Advanced Approach in the Research Methods of the Mathematical Models of Strongly Nonlinear Physical Systems Petro Pukach 1 , Volodymyr Il’kiv 1 , Zinovii Nytrebych 1 , Myroslava Vovk 1 , Pavlo Pukach 2 1 Dept. of Mathematics, 2 Dept. of Applied Mathematics Lviv Polytechnic National University Lviv, Ukraine [email protected] Abstract -- The approximation methods to investigate the dynamical processes in the strong nonlinear single- or multidegree physical systems are studied in the paper. The developed procedure of the investigation of the oscillations in the nonlinear systems with the concentrated mass can solve not only the analysis problems but the problems of the cyberphysical oscillation system synthesis on the projective phase choosing such system characteristics enabled the resonance phenomena as well. There is described the real application of the obtained results to analyze the technical systems using to protect the equipment from the vibration load. Keywords—mathematical model; approximaiton physical system;resonance; dynamical process

I.

method;

INT RODUCT ION

The mathematical modeling is the important tool to solve the most of the scientific and engineering problems. It also helps to understand the kernel of the processes that are investigating in the modern physics, to interpret the known phenomena and to predict the new ones. The analysis and optimization of the physical, chemical dynamical processes in the technical systems in the cases of high speed, high pressure and energy demands the investigation of the complicated processes in the nonlinear oscillation theory. The strong nonlinear oscillation physical systems are widely applied in the engineering. They are well studied in the case while the ordinary linear or with the partial derivatives differential equations are used as the mathematical models. In the first case these equations describe the processes in the systems with the concentrated masses, in the second one in the system with the distributed parameters. Moreover, for the linear computing models of the oscillations physical systems basing on the superposition principle are developed the analytical methods in virtue of which one can state the reaction on the one- or multi-frequency periodical perturbations. Quasilinear systems i.e. systems with the small maximum values of the nonlinear forces comparing to the linear component of reducing force are similar to the linear systems. The analytical investigating methods for the most of the classes of the quasilinear systems are based on the different modifications of the perturbation methods in

particular asymptotic methods of the nonlinear mechanics. They give the possibility to describe and to state the special attributes of the oscillations in the quasilinear systems that are not proper to the linear systems with the constant in time physical-mechanical characteristics. On can speak firstly about resonance phenomena [1, 2] (main and combinational), process stability [3-5]. As to the more complicated systems , namely the strong nonlinear systems one can use the analytical investigating methods only in some cases . The nonlinear systems with the reducing force described by the power or close to power nonlinearity can be treated to this case [6, 7]. In reference to non-autonomous strong nonlinear systems, i.e. the systems under the external periodic perturbation (even small on value), it is necessary to realize the additional investigation. Their characteristics can be described solely. The main issue of the developed in the paper procedure of the investigating the resonance phenomena in the discussed class of the systems is based on idea to use the special periodic Ateb-functions constructing the solutions of the unperturbed analogue systems and to adopt the general ideas of the perturbation methods – in the case of the non-autonomous systems. II.

THE MAT HEMAT ICAL MODEL OF T HE ST RONG NONLINEAR SINGLE – DEGREE PHYSICAL SYST EM

In this section there is described the procedure of the investigation of the oscillations in the systems with one d egree freedom while the reducing force is approximated close to the power law. In this case the differential equation that describes such oscillations can be written as

mx cx 1 f x, x, where m is the mass of the material point; x is its coordinate at the arbitrary time moment; F ( x) cx 1 is function, describing the main part of the reducing force; c , are 2 p 1 constants, provided 1 ( p, q 0,1,2,... ); 2q 1

f x, x, is analytical 2 – periodic on t function,

is frequency of the external periodic perturbation acting on

the system; is the small parameter indicating on the small value of the external periodic perturbation and inconsiderable deviation of the restoring force from the power law. The restrictions concerning the parameter value are caused by the symmetry with respect to the coordinate origin of the function F ( x) cx 1 , describing the restoring force. More general representation of the reducing nonlinear force in the form

p

F ( x) c x x always with the necessary exactness

power can be approximated by the upper written depend ence, thereby all obtained results, one can transfer also on the systems with such type restoring force. Let’s notice, that function f x, x, can describe the nonlinear forces with the maximumе value being the small quantity comparing to the maximum value of the restoring force F ( x) cx 1 , that is

F ( x) max f x, x, . The last inequality can use the general ideas of the perturbation methods to construct the solution of the nonlinear equation (1). As per usual, it is necessary to describe the dynamical process of the unperturbed ( 0 ) or generating system, corresponding to (1). The next nonlinear equation is its analogue mx cx 1 0 .

(2)

The solution of the equation (2) is represented via periodic Ateb- functions as (3) x aca 1,1, (t ) ,

Fig. 1. Natural frequency’s dependence on the amplitude and the small nonlinear parameter

where a is the amplitude, (t ) is the oscillation phase of the unperturbed motion, (a) is the oscillation frequency equals

(a)

с 2 2m

a2 .

(4)

Hence, even for the unperturbed motion natural frequency of the dynamical process depends on amplitude. This is the first conceptual difference between the strong nonlinear and quasilinear systems. Thus, in case of the unperturbed motion natural frequency (a) depends on the mass of the material body and on the proportionality constant in the restoring force (analogue constant inelasticity), and on the amplitude and the nonlinear parameter also. This is the main difference of the considered class of the systems from the linear one. The issue is to construct the solutions of such strong nonlinear systems, taking into account all upper dynamical peculiarities , and to answer on the main questions of the investigating of the resonant oscillations. Fig. 1 demonstrates the dependence of the frequency (a) on the parameters defining the dynamical process of th e unperturbed motion, fig. 2 demonstrates the ratio of the periods of natural nonlinear to the linear 0 oscillations on the 1, 1 amplitude and on the parameter , i.e. . (a)

Fig. 2. T he ratio of the oscillation period of the nonlinear model to the oscillation period of the linear model

The parameters a and for the unperturbed motion are constants and are defined from the initial conditions x 0 x0 , satisfying the relationships 2

x0 a

2

V ( 2) 0 1 , 2a a

cta 1,1,

1, 1 . (a)

2 ( a ) . 2

Figures 1 and 2 demonstrate: 1) for the systems with small stiffness 1 0 the oscillation natural frequency of the physical system for the greater amplitude’s values is less; 2) while the nonlinear parameter converges to zero process in the system is considered as similar to the isochronous; 3) for the systems of the considerable stiffness 0 and for the oscillation amplitude values less than 1 for the greater values of nonlinear parameter oscillation natural frequency is less; for the oscillation amplitude values greater than 1 for its greater values correspond the greater values of the natural frequencies; 4) at the equal values of the parameters c and m the oscillation period of the strong nonlinear system is greater than the oscillation period of the system’s linear analogue 1 0 and a 1 and less at 0 і a 1 . It is necessary to take into account all these peculiarities in the resonance phenomena investigation. III.

RESONANCE PHENOMENA IN T HE ST RONG NONLINEAR PHYSICAL SYST EMS WIT H T HE DISCRET E ST RUCT URE

To develop the investigating methods for the reaction of the strong nonlinear system on the periodic non-autonomous type perturbations it is necessary to study the resonance conditions considering equation (2) as the model. The classic definition of the resonance phenomena can be applied for the considered systems noting the difference, that concept “frequency” treating as the meaning of the function (a) , is wider, than in the quasilinear systems. At first, the oscillation natural frequency of the system is defined not only by the phys ical and mechanical characteristics (parameters m, c ), but also by the amplitude a ; secondly, the periodic process of the unperturbed motion is described by the periodic Atebfunctions. The period 2 for the last ones by the argument (a)t depends on the nonlinear parameter and is defined by the relationship

1 2 . 1, 1 1 1 2 2

On account of this for the considered system class one can accept the derivative from the classic meaning instead of the main resonance, in other words – the natural oscillation period coincides with the perturbation period. Then the resonance condition transforms as

(5)

(6)

Hereafter the parameter value means the frequency of the external periodic perturbation (force). Taking into account the function, describing “natural frequency”, one can obtain the parameter value a* , responsible for the resonance phenomena: 2

1, 1 2 a . k 2 *

(7)

c . Thus, the resonance phenomena in the m investigated dynamical systems appears when the oscillation where k

amplitude is close to a* . If the amplitude is greater than a* , then existing resistant forces in the real systems produce the decrease of the oscillation amplitude to the value a* , causing the resonance phenomena in the future. This explains the second principle difference of the resonance phenomena in the strong nonlinear systems. If there exist one or several stationary oscillation modes in the perturbed strong nonlinear system, then the investigation of the physical system reaction on the periodic perturbation must be considered separately. Fig. 3 demonstrates the dependence of the resonance amplitude on the parameters , that characterized the physical and mechanical system properties and forced periodic perturbation frequency. In particular, from the obtained in the paper plotted dependences follows, that for the «soft» systems ( 1 0 ) the resonance phenomena is possible in the case, when the natural frequency oscillation of the system linear analogue is less than the frequency of the perturbing force; while the parameter tends to 1 the resonance amplitude a* extremely increases. In the case of the «rigid» systems ( 0 ) the resonance phenomena is possible when the frequency of the linear analogue of the system is greater than the frequency of the perturbing force ; for the greater values of the nonlinear system parameter and the frequency of the perturbing force the resonance amplitude is greater. IV.

THE EXAMPLE OF T HE APPLICAT ION OF T HE DEVELOPED MET HODS FOR T HE INVEST IGAT ION OF T HE MAT HEMATICAL MODELS OF T HE ST RONG NONLINEAR VIBROPROT ECT ION SYST EMS. THE NUMERICAL SIMULAT ION In the last decade there is very popular anti-vibrating procedure, using the systems, known as quasi-zero stiffness vibro-protection systems [8]. In spite of the wide spectrum of such type systems, their mathematical models are described by

the ordinary one-type differential equations with strong nonlinearity, exactly

mx x x x3 f (t ) .

(14)

In the equation (14): m is the mass of the body under the action of the vibrating load, caused by the motion of the base in accordance with the law g (t ) . The function f (t ) is

f (t ) mg (t ) ; is the constant of proportionality in the resistant force, is taken for the sake of simplicity proportional to the motion speed of the body ; , are constants defined by stiffness of the system [8]; the function g (t ) and f (t ) too, are periodic on time and the maximum value of the periodic perturbation is small comparing to the restoring force, i.e.

max f (t ) max x x3 . Let note, that the constant can be the positive or negative, and can be equal to zero also. The numerical integration and the respective analysis of the results based on it was realized for some cases of the considered equation in [9]. To investigate the dynamical processes of the considered vibroprotection system is effectually to use the results of the sections II, III in this paper. In accordance with them the first-order approximation of the non-resonant process is described by the relationship

2 x(t ) aca 3,1, at m

,

2m

3 a. 5 4

4

The resonance oscillations on the frequency appear when the amplitude is close to a* 0.425

m

REFERENCES [1]

[2]

[3]

[4]

[6] [7]

[8]

[9]

(14)

and can be

described by some system of the differential equations and numerically found.

CONCLUSIONS

There is developed the procedure to investigate the dynamical processes in the strong nonlinear oscillation singleand multi-degree physical systems. The attribute of the considered systems is the next: the oscillation process of the unperturbed analogues can be described via the special periodic Ateb- functions; the frequency (period) of this process depends on the amplitude. The procedure of the investigation oscillations of the so called systems with quasi-zero stiffness, used to protect the equipment from the vibration load is developed. There are obtained the resonance conditions for such type systems and also are described the laws of the resonant and non-resonant amplitude changes. The proposed method of the investigating the oscillation processes of the strong nonlinear physical discrete structure systems helps to solve not only the analysis problems, but also the important problems of the oscillation systems synthesis on the projective phase, and to choose such dynamical systems characteristics that make the resonance phenomena impossible.

[5]

where the parameters a and are determined from the system: 3 1 2 2 4, a m (1,3) 7 4

3 4 1 4

V.

N. N. Bogolyubov, and Yu. A. Mitropol'skii. Asymptotic Methods in the T heory of Nonlinear Oscillations [Asimptoticheskiye metody v teorii nelineynykh kolebaniy]. Moscow: Nauka, 1974. [in Russian] E. A. Grebennikov, and Yu. A. Ryabov. Constructive Methods for the Analysis of Nonlinear Systems [Konstruktivnyye metody analiza nelineynykh sistem]. Moscow: Nauka, 1979. [in Russian] R. Bellman. A T heory of the Stability of Solutions of Differential Equations [Teoriya ustoychivosti resheniy differentsial'nykh uravneniy]. Moscow: Foreign Literature Publishing House, 1954. [in Russian] M.O. Perestyuk, and O. S. Chernihiv. Theory of the stability [T eoriya stiykosti]. Kyiv: P ublishing House "Kyiv", 2002. [in Ukrainian] Y. O. Kyrychenko, V. I. Samusya, V. Y. Kyrychenko, and A. V. Romanyukov, “Experimental investigation of aero-hydroelastic instability parameters of the deepwater hydrohoist pipeline,” Middle East Journal of Scientific Research, No. 18(4), pp.530–534, 2013. A. H. Nayfeh. Perturbation methods [Metody vozmushcheniy]. Moscow: Mir, 1976. [in Russian] P. Ya. Pukach, and I. V. Kuzio, “Nonlinear transverse vibrations of semiinfinite cable with consideration paid to resistance,” [Nelíníyní poperechní kolivannya napívneobmezhenogo kanata z urakhuvannyam oporu], Naukovyi Visnyk Natsіonalnoho Hіrnychoho Unіversitetu, No. 3, pp.82–86, 2013. [in Ukrainian] P. Ya. Pukach, and I. V. Kuzio, “Resonance fenomena in quasi-zero stiffness vibration isolation systems,” Naukovyi Visnyk Natsіonalnoho Hіrnychoho Unіversitetu, No. 3, pp. 62–67, 2015. T . Hayashi. Nonlinear oscillations in physical systems [Nelineynyye kolebaniya v fizicheskikh sistemakh]. Moscow: Mir, 1968. [in Russian]

I.

method;

INT RODUCT ION

The mathematical modeling is the important tool to solve the most of the scientific and engineering problems. It also helps to understand the kernel of the processes that are investigating in the modern physics, to interpret the known phenomena and to predict the new ones. The analysis and optimization of the physical, chemical dynamical processes in the technical systems in the cases of high speed, high pressure and energy demands the investigation of the complicated processes in the nonlinear oscillation theory. The strong nonlinear oscillation physical systems are widely applied in the engineering. They are well studied in the case while the ordinary linear or with the partial derivatives differential equations are used as the mathematical models. In the first case these equations describe the processes in the systems with the concentrated masses, in the second one in the system with the distributed parameters. Moreover, for the linear computing models of the oscillations physical systems basing on the superposition principle are developed the analytical methods in virtue of which one can state the reaction on the one- or multi-frequency periodical perturbations. Quasilinear systems i.e. systems with the small maximum values of the nonlinear forces comparing to the linear component of reducing force are similar to the linear systems. The analytical investigating methods for the most of the classes of the quasilinear systems are based on the different modifications of the perturbation methods in

particular asymptotic methods of the nonlinear mechanics. They give the possibility to describe and to state the special attributes of the oscillations in the quasilinear systems that are not proper to the linear systems with the constant in time physical-mechanical characteristics. On can speak firstly about resonance phenomena [1, 2] (main and combinational), process stability [3-5]. As to the more complicated systems , namely the strong nonlinear systems one can use the analytical investigating methods only in some cases . The nonlinear systems with the reducing force described by the power or close to power nonlinearity can be treated to this case [6, 7]. In reference to non-autonomous strong nonlinear systems, i.e. the systems under the external periodic perturbation (even small on value), it is necessary to realize the additional investigation. Their characteristics can be described solely. The main issue of the developed in the paper procedure of the investigating the resonance phenomena in the discussed class of the systems is based on idea to use the special periodic Ateb-functions constructing the solutions of the unperturbed analogue systems and to adopt the general ideas of the perturbation methods – in the case of the non-autonomous systems. II.

THE MAT HEMAT ICAL MODEL OF T HE ST RONG NONLINEAR SINGLE – DEGREE PHYSICAL SYST EM

In this section there is described the procedure of the investigation of the oscillations in the systems with one d egree freedom while the reducing force is approximated close to the power law. In this case the differential equation that describes such oscillations can be written as

mx cx 1 f x, x, where m is the mass of the material point; x is its coordinate at the arbitrary time moment; F ( x) cx 1 is function, describing the main part of the reducing force; c , are 2 p 1 constants, provided 1 ( p, q 0,1,2,... ); 2q 1

f x, x, is analytical 2 – periodic on t function,

is frequency of the external periodic perturbation acting on

the system; is the small parameter indicating on the small value of the external periodic perturbation and inconsiderable deviation of the restoring force from the power law. The restrictions concerning the parameter value are caused by the symmetry with respect to the coordinate origin of the function F ( x) cx 1 , describing the restoring force. More general representation of the reducing nonlinear force in the form

p

F ( x) c x x always with the necessary exactness

power can be approximated by the upper written depend ence, thereby all obtained results, one can transfer also on the systems with such type restoring force. Let’s notice, that function f x, x, can describe the nonlinear forces with the maximumе value being the small quantity comparing to the maximum value of the restoring force F ( x) cx 1 , that is

F ( x) max f x, x, . The last inequality can use the general ideas of the perturbation methods to construct the solution of the nonlinear equation (1). As per usual, it is necessary to describe the dynamical process of the unperturbed ( 0 ) or generating system, corresponding to (1). The next nonlinear equation is its analogue mx cx 1 0 .

(2)

The solution of the equation (2) is represented via periodic Ateb- functions as (3) x aca 1,1, (t ) ,

Fig. 1. Natural frequency’s dependence on the amplitude and the small nonlinear parameter

where a is the amplitude, (t ) is the oscillation phase of the unperturbed motion, (a) is the oscillation frequency equals

(a)

с 2 2m

a2 .

(4)

Hence, even for the unperturbed motion natural frequency of the dynamical process depends on amplitude. This is the first conceptual difference between the strong nonlinear and quasilinear systems. Thus, in case of the unperturbed motion natural frequency (a) depends on the mass of the material body and on the proportionality constant in the restoring force (analogue constant inelasticity), and on the amplitude and the nonlinear parameter also. This is the main difference of the considered class of the systems from the linear one. The issue is to construct the solutions of such strong nonlinear systems, taking into account all upper dynamical peculiarities , and to answer on the main questions of the investigating of the resonant oscillations. Fig. 1 demonstrates the dependence of the frequency (a) on the parameters defining the dynamical process of th e unperturbed motion, fig. 2 demonstrates the ratio of the periods of natural nonlinear to the linear 0 oscillations on the 1, 1 amplitude and on the parameter , i.e. . (a)

Fig. 2. T he ratio of the oscillation period of the nonlinear model to the oscillation period of the linear model

The parameters a and for the unperturbed motion are constants and are defined from the initial conditions x 0 x0 , satisfying the relationships 2

x0 a

2

V ( 2) 0 1 , 2a a

cta 1,1,

1, 1 . (a)

2 ( a ) . 2

Figures 1 and 2 demonstrate: 1) for the systems with small stiffness 1 0 the oscillation natural frequency of the physical system for the greater amplitude’s values is less; 2) while the nonlinear parameter converges to zero process in the system is considered as similar to the isochronous; 3) for the systems of the considerable stiffness 0 and for the oscillation amplitude values less than 1 for the greater values of nonlinear parameter oscillation natural frequency is less; for the oscillation amplitude values greater than 1 for its greater values correspond the greater values of the natural frequencies; 4) at the equal values of the parameters c and m the oscillation period of the strong nonlinear system is greater than the oscillation period of the system’s linear analogue 1 0 and a 1 and less at 0 і a 1 . It is necessary to take into account all these peculiarities in the resonance phenomena investigation. III.

RESONANCE PHENOMENA IN T HE ST RONG NONLINEAR PHYSICAL SYST EMS WIT H T HE DISCRET E ST RUCT URE

To develop the investigating methods for the reaction of the strong nonlinear system on the periodic non-autonomous type perturbations it is necessary to study the resonance conditions considering equation (2) as the model. The classic definition of the resonance phenomena can be applied for the considered systems noting the difference, that concept “frequency” treating as the meaning of the function (a) , is wider, than in the quasilinear systems. At first, the oscillation natural frequency of the system is defined not only by the phys ical and mechanical characteristics (parameters m, c ), but also by the amplitude a ; secondly, the periodic process of the unperturbed motion is described by the periodic Atebfunctions. The period 2 for the last ones by the argument (a)t depends on the nonlinear parameter and is defined by the relationship

1 2 . 1, 1 1 1 2 2

On account of this for the considered system class one can accept the derivative from the classic meaning instead of the main resonance, in other words – the natural oscillation period coincides with the perturbation period. Then the resonance condition transforms as

(5)

(6)

Hereafter the parameter value means the frequency of the external periodic perturbation (force). Taking into account the function, describing “natural frequency”, one can obtain the parameter value a* , responsible for the resonance phenomena: 2

1, 1 2 a . k 2 *

(7)

c . Thus, the resonance phenomena in the m investigated dynamical systems appears when the oscillation where k

amplitude is close to a* . If the amplitude is greater than a* , then existing resistant forces in the real systems produce the decrease of the oscillation amplitude to the value a* , causing the resonance phenomena in the future. This explains the second principle difference of the resonance phenomena in the strong nonlinear systems. If there exist one or several stationary oscillation modes in the perturbed strong nonlinear system, then the investigation of the physical system reaction on the periodic perturbation must be considered separately. Fig. 3 demonstrates the dependence of the resonance amplitude on the parameters , that characterized the physical and mechanical system properties and forced periodic perturbation frequency. In particular, from the obtained in the paper plotted dependences follows, that for the «soft» systems ( 1 0 ) the resonance phenomena is possible in the case, when the natural frequency oscillation of the system linear analogue is less than the frequency of the perturbing force; while the parameter tends to 1 the resonance amplitude a* extremely increases. In the case of the «rigid» systems ( 0 ) the resonance phenomena is possible when the frequency of the linear analogue of the system is greater than the frequency of the perturbing force ; for the greater values of the nonlinear system parameter and the frequency of the perturbing force the resonance amplitude is greater. IV.

THE EXAMPLE OF T HE APPLICAT ION OF T HE DEVELOPED MET HODS FOR T HE INVEST IGAT ION OF T HE MAT HEMATICAL MODELS OF T HE ST RONG NONLINEAR VIBROPROT ECT ION SYST EMS. THE NUMERICAL SIMULAT ION In the last decade there is very popular anti-vibrating procedure, using the systems, known as quasi-zero stiffness vibro-protection systems [8]. In spite of the wide spectrum of such type systems, their mathematical models are described by

the ordinary one-type differential equations with strong nonlinearity, exactly

mx x x x3 f (t ) .

(14)

In the equation (14): m is the mass of the body under the action of the vibrating load, caused by the motion of the base in accordance with the law g (t ) . The function f (t ) is

f (t ) mg (t ) ; is the constant of proportionality in the resistant force, is taken for the sake of simplicity proportional to the motion speed of the body ; , are constants defined by stiffness of the system [8]; the function g (t ) and f (t ) too, are periodic on time and the maximum value of the periodic perturbation is small comparing to the restoring force, i.e.

max f (t ) max x x3 . Let note, that the constant can be the positive or negative, and can be equal to zero also. The numerical integration and the respective analysis of the results based on it was realized for some cases of the considered equation in [9]. To investigate the dynamical processes of the considered vibroprotection system is effectually to use the results of the sections II, III in this paper. In accordance with them the first-order approximation of the non-resonant process is described by the relationship

2 x(t ) aca 3,1, at m

,

2m

3 a. 5 4

4

The resonance oscillations on the frequency appear when the amplitude is close to a* 0.425

m

REFERENCES [1]

[2]

[3]

[4]

[6] [7]

[8]

[9]

(14)

and can be

described by some system of the differential equations and numerically found.

CONCLUSIONS

There is developed the procedure to investigate the dynamical processes in the strong nonlinear oscillation singleand multi-degree physical systems. The attribute of the considered systems is the next: the oscillation process of the unperturbed analogues can be described via the special periodic Ateb- functions; the frequency (period) of this process depends on the amplitude. The procedure of the investigation oscillations of the so called systems with quasi-zero stiffness, used to protect the equipment from the vibration load is developed. There are obtained the resonance conditions for such type systems and also are described the laws of the resonant and non-resonant amplitude changes. The proposed method of the investigating the oscillation processes of the strong nonlinear physical discrete structure systems helps to solve not only the analysis problems, but also the important problems of the oscillation systems synthesis on the projective phase, and to choose such dynamical systems characteristics that make the resonance phenomena impossible.

[5]

where the parameters a and are determined from the system: 3 1 2 2 4, a m (1,3) 7 4

3 4 1 4

V.

N. N. Bogolyubov, and Yu. A. Mitropol'skii. Asymptotic Methods in the T heory of Nonlinear Oscillations [Asimptoticheskiye metody v teorii nelineynykh kolebaniy]. Moscow: Nauka, 1974. [in Russian] E. A. Grebennikov, and Yu. A. Ryabov. Constructive Methods for the Analysis of Nonlinear Systems [Konstruktivnyye metody analiza nelineynykh sistem]. Moscow: Nauka, 1979. [in Russian] R. Bellman. A T heory of the Stability of Solutions of Differential Equations [Teoriya ustoychivosti resheniy differentsial'nykh uravneniy]. Moscow: Foreign Literature Publishing House, 1954. [in Russian] M.O. Perestyuk, and O. S. Chernihiv. Theory of the stability [T eoriya stiykosti]. Kyiv: P ublishing House "Kyiv", 2002. [in Ukrainian] Y. O. Kyrychenko, V. I. Samusya, V. Y. Kyrychenko, and A. V. Romanyukov, “Experimental investigation of aero-hydroelastic instability parameters of the deepwater hydrohoist pipeline,” Middle East Journal of Scientific Research, No. 18(4), pp.530–534, 2013. A. H. Nayfeh. Perturbation methods [Metody vozmushcheniy]. Moscow: Mir, 1976. [in Russian] P. Ya. Pukach, and I. V. Kuzio, “Nonlinear transverse vibrations of semiinfinite cable with consideration paid to resistance,” [Nelíníyní poperechní kolivannya napívneobmezhenogo kanata z urakhuvannyam oporu], Naukovyi Visnyk Natsіonalnoho Hіrnychoho Unіversitetu, No. 3, pp.82–86, 2013. [in Ukrainian] P. Ya. Pukach, and I. V. Kuzio, “Resonance fenomena in quasi-zero stiffness vibration isolation systems,” Naukovyi Visnyk Natsіonalnoho Hіrnychoho Unіversitetu, No. 3, pp. 62–67, 2015. T . Hayashi. Nonlinear oscillations in physical systems [Nelineynyye kolebaniya v fizicheskikh sistemakh]. Moscow: Mir, 1968. [in Russian]