=Paper=
{{Paper
|id=Vol-2488/paper3
|storemode=property
|title=On the Mathematical Model of Nonlinear Vibrations of a Biologically Active Rod with Consideration of the Rheological Factor
|pdfUrl=https://ceur-ws.org/Vol-2488/paper3.pdf
|volume=Vol-2488
|authors=Petro Pukach,Volodymyr Il'kiv,Myroslava Vovk,Olha Slyusarchuk,Yulia Pukach,Yuriy Mylyan,Winfried Auzinger
|dblpUrl=https://dblp.org/rec/conf/iddm/PukachIVSPMA19
}}
==On the Mathematical Model of Nonlinear Vibrations of a Biologically Active Rod with Consideration of the Rheological Factor==
https://ceur-ws.org/Vol-2488/paper3.pdf
On the Mathematical Model of Nonlinear Vibrations of a
Biologically Active Rod with Consideration of the
Rheological Factor
Petro Pukach1[0000-0002-0359-5025] , Volodymyr Il'kiv 2[0000-0001-6597-1404],
Myroslava Vovk 3[0000-0002-7818-7755], Olha Slyusarchuk 4[0000-0003-3464-0252],
Yulia Pukach 5[0000-0002-6358-8396] Yuriy Mylyan 6[0000-0002-5518-0956]
and Winfried Auzinger 7[0000-0002-9631-2601]
1,2,3,4,5 Lviv Polytechnic National University, 79013, Lviv, Ukraine
6 Danylo Halytsky Lviv National Medical University, 79010, Lviv, Ukraine
1ppukach@gmail.com, 2ilkivv@i.ua, 3mira.i.kopych@gmail.com,
4olga_slusarchuk@ukr.net, 5ilpach@yahoo.com.ua,
6myp.ct2019@gmail.com
7 Vienna University of Technology, Karlsplatz 13, 1040 Vienna, Austria
w.auzinger@tuwien.ac.at
Abstract. Qualitative and numerical methods of researching nonlinear vibration
systems are used to study the mathematical model of nonlinear vibrations of a
biologically active rod. This model is widely used in biomechanics and medical
research for designing new materials with biofactor elements that possess cer-
tain preset features. Conditions are established for the existence of a unique so-
lution of the boundary value problem for the beam vibration nonlinear differen-
tial equation, in which there is an integral summand with the fourth derivative
by the spatial variables. This summand models the rheological factor in the sys-
tem. The existence of classes of nonlinear rheological vibration systems with
dissipation that have blow-up regimes is stated theoretically. The relation be-
tween nonlinearity indices in such regimes is obtained. The theoretical possibil-
ity of using the Runge-Kutta method for numerical solution of the correspond-
ing boundary value problem is shown. The results are illustrated by a model ex-
ample. The importance of the obtained theoretical assumptions for the practical
modeling, analysis, and synthesis of parameters of technological vibration sys-
tems is shown.
Keywords: Mathematical Model, Nonlinear Vibrations, Galerkin
Method, Biofactor, Rheological System.
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution
4.0 International (CC BY 4.0)
2019 IDDM Workshops.
2
1 Introduction
Mathematical modeling of both normal physiological and pathological processes is
one of the current trends of modern medical research. It is especially important to note
that modern medicine is largely an experimental science with a vast empirical experi-
ence of affecting different diseases with a variety of means. However, more often than
not searching for experimental means of studying different process in biological me-
dia has many flaws due to our inability to limit ourselves to experiment only. There-
fore, mathematical modeling is often the most effective way of studying processes in
living organisms (or their parts).
In medical practice, numerical modeling of biomechanical processes is carried out
on the basis of the continuous media mechanics models and numerical methods of
solving corresponding systems of partial differential equations.
Mathematical modeling methods can narrow down the search of optimal system
parameters significantly. After such parameter optimization, experimental research
can be carried out with much more information about the functioning of a biological
system. The development of the mathematical modeling framework involves build-
ing a closed mechanical-mathematical model of the process that describes the be-
havior of a biological medium on the basis of equations in partial derivatives and the
continuous medium mechanics principles. In addition, mathematical modeling in-
volves calculating constitutive relations between the components of stress tensors and
deformation tensors. Correct mathematical formulation of the problem and the preset-
ting of initial and boundary conditions are necessary for effective research. The de-
velopment and software implementation of numerical algorithms adapted to the spe-
cifics of the problem under consideration and the visualization of the obtained numer-
ical results are also important.
During the study of biomedical issues, we may come across processes, for whose
mathematical description we use the frameworks of ordinary differential equations,
mathematical physics equations, algebraic nonlinear equation systems, difference
equations, the theory of bifurcations, chaos and order, etc. Examples of a successful
use of such mathematical frameworks are presented in [1] for prognosing disease
development, in [2-5] - for solving nonlinear dynamics problems in biology, chemical
kinetics, etc. The development of numerical methods for solving problems in biome-
chanics also allowed solving problems in the physics of plasmas, the mechanics of
deformable solids, etc. It is known that certain mathematical methods have evolved
under the influence of biomedical problems, for example, the methods of mathemat-
ical statistics, Volterra equations, neural networks, methods of solving rigid differen-
tial equations, etc.
The problems of researching mathematical models of linear and nonlinear dynam-
ic systems have become widespread in recent decades. We are talking about qualita-
tive approaches [6-11], analytical approaches [12-18], and combinations of such ap-
proaches and approximate research methods [19].
The biological, medical, and sport problems that require research and numerical
solution of partial equations have been formulated relatively recently. They are pre-
3
sented in [20-22]. Rheological relations for biological continuous media have been
developed in [23-24]. The range of tasks considered in this area is quite wide.
The most important area in traumatology is the problem of mathematical model-
ing of human leg movement while walking in order to build orthopedic prostheses
that imitate this movement. To model the distribution of dynamic loads and defor-
mations at the time of movement of the entire foot, it is necessary to use the frame-
work of partial differential equations, in particular the system of equa-tions of the
mechanics of deformable solid body. Creating such models for the needs of trauma-
tology and orthopedics is a new and relevant task for computational biolo-gy and
medicine. Computer-assisted implementation of virtual surgeries and predic-tion of
their consequences is another prospective area. This is a very complex area of re-
search that is just beginning to emerge. The formulation of certain mathematical mod-
els and methods of their research are not totally clear. However, the implementation
of some virtual surgeries is a real task. Thus, in [25], numerical modeling of lithotrip-
sy surgeries (fragmentation of renal stones with acoustic waves initiated by a spark
discharge or a laser pulse) is presented. The purpose of such studies is to find litho-
tripter operating modes (pulse duration and intensity, number of pulses), at which
fragments of destroyed stone would be small enough for natural excretion. For this
purpose, the picture of acoustic pulse propagation in the body and the stone was in-
vestigated numerically, and the problem of its destruction was solved.
The problems of biomechanics, as well as the tasks of controlling and regulating
vibration processes in structural systems, are largely related to the problem of contact
interactions with the medium, whose response to external influence depends on the
prehistory or the history of load. In other words, external influence turns into the re-
sponse of the coupling medium. This feature of the medium is called self-regulation.
Models of self-regulatory systems in biomechanics are models of bioactive materials,
or materials with biofactor. Similar models have been developed, for example, in [26-
28]. A model of a self-regulatory medium, whose response to force impact is de-
scribed by a hereditary-type biofactor model [27], is used in this case. The solution of
the corresponding mixed problem for the fifth-order equation is built and the impact
of the biofactor and material viscosity on the vibration process is investigated.
The aim of the presented studies is to develop qualitative approaches and on their
basis to theoretically substantiate the possibility of creating proper computational
methods for solving problems in biomechanics. These tasks arise in the process of
creating new orthopedic materials, as well as the modeling, synthesis and optimiza-
tion of parameters of corresponding orthopedic systems.
4
2 Investigation of the mathematical model of a nonlinear
vibration system that generalizes the rheological vibration
model with consideration of the effect of the biofactor
2.1 Problem statement. The main result
Let us denote QT (0, l ) (0, ) , 0,T , 0 l , T . In the domain QT , we
consider the first mixed problem for the nonlinear equation with real coefficients
U tt a2 ( x)U xxt xx b2 ( x)U xx xx b1 ( x) U xx
q2
U xx
xx
t
g (t ) d ( x)U xx ( x, ) xx d c0 ( x) U
p2
U f ( x, t ) (1)
0
with the initial conditions
U ( x,0) U 0 ( x) , (2)
Ut ( x,0) U1 ( x) (3)
and the boundary conditions
U (0, t ) U xx (0, t ) 0 , U (l , t ) U xx (l , t ) 0 . (4)
The mixed problem for the fifth-order nonlinear evolution equation considered here
describes the vibrations of an elastic bioactive rod with consideration of the
"memory" effect. The aim of this article is to conduct a qualitative study of the solu-
tion to the problem (1) - (4) in a limited range and obtain sufficient conditions for the
existence of a generalized solution of the mixed problem in Sobolev spaces for the
fifth-order differential equation (1), in which there is an integral summand with the
fourth derivative according to the spatial variable that models the effect of "memory"
in the vibration system. The obtained results will make it possible to apply adequate
computational methods and computer simulation to the above problem for the optimal
synthesis of the parameters of a vibration system whose mathematical model is the
problem (1)-(4). Let us assume the following conditions are true:
(1) functions a2 ( x), a2 ( x) xx are bounded on (0, l ) ; a2 ( x) A2 , a2 ( x) xx A2 ,
A2 0 ;
(2) functions b2 ( x), b2 ( x) xx are bounded on (0, l ) ; b2 ( x) B2 , b2 ( x) xx B2 , B2 0 ;
(3) functions b1 ( x), b1 ( x) x are bounded on (0, l ) ; b1 ( x) b0 0 ;
(4) function c0 ( x) is bounded on (0, l ) ;
(5) g (t ) 0 , g (t ) 0 for all t 0, , 0 g (t )dt G ;
0
(6) function d ( x) is bounded on (0, l ) , d ( x) d2 0 ;
(7) p 2 , q 2 ;
(8) functions f ( x, t ) , ft ( x, t ) are integrable with square according to Lebesgue in the
domain Q 0 for any 0 0 ;
5
(9) the initial deviation has the following features: U 0 ( x) is a function integrable with
power 2 p 2 on (0, l ) , the second derivative U 0 ( x) is a function integrable with
power q on (0, l ) , the fourth derivative U 0 ( x) is a function integrable with square
U
q 3
U xxx , U 0 xx
q 3 2 2
on (0, l ) , U xx 0 xxx are the functions integrable with
square on (0, l ) , while U 0 ( x) satisfies the conditions (4);
(10) the initial deviation has the following features: the second and the fourth deriva-
tives of U1 ( x) are functions integrable with square on (0, l ) , while U1 ( x) satisfies the
conditions (4).
The function U : (0, l ) 0, T ( T is a positive number or ) is called the
generalized solution to the problem (1)-(4) in the domain QT if it satisfies the initial
conditions (2) and the integral equality
l
U V a ( x)U V b ( x)U V b ( x) U
q2
tt 2 xxt xx 2 xx xx 1 xx U xxVxx
0
t
UV f ( x, t )V dxdt 0
p2
g (t )d ( x)U xx ( x, )Vxx ( x)d c0 ( x) U (5)
0
for almost all t 0, T and for all testing functions V , for which the equality (5) is
correct.
The solution U ( x, t ) has the following features:
- the functions U , U t are continuous on 0,T0 according to the time variable,
the second derivative Utt is bounded on 0,T0 according to the time variable for an
arbitrary number T0 from the interval 0,T ;
- the function U is integrable according to the spatial variable with power q on
0,l ; the function U t is integrable with square according to the spatial variable on
0,l ; the function Utt is integrable with square together with the second derivative
according to the spatial variable on 0,l .
The main result. Let the conditions (1), (2), (3), (4), (5), (6), (7), (8), (9), (10) be
satisfied. Then the finite time T , which depends on the coefficients, the right-hand
side of the equation, and the initial data, can be specified, at which a generalized solu-
tion U of the problem (1) - (4) exists in the domain QT .
6
2.2 Galerkin method
Because the space Vˆ (0, l ) W 2, r (0, l ) H 4 (0, l ) L2 p 2 (0, l ) with r max q, 2q 4 is
a separable Banach, there is a countable set in it k k , where any finite number of
elements is linearly independent and the closure of its linear shell in Vˆ (0, l ) coincides
with Vˆ (0, l ) . Let us note that k
k
can be selected orthonormal in the
N
space L2 (0, l ) . Let’s consider the functions U N ( x, t ) ckN (t ) k ( x) , N 1, 2,... ,
k 1
where c1N , c2N ,…, cNN are solutions of the corresponding Cauchy problems
l
N q2
U a ( x)U b ( x)U b ( x) U U xxN xxk
N k N k N k
tt 2 xxt xx 2 xx xx 1 xx
0
t
U N k f ( x, t ) k dxdt 0 ,
p2
g (t )d ( x)U xxN ( x, )xxk d c0 ( x) U N (6)
0
ckN (0) U0,Nk , c (0) U ,
N
k t
N
1, k (7)
N N
where U 0N ( x) U 0,Nk ( x) k , U1N ( x) U1,Nk ( x) k ,
k 1 k 1
U 0N U 0 ˆ 0 , U1N U1 0,
V (0, l ) H 02 (0, l ) H 4 (0, l )
N . On the basis of the Karatheodori theorem [29] there exists an absolutely
continuous solution to the problem (6), (7), determined in a certain interval 0,t0 .
From the evaluations obtained below, it follows that t0 T , while number T will be
determined later.
Let us multiply (6) by ckN t , sum it up by k from 1 to N and integrate it by t
from 0 to T . We will obtain
l
1
U tN ( x, ) dx a2 ( x) U xxt
q2
2 N 2
b2 ( x)U xxNU xxt
N
b1 ( x) U xxN U xxNU xxt
N
20 Q
t l
U NU tN f ( x, t )U tN dxdt U 1N dx . (8)
1
p2 2
g (t )d ( x)U xxN ( x, )U xxt
N
d c0 ( x) U N
0
20
Let us evaluate the summands of the equality (8). Based on condition (1)
I1 a2 ( x) U xxt dxdt A2 U xxtN dxdt .
N 2 2
Q Q
According to condition (2)
B l
l
B2
2
I 2 b2 ( x)U xxNU xxt dxdt 2 U xxN ( x, ) dx U 0N ( x, ) dx, B 2 sup b2 ( x) .
N 2 2
Q 2 0 2 0 xx x(0,l )
Using condition (3), we will obtain
7
q2 b0 C1 l N
C2 l
U 0N ( x, )
q
I 3 b1 ( x) U xxN
2
U xxN U xxt
N
U xx ( x , ) dx dx ,
Q q 0 q 0 xx
at that C1 0 , the positive constant C2 depends on b 0 sup b1 ( x) .
x(0, l )
Based on conditions (5), (6),
t
C3G1
I 4 g (t )d ( x)U xxN ( x, )U xxt N 2
dxdt 4 U xxN dxdt ,
C 2
N
d dxdt U xxt
Q 0 2 Q 21 Q
where 1 0 is an arbitrary constant, while positive constants C3 , C4 depend on
d 0 sup d ( x) , T . According to condition (4),
x(0, l )
p2
I 5 c0 ( x) U N U N U tN dxdt C 0C5 U N dxdt C6 U tN
p p
dxdt
Q Q Q
t p l
C C5 U ( x, 0) U ( x, ) dxdt C6 U tN dxdt C7 U 0N dx
0 N N p p
t
Q 0 Q 0
p
l l
dx C9 U tN ( x, ) dx dt , C 0 sup c0 ( x) ,
2
C8 U tN dxdt C7 U 0N
p p 2
x(0, l )
Q 0 0 0
positive constants C5 C9 are independent from N .
Using condition (8), one can get
1 2
I 6 f ( x, t )U N dxdt f 2 ( x, t ) U N dxdt .
Q
2 Q
Taking into account the evaluation of integrals I1 I 6 , after proper choice of a
sufficiently small constant 1 the next inequality is true:
l
1 N
U t ( x, ) U xxN ( x, ) U xxN ( x, ) dx U xxt ( x, ) dxdt
2 q 2 N 2
20
Q
l
C10 U tN U xxN dxdt C11 U 0N U1N U 0N U 0N dx
2 2 p 2 q
Q
0
xx xx
p
l
C12 f ( x, t ) dxdt C13 U ( x, ) dx dt , 0,T ,
2 2
2 N
t (9)
Q 0 0
where C10 C13 are positive constants independent on N . Using the Grönwall-
Bellman inequality, from (9) we obtain
l
1 N
U t ( x, ) U xxN ( x, ) U xxN ( x, ) dx U xxt
2 q 2 2
N
( x, ) dxdt
20 Q
p
l
M 1 M 2 U ( x, ) dx dt , 0,T ,
2 2
N
t
(10)
0 0
8
while positive constants M1 , M 2 depend on the coefficients, the right-hand side of
the equation, and the initial data and are independent of N .
The Bihari lemma can be applied to inequality (10) [30, p. 110].
l
1 N
U t ( x, ) U xxN ( x, ) U xxN ( x, ) dx U xxt
2 q 2 2
N
( x, ) dxdt
20 Q
2M1
2 p 2
(11)
2 ( p 2) M 1 p 2 2 M 2T
2
at T . Therefore, from (11) it follows
p 2 M1 p 2 2 M 2
UN M3
L 0,T1 ;W02,q (0, l )
, (12)
U tN M3
L 2
0,T1 ; H 02 (0, l ) L 0,T ; L (0,l )
1
2
where positive constant M 3 is independent on N , T1 0, T .
Let us further differentiate (6) according to variable t , multiply the obtained
equality by ckN tt , sum up all the equations according to k from 1 to N and inte-
grate the result according to the variable t from 0 to , (0, T1 ] . Let us evaluate the
summands of the obtained equality using conditions (1)-(10) just as the previous
evaluations were obtained. Based on the above evaluations, on can get
l
0 U tt ( x, ) U xxt ( x, ) dx Q U xxtt ( x, ) dxdt 2 (2q 2)M q 1M T 1 q 1 (13)
N 2 N 2 N 2 2M 4
4 5
1
at T . From inequality (13) we conclude that
q 1 M 4q 1M 5
U tN M6
L 0,T2 ; H 02 (0, l )
,
U ttN M6
L 2
0,T2 ; H 02 (0, l ) L 0,T ; L (0,l )
2
2
where the positive constant M6 is independent on N , T2 0, T . Let
2 1
T min , . After performing additional a priori
p 2 M 1
p 2 2
M 2 q 1 M 4 M 5
q 1
evaluations and conclusions, for the arbitrary T0 0, T one can obtain
t
( x)U ttU a2 ( x)U xxU xxt b2 ( x)U xxU xx g (t )d ( x)U xx ( x, )dU xxU xx ( x, t )
QT0 0
c0 ( x) U f ( x, t )U dxdt b1 ( x) U xx dxdt 0 .
p q
(14)
QT0
9
Given the arbitrariness of T0 , it follows from (14) that U satisfies equation (1) in
terms of distributions. Taking into account the smoothness of the obtained function,
we conclude: U is a generalized solution of the problem (1)-(4) in QT .
3 Model example. Results of numerical integration
The following equation can serve as the simplest model example (1)
p2
U tt aU xxxxt bU xxxx a0U t b0U c0 U U f ( x, t ) , p 2 . (15)
In equation (15), the function U ( x, t ) is transverse movement of beam cross-section
with the coordinate x at any given time t ; a 0 , b 0 , b0 0 are constants that are
expressed through geometric and physical-mechanical parameters of the beam, con-
stant a0 0 characterizes the effect of resistance forces in the vibration system (linear
case), constant c0 describes nonlinearly elastic forces affecting the system, f ( x, t ) is
external driving force. Boundary conditions (4) correspond to the model of the beam
with fixed pivot bearings at the ends x 0 and x l . In case of the mixed problem
(15), (2)-(4) can be obtained using the above considerations, the value of the critical
time T0 , at which the vibration system functions in a regime without blow-up at
t T0 , and goes into the blow-up regime at t T0 . It is easy to show that value T0
satisfies the condition
2
T0 ,
p 2 / 2
p 2 M Mˆ
while M , M are some generalized parameters of the vibration system which de-
pend on the constant of equation (15) and the initial data.
Fig. 1 shows the dependence of the critical value of T0 on generalized parameters
of the vibration system M , M at nonlinearity index p 3 which characterizes non-
linearly elastic features of the environment.
The qualitative results obtained in the previous section make it possible to investi-
gate with the help of numerical methods the dynamic regimes of vibrations for equa-
2x l, 0 x l 2
tion (15) in case of the problem with initial deviation U 0 ( x) and
2 2 x l , l 2 x l
zero initial velocity of deflection of the pivot points and zero boundary conditions.
The problem set describes natural transverse vibrations of the rod, which at the initial
moment of time is loaded by concentrated force at the point with coordinate x l 2 .
The above problem is a problem of the same form as (15), (2)-(4). As shown above,
there is a single generalized solution to this problem. Therefore, for numerical inte-
gration of motion equations, the choice of method is important only from the compu-
tational point of view. Numerical solution of the problem is carried out using the
fourth-order Runge-Kutta method. Figure 2 presents the law of time deviation of the
rod midpoint, depending on the correlation between the frequencies of natural and
10
forced vibrations under the following conditions: l 1 , a 0,001 , b 1 ,
a0 b0 0 , c0 100 , p 5 , f ( x, t ) 300sin9,48 t .
Fig. 1. Dependence of value T0 on the generalized parameters of the vibration system at
p 3.
Fig. 2. The law of rod midpoint deviation time change (resonant regime)
Figure 3 shows the same law provided f ( x, t ) 300sin9,48 t 2 .
11
Fig. 3. The law of rod midpoint deviation time change (non-resonant regime)
4 Conclusions
The mathematical model of nonlinear vibrations of a bioactive rod was investigated
using combined qualitative and numerical approaches with consideration of the self-
regulation phenomenon. This mathematical model is used in biomechanical studies of
new materials and to synthesize vibration system parameters. This, in turn, is an im-
portant issue in current medical research. The mathematical model of a vibration sys-
tem is presented as a mixed problem for a fifth-order equation with memory. Subcriti-
cal and critical system operation regimes were evaluated. Analytical correlations that
characterize the moment of process transition to the blow-up regime were established.
The qualitative and numerical results are the next:
physical and mechanical parameters of a vibration system determine the critical
value of the time parameter, up to which the system is in the blow-up-free regime;
the attenuation rate does not depend much on the degree of nonlinearity of the
resistance force, while the effect of the resistance force on the vibration period at
small values of a , p is minor;
depending on the correlation of frequencies of natural and induced vibrations in
the system, there will be a time increase of vibration amplitude (resonance) or
vibration beating.
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