Journal of Sound and Vibration 430 (2018) 214e230

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Journal of Sound and Vibration

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

Numerical and experimental study of a double physicalpendulum with magnetic interaction

Mateusz Wojna, Adam Wijata, Grzegorz Wasilewski, Jan Awrejcewicz*

Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland

a r t i c l e i n f o

Article history:Received 30 March 2018Accepted 23 May 2018Available online 31 May 2018

Keywords:Double pendulumMagnetsModellingExperimentBifurcationChaos

* Corresponding author.E-mail address: jan.awrejcewicz@p.lodz.pl (J. Aw

https://doi.org/10.1016/j.jsv.2018.05.0320022-460X/© 2018 Elsevier Ltd. All rights reserved.

a b s t r a c t

Chaotic and regular behavior of the system consisting of a double physical pendulum withtwo repulsive permanent magnets is studied. We are focused on mathematical modelling,numerical simulations, experimental measurements and in particular on a novel magneticinteraction modelling. System parameters are identified by matching the output signalsfrom experiments and numerical solutions to the developed mathematical model gov-erned by a strongly non-linear set of two second order ODEs including the friction and themagnetic interaction torques. Considered system shows chaotic and periodic dynamics.Few chaotic zones have been detected numerically and confirmed experimentally. Sce-narios of transition from regular to chaotic motion and vice versa, as well as the bifurcationdiagrams are illustrated and discussed. Good agreement between the numerical simula-tion and experimental measurement is achieved.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

A rapid development of science and technology requires their matching and mutual feedback for detection and under-standing of real-world non-linear phenomena and to guarantee the useful application of non-linear behavior aimed atimproving the quality and safety of the numerous engineering processes. Complexity of evolution of real-world non-linearprocesses requires novel methodological approaches spanned by interdisciplinary exchange of ideas, combining differentsciences (mechanics, physics, applied mathematics, electrical and electronical engineering) and developing dedicated ap-proaches aimed at accurate modelling of the studied phenomena.

Our approach which is presented in this paper is focused on mathematical modelling, numerical and experimentalinvestigation of a mechanical system (double pendulum) subjected to influence of magnetic and electric fields. The carriedout research fills the gaps in insufficient knowledge and uncertainties between mechanics, mechatronics and applied physicsspanned by non-linear phenomena.

It is well known and documented that even simple configuration of the pendula may exhibit almost all features of non-linear phenomena detected and reported so far, in the field of non-linear dynamical systems [1e12]. The carried out overviewof the up to date research devoting to mathematical modelling based on fundamental mechanical-electro-magnetic laws andmeticulous experimental validations highlights a need to refresh and extend the initiated investigations of vibrations of thependulum-type systems with and account of the magnetic, electric and mechanic fields.

rejcewicz).

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 215

On the other hand, the magneto-electro-mechanical pendulum-type systems may serve as archetypical members in theengineering systems including mechanisms and machines construction, space exploration, manufacturing estimation, con-trol and automation, robotics, sensors and servo-motor construction and fabrication, as well as numerous MEMS and NEMSdevices (including sensors and gyroscopes).

In what follows we briefly describe state-of-the-art of research results in this field embracing a group of comprehensiveand up to date accounts of non-linear vibrations of the mechanical systems under the magnetic and electric fields.

Nana et al. [13] carried out both theoretical and experimental study of an electromechanical system composed of aphysical pendulumwith repulsivemagnets and a DCmotor. The employed analytical/numerical prediction has been validatedby modelling, simulation and experimental investigation of the introduced mathematical model. A few novel non-linearfeatures have been illustrated and discussed including hysteresis, chaotic and regular dynamics, as well as jump phenomena.

Schmidt and Childers [14] studied first- and second-order phase transitions and tri-critical points exhibited by a magneticpendulum controlled by the currents. Siahmakoun et al. [15] reported regular and chaotic behavior, as well as amplitudejumps, hysteresis and bistable states of a sinusoidally driven pendulum embedded into a repulsive magnetic field. Fradkovet al. [16] investigated more complex laboratory rig including mechanical and electromagnetic devices and consisting of twosimilar subsystems, where each of them consists of two coupled pendula.

Dannagain and Rasskazov [17] reported numerical investigation of themodified Duffing oscillatormodelling a periodicallydriven iron pendulum embedded into a non-uniform magnetic field. In particular, the Preisach nonlinearity to modelmagnetization effects have been used and the hysteretic/chaotic effects have been illustrated. Experimental investigationswith a magnetically controlled pendulummodified by positive/negative feedback with an emphasis on the chaotic vibrationsand data acquisition system have been considered by Kraftmakher [18]. Energy-based criterion to predict escapes for eitherexternally forced or parametrically excited bistable magnetic pendulum has been proposed by Mann [19].

Kitio Kwuimy et al. [20] considered the effect of tilted harmonic excitation and parametric damping on chaotic vibrationsof an asymmetric magnetic pendulum. Both regular and fractal shapes of the basin of attraction have been employed tovalidate the Melnikov-type analytical prediction. It has been illustrated how increase of the tilt angle of the excitation yieldsincrease of the lower part of chaotic domain. It has been also demonstrated that the parametric damping may enhance orsuppress chaos. Lima et al. [21] analyzed the impact behavior of an embarked pendulum in a vibro-impact electromechanicalsystem. A continuous contact dynamic model has been proposed, whereas the impact has been modeled by the spring-dashpot system. The maximum entropy principle has been employed to derive the probability model, and the MonteCarlo simulations have been carried out to validate the confidence interval of the pendulum displacements and the angularspeed of the motor shaft. Tran et al. [22] predicted and studied the behavior of the chaotic physical-magnetic pendulumwithvariable interaction potential. Standard tools of theory of non-linear dynamical systems have been used and it has beenillustrated that the large scale system dynamics encompassing different attractors can be predicted by employing only asingle region of chaotic behavior.

Skubov and Vavilov [23] considered dynamics of the conductivity bodies of pendulum types under alternating magneticfield. Kadjie and Woafo [24] investigated an energy harvester consisting of an electromechanical pendulum supported bynon-linear springs. It has been illustrated by an appropriate choice of the springs, that output power attains higher mag-nitudes than in the case without springs. A simple instance of the electro-mechanical system (EMS) is considered in Ref. [25].The paper contains an analysis of a dynamical systemwith a pendulum, an AC electromagnet and a permanent magnet. Themagnet is mounted at the end of the pendulum and determines its bob. The coexistence and relative relations between chaosand parametric resonances are studied. Analytical results are compared against numerical simulations and experimentalstudies.

Over the past three decades, a great attention has been paid to modelling, theoretical, numerical and experimental in-vestigations of the active magnetic bearing employed to support rotating machinery and rotors in particular, due to theiradvantages including much lesser friction, absent lubrication and high rotating speeds. Ji [26,27] studied both theoreticallyand experimentally non-linear dynamic phenomena exhibited by a Jeffcott rotor-magnetic bearing system with time delays.Zhang and Zhan [28] investigated regular and chaotic dynamics of a rotor activemagnetic bearings with non-linear terms andtime-varying stiffness, and with 8-pole legs. Zhang et al. [29] carried out investigations of multi-pulse Shilnikov-type chaoticvibrations of a rotor-acting magnetic bearing system with time-varying stiffness. Chen and Hegazy [30] reported non-lineardynamics of a rotor active magnetic bearing, whereas Kamel and Bauomy [31] analyzed non-linear behavior of a rotor underactive magnetic bearing withmulti-parameter excitations. More recently, Fang et al. [32] proposed an active vibration controlof rotor imbalance in active magnetic bearing system, whereas Yang et al. [33] implemented control of elliptic motions ofrotors suspended in active magnetic bearings. Chen et al. [34] employed a recurrent wavelet fuzzy-neural network to thepositioning control of a magnetic-bearing mechanisms. In 1996 Hansen et al. [35] studied oscillations of two nearly identicalresonant series of LC circuits, where two coaxial coils were placed nearby and when one of themwas fixed and the other onewas movable. New electromagnetic phenomena associated with induction and magnetic hysteresis have been detected andstudied including estimation of the Lyapunov exponent based on reference [36]. Foerster et al. [37] considered mechanicalvibrations of the gradient coil system during readout in echo-planar imaging under the magnetic field distribution duringfunctional magnetic resonance imaging. The frequency drift correction method has been proposed. Pulse-modulated controllaw for both synchronized and non-synchronized vibrations has been employed based on the speed-gradient method. Thesimulation and laboratory experiments showed a good coincidence.

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230216

Przybyłowicz and Schmidt [38,39] studied the electromagnetic damping and chaotic dynamics of a harmonically drivenoscillator in a magnetic field. It has been shown that this kind of damping is efficient and it has been suggested to employ itfurther in order to study lumped mass mechanical system with a few DoFs as well as continuous structural members likebeams, plates and shells or pipes convening fluids.

Donoso et al. [40] investigated a pair of magnets vertically oscillating inside hollow magnetic coils linked in series. It hasbeen found that the electro-magnetic coupling is not directly proportional to the number N of turns in the coils. Instead, anon-linear dependence has been observed and the coupling tended to maximum as N was varied.

It should be emphasized that magneto-mechanical vibrations are exhibited by oscillators fabricated from small plates ofquartz crystals used in various electronic devices including cell phones, microwave transmitters, and computers being underaction of magnetic fields. An important role in physics plays solution to 3D problems of harmonic oscillator embedded into astationary magnetic field and oscillating electric radiation field. In particular, splitting of energy levels known as Zeemaneffect and non-resonant interaction with radiation incident parallel to the applied magnetic field generating the Faradayeffect (known also as the magnetic circular birefringence) as well as the electric polarization effects, found in classicalelectrodynamics, still require more sophisticated modelling of the experimentally observed phenomena.

Magnetic field effects play a crucial role in MEMs (microelectromechanical systems), NEMs (nanoelectromechanicalsystems), spintronics, nanocomposites and/or nano-sensors and nano-actuators. A study of magneto-elastic vibrations ofplates and shells has been initiated by Ambartsumyan et al. [41]. Later on, a decrease has been detected in the vibrationfrequencies of cylindrical shells under a longitudinal magnetic field, and in particular, the influence of a transverse magneticfield on the natural frequencies of shallow shell vibrations has been investigated in Ref. [42]. Chang and Lue [43] studiedmagnetic properties of multiwall carbon nanotubes using the electron paramagnetic resonance. Dynamic characteristics ofmulti-walled nanotubes subjected to magnetic field have been analyzed by Li et al. [44]. The Lorentz magnetic forces havebeen employed to derive the dynamic equations to follow the wave propagation of carbon nanotubes by Wang et al. [45].Kibalchenko et al. [46] investigated the magnetic response of single-walled carbon nanotubes induced by an externalmagnetic field. On the other hand, vibrations of the single-walled and multi-walled nanotubes under magnetic fields haverecently attracted certain attention. Murmu and Adhikari [47] studied vibration and stability of double nano-beam system,and more recently Murmu et al. [48] reported the effects of a longitudinal magnetic field action on the vibration of amagnetically sensitive double single-walled carbon nanotube system. Liang Ke et al. [49] brought the effect of magneto-electro-elastic characteristics of nanoplate employing the Kirchhoff-s plate theory. In particular, they compared resultsregarding natural frequency estimation with thermal field and with mechanical, elastic and magnetic loading.

More recently, Satish et al. [50] investigatedmagneto-thermal properties of nanoplates based on Eringen's nonlocal theorywith inclusion of magnetic field and surface elasticity effects. They found that the presence of surface layers on a nanoplateyields an increase in the natural frequency and structure stiffness, in contrary to the magnetic field effects.

Our paper consists of numerical and experimental study of a unique real-world magneto-mechanical system. The object ofresearch is a double physical pendulum with magnetic interaction caused by two neodymium permanent magnets. Therelative orientation of the magnets is such that a repulsive force exists between them. In the considered system the firstmagnet is installed at the end of the second pendulum, whereas the second one is attached to the immovable body of the builtexperimental rig. Magnets are in their vicinity when a pendulum is oriented downwards. The considered system is unique fora few reasons. Firstly, there are no papers describing dynamics of a double physical pendulum with magnets or electro-magnets. Additionally, the analyzed pendulum is constructed in a way which allows it to perform multiple revolutions.Furthermore, a motion of the first link of pendulum is excited by a specially constructed motor. Since this paper concernsmathematical modelling, a great emphasis is put on friction torque, excitation and magnetic torque modelling. Additionally,the system parameters identification and comparison between numerical simulations with experimental study are carriedout. Bifurcation diagrams and phase portraits are provided. In fact, this research extends the results obtained by the authors intheir earlier work [51].

As it has been alreadymentioned, the analyzed systempossesses a unique construction and there is a lack of investigationsfocused on double pendulums dynamics with magnetic interaction. There are certain published works where only singlependulums are studied.

The paper is organized in the following way. The experimental rig is described. Section 3 is devoted to derivation of thegoverning equations of motionwith an emphasis put on themagnetic interaction, friction and excitation modelling. Section 4includes the system parameters identification. In section 5, the non-linear dynamical phenomena are detected and followedusing numerical, experimentally validated, simulations. Our conclusions are presented in section 6.

2. Experimental rig

The double pendulum experimental rig system, shown in Fig. 1, consists of the following main parts: a double pendulum, adriving unit and a measuring subsystem.

A pendulum's frame is amassive, symmetrical andwelded constructionmade of steel. The links of the pendulum aremadeof small aluminum blocks and rods. The blocks also function as a shaft fixation at the bearing location. The bearings that areused can sustain a big load, and they guarantee the proper elimination of backlashes. The magnets used during the exper-imental study (see (3) in Fig. 1) are the two neodymium permanent magnets, with a diameter of f33mm and a height of10mm.

Fig. 1. Experimental rig views: 1, 2 e links, 3 emagnets, 4 emagnetic interaction measurement system support, 5 e extensometer beam, 6 e ties system, 7 e tiesguide, 8 e links relative coupling system, 9 e rotor, 10 e stator.

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 217

The pendulum is assembled as a modular construction in order to ease the modification of its configuration variant. Aphysical pendulum setup with varying configurations, i.e. with single, double or triple pendulum, was an object of researchfor many works, e.g. Ref. [5]. Additionally, the system parameters, such as the links mass, the positions of mass centers, themoments of inertia and the links length can be changed. The considered, experimental rig of double physical pendulumwithmagnets is augmented with a novel magnetic interaction torque measurement system (see (4)e(8) in Fig. 1). The systemmakes it possible to measure how the force depends on the links relative angle.

The original design and authors construction of the pendulum driving subsystem consists of two structurally symmetricaland electrically coupled electric motors of optoelectronic commutation and slow alternating currents (for more details seeRef. [5]). Such symmetrical driving unit and symmetrical pendulum's links prevent the skewing of the construction andeliminate the possibility of forces and torques acting in planes which are different from the assumed ones. On the other hand,such construction allows for themultiple full revolutions of the pendulum's links. The square-shape forcing torque function ofthe first link with adjustable frequency is achieved. More details about the excitation are provided in subsection 3.2.

All signals coming from measuring devices are processed in LabView™ measurement software. The angular positionsensors which are used are the precise rotational potentiometers. Additionally, the experimental rig measurement subsystemconsists of an extensometer beam (see (5) in Fig. 1) which measures the force generated by magnetic interaction.

3. System modelling

This section concerns the modelling process of the considered system. It consists of physical and mathematical models ofthe entire system with the emphasis put on the components of the friction, excitation and magnetic interaction torques.Physical model of the considered system is presented in Fig. 2. The geometric quantities ei and l1 stand for the distance of thecenters of gravity zi counted from centers of rotation Oi and the pendulum's link length, respectively. In addition, mi and Jistand for mass and moment of inertia respectively.Me denotes excitation torque acting on the first link of the pendulum dueto the motor, and j1;j2 are the variables of the angular position.

The governing equations of motion have the following form

MðjÞ�€j1€j2

�þ NðjÞ

8><>:

_j21_j22

9>=>;þ r

�j; _j; €j

�þ�M1 sin j1M2 sin j2

�þ�M1mag

M2mag

�¼

�Me0

�; (1)

where

Fig. 2. Physical model of the double pendulum system.

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230218

MðjÞ ¼�

B1 N12c12N12c12 B2

�; (2)

�0 N12s12

�

NðjÞ ¼ �N12s12 0 ; (3)

�_ €

� �MR1 �MR2

�

r j;j;j ¼

MR2; (4)

and

cij ¼ cos�ji � jj

�;

sij ¼ sin�ji � jj

�:

(5)

Other quantities are expressed as follows (see also Fig. 2)

B1 ¼ J1 þ e21m1 þ l21m2;

B2 ¼ J2 þ e22m2;N12 ¼ m2e2l1;M1 ¼ m1ge1 þm2gl1;M2 ¼ m2ge2:

(6)

Torques of friction MRi are described in more detail in the following subsection, and the forcing torque Me ¼ Meðt; f Þapplied to the first link of pendulum is a rectangular time functionwith frequency f (detailed description of this component isdetermined in a separate subsection in the same way as the torque generated by magnets interaction Mimag

).

3.1. Friction torque

Friction torques MRi occurring in bearings are governed by the following equations

MRi ¼ ðTi þ mNiÞ2patan

�εurelðiÞ

�h�1� m

0�e�c

0 jurelðiÞj þ m0iþ ciurelðiÞ; i ¼ 1;2: (7)

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 219

The proposed function models a few frictional effects, i.e. the Coulomb friction, viscous friction and the Stribeck effect [5].Function 2

p atanðεurelðiÞÞ is used instead of sgnðurelðiÞÞ function in numerical simulations (ε plays the role of regularizationparameter which keeps quick simulation with a simultaneous high accuracy of the obtained results). In Eq. (7) Ti stands forconstant drag torque independent of the load, m is a coefficient of friction in the bearings giving a torque proportional to theradial loads Ni, whereas m

0and c

0represent parameters responsible for modelling of the Stribeck effect (see Fig. 3).

Angular velocities urelðiÞ follow

urelð1Þ ¼ _j1;

urelð2Þ ¼ _j2 � _j1:(8)

Viscous friction coefficients ci are assumed based on the value of one common coefficient c, i.e. we have

c1 ¼ 2c;c2 ¼ c:

(9)

The considered construction has all identical radial ball bearings. The joint of first degree of freedom contains two timesmore bearings than the second one, what is the reason of an assumption expressed by Eq. (9).

Reaction forces Ni in bearings in Eq. (7) are calculated from normal and tangent directions components, and they have thefollowing form

Ni ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN2in þ N2

it

q; i ¼ 1; 2: (10)

Using Eq. (5) and notation employed in Fig. 2, components of reaction forces are described in the following way [5]

N1n ¼ m1

g cos j1 þ e1 _j1

2þ N2nc12 � N2ts12;

N1t ¼ m1

g sin j1 þ e1 €j1

þ N2ns12 þ N2tc12;

N2n ¼ m2

�g cos j2 þ e2 _j2

2 þ l1

€j1s12 þ _j1

2c12

�;

N2t ¼ m2

�g sin j2 þ e2 €j2 þ l1

€j1c12 � _j1

2s12

�:

(11)

Observe strongly non-linear behavior of the normal forces including their dependence on the velocities and acceleration.

3.2. Excitation torque

Electric motor, which is a source of an external torque in our system, is powered through a squarewave signal generator. Inorder to model and numerically simulate the system we need a smooth and continuous model of the excitation signal. Theapplied model approximates a square wave shape as a value of a sign function of a sine signal. In order to make the modelcontinuous, sign function has been replaced by typical arctangent approximation

Me ¼ q2p

arctanðε sinð2pf tÞÞ; (12)

where q stands for the torque amplitude, ε is a shape coefficient and f is a signal frequency in [Hz]. Square wave described by

Eq. (12) has a 50% duty cycle, which means that a high state of the signal lasts for 50% of a one period time. A detailed

Fig. 3. Friction torque model exhibiting the Stribeck effect.

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230220

examination of a square wave signal from our laboratory generator revealed that the real duty cycle of the signal is actually49.27%. In order to incorporate this observation in excitation signal model, a constant value a has been added to sine value,and it has distorted the previously balanced 50% duty cycle. The value of the constant a has been calculated according to thefollowing formula

a ¼ �sin12

p� 2pd

100%

; (13)

where d stands for the percentage duty cycle. Although this action has improved the duty cycle, it has also caused shifting of

the square wave phase, i.e. high state no longer starts for t ¼ 0. In order to fix this problem, a phase shift should be added tothe sine wave in a way which enables it to cross the point ð0;0Þ. That needed phase shift has been calculated in the followingway

b ¼ p� 2pd100%

2pf: (14)

Finally, the excitation torque model has the following form

Me ¼ q2p

arctanðε sinð2pf ðt þ bÞÞ þ aÞ: (15)

Fig. 4 shows the plot of excitation torque for different values of ε.One can tune this parameter to obtain realistic time delay for a shift from high to low state.

3.3. Magnetic interaction modelling validated experimentally

Here we deal with a description of a magnetic interaction modelling. First, the method of experimental measurements ispresented. Second, mathematical description of torques due to magnets relative interaction is provided and described indetail.

In order to get the experimental torque characteristics generated by the employed magnets, a special measurementsystem has been constructed (see Fig. 5a), where the extensometer beam (see (5) in Fig. 5a) is used as a force sensor. Fig. 5beeshows possible link configurations, depending on the side (right or left) where movable pendulum's magnet is locatedrelatively to the immovable, and the sign of the links relative angle jR is presented as well.

The first pendulum's link is mechanically coupled with extensometer beam by ties system (see (6) and (7) in Fig. 5a). Inorder to measure the way inwhich the force acts on the extensometer beam and to estimate the magnitude of the torque, thesecond link was stiffly coupled relatively to the first link in known angular position. Our setup allows for investigation of 17different case studies, for fixed relative angles ranging from �35� to þ 35�. During the measurements, the extensometerbeam has been moved downwards pulling the tie at the same time. In this way a torque acts on the first link of pendulumtrying to overcome the repelling force of approachingmagnets. Suchmeasurements have been carried out by approaching themagnets from the right and the left side of the experimental setup, and including every possible relative angular coupling oflinks (see Fig. 5bee). Observe that the extensometer beam force has the constant arm r. The last property allows for the torqueestimationwhich acts on the first link of pendulum due to magnets interaction. In addition, the obtained values of the torquehave beenmodified and supplemented by known values of theweight of the tie system and the gravity force of the pendulum.Therefore, only the torque caused by the magnets interaction has been considered. The similar investigation concerns themeasurements from the right side where the torque acts in the positive direction and achieves positive values.

The measurements method described above, enables us to obtain experimental characteristics of the magnetic interactiontorque presented in Fig. 6. It should be noticed that the obtained characteristics are the function of the first link angular

Fig. 4. Excitation torque signal for 30% duty cycle and different values of ε.

Fig. 5. Magnetic interaction measurement system (a) with possible links configuration: left side measurement and positive jR (b); left side measurement andnegative jR (c); right side measurement and positive jR (d); right side measurement and negative jR (e); elements notation is identical as introduced in Fig. 1 andstands for: 4 e magnetic interaction measurement system support, 5 e extensometer beam, 6 e ties system, 7 e ties guide.

Fig. 6. Experimental characteristics of magnetic interaction torque versus links configuration (see text for more elaborate description).

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 221

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230222

position j1 and the relative links angle jR. In Fig. 6 the used curves notation stands for the measurement setup, i.e. anapproximate value of jR and the side fromwhich the measurements have been carried out (e.g. “15R” indicates jRz15� andright side measurement).

The preliminary experimental results that were obtained, serve for the mathematical description of the magnetic inter-action. A typical approach for modelling magnetic interactions in mechanical systems is to treat magnets as particles whichattract or repel each other. The interaction force is directed from one point to the other and its magnitude is calculated via thefollowing simple formula [15,25]:

FM ¼ m04p

m1m2

r2; (16)

where r is a distance between point magnets, m0 stands for a vacuum permeability and m1, m2 are the used magnets dipolemoments. However, this simple approximation governed by Eq. (16) is not feasible for our system. In other words, it isimpossible to find m1, m2 that fit the function from Eq. (16) and could validate the values yielded by our experimentalmeasurements. Possible reasons for that can be the size and the magnitude of distance between the magnets, which do notallow to reduce them to points. This is why we have decided to find a formula which can describe our measurements as afunction of j1 and jR angles. For a purpose of numerical simulation, the function should also be smooth and continuous.

Firstly, we have used the following function

f ðj1Þ ¼ a exp �"

j1 þ bc

2#

(17)

to fit measurements for different jR angles. Observe that in Eq. (17) a and b parameters stand for the coordinates of a peakvalue of the function, and c can be treated as a shape coefficient. Plot of the function f ðj1Þ and the parameters indicators areshown in Fig. 7a. However, the second difficulty occurred. Namely, it was possible to fit the function f ðj1Þ into all sets of themeasured values with the best fit for “-30R” setup (see Fig. 6) (R2¼ 0.999986) and the worst for “-30L” setup (see Fig. 6)(R2¼ 0.996736). An exemplary fit is shown in Fig. 7b.

After having all data sets described by the function governed by Eq. (17), the parameters a, b, c change with respect to thejR angle has been investigated, and the validated approximations have been proposed. The function aiðjR Þ has the same formas Eq. (17), biðjR Þ is linear and ciðjR Þ has been set to a constant value (the parameters approximations are shown in Fig. 8).

Substituting parameters a, b, c appearing in Eq. (17) by the functions aiðjRÞ, biðjRÞ, ciðjR Þ, i ¼ 1;2, one can obtaindescription for positive (i ¼ 1) and negative (i ¼ 2) values of the measured magnetic interaction torque, i.e. we have

fiðj1;jRÞ ¼ aiðjR Þexp �"

j1 þ biðjR ÞciðjR Þ

2#; i ¼ 1;2: (18)

The remaining feature to be determined is a transition point from one characteristic ðf1Þ to the other ðf2Þ. We have assumedthat this transition occurs in the middle between positive and negative peak value for a given relative angle jR. Since biðj2 Þdoes describe how the peak coordinates j1 changeswith regard to jR, a transition line j1 ¼ m1jR þm0 has been calculated asan average between b1ðj2 Þ and b2ðj2 Þ. Finally, the function governing the magnetic interaction torque has the followingform

MMðj1;jRÞ ¼f1 � f2

2signðj1 � m1jR �m0Þ þ

f1 � f22

: (19)

Fig. 7. Function governed by Eq. (17): parameters influence (a) and exemplary fits into experimental data for “-15R” and “-30R” (see Fig. 6) setups (b).

Fig. 8. Approximations for parameters aiðjR Þ, biðjR Þ and ciðjR Þ, i ¼ 1;2. (red dots indicate the experimental data and blue lines show their approximations).(For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 223

A sign function which occurred in Eq. (19) has been replaced with arctangent approximation for the simulation purposes(it is left here for a description clarity), whereas a plot of the MMðj1;jRÞ is presented in Fig. 9.

Themeasurements have been carried out on the rig setup with a fixed jR angle. Torque frommagnetic interaction forces isdifferent for the systemwith a joint and that is the factor whichmotivated us to recalculate the torque from ourmeasurementinto a magnetic interaction force. One important assumption has been introduced here, namely that the magnetic force isdirected from the center of onemagnet to the center of the other one. Force components are estimated based on the followingrelations

FmagX ¼ MM

h; FmagY ¼ MM

hx2y2

: (20)

Symbols x2; y2;h used in Eq. (20) are defined in Fig. 10. Forces in Eq. (20), imply occurrence of the torques in the first andthe second joint M1mag

, M2mag, respectively.

4. Parameters identification

The derived mathematical model consists of 15 parameters. Vector W containing these parameters can be expressed asfollows

Fig. 9. Function describing the experimentally gathered data for magnetic torque with respect to j1 and jR .

Fig. 10. Magnetic force calculated from measured torque.

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230224

W ¼hg; l1;m1;m2; e1; e2; J1; J2; T1; T2; c;m;m

0; c

0; εi: (21)

The values of parameters such as g; l1;m1;m2 have been determined independently by measurements. Other parametershave been identified by comparing real time series of the double pendulum without magnets with the proposed mathe-matical model (excluding the torque generated by the magnets). This approach has been adopted due to the fact that thedouble pendulummodel is well known and described in detail in the literature. There are also some well-known methods ofglobal minimum finding for the used criterion function. Identification process has been repeated a few times yielding manysets of parameters. The best correlation coefficient value has been achieved basing on a set of time series containing a free anda forced pendulum's motion.

The identified parameters used in our further numerical analysis are presented in Table 1.Observe that a few parameters, such as T1; T2; c;m concerning static and viscous friction are relatively small. However,

when omitting them and using a simplified friction model, the results are much less accurate [5]. The recorded time series ofthe double physical pendulum system compared to the simulated time series with the determined parameters set arejuxtaposed in Fig. 11.

5. Numerical and experimental study

The modeled system has been investigated by numerical bifurcation analysis. The excitation frequency f serves as abifurcation parameter. Bifurcation diagrams are constructed for increasing value of the parameter in the range of (0.1 ÷ 3) Hz.Fig. 12 presents bifurcation diagrams for variables j1 and j2 of the pendulum. Surprisingly the diagrams show wide zones ofchaotic behaviors and relatively narrow zones of a regular motion. The most interesting range of frequency in the presenteddiagrams is a band between (2 ÷ 3) Hz. In order to explore the aforementioned frequencies interval more thoroughly, thebifurcation diagrams for decreasing frequency have been shown (see Fig. 13). They are similar to the previous cases ofincreasing frequency, but the dynamic phenomena are more easily visible and legible. One may see that in the central regionof the frequency range a periodic solution has got a negative instead of a positive value, as it is shown in Fig. 12. The reason isthat the pendulum exhibits two stable configurations, i.e. when it is located on the right or on the left side of the magnetinstalled on the body rig (the reported diagrams present only one of them). The numerical bifurcation diagrams constructionmethod which was used, assumes the initial conditions in particular points of simulation as a solution of the previous one.Thus, only one periodic solution exists in the diagrams (at zones of single periodic motion), positive or negative in the samesimulation report.

All numerical analysis has been performed in Wolfram Mathematica™ software environment. The code which generatesthe numerical bifurcation diagrams has been described in the provided Appendix.

Table 1The parameters estimated experimentally.

Parameter [Unit] Identified Value Parameter [Unit] Identified Value Parameter [Unit] Identified Value

g ½m s�2� 9:812 e2 ½m� 88:292$10�3 c ½N m s� 0:759$10�3

l1 ½m� 0:174 J1 ½kg m2� 45:381$10�3 m ½m� 0:461$10�3

m1 ½kg� 4:275 J2 ½kg m2� 14:308$10�3 m0 ½ � � 0:631

m2 ½kg� 1:365 T1 ½N m� 86:004$10�3 c0 ½s rad�1� 13:447

e1 ½m� 64:175$10�3 T2 ½N m� 18:824$10�3ε ½s� 1000

Fig. 11. The comparison of real (dashed green lines) and simulated (solid red for j1 and solid blue for j2 lines) time series of the first (a) and the second (b)pendulum's variable (the time series overlap and that indicates very good agreement between the experimental and simulation results). (For interpretation of thereferences to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 12. Bifurcation diagrams for frequency in the range of (0.1 ÷ 3) Hz: the first (a) and the second (b) pendulum's variable (increasing frequency).

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 225

Bifurcation diagrams presented in Fig. 13 show that at the range of (2 ÷ 3) Hz chaos for both variables of pendulum occursfirst. The act of leaving the chaotic zone is sharp and an ordered single periodic behavior is stable for approximately 0.25 Hz.The order leads to the next chaotic zone where the entry is also sharp. Next period-halving bifurcations implies a tendency tomove to regular dynamics. It should be emphasized that in very narrow range of frequency, four periodic solutions exist. Atthe described range of frequency an attempt at experimental verification has been carried out.

Experimentally obtained bifurcation diagrams have been compared to the simulation results (see Fig. 14). Experimentaldiagrams reported in Fig. 14c and d presents the first zone of a regular motion which is wider than those obtained via nu-merical simulations. It is worth mentioning that the values of angles in the first order zone (e.g. at 2.2 Hz) are very similar.

I-

Fig. 13. Bifurcation diagrams for frequency in the range of (2 ÷ 3) Hz: the first (a) and the second (b) pendulum's variable (decreasing frequency).

Fig. 14. Bifurcation diagrams for increasing frequency in the range of (2 ÷ 3) Hz for the first and the second pendulum's variables: numerical simulations (a,b) andlaboratory experiments (c,d).

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230226

nstead of sharp entering to the chaotic zone, experiment exhibits the period doubling bifurcations. Comparing to the sim-ulations, the experimental chaotic zone in the neighborhood of frequency 2.4 Hz is narrower. It is worth highlighting that thischaotic zone ends at almost the same frequency value in both diagrams. Similarly to the simulation results a range of theperiod-halving bifurcations exists, and four periodic motion is visible. Period doubling bifurcations are robust. Regular motionzones are localized at the same value for both simulation and experimental diagrams. It should bementioned that in the (2.65÷ 2.7) Hz range of frequency, there are period doubling bifurcations exhibited by the first pendulum's variable, while thesecond pendulum's link angle does not tend to any stable configuration during the measurement time of evolution. It couldmean that second link has much longer transitional state or there is a coexistence and an interplay of regular (of the first link)and chaotic behavior of the second pendulum's link.

Fig. 15. Numerical (aed) and experimental (eeh) results for forcing frequency f ¼ 2:2 Hz.

Fig. 16. Numerical (aed) and experimental (eeh) results for forcing frequency fy2:48 Hz.

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 227

For further analysis and comparison between simulation and experimental results, the phase portraits for few excitationfrequencies are reported (see Fig. 15, Fig. 16, Fig. 17). Besides the phase portraits, the lateral projections of the end of secondpendulum's link are provided.

Fig. 15 shows that for the bifurcation parameter f ¼ 2:2 Hz both simulation and experimental results are qualitativelysimilar. There is a regular zone of the system behavior, and the ranges of angular displacements and velocities are very similar.

At frequency fy2:48 Hz, period doubling bifurcations appear (see Fig. 16). The bifurcation diagrams indicate an exit fromthe chaotic zone. Shapes of simulation and experimental phase portraits are similar, but not identical. It points out thatapplied mathematical model with the estimated parameters set should be verified and eventually improved.

It can be seen that in phase portraits the non-differentiable points exist (e.g. in Fig. 16a), especially for the first pendulum'slink portraits. The cause of their existence is the occurrence of the rectangular signal of the excitation. To eliminate suchpoints a dynamic model of an excitation torque can be implemented.

The last set of phase portraits that is shown (Fig. 17) presents very good agreement between the numerical simulation andexperimental measurements.

Fig. 17. Numerical (aed) and experimental (eeh) results for forcing frequency f ¼ 3:0 Hz.

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230228

6. Concluding remarks

The system consisting of a double physical pendulum with permanent magnets has been studied numerically andexperimentally. Good agreement between numerical and experimental results has been achieved and presented. Addition-ally, there is a novel solution of magnetic interaction due to repelling magnets modelling proposed and verified. Consideredinquiries point out that the offeredmathematical description of studied systemwith its parameters estimated experimentallycan be applied as a tool for a primary and rapid examination of the rich phenomena of non-linear dynamics exhibited by a realconstructed pendulum with neodymium magnets.

A few divergences are also detected and discussed. An insufficiently complexmathematical model could be the first reasonfor differences between numerical and experimental results. Additionally, the considered system has often exhibited a verylong transitional processes, which could also be the source for a few discrepancies between simulation and experimentalobservations. Furthermore, an identification process has been carried out using global minimum finding method which doesnot belong to the perfect ones as far as multi-dimensional optimization is concerned.

Acknowledgments

This work has been supported by the Polish National Science Centre under the grant OPUS 14 No. 2017/27/B/ST8/01330.

Appendix

This part of the paper concerns a code for generation numerical bifurcation diagrams inWolframMathematica™ language.In the following code:

� eq (1), eq (2) e equations of motion written as a function of a bifurcation parameter e in our case the forcing frequency.� j1[t],j2[t] e variables in our system.� TDrop e count of periods to omit, as a transient motion.� TPlot e count of periods to include in bifurcation diagram.� parChange e table with consecutive values of the bifurcation parameter, e.g. parChange ¼ Table[i, {i, 2.0, 3.0,

0.01}].� init e initial conditions for the NDSolve procedure.

(*Excitation period*)

TB[f_]:¼ 1/f;

(*Event for getting Poincare sections*)

eventB[f_]:¼ WhenEvent[Mod[t,TB[f]]¼¼0, Sow[{j1[t],j2[t],t}]];

(*function for iterating in FoldList*)

func[arg_,par_]:¼ (

{solBif,dataBif}¼Flatten[Reap[NDSolve[{eq1[par], eq2[par], j1[0]¼¼arg[[-1,2,1]], j1'[0]¼¼arg[[-1,2,2]], j1''[0]¼¼arg[[-1,2,3]], j2[0]¼¼arg[[-1,2,4]], j2'[0]¼¼arg[[-1,2,5]], j2''[0]¼¼arg[[-1,2,6]], eventB[Echo[par]]}, {j1,j2},

{t,0,TB[par]*(TDdropþTPlot)}, Method->{"DiscontinuityProcessing"- > False, "EquationSimplification"->"Residual"},MaxSteps->∞]], 1];

{par,#}&/@(({#[[1]], j1'[#[[3]]], j1''[#[[3]]],#[[2]], j2'[#[[3]]], j2''[#[[3]]]})/.solBif&/@Drop[dataBif,TDrop])

)

(*actual function for calculating bifurcation diagram*)

bifurcation[parChange_,init_]:¼(

bifData ¼ Drop[Flatten[FoldList[func, init, parChange], 1], 1];

(*Saving to a file*)

Export[ToString[NotebookDirectory[]]<>"biff_" <> ToString[parChange[[1]]] <> "_" <> ToString[parChange[[-1]]] <>"Hz_" <> ToString[TDrop] <> "_" <> ToString[TPlot] <> "_" <> ".m",bifData]

)

M. Wojna et al. / Journal of Sound and Vibration 430 (2018) 214e230 229

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