7] H. Sol. Identification of anisotropic plate rigidities using free vibration data. PhD thesis, Vrije Universiteit
Brussel, Brussels, Belgium, 1986.
[8] H.Sol and C.W.J. Oomens. Material identification using mixed numerical-experimental methods. Kluwer
st
Academic Publishers, Kerkrade, The Netherlands, 1 edition, 1997.
[9] L. Meirovitch, Fundamentals of vibrations. McGraw-Hill, International edition 2001.
[10] Carne T.G., Dohrmann C.R., Support Conditions, their effect on measured modal parameters, Proc.
SPIE Vol. 3243, Proceedings of the 16th International Modal Analysis Conference, p. 477, 1998
Figures
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Figure5
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Figure8
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Figure10
Figure11
Figure12
Figure13
Figure14
Figure14