Mechanical Systems and Signal Processing 154 (2021) 107560

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Mechanical Systems and Signal Processing

journal homepage: www.elsevier .com/locate /ymssp

Nonlinear oscillations of coupled pendulums subjectedto an external magnetic stimulus

https://doi.org/10.1016/j.ymssp.2020.1075600888-3270/� 2020 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: krystian.polczynski@dokt.p.lodz.pl (K. Polczynski), skurserg@gmail.com (S. Skurativskyi), maksymilian.bednarek@dokt

(M. Bednarek), jan.awrejcewicz@p.lodz.pl (J. Awrejcewicz).

Krystian Polczynski a, Sergii Skurativskyi b, Maksymilian Bednarek a, Jan Awrejcewicz a,⇑a Lodz University of Technology, Department of Automation, Biomechanics and Mechatronics, Lodz, Polandb Subbotin Institute of Geophysics, NAS of Ukraine, Kyiv, Ukraine

a r t i c l e i n f o

Article history:Received 17 September 2020Received in revised form 7 December 2020Accepted 20 December 2020Available online 8 January 2021

Keywords:VibrationChaosHarmonic balance methodCoupled pendulumMagnetic field

a b s t r a c t

The nonlinear forced vibrations of two pendulums coupled by an elastic element are con-sidered. One of the pendulums is subjected to the magnetic stimulus produced by a pair ofa periodically powered coil and a permanent magnet mounted at the end of the pendulum.We extend our previous experimental and numerical results presented in Polczynski et al.(2019) and proposed a new rational approximation of a magnetic torque to enable theapplication of analytical methods. Based on this approximation, the numerical and analyt-ical studies of equations of motion are carried out. In particular, the resonant phenomenaand periodic regimes are studied in detail. The procedure to approximate the subharmonicand almost harmonic regimes is developed using the balance harmonic method. The suc-cessive maxima map has been employed to study more complicated regimes. This tech-nique allows one to reveal changes in profile’s shapes for various driving frequenciesand shows the existence of strongly chaotic trajectories.

� 2020 Elsevier Ltd. All rights reserved.

1. Introduction

Pendulums are one of the most popular objects in physics and were studied many times in terms of different kinds ofexcitation sources. Pendulum systems are mostly forced by mechanical excitations such as motors [1,2] or unbalancedmasses [3], apart from that, a large group of mechanical systems used a magnetic interaction as a forcing source. Magneticexcitations are usually based on an interaction between the pair of magnets or various magnet-coil configuration. One of thefirst researchers that analysed such a system was Bethenod [4,5]. Sustained and undamped oscillations of a pendulum wereanalysed analytically taking into account both mechanical and electrical parts of a system. A different look at the same sys-tem is presented in works [6,7]. Authors considered the magnetic pendulum as two conjugated autonomous systems insteadof one nonautonomous. Khomerik [8] detected that parametric resonance induced chaos in a magnetic pendulum and thecondition of its appearance has been given. Dynamics of a special magnetic pendulum called Duboshinskiy pendulum havebeen studied in [9–11]. Argumental oscillations and subharmonic frequency responses were observed and discussed. Ponteset al. [12] investigated the contribution of electrical parameters on the dynamics of the oscillating electromechanical system.The system has a stiffness of Duffing type and an RLC circuit acted as a damper. Extended studies of bifurcation phenomenahave been performed in terms of different values of the capacitance. Bednarek et al. [13] have studied force interactions

.p.lodz.pl

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

between magnets as well as active damping in oscillating system realised by a magnetic field. Magnet-magnet and magnet-coil interactions have been investigated on the experimental stand with aerostatic supports.

Two and more degrees of freedom systems consist of coupled magnetic pendulums have been investigated in [14–17].The problem of synchronization of oscillations in a system of two coupled magnetic pendulums has been considered by Frad-kov et al. [14]. Pulse-modulated control law has been derived based on the speed-gradient method. Polczynski et al. [15,16]have focused on a bifurcation analysis of two torsionally coupled pendulums subjected to an alternating magnetic field. Peri-odic, quasi-periodic and chaotic solutions have been investigated numerically and experimentally. Double pendulum with amagnetic interaction has been studied by Wojna et al. [17]. The mathematical model that includes complex friction torqueand elaborate magnetic torque has been verified by the bifurcation analysis of the motion and confronted with experiments.Normal modes of a double pendulum at low energy levels have been studied by Kovacic et al. [18]. Relationships betweeninitial positions of the pendulums’ links and frequencies of the linear and nonlinear modes have been examined. Two degreesof freedom mechanical system of coupled oscillators equipped with strongly nonlinear magnetic springs has been examinedexperimentally and numerically by Witkowski et al. [19]. Chaotic attractor crisis, as well as quasi-periodic regimes and finiteperiodic doubling cascades, have been noted and discussed. Extensive bifurcation and dynamical analysis of a similar systemwith impacts but reduced to the one degree of freedom has been conducted in [20,21]. Statistical properties of chaotic impactevents have been studied in detail. Furthermore, the impact sequence has been confirmed as a stationary but non-Poissonianand contained temporal scales. Darula and Sorokin [22] have analysed linear and weakly nonlinear dynamics of a coupledelectro-magneto-mechanical system. Multiple scales method has been employed to obtain a steady-state response of thesystem subjected to a harmonic close-resonance excitation.

Coupled magnetic pendulums might be also used as an energy harvesting systems [23–25]. Double pendulum systemwith a magnet moving in vicinity of electric coils [23] has shown that harvested power increases when the system motionundergoes a chaotic regime. Malaji et al. [24] have reported that harvesting capabilities may be enhanced by finding the opti-mal initial conditions in a harvester consists of two torsionally coupled pendulums. Moreover, in other similar pendula sys-tem [25], they have observed that nonlinear responses improve individual harvester performance due to adding mechanicalcoupling to the system which had only a magnetic one.

It is evident that the coupled pendulum systems incorporating different types of interactions are widespread in technicalapplications. Investigations of these systems can be an essential challenge due to the presence of strong nonlinearities. Thefine effects related to the details of the interaction description and mutual influence of force fields can appear in experi-ments, in particular as it is observed in our previous works [15,16,27,28]. In this paper, to a deeper understanding of thedynamics of a complex pendulum system with magnetic interaction, we modified our numerical studies of the model andsupplemented them by analytical approaches.

This paper is constructed in the following way: Section 2 contains the statement of problems and concise description ofthe experimental rig. Section 3 gives the results of experimental and numerical studies of the mathematical model. Section 4deals with the analytical considerations of steady solutions of the unperturbed system. The two-harmonic approximation ofthe system solutions are presented in Section 5. The analysis of successive maxima map is outlined in Section 6. Section 7contains general conclusions.

2. Experimental equipment and modified mathematical approach

At the beginning of this section, we introduce the results of our previous work [15] such as experimental rig and corre-sponding mathematical description. After that, we present modifications of the mathematical models which will be studiedlater.

Fig. 1a shows the experimental stand of two coupled pendulums. The pendulums are marked as (1) and (2). The pendu-lum (2) has a neodymium magnet (3) at the end of its arm. An electric coil (4) is placed under the pendulum (2). The coillocated under the pendulum (1) was not used in the experiments presented in this article. The end of the pendulum (1)has a brass element characterized by the same dimensions and mass as the magnet of the pendulum (2). Both pendulumsare suspended on shafts (6) which are joined by a flexible element (7). Each of the shafts is supported by two rolling bearingswhich pillow block is attached to a frame (8). The pendulum (2) is excited by the magnetic field produced by the coil. Thedetailed characteristics of experimental equipment can be found in [7,15,16,27,28]. Fig. 1b displays a scheme of the physicalmodel of the studied system. The explanation of the moments affecting the physical model is provided later in this section.The shape of a dimensionless current signal send to the coil by the electronic system is shown in Fig. 2. In practice, both fre-quency f ¼ 1=tp and duty cycle d ¼ t2=tp of the rectangular waveform may be controlled, however, due to analyticalapproaches employed to an analysis of the system, we set the value of duty cycle on d ¼ 50%. The amplitude of the currentalso was fixed on 1 A during dynamical experimental tests. The polarization of the magnetic interaction between the magnetand the coil causes their repelling.

In the cases when sources of a magnetic force cannot be considered as a point objects, the understanding of phenomenainduced by the magnetic interactions and mathematical description of them become a complicated problem. As it has beenshown in [15], the experimental data of the magnetic moment were fitted by the following function

2

Fig. 1. The experimental stand of two axially coupled pendulums (a), where: (1) – pendulum 1, (2) – pendulum 2, (3) – magnet, (5) – brass element,(4) – electric coil, (6) – shaft, (7) – flexible element, and (8) – frame. Scheme of the studied system with torques acting on it (b).

Fig. 2. The shape of the current signal (a) delivered to the coil during the experiment. The frequency is defined as f ¼ 1=tp and the duty cycle is described asd ¼ t2=tp . The rectangular signal used in numerical simulations and its approximation by three terms of Taylor series (b) at f ¼ 3=2 Hz and d ¼ 50%.

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

MmagðxÞ ¼ asgnðxÞ exp � sgnðxÞxþ bc

� �2" #

; ð1Þ

where a; b; c are parameters defined during the fitting process. Then the mathematical model describing the dynamics of thecoupled pendulums (see Fig. 1b) is fully defined by the following system [15]

J1X00 ¼ �ðc1X0 þ ceðX0 � Y 0Þ þ cB1sgnX

0Þ �mgs � sinX � keðX � YÞ;J2Y

00 ¼ MmagðYÞ � IðtÞ � ðc2Y 0 þ ceðY 0 � X0Þ þ cB2sgnY0 Þ �mgs � sinY � keðY � XÞ; ð2Þ

where X and Y are the angular displacements of the pendulum 1 and 2, respectively. J1 and J2 are the moments of inertia,terms c1 and c2 denote pendulums’ viscous frictions, whereas cB1 and cB2 define the static friction torques. Term mgs is asso-ciated with the gravitational force mg acting in the centre of pendulums’ mass placed in the distance of s from the rotationaxis. Stiffness of the elastic element is defined by ke, while ce represents its viscous damping factor. Furthermore, the dimen-sionless rectangular current signal shown in Fig. 2 is expressed by a formula

IðtÞ ¼ 12

1þ sgn½sin 2pf ðt þ t0Þ � sin 2pft0�ð Þ; t0 ¼ 1� 2d=1004f

; ð3Þ

where parameter f stands for frequency and d defines the percentage duty cycle.According to the previous experimental and numerical studies [15,16,27,28], the system manifests different types of

dynamical behaviour such as periodic, multiperiodic, quasiperiodic, chaotic regimes, chains of bifurcations, etc. Some ofthose phenomena can be studied analytically in more detail to get a deep understanding of the influence of a magnetic fieldincorporated in the model. In this paper, we supported experimental and numerical results by analytical investigations. Inthe first step of our analytical investigation, we need to simplify Eq. (2).

3

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

One can note that this model is invariant with respect to the transformation fX;Yg ! f�X;�Yg. From this, it follows thatif the model admits the solution fXc; Ycg then it admits the solution f�Xc;�Ycg as well. Since Eq. (2) contains the nonsmoothfunctions IðtÞ and non-polynomial functionsMmagðxÞ and sinðxÞ, the application of analytical methods is essentially restricted.Thus, to adjust the model suitable for further analytical studies let us modify Eq. (2). The rational function has beenemployed to approximate moment of the magnetic force instead of Eq. (1) and it reads

Fig. 3.coil cur

MRmagðx;p; qÞ ¼ px=q

1þ ðx=qÞ4; ð4Þ

where parameters p and q are estimated in such a way to fit best the experimental data. The results of approximation andexperimental data are presented in Fig. 3. The experiment has been conducted for different steady values of the coil currentI ¼ 0:25; 0:5; 0:8; 1 A. It turned out that the parameter q is changed by a very small value with regard to the coil current.This allows us to assume that parameter q ¼ 0:127 and one can approximate experimental data by means of p variation only.The Mathematica fit procedure provides a little deviation of q values from 0.127, therefore, for our dynamical studies atI ¼ 0:25 A, values of the mentioned parameters during investigations were fixed on q ¼ 0:121092; p ¼ 0:1198114. WhenI ¼ 1 A, the parameters are as follows q ¼ 0:124767; p ¼ 0:493017.

Since the amplitudes of considered oscillations are quite small, the sin x function appearing in Eq. (2) was replaced by twofirst terms of the Taylor series

sin x ¼ x� x3=6; ð5Þ

which is valid for small deviations of x (pendulum position).Taking into account the suggestions mentioned above, Eq. (2) can be presented in the modified form

J1X00 ¼ �c1X

0 � ceðX0 � Y 0Þ � cB1sgnX0 �mgsðX � X3=6Þ � keðX � YÞ;

J2Y00 ¼ MRmagðYÞ � IðtÞ � c2Y

0 � ceðY 0 � X 0Þ � cB2sgnY0 �mgsðY � Y3=6Þ � keðY � XÞ:

ð6Þ

The aim of the studies is to elucidate the properties of observed regimes in experiments and numerical simulations of Eq. (6)by applying analytical methods.

3. Experimental and numerical bifurcation diagrams of the dynamics of coupled pendulums

Numerical computations have been carried out for parameters obtained in the paper [15], their values are as follow:

J1 ¼ 6:8025 � 10�4 kgm2; J2 ¼ 6:7101 � 10�4 kgm2,mgs ¼ 0:0578 Nm, ke ¼ 145:073 � 10�4 Nm/rad, ce ¼ 13:7 � 10�5 Nms/rad,

c1 ¼ 3:1 � 10�5 [Nms/rad], c2 ¼ 7:2 � 10�5 [Nms/rad], cB1 ¼ 27:523 � 10�5 [Nm], cB2 ¼ 27:888 � 10�5 [Nm]. The experimentalbifurcation diagrams are presented in Figs. 4,5a, b whereas the numerical ones are depicted in Figs. 4,5c, d. Bifurcation stud-ies have been derived for two values of current: 0.25 A and 1 A.

Comparison of experimental and numerical bifurcation diagrams shows that qualitatively their structure is similar. Atfirst glance, the system may seem simple, but in fact, it is stiff and therefore complex for studying even by the numericalmethods.

The experimental data of the moments of the magnetic forces (dots) and their approximations (solid curves) by the function (4) obtained for variousrents I ¼ 0:25; 0:5; 0:8; 1 A, and related parameters q ¼ 0:127; p 2 f0:1198114; 0.23853; 0.389356; 0:4930167g.

4

Fig. 4. The experimental (a, b) and numerical (c, d) bifurcation diagrams at increasing frequency f, when the coil current is 0.25 A. The labels ‘‘1” and ‘‘2”mark the bifurcation curves for the symmetric periodic regimes coexisting in the system’s phase space. They correspond to the different initial dataspecified near the bifurcation point f � 3:1 Hz.

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

4. Steady solutions of the unforced pendulum system

To explain the phenomena observed during the studies, let us derived some analytical solutions of Eq. (6). At first, con-sider the steady solutions of an autonomous counterpart of Eq. (6), i.e., when IðtÞ ¼ const. Such solutions are important tounderstand the coexistence of attractors and resonant phenomena studies.

Forcing signal IðtÞmight be considered as a perturbation of the steady state IðtÞ ¼ 1=2. Because of that, we have dealt withEq. (6) at IðtÞ ¼ 1=2 considering the steady solutions ðX;YÞ ¼ ðA; BÞ ¼ const. Due to the system’s symmetry, the numbersð�A;�BÞ also provide the solution.

These solutions defining the symmetrically located fixed points in the system’s phase space satisfy the following algebraicsystem

mgsðA� A3=6Þ þ keðA� BÞ ¼ 0;

½mgsðB� B3=6Þ þ keðB� AÞ� � ðq4 þ B4Þ � pq3

2B ¼ 0:

ð7Þ

To understand the structure of the system’s solutions, at first, we assume that p ¼ 0, hence Eq. (7) is solved in quadratures.Among the possible solutions, there are roots such that A ¼ B and A ¼ �Bwhich location in phase space is depicted in Fig. 6a.The maximal number of real-valued roots is 9.

When p increases to p ¼ 0:493017, the curve of the second equation becomes more complex, therefore, the number ofroots increases up to 15 (Fig. 6b). As experiments have shown, the red points Q� ¼ ð�0:0575048;�0:286489Þ play the nota-ble role in the solution formation.

Studies on the stability of fixed points yield resonant frequencies. Analyzing the fixed point at the origin, the linearizedsystem reads

J1X00 þmgsX þ keðX � YÞ ¼ 0;

J�2Y

00 þmgsY þ keðY � XÞ � p2q

Y ¼ 0:ð8Þ

It is well known [29,30] that the periodic solutions of this system define the resonant frequencies of the initial system.Applying the conventional approach, let us seek the solution of Eq. (8) in the form fX;Yg � exp ks that leads us to thecharacteristic equation

5

Fig. 5. The experimental (a, b) and numerical (c, d) bifurcation diagrams at increasing frequency f, when the coil current is 1 A. The labels ‘‘1” and ‘‘2” markthe bifurcation curves for the symmetric periodic regimes coexisting in the system’s phase space.

Fig. 6. Graphical solution of Eq. (7) at p ¼ 0 (a) and p ¼ 0:493017 (b). The curves labelled as 1 and 2 correspond to the first and second equations of thesystem (7), respectively.

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

J1k2 þmgsþ ke �ke

�ke J2k2 þmgsþ ke � p=ð2qÞ

���������� ¼ 0:

The complex-valued eigenvalue providing the resonant frequency existence is as follows

k2 ¼ � 12J1J2

ðke þmgsÞðJ1 þ J2Þ � J1p2q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðke þmgsÞðJ1 þ J2Þ � J1

p2q

� �2

� 4J1J2 2kemgsþmgs2 � p2q

ke þmgsð Þ� �s2

435:

6

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

In fact, the resonant frequency is described by f res ¼ 12p

ffiffiffiffiffiffiffiffiffi�k2

p. When the parameters of the system correspond to the cur-

rent of 1 A, the resonant frequency is f res ¼ 1:64209 Hz. Note that, k2 ! � 12J1 J2

ðke þmgsÞðJ1 þ J2Þ for p ! 1. Within the

accepted approximation, the limiting resonant frequency is f lim ¼ 12p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkeþmgsÞðJ1þJ2Þ

2J1J2

q¼ 1:64652 Hz which is close to exact value

of f res.Using the predicted frequency value of resonant growth of solution’s amplitude, the experimental data ðf ;XmaxÞ (Fig. 7,

black points) were obtained. Numerical amplitude curve (Fig. 7, blue points) was derived by means of Eq. (6). It is evidentthat in the vicinity of f res, the amplitude peak is observed. As numerical treatment has shown, the shape of curve is sensibleto the variation of parameters related to friction and elastic element.

5. Two-harmonic approximation of model’s solutions

According to the numerical studies, Eq. (6) possesses periodic solutions which can be approximated by the finite set ofharmonics. Before the analytic treatments, let us compare the quantities describing different effects in the model. Introduc-ing the transformation of time variable t ¼ ffiffiffiffi

J1p

s to Eq. (6), the system can be given in the quasi-nondimensional form with

new parameters c�1;2 ¼ c1;2=

ffiffiffiffiJ1

p; J

�2 ¼ J2=J1; c

�e ¼ ce=

ffiffiffiffiJ1

p. Taking into account the specified values of ‘‘old” parameters, the

new recalculated parameters are fJ�2; c

�1; c

�2; c

�eg = f0:986417, 0.00118858, 0.00276057, 0.00526655g. Since

cBi � 10�4 cj � 10�3; i; j ¼ 1;2, this allows us to neglect dry friction in comparison with viscous friction. We are lookingfor periodic solutions of the model without dry friction terms. Thus, the governing equations take the following form

J1X00 þ c1X

0 þ ceðX0 � Y 0Þ þ keðX � YÞ þmgs X � X3

6

!¼ 0;

J2Y00 þ c2Y

0 þ ceðY 0 � X0 Þ þ keðY � XÞ þmgs Y � Y3

6

!( )� ðq4 þ Y4Þ ¼ pq3Y � IðtÞ:

ð9Þ

The periodic solutions were obtained using the harmonic balance method [26]. According to the method, the solutions sup-posed to have the following form

X ¼ A1sin 12xt þ B1cos 1

2xt þ A3sin 32xt þ B3cos 3

2xt ;

Y ¼ R1sin 12xt þ Q1cos 1

2xt þ R3sin 32xt þ Q3cos 3

2xt; x ¼ 2pf :ð10Þ

To get the algebraic equations with respect to the coefficients, we have inserted the relations expressed by Eq. (10) into Eq.(9). Furthermore, the trigonometric expressions have been transformed into the sum of trigonometric functions with a com-bined argument using the Mathematica command TrigReduce. Finally, we equate the coefficients ofsin kxt; cos kxt; k ¼ 1=2;3=2 to zero.

During these manipulations, the rectangular signal IðtÞ has been reduced to the Fourier series (Fig. 2b)

IðtÞ ¼1; 0 < t < T=20; T=2 < t < T

Iðt � TÞ; t > T

8><>: ¼ 1

2þ 2pXNj¼0

sinxtð2jþ 1Þ2jþ 1

; T ¼ 1f; N ¼ 1:

The resulting algebraic system is composed of eight equations, which number can be reduced to the four via the exclusion ofthe unknown variables R1;3;Q1;3 using the relations presented in A.

Fig. 7. The experimental (black dots) and numerical (blue dots) amplitude curves at increasing frequency f, when the current is 1 A.

7

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

The rest of equations forms a highly nonlinear system that requires the proper initial data to get the correct result.We inves-tigated systembehaviour forfixeddriving frequency f ¼ 3=2Hz. Thenumerically derivedprofiles are shown inFig. 8(a-c). Four-ier analysis (Fig. 8a) has shown that the regular regime is characterized by two prevailingmodes located at frequencies f=2 and3f=2. The coefficients of the solution of Eq. (10) are derived from the algebraic system by the iteration procedure subjected tothe initial data, for instance, A1¼B1¼A3¼B3¼0:01. Then, we get the roots A1¼0:0707558;B1¼0:0746015;A3¼0:00927566;B3¼�0:00730256, R1¼0:287187;Q1¼0:284754;R3¼�0:0345941;Q3¼0:038385. The comparison of approxi-mated and numerical solutions is presented in Fig. 8a, c. One can note that the two-mode solution of Eq. (10) provides a goodapproximation of the actual regime. The first solution’s component, i.e. X, is described more accurate than the other one. Thistestifies about more nonlinear character of oscillations of the second pendulum.

According to the bifurcation diagram (Fig. 5), there is a wide window corresponding to the periodic regime existence.Numerical computations of Eq. (9) obtained for f ¼ 4 Hz have yielded a periodic solution which phase portrait is shownin Fig. 9a. Fourier analysis of X solution component (see Fig. 9a, inset) contains only two prevailing maxima, located at zerofrequency and forcing frequency of 4 Hz (doubled frequency is negligible). This suggests looking for the analytic solution inthe form

Fig. 8.frequencomponversion

X ¼ B0 þ A1 sinxt þ B1 cosxt; Y ¼ Q0 þ R1 sinxt þ Q1 cosxt: ð11Þ

Applying to Eq. (11) similar mathematical manipulations as for Eq. (10), we obtain six-dimensional algebraic system withrespect to B0;A1;B1;Q0;R1;Q1. It turned out the system possesses several sets of roots. Among them, there is a root providingthe most closest approximation of the numerical solution (Fig. 9b, c), namely B0 ¼ �0:0651404;A1 ¼ �0:0032589;B1 ¼ �0:0009147;Q0 ¼ �0:3244887;R1 ¼ 0:08111946;Q1 ¼ 0:0022859. Note that the approximation of Y component is bet-ter than X. It is also worth noting that the derived solution of Eq. (11) is associated with the fixed point Q�.

6. Some remark on the complex model’s regimes

As shown in Section 3, in addition to model’s solutions which can be presented as a relatively simple combination oftrigonometric functions, there are also other more complex trajectories. Those trajectories appear when subharmonicsand several high-frequency components are excited. Moreover, it is observed chaotic attractors characterized by the contin-

The solution of the X-component of Eq. (6) (red solid curve), its Fourier spectrum (b) and the Y-component solution (c), when the current is 1 A andcy f ¼ 3=2 Hz. Dashed curves depict the solution’s profiles of Eq. (6) when the dry friction is omitted. The solid black curves correspond to the two-ent analytic solutions represented by Eq. (10). (For interpretation of the references to colour in this figure legend, the reader is referred to the webof this article.)

8

Fig. 9. The phase portrait of Eq. (6) (a) and profiles of X;Y components at the frequency f ¼ 4 Hz. Inset shows the Fourier spectrum of X solution component.Dashed curves show the approximate solution of Eq. (11).

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

uous Fourier spectrums. The analysis of that solutions is based on the successive maximamap. This approach is useful in casewhen high-frequency harmonics increase and change the solution’s profile over a period. Since the solution’s period does notchange, the traditional Poincaré section technique misses these changes of profile shape.

Thus, to construct the map Tj ¼ maxfXðtÞg; j ¼ 1; . . ., we have fixed the driving frequency on f ¼ 0:87 Hz. The system’sprofile is presented in Fig. 10a and is characterised by two different successive maxima. As shown in the inset in Fig. 10b,these maxima produce two different fixed points in the map. Note that, the Poincaré section provides only a single point.When the frequency f ¼ 0:9 Hz the map possesses four fixed points that tells us about the existence of four different maximain the X-profile. Further growth of f leads to the chaotic attractor formation. The corresponding map derived at f ¼ 0:9015 Hz(Fig. 10b) shows the ”cloud” of points without clearly visible one-dimensional structure like in the Rössler attractor [36] orLorenz system [35]. However, in the figure one can distinguish several zones with more dense filing which relate to prevail-ing values of profile deviations during the chaotic mode.

Fig. 10. Map construction (a) for the X-profile derived at the frequency f ¼ 0:87 Hz (labels ‘‘}”, inset) and resulting successive maxima maps (b) atf ¼ 0:9015 Hz (black points), 0.9 Hz (labels ‘‘�”, inset).

9

Fig. 11. The Lameray diagrams of the map Gðy;0:02Þ (a) and Gðy;�0:05Þ (b).

K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

In particular, we can assume that chaotic sequence Tj incorporates the subsequence Gj Tj producing the almost straightlines in the diagram of Fig. 10b. The simplest way to define this map is to consider the explicit piecewise linear map in theform

Gðy; rÞ ¼�0:4725þ 6:25y for y < 0:17;ð1� rÞy for 0:17 6 y < 0:4;0:01þ 0:2y for 0:4 6 y < 0:7:

8><>: ð12Þ

The parameter r describes the small deviation of the middle segment from the bisector. Fig. 10b shows the maps Gðy;0:02Þand Gðy;�0:05Þ. To study the properties of the sequences generated by function (12), the Lameray diagram technique is used.After omitting the transient processes, the chaotic attractors are revealed in both cases. Moreover, the similar chaotic beha-viour is remained, when the parameter r varies in more wide range. Certain kind of maps represented by Eq. (12) and similarones [31–34,36], gives the possibility to obtain rigorous results concerning system’s evolution, e.g., bifurcation scenarios,sources, properties, and prediction of chaos. Therefore, the map (12) should be studied deeper to find out its behavior withrespect to the variation of r, number of segments, their lengths, and slop. (see Fig. 11)

7. Concluding remarks

This research is concerned with the nonlinear dynamics of coupled pendulums subjected to the magnetic excitation. Bothexperimental and analytical studies of this model were described. The special attention is paid to the correct description ofstrong magnetic field and understanding its influence on the system’s evolution. Since the use of traditional dipole approx-imation for magnetic force modelling leads to cumbersome calculations, the direct approximation of magnetic torque wasperformed. To ensure the necessary accuracy without losing the possibility of analytical methods application, the two-parametric rational functions were chosen for these purposes.

On the base of these rational approximations, the pendulums equations of motion were modified and solved, both numer-ically and analytically. According to the numerical approach, the bifurcational diagrams describing the development of theperiodic, multiperiodic and chaotic regimes were constructed. The analytical studies allowed us to approximate the subhar-monic periodic regimes at low values of driving frequencies. It is worth noting that these regimes do not contain harmonicscorresponding to values of the driving frequency. When driving frequency increased, the appearance of another periodicregime was observed. Unlike the aforementioned solution, this regime was approximated by the harmonics correspondingto values of the driving frequency.

Besides periodic solutions, chaotic attractors exist as well. Note that the scenarios of a chaos formation are not accompa-nied by period doubling cascades. The chaotic zones in bifurcation diagrams (their width, location, amplitude) are sensitiveto the details of the model, in particular to friction coefficient variation. In the system, the developed chaos is encountered.Moreover, the prevailing amplitude of oscillations can be selected during the analysis of successive maxima map.

CRediT authorship contribution statement

Krystian Polczynski: Data curation, Methodology, Conceptualization, Investigation, Software, Writing - original draft,Writing - review & editing. Sergii Skurativskyi: Software, Validation, Investigation, Methodology, Writing - original draft,Writing - review & editing. Maksymilian Bednarek: Software, Writing - original draft, Writing - review & editing. Jan Awre-jcewicz: Conceptualization, Supervision, Project administration.

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K. Polczynski, S. Skurativskyi, M. Bednarek et al. Mechanical Systems and Signal Processing 154 (2021) 107560

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could haveappeared to influence the work reported in this paper.

Acknowledgments

This work has been supported by the Polish National Science Centre under the grant OPUS 14 No. 2017/27/B/ST8/01330.The first and third authors would like to thank for the financial support from the project POWR.03.02.00-00-I042/16-00 ofthe National Center for Research and Development.

Appendix A

The coefficients R1;3 and Q1;3 of solution (10) are derived by means of the following relations:

R1 ¼ cexD1þ2keD2Da

; Q1 ¼ 2keD1�cexD2Da

;

R3 ¼ 3cexD3þ2keD4Db

; Q3 ¼ 2keD3�3cexD4Db

;

where Da ¼ 4ðc2ex2 þ 4k2e Þ,

Db ¼ 12ð9c2ex2 þ 4k2e Þ,D1 ¼ mgsfB1ðA2

1 þ B21 þ 2A1A3 þ 2ðA2

3 þ B23 � 4ÞÞ � B3ðA2

1 � B21Þg þ B1ð2I1x2 � 8keÞ � 4A1xðc1 þ ceÞ,

D2 ¼ mgsfA1ðA21 þ B2

1 � 2B1B3 þ 2ðA23 þ B2

3 � 4ÞÞ � A3ðA21 � B2

1Þg þ A1ð2I1x2 � 8keÞ þ 4B1xðc1 þ ceÞ,D3 ¼ mgsfB1ðB2

1 � 3A21Þ þ 3B3ð2A2

1 þ 2B21 þ A2

3 þ B23 � 8Þg � 36A3ðc1 þ ceÞxþ 3B3ð18I1x2 � 8keÞ,

D4 ¼ mgsfA1ð3B21 � A2

1Þ þ 3A3ð2A21 þ 2B2

1 þ A23 þ B2

3 � 8Þg þ 36B3ðc1 þ ceÞxþ 3A3ð18I1x2 � 8keÞ.

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