INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
TRANSIENT RESPONSE ANALYSIS OF STRUCTURES
MADE FROM VISCOELASTIC MATERIALS
EVGENY BARKANOV*
Institute of Computer Analysis of Structures, Riga ¹echnical ºniversity, Kalku St. 1, ¸»-1658, Riga, ¸atvia
SUMMARY
The response of structures made from viscoelastic materials to transient excitations is studied using the finite
element method. The viscoelastic material behaviour is represented by the complex modulus model. An
efficient method using fast Fourier transform has been developed. This method is based on the trigonometrical representation of the input signals and matrix of the transfer functions. The present implementation gives
the possibility to preserve exactly the frequency dependence of the storage and loss moduli of materials. On
this reason this time-domain representation is a mathematically correct way to avoid the non-causal effect.
Test problems and numerical examples are given to demonstrate the validity and effectiveness of the
approach suggested in this paper. Copyright 1999 John Wiley & Sons, Ltd.
KEY WORDS: transient response analysis; fast Fourier transform; finite element method; viscoelastic damping;
complex modulus model
1. INTRODUCTION
Recent decades, an important area of computer simulation in mechanics is the dynamic analysis
of structures made from viscoelastic materials in the frequency domain.— This is connected with
further improvements of commercially available damping composite materials and its increasing
applications in a various fields of engineering: aerospace technology, shipbuilding, automotive
industry, mechanical engineering, etc. However, a smaller quantity of papers— is devoted to the
solution in the time domain. At present, for the viscoelastic models, most commonly applied in
structural analysis, one of the following approaches is employed. The first is to relate timedependent stresses to time-dependent strains through a differential or an integral operators.
The second is based on the complex modulus model.— Using the last, the constitutive equations
are expressed in the frequency domain.
In the first approach, for viscoelastic materials having storage and loss moduli that are strongly
frequency dependent over many decades of frequency, the number of time derivatives is large.
Consequently, the number of empirical parameters in the model is large too. As a result, the
* Correspondence to: Evgeny Barkanov, Institute of Computer Analysis of Structures, Riga Technical University, Kalku
St 1, LV-1658, Riga, Latvia
Contract/grant sponsor: Konferenz der Deutschen Akademien der Wissenschaften
Contract/grant sponsor: Latvian Council of Science; Contract/grant number: 96.p.III-1.04
CCC 0029—5981/99/030393—11$17.50
Copyright 1999 John Wiley & Sons, Ltd.
Received 28 August 1997
Revised 24 March 1998
394
E. BARKANOV
model is time consuming to manipulate and, when put into the equations of motion, produces
high-order differential equations. However, progress is being made in the papers, where
a fractional derivative model of the frequency-dependent mechanical properties requires only five
empirical parameters. This is fewer parameters than are usually required with the corresponding
standard linear viscoelastic model.
The constitutive equation can be imagined physically in terms of an interlocking network of
spring and dash-pots elements, each of them possesses different properties. The problem in this
case is the large number of terms needed to model real material behaviour. This model is used in
the paper to describe the viscoelastic properties of Timoshenko beams in the quasi-static and
dynamic analysis. The beam structure is considered as a Maxwell fluid and the three parameter
solid model. For simplicity, the experimental data for real material is replaced by the assumed
values of spring constants and viscosity coefficients in order to compute the relaxation modulus
in the form of the Prony series.
The second approach is quite well accepted in the frequency domain to simulate the linear
viscoelastic material behaviour. Long time, it was believed that the linear viscoelastic model
represented by complex modulus leads to some difficulties in calculation of the transient response,
but they are more apparent than real. If, for example, one assumes that storage and loss moduli of
material are constant, the transient response obtained by the Fourier analysis is non-causal,
with response before application of force, but if these moduli are allowed to vary in accordance
with the behaviour observed in real materials, the non-causality disappears, as explained in the
literature. The time-domain behaviour of a hysteretically damped structures was obtained from
the frequency-domain response by the Fourier transform technique in the papers. Same
technique in a form appropriate for a non-complex notation was applied in the paper. In the
paper, a consistent time-domain representation for linear hysteretic damping was presented
using the Hilbert transform. An iterative technique was used in situations in which a software
package for time-domain computation of transient response of linear or non-linear causal models
is available and the desired mathematical model of the structure contains linear hysteretic
elements. In the calculation models used in these papers, the material loss factors were approximated as piecewise linear functions or as constant values and storage moduli were taken
into consideration as constant values.
The objective of the present study is to obtain the transient response of structures made from
viscoelastic materials using the finite element method. The present implementation gives the
possibility to preserve the frequency dependence for the storage and loss moduli of materials
exactly and so to avoid the non-causal effect. Some test problems and numerical examples are
given.
2. TRANSIENT RESPONSE ANALYSIS
One of the most used models to describe the rheological behaviour of viscoelastic materials in the
finite element analysis is the complex modulus representation. Since this model is solvable, that
is making use of existing computing facilities, and the results of such a theoretical analysis show
sufficiently good agreement with experiments. Using this model the constitutive relations will
be expressed in the frequency domain as follows:
p "E*(u)e "E(u)[1#ig(u)]e ,
Copyright 1999 John Wiley & Sons, Ltd.
E(u)
g(u)"
E(u)
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
TRANSIENT RESPONSE ANALYSIS OF STRUCTURES
395
where p and e are the amplitude of the harmonically time-dependent stress and strain,
respectively, E* is the complex modulus of elasticity, E, E are the real and imaginary parts of the
complex modulus of elasticity, g is the loss factor and u is the frequency. It is necessary to note
that the storage and loss moduli in this case are defined directly in the frequency domain by
experimental technique for each material and can be used without any transformations in the
numerical analysis.
The forced vibration equation of a structure with viscoelastic damping using the complex
modulus model appears as follows in matrix form:
MX® *#K*(u)X*"F(t)
where M is the mass matrix, K*(u)"K(u)#iK(u) is the complex stiffness matrix. K(u) is
determined using the elastic E(u) and shear G(u) moduli, while K(u) is found using the
imaginary parts of the complex moduli E(u)"g (u)E(u) and G(u)"g (u)G(u), where
#
%
g (u) , g (u) are the material loss factors in tension and shear, respectively and u is the frequency.
#
%
X*, X® * are the complex vectors of displacements and accelerations, F(t) is the load vector. The
transient response of the system, described above, cannot be obtained effectively applying direct
integration methods or modal superposition method, because in this case it is not possible to
determine the variation of the material properties E*(u) and G*(u) with respect to time. The
time-domain behaviour of a structure may be obtained from the frequency-domain response by
the Fourier transform technique.
The method proposed is based on the assumption that any complex input signal can be
interpolated by trigonometric polynomials. It is more convenient to use for this purpose the
Fourier transform to find the frequency spectra of excitation
F*(u )"F[F(t )]
H
I
where t is a set of discrete times for the excitation F(t) and for the response X*(t), u is a set of
I
H
discrete frequencies for the frequency spectra of excitation F*(u) and for the frequency response
X*(u). The response of the structure for each trigonometric component is calculated exactly using
the matrix of transfer functions. Incidentally it is necessary to solve the following system of
complex linear equations:
[K*(u )!uM]X*(u )"F*(u )
H
H
H
H
The displacements of structure in the time domain can be obtained by the inverse Fourier
transform
X*(t )"F\[X*(u )]
I
H
3. DISCRETE FOURIER TRANSFORM
Numerical realization of the Fourier transform is performed by the routine using a variant of the
fast Fourier transform algorithm known as the Stockham self-sorting algorithm, which takes
advantage of the cyclic repetition of the complex exponentials in the discrete Fourier transform
and drastically reduces the number of calculations required. This routine is designed to be
particularly efficient on vector processors. The transform of m sequences, each containing
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
396
E. BARKANOV
n complex values, is defined by
1 L\
z*N e\LJQL
z*N"
Q
(n J J
where s"0, 1, . . . , n!1; p"1, 2, . . . , m. The inverse Fourier transform is defined using
a positive sign in the exponential term.
Before using the program to evaluate transient response of structures, several calibration
functions, having theoretically exact Fourier transforms, were used to develop confidence in the
results. To produce equivalence between the continuous and discrete transforms it is necessary to
change the scale factors. Then the discrete Fourier transform pair for a separate sequence will has
the following form:
*u ,\
F*(u )"F[F(t )]"
F*(t ) e\LHI,
H
I
I
2n
I
,\
X*(t )"F\[X*(u )]"*t X*(u ) eLHI,
I
H
H
H
where N is the number of samples. Obviously, the accuracy of the discrete Fourier transform
depends on the number of samples N and the sampling interval *t. The choice of *u and
N depends on the frequency response shape, the accuracy needed and the computing capacity
available. The frequency interval *u for the inverse transform must be the reciprocal of the total
time record length and equals to *u"2n/N*t.
It is necessary to note that the value of function at a discontinuity must be defined as the
midvalue if the inverse Fourier transform is to hold. Moreover, using discrete Fourier transform,
it is necessary to remember that it is based on the assumption about periodicity of load applied.
For periodic functions with known periods, it is necessary to choose N*t interval equal to
a period or integer multiple of a period. For those cases, where the period of a periodic function is
not known, the concept of a data-weighting function or data window must be employed. For
the non-periodical loads, the period of load can be expanded by addition of long interval for
a zero loading.
4. NUMERICAL RESULTS
Some test problems and simple finite element examples are given to illustrate the application of
the method.
4.1. Test problems
As a test problem, the transient response of a single-degree-of-freedom structure (Figure 1) with
high damping under impulse loading F(t)"Fd(t) is examined. At this case it is possible to obtain
the analytical solution of the problem:
X(t)"
F (k!mu)cos ut#kg sin ut
du
n
(k!mu)#kg
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
TRANSIENT RESPONSE ANALYSIS OF STRUCTURES
397
Figure 1. A single-degree-of-freedom structure
As in the literature, the amplitude of impulse is F"1 N and the mass of structure is
m"0)007382 kg. Two variants with different stiffness and loss factor are studied, where for
the first case is taken the following values: k"2910 N/m and g"1)40, but for the
)
second—k"113 (4#f )) and g"1)4 e\)D " . Last expressions are convenient approximation of the stiffness and loss factor in the frequency range f"0, . . . , 10 000 Hz for the
3M-647 material. With the same success, another expressions describing the experimental
dependencies can be used. The first case presents values calculated at the frequency f"100 Hz.
The integral examined is taken numerically. In our algorithm, N"20 000 and *t"0)0001 s are
chosen resulting in *u"p s\.
The transient response of a single-degree-of-freedom structure for two cases is presented in
Figure 2, where the present solutions are in a very good agreement with the analytical results. It is
found from the Figure 2 that only case, where stiffness and loss factor of structure are taken as
constant values, has the non-causal effect. It can be explained that for all, physically real materials
and systems, the stiffness and loss factor should be dependent from the vibration frequency in
a sufficiently wide frequency range and never be constant values. Therefore, effect examined do
not concerns to the real problems and appears only as a consequence of extraordinary strong
approaches. If stiffness and loss factor is presented by functions obtained from the experiments,
then non-causal effect is absent with the exception of case, when the selected functions or the
experimental data are themselves in error.
In all likelihood, the correct approach for a constant complex modulus model consists from the
solution of initial problem and using of the Laplace transform. At this case, the boundary of
integration in the inverse Laplace transform is removed to the upper semiplane of the complex
variable so, that all peculiarities of the integral expression lie lower of this boundary.
4.2. Numerical examples
As a numerical example illustrating the efficiency of the method, a transient response analysis
of a sandwich cantilever (Figure 3) with width b"12 mm and length ¸"720 mm has been
presented. The thickness of layers: h "4 mm, h "1)7 mm, h "4 mm. The external layers are
aluminium and have the following properties: E"71 000 N/mm, t"0)32, o"0)28;10\
Ns/mm. The damping materials C-1002 (EAR Corporation) or ISD-112 (3M Company) with
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
398
E. BARKANOV
Figure 2. Transient response of a single-degree-of-freedom structure: (a) constant complex modulus model; (b) frequencydependent complex modulus model
Figure 3. A sandwich cantilever tested (MP—measurement point)
t"0)49 and o"0)13;10\ Ns/mm are chosen from the literature to simulate the sandwich
core. Expressions describing the shear modulus and loss factor of these materials for the
frequency range f"0, . . . , 1000 Hz are given below:
(1) frequency-independent model of the damping material C-1002
G"3)5 N/mm
g "g "1)436
%
#
(2) piecewise linear frequency-dependent model of the damping material C-1002
f)80 Hz: G"0)4#0)0336 f N/mm; g "g "0)02 f
%
#
f'80 Hz: G"1)46#0)0201 f N/mm; g "g "1)6
%
#
(3) frequency-dependent model of the damping material C-1002
G"44)4!17)6/z N/mm,
where z"0)4#0)0003f
g "g "1)643!0)6025z!0)2557;10\/z#0)1260;10\/z!0)1959;10\/z
%
#
where z"0)05#0)000475f
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
TRANSIENT RESPONSE ANALYSIS OF STRUCTURES
399
(4) frequency-dependent model of the damping material ISD-112
G"e) D \ ) N/mm
g "g "e\) D > )
%
#
Three first cases present approximations of the experimental curves with different accuracy,
third dependencies describe them more exactly. Two first models remind expressions describing
the damping materials used in the papers\ with the exception of the second case where the
shear modulus is examined as frequency-dependent value. Fourth dependencies based on the
experimental data is taken from the paper. Shear modulus and material loss factor for all
cases are shown in Figure 4. The excitation point and response measurement point are located at
the centre of the beam. The same load, rectangular shape impulse (Figure 5(a)), was applied in all
cases. The cantilever beam is discretized with four sandwich beam finite elements.
In the numerical treatment, N"2000 and *t"0)001 s are chosen resulting in *u"n s\.
The time interval is selected in order that the response of the system vanishes at this time.
Exception presents only case 2, where the vanished response of the structure was obtained at the
Figure 4. Material damping characteristics: (a) dependence of shear modulus on frequency; (b) dependence of loss factor
on frequency
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
400
E. BARKANOV
Figure 5. Applied load and transient response of sandwich cantilever: (a) rectangular shape impulse; (b) frequencyindependent model of the damping material C-1002; (c) piecewise linear frequency-dependent model of the damping
material C-1002; (d) frequency-dependent model of the damping material C-1002; (e) frequency-dependent model of the
damping material ISD-112
time 2)5 s, what requires the following values: N"2500; *t"0)001 s; *u"0)8n s\. The load
and the transient response are shown in Figure 5. Some non-causal effect can be seen for
case 1 (Figure 5(b)), where a very considerable approach, describing the material properties, is
used. However, already for the case 2 (Figure 5(c)), this non-causal effect is very small. But for
cases 3 and 4 (Figures 5(d) and 5(e)), more exactly presenting the real experimental dependencies
of shear modulus and material loss factor, this non-causal effect do not appears. It is necessary to
note that the static component (u "0) is deleted from the results presented.
H
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
TRANSIENT RESPONSE ANALYSIS OF STRUCTURES
401
Figure 6. Applied load and transient response of sandwich cantilever: (a) trapezium shape impulse; (b) frequencydependent model of the damping material C-1002
Figure 7. Applied load and transient response of sandwich cantilever: (a) saw graphic impulse; (b) frequency-dependent
model of the damping material C-1002
The problem of non-causal effect was studied in addition. Its value was defined as ratio of the
value of transient response before beginning of the load applied to the maximum displacement of
transient response. It was established that the major source of non-causality is approximate
mathematical modelling of material properties: shear modulus and loss factor. Also, it was
defined that the value of non-causal effect for the same constant shear modulus and loss factor is
different for various structures and load applied. It can be explained only by the accuracy of the
discrete Fourier transform, which is connected with the number of samples and sampling interval.
Since, to change these values only, it can be obtained a more accurate solution for the frequency
spectra of excitation and more accurate description of the frequency response of structures.
Results of transient response analysis for another types of loading using the frequencydependent model of the damping material C-1002 are shown in Figures 6 and 7. At these cases the
response measurement point is located at the tip of the beam. Also, as at the previous numerical
examples (Figures 5(d) and 5(e)), the non-causal effect is absent now (Figures 6(b) and 7(b)). So to
preserving the exact experimental dependence of shear modulus and material loss factor from the
frequency it can be possible to obtain the causal transient response of structure made from
viscoelastic material.
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)
402
E. BARKANOV
5. CONCLUSIONS
The present approach was developed with the aim to use his as a universal tool in the transient
finite element analysis of different type structures made from viscoelastic materials. Examples are
transient vibrations of aeronautical, ship and automobile structures and its parts made from
sandwich and laminated composite panels with viscoelastic layers. This technique can be used
successfully also to high- and low-velocity impact problems, since gives the possibility to take into
consideration a sufficiently wide frequency range, what is very important in the impact analysis.
Material data in the frequency domain are taken into consideration, what gives the possibility to
use straight data from experiments without any transformations. The major source of noncausality—the approximate mathematical modelling of material damping properties is successfully overcome by using the exact mathematical representation not only for a loss moduli, but
also for a storage moduli in the present implementation. Test problems and numerical results
indicate the efficiency of the present technique.
ACKNOWLEDGEMENTS
The author gratefully acknowledge support of this research work from the ‘Konferenz der
Deutschen Akademien der Wissenschaften’ and from the Latvian Council of Science under
Program No. 96.p.III-1.04.
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Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 44, 393—403 (1999)