Nonlinear power flow analysis of the Duffing oscillator

Mechanical Systems and Signal Processing 45 (2014) 563–578

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Nonlinear power flow analysis of the Duffing oscillator J. Yang, Y.P. Xiong n, J.T. Xing Faculty of Engineering and the Environment, Fluid Structure Interactions Research Group, University of Southampton, Highfield, Southampton SO17 1BJ, UK

a r t i c l e i n f o

abstract

Article history: Received 24 February 2013 Received in revised form 8 November 2013 Accepted 9 November 2013 Available online 13 December 2013

Power flow characteristics of different forms of the Duffing oscillator, subject to harmonic excitations, are studied in this paper to reveal the distinct power input and dissipation behaviour arising from its nonlinearity. Power flow variables, instead of the displacement and velocity responses, are used to examine the effects of nonlinear phenomena including sub-/super-harmonic resonances, non-uniqueness of solutions, bifurcations and chaos. Both analytical harmonic balance approximations and Runge–Kutta numerical integrations are adopted to effectively address instantaneous/time-averaged power flows of the system with periodic/chaotic motions without losing the essential nonlinear characteristics. It is demonstrated that only the in-phase velocity component with the same frequency as the excitation contributes to the time-averaged input power (TAIP). It is shown that super-/sub-harmonic resonances may result in substantial increases in TAIP and the nonlinearity leads to varying time-averaged power flow levels sensitive to the initial conditions. The study reveals that bifurcations may cause large jumps in timeaveraged input power. However, for bifurcations of periodic to chaotic motions encountered in the low-frequency range, the corresponding variations in TAIP of the double-well potential systems are small. For a chaotic response, the associated TAIP is insensitive to the initial conditions but tends to an asymptotic value as the averaging time increases, and thus can be used as a measure to quantify chaotic responses. The paper concludes some inherently nonlinear power flow characteristics which differ greatly from those of the linear systems, and provides useful information for applications. Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.

Keywords: Power flow analysis Duffing oscillator Nonlinear stiffness Super-/sub-harmonic resonances Bifurcation Chaos

1. Introduction Vibrational power flow analysis (PFA) approach has become a widely accepted tool to investigate dynamic behaviour of coupled systems and complex structures. Compared with individual measures such as force and displacement transmissibility, vibrational power flow combines the effects of force and velocity amplitudes as well as their relative phase angle in a single quantity, and thus can better reflect the transmission of vibration energy between various sub-systems of an integrated structure. The PFA fundamental concepts were discussed by Goyder and White [1]. In recent years, various approaches, such as a dynamic stiffness method [2], a receptance method [3], a mobility method [4], a wave intensity method [5], a finite-element based energy flow modelling technique [6] and progressive approaches [7] were developed and applied to investigate vibration control systems [8]. Instead of investigating individual structures such as coupled beam/ plate-like structures or periodic systems, Xing and Price [9] proposed a more general PFA approach based on the fundamental

n

Corresponding author. Tel.: þ44 2380596619; fax: þ 44 2380 597744. E-mail addresses: [email protected], [email protected] (Y.P. Xiong).

0888-3270/$ - see front matter Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2013.11.004

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principles of continuum dynamics, in which the energy-flow vector, energy-flow potential and energy-flow lines describing energy flow distributions were defined. Xiong et al. [10] developed a power flow mode theory to reveal the inherent power flow behaviour of a dynamic system based on its damping distribution, from which energy flow design approaches were proposed to achieve specific power flow patterns satisfying vibration control requirements. Although significant advances on power flow characteristics of linear dynamical systems have been reported, investigations on power flows of nonlinear vibrating systems are limited. However, dynamic systems in practice are inherently nonlinear, and strong nonlinear effects have been encountered in many applications. For example, nonlinear models for both damping and stiffness were needed for accurate dynamic analysis of hydraulic engine mounts used in the automobile industry [11]. The damping coefficient of orifice-type dampers varies with their internal geometry, frequency of flow oscillation and the Reynolds number [12]. There may also be significant nonlinearity in structural joints [13]. Moreover, introducing nonlinear elements to a design may bring benefits which could not be achieved by linear systems. For instance, it has been shown that nonlinear vibration absorbers can successfully enlarge the effective working frequency range of their linear counterparts [14]. A nonlinear negative stiffness mechanism can be used in parallel with linear isolators to improve the effectiveness of vibration isolators in the low-frequency range [15]. In view of these facts, there has been a growing interest in studying nonlinear dynamical systems from the perspective of power flows in past few years. Royston and Singh [16] employed vibratory power transmission as a performance index in the optimisation of multiple degrees-of-freedom nonlinear mounting systems, and examined automotive hydraulic engine mounts by investigating vibratory power flows from an excited rigid body through a nonlinear path into a resonant receiver [17]. Xiong et al. [18] studied a nonlinear coupling system consisting of a machine, a generic nonlinear isolator and a flexible beam-like ship travelling in seaway. The nonlinearity was characterised by a pth power damping term and a qth power stiffness term, and was shown to have a significant influence on the system's power flows, especially when the excitation frequency is close to resonant frequencies. Oscillators with essentially nonlinear stiffness may exhibit the phenomenon of targeted energy transfer (TET), which corresponds to one-way channelling of the vibrational energy from a primary structure to a passive nonlinear attachment [19]. Based on the time-averaged input power information associated with free oscillations of conservative systems, a frequency-energy plot (FEP) can be used to represent nonlinear normal modes and the frequency-energy dependence of nonlinear systems [20]. Yang et al. [15] used time-averaged power flow quantities instead of traditional force transmissibility to assess vibration isolation performance and attempted to quantify the nonlinear responses of the Duffing oscillator by power flow analysis [21]. Previous research has clearly shown that a better understanding of power flow patterns in nonlinear dynamical system can bring valuable benefits for science and engineering. However, due to a lack of power flow theory and effective modelling and simulation methods to deal with systems involving complex nonlinear phenomena, the influences of nonlinearity on system power flows remain unclear. Fundamental studies are still needed to reveal the basic principles governing vibration power generation, dissipation and transmission in nonlinear dynamical systems. In this paper, the power flow behaviour of a typical nonlinear system, the Duffing oscillator, is investigated as an attempt to address the above problem. This system has been extensively studied with focus on its displacement/velocity responses characteristics [22–24]. Nevertheless, new information can still be obtained by examining it from another perspective of vibrational power flows. Such examination is necessary considering that the influence of the stiffness nonlinearity on power flows has not been clarified and also the findings may provide promising applications to vibration control and energy harvesting. Emphasis of the present study will be placed on revealing the associated power flow behaviour of the system when it exhibits complex nonlinear phenomena, such as sub-/super harmonic resonances, bifurcations and chaotic motions. Following the derivations power flow formulations, the solution methods used in the paper are briefly described. The harmonic balance method is used for analytical approximations of the power flow variables of the system undergoing periodic motions. Numerical simulations are conducted to investigate the instantaneous power flows, to verify the analytical approximations and to examine the effects of chaotic motion on system power flows. Conclusions and some suggestions for applications are provided at the end of the paper. 2. Power flow formulations and solution approaches 2.1. Power flow formulations The Duffing oscillator is governed by the equation x€ þ 2ξx_ þ αx þ βx3 ¼ f cos ωt;

(1)

in which the restoring force is characterised by a linear term αx and a cubic nonlinear term βx . It may be referred to as a softening stiffness system (Case I) when α4 0; β o0; a hardening stiffness system (Case II) when α 40; β 4 0; or a doublewell potential system (Case III) when α o 0; β 4 0. A system with α o 0; β o 0 has non-positive stiffness, thus it is unstable and will not be investigated in this paper. Multiplying by the velocity x_ on both sides of Eq. (1), we derive the power flow balance equation of the system in the form: 3

K_ þ U_ þpd ¼ pin ;

(2)

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where K_ ¼ x€ x_ ; U_ ¼ αxx_ þβx3 x_ ; pd ¼ 2ξx_ x_ ; pin ¼ x_ f cos ωt;

(3a–3d)

are time change rates of kinetic and potential energies, and dissipated and input powers, respectively. Assuming that the displacement and velocity of the system are xi and x_ i at t ¼ t i ; and x and x_ at t ¼ t i þt p , respectively, an integration of Eq. (3) over the time period leads to ΔK þΔU þ Ed ¼ Ein ;

(4)

where Z ti þ tp αðx2  x2i Þ βðx4 x4i Þ x_ 2  x_ 2i ; ΔU ¼ þ ; ðαxx_ þβx3 x_ Þdt ¼ 2 2 4 ti Z ti þ tp Z ti þ tp Ed ¼ x_ f cos ωt dt; 2ξx_ 2 dt; Ein ¼ ΔK ¼

ti

(5a–5d)

ti

represent the changes in kinetic and potential energies, and total dissipated and input energies of the system, respectively; t p is the considered time span. The nonlinearity of the system is demonstrated by Eq. (5b), with the potential energy being affected by the nonlinear parameter β. For cases I and II systems, only one minimum point in potential energy exists at x ¼ 0. However, for ffi a Case III system, the potential energy at x ¼ 0 refers to a local maximum, with two local minima at pffiffiffiffiffiffiffiffiffiffiffiffi x ¼ 7  α=β, as shown in Fig. 1. Supposing the reference potential energy at x ¼ 0 is zero, the difference in the potential energies of the local maximum and minimum points is rffiffiffiffiffiffiffiffiffi2 rffiffiffiffiffiffiffiffiffi4 ! α α β α α2 ΔU ¼ 0  ¼   : (6) þ 2 β 4 β 4β which becomes more significant as α2 increases or β decreases. This quantity is important in affecting the dynamic and power flow behaviour of the Case III systems. As shown in Fig. 1, the curve of potential energy forms two wells around the local minimum points. If the system firstly oscillates in one of the wells, in order to reach the other it will need enough energy to overcome the potential barrier at x ¼ 0. The time-averaged dissipated and input powers are formulated as Z Z E 1 ti þ tp E 1 ti þ tp x_ f cos ωt dt: (7a and b) pd ¼ d ¼ 2ξx_ 2 dt; pin ¼ in ¼ t p ti t p ti tp tp For a linear system subject to a harmonic excitation, the steady-state response will also be harmonic. Over a cycle of oscillation, there will be no net change in kinetic and potential energies ðΔK ¼ 0; ΔU ¼ 0Þ and the time-averaged dissipated and input powers will be equal. However, this is not generally true for nonlinear systems, as the steady-state response may become non-periodic. For example, they may exhibit quasi-periodic motion when the response frequency components are incommensurate with each other [25]. Chaotic motions may also be encountered containing infinite frequency components. 2.2. Solution approaches The harmonic balance (HB) method [26] can be used to obtain the power flow characteristics of the system exhibiting periodic solutions. This method represents the steady-state displacement and velocity in a form of the

Fig. 1. Potential energy of the double-well potential system. Solid line: α ¼  1:0; β ¼ 0:5; dashed line: α ¼  1:0; β ¼ 1 and dash-dot line: α ¼  0.5, β ¼1.

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truncated Fourier series: N

x ¼ x^ 0 þ ∑ ðx^ 2n  1 cos nω1 t þ x^ 2n sin nω1 tÞ;

(8a)

n¼1

N

x_ ¼ ω1 ∑ ð  nx^ 2n  1 sin nω1 t þnx^ 2n cos nω1 tÞ;

(8b)

n¼1

respectively, where ω1 is the fundamental frequency of oscillation, x^ j ðj ¼ 0; 1; ⋯; 2NÞ are the coefficients of the related harmonic functions and N is the highest order of harmonics considered. Using a Fourier series expansion, the cubic term in Eq. (1) is expressed as N

x3  q^ 0 þ ∑ ðq^ 2n  1 cos nω1 t þ q^ 2n sin nω1 tÞ;

(9)

n¼1

where the coefficients are calculated by Z Z Z ω1 ð2π=ω1 Þ 3 ω1 ð2π=ω1 Þ 3 ω1 ð2π=ω1 Þ 3 q^ 0 ¼ x dt; q^ 2n  1 ¼ x cos nω1 t dt; q^ 2n ¼ x sin nω1 t dt: 2π 0 π 0 π 0 Using Eqs. (3d) and (8b), the expression of the instantaneous input power may be transformed into f ω1 N  ^ pin ¼ ∑ n x2n ½ cos ðnω1 þωÞt þ cos ðnω1  ωÞt 2 n¼1   x^ 2n  1 ½ sin ðnω1 þ ωÞt þ sin ðnω1  ωÞt ;

(10a–c)

(11)

containing frequency components of nω1 7ω ðn ¼ 0; 1; ⋯; NÞ: It can be seen that pin is periodic with its period being the least common multiple of that of each individual component. Averaging Eq. (11) over its period, only the stationary component with nω1 ω ¼ 0 gives a non-zero value, i.e., pin ¼ 12 f ω1 nx^ 2n ;

(12)

where n ¼ ω=ω1 . In other words, only the in-phase component of the velocity with the same frequency as that of the excitation will contribute to TAIP. Alternatively, the governing Eq. (1) may be firstly transformed into two first-order differential equations, which are then numerically integrated to determine displacement/velocity responses and subsequently the instantaneous and timeaveraged power flow variables. Fourier transformations may be employed to provide the corresponding frequency spectrum of a response, so that its dominant components can be identified and included in the analytical model. Comparing these two approaches, the numerical one provides more accurate results incorporating the effects of different frequency components on power flows. It is a generalised approach suitable to solve various systems exhibiting either periodic/quasi-periodic or even chaotic motions. However, it is computationally more expensive. The harmonic balance method, in contrast, provides approximate solutions with less computational cost and more physical insight. Therefore, both the HB method and numerical integrations will be used in this paper. The former will be used to derive analytical approximations of power flow variables when the system exhibits periodic motions, while the latter is employed to verify the HB results and obtain the power flow behaviour of the system exhibiting chaotic motions.

3. Power flow behaviour associated with periodic motions 3.1. A first-order approximation 3.1.1. Basic characteristics of the time-averaged input power (TAIP) In the frequency range of primary resonances, a harmonic response with the same frequency as that of the excitation may be assumed to obtain analytical approximations of power flow variables: x ¼ x^ 0 þ x^ 1 cos ωt þ x^ 2 sin ωt;

(13a)

x_ ¼ ωx^ 1 sin ωt þ ωx^ 2 cos ωt:

(13b)

Then the nonlinear term x3 may be found by using Eqs. (9) and (10). Substituting the terms in Eq. (1) with their approximate expressions and equating the coefficients of the corresponding harmonic terms, we obtain 2

2

2

(14a)

ðα ω2 Þx^ 2  2ξωx^ 1 þ 34 βðx^ 1 þ x^ 2 þ 4x^ 0 Þx^ 2 ¼ 0;

2

2

2

(14b)

  2 2 2 αx^ 0 þβx^ 0 x^ 0 þ 32 ðx^ 1 þ x^ 2 Þ ¼ 0:

(14c)

ðα ω2 Þx^ 1 þ 2ξωx^ 2 þ 34 βðx^ 1 þ x^ 2 þ 4x^ 0 Þx^ 1 ¼ f ;

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A manipulation of Eqs. (14a) and (14b) leads to 2

(15a and b) ðα  ω2 Þr 2 þ 34 βr 2 ðr 2 þ 4x^ 0 Þ ¼ f x^ 1 ; 2ξωr 2 ¼ f x^ 2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where r ¼ x^ 1 þ x^ 2 , i.e., the amplitude of the oscillation. For clarity, we define Type-1 oscillations to represent harmonic vibrations with x^ 0 ¼ 0, i.e., oscillations around the static equilibrium point of x ¼ 0, so that Eqs. (15a and b) are further simplified into  2 2 ðα ω2 Þr þ 34 βr 3 þ ð2ξωrÞ2 ¼ f : (16) Similarly, Type-2 oscillations are used to denote harmonic motions with x^ 0 a 0, i.e., a double-well potential system oscillating in one of its potential wells. In this case, Eqs. (14c) and (15a and b) can be transformed into 2

½ðω2 þ2αÞr þ3:75βr 3 2 þ ð2ξωrÞ2 ¼ f :

(17)

Based on the previous formulations and a first-order approximation, the following characteristics are identified. Characteristic 1: The time-averaged input power pin is proportional to the maximum kinetic energy K max as well as the damping coefficient ξ. To show this, TAIP over a cycle of oscillation T ¼ 2π=ω is formulated by using Eqs. (12) and (15b), i.e., pin ¼ ξr 2 ω2 :

(18)

In steady-state motion, the maximum velocity x_ max ¼ rω of the system corresponds to the maximum kinetic energy: K max ¼ 12 r 2 ω2 :

(19)

Therefore, pin ¼ 2ξK max ;

(20)

and the Characteristic 1 is valid. 2 Characteristic 2: There exists an upper bound P in ¼ f =4ξ of TAIP, independent of stiffness parameters a and b and the excitation frequency. To demonstrate this, we rewrite Eqs. (16) and (17) as r2 ¼

f

2

ðω2  α 0:75βr 2 Þ2 þð2ξωÞ2

; r2 ¼

f

2

ðω2 þ2α þ3:75βr 2 Þ2 þ ð2ξωÞ2

;

(21a and b)

respectively. Using Eqs. (18),(19) and (21), we have pin ¼

ξf

2 2 2

ðω  α 0:75βr Þ þ 4ξ2 ω2 2

; K max ¼

0:5f

2

ðω  α 0:75βr 2 Þ2 þ 4ξ2 ω2 2

;

(22a and b)

for Type-1 oscillations and pin ¼

ξf

2 2 2

ðω2 þ 2α þ 3:75βr Þ þ4ξ2 ω2

; K max ¼

0:5f

2

ðω2 þ2α þ3:75βr 2 Þ2 þ 4ξ2 ω2

;

(23a and b)

for Type-2 oscillations. Therefore, we have 2

pin r

2

f f ¼ P in ; K max r 2 ; 4ξ 8ξ

(24a and b)

which confirms Characteristic 2. This implies that the upper bound of TAIP depends only on the excitation amplitude f and the damping coefficient ξ. For a Type-1 oscillation to reach the upper bound values of pin and K max , the values of ω2 and r 2 should satisfy ω2 α  34 βr 2 ¼ 0; which corresponds to the backbone curve of the undamped system. Together with Eq. (16), we obtain pffiffiffiffiffiffi pffiffiffiffiffiffi  2α 7 Δ1 2α7 Δ1 ; ω2 ¼ ; r2 ¼ 3β 4

(25)

(26a and b)

2

where Δ1 ¼ 4α2 þ3βf =ξ2 . Similarly, for a Type-2 oscillation to achieve the upper bound, it requires ω2 þ2α þ3:75βr 2 ¼ 0: Together with Eq. (17), we obtain pffiffiffiffiffiffi pffiffiffiffiffiffi  4α 7 Δ2 4α 8 Δ2 ; ω2 ¼ ; r2 ¼ 15β 4

(27)

(28a and b)

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where Δ2 ¼ 16α2  15βf =ξ2 . Table 1 lists the conditions and positive solutions of r 2 and ω2 to reach the upper bound. For a Case II system, only one set of ðr 2 ; ω2 Þ leads to the maximum TAIP, whereas for Cases I and III systems, the peak value may be encountered at different locations. Also, it can be shown that the upper bound values are found at saddle-node bifurcation points. 2 Characteristic 3: In the high-frequency range, pin  ξf =ðω2 þ 4ξ2 Þ, independent of parameters α and β, so that power flows are not sensitive to the stiffness nonlinearity at high excitation frequencies. In the high-frequency range where ω2 is large, pin and K max can be approximated by pin 

ξf

2

ω2 þ 4ξ2

2

; K max 

0:5f ; ω2 þ4ξ2

(29a and b)

where Eqs. (22a and b) and (23a and b) are used for Type-1 and Type-2 oscillations, respectively. This shows that TAIP of the system is not sensitive to variations in stiffness parameters α and β in the high-frequency range. 3.1.2. Effects of different parameters on power flows Solving Eq. (16) or (17) with a bisection method, the response amplitude, time-averaged input power as well as the maximum kinetic energy of the system can be obtained. In this section, the influence of different parameters on system power flows is investigated. The default dB reference for pin and K max is set as 10  12 throughout the paper. In Fig. 2, the effects of parameter β on pin and K max are examined with other parameters fixed as ξ ¼ 0:01; α ¼ 1:0; f ¼ 0:1. The system changes from Case I with softening stiffness ðβ ¼  0:1;  0:01Þ, to linear ðβ ¼ 0Þ, and then Case II ðβ ¼ 0:5Þ with hardening stiffness. For completion, both stable and unstable solution branches are shown, though the latter ones are not physically realisable. From the figure, the following power flow characteristics are observed. (1) The nonlinear parameter β has a strong influence on power flows when the excitation frequency is close to the resonance frequency. When the excitation frequency ω is far from the resonance frequency, the power flow variables are not sensitive to variations in β as the curves coincide. Table 1 Conditions and locations for reaching the upper bound (peak value) of TAIP.

2

40 Type-1 oscillations with conditions Δ1 ¼ 4α2 þ 3βf ξ2

System category

Number of peaks

The value of ðr2 ; ω2 Þ

Case I systems ðα 4 0; β o 0Þ

0, when Δ1 o 0

Not applicable    2α α 3β ; 2  pffiffiffiffiffi pffiffiffiffiffi  2α 7 Δ1 ; 2α 74 Δ1 3β  pffiffiffiffiffi pffiffiffiffiffi  2α þ Δ1 2α þ Δ1 ; 3β 4  pffiffiffiffiffi pffiffiffiffiffi  2α þ Δ1 2α þ Δ1 ; 3β 4

1, when Δ1 ¼ 0

2

r 4 0, ω 40. 2

2, when Δ1 4 0

2

40 Type-2 oscillations with conditions Δ2 ¼ 16α2  15βf ξ2

Case II systems ðα 4 0; β 4 0Þ

1, as Δ1 4 0 holds

Case III systems ðα o 0; β 4 0Þ

1, as Δ1 4 0 holds

Case III systems ðα o 0; β 4 0Þ

0, when Δ2 o 0

r2 40, ω2 40. 1, when Δ2 ¼ 0 2, when Δ2 4 0

Not applicable 

 α  pffiffiffiffiffi  4α 7 Δ2 ; 15β  4α 15β ;

 4α 8 4

pffiffiffiffiffi Δ2

Fig. 2. Effects of nonlinear stiffness coefficient β on (a) pin and (b) K max . First-order HB approximations: dash-dot line ðβ ¼  0:1Þ, dashed line ðβ ¼  0:01Þ, solid line ðβ ¼ 0Þ and dotted line ðβ ¼ 0:5Þ. Numerical integration results: circles ðβ ¼  0:1Þ, squares ðβ ¼  0:01Þ and triangles (β¼ 0.5).

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(2) For the softening, linear and hardening stiffness systems with β ¼ 0:01; 0 and 0:5, respectively, the peaks in pin and K max curves are of the same height and are found at saddle-node bifurcation points. This is similar to the influence of the nonlinear stiffness parameter β on peak response amplitude [24]. However, the peak value for the softening system with β ¼ 0:1 is much smaller. For this case, response curves obtained using the frequency response relationship are connected and extend to zero. Using the analysis results in Section 3.1.1, it can be shown that the upper bound of power flows can be realised for the former three cases, but not for the latter with β ¼  0:1 for which Δ1 ¼  26o 0. For β ¼  0:01 case, there are two solution branches coincided in the upper left corner of the figure and disconnected from the main part of the resonance curve. This is due to the fact that in the frequency band between A′ and B′, there is only a single solution at each excitation frequency. The results suggest an approach to reduce the peak power flow values by adding nonlinear softening stiffness into a linear isolator. (3) The figure shows that at points A′; A″ and B′; B″ on the curves of β ¼ 0:01, as well as at points D′ and D″ on the curves of β ¼ 0:5, there are two values of pin and K max at each corresponding frequency. Bifurcation occurs at these critical bifurcation points with possible jumps in the values of pin and K max when the excitation frequency changes. In some frequency ranges, the time averaged power flow become multivalued and sensitive to initial conditions. (4) A nonlinear stiffness of β a0 introduces a wide frequency band with a larger pin , compared with the linear system. This feature can be used for broadband vibration energy harvesting. However, due to the existence of non-unique solutions, proper initial conditions have to be chosen such that they locate in the basin of attraction of the large amplitude motion. In Fig. 3, the effects of the coefficient of the linear restoring force term α on the power flow variables are examined with the other parameters set as ξ ¼ 0:01; f ¼ 0:1; β ¼ 0:5. In this way, the system changes from Case II when α ¼ 5; 3 or 1 to Case III when α ¼  1. When α 4 0, it shows that an increase in α shifts the power flow curves to the high frequencies. This is due to the fact that parameter α determines the natural frequency of the linearised system. At low frequencies, both pin and K max increase as α reduces from 5 to 1. However, at frequencies higher than the resonance frequency, a larger α results in larger power flows into the hardening stiffness system. This figure again demonstrates that there is a uniform upper bound (peak value) for both pin and K max of the system regardless of the variations in α. For the case with α ¼  1, numerical integration results indicate that the system exhibits Type-2 oscillations in one of the potential wells. Therefore, Eq. (17) is used to obtain the analytical approximations of power flows. The results in Fig. 3 shows that the corresponding power flow curves bend towards the low frequencies, which is similar to the characteristics of the softening stiffness system (Case I). Also, using the results from Section 3.1.1, it can be shown that upper bound of pin cannot be reached as correspondingly we have Δ2 ¼  734 o 0. The figure also shows that as ω increases towards high frequencies, the power flow curves of different α values tend to merge with each other. This demonstrates Characteristic 3 in Section 3.1.1. 3.2. A second-order approximation In first-order HB approximations, only the primary response component with the same frequency as the excitation is considered. However, for a nonlinear system, the other frequency components may become large and dominant when away from the primary resonance region. To reveal this phenomenon, a second-order HB approximation is adopted to obtain more accurate solutions of power flow variables. 3.2.1. Power flows due to super-harmonic resonances For the system with α ¼ 1:0; β ¼ 0:1; ξ ¼ 0:02; f ¼ 2, Fig. 4 compares the first-order HB approximations of pin and K max with those obtained using numerical integrations. In the low-frequency region, local peaks are encountered at points A

Fig. 3. Effects of the linear stiffness coefficient α on (a) pin and (b) K max . First-order HB approximations: solid line ðα ¼ 5Þ, dashed line ðα ¼ 3Þ, dash-dot line ðα ¼ 1Þ and dotted line ðα ¼  1Þ. Numerical integration results: circles ðα ¼ 5Þ, triangles ðα ¼ 3Þ, squares ðα ¼ 1Þ and pentagons (α ¼  1).

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and B in pin and at A′ and B′ in K max , resulting from the occurrences of super-harmonic resonances. Around these local peaks, the response amplitude is found to be larger than 1 so that the nonlinear term in the governing equation becomes significant. Fig. 5 examined the power flow characteristics when the excitation frequency ω ¼ 0:41 and 0:24, locating at peaks A and B in Fig. 4, respectively. Fig. 5(a) and (d) shows the time histories of pin . The frequency spectra ωr of the instantaneous input power and the displacement response are presented by Fig. 5(b), (e) and (c), (f) respectively. When ω ¼ 0:41, Fig. 5(c) shows that apart from the primary response component at ωr ¼ ω, there is a large super-harmonic component at ωr ¼ 3ω. As a result, there are significant signatures in pin at 2ω and 4ω. Similarly, when ω ¼ 0:24, Fig. 5(f) shows that major response components locate at ω, 3ω, 5ω and 7ω. Consequently, the input power pin is dominated by components at 2ω, 4ω, 6ω and 8ω.

Fig. 4. Effects of super-harmonic resonances on (a) pin and (b) K max . Solid lines: first-order HB approximations; dots: numerical integration results. Parameters are set as α ¼1.0, β ¼ 0.1, ξ ¼0.02, and f ¼2.

Fig. 5. Power flows of the system when super-harmonic resonances occur ðα ¼ 1:0; β ¼ 0:1; ξ ¼ 0:02; f ¼ 2:0Þ : (a)–(c), ω ¼ 0:41; (d)–(f),ω ¼ 0:24. (a, d) Instantaneous input power; (b, e) frequency spectra of pin ; (c, f) frequency spectra of displacements.

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To reveal the effects of super-harmonic resonances on power flows, we use a second-order HB approximation containing two frequency components ω1 and 3ω1 : x ¼ x^ 1 cos ω1 t þ x^ 2 sin ω1 t þ x^ 5 cos 3ω1 t þ x^ 6 sin 3ω1 t;

(30a)

x_ ¼  ω1 x^ 1 sin ω1 t þω1 x^ 2 cos ω1 t 3ω1 x^ 5 sin 3ω1 t þ 3ω1 x^ 6 cos 3ω1 t;

(30b)

where ω1 equals the excitation frequency ω. Then, the nonlinear term x3 is expressed with harmonic functions using Eqs. (10) and (11). A second-order harmonic balance condition of Eq. (1) yields h i 3 2 2 2 2 2 (31a) ðα  ω21 Þx^ 1 ¼ f  2ξω1 x^ 2  34 β x^ 1 þ x^ 1 x^ 2 þðx^ 1  x^ 2 Þx^ 5 þ 2x^ 1 x^ 2 x^ 6 þ 2ðx^ 5 þ x^ 6 Þx^ 1 ; h i 3 2 2 2 2 2 ðα  ω21 Þx^ 2 ¼ 2ξω1 x^ 1  34 β x^ 2 þ x^ 1 x^ 2 þ ðx^ 1  x^ 2 Þx^ 6  2x^ 1 x^ 2 x^ 5 þ2ðx^ 5 þ x^ 6 Þx^ 2 ;

(31b)

h i 3 2 2 2 3 2 ðα  9ω21 Þx^ 5 ¼  6ξω1 x^ 6  14 β x^ 1  3x^ 1 x^ 2 þ 6ðx^ 1 þ x^ 2 Þx5 þ3x^ 5 þ 3x^ 5 x^ 6 ;

(31c)

h i 3 2 2 2 3 2 ðα  9ω21 Þx^ 6 ¼ 6ξω1 x^ 5  14 β  x^ 2 þ 3x^ 1 x^ 2 þ 6ðx^ 1 þ x^ 2 Þx6 þ3x^ 6 þ 3x^ 5 x^ 6 :

(31d)

A Newton–Raphson technique is used to solve this set of nonlinear equations. Subsequently, TAIP is obtained from pin ¼ 0:5f ω1 x^ 2 . When α ¼ 1:0; β ¼ 0:1; ξ ¼ 0:02; f ¼ 2:0; ω ¼ 0:41, the value of x^ 2 is calculated to be 0:2038 so that pin equals 218.44dB, agreeing well with the numerical integration results shown in Fig. 4. Fig. 6 examines the effects of the nonlinear stiffness parameter β on the system's power flows in the super-harmonic region. The system parameters are set as ξ ¼ 0:02; α ¼ 1:0; f ¼ 2:0. It shows that the numerical integration results agree relatively well with the second-order HB approximations. The peaks of both pin and K max shift to the higher frequencies as the system nonlinearity becomes stronger. In super-harmonic resonant regions, TAIP and kinetic energy of the nonlinear system is larger than that of the corresponding linear system. However, when the excitation frequency ω becomes larger than 0:5, the value of pin and K max reduces with an increasing nonlinear stiffness parameter β. 3.2.2. Power flows due to sub-harmonic resonances For a system subject to strong excitations, sub-harmonic resonances may appear when the excitation frequency is larger than the natural frequency of the linearised system [15,23]. Thus, it is necessary to study their effects to obtain full power flow information for effective designs of passive nonlinear vibration isolators. Fig. 7 shows the variations of pin and K max with the excitation frequency for a system with α ¼ 1:0; β ¼ 0:2; ξ ¼ 0:01; f ¼ 10. The results are obtained by both the RK method and a first-order HB method. There are local peaks in the lower branches of pin and K max obtained by numerical integrations, with the local peak values much larger than the first-order HB predictions. Therefore, to obtain a better analytical approximation, the dominant sub-harmonic components should be included in the HB model. The figure also shows a super-harmonic peak in the numerical results at ω ¼ 1:14, with a strong super-harmonic component in the response. Fig. 8 shows the power flow time histories, as well as frequency spectra of input power and displacement at two excitation frequencies of ω ¼ 3:5 and 2:8. Fig. 8(a)–(c) for ω ¼ 3:5 shows a large sub-harmonic response component at ωr ¼ ω=3 and a primary one at ωr ¼ ω. Correspondingly, significant components in pin exist at 2ω=3, 4ω=3 and 2ω. When the excitation frequency is 2:8, there is a large sub-harmonic response component at ωr ¼ ω=2, as shown in Fig. 8(f). As a result, the input power is dominated by components at ω=2, 3ω=2 and 2ω. Clearly, sub-harmonic resonances occur in both cases.

Fig. 6. Effects of nonlinear parameter β on (a) pin and (b) K max , in super-harmonic resonance region. Numerical integrations results: solid line ðβ ¼ 0Þ, dashed line ðβ ¼ 0:05Þ, dash-dot line ðβ ¼ 0:1Þ and dotted line ðβ ¼ 0:2Þ. Second-order HB approximations: circles ðβ ¼ 0:05Þ, squares ðβ ¼ 0:1Þ and triangles (β ¼0.2).

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Fig. 7. Effects of sub-harmonic resonances on (a) TAIP and (b) K max . Solid lines: first-order HB approximations; dots: numerical integration results. Parameters are set as: ξ ¼0.01, α¼ 1.0, β ¼0.2, and f ¼10.

Fig. 8. Power flows of the system when sub-harmonic resonances occur ðα ¼ 1:0; β ¼ 0:2; ξ ¼ 0:01; f ¼ 10Þ. For (a)–(c) ω ¼ 3:5; (d)–(f), ω ¼ 2:8; (a, d) Instantaneous input power; (b, e) frequency spectra of pin ; (c, f) frequency spectra of displacements.

Here again, a second-order HB method can be used to approximate TAIP of the system exhibiting sub-harmonic resonances by considering the primary and sub-harmonic frequency components in the response. 4. Power flow behaviour associated with bifurcations and chaos The previous section studied the power flows of the system with periodic responses containing discrete frequency components. However, chaotic motions may also be encountered, containing a broadband frequency signature. The corresponding power flow behaviour of the oscillator is of particular interest and investigated herein.

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4.1. Effects of bifurcations To reveal the dynamic characteristics of the system over a frequency range, figures can be plotted by realizing a Poincaré section, with a stroboscopy at the forcing frequency. For this, numerical integrations are conducted for each value of ω. The sampling displacement xs ðt s Þ at sampling time t s ¼ t i þ ðn 1ÞT; ðn ¼ 1; 2; …Þ is recorded, starting from t i and using the excitation period T ¼ 2π=ω as the sampling interval. In this way, for a period-1 response with a period of T, the samplings coincide at one point for a specific value of ω. Similarly, for a period-q response with its period being qT, there are q points shown for a single value of ω. If the motion is not periodic, i.e., quasi-periodic or chaotic, theoretically, there will be infinite points shown for a particular ω. The corresponding Lyapunov exponents [27] can be used to identify chaotic responses. Bifurcation occurs at frequencies where there is a sudden change in the number of shown sampling points. Similar sampling diagrams for instantaneous input power ps ðt s Þ at time t s can be plotted to investigate input power behaviour. For the system with α ¼  1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5 with t i ¼ 500 T, Figs. 9 and 10 provide bifurcation diagrams by sampling x and pin , as well as variations of TAIP based on averaging time t p ¼ 1000 T. To reveal different power flow branches, continuations are used by varying the excitation frequency from 0:1 to 10 (Fig. 9), and from 10 to 0.1 (Fig. 10), respectively. First-order HB approximations of TAIP are also shown in the figures for comparisons. Comparing these figures, it is seen that different stable motions co-exist when 1:58 oω o4:21. In this situation, the TAIP will be dependent on the initial conditions. Investigating the variations of ps and TAIP with references of the bifurcation diagrams xs , it is found that the same number of sampling points is shown for xs and ps . The following power flow characteristics are observed. As shown in Fig. 9, the bifurcation at ω ¼ 4:21 from Type-1 (solid line) to Type-2 (dashed line) oscillations leads to a substantial jump-down in the numerical value of pin . To examine the reason, Fig. 11(a) and (d) presents the variations of displacement and dissipated power at ω ¼ 4:21 and 4:22, respectively. It is shown that the former type of oscillation is of large amplitude around the point x ¼ 0, while the latter case with ω ¼ 4:22 yields only a small-amplitude vibration in a single potential well. Therefore, there is more power dissipation for the former and a substantial change in TAIP due to the bifurcation is observed. Similarly in Fig. 10, the system bifurcates from a period-2 oscillation to a chaotic response at ω ¼ 1:58, and then to a Type-1 oscillation at ω ¼ 1:56. Correspondingly, TAIP increases significantly as a result of these bifurcations. As shown in Fig. 11(b) and (e), the period-2 oscillation at ω ¼ 1:60 is in a single potential well, whereas the chaotic motion at ω ¼ 1:58 oscillates in both wells with larger amplitude, so that more power is dissipated in the latter case. Moreover, Fig. 11(c) shows that the amplitude of Type-1 oscillation at ω ¼ 1:56 is larger than that of the chaotic response at ω ¼ 1:58. Also, the Type-1 motion moves across two potential wells more frequently than the chaotic motion, and thus requires more energy dissipation, as shown in Fig. 11(f). In the low-frequency range, Figs. 9 and 10 show that the changes in time-averaged input power due to bifurcations are smaller than those at high frequencies. Fig. 12 provides an enlarged and more detailed view of the variations of TAIP in the low-frequency range for better clarity. This figure shows that TAIP of chaotic motions indicated by the dots only varies slightly with the excitation frequency. In contrast, the TAIP corresponding to period-1 motions marked by squares changes more

Fig. 9. Effects of bifurcation on TAIP when varying ω from low to high frequencies ðα ¼  1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5Þ. (a) Bifurcation diagram, (b) sampling points ps and (c) TAIP. First-order HB approximations: solid line (Type-1 oscillations); dashed line (Type-2 oscillations). Numerical integration results: red dots (chaotic response), black dots (periodic response). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 10. Effects of bifurcation on TAIP when varying ω from high to low frequencies ðα ¼  1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5Þ. (a) Bifurcation diagram, (b) sampling points ps and (c) TAIP. First-order HB approximations: solid line (Type-1 oscillations); dashed line (Type-2 oscillations). Numerical integration results: red dots (chaotic response), black dots (periodic response). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Bifurcations at high frequencies ðα ¼  1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5Þ. (a) and (d): solid line: ω ¼ 4:21 and dotted line: ω ¼ 4:22; (b) and (e): solid line ω ¼ 1:70 and dotted line ω ¼ 1:60; (c) and (f): solid line ω ¼ 1:58 and dotted line ω ¼ 1:56. (a)-(c): displacement response, (d)–(f): dissipated power.

abruptly at some frequencies due to bifurcations. For example, the response changes from a chaotic motion at ω ¼ 0:12 to a period-1 motion at ω ¼ 0:125, and then resumes being chaotic at ω ¼ 0:13. Correspondingly, TAIP for the period-1 motion is smaller than that for the chaotic motions. Fig. 13 shows the frequency components in the input power at these three excitation

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Fig. 12. Variations of TAIP at low excitation frequencies using a high-to-low frequency sweep ðα ¼  1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5Þ. Dots: chaotic motions; squares: period-1 motions; triangles: period-2 motions and crosses: period-3 motions.

Fig. 13. Frequency spectra at low excitation frequencies ðα ¼  1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5Þ. (a) ω ¼ 0:120 with the largest Lyapunov exponent: 0.039, (b) ω ¼ 0:125, period-1 motion and (c) ω ¼ 0:130 with the largest Lyapunov exponent: 0.04.

frequencies. Fig. 13(b) shows that the period-1 motion contains discrete frequency signatures, but both chaotic motions with the largest Laypunov exponent being positive contain broadband frequency spectra, as shown by Fig. 13(a) and (c). 4.2. Effects of chaotic responses Here we focus on the instantaneous and time-averaged input power of the system when it exhibits chaotic motions. Figs. 9 and 10 show that given an infinite sampling time span, there would be infinite sampling points for input power ps ðt s Þ when a chaotic motion occurs as the input power become non-periodic. The following behaviour of power flows due to the occurrences of chaotic motions is observed. Chaos effect 1: In the low-frequency range, the numerical value of TAIP is larger than the corresponding HB approximation of Type-1 motions. In comparison, in the high-frequency range, the HB approximations of Type-1 and Type-2 motions provide both upper and lower bounds for TAIP of a chaotic response, based on the same considerations. To explain this characteristic, Fig. 14 compares the power flows of the system with α ¼  1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5, excited at ω ¼ 0:13 and ω ¼ 1:58, respectively. For the former case, as shown in Fig. 14(a)–(c), the HB approximation underestimates power flow levels as it neglects the power dissipation caused by super-harmonic oscillations in the potential wells. This is in contrast to the latter case with a larger excitation frequency of ω ¼ 1:58. As shown in Fig. 14(d)–(f), the system moves around the unstable equilibrium at x ¼ 0 less frequently with a smaller amplitude than the HB approximation. Correspondingly, the dissipated and input powers of the chaotic response are much smaller than the HB predictions obtained by assuming Type-1 motions. When the system exhibits chaotic motions, the influences of averaging time t p on TAIP should be clarified. To examine this, we set t p as N cycles of the excitation, i.e.,t p ¼ NT and plot the variations of pin against t p in Fig. 15. For a chaotic response with the largest Lyapunov exponent of 0:07, the figure shows that the variations in pin are smaller than 4dB when 10 o N o100. When N 4 100, the value of TAIP remains in between 200dB and 202dB. Moreover, as N increases, the fluctuation becomes smaller and pin tends asymptotically to 200.6dB. This suggests a conclusion as follows. Chaos effect 2: For a chaotic response, there exists an asymptotic value of TAIP as the averaging time increases. The asymptotical behaviour of TAIP may be an important characteristic of chaotic motions of a system. As we have learnt, for periodic vibrations with a periodic of T, TAIP based on t p ¼ NT (N is a positive integer) is constant, independent of N. A chaotic motion could be considered as a periodic one with an infinite period, and also by definition chaotic responses are bounded in phase space [25], so that its TAIP tends to a constant with increasing averaging time. Chaos effect 3: The TAIP for chaotic motions is not sensitive to initial conditions if the averaging time is large enough. A chaotic response is characterised by a sensitivity of end time displacement to initial conditions [25]. It is thus of interest to examine the effects of initial conditions on the associated TAIP when chaos occurs. Fig. 16 provides the numerical results

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Fig. 14. Comparisons of power flows for chaotic motions with the corresponding HB approximations. (a)–(c): ω ¼ 0:13 and (d)–(f): ω ¼ 1:58. (a) and (d): displacement; (b) and (e): dissipated power; (c) and (f): input power. Solid line: chaotic motions; dashed line: first-order HB approximations.

Fig. 15. Effects of averaging time on time-averaged input power of a chaotic response. Parameters and initial conditions are set as α ¼  1, β ¼1, ξ ¼0.01, f ¼0.5, ω ¼0.4, (x0,y0) ¼ (0,0).

of TAIP of a system with prescribed parameters of α ¼ 1; β ¼ 1; ξ ¼ 0:01; f ¼ 0:5; ω ¼ 0:4, but with different initial conditions. The initial time t i and averaging time t p are set as 500T and 1000 T, respectively. With the initial displacement x0 and velocity y0 varying between  1 and 1, it shows that the variations in TAIP are smaller than 1dB. It should be noted that the chaotic motion considered here is the only stable solution in the examined region of initial conditions. Otherwise, the value of TAIP may change significantly if the system response evolves to other stable solutions.

5. Conclusions The Duffing oscillator was studied to explore its intrinsic power flow behaviour due to the stiffness nonlinearity. The effects of different nonlinear phenomena, such as sub-/super-harmonic resonances, bifurcation and chaos on power flows were investigated. The influences of different parameters, initial conditions and averaging time on the time-averaged input

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Fig. 16. Effects of initial conditions on time-averaged input power of chaotic motions. Parameters are fixed as α¼  1, β¼ 1, ξ ¼ 0.01, f¼ 0.5, and ω¼ 0.4.

power (TAIP) were examined. Both the instantaneous and time-averaged power flow characteristics of the system were analysed analytically and numerically. Based on the investigations, the following power flow characteristics are found. (1) When the system exhibits periodic motions, only the in-phase velocity component with the same frequency as the excitation contributes to the TAIP averaged over the response period. (2) Based on a first-order approximation, it was found that the stiffness nonlinearity has a significant influence on TAIP of the hardening/softening systems ðα 40Þ near the resonance frequency. When away from resonance frequencies, TAIP of these types of systems becomes not sensitive to variations of the nonlinear stiffness coefficient β. Also, the TAIP is proportional to the maximum kinetic energy K max of the system if the damping coefficient remains unchanged. It was shown that there exist upper bounds to time-averaged power flows, regardless of variations of the stiffness parameters α and β. The upper bound value of time-averaged input power increases with the excitation amplitude but decreases with an increasing damping coefficient. Moreover, the time-averaged power flow levels become independent of stiffness parameters α and β when the excitation frequency is large. (3) The occurrences of sub-/sub-harmonic resonances can lead to local peaks in time-averaged input power and in the maximum kinetic energy at some excitation frequencies. (4) When bifurcation occurs, the time-averaged input power into the system may change significantly. However, for bifurcations of periodic to chaotic motions encountered in the low-frequency range, the associated variations of TAIP of double-well potential systems were found to be small. (5) The chaotic motions encountered at low excitation frequencies were shown to contain large super-harmonic components, and as a result, the associated time-averaged input power is much larger than the corresponding firstorder HB approximations. In comparison, a chaotic motion occurred at high excitation frequency is with a lower timeaveraged input power than the HB approximation of the corresponding Type-1 motion. (6) When the system exhibits chaotic motion, the amount of time-averaged input power tends to an asymptotic value as averaging time increases. Also, this value is not sensitive to the initial conditions when there is a single chaotic attractor in the examined region of initial conditions.

This study reveals some inherently nonlinear power flow behaviour of a nonlinear dynamical system that is absent from the linear systems. With an enhanced understanding of such behaviour, nonlinearity can be introduced to improve the performance of dynamical systems for different design purposes. For instance, the double-well potential system with possible chaotic motions may be employed for efficient energy harvesting from low-frequency vibrations. Also, softening stiffness may be used for nonlinear vibration isolations so that the peak values of power flows can be greatly reduced. More importantly, the study successfully showed that major benefits of power flow analysis lie in its ability to quantify both periodic and chaotic responses using time-averaged power flow variables, in comparison to the conventional frequency/ displacement analysis. These variables can be used as uniform indices to assess the long-time behaviour of a nonlinear system exhibiting either periodic or chaotic motions. Also, it is shown that TAIP plots can reveal intrinsic nonlinear dynamic information of systems with nonlinear phenomena such as bifurcation and chaos. The power flow behaviour associated with other nonlinear phenomena such as internal resonances and modal interactions in multiple degrees-of-freedom nonlinear systems are of interest and needs further investigations.

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