Free vibrations of the single-mass system with variable parameters that characterize inertia, friction, rigidity are considered. For an exact analytical solution the method of constant coefficients under derivatives with a new variable is propose

A. Ivanovskaya

English

Journal of Vibroengineering

 585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. 
A. IVANOVSKAYA 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 
494 
585. The research of dynamics of mechanical system  
        with variable parameters 
 
A. Ivanovskaya 
Kerch State Marine Technological University, Kerch, Ukraine 
e-mail: inv07@ukr.net 
 
(Received  20 September  2010;  accepted 9 December 2010) 
 
 
Abstract. Free vibrations of the single-mass system with variable parameters that characterize 
inertia, friction, rigidity are considered. For an exact analytical solution the method of constant 
coefficients under derivatives with a new variable is proposed. 
 
Keywords: vibrations, variable parameter, differential equation. 
 
Introduction 
 
As it is known, the main object of dynamics is the definition of the law of the body motion 
with respect to known forces exerted on it. For the solution of this problem the differential 
equation should be derived and its integration will lead to the desired law of motion. In the case 
when the mathematical system has fixed parameters of motion, modern mathematical tool al-
lows convenient fulfillment of the corresponding operations. But, due to onrush of science and 
engineering, generation of more precise solutions is required. There is a need in consideration 
of dozens of factors that affect the dynamics of a mechanical system. Therefore it is common 
that in the considered differential equations the parameters of motion (namely, inertial coeffi-
cient, quasi-elastic coefficient, environmental resistance coefficient and so on), are various 
functions. Thus, there is a necessity to integrate differential equation with variable coefficients.  
Currently, there are methods of solution of separate classes of differential equations with 
variable coefficients, for instance: equation of Euler, Mathieu-Hill equation, equation of 
Meshcherskoi. Also the formula of Liouville-Ostgorskii, Kulikow method of solution of linear 
and nonlinear equations with variable coefficients in total derivatives, grid method of solution 
of linear differential equations with variable coefficients based on Taylor development are 
known. But these methods are not always acceptable in solution of problems of dynamics as 
they impose strict limits. And the obtained solution of the Mathieu-Hill equation does not de-
scribe the movement characteristics: such as amplitude, oscillation period, speed and accelera-
tion - only stability and instability zones of mechanical system are examined. 
The aim of this work is to develop the method of analytical solution of a separate class 
of linear differential equation with variable coefficient.  
Let’s examine linear homogeneous differential equation of the second kind with variable 
coefficient and find its terms of integrability 
( ) ( ) ( )
2
0 1 22 0
d x dx
a t a t a t x
dtdt
+ + =
 (1) 
Let’s choose arbitrary substitution ( )z f t=  (2) 
and name ( )f t  the defining function. Let’s suppose that given function is continuous and de-
fined on the interval ( )0 ; kt t t∈ , twice differentiated and ( ) 0f t′ ≠ . 
 
 585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. 
A. IVANOVSKAYA 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 
495 
Denote ( ) ( )dz f t t
dt
ϕ′= = . (3) 
We need to find the first and the second derivative of the equation (1), we substitute 
them in the initial equation, and, grouping together the summands with identical derivatives, we 
present the equation (1) as 
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2
2
0 0 1 22 0
d td x dx
a t t a t t a t t a t x
dz dzdz
ϕ
ϕ ϕ ϕ
 
  + + + =  
 
 (4) 
The received equation can always be integrated, if coefficients of unknown x , dx
dz
, 
2
2
d x
dz
 
will be invariable. So let’s suppose that  
( ) ( )20 0a t t Cϕ =  (5) 
( ) ( ) ( ) ( ) ( )0 1 1
d t
a t t a t t C
dz
ϕ
ϕ ϕ+ =  (6) 
( )2 2a t C=  (7) 
where 0C , 1C , 2C  are invariable. 
Considering the obtained formula, we can get a new differential equation with constant 
coefficients, with derivatives with respect to z  
2
0 1 22 0
d x dxC C C x
dzdz
+ + = . (8) 
The solution of the differential equation (8) can be written as 
( ) 1 21 2r z r zx z K e K e= + ,  or ( ) ( )1 2rzx z e K K z= + , (9) 
in case if the roots of characteristic equation are real numbers 1 2,r r  or 1 2r r r= = .  
If the roots are complex numbers 1,2r iα β= ± , the solution of equation (8) will be ex-
pressed as  
( ) ( )1 2cos sinzx z e K z K zα β β= +  (10) 
Keeping in mind substitution ( )z f t= , let’s come over to original variable t , thus will 
get the solution of the equation (1) 
( ) ( ) ( )1 21 2
r f t r f t
x t K e K e= + ,  ( ) ( ) ( )1 2
rf t
x t e K K f t= +     
or ( ) ( ) ( ) ( ){ }1 2cos sinf tx t e K f t K f tα β β= +        (11) 
where 1K  and 2K  are constant values, defined from initial data. 
Now we need to define the terms of integrability of the equation (1). For this purpose, 
using the equation (5)-(7) and (3), one can find what coefficient ( )0a t , ( )1a t , ( )2a t  should be 
( )
( ) ( )
0 0
0 2 2
C C
a t
t f tϕ
= =
′  
. (12) 
( )
( )
( )
( )
01
1 3
C f tC
a t f t f t
′′⋅
= −
′ ′  
. (13) 
And, finally, it was previously defined that coefficient ( )2 2a t C= . (14) 
Substituting obtained equations (12), (13), (14) into initial differential equation (1), one 
will receive the following 
 585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. 
A. IVANOVSKAYA 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 
496 
( ) ( )
( )
( )
2
00 1
22 2 3 0
C f tC Cd x dx C xf t dtdtf t f t
 ′′⋅ 
+ − + = ′′ ′        
. (15) 
The above equation represents linear homogeneous differential equation of the second 
kind with constant coefficients, which always allows analytical integration. Arbitrary constant 
0C , 1C  and 2C  take the form of coefficients of equation with new derivative z , and ( )f t  - 
defining function. 
Let’s determine sufficient terms of integrability of the initial differential equation with vari-
able coefficients.  
From equation (5)-(7) it follows that 
( )
( )
( ) ( ) ( )
( )
0
0
01
1 0
0
2 2
2
C
t
a t
a tC
a t a t
C
a t C
ϕ

=



= +

 =

ɺ
 (16) 
It follows herefrom that linear differential equation of the second kind with variable co-
efficient can be integrated in quadratures only if there was a definite dependence between the 
coefficients ( )0a t  and ( )1a t  and coefficient ( )2a t  was constant (16). 
In this case the defining function is determined as 
( ) ( )
( )
0
0
dtf t t dt C C C
a t
ϕ= + = +∫ ∫ . (17) 
Similarly, one can get the terms of integrability for homogeneous linear differential 
equation with variable coefficient of the third and the fourth kind [1-2]. 
In order to solve inhomogeneous linear differential equation with variable coefficient 
one should additionally make a substitution in the right part of the equation ( )1t f z=  and solve 
the obtained equation with the constant coefficients and derivatives with new variable, with the 
help of commonly known methods. 
Linear equation as 
( ) ( ) ( ) ( ) ( ) ( )1 11 1...
n nn n
n n
ax b y A ax b y A ax b y A y f x− − − ′+ + + + + + + = , (18) 
where 1, , ,..., na b A A  are constants and called the equation of Euler. As it is known, for the solu-
tion of the equation of such kind one should introduce new independent variable t , supposing 
ax b t+ = . 
We can show that while solving the equation of Euler with the method we introduced, we 
will come to the same substitution and will get the same result. 
Example 1.  To solve the equation of Euler 
3 2
3 2 3
3 2 2 2
d x d x dx
t t t x t
dtdt dt
− + − =  
Solution. Let’s check whether the coefficients of the equation correspond to the received terms 
of integrability. Assume 0 1C = ; ( ) 30a t t= . 
Therefore ( ) 20 3a t t=ɺ ; ( )0 6a t t=ɺɺ . 
 585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. 
A. IVANOVSKAYA 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 
497 
1) On sufficient condition ( ) ( ) ( )21 31 0 023
0
C
a t a t a t
C
= + ɺ . 
In our example ( ) ( )23 2 2 231 1 13 3a t C t t C t t= + = + . 
As in initial equation ( ) 21a t t= − , so 2 2 21 3t C t t− = +  
Therefore, 1 4C = − . 
2) On sufficient condition ( ) ( ) ( )
( )
( )
( )
( )
2
00 02 13
2 0 23 3 3 00 00
9 33
a ta t a tC C
a t a t
a tC a tC
  = + − +
ɺɺ ɺɺ
 
In our case ( )
222
3 3
2 2 233 3
34 3 6 4 2
3 39
tt t
a t C t C t t t t
tt
  = − − + = − − + . 
In initial equation ( )2 2a t t= , that is why 22 4 2t C t t t t= − − +  
Therefore, 2 5C = . 
3) On sufficient condition ( )3 3a t C= . 
As by the data ( )3 2a t = − , that is why 3 2C = − . 
As it is shown, all variable coefficients meet the sufficient terms of integrability (16), 
where 0 1C = ; 1 4C = − ; 2 5C = ; 3 2C = − . Let’s solve the initial equation by means of the pro-
posed method. 
As ( ) 30a t t=  and 0 1C = , then we arrive at 
( )
( )
0
3
0
C
t
a t
ϕ = , that ( ) 1t
t
ϕ = . 
Therefore, ( ) ( ) lndtf t t dt C C x C
t
ϕ= + = + = +∫ ∫ . 
Assume ( ) lnz f t t= = , hence ( )1 zt f z e= = .  
Now one should come to a new differential equation, but already with constant coeffi-
cients and derivatives with respect to z . 
 
3 2
3
3 24 5 2
zd x d x dx x e
dzdz dz
− + − = . 
So the roots of the characteristic equation of the corresponding homogeneous equation 
are equal to  1 2 1r r= =  and 3 2r = , then the general solution of the homogeneous equation can 
be written as 
( ) ( ) 21 1 2 3z zx z e K K z K e= + + . 
Complementary function will be searched in the form of 
( ) 32 zx z Ae= . 
As its derivatives are equal to 
( ) 32 3 zx z Ae=ɺ ; ( ) 32 9 zx z Ae=ɺɺ ; ( ) 32 27 zx z Ae=ɺɺɺ , then we obtain 
3 3 3 3 327 36 15 2z z z z zAe Ae Ae Ae e− + − =  
therefore 1 4A = . 
Consequently, the general solution of the differential equation with constant coefficients  
 585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. 
A. IVANOVSKAYA 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 
498 
( ) ( ) 2 31 2 3 1 4z z zx z e K K z K e e= + + + . 
Considering the fact, that lnz t= , complete integral with variable coefficients  
( ) ( ) 2 31 2 3ln 1 4x t t K K t K t t= + + + . 
Solving the given equation by Euler method with the help of substitution zt e= , the solu-
tion will be identical.  
Therefore one can make a conclusion, that the equation of Euler is a subclass of equations 
that can be solved with the help of proposed method of constant coefficients under derivatives 
with a new variable.  
As it was stated above, the necessity of integration of differential equation with variable 
coefficients appeared in the context of solution of the main problem of dynamics.   
Example 2. To study the motion of the mechanical system, described by the following differen-
tial equation, if it is known, that in the start time the body was shifted out of equilibrium on 0,1 
m and  left without initial speed. 
( ) ( )
2
2 2 3
1 2 cos 5 0
3 sin3 sin 3 sin
d x t dx
x
t dtdtt t
  
+ + + = 
−− −  
 
Solution. Assume a ( )
( )0 2
1
3 sin
a t
t
=
−
; 0 1C = . 
Then ( )
( )0 3
2cos
3 sin
t
a t
t
=
−
ɺ . 
1) On sufficient condition 
( ) ( ) ( )011 0
0 2
a tC
a t a t
C
= +
ɺ
 
In the given problem ( )
( )
1
1 3
cos
3 sin 3 sin
C t
a t
t t
= +
− −
 
Under the statement of problem ( )
( )1 3
2 cos
3 sin 3 sin
t
a t
t t
= +
− −
. 
Therefore, 1 2C = . 
2) On sufficient condition ( )2 2a t C=  
As by data ( )2 5a t = , then 2 5C = . 
It is obvious, that all the variable coefficients correspond to the terms of integrability 
(16), where 0 1C = ; 1 2C = ; 2 5C = . Let’s find the analytical solution of the initial equation 
according to the proposed method. 
As ( )
( )0 2
1
3 sin
a t
t
=
−
 and 0 1C = , so 
( )
( )
0
0
C
t
a t
ϕ = , that ( ) 3 sint tϕ = − . 
Hence, defining function  
( ) ( )3 sin 3 cosz f t t dt t t= = − = +∫ . 
Let’s come to the differential equation with constant coefficients and derivatives with re-
spect to z  
 585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. 
A. IVANOVSKAYA 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 
499 
2
2 2 5 0
d x dx
x
dzdz
+ + = . 
The roots of the characteristic equation are equal to 1,2 1 2r i= − ± . That is why the gen-
eral solution can be written as 
( ) ( )1 2cos 2 sin 2zx z e K z K z−= + . 
or ( ) ( )sin 2zx z e a z γ−= + , 
where a , γ  are constant values, detected out of initial data 
0
0
0,1,  0t
t
dx
x
dt= =
= = . 
Considering the fact that 3 cosz t t= + , we will receive that the law of motion of re-
quired mechanical system can be written as 
( ) ( ) ( )3 cos0,11 sin 2 3 cos 1,1t tx t e t t− += + +   . 
Given equation describes free attenuation oscillation of the system. Graphic of such os-
cillation is illustrated in Fig. 1. 
 
 
Fig. 1. Graphics of free oscillation of the system with variable parameters 
 
 
Conclusions  
 
1. In this work the method of constant coefficients under derivatives with a new variable is 
proposed, which allows integration in quadratures of separate class of the linear differential 
equations with variable coefficients.  
2. Sufficient terms of integrability (16) are established by means of the developed method of 
differential equations of the second, the third and the fourth kind.  
3. Practical application of the method is presented in terms of research of dynamics of the me-
chanical system. 
 
References 
 
[1]  Kisly А., Ivanovskiy N., Ivanovskaya А. Integration of linear differential equations of the third 
order with variable coefficients. Collection of scientific papers of KSMTU. Kerch, 2008. Issue 9, p. 
38-41. 
[2]  Kisly А., Ivanovskiy N., Ivanovskaya А. Integration of linear differential equations of the fourth 
order with variable coefficients. Collection of scientific papers of KSMTU. Kerch, 2008. Issue 9, p. 
42-46. 
0.8 1.6 2.4 3.2 4 
-0.021 
-0.016 
-0.011 
-0.006 
0.004 
 t, с 
x(t), м 
0 


The research of dynamics of mechanical system with variable parameters

Ivanovskaya, A.

JVE International

 585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. A. IVANOVSKAYA      VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 494 585. The research of dynamics of mechanical system          with variable parameters  A. Ivanovskaya Kerch State Marine Technological University, Kerch, Ukraine e-mail: inv07@ukr.net  (Received  20 September  2010;  accepted 9 December 2010)   Abstract. Free vibrations of the single-mass system with variable parameters that characterize inertia, friction, rigidity are considered. For an exact analytical solution the method of constant coefficients under derivatives with a new variable is proposed.  Keywords: vibrations, variable parameter, differential equation.  Introduction  As it is known, the main object of dynamics is the definition of the law of the body motion with respect to known forces exerted on it. For the solution of this problem the differential equation should be derived and its integration willlead to the desired law of motion. In the case when the mathematical system has fixed parameters of motion, modern mathematical tool al-lows convenient fulfillment of the corresponding operations. But, due to onrush of science and engineering, generation of more precise solutions is required. There is a need in consideration of dozens of factors that affect the dynamics of a mechanical system. Therefore it is common that in the considered differential equations the parameters of motion (namely, inertial coeffi-cient, quasi-elastic coefficient, environmental resistance coefficient and so on), are various functions. Thus, there is a necessity to integrate diff rential equation with variable coefficients.  Currently, there are methods of solution of separate classes of differential equations with variable coefficients, for instance: equation of Euler, Mathieu-Hill equation, equation of Meshcherskoi. Also the formula of Liouville-Ostgorskii, Kulikow method of solution of linear and nonlinear equations with variable coefficients i  total derivatives, grid method of solution of linear differential equations with variable coefficients based on Taylor development are known. But these methods are not always acceptable in solution of problems of dynamics as they impose strict limits. And the obtained solution f the Mathieu-Hill equation does not de-scribe the movement characteristics: such as amplitude, oscillation period, speed and accelera-tion - only stability and instability zones of mechanical system are examined. The aim of this work is to develop the method of analytical solution of a separate class of linear differential equation with variable coefficient.  Let’s examine linear homogeneous differential equation of the second kind with variable coefficient and find its terms of integrability ( ) ( ) ( )20 1 220d x dxa t a t a t xdtdt+ + =  (1) Let’s choose arbitrary substitution ( )z f t=  (2) and name ( )f t  the defining function. Let’s suppose that given fuction is continuous and de-fined on the interval ( )0; kt t t∈ , twice differentiated and ( ) 0f t′ ≠ .   585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. A. IVANOVSKAYA      VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 495 Denote ( ) ( )dz f t tdtϕ′= = . (3) We need to find the first and the second derivative of the equation (1), we substitute them in the initial equation, and, grouping together t  summands with identical derivatives, we present the equation (1) as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )220 0 1 220d td x dxa t t a t t a t t a t xdz dzdzϕϕ ϕ ϕ   + + + =    (4) The received equation can always be integrated, if coe ficients of unknown x , dxdz, 22d xdz will be invariable. So let’s suppose that  ( ) ( )20 0a t t Cϕ =  (5) ( ) ( ) ( ) ( ) ( )0 1 1d ta t t a t t Cdzϕϕ ϕ+ =  (6) ( )2 2a t C=  (7) where 0C , 1C , 2C  are invariable. Considering the obtained formula, we can get a new differential equation with constant coefficients, with derivatives with respect to z  20 1 220d x dxC C C xdzdz+ + = . (8) The solution of the differential equation (8) can be written as ( ) 1 21 2r z r zx z K e K e= + ,  or ( ) ( )1 2rzx z e K K z= + , (9) in case if the roots of characteristic equation are real numbers 1 2,r r  or 1 2r r r= = .  If the roots are complex numbers 1,2 iα β= ± , the solution of equation (8) will be ex-pressed as  ( ) ( )1 2cos sinzx z e K z K zα β β= +  (10) Keeping in mind substitution ( )z f t= , let’s come over to original variable t , thus will get the solution of the equation (1) ( ) ( ) ( )1 21 2r f t r f tx t K e K e= + ,  ( ) ( ) ( )1 2rf tx t e K K f t= +     or ( ) ( ) ( ) ( ){ }1 2cos sinf tx t e K f t K f tα β β= +        (11) where 1K  and 2K  are constant values, defined from initial data. Now we need to define the terms of integrability of the equation (1). For this purpose, using the equation (5)-(7) and (3), one can find what coefficient ( )0a t , ( )1a t , ( )2a t  should be ( )( ) ( )0 00 2 2C Ca tt f tϕ= =′  . (12) ( )( )( )( )011 3C f tCa tf t f t′′⋅= −′ ′  . (13) And, finally, it was previously defined that coefficient ( )2 2a t C= . (14) Substituting obtained equations (12), (13), (14) into i itial differential equation (1), one will receive the following  585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. A. IVANOVSKAYA      VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 496 ( ) ( )( )( )200 122 2 30C f tC Cd x dxC xf t dtdtf t f t ′′⋅ + − + = ′′ ′        . (15) The above equation represents linear homogeneous differential equation of the second kind with constant coefficients, which always allows analytical integration. Arbitrary constant 0C , 1C  and 2C  take the form of coefficients of equation with new derivative z , and ( )f t  - defining function. Let’s determine sufficient terms of integrability of the initial differential equation with vari-able coefficients.  From equation (5)-(7) it follows that ( )( )( ) ( ) ( )( )00011 002 22Cta ta tCa t a tCa t Cϕ== + =ɺ  (16) It follows herefrom that linear differential equation of the second kind with variable co-efficient can be integrated in quadratures only if there was a definite dependence between the coefficients ( )0a t  and ( )1a t  and coefficient ( )2a t  was constant (16). In this case the defining function is determined as ( ) ( )( )00dtf t t dt C C Ca tϕ= + = +∫ ∫ . (17) Similarly, one can get the terms of integrability for homogeneous linear differential equation with variable coefficient of the third and the fourth kind [1-2]. In order to solve inhomogeneous linear differential equation with variable coefficient one should additionally make a substitution in the right part of the equation ( )1t f z=  and solve the obtained equation with the constant coefficients and derivatives with new variable, with the help of commonly known methods. Linear equation as ( ) ( ) ( ) ( ) ( ) ( )1 11 1...n nn nn nax b y A ax b y A ax b y A y f x− −− ′+ + + + + + + = , (18) where 1, , ,..., na b A A  are constants and called the equation of Euler. As it is known, for the solu-tion of the equation of such kind one should introduce new independent variable t , supposing ax b t+ = . We can show that while solving the equation of Euler with the method we introduced, we will come to the same substitution and will get thesame result. Example 1.  To solve the equation of Euler 3 23 2 33 22 2d x d x dxt t t x tdtdt dt− + − =  Solution. Let’s check whether the coefficients of the equation c rrespond to the received terms of integrability. Assume 0 1C = ; ( ) 30a t t= . Therefore ( ) 20 3a t t=ɺ ; ( )0 6a t t=ɺɺ .  585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. A. IVANOVSKAYA      VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 497 1) On sufficient condition ( ) ( ) ( )21 31 0 0230Ca t a t a tC= + ɺ . In our example ( ) ( )23 2 2 231 1 13 3a t C t t C t t= + = + . As in initial equation ( ) 21a t t= − , so 2 2 21 3t C t t− = +  Therefore, 1 4C = − . 2) On sufficient condition ( ) ( ) ( )( )( )( )( )200 02 132 0 23 3 300 009 33a ta t a tC Ca t a ta tC a tC  = + − +ɺɺ ɺɺ In our case ( )2223 32 2 233 334 3 64 23 39tt ta t C t C t t t ttt  = − − + = − − + . In initial equation ( )2 2a t t= , that is why 22 4 2t C t t t t= − − +  Therefore, 2 5C = . 3) On sufficient condition ( )3 3a t C= . As by the data ( )3 2a t = − , that is why 3 2C = − . As it is shown, all variable coefficients meet the sufficient terms of integrability (16), where 0 1C = ; 1 4C = − ; 2 5C = ; 3 2C = − . Let’s solve the initial equation by means of the pro-posed method. As ( ) 30a t t=  and 0 1C = , then we arrive at ( )( )030Cta tϕ = , that ( ) 1ttϕ = . Therefore, ( ) ( ) lndtf t t dt C C x Ctϕ= + = + = +∫ ∫ . Assume ( ) lnz f t t= = , hence ( )1 zt f z e= = .  Now one should come to a new differential equation, but already with constant coeffi-cients and derivatives with respect to z .  3 233 24 5 2 zd x d x dxx edzdz dz− + − = . So the roots of the characteristic equation of the corresponding homogeneous equation are equal to  1 2 1r r= =  and 3 2r = , then the general solution of the homogeneous equation can be written as ( ) ( ) 21 1 2 3z zx z e K K z K e= + + . Complementary function will be searched in the form of ( ) 32 zx z Ae= . As its derivatives are equal to ( ) 32 3 zx z Ae=ɺ ; ( ) 32 9 zx z Ae=ɺɺ ; ( ) 32 27 zx z Ae=ɺɺɺ , then we obtain 3 3 3 3 327 36 15 2z z z z zAe Ae Ae Ae e− + − =  therefore 1 4A = . Consequently, the general solution of the differential equation with constant coefficients   585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. A. IVANOVSKAYA      VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 498 ( ) ( ) 2 31 2 3 1 4z z zx z e K K z K e e= + + + . Considering the fact, that lnz t= , complete integral with variable coefficients  ( ) ( ) 2 31 2 3ln 1 4x t t K K t K t t= + + + . Solving the given equation by Euler method with the help of substitution zt e= , the solu-tion will be identical.  Therefore one can make a conclusion, that the equation of Euler is a subclass of equations that can be solved with the help of proposed method of constant coefficients under derivatives with a new variable.  As it was stated above, the necessity of integration of differential equation with variable coefficients appeared in the context of solution of the main problem of dynamics.   Example 2. To study the motion of the mechanical system, described by the following differen-tial equation, if it is known, that in the start time the body was shifted out of equilibrium on 0,1 m and  left without initial speed. ( ) ( )22 2 31 2 cos5 03 sin3 sin 3 sind x t dxxt dtdtt t  + + + = −− −   Solution. Assume a ( )( )0 213 sina tt=−; 0 1C = . Then ( )( )0 32cos3 sinta tt=−ɺ . 1) On sufficient condition ( ) ( ) ( )011 002a tCa t a tC= +ɺ In the given problem ( )( )11 3cos3 sin 3 sinC ta tt t= +− − Under the statement of problem ( )( )1 32 cos3 sin 3 sinta tt t= +− −. Therefore, 1 2C = . 2) On sufficient condition ( )2 2a t C=  As by data ( )2 5a t = , then 2 5C = . It is obvious, that all the variable coefficients correspond to the terms of integrability (16), where 0 1C = ; 1 2C = ; 2 5C = . Let’s find the analytical solution of the initial equation according to the proposed method. As ( )( )0 213 sina tt=− and 0 1C = , so ( )( )00Cta tϕ = , that ( ) 3 sint tϕ = − . Hence, defining function  ( ) ( )3 sin 3 cosz f t t dt t t= = − = +∫ . Let’s come to the differential equation with constant coefficients and derivatives with re-spect to z   585. THE RESEARCH OF DYNAMICS OF MECHANICAL SYSTEM WITH VARIABLE PARAMETERS. A. IVANOVSKAYA      VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   DECEMBER 2010. VOLUME 12, ISSUE 4. ISSN 1392-8716 499 222 5 0d x dxxdzdz+ + = . The roots of the characteristic equation are equal to 1,2 1 2r i= − ± . That is why the gen-eral solution can be written as ( ) ( )1 2cos 2 sin 2zx z e K z K z−= + . or ( ) ( )sin 2zx z e a z γ−= + , where a , γ  are constant values, detected out of initial data 000,1,  0ttdxxdt= == = . Considering the fact that 3 cosz t t= + , we will receive that the law of motion of re-quired mechanical system can be written as ( ) ( ) ( )3 cos0,11 sin 2 3 cos 1,1t tx t e t t− += + +   . Given equation describes free attenuation oscillation of the system. Graphic of such os-cillation is illustrated in Fig. 1.   Fig. 1. Graphics of free oscillation of the system with variable parameters   Conclusions   1. In this work the method of constant coefficients under derivatives with a new variable is proposed, which allows integration in quadratures of separate class of the linear differential equations with variable coefficients.  2. Sufficient terms of integrability (16) are established by means of the developed method of differential equations of the second, the third andthe fourth kind.  3. Practical application of the method is presented in terms of research of dynamics of the me-chanical system.  References  [1]  Kisly А., Ivanovskiy N., Ivanovskaya А. Integration of linear differential equations of the third order with variable coefficients. Collection of scientific papers of KSMTU. Kerch, 2008. Issue 9, p. 38-41. [2]  Kisly А., Ivanovskiy N., Ivanovskaya А. Integration of linear differential equations of the fourth order with variable coefficients. Collection of scientific papers of KSMTU. Kerch, 2008. Issue 9, p. 42-46. 0.8 1.6 2.4 3.2 4 -0.021 -0.016 -0.011 -0.006 0.004  t, с x(t), м 0 

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The research of dynamics of mechanical system with variable parameters

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