This document summarizes a study on the vibration analysis of an airfoil model with nonlinear hereditary deformable suspensions. The airfoil is modeled as a two-degree-of-freedom structure with pitch and plunge motions. Weakly singular integro-differential equations are derived to model the nonlinear hereditary behavior and numerically solved using an integration method. Numerical results are presented analyzing the creep response and resonance behavior of the viscoelastic materials in the perfect elastic and viscoelastic suspension cases.

1. International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.1, February 2023
DOI:10.14810/ijmech.2023.12101 1
VIBRATION ANALYSIS OF AIRFOIL MODEL
WITH NONLINEAR HEREDITARY
DEFORMABLE SUSPENSIONS
Botir Usmonov
Tashkent Institute of Chemical Technology, Tashkent, Uzbekistan
ABSTRACT
In the present work, the vibratory behavior of an airfoil is discussed. The airfoil is considered as two-
degree-of-freedom structure with hereditary deformable suspensions. The weak singular integro-
differential equation is numerically solved using numerical integration method. Finally, numerical results
for the creep response and resonance behavior of the viscoelastic materials were analyzed. These results
are obtained for the perfect elastic and viscoelastic suspensions with the nonlinearity feature. As
demonstrated in the airfoil model, the equation of motion with hereditary and nonlinear terms successfully
illustrate realistic vibratory characteristics of two-dimensional viscoelastic problems.
KEYWORDS
Vibration analysis, hereditary deformable, integro-differential equation, viscoelastic
1. SYMBOLS
U Potential energy
T Kinetic energy
L Laplace description
Q Generalized coordinate
a No dimensional distance from the mid chord to the elastic axis
b Semi chord of wing (reference length)

c pitch degree of freedom structural damping coefficient
h
c Plunge degree of freedom structural damping coefficient
z No dimensional distance from mid chord to control surface leading edge
g Acceleration due to gravity
h Plunge displacement coordinate

I mass moment of inertia about the elastic axis

k pitch degree of freedom structural spring constant
h
k Plunge degree of freedom structural spring constant
)
(t
L Lift of the wing
)
(t
M Moment of the wing about the elastic axis
m Mass of the wing
u free stream velocity

x no dimensional distance between elastic axis and the center of mass

2. International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.1, February 2023
2
 pitch displacement coordinate
h
 Static coefficient of friction in the plunge direction
 static coefficient of friction in the pitch direction
 density of air
ab Distance between mid-chord and rotation point (elastic axis)

X distance between elastic axis and c.g. point
2. INTRODUCTION
Due to strong implications of various nonlinearities and “hereditary deformable” nature of the
airfoil model, the stability of the airfoil cannot longer be analyzed within linearized theory. In
order to investigate this behavior, the governing equations have to be considered in a nonlinear
form. The Volterra series-based theory gives important information about the effects of
nonlinearities of the structures (see Volterra, V., 1959; Lee, Y.M. et al., 1964; Schetzen, M.,
1980; Silva, W.A., 1991). A number of fundamental contributions related to the application of
Volterra’s series were developed by several researchers (Volterra, V., 1959; Silva, W.A., 1991;
Badalov, F.B., 1987; Rabotnov, Yu. N., 1977; Volmir, A. S., 1972; Koltunov, M. A., 1976).
There are two schools of thoughts in solving viscoelastic problems, where involved the hereditary
relation. First is based on numerical integrations in time domain, and the second is involving
Laplace transform to the frequency domain (see Muravyov A., 1998). The basic idea is using
numerical integration method to introduce elimination of singularities based on the method of
Badalov (Badalov, F.B., 1987).
In present work is developed a mathematical model, where rheological and heredity property of
materials are taken in account for the deformation-stress analysis of structure. In the theory of
linear viscoelasticity, one of the hereditary deformable models is a constitutive equation of the
form (a one-dimensional structure is taken as a demonstrative example)
       
0
t
t E t R t d
     
 
  
 
 
 , (1)
where  is the stress,  is the strain, E is the instantaneous Young’s modulus, and  
R t 
 is
referred to as the relaxation kernel. Viscoelastic properties have been addressed by using
exponentialrelaxation kernels [1], [2] and [5] as
   
t
R t Ae  
  
 
In this work the vibratory processes based on application of Volterra principle, which allows
solve viscoelastic problems with heredity properties of material. where A is the viscosity and 
is the relaxation parameter. The kernel (6) may cause some errors in initial stage when small 
is used in the integral constitutive equation (2). The errors in the initial stage results impact on
final result. Even if the errors exist, the kernel has been frequently used due to easy numerical
implementation such as a Laplace transform based method [10].
In order to remove the errors, researchers have used weak singular kernels of Koltunov,
Rzhanitsyn, Abel, Rabotnov [13] [14].
   
  1
0
,
0
,
0
,
1













 


A
t
Ae
t
R t
,

3. International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.1, February 2023
3
where  is the singularity parameter. Advantages of these kernels are known to describe
simultaneously the creep deformation and stress relaxation of viscoelasticity with difference three
parameters. However, its numerical implementation has been an issue
In this work, we present a numerical solution technique based on a numerical integration method
for solving a hereditary equation with a weakly singular kernel. Recurrent algebraic equations of
the hereditary deformable system are formulated and eliminated weak singularity. The vibratory
responses by the present approach with weakly singular kernel are compared with those by an
exponential kernel in a simple wing model and the influence of three parameters is explored as
well.
3. EQUATIONS OF MOTION
Consider coupled bending-torsion vibration of an airfoil model on nonlinear hereditary deformable
suspensions. Assumed, that an airfoil motion is characterized by two generalized coordinates, such as pitch
angle  and plunging h . Airfoil is excited under initial load. Motion of the airfoil model can be
represented as in Figure 1, where the model of an airfoil has a translational spring with
coefficient stiffness- 1
c and a torsion spring with a coefficient of stiffness - 2
c . The virtual
springs are located in the elastic center. Therefore, the model of airfoil has two degrees of
freedom-( , h ).
Potential energy of the airfoil model. If 2
1, c
c are respectively, the spring coefficients of the
bending and torsion springs, then the potential energy is
 
 
)
2
1
(
2
1
2
1 2
2
2
1





 R
h
c
R
c
P  (5)
Kinetic energy of the airfoil model. The kinetic energy of the airfoil can be presented as
follows:
 
2
22
12
2
11 2
2
1
h
a
h
a
a
K 


 

 
 , (6)
where  
2
2
22
21
12
11 ,
, b
r
m
a
mb
a
a
m
a 




 .
Then Lagrange’s equations in generalised coordinates can be written as
2
1
q
h
L
dt
d
h
L
q
L
dt
d
L















 , (7)
where K
P
L 
 .
Representing equation (7) in classical form, one obtains
2
3
2
22
21
1
3
1
12
11
)
)(
1
(
)
)(
1
(
q
h
h
R
c
h
a
a
q
R
c
h
a
a

























(8)
From system of integro-differential equations (8) follows, that coupling of bending-torsion

4. International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.1, February 2023
4
vibration determined by coefficient b . When center of gravity coincides with axis of stiffness ( 0

b ),
then system (8) uncoupled into two independent equations. First equation will describe vertical vibration,
and second equation describes torsional vibration of the airfoil of wing.
It is necessary solve equation (8) under given initial conditions.
Assuming 21
12
1
11 , c
c
с
с 
 and 2
22 c
c  the solution of Eq. (8) is obtaining by numerical
integration method. We use following parameters for dimensioning the Eq. (8)
2
1
2
1
2
2
2
1 ,
,
;
,
2
,
2
1
b
c
c
t
m
c
t
w
u
bh
b
b


















 



 .
Thus, equation of motion (8) can be described as following
   0
)
(
)
(
1
0
)
(
)
(
)
1
(
3
3
2
3
1
2
3
1


























u
u
w
w
R
r
b
u
u
u
w
w
R
w












, (9),
Theinitial conditions are 0
|
,
|
,
0
|
,
| 0
0
0
0
0
0 


 


 t
t
t
t u
u
u
w
w
w 
 .
Then applying the method described in [2] into system of equations (9), one obtains
 
 


























































j
k
k
j
k
j
k
j
k
j
t
k
j
j
j
j
n
j
j
n
j
n
j
k
k
j
k
j
k
j
k
j
t
k
j
j
j
j
n
j
j
n
j
n
u
u
w
w
e
B
a
u
u
w
w
t
t
C
b
u
u
u
u
w
w
e
B
a
u
u
w
w
t
t
C
w
w
k
k
0
3
3
3
3
1
1
0
2
0
0
3
2
3
1
3
2
3
1
1
0
0
)
(
)
(
)
(
)
(
)
(
2
)
(
)
(
)
(
)
(
)
(



















(10),
where:
 
 
 
 
a
a
a
j
a
a
a
a
k
n
j
n
n
n
n
n
j
j
t
B
t
B
j
k
k
k
t
B
t
C
C
n
j
t
C
u
t
u
w
t
w
t
n
t
1
2
,
2
1
,
1
,
)
1
(
1
2
,
2
1
,
1
,
,
)
(
,
)
(
,
0
0
























The Eq. (10) is the mathematical model, which describes coupled bending-torsion vibration of the airfoil
model. Calculation of the Eq. (10) is providing as for linear/nonlinear perfect-elastic, as well linear/
nonlinear viscoelastic cases.
4. NUMERICAL EXAMPLES
Numerical implementation of the problem illustrated in Fig.1 is performed for linear/nonlinear elastic and
linear/nonlinear viscoelastic cases.

5. International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.1, February 2023
5
4.1. Free Vibration Analysis without Hereditary Form
  0
)
(
)
(
0
)
(
)
(
3
3
2
3
1
2
3
1






















u
u
w
w
r
b
u
u
u
w
w
w












Theinitial conditions and parameters aregiven by
0
|
,
|
,
0
|
,
| 0
0
0
0
0
0 


 


 t
t
t
t u
u
u
w
w
w 
 ,
1
.
0
;
1
;
0
.
0
;
00025
.
0
;
0
.
0
;
06
.
0
,
5
.
3
;
25
.
0
;
1
.
0
2













 t
a
b
r



 .
Figure 1A. Linear elastic case
Figure 1B. Nonlinear elastic case

6. International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.1, February 2023
6
4.2. Free Vibration Analysis with Hereditary Form
Theinitial conditions and parameters aregiven by
0
|
,
|
,
0
|
,
| 0
0
0
0
0
0 


 


 t
t
t
t u
u
u
w
w
w 

1
.
0
;
1
;
05
.
0
;
25
.
0
;
1
.
0
;
06
.
0
,
5
.
3
;
25
.
0
;
1
.
0
2













 t
a
b
r




Figure 1C. Linear viscoelastic case
Figure 1D. Nonlinear viscoelastic case
5. CONCLUSIONS
Proposed weak singular integro-differential equation provides a good solution of the nonlinear
viscoelastic (hereditary) system. Through numerical implementation, the dynamic response
explored for viscoelastic materials. Also compared the vibratory behaviors of perfect elastic and
viscoelastic material with the nonlinearity feature. As demonstrated in the example of the airfoil

7. International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.1, February 2023
7
model, the mathematical equations with hereditary and nonlinear terms successfully analyze
realistic vibratory characteristics of two-dimensional viscoelastic problems.
ACKNOWLEDGEMENTS
The authors would like to thank everyone, just everyone!
REFERENCES
[1] Badalov, F.B., (1980) Method of series in nonlinear hereditary theory of viscoelasticity, Fan,
Uzbekistan. (in Russian)
[2] Badalov, F.B., (1987) Methods of solution of integral and integro-differential equations of hereditary
viscoelasticity, Mexnat, Uzbekistan, 1987. (in Russian)
[3] Badalov, F.B., Ganikhanov, Sh.F., (2000) Vibration of hereditary-deformable elements of structure of
flying vehicles, TSAI, Uzbekistan. (in Russian)
[4] Badalov, F. B., Usmonov, B. Sh., (2004) “New solution setting for bending-aileron flutter of
vehicle,” Reports of Academy of Science of Uzbekistan No. 6, pp30-33. (in Russian)
[5] Badalov, F. B., Eshmatov, Kh., Yusupov, M., (1987) "About some methods of the decision of
systems integro-differential equations meeting in problems viscoelasticity", Soviet Appl. Math. Mech.
No. 51, pp867-871.
[6] Goroshko, O.A., Pushko, N.P., (1997) “Lagrangian equations for the multi body hereditary systems,”
Facta Universitatis. Series: Mechanics, Automatic, Control and Robotics Vol.2, No 7, pp209-222.
[7] Lee, Y.M. et al., (1964) Selected Papers of Norbert Wiener, Cambridge, Mass.: M.I.T. Press.
[8] Il'yushin, A. A., Pobedrya, B. E., (1970) Fundamentals of the mathematical theory of
thermoviscoelasticity, Nauka, Moscow. (in Russian)
[9] Muravyov, A., (1998) “Forced vibration responses of a viscoelastic structure.” J Sound Vib 218(5),
pp892-907.
[10] Koltunov, M. A., (1976) Creep and relaxation, Visshaya shkola, Moscow. (in Russian)
[11] Potapov, V. D., (1985) Stability of viscoelastic elements of designs, Stroyizdat, Moscow. (in Russian)
[12] Rabotnov, Yu. N., (1977) Elements of hereditary mechanics of solids. Moscow. (in Russian)
[13] Rzanitsyn, A. R., (1968) Theory of creep, StoryIzdat., Moscow. (in Russian)
[14] Schetzen, M., (1980) The Volterra and Wiener Theories of Nonlinear Systems, John Wiley & Sons
Interscience, New York.
[15] Silva, W.A., (1991) “A methodology for using nonlinear aerodynamics in aeroservoelasticity analysis
and design,” AIAA paper 91-1110-CP.
AUTHOR
Professor of Tashkent Institute of Chemical Tehnology, PhD in education management,
MSc..Mechatronics and Space Technology. Published more than 35 scientific papers