PROCEEDINGS OF THE LATVIAN ACADEMY OF SCIENCES. Section B, Vol. 69 (2015), No. 6 (699), pp. 326–330.
DOI: 10.1515/prolas-2015-0048
A NUMERICAL SOLUTION OF HEREDITARY EQUATIONS
WITH A WEAKLY SINGULAR KERNEL FOR VIBRATION
ANALYSIS OF VISCOELASTIC SYSTEMS
Botir Usmonov
Faculty of Aviation, Tashkent State Technical University, Universitetskaya 2, 100095, Tashkent, UZBEKISTAN;
[email protected]
Communicated by Isaak Rashal
Viscoelastic, or composite materials that are hereditary deformable, have been characterised by
exponential and weakly singular kernels in a hereditary equation. An exponential kernel is easy to
be numerically implemented, but does not well describe complex vibratory behaviour of a hereditary deformable system. On the other hand, a weakly singular kernel is known to describe the
complex vibratory behaviour, but is nontrivial to be numerically implemented. This study presents
a numerical formulation for solving a hereditary equation with a weakly singular kernel. Recursive
algebraic equations, which are numerically solvable, are formulated by using the Galerkin method
enhanced by a numerical integration and elimination of weak singularity. Numerical experiments
showed that the present approach with a weakly singular kernel is well fitted into a realistic vibratory behaviour of a hereditary deformable system under dynamic loads, as compared to the same
approach with an exponential kernel.
Key words: viscoelastic system, vibration analysis, hereditary equation, weakly singular kernel,
Galerkin method.
INTRODUCTION
With development in aeronautical, civil, and automotive engineering, the mechanics of composite materials have made
remarkable progress. Strong demand for light and reliable
structural elements made from advanced composite materials motivates the development of better mathematical models of viscoelastic systems. Also, the development of numerical solutions for vibratory analysis of viscoelastic
structures is an important issue.
Many properties of a composite material are described by
distinct viscoelastic properties (Flugge, 1975; Cupial and
Niziol, 1995; Usmonov et al., 2007). Viscoelastic systems
made of composite materials in a different structure have
been described previously (Christensen, 1982; Chandiramani, 1989; Zenkour, 2004; Menon and Tang, 2004). Even
if the problems were solved concerning viscoelasticity,
most of the viscoelastic characteristics were only taken into
account in a restricted context (DiTarano, 1965; Muravyov,
1997; 2004). In those studies, a viscoelastic material was
characterised by a hereditary equation with an exponential
kernel. The mathematical equation with the kernel is limited
as it does not well describe a realistic vibratory behaviour
under dynamic loads (Badalov and Ganikhanov, 2002;
Usmonov and Badalov, 2004). A hereditary equation with
weakly singular kernels of Koltunov, Rzanitsyn, Abel, and
326
Rabotnov (Badalov and Usmonov, 2004) has shown to
overcome the limitation, but their numerical implementations are known to be nontrivial. The theory introduced by
Badalov and Usmonov (2004) provides an idea to numerically solve the hereditary equation with an Abelian-type
weakly singular kernel.
Such a hereditary behaviour can be characterised by the following constitutive equation in an integral form
t
é
ù
s(t) = E êe(t) - ò R (t - t) e(t) dtú,
0
ë
û
(1)
where t is the present time, t is the past time, s is the stress,
e the strain, E is the instantaneous or prolonged modulus of
elasticity, and R(t – t) is referred to as the relaxation kernel.
Viscoelastic properties have been addressed by using exponential relaxation kernels (Usmonov at al, 2007) as
R (t - t) = Ae - b ( t - t ) ,
(2)
where A is the viscosity parameter and b is the relaxation
coefficient. The kernel (2) may cause some errors in the initial stage when small b is used in the integral constitutive
equation (1). The errors in the initial stage affect the final
result. Even if errors exist, this kernel has been frequently
used due to easy numerical implementation, such as a
Laplace transform based method (Menon and Tang, 2004).
Proc. Latvian Acad. Sci., Section B, Vol. 69 (2015), No. 6.
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In order to remove the errors, researchers have used weak
singular kernels of Abel (Potapov, 1985; Usmonov, 2009).
R (t - t) = Ae - b ( t - t ) (t - t) a -1 , A > 0, b > 0, 0 < a < 1
(3)
where a is the singularity coefficient, b is the relaxation coefficient and A is a viscosity parameter and where t is the
present time and t is the past time. The advantage of this
kernel is ability to describe the creep deformation and stress
relaxation of viscoelasticity with the three parameters. The
three rheological parameters in the weakly singularity kernel are utilised to provide realistic behaviour of a hereditary
deformable system. However, its numerical implementation
has been nontrivial when this kernel is used in a hereditary
equation.
In this study, we present a numerical solution based on a
Galerkin method for solving a hereditary equation with a
weakly singular kernel. The Galerkin method can formulate
recursive algebraic equations, which is numerically solvable
by using the numerical integration and an elimination technique of weak singularity. The vibratory responses by the
present approach with a weakly singular kernel are compared with that with an exponential kernel by a simple wing
model and the singular parameter among the three parameters is explored in detail.
NUMERICAL FORMULATION OF A HEREDITARY
EQUATION
This section describes the numerical formulation of a hereditary equation with a weakly singular kernel for the
viscoelastic system of a wing model. The wing in Figure 1
is modelled as a console structure with variable width (or a
span of wing) b(x ) and thickness h(x ). Considering bending
moment M of the wing
M = -EI (x )Wxx , I (x ) =
b(x ) h 3 (x )
,
12
(4)
where W (x , t) is the in-plane bending deformation, Wxx is
bending, E is elastic modulus and I(x) is the moment of inertia.
When we consider the wing model as a viscoelastic structure, Eq. (4) is rewritten based on the constitutive law of (1)
as
t
é
ù
M = -EI (x ) êWxx - ò R (t - t)Wxx (x , t) dtú
0
ë
û
(5)
Fig. 1. Wing 3-D model.
t
mWtt + LW - ò R (t - t)LW (x , t) dt = f (x , t),
0
LW =
¶2
¶x 2
é ¶2 W ù
êEI ¶x 2 ú,
û
ë
where R (t - t) is chosen as the weakly singular kernel in (3).
This equation is utilised for solving the viscoelastic system,
which is hereditary deformable, and then is referred to as
the hereditary equation. The boundary conditions and initial
conditions are
W = Wx = 0, at x = 0, M = M x = 0 at x = l
W
t=0
= a(x ), W&
t=0
(8)
= b(x ) ,
(9)
where a(x ) and b(x ) are given functions. Instead of employing an exponential kernel for the viscoelastic systems
(Muravyov, 1998), Menon, Tang 2004; Eq. (7) uses a
weakly singular kernel. Due to the weak singularity, it has
been a challenging problem to solve the equation numerically. Based on the theory of Badalov (Badalov and
Usmonov, 2004), we present a numerical formulation by using the Galerkin method enhanced by the numerical integration and an elimination technique of weak singularity. First,
an approximate solution of equation (7) is sought in the
form
N
W (x , t) = å Uk (t) j k (x ) ,
(10)
k =1
where Uk (t) is an unknown time function, which must be
computed; j k (x ) is a shape function, which satisfies the
boundary conditions of the equation (8). N is the number of
the given functions. Substituting expression (10) into (7)
and multiplying the equation integrated with respect to x on
the interval [0,l] by j j yields
N
ì
å ím
k =1
î
ki
t
ü
&& + w2 éU (t) - R (t - t)U (t) dtù ý = f (t), (11)
U
k
ki ê k
k
i
ú
ò
0
ë
ûþ
l
l
0
0
It is known that the equation of bending motion is described
by the following expression
where m ki = ò m (x )j k j i dx, and w2ki = ò (Lj k ) j i dx.
M xx + f (x , t) = m (x )Wtt ,
Choosing the shape function such that
(6)
where m (x ) is wing mass, f (x , t) is a given external load
vector, Mxx is momentum along axis x, and Wtt is bending.
Then equation of motion for the viscoelastic wing model
should be described as
(7)
ì w2
w2k ,i = í k
î 0
if
if
i=k
ì 1 if
and m k ,i = í
i¹k
î 0 if
i=k
i¹k
(12)
then the system of integral equations (11) becomes
327
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t
&& t + w2 éU (t) - R (t - t)U (t) dtù = f (t) k = 1,.. . ,N, (13)
U
k
ki ê k
k
k
ú
ò0
ë
û
and initial conditions become
Uk (0) = a k and U& k (0) = bk .
(14)
Hence unknown functions Uk (t) are computed by the integral equations (13). Solving exact solutions of the integral
equations (13) in the existence of the weakly singular kernels faces mathematical difficulties. Therefore, we solve the
equations (13) at initial conditions (14) by twice integrating
at t.
t
s
é
ù
Uk (t) + w2k ò (t - s) êUk (s) - ò R (t - t)Uk (t) dtú ds =
0
0
ë
û
t
= ò (t - s) f k (s) dt + a k + bk t. (15)
0
By introducing terms t = t i , t i = iDt, i = 0,1,..., where Dt
is
the time increment and inserting Uk (t i ) = Uk ,i and
ti
(
t
ò i - s) f k (t) dt = f k ,i into Eq. (15), we have
i -1
A i
é
ù
Uk ,i = -w2k å C j (t i - t) êUk , j (s) - å Ds e - bts Uk , j - s ú +
a s =1
j=0
ë
û
+ f k ,i + a k + bk t i (19)
The recursive algebraic formula (19) computes Uk ,i ,
i = 0 ,1,2 ,... . Finally, the computed Uk , j are substituted into
Eq. (10) to obtain the approximate solution of the hereditary
equation.
NUMERICAL EXPERIMENTS
The numerical formulation will be verified and the hereditary equation with an Abelian-type weakly singular kernel
will be compared to that with exponential kernel in this section. All results were obtained by using MATLAB14.
Example. We chose a one-dimensional viscoelastic system
in Figure 2 as a special problem to validate the proposed numerical formulation. The thickness and the width are assumed to be constant in the example and then the mass and
the moment of inertia become constant. The load at end of
the tip of the beam is given by f (t) = q cos wt, where q is a
amplitude and w is a frequency.
0
i
Uk ,i + w2k å
j =1
tj
s
é
ù
(
)
(
)
t
s
U
s
òt -1 i êë k ò0 R (t - t)Uk (t) dtúû ds =
j
= f k ,i + a k bk + bk t i (16)
As observed, equation (19) can have general solution characteristics. Namely, the solution of the perfect elastic sysf (t)
In Eq. (16) the external integral of the second term is computed by the trapezoid numerical integration and then we
have
s
i -1
é
ù
Uk ,i + w2k å C j (t i - t) êUk , j (s) - ò R (t i - t)Uk (s) dtú ds =
j=0
0
ë
û
= f k ,i + a k + bk t i (17)
Dt
and C j = jDt, j = 1,... , i. The integral term in
2
Eq. (17) can be described by
where C 0 =
ti
t1
t2
ti
0
t0
t1
ti- 1
ò R (ti - t)Uk (t) dt = ò ( ) + ò ( ) +...+ ò ( ) =
i
=å
ts
ò R (t
i
- t)Uk (t) dt.
s =1 t
s- 1
An elimination technique of week singularity is applied with
1
a
replacing t j - t = z and each integral term is computed by
using the trapezoid numerical integration (Badalov, 1980).
Then Eq. (17) becomes
i -1
A
é
ù
Uk ,i + w2k å C j (t i - t) êUk , j (s) - å Ds e - bts Uk , j - s ú =
a s =1
j=0
ë
û
= f k ,i + a k + bk t i , (18)
where Ds =
328
i
(Dt) a
[(s + 1) a - (s - 1) a ]. Eq. (18) is rewritten by
2
Fig. 2. One-dimensional viscoelastic system.
tem is obtained by setting A = 0. If A ¹ 0, solution for the
exponential kernel is computed by setting a = 1 and that of
Abelian kernel is obtained by choosing value in 0 < a < 1.
Previous research by Badalov, 2002 showed that the influence of variation of b is not considerable on the damping effect so that we fixed the value of b in subsequent experiments. A study by Menon (Menon and Tang, 2004) used
only an exponential kernel by choosing b as 1 and showed a
reasonable solution. Here we chose a smaller value, 0.05 as
a special case for the next comparison.
CONSTANT LOAD
In case of constant load, initial conditions and parameters
are given by a = 0, b = 0.0, b = 0.05, w = 0.0, q = 1.
As it is well known, in viscoelasticity the hereditary deformable systems under constant load follow the curve of
the creep response that is damped by time. Also, the period
becomes longer. These facts have been observed for the
viscoelastic system (Chandiramani at al., 1989).
First, the vibratory behaviour of an exponential kernel by
increasing A from 0 to 0.1 were depicted in Figure 3.
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a
Fig. 3. Curves of vibratory responses with exponential kernel.
Figure 3 shows that the vibration is more damped as A increases. However, the periods are not changed and the displacement diverges as A increases. Therefore, an exponential kernel shows limitation in analysis of the realistic
vibratory behaviour of viscoelasticity as afore-mentioned.
Next, we picked up the singularity parameters a (alpha) by
1.5, 2, and 5 times of A when A is given by 0.1 and 0.05 in
order to investigate the vibratory behaviour of an Abelian
kernel. The results for A = 0.1 are depicted in Figure 4a and
those for A = 0.05 in Figure 4b.
Figure 4 indicates that the creep response becomes steeper,
the vibration more damped and the period becomes longer
as a decreases. These characteristics of vibratory behaviour
are very close to realistic responses of viscoelastic material.
Therefore, the weakly singular kernel can provide more realistic responses for a hereditary deformable system. However, as shown in A = 0.1 and a = 0.15 of Figure 4a, the
motion can diverge as the singularity parameter is close to
A. Thus, one needs much caution to choose the singularity
parameter when it is close to A in order to obtain steeper
and damped creep response.
b
Fig. 4. Curves of vibratory responses when (a) viscosity parameter = 0.1
and (b) viscosity parameter = 0.05.
DIFFERENT LOADS
Here, we explore the vibratory behaviour of the hereditary
equation with an exponential kernel and Abelian kernel in
free vibration with initial conditions and vibration by harmonic excited load.
The initial conditions and parameters are given by
a = 1, b = 0.0, b = 0.05, w = 0.0, q = 0.0.
for free vibration and
a = 0, b = 0.0, b = 0.05, w = 1.0, q = 1.0.
for harmonic load with resonant frequency. In Figure 5,
case 1 (A = 0) represents perfect elastic condition, case 2
(A = 0, a = 1) utilizes an exponential kernel and finally case
3 (A = 0, a = 0.2) uses the Abelian kernel.
Fig. 5. Response curves of the free vibration for all cases.
The vibratory behaviour for free vibration are simpler than
those of constant load. An exponential kernel makes the vibration damped by the given A and the period, compared to
the elastic case. When an Abelian kernel with the given alpha is applied, the vibration is more damped and the period
is longer. This shows that the Abelian kernel provides complex, realistic vibratory responses for free vibration.
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Badalov, F. B., Usmonov, B. Sh. (2004). New solution setting for bending-aileron flutter of vehicle. Rep. Acad. Sci. Uzbekistan, 6, 30–33 (in Russian).
Chandiramani, N. K., Librescu, L, Aboudi, J. (1989). The theory of
orthotropic viscoelastic shear deformable composite flat panels and their
dynamic stability. Int. J. Solids Struct., 25 (5), 465–482.
Chandiramani, N. K., Librescu, L. (1989). Dynamic stability of unidirectional fiber-reinforced viscoelastic composite plates. Appl. Mech. Rev., 42
(11), 39–47.
Christensen, R. M. (1982). Theory of Viscoelasticity. Academic Press, New
York. 365 pp.
Cupial, P., Niziol, J. (1995). Vibration and damping analysis of a
three-layered composite plate with a viscoelastic mid-layer. J. Sound Vibr.,
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DiTaranto, R. A. (1965). Theory of vibratory bending for elastic and
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Fig. 6. Response curves of the harmonic excitation for all cases.
Flugge, W. (1975). Viscoelasticity. 2nd revised edition, Springer-Verlag,
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Next, vibratory behaviour under harmonic load are depicted
in Figure 6 for case 1 (A = 0), case 2 (A = 0.1, a = 1) and
case 3 (A = 0.1, a = 0.4).
Flory, A., McKenna, G. B. (2004). Finite step rate correction in stress relaxation experiments: A comparison of two methods. Mech. Time-Dependent
Mater., 8, 17–37.
The characteristics of vibratory behaviour for harmonic load
are the same as those of the free vibration. Therefore, the
Abelian-type weakly singular kernel provides more realistic
vibratory response than the exponential kernel for dynamic
load.
In conclusion, we presented a numerical solution based on
the Galerkin method for the hereditary equation of a
viscoelastic system and then explored the creep response for
viscoelastic materials under a constant load. As demonstrated in the example of the one-degree-of-freedom
viscoelastic system, the proposed solution for the hereditary
equation with weakly singular kernel successfully provided
well-fitted responses to realistic vibratory characteristics of
one dimensional viscoelastic problems. The solution also
provided close to realistic responses in cases of harmonic
excited load and free vibration. Therefore, solutions of a hereditary equation with weak singular kernels are more preferable for obtaining realistic responses for dynamic problems of the viscoelasticity.
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Received 22 April 2015
VIENÂDOJUMU AR VÂJO SINGULÂRO KODOLU SKAITLISKAIS RISINÂJUMS IEDZIMTO VISKOELASTÎGO SISTÇMU
VIBRÂCIJU ANALÎZEI
Pçtîjums sniedz vienâdojumu ar singulâro kodolu skaitlisko risinâjumu iedzimto viskoelastîgo sistçmu vibrâciju analîzei. Izmantota
Galerkina metode cieðâkai skaitliskai integrâcijai un vâjas singularitâtes pastiprinâðanai.
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