Analysis of dynamics and controllability of a system described by the equation of a pipeline with a moving fluid†

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Analysis of dynamics and controllability of a system described by the equation of † Analysis of dynamics andacontrollability a system described by the equation of pipeline with aofmoving fluid † a pipeline with a moving fluid Vitaly V. Chernik* 

Vitaly V. Chernik* * Ishlinsky Institute for Problems in Mechanics of the Russian  Academy of Sciences, Prospekt Vernadskogo 101-1, Moscow, 119526, Russia (Tel:+7-915-329-62-45; e-mail:[email protected]). * Ishlinsky Institute for Problems in Mechanics of the Russian Academy Sciences, Prospekt Vernadskogo 101-1, Moscow, 119526, Russia (Tel:+7-915-329-62-45; e-mail: [email protected]). Abstract: The conditions for the loss of stability and the character of the loss of stability of the system described by the equation for withthe integral are investigated. dynamic system is Abstract: The conditions loss ofoperator stabilitywith andaftereffect the character of the loss ofSuch stability of the system the model of transverse oscillations of a pipeline with a uniformly flowing fluid, taking into account the described by the equation with integral operator with aftereffect are investigated. Such dynamic system is inertia forces of the pipe oscillations and liquid, the Coriolis forces and centrifugal forces,into and account the Kelvinthe model of transverse of amoments pipeline of with a uniformly flowing fluid, taking the Voigt endsand of the pipeline are fixedofinCoriolis a hinged manner, and the elastic characteristics are inertia friction. forces ofBoth the pipe liquid, the moments forces and centrifugal forces, and the Kelvinconstant. It is shown that as the velocity of the fluid increases, the system loses stability, and the nature of Voigt friction. Both ends of the pipeline are fixed in a hinged manner, and the elastic characteristics are the loss of stability is divergent. The presence of points of accumulation of the spectrum indicates the lack constant. It is shown that as the velocity of the fluid increases, the system loses stability, and the nature of of byisadivergent. control function, concentrated on of one of the bounds. thecontrollability loss of stability The presence of points accumulation of the spectrum indicates the lack of controllability by afrequency control function, onanalysis, one of the bounds. © 2018, IFAC (International Federation ofconcentrated Automatic Control) Hosting by Elsevier All rights reserved. Keywords: pipelines, spectrum, eigenmode stability analysis,Ltd. controllability Keywords: pipelines, frequency spectrum, eigenmode analysis, stability analysis, controllability  

1. INTRODUCTION

1. INTRODUCTION The first studies of the interaction of a liquid or gas flow with aThe pipeline were carried by Ashley Heviland and first studies of the out interaction of aand liquid or gas(1950) flow with Feodos’ev (1951). Theout equations, that most complete in a pipeline were carried by Ashley andare Heviland (1950) and terms of completeness of loads considered, are given in works Feodos’ev (1951). The equations, that are most complete in of Padossius and Issidof(1973), Zefirov et are al. given (1985). Small terms of completeness loads considered, in works transverse oscillations in the   u  u ( t , x ) 0  x  2  , t  0 of Padossius and Issid (1973), Zefirov et al. (1985). Small direction in the transverseperpendicular oscillations to u the u(taxis , x) of0the  xpipeline  2 , t in0a certain plane of viscoelastic pipe in dimensionless variables is direction perpendicular to the axis of the pipeline in a certain described by equation plane of viscoelastic pipe in dimensionless variables is

1.1 Sustainability issues 1.1 Sustainability issues As a criterion for system stability, we will use the absence of spectrum points positive realwe part. is, the system As a criterion forwith system stability, willThat use the absence of described by (1) and (2) is asymptotically stable if spectrum points with positive real part. That is, the system described by (1) and(2) is {asymptotically   | Re   0}  stable  , if 

    { | Re   0}   ,

where  is set of eigenvalues (spectrum) of the system. It will be shown that, for sufficiently the set of eigenvalues (spectrum)small of the values system.of It will where  isbelow parameter , the system is stable, however, with increasing of v be shown below that, for sufficiently small values of the this parameter, some points of the spectrum are shifted from parameter v , the system is stable, however, with increasing of the half-plane There are are two different Re points  0 toofthe thisleft parameter, some theright. spectrum shifted from forms of stability loss, it is divergence and flutter. Flutter is a the left half-plane Re   0 to the right. There are two different dynamic instability of an elastic structure in a fluid flow, forms of stability loss, it is divergence and flutter. Flutter is a caused byinstability positive feedback between the body's and dynamic of an elastic structure in deflection a fluid flow, the force exerted by the fluid flow, which leads to oscillations caused by positive feedback between the body's deflection and withforce amplitude infinity. is a the exertedrapidly by the increasing fluid flow,towhich leadsDivergence to oscillations phenomenon in which the elastic deformations of the object with amplitude rapidly increasing to infinity. Divergence is a monotone tends infinity, causing destruction of the phenomenon in to which the typically elastic deformations of the object object. Both options mean that the motion of the system is monotone tends to infinity, typically causing destruction of the unstable and beyond the scope of the model of small object. Both options mean that the motion of the system is oscillations. the point of viewofof the spectral analysis, the unstable andFrom beyond the scope model of small loss of stability in the form of a flutter occurs when two oscillations. From the point of view of spectral analysis, the conjugate points of loss of stability in the the spectrum form of simultaneously a flutter occursegress when from two left half plane into the open half-plane . The of Re   0 conjugate points of the spectrum simultaneously egressloss from stability in the into formthe of aopen divergence arises realloss point . The of left half plane half-plane Rewhen   0the of the spectrum, along the realwhen axis,thegoes from stability in the formmoving of a divergence arises real point negative to its positive part. along the real axis, goes from of the spectrum, moving

described '''' by equation '

u   u  2 vu  k0u  u ''''  v2u ''  u   u ''''t  2 vu '  k0u  u ''''  v2u ''   t at   u ''''  , x d  0  0 at   u ''''  , x d  0

(1)

(1) Equation (1) is expressed in dimensionless terms. Definitions 0 of dimensionless quantities is given in Padossius and Issid Equation (1) is expressed in dimensionless terms. Definitions (1973) and Akulenko et al. (2011) work in such way Issid that of dimensionless quantities is given in Padossius and parameters are positive, is proportional to the v  ,  , k , v 0 (1973) and Akulenko et al. (2011) work in such way that flow velocity in the pipe, is proportional to are positive, to the parameters v isproportional  ,of  , the k 0 , vliquid internal friction in the pipe, is proportional to the density of  flow velocity of the liquid in the pipe,  is proportional to the moving fluid, is proportional to thetofriction of the k 0pipe, the density of internal friction in the  is proportional external environment, term to models the hereditary is integral proportional the friction of the the moving fluid, k 0 the properties of the medium. that the ends of pipe are external environment, the Suppose integral term models thethe hereditary hinged: properties of the medium. Suppose that the ends of the pipe are hinged: '' u(t,0)  u (t ,0)  u(t ,2 )  u '' (t ,2 )  0

(2) u(t,0)  u '' (t ,0)  u(t ,2 )  u '' (t ,2 )  0 The initial conditions for equation (1) are not posed, (2) since for the analysis of stability and controllability it is required to The initial conditions for equation (1) are not posed, since for determine theofspectrum equation. the analysis stabilityofand controllability it is required to

negative to its positive part.

determine the spectrum of equation. †

This work was founded with Russian Science Foundation under grant 16-11-10343

2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. † This workunder was responsibility founded with Science Foundation underControl. grant 16-11-10343 Peer review of Russian International Federation of Automatic Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.03.027 Copyright © 2018 IFAC

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1.2 Controllability issues

assuming that the kernel in the integral term is exponential, that is, a   a1e 0 , a1 , 0  0 , we obtain equation in following form:

The controllability of the system under consideration is investigated in the following sense: if for any initial states u  u0 ( x), u  u1 ( x) (t  0,0  x  2 ), it is possible to bring the system into a given state (for example, in complete rest) in a finite time, applying a limited control u(t ,0)   (t ) to the boundary of the segment on which the system is defined, then the system is considered as controllable. Otherwise, the system is uncontrollable. A study of the controllability of systems with integral delay was carried out by Romanov and Shamaev (2016) based on the spectrum density characteristic. Let n(t ) is the number of spectrum  points inside circle with radius

t

u   u  k0u  u  v u  a1e  0 (t  )u ''''  , x  d  0 ''''

a *  lim

r 

 r

 o

nt  dt t



(3)

4

2 2

equations is the spectrum of the system (5). Coefficients of 2 and  are polynomials of the fourth power of the parameter

k with the coefficients of the highest degree k 4 are positive. It follows that the criterion for the absence of infinite number of positive points of the spectrum is condition 0  a1 . If this condition is satisfied, then starting with a certain value of k all coefficients of equation (6) are positive and it cannot have positive roots. Otherwise, starting with some value of k , all the coefficients except the free term of equation (6) are positive, therefore all equations starting from some will have positive roots. Even if 0  a1 , a finite number of positive points of the spectrum may appear because of free term 0 k 4  a1k 4  v 2 k 20 can be negative for sufficiently small

0

u0, x   ( x) , ut ,0  v(t ), u(t ,  )  0 . Here v(t ) is bounded function of control. For the proof, the method of moments was used. An analogous method can be used to prove the same result for the problem of boundary control of the system described by equations (1) and (2) in the case when the parameter   0 :

values of k . To exclude this possibility, we require the positivity of the free term of equation for k  1 . Namely,

v  1

t

u   u  k0u  u  v u  a  t    u  , x  d  0 ''''

a1

0

(7)

Hence it follows that it is impossible to increase the parameter of the fluid velocity in the pipeline arbitrarily, without loss of stability of the system. At the same time, it is possible to select the parameters of the system so that it will be stable.

0

u(t ,0)   (t )



(6).  0 k  a1k  v k 0  0 We note that the roots of a given set of equations are analytic functions of the equations coefficients, including the parameter v . The union of all the roots of a given set of 4





 

3  ak 4  k0  0 2  ak 4 0  k 4  v 2 k 2  k00  

t

2 ''



Considering the boundary conditions (2) we obtain, that the system of eigenfunctions sin kx, k  N is a complete basis in the solution space. Let us write down the characteristic equation for the spectral parameter  for equation (5):

u t , x   at   u ''  , x d  0 ,

''''

2 ''

0

In the work of Romanov and Shamaev (2016), it was proved that characteristic a * of the spectrum density is the lower bound for the optimal time to bring to rest the system described by following equations:

''''

''''

(5)

t . Then the characteristic of spectrum  density a * is defined as follows: r

157

(4)

u(t ,2 )  u '' (t ,2 )  0

3. SPECTRAL PICTURES OF THE SYSTEM (4) If the density parameter is zero, we proceed to a new dynamical system (4). Analysis of spectrum of that system will allow us to make conclusions about the properties of the original system (1). For this we rely on the Miloslavskiy theorem given in the fourth part of this paper and on the fact that the points of the spectrum analytically depend on the parameters.

Consider the spectral picture of the equation (4) with the following values of the parameters a1  0  k0    v  1 (Fig. 1). There are two conjugate complex roots with negative real part (when k  1 ), three accumulation points: 0, -2, -∞. Positive real roots tend to zero. The real roots on the right of 2 tend to the right. At such parameter values, the system is unstable and uncontrollable, which fact is in accordance with criterion (7). This spectral picture, consisting of several pairs of conjugate points, two real accumulation points and

2. CHARACTERISTIC EQUATIONS OF THE SYSTEM (4) When excluding the term 2vu ' from equation (1), containing the first derivative with respect to the spatial variable, and also 2

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sequence of points approaching to  , is common for equation (5).

Fig. 2. Picture of the spectrum of equation (5) with

a1  0,1; 0  9; k0  0,05;  0,2; v  0,9 We should also pay attention to the fact that the presence of accumulation points l1 and l 2 of the spectrum and their location is not dependent on the flow velocity v . Therefore, all further conclusions about controllability associated with these points are internal properties of the system, namely, with the presence and coefficient of Kelvin-Voigt friction and with the kernel of integral operator describing the hereditary properties.

Fig. 1. Picture of the spectrum of equation (1) with a1  0  k0    v  1 . With the help of Cardano's formulas it can be established that the points of the spectrum tend to -∞ with asymptotic  k 4 . Two finite points of accumulation l1 and l 2 satisfy the system of equations 1  l  l  0   1 2    l l  0  a1 1 2   

Consider the spectral picture of equation (5) with the following values of the dimensionless parameters a1  0,1; 0  9; k0  0,05;  0,2; v  0,9 (Fig. 2). Points that tend to infinity are not shown on this picture, to be able to pay attention to the first eigenvalues and high modes converging at the accumulation points. As the flow velocity increases, the imaginary parts of the complex roots will decrease to zero while real parts almost do not change (Fig. 3). For some value of the parameter v conjugate complex roots become one real root of second order, with a further increase of parameter v , roots bifurcate again and move in different directions along the real axis. One of this pair of roots inevitably becomes positive for a certain value of v . The general property of the spectrum picture of equation (5) is also the presence of two real accumulation points and a sequence of real roots tending to -∞.

Fig. 3. The nature of the change in the eigenvalues of the first three modes as a function of the parameter 𝑣𝑣 4. TYPE OF STABILITY LOSS OF THE INITIAL SYSTEM Let us consider the nature of the stability loss of the initial system, the equation for which contains the term 2vu ' . Spectral analysis of the initial equation seems to be a much more complicated problem because of the presence of a term with the first derivative with respect to the spatial variable. Miloslavskiy (1991) showed that the number of points of a spectrum with a positive real part is independent of the parameter β.

With none of the considered sets of parameters, a pair of complex roots with a positive real part was observed. These observations indicate that the loss of stability of system (5), caused by flow velocity increase, occurs in the form of a divergence.

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

From the previous results it is clear that as the parameter of flow velocity in the pipeline increases, points of the spectrum go to the right half-plane Re   0 one by one along the real axis, and the system loses stability in a divergent manner. To change the nature of the loss of stability, it is required that the points of the spectrum pass into the right half-plane by conjugate pairs. The number of points of the spectrum from the result of Miloslavskiy (1991) for systems described by equations (1) and (5) with hinge fixation conditions coincide for the same set of parameters. Therefore, it can be concluded that the system (1) loses stability in the same way and at the same instant as system (5) without the term 2vu ' .

Consider the characteristic spectral picture of the resulting system (Fig. 5). It can be seen that the spectrum consists of a finite number of pairs of conjugate complex roots with negative real roots corresponding to vibrational modes with damping and an infinite number of negative real roots. We can also note the presence of two points of accumulation:  

aˆ 0  1,   Im aˆ  i   0   0 ,

2 with asymptotic  k 4 and  1 with asymptotic v .  k2

where a is the coefficient of Kelvin-Voigt friction term au '''' , at  is the kernel of integral operator, aˆ   is it’s Laplacian image, then the number of points of the system (1) spectrum lying in the right half-plane Re   0 , taking into account the order, coincides with the number of negative points of the

 v2

k 

Obviously, because of the positivity of the coefficients of the system, any complex roots of a given family of equations will have a negative real part. Positive real roots will be for a given system only under the condition v  1 , whence the condition of stability of the system is the inequality v  1 . That is, as the velocity parameter of the fluid increases, the system inevitably loses stability. The type of loss of stability, as well as in the case of presence of delay, is the divergence. This follows from the fact already mentioned that no equation of the family (6) can have a pair of complex roots with positive real parts because of the positivity of the parameters on which the coefficients of the equations depend.

a '  0    , at  L1[0,], tat   L1[0,],

4



 2   k 4  k0   k 4  v 2 k 2  0 ,

In more details, Miloslavskiy's (1991) result is in the following theorem. If the kernel of the integral operator describing the hereditary properties of system (1) satisfies conditions:

spectrum of the operator 1  aˆ (0) 

159

2

on the set of x x 2 functions satisfying the boundary conditions (2). The conditions imposed on the kernel at  of the integral operator in this theorem provide that all solutions of equation (1) with k0  v    0 tend to 0 when t  0 , as well as the decrease of the at  . 4

4.1 Effect of integral term with delay In the work Akulenko et al. (2016) a system of a pipeline with a moving fluid was considered, however, in the equation describing the system there is no integral term with delay corresponding to the hereditary properties of the medium, there is also no term corresponding to the Kelvin-Voigt friction and the boundary conditions correspond to rigid fixation of the pipe ends. Let us consider a similar version of the problem without hereditary properties. It will differ from the problem in the above-mentioned work only by boundary conditions in which the rigid fixation is replaced by the hinged one and by the presence of Kelvin-Voigt friction:

u   u  k0u  u  v u  0 ''''

''''

2 ''

Fig. 4. Picture of the spectrum of equation (8) with k0  0,05; a  0,03; v  0,75 4.2 Effect of Kelvin-Voigt friction Consider a system in the absence of Kelvin-Voigt friction, described by the following equation t

u  k 0 u  u ''''  v 2 u ''  a1e 0 (t  ) u ''''  , x d  0 .



(8)

(9)

0

The set of characteristic equations for a given problem with boundary conditions (2) is analogous to the set of equations (6) under the condition   0 . Picture of the spectrum of equation (9) significantly differs from the cases considered earlier (Fig.

u(t,0)  u '' (t ,0)  u(t ,2 )  u '' (t ,2 )  0 Characteristic equations for finding the points of the spectrum of a given system are square:

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where aˆ   is Laplacian image of kernel at  , parameters  and k0 are assumed to be equal to zero for simplicity, which does not affect the generality of the subsequent arguments. Consider equation

5).

a  1  aˆ ( )  0,

(11)

P ( ) , where deg P( )  deg Q( ) and this Q ( ) fraction is irreducible. Then equation (11) takes the form

and let aˆ ( ) 

(  1)Q( )  P( ) 0 Q ( ) The degree of the numerator of this fraction is by 1 greater than the degree of the denominator and it is also irreducible. Fig. 5. Picture of the spectrum of equation (9) with a1  0,1; 0  9 ; k0  0,05; v  1,1 .

Therefore, this equation will have at least one root. Let ˆ is the root of this equation (11). Then expression

In this case, an infinite number of oscillating modes are observed, corresponding to an infinite set of conjugate complex eigenvalues whose imaginary parts tend to infinity. Stability region and the nature of the loss of stability are similar to the previous cases considered. There is also one real accumulation point 0  k0 .

( (ˆ   )  1)Q( )  P(ˆ   )   ( ) is polynomial with variable  and of degree not less than 2, and besides   0 is the root of this polynomial, which means that the free term of this polynomial is equal to zero. Let us rewrite equation (10) in the following form:

5. CONTROLLABILITY OF INITIAL SYSTEM In the work Romanov and Shamaev (2016), the problem of boundary control, focused on one of the ends of a system with aftereffect, is considered. It is shown that with the help of the spectrum density characteristic it is possible to estimate the boundary controllability and the time of optimal control.

(ˆ   k ) 2  k 4

Here deviation of the root of equation (10) from the root ˆ of equation (11) for some value of parameter k is designated by k .

For the systems considered above, both with the aftereffect described by the simplest kernel a   a1e 0 and in the absence of aftereffect, it is possible to observe the existence of finite points of accumulation of eigenvalues in the spectrum picture. In this case, the characteristic of the density of the spectrum (3) is equal to infinity, since, starting from some nr  takes on the value of finite value of r , the integrand r infinity. This means that such systems are uncontrollable by a control function, concentrated on one of the ends (problem (4)).

We multiply both sides of the latest equation by

Q(ˆ   k ) k4

,

we get

 ( k ) 

(ˆ   k ) 2 Q(ˆ   k ) k4



v 2Q(ˆ   k ) k2

0.

Considering this equation as an algebraic equation with variable  k we obtain that this equation is of at least third

We now consider the more general case, when the Laplacian image of the kernel of the integral aftereffect operator is a regular algebraic fraction. This case covers a fairly wide range of mechanical problems. Including, the case when the kernel of an integral operator is represented in the form of exponential ai e  i . The secular equation for finding expansion a  

degree, and the coefficients of the higher powers depend on k , but are separated from zero, and the free term tends to zero when k   . By Vieta's theorem, the free term of the equation is equal to the product of its roots. Therefore, there is a sequence {ˆ } of the roots tending to zero for k   . This



k

the eigenvalues of equation (1) will have the following form:

 2  k 4 (  1  aˆ ( ))  v2 k 2  0,

 ( k )  v2k 2  0 . Q(ˆ   k )

means that ˆ is a point of accumulation of the spectrum of the given system (1) with kernel a  . It follows that there is no controllability by a control function, concentrated on one of the ends of the system described by equation (1) under the

(10)

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Vitaly V. Chernik / IFAC PapersOnLine 51-2 (2018) 156–161

condition that the Laplacian image of the kernel a  in the integral operator with aftereffect is a regular algebraic fraction. 6. CONCLUSIONS In the work with the help of numerical modeling it is shown that when the velocity of fluid in the pipeline increases with the hinged fastening of the ends, the stability of the system is lost in the divergence type, i.e. one real eigenvalue appears on the left half-plane. The criterion of stability in terms of the parameters of the original problem is obtained. The presence of the Kelvin-Voigt friction in the friction system does not change the stability zones in the parameter space of the problem, and the presence of an integral aftereffect only reduces the indicated zone, which at first glance may seem paradoxical. It is also shown that, assuming   0 , with both the aftereffect in the system and the Kelvin-Voigt friction, the total boundary controllability of the system is lost. Further investigations, as we assume, will be related to the construction of the spectrum picture for the complete equation (1), and also to the proof of the absence of complete boundary controllability of the system given by equation (1). The simplest considerations based on the fact that the roots of the characteristic equations considered in this paper depend analytically on the parameters of the problem indicate that the points of accumulation of the spectrum must remain for nonzero values of the parameter  , which in turn means the absence of boundary controllability. This work was founded with Russian Science Foundation under grant 16-11-10343

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