on the chaotic and hyper-chaotic dynamics of nanobeams...
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Chaos 31, 023107 (2021); https://doi.org/10.1063/5.0032069 31, 023107
© 2021 Author(s).
On the chaotic and hyper-chaotic dynamicsof nanobeams with low shear stiffnessCite as: Chaos 31, 023107 (2021); https://doi.org/10.1063/5.0032069Submitted: 06 October 2020 . Accepted: 06 January 2021 . Published Online: 02 February 2021
T. V. Yakovleva, J. Awrejcewicz, V. S. Kruzhilin, and V. A. Krysko
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Paper published as part of the special topic on Recent Advances in Modeling Complex Systems: Theory and
Applications
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Chaos ARTICLE scitation.org/journal/cha
On the chaotic and hyper-chaotic dynamics ofnanobeams with low shear stiffness
Cite as: Chaos 31, 023107 (2021); doi: 10.1063/5.0032069
Submitted: 6 October 2020 · Accepted: 6 January 2021 ·
Published Online: 2 February 2021 View Online Export Citation CrossMark
T. V. Yakovleva,1,a) J. Awrejcewicz,2,b) V. S. Kruzhilin,1,c) and V. A. Krysko1,d)
AFFILIATIONS
1Department of Mathematics and Modeling, Saratov State Technical University, 77 Politehnicheskaya St., 410054 Saratov, RussianFederation2Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz,Poland
Note: This paper belongs to the Focus Issue, Recent Advances in Modeling Complex Systems: Theory and Applications.a)Electronic mail: [email protected])Author to whom correspondence should be addressed: [email protected])Electronic mail: [email protected])Electronic mail: [email protected]
ABSTRACT
We construct a mathematical model of non-linear vibration of a beam nanostructure with low shear stiffness subjected to uniformly dis-tributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elasticmaterial obey the Hooke law and are governed by the kinematic third-order approximation (Sheremetev–Pelekh–Reddy model). The von Kár-mán geometric non-linear relation between deformations and displacements is taken into account. In order to describe the size-dependentcoefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations, aswell as the initial and boundary conditions. A solution to the dynamical problem is found via the finite difference method of the secondorder of accuracy, and next via the Runge–Kutta method of orders from two to eight, as well as the Newmark method. Investigations of thenon-linear nanobeam vibrations are carried out with a help of signals (time histories), phase portraits, as well as through the Fourier andwavelet-based analyses. The strength of the nanobeam chaotic vibrations is quantified through the Lyapunov exponents computed based onthe Sano–Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling proce-dure allowed us to achieve reliable results. In particular, we have detected chaotic and hyper-chaotic vibrations of the studied nanobeam, andour results are authentic, reliable, and accurate.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0032069
There are numerous articles devoted to studying deterministicchaotic and hyper-chaotic dynamics, but a majority of them eitherdeal with a few non-linear partial differential equations (PDEs)or employ strong modes truncation while analyzing non-linearPDEs. Our paper, despite using the real-world applications takenfrom mechanical engineering based on the introduced mathe-matical model (non-linear PDEs) of nanobeams made from anisotropic and elastic material, takes into account novel theoreti-cal contributions with an account of the size-dependent effects inthe frame of the modified couple stress theory. Particular atten-tion is paid to the reliability, validity, and authenticity of theobtained numerical results. We have employed numerous numer-ical approaches beginning with the finite difference method(FDM) of the second-order accuracy, Runge–Kutta methods,
Newmark method, Fourier, and wavelet type analysis, and end-ing on the different algorithms devoted to the estimation of theLyapunov exponents (LEs). In addition, a few novel results deal-ing with the transition from regular to chaotic dynamics as wellas from chaotic to hyper-chaotic dynamics exhibited by the non-linear nanobeam with low shear stiffness have been detected,illustrated, and discussed.
I. INTRODUCTION
The size-dependent behavior of structural nanometers (rods,beams, plates, and shells) has attracted huge attention of researchers
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and has forced engineers to fabricate non-traditional materials thatfound applications in various branches of mechanical and electricalengineering as well as in medicine.1–3
The immense use of the nano-structural members in thedesign, monitoring, and control of various devices, mechanisms,and machines related to micro/nanoelectromechanical systems(MEMS/NEMS) implied a review of the classical elasticity theo-ries to fit the size-dependent effects. It occurred that the traditionaltheories should be supplemented by the small-scale effects, whichresulted in developing novel non-traditional approaches includingcouple (modified) stress theory,4 non-local elasticity theory,5–7 non-local strain gradient theory,8–10 enhanced non-local formulations,11
homogenization theories,12,13 and, more recently, the stress-drivennon-local integral theory.14
It is already well recognized and approved that the size-dependent phenomena play a crucial role in describing and model-ing of mechanical behavior of nanostructures. Because it is difficultto estimate the small-scale parameters experimentally, the theoret-ical modeling matched with numerical simulations with emphasesput to the results’ reliability, validity, and accuracy may play a crucialrole in advanced analysis of non-linear dynamics of nanostructures.
The recent advances in material science including smart mate-rials and functionally graded materials allowed us to fabricatemicro/nanostructures with a priori assumed properties that can varyalong a chosen direction. However, in this paper, the fabricationof the advanced materials may yield unexpected non-linear behav-ior (chaos) exhibited by nano-structural members, which should beeither avoided (in the majority of cases) or employed (in the case ofnano-resonators and nano-gyroscopes).
In what follows, we briefly review the state-of-the-art of recentresults regarding modeling and analysis of nanobeams vibrations.
The modified couple stress theory was successfully employedto the Euler–Bernoulli and Timoshenko beams.15–18 Barretta et al.19
solved the electrostatic problem of functionally graded circularnanobeams with non-local elastic behavior made of periodic fiber-reinforced materials. An exact solution was found based on theLaplace transform. Barretta et al.20 employed Eringen’s first-gradientnon-local model to study the bending of Timoshenko function-ally graded nanobeams. The authors derived the analytical solutionsfor a simply supported beam regarding rotations and transversedisplacements. Shafiei et al.21 investigated dynamic features of the2D functionally graded nanobeams and microbeams made fromporous materials within the Timoshenko beam model. Eringen’snon-local elasticity (nanobeams) and modified couple stress the-ories (microbeams) were adopted, and the governing equationswere solved via the generalized differential quadrature method.The studied beams included those that were functionally gradedalong with thickness and length, porous and perfect, and sub-jected to a variety of boundary conditions. Sahmani and Aghdam22
modeled and investigated non-linear vibrations of pre-buckled andpost-buckled multilayer functionally graded nanobeams. Both theHalpin–Tsai model and Hamilton’s principle were employed toderive size-dependent PDE. Matching of a perturbation method andthe Galerkin method yielded the explicit analytical solution. Bothpre-buckling and post-buckling regimes were studied.
Apuzzo et al.23 analyzed free vibrations of the Euler–Bernoullinanobeams based on the stress-driven non-local integral model. It
was shown that the latter approach was effective in estimating thenon-local phenomena of the studied nanobeams. Barretta et al.24
utilized both strain gradient and stress-driven non-local models tostudy size-dependent vibrational behavior of functionally gradedelastic Timoshenko nanobeams. Talebitooti et al.25 carried out acomprehensive semi-analytical vibration analysis of rotating taperedaxially functionally graded nanobeam. The characteristic equationwas derived to get the non-dimensional frequencies.
As the brief so far described literature review shows, the beammodulus is governed by PDEs, i.e., the associated problem should besolved in an infinite dimension.
However, very often, low-dimensional models have been intro-duced via single-mode or multi-mode (few) models. The authorsclaim that the low-dimensional models possess several advantagesby giving a basic understanding of the non-linear phenomena, butthe hidden idea of this order reduction procedure is to limit theconsideration to mechanical/dynamic properties of the archetypalsystems like Duffing, van der Pol, and/or Helmholtz–Duffing oscil-lators. The latter approach allows for capturing of the main non-linear features like bifurcation and chaotic phenomena well knownand described through simple non-linear oscillators.
In some cases, the results can be validated with regard to thehigh-dimensional models via finite difference methods (FDMs),finite element methods (FEM), or the Bubnov–Galerkin higher-order methods (BGM) because the associated chaotic dynamics canbe embedded into the space of finite dimension, and hence only afew active modes can be taken into account.
However, as discussed in the following, in many cases such anapproach cannot provide reliable results.
Moon and Holmes26 are the first ones who reported the Smalehorseshoe strange attractor based on the reduction of non-linearPDEs governing vibration of laterally excited cantilever beam buck-led by two magnets into a single-dot Duffing oscillator with negativestiffness.
A similar somehow drastic reduction to single-mode approx-imation yielding the archetypal Duffing equation was offered byMoon and Shaw,27 though their investigations were supported bylaboratory experiments.
Baran28 considered a variety of beam models includingEuler–Bernoulli, Rayleigh, and Timoshenko beams, and illustratedhow homoclinic bifurcations generated the birth of chaotic dynam-ics.
Luo and Han29 predicted analytical chaotic vibrations ofa simply supported planar, non-linear rod via the mth modeGalerkin approximation based on the Duffing equation. Lenci andTarantino30 and Lenci et al.31 employed third-order and fifth-orderterms of the Galerkin one-mode approximation of the non-lineardynamics of an elastic cantilever beam on Winkler-type soil sub-jected to transversal harmonic excitation. Then, the authors appliedthe analytical Melnikov method for the detection of the intersectionsof homoclinic/heteroclinic orbits. The one-mode reduced model(Duffing equation) was derived by Santee and Goncalves,32 andthe lower bound of the dynamic buckling loads yielding chaos wasestimated via the Melnikov method.
The chaotic dynamics of a cantilever beam with axialand transversal harmonic excitations in 1:2 internal resonanceconditions with emphases put to the Shilnikov-type single-pulse
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homoclinic orbit and higher-dimensional Melnikov theory wasaddressed in Refs. 33 and 34.
Numerous examples of the order-reduced models and beyondare reported in the review paper by Rega et al.,35 and they are omittedhere.
As we have illustrated, in the majority of the publishedpapers, the problem of non-linear vibrations of structural mem-bers usually have been strongly reduced via the Bubnov–Galerkin,Faedo–Galerkin, and Ritz methods to that of dynamics of the so-called reduced-order systems usually approximated by a systemconsisting of a few degrees-of-freedom. In other words, the systemsof infinite dimensions have been reduced to that of finite dimen-sion through a strong truncation of the original PDE (PDEs) oftento one or two degrees-of-freedom. However, numerous papers andmonographs recently published by Awrejcewicz and Krysko showedthat this approximation can result in achieving non-reliable andsometimes erroneous results and conclusions.36–53
Another observation followed from the review of the state-of-the-art literature devoted to studying vibrations of structuralmembers governed by non-linear PDEs is that there exist chaoticdynamics exhibited by rods and beams (1D structural members),as well as by plates and shells (2D structural members) generatedby various non-linear effects implied by the geometric, physical,and design non-linearity. In this regard, it should be emphasizedthat in spite of “classical” chaotic regimes, the structural membersmay exhibit hyper-chaos [two positive largest Lyapunov exponents(LLEs)], hyper–hyper-chaos (three positive LLEs), and as the deepchaos (four positive LLEs) (see Refs. 45, 54, and 55). However,the mentioned results do not concern the micro/nano-structuralmembers.
Experimental studies of MEMS/NEMS devices are carried outin a number of publications.56–62 For example, NEMS actuators havebeen experimentally investigated in Ref. 60, which include statorelectrodes on both sides of the rotor electrode, designed for actu-ation of the tuning beam. Fabricated slots between the optical andmechanical beams range from about 70 nm to 90 nm. Tuning gapsfor all devices are about 200 nm. The released distance between theNEMS actuator’s electrodes ranges from about 90 nm to 140 nm.Reference 61 is devoted to the experimental study of MEMS archresonators with large vibration amplitudes. A single crystal siliconin-plane arch microbeam is fabricated such that it can be excited axi-ally from one of its ends by a parallel-plate electrode. The arch beamis of 1000µm length, 25µm width, and 2µm thickness. The ini-tial curvature of the arch is 1.7µm. The gap between the arch beamand the transverse actuating electrode is 10.6µm. The gap betweenthe side electrode and the stationary electrode of the parametricactuation is 5µm. An experimental investigation is conducted com-paring the response of the arch near primary resonance using theaxial excitation to that of a classical parallel-plate actuation wherethe arch itself forms an electrode. The results show that the axialexcitation can be more efficient and requires less power for primaryresonance excitation. Reference 62 theoretically and experimentallyinvestigated the two-to-one internal resonance in micromachinedarch beams, which are electrothermally tuned and electrostaticallydriven. The non-linear response of the arch beam during the two-to-one internal resonance is simulated. The solutions for perturbationstaking into account the influence of quadratic non-linearities, cubic
non-linearities, and two simultaneous excitations at higher AC volt-ages are compared with the solutions obtained using the multi-modeGalerkin procedure and with experimental data based on a speciallyfabricated silicon arc beam. The perturbation solutions consider-ing the influence of the quadratic non-linearities and cubic non-linearities are compared with those obtained from a multi-modeGalerkin procedure and with experimental data based on the delib-erately fabricated silicon arch beam. A good agreement is foundamong the results. Results indicate that the system exhibits differ-ent types of bifurcations, such as saddle-node and Hopf bifurcations,which can lead to quasi-periodic and potentially chaotic motions.
It should be noticed that the mentioned results cannot bevalidated using the classical macro beam theories and the size-dependent behavior should be taken into account to achieve reliableresults.
The present paper offers the following novel contributions.First, they are not papers devoted to the study of chaotic vibrationsof flexible nanobeams. Second, the derived governing evolution-ary PDEs are original. Third, the employed methodology and usedtools for quantifying the results have no limitation to that associatedwith the reduced-order modeling, since the studied nano-objects aretreated as systems with an infinite number of degrees-of-freedom.Fourth, the validity, reliability, and accuracy of the obtained results(solutions) have been guaranteed not only because of using a com-bination of the finite difference method and Runge–Kutta typemethods but also via qualitatively different approaches associatedwith numerical analysis of the obtained results like analysis of thesignals, phase portraits, FFT and wavelet-based algorithms, as well asthe Lyapunov exponents estimated by qualitatively different compu-tational schemes. Fifth, chaotic and hyper-chaotic vibrations of thenanobeam with low shear stiffness have been reported for the firsttime.
The paper is organized in the following way. The governingequations are formulated in Sec. II. Methods of the solution aredescribed in Sec. III. Non-linear dynamics of the nanobeam is stud-ied in Sec. IV, where the influence of a few control parameters onthe nanobeam vibration is investigated. Section V yields concludingremarks of the carried out study.
II. PROBLEM FORMULATION
In this section, we derive the mathematical model of non-linear vibrations of a nanobeam with low shear stiffness subjected
FIG. 1. Computational scheme of the nanobeam.
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to the external transversal load of the following form q(x, t) = q0 +
q1 sin(ωpt) (Fig. 1).We consider the 2D area � = {x ∈ [0, a]; −h/2 ≤ z ≤ h/2},
0 ≤ t ≤ ∞ of the space R2 with the attached Cartesian coordi-nates. Recall that the composite materials possess low shear stiffnesswhen G/E ∈
[
17; 1
100
]
, where G/E is the shear Young modulus.63 Thisimplies the employment of the kinematic models of higher ordersincluding the Timoshenko model (second order) and the Shereme-tev–Pelekh–Reddy (third order) instead of the Euler–Bernoullimodel (first order) (higher-order modeling of structural membersis widely described in the book).51 The following hypotheses aretaken:
(1) the nanobeam obeys the Sheremetev–Pelekh–Reddy model,64,65
i.e., a normal to the middle beamline is not normal after defor-mation and is curved;
(2) the nanobeam material is homogeneous, elastic, and obeysHooke’s law;
(3) the longitudinal nanobeam dimension is much higher than itstransversal dimensions;
(4) the nanobeam axis is a straight line;(5) the load q(x, t) = q0 + q1 sin(ωpt) acts along the OZ axis
and the external forces do not change their direction undernanobeam deformation;
(6) the geometric deformation follows the von Kármán assumption,66
(7) size-dependent effects are taken into account with a help of themodified couple stress theory.4
The nanobeam governing equation as well as both bound-ary and initial conditions are yielded by the following energeticHamilton principle:
∫ t1
t0
(δK − δ5+ δW)dt = 0,
where K is the kinetic Energy of the nanobeam, 5 is the poten-tial Energy of the nanobeam, and W is the sum of all works of theexternal nanobeam forces.
The governing equations of the nanobeam written in the non-dimensional form are as follows:
p1
{
∂ 2u
∂ x2+ L1(w, w)
}
=∂2u
∂ t2, B3(w)− γ1
2 B4(ψ) =∂2ψ
∂ t2,
p1
γ12{L2(u, w)+ L3(w, w)} + B1(w)+ B2(ψ)+ q(x, t)
=∂2w
∂ t2+ ε
∂w
∂ t, (1)
and bars over the non-dimensional quantities are already omitted,and where the following notation is employed,
p1 =n + (1 − 2n)ν
(1 + ν)(1 − 2ν), p2 =
n
2(1 + ν),
B1(w) =5 + 5ν
6 + 5ν
p2
15(8 + 5 γ2
2)∂2 w
∂ x2−
{
5
315
p1
γ12
+147
315p2
γ22
γ12
}
∂4 w
∂ x4,
B2(ψ) =5 + 5ν
6 + 5ν
p2
15(8 + 5 γ2
2)∂ ψ
∂ x+
{
16
315
p1
γ12
+63
315p2
γ22
γ12
}
∂3 ψ
∂ x3,
B3(w) =5 + 5ν
6 + 5ν
21
68p2(8 + 5 γ2
2)∂ w
∂ x−
{
16
68p1 +
63
68p2 γ2
2
}
∂3 w
∂ x3,
B4(ψ) =
{
p1 +42
68p2 γ2
2
}
∂2 ψ
∂ x2+
5 + 5ν
6 + 5ν
21
68p2(8 + 5 γ2
2) ψ ,
L1(w, w) =∂ w
∂ x
∂2 w
∂ x2, L2(u, w) =
∂ 2u
∂ x2
∂ w
∂ x+∂ u
∂ x
∂2 w
∂ x2, L3(w, w) =
3
2
(
∂ w
∂ x
)2∂2 w
∂ x2.
The relations between dimensional and non-dimensionalparameters are as follows: w = w
h, u = ua
h2 , ψ =ψ ah
, x = xa,
γ1 = ah, γ2 = l
h, q = q a4
h4E, t = tc
a, c =
√
Eρ
, ε = ε ac
, and l is the
size-dependent parameter (observe that for n = 1, one gets theisotropic material). The derived non-linear PDEs (1) require asupplement of the boundary and initial conditions.
III. METHODS OF SOLUTION
The system of non-linear PDEs (1) is reduced to the Cauchyproblem through the finite difference method (FDM) of the secondorder of accuracy with regard to the spatial coordinate. In order to
get the validated results, we have investigated convergence of theFDM against a number of beam length partition N. It has beenfound that convergence is achieved for N = 40. Then, the Cauchyproblem has been solved using the Runge–Kutta methods from thesecond to eight order and via the Newmark method. It has beenfound that all of the methods give almost the same results. Appli-cation of a few numerical methods on each stage of the modelingprocedure refers to the problem of the results reliability, and in addi-tion, one needs to remove (decrease) the occurred numerical errors.This is why we have constructed the phase portraits as well as theFourier and wavelet spectra in order to check the results’ reliability.We have employed the Morlet, Gauss8, Gauss32, and Haar wavelets,
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and it has been also found that the Morlet and Gauss32 waveletsbelong to the most informative and allow us to achieve good fre-quency localization at any time instant. A type of chaotic vibrationis quantified based on the Gulick criterion67 as well as the signs ofthe Lyapunov exponents computed via the Sano–Sawada method.68
Because one exact method for the LE computation does not exist, inorder to get reliable results, the LLEs are additionally estimated by afew methods: the Wolf,69 Kantz,70 Rosenstein71, and Sano–Sawada68
algorithms.
IV. NON-LINEAR DYNAMICS OF THE NANOBEAM
WITH LOW SHEAR STIFFNESS
It is known that nowadays glass–plastic materials can befabricated with a priori given properties. Let us consider the flex-ible beam with the designed material with the following parame-ters: γ1 = 100 , γ2 = 0, ε = 0.5, G/E =1/10, under the transversalload q(x, t) = q0 + q1 sin(ωpt) with the amplitude q1 and frequencyωp = 3.5.
TABLE I. Dynamical nanobeam characteristics (γ1 = 100 , γ2 = 0, ε = 0.5, q1 = 50 000).
t ∈ [600; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
LE–-LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
Lyapunov exponents (LEs) 0.048 585; 0.0197 87; −0.248 23 0.05737; −0.280 55 0.054 414; −0.245 19−0.021 963; −0.278 09 0.023 804; −0.246 12 0.020 882; −0.267 84−0.082 835; −0.256 81 −0.021 023; −0.230 86 −0.020 439; −0.267 13−0.504 11 −0.085 031; −0.082 368;
−0.609 57 −0.599 54
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wav
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We consider the following boundary conditions:
w(0, t) = w(1, t) =∂w(0, t)
∂x=∂w(1, t)
∂x= u(0, t) = u(1, t)
= ψ(0, t) = ψ(1, t) = 0,
and the following initial conditions:
w(x, 0) =∂w(x, 0)
∂t= u(x, 0) =
∂u(x, 0)
∂t= ψ(x, 0)
=∂ψ(x, 0)
∂t= 0.
We begin with the case when q1 < 5 · 104, where periodicnanobeam vibrations appear with ωp = 3.5. The Lyapunov expo-nent spectra contain five LEs from LE1 to LE5 being estimated using
the Sano–Sawada method, as well as the LLEs found through themethods of Rosenstein, Kantz, and Wolf (all of them have the neg-ative real part). When q1 = 5 · 104 and beginning with time instantt = 850, the change of the character of the vibration appears, i.e.,the Hopf bifurcation yields period doubling ω1 = ωp/2 = 1.75, andin addition, the linearly independent frequency ω2 = 0.3 occurs.The spectrum of the Lyapunov exponents takes the following val-ues LE1, LE2 > 0; LE3, LE4,LE5 < 0, which imply occurrence ofhyper-chaos (Table I).
Further increase of the load amplitude q1 > 6.5 · 104 yieldedmore sub-harmonics (Table II, see Fourier spectra) in spite of thefundamental three frequencies, and the vibrations are spanned overthe whole frequency interval ω ∈ [0; 4]. In addition, the Morletwavelet spectra (Table II) exhibit windows of switching on andoff of the frequencies. Owing to the Lyapunov exponent spectra
TABLE II. Dynamical nanobeam characteristics (γ1 = 100 , γ2 = 0, ε = 0.5, q1 = 85 000).
t ∈ [600; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
Lyapunov exponents (LEs) LE1–LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
0.070 239; 0.016 27; −0.309 57 0.069 384; 0.012 805; −0.316 77 0.060 39; −0.314 82−0.026 736; −0.312 24 −0.027 595; −0.297 38 0.012 657; −0.309 23−0.063 076; −0.228 76 −0.062 996; −0.325 14 −0.029 13; −0.347 12−0.119 14 −0.114 15 −0.059 44;
−0.112 82
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wavelet spectrum (Morlet)
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TABLE III. Dynamical nanobeam characteristics (γ1 = 50 , γ2 = 0, ε = 0.5, q1 = 3000).
t ∈ [600; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
Lyapunov exponents (LEs) LE1–LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
0.000 128 24; −0.3212 0.000 169 79; −0.3451 0.017 225; −0.3211−0.004 717 1; −0.4102 −0.004 345 6; −0.3678 0.007 492 3; −0.3512−0.287 75; −0.2907 −0.303 91; −0.3541 −0.008 650 4; −0.3651−0.325 69; −0.352 01; −0.034 895;−0.442 15 −0.441 54 −1.0438
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wavelet spectrum (Morlet)
LE1, LE2 > 0; LE3, LE4, LE5 < 0, surprisingly, we have detectedagain the hyper-chaotic nanobeam vibrations.
Therefore, for fixed γ1 = 100 , ε = 0.5, the transition of thenanobeams vibrations to chaos follows the Ruelle–Takens–Newhauseand the Feigenbaum scenarios.
A. Influence of the geometric parameter γ1 = ah
Now, we consider non-linear nanobeam dynamics for γ1 = 50and for the same remaining previously fixed parameters. Forq1 < 2637, periodic vibrations with the excitation frequencyωp = 3.5 are observed. The Lyapunov exponents LE1--LE5 havenegative values. For q1 = 0.3 · 104 and for t < 1500, the vibrationsare again periodic though the LLE computed via the Sano–Sawada
method as its magnitude is close to zero. However, the other threemethods, i.e., the Wolf, Kantz, and Rosenstein, yield negative LLEs,which validate the nanobeam periodic vibrations (Table III). Forthe given load and in time interval t ∈ [1500; 2000], the nanobeamsvibrations are spanned on three frequencies ωp = 3.5,ω1 = ωp/2 = 1.75,ω2 = 0.3. Again, since LE1, LE2 > 0; LE3, LE4, LE5 < 0,we detected the hyper-chaos (Table III).
Further increase of the external load amplitude implies theoccurrence of chaotic behavior earlier than in the previous case. Forq1 = 0.5 · 104, the interval t ∈ [600; 900] displays the Hopf bifur-cations ω1 = ωp/2 = 1.75, where LE1 = 0, while in the time inter-val t ∈ [900; 2000] the hyper-chaotic vibrations appear (LE1, LE2> 0)—see Table IV. Remarkably, 0.5 · 104 < q1 ≤ 1.8 · 104, thehyper-chaos (LE1, LE2 > 0) occupies the whole considered timeinterval.
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TABLE IV. Dynamical nanobeam characteristics (γ1 = 50 , γ2 = 0, ε = 0.5, q1 = 5000).
t ∈ [600; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
Lyapunov exponents (LEs) LE1–LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
0.000 112 34; −0.3152 0.034 853; −0.3341 0.046 782 7; −0.3572−0.003 567; −0.4678 0.011 311; −0.3219 0.011 679; −0.4224−0.387 52; −0.3519 −0.025 038; −0.3274 −0.026 781; −0.3987−0.322 76; −0.069 398; −0.081 278;−0.451 25 −0.573 16 −0.615 78
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wavelet spectrum (Morlet)
Further increase of the load amplitude, i.e., for 1.9 · 104 < q1
≤ 2.1 · 104 causes change of the character of the vibration. Namely,a periodicity window is observed in the initial time intervalt ∈ [600; 900] and the nanobeam exhibits a periodic regime(LE1 < 0), whereas in the time interval t ∈ [900; 1500] and begin-ning from the time instant t = 1120, a sudden increase ofthe nanobeam vibration is associated with the Hopf bifurca-tion ω1 = ωp/2 = 1.75 is observed. Finally, in the time intervalt ∈ [1500; 2000], the nanobeam is in the hyper-chaotic regime(LE1, LE2 > 0), though the 2D Morlet wavelets report the windowsof intermittency (Table V).
For 2.1 · 104 < q1 ≤ 2.8 · 104, the nanobeam vibrates in aperiodic manner with ωp = 3.5 in the whole considered time
interval. The Lyapunov exponents (LE1 − LE5) are negative. Forq1 ≥ 28250, again the qualitative change of the nanobeamsvibrations is observed, and in the time intervalt ∈ [600; 900], one-frequency vibrations occur. Next, in the timeinterval t ∈ [900; 1500], two-frequency vibrations take place (quasi-periodicity), while later in the time interval t ∈ [1500; 2000], hyper-chaos appears. Further increase of the excitation load amplitudeimplies an extension of the hyper-chaotic vibrations into the wholetime interval. Therefore, for γ1 = 50 , ε = 0.5, the investigatednanobeam vibrations transit into hyper-chaos via a combination ofthe Ruelle–Takens–Newhouse and Feigenbaum scenarios with theoccurrence of the windows of periodicity. However, the most impor-tant observation (often ignored and not widely recognized in the
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TABLE V. Dynamical nanobeam characteristics (γ1 = 50 , γ2 = 0, ε = 0.5, q1 = 19 000).
t ∈ [600; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
Lyapunov exponents (LEs) LE1–LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
−0.000 390 56; −0.3321 0.070 919; −0.3512 0.087 345; −0.4277−0.047 779; −0.4096 0.009 063 3; −0.3208 0.021 514; −0.4651−0.368 99; −0.3527 −0.043 272; −0.3456 −0.033 336; −0.4458−0.641 56; −0.078 421; −0.075 67;−0.975 59 −0.136 07 −0.139 73
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wavelet spectrum (Morlet)
existing available literature) is that nanobeam non-linear vibrationsundergo even qualitative changes with an increase of the simulationtime, i.e., with regard to the independent variable.
In what follows, we consider non-linear vibrations of thenanobeam for fixed γ1 = 30 and the same remaining parame-ters. For q1 < 0.1 · 104, the quasi-periodic vibrations are spannedon two frequencies, i.e., on the excitation frequency ωp = 3.5 andthe independent frequency ω1 = 2. For 1000 ≤ q1 < 4480, peri-odic vibrations with one-frequency ωp = 3.5 appear. For 4480 ≤ q1
< 4500, we observe a qualitative change of the character of thevibration, i.e., in the time interval t ∈ [600; 900] one-frequencyωp = 3.5 vibration takes place, whereas in the time intervalt ∈ [900; 1500], two-frequency vibrations (ωp = 3.5 and ω1 =
ωp/3 = 1.16) appear; in the time interval t ∈ [1500; 2000], the
period tripling bifurcation appears, which implies the nanobeamhyper-chaotic state (LE1, LE2 > 0)—see Table VI.
For the load q1 ≥ 0.5 · 104, the nanobeam vibrations arechaotic in the whole considered time interval. Therefore, forγ1 = 30, ε = 0.5, the transition from periodic vibrations to hyper-chaotic vibrations of the investigated nanobeam is realized viaperiod tripling bifurcation.
B. Influence of the dissipation ε
In this section, we consider the flexible nanobeam madefrom the material with the following parameters: γ1 = 100 , γ2 = 0,ε = 0.25.
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TABLE VI. Dynamical nanobeam characteristics (γ1 = 30 , γ2 = 0, ε = 0.5, q1 = 44 995).
t ∈ [600; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
Lyapunov exponents (LEs) LE1–LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
3.4408 × 10−5 −0.13 8965 0.000 825 641, 0.014 426 0.074 073 0.036 548−0.289 145, −0.320 211 −0.026 197, 0.012 337 0.031 438 0.062 417−0.220 286, −0.271 568 −0.105 121, 0.011 703 −0.001 077 0.043 158−0.365 475 −0.008 657, −0.062 927−0.065 547 −0.014 044
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wavelet spectrum (Morlet)
For the load amplitude q1 ≤ 1.5 · 104, the nanobeam exhibitsquasi-periodic vibrations spanned on the frequencies ωp = 3.5and ω1 = 2. For the load amplitude 15 000 < q1 ≤ 30 055.7, thenanobeam vibrates harmonically with the frequency ωp = 3.5. TheLEs spectrum composed of five LEs ((LE1–LE5) is computed withthe use of the Sano–Sawada method, whereas the LLEs (here LE1)are additionally computed via the Rosenstein, Kantz, and Wolfmethods, and they yield negative values of the LLEs. The excita-tion amplitude q1 = 30 055.8 implies the qualitative change of thenanobeam vibration character in the time interval t ∈ [520; 580],i.e., the beam deflection suddenly increases and simultaneouslythe LLE changes its sign from negative to positive. This canserve as the dynamic stability loss of the plane nanobeam vibra-tions under the uniformly distributed sinusoidal load. Remarkably,
this phenomenon has been already reported by Awrejcewicz andKrysko in their previous works.46,51 In addition, the intermittencybehavior is observed. Namely, the vibrations became chaotic andalso the Hopf bifurcation ω1 = ωp/2 = 1.75 and independent fre-quency ω2 = 0.3 appeared (see Table VII). Furthermore, in thetime interval t ∈ [580; 2348], nanobeam vibrations are decreasedtwo times though the nanobeam vibrations remain chaotic. Thespectrum composed of five Lyapunov exponents takes the valuesLE1 > 0; LE2, LE3, LE4, LE5 < 0 (LLEs computed via the methodsof Rosenstein, Kantz, and Wolf have negative values).
In the interval 30 056 < q1 ≤ 35 000, the nanobeam vibrationsare chaotic spanned on the two frequencies ω1 = ωp/2 = 1.75 andω2 = 0.3. Besides, in the time interval t ∈ [900; 1324], we havehyper-chaos since LE1, LE2 > 0; LE3, LE4, LE5 < 0, whereas in the
Chaos 31, 023107 (2021); doi: 10.1063/5.0032069 31, 023107-10
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TABLE VII. Dynamical nanobeam characteristics (γ1 = 100 , γ2 = 0, ε = 0.25, q1 = 30 055.8).
t ∈ [400; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
Lyapunov exponents (LEs) LE1–LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
−0.000 069 7; −0.285 167 0.021 874; −0.281 671 0.0254 09; −0.316 788−0.028 855; −0.291 568 −0.013 649; −0.321 096 −0.016 394; −0.326 261−0.045 959; −0.278 772 −0.045 119; −0.328 708 −0.078 418; −0.318 178−0.065 668 −0.081 977; −0.135 53;
−0.146 36 −0.238 19
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wavelet spectrum (Morlet)
time intervals t ∈ [400; 900] and t ∈ [1500; 2000], only LE1 > 0 (seeTable VIII). For the load amplitude 35 000 < q1 ≤ 60 000, the sys-tem vibrations are chaotic, and they are spanned on the frequen-cies ω1 = ωp/2 = 1.75 and ω2 = 0.3 in the whole considered timeinterval.
Therefore, for γ1 = 100 and ε = 0.25, the nanobeam periodicvibrations transit to chaotic vibrations via combinations of theRuelle–Takens–Newhouse and the Feigenbaum scenarios inter-played with the occurrence of the windows of periodicity.
V. CONCLUDING REMARKS
We have derived a mathematical model of non-linear vibra-tions of the nanobeam with low shear stiffness subjected to external
uniformly distributed transversal sinusoidal load. A numerical studyhas been carried out using the FDM and its convergence has beenvalidated via numbers of nanobeams length partition. The Cauchyproblem has been solved by a few methods in order to keep theresults’ reliability. It includes the choice of the most suitable waveletsas well as estimation of the Lyapunov exponents through four quali-tatively different methods. In addition, the character of non-linearnanobeam vibrations has been investigated and transitions intovarious chaotic regimes have been illustrated.
We have shown that the control parameters such as the geo-metric γ1 and size-dependent γ2, the amplitude of the sinusoidalload q1, and the dissipation coefficient ε have an important impacton the chaotic vibrations including chaotic and hyper-chaoticregimes. We have also detected and analyzed the spatiotemporal
Chaos 31, 023107 (2021); doi: 10.1063/5.0032069 31, 023107-11
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TABLE VIII. Dynamical nanobeam characteristics (γ1 = 100 , γ2 = 0, ε = 0.25, q1 = 35 000).
t ∈ [400; 900] t ∈ [900; 1324] t ∈ [1500; 2000]
Lyapunov exponents (LEs) LE1–LE5 LE1 LE1–LE5 LE1 LE1–LE5 LE1
0.000 143 11; −0.302 79 0.061 983;| −0.287 27 0.0292 86; −0.326 27−0.0106; −0.301 72 0.031 157; −0.280 98 −9.8363 × 10−5; −0.311 79−0.33 188; −0.316 83 −0.008 573 1; −0.291 12 −0.036 497; −0.320 98−0.39 846; −0.062 191; −0.055 326;−0.60 282 −0.180 56 −0.076 457
Signal w(0.5,t)
Phase portrait w(w)
Fourier spectrum
2D wavelet spectrum
chaos, i.e., the case when for the same intensity of the excitationparameters and fixed other parameters the nanobeam may exhibitdifferent vibration regimes like periodicity, quasi-periodicity, chaos,and hyper-chaos when the time variable increases.
The results obtained in this work are in qualitative agreementwith the results obtained experimentally in Refs. 60–62, where var-ious types of bifurcations were found, such as a saddle-node andHopf bifurcations, which can lead to quasi-periodic and poten-tially chaotic motions. These effects were also found in the presentwork.
The transition from periodic vibrations to chaos for fixedε = 0.5, γ1 = 100 or γ1 = 50 parameters has been realized via acombination of the Ruelle–Takens–Newhouse and the Feigenbaum
scenarios. In addition, for γ1 = 50 , we have detected the occurrenceof the windows of periodicity. For γ1 = 30, the transition of peri-odic vibrations to hyper-chaos has been realized via period triplingassociated with the Hopf bifurcation.
Decrease of at least one of the parameters, i.e., either γ1 (ratioof the length and thickness of the nanobeams) or the dissipationcoefficient ε implies that the nanobeam vibrations transit to hyper-chaos for the less intensity of the external load. Besides, for thecase of ε = 0.25, γ1 = 100 , we have detected dynamical stabilityloss of the nanobeam combined with the system transition to hyper-chaotic vibrations. This can be recognized as our novel contributionto the problem of dynamic stability loss of the nanobeam under theuniformly distributed sinusoidal load.
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The carried out investigations as well as the developed method-ology and obtained results yield also an important observationthat in order to get reliable and validated results of the nanobeamchaotic vibrations, there is a need to employ qualitatively differentapproaches. In other words, analysis based on only one employedmethod may lead to erroneous results. In our work, to avoidthis problem, we have computed the largest Lyapunov exponentsbased on the few qualitatively different algorithms of Sano–Sawada,Rosenstein, Kantz, and Wolf while estimating the values of the Lya-punov exponents. Furthermore, our experience supported by theobtained results indicates that their reliability requires the investi-gation of the dynamic characteristics including time series, phaseportraits, Fourier and wavelet spectra, as well as spectra of theLyapunov exponents.
AUTHORS’ CONTRIBUTIONS
All authors contributed equally to this work.
ACKNOWLEDGMENTS
The work was supported by the RFBR (No. 19-31-90131).The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared toinfluence the work reported in this paper.
DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
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