Oscillation Analysis of Some Hybrid Dynamical Systems of Transmission Pipelines

Oscillation Analysis Oscillation Analysis of Some Hybrid Dynamical of Some Hybrid Dynamical Systems of Transmission Systems of Transmission

PipelinesPipelinesOlena Mul Olena Mul

((jointly withjointly with Volodymyr Kravchenko) Volodymyr Kravchenko)Ternopil Ivan Pul'uj National Technical UniversityTernopil Ivan Pul'uj National Technical University

Ternopil, UkraineTernopil, Ukraine

ESF-EMS-ERCOM/INI Mathematical Conference "Highly Oscillatory Problems From Theory to Applications“,

Isaac Newton Institute for Mathematical Sciences, Cambridge, September 12-17, 2010

Cambridge, Sept.16, 2010 Cambridge, Sept.16, 2010 22

1. THE PROBLEMS1. THE PROBLEMS

It is necessary to investigate possible oscillations in the transmission pipelines, which always have negative influence on systems functioning.

The main problems are:

to determine conditions of oscillations self-excitation; to determine frequencies and amplitudes of possible

oscillations; to find ways how to decrease negative influence of

oscillations or even to avoid them.


2. SOME REFERENCES2. SOME REFERENCES

K. Ya. Kuhta, V. P. Kravchenko, and V. A. Krasnoshapka, K. Ya. Kuhta, V. P. Kravchenko, and V. A. Krasnoshapka, The qualitative theory of The qualitative theory of controllable dynamical systems with continuous-discrete parameterscontrollable dynamical systems with continuous-discrete parameters, ''Naukova Dumka'', Kiev, , ''Naukova Dumka'', Kiev, 1986 (Russian). 1986 (Russian).

A. Samoilenko and R. Petryshyn, A. Samoilenko and R. Petryshyn, Multifrequency oscillations of nonlinear systemsMultifrequency oscillations of nonlinear systems, , Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (2004), no. Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (2004), no. 567. 567.

O. O. Mul and V. Kravchenko, Mul and V. Kravchenko, Investigations of Vibrations in the Complex Dynamical Systems of Investigations of Vibrations in the Complex Dynamical Systems of Transmission PipelinesTransmission Pipelines, "Interface and Transport Dynamics. Computational Modelling", , "Interface and Transport Dynamics. Computational Modelling", Lecture Notes in Computational Science and Engineering, Springer-VerlagLecture Notes in Computational Science and Engineering, Springer-Verlag Berlin Heidelberg Berlin Heidelberg (2003), no.32, 295-300.(2003), no.32, 295-300.

O.V. Mul,, D.F.M. Torres, O.V. Mul,, D.F.M. Torres, Analysis of Vibrations in Large Flexible Hybrid Systems, Analysis of Vibrations in Large Flexible Hybrid Systems, Nonlinear Nonlinear Analysis, Elsevier, vol. 63, 2005, USA, pp. 350-363. Analysis, Elsevier, vol. 63, 2005, USA, pp. 350-363.


3. THE SCHEME OF TRANSMISSION 3. THE SCHEME OF TRANSMISSION PIPELINESPIPELINES

X

M

m

L

) t

) t , L ( u ( F


4.1. THE MATHEMATICAL MODEL4.1. THE MATHEMATICAL MODEL

0)t

uu(

xa

t

u2

22

2

2

0x ,

))t

)t,0(u)t,0(u(

x(ES

t

)t,0(u

t

)t,0(um

2

2

(1)

Lx , )t

)t,L(u)t,L(u(

xES

t

)t,L(uM

2

2

331 )

t

)t,L(u(

t

)t,L(u

where: )t,x(u is the longitudinal displacement of pipe points; E is the

material elasticity modulus; is the material volume unit mass;

E

a 2 ; is

the factor of material internal friction; M is the platform mass; S is the pipe

cross-section area; L is the pipe length; m is the Lanchester damper mass;

is the damper energy dissipation factor; 1 , 3 are constants characterizing

resistant environment.


4.2. THE DIMENSIONLESS 4.2. THE DIMENSIONLESS MATHEMATICAL MODELMATHEMATICAL MODEL

2

3

12

2

2

2

x

u

x

uu

0x ,

uu

x

u

x

u32

2

1

2

1 (2)

1x ,

u

x

u

x

uu2

2

12

2

))u

( 34

where: L

xx ,

L

uu ,

L

at are dimensionless variables;

and L

a1

,

Ma

L12

,

ES

a3

,

ML

a34

,

M

SL ,

SL

m1 are

dimensionless parameters.


5. PERTURBATION THEORY METHOD5. PERTURBATION THEORY METHOD

Perturbation theory comprises mathematical methods to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem.

Applicability: if the problem can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.

The desired solution will be expressed in terms of a power series in some "small" parameter that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, and further terms describe the deviation in the solution, due to the deviation from the initial problem.


5.1. THE IDEA5.1. THE IDEA

Formally, for the approximation to the full solution A we have a series in the small parameter ::

...AAAA 22

10 ,

where 0A would be the known solution to the exactly solvable initial problem, and 1A , 2A , … represent the "higher orders" which are found iteratively by some systematic procedure. For small these higher orders become successively less important. For our problem: we should choose systems of equations of higher approximations whose solutions approximate the solutions of the original system of equations to within values proportional to integer powers of some small parameter.


5.1. THE IDEA5.1. THE IDEA

Equations (2) contain five dimensionless parameters 1 , 2 ,

3 , 4 and 1 , which as a rule are less than unit, although they

have different orders of smallness.

The main assumption is: we assume that dimensionless

parameters characterizing dissipation of different physical nature

are proportional to some common small parameter :

11 , 22 , 33 , 44 , 11 . (3)


5.2. THE FIRST APPROXIMATION5.2. THE FIRST APPROXIMATION

We neglect all the dissipative factors of the mathematical model, i.e. put 0 . Then the problem (2) simplifies to

0x

uu2

2

2

2

0x , 0x

u

(4)

1x , 0x

uu2

2

We have the next transcendental equation for exact determination of the eigenvalues of the boundary problem (4):

)sin()cos( (5)


5.3. THE FIRST 5.3. THE FIRST TECHNIQUETECHNIQUE

We start from the problem of determination of oscillation

conditions. For that, it’s enough to consider the linear problem, i.e. to

put 04 .

We look for the solution of the boundary problem (2) in the form:

)i

2(

2

1

exsinBxcosA,xu , (6)

where is an eigenvalue of the boundary problem (2), e is the base of

the natural logarithms, A and B are arbitrary constants, and 1i .


Satisfying equations (2) accurate to 2 , we will obtain:

2

2232

1

)(i)()cot(

. (7)

The solution of transcendental equation (7) at 0 is known, and

it determines eigenvalues of the boundary problem (4).

Accounting smallness of damping forces, we assume that the

correction factor for the system nonconservatism is a quantity of order :

1 , (8)

where 1 is the correction factor for the problem nonconservatism.



The approximate formula for complex eigenvalues of the

nonconservative problem (2) is:

22

2232

22

2

1

)(i

. (9)

Then the condition, whose satisfaction leads to self-excitation of

oscillations and self-oscillations in case of equality, is as follows:

)()(2

223

222

12 . (10)



5.4. THE 5.4. THE SECOND SECOND TECHNIQUETECHNIQUE

Now we assume that only

11 , 22 , 33 , 44 . Then the problem may be written as follows:

2

3

12

2

2

2

x

u

x

uu

0x ,

u

x

uu

x

u3

2

12

2

1 (11)

1x ,

u

x

u

x

uu2

2

12

2))

u( 3

4



In a conservative system (at 0 ) periodic motions appear with

any amplitude dependent on initial conditions.

But in the nonconservative system (at 0 ) they have only fixed

amplitudes, which conform to equality of an energy present at the

expense of negative resistance and energy dissipation.

The solution of the problem (11) is thought for in the form:

...,xu,xu,xu,xu 22

10 , (12)

where ,xu0 is one of the solutions of the problem (11) at 0 .



The first approximation of the solution (9) will be:

cos)xsinx(cosA,xu 10 , (13)

where A is a sought-for amplitude.

Then we receive the following boundary problem (14), which

may be considered as a mathematical model of the conservative

system with natural frequency , which is affected by external force

of resonance frequency.



sin)xsinx(cosA

x

uu1

312

12

21

2

0x ,

sin)(Au

x

u3

2112

12

11 (14)

1x ,

2

211

121

2(sincosA

x

uu

sin))sin(cosA75,0 21

224

A stationary solution exists only on the condition of an

orthogonality of an external force and the normal mode of a system.

This condition determines an amplitude, not increasing with time.


Let us choose the solution in the form

sin)x(U,xu1 , (15)

which reduces the problem to ODE:

)xcosxsin(AUxd

Ud1

31

22

2 ,

0x , )(AUxd

dU3

211

21 (16)

1x , 22

112 )sin(cosAU

xd

dU

]sincosA75,0 21

224



Then the quadrate of amplitude A of stationary oscillations is:

21

22122322

13

4

212 5,0)1([

sin)1(3

)1(4A (17)

)]())1)(()(( 223

221

22221

The condition for self-oscillation excitation is:

)1( 2212 )((5,0 22

12

1 (18)

)())1)(( 223

221

22




The first approximation of the stationary solution is

)cos()xsin()xcos(Au 1

)xsin()AA5,0B()xcos(B( 32

11111 (19)

xA5,0 21 )sin()))xsin()xcos(( 1 .

Analysis of the condition (18) allows to make an important

conclusion: for the considered systems some boundary frequency

always exists such that on the greater frequencies excitation of

oscillations is impossible.


6. THE NUMERICAL APPROACH6. THE NUMERICAL APPROACH The numerical method of the normal fundamental system of

solutions is used for determination of the eigenvalues of the problem.

The main idea of the method is: to reduce solving boundary value

problem with complex boundary conditions to multiple solving Cauchy

problems with initial conditions equal to 1 or 0.

We will search for solution of the problem (2) in the form

iq21 exuixu,xu , (20)

where q, are, respectively, the real and the imaginary parts of the

eigenvalues.


In new functions

11 u , 22 u , 33 u , 44 u , (21)

the problem is a system of ODEs of the 1-st order in the normal form

31 ,

42 ,

22113 KK , (22)

21124 KK .

which satisfy the next boundary conditions:

0x , 0TTTT 44332211 , 0TTTT 43342112 ,

1x , 0DDDD 44332211 , (23) 0DDDD 43342112 .

6. THE NUMERICAL APPROACH6. THE NUMERICAL APPROACH


The coefficients 1K and 2K from (22) are defined as

221

21

221

22

1q1

qqqK

,

221

21

221

2q1

qq2K

,

whereas the coefficients from (23) are the next:

1T q)q( 322

1 , 2T )q2( 13 ,

3T q1 1 , 4T 1

1D )qq( 2

22 , 2D )q2( 2 ,

3D )q1( 1 , 4D 1 .



6. THE NUMERICAL APPROACH6. THE NUMERICAL APPROACH The accuracy of determination of the eigenvalues depends on the length of the integration interval for system (22). We divide the integration interval [0,1] into subintervals by points ixx , where 1n,1i , 0x0 , 1x n .

Solutions q,,xk of boundary problem (22)-(23) are formed on

corresponding subintervals by solutions q,,x)i(k :

q,,xq,,x )i(kk , i1i x,xx

of equations systems

i3

i1 ,

i4

i2 ,

i22

i11

i3 KK , n,1i (24)

i21

i12

i4 KK ,

where subscript and superscript are function and subinterval numbers.


6. THE NUMERICAL APPROACH6. THE NUMERICAL APPROACH The solutions q,,x)i(

k of (24) must satisfy boundary conditions (23) and, besides, the conjugation conditions

q,,xq,,x i)1i(

ki)i(

k , (25)

q,,xq,,x i)1i(

ki)i(

k which link them at the ends of the subintervals.

By any numerical method we solve the Cauchy problem for the set of differential equations (24) four times for each subinterval with the initial conditions:

jk,0

jk,11i

)i(k,j q,,x , 4,3,2,1k,j , (26)

where the first subscript of the function is a solution number, and the second one is a function number.


Thus, the normal fundamental system of solutions q,,x)i(k,j for set

(24) may be generated and general solution of set (24) is

4

1j

)i(k,j

)i(j

)i(k q,,xCq,,x (27)

Coefficients )i(jC from general solution (27) may be presented as

)i(j

)1(4

)i(j

)1(3

)i(j QCRCC .

From the condition (23) at 0x and conjugation conditions (25), we

can calculate all coefficients )i(jR and )i(

jQ . From the condition (23) at

1x we have the set of two linear algebraic equations

0Сq,,xEСq,,xE )1(4n2

)1(3n1 , (28)

0Сq,,xEСq,,xE )1(4n4

)1(3n3 ,



The relation

0

q,,xEq,,xE

q,,xEq,,xEq,

n4n3

n2n1

(29)

is a necessary and sufficient condition for the existence of a non-trivial solution of the boundary value problem (2). It is an equation for a pair of numbers q, , which may be determined as its roots. Thus, from equation (29) it is possible to determine complex eigenvalues iq of the boundary value problem.



7. SOME RESULTS7. SOME RESULTS

025,01 , 5,22 , 375,03

125,04 , 25,01

0 2 4 6 8 10

0

1

2

3

4

5

2

1 b

0 2 4 6 8 10

0.81.21.62.02.42.83.23.64.04.44.8

b2

1

Rigidly attached With Lanchester damper


0 2 4 6 8 10012345678

R1А

L

EV1А,E

1

1

1А

А

А


025,01 , 5,22 , 375,03

125,04 , 25,01



025,01 , 5,22 , 125,04

5,1 , 25,01

0,0 0,2 0,4 0,6 0,8 1,0

0

1

2

3

4

3

1A

0,0 0,5 1,0 1,5 2,0

0

1

2

3

4

5

21

b

3



025,01 , 5,22 , 375,03

25,01

0 2 4 6 8 101,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

2

1

0 2 4 6 8 10

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

2

1q

q

q


CONCLUSIONSCONCLUSIONS

A mathematical model of transmission pipelines with a Lanchester damper, intended for lifting of minerals from great depth, was formulated, which was PDE satisfying nonlinear boundary conditions.

For the analysis of possible oscillations a perturbation method was applied. The conditions of oscillation excitation as well as a formula for oscillations amplitudes were obtained in analytical form.

The results of the approximate method of small parameter were verified with the help of numerical method of normal fundamental systems of solutions.

The important conclusion was drawn that for considered transmission pipelines some boundary frequency always existed such that on the greater frequencies excitation of oscillations was impossible.

It is possible to change purposefully oscillation frequencies and amplitudes in order to avoid the excitation of undesirable oscillations.

Oscillation Analysis of Some Hybrid Dynamical Systems of Transmission Pipelines

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