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Dynamic responses and mitigation of limit cycle oscillations in Van der Pol–Duffing oscillator with nonlinear energy sink
Journal of Sound and Vibration 332 (2013) 5489–5507
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Dynamic responses and mitigation of limit cycle oscillations in Van der Pol–Duffing oscillator with nonlinear energy sink E. Domany, O.V. Gendelman n Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e i n f o
abstract
Article history: Received 24 September 2012 Received in revised form 28 April 2013 Accepted 1 May 2013 Handling Editor: L.N. Virgin Available online 15 June 2013
The paper considers dynamics of Van der Pol–Duffing (VdPD) oscillator with attached nonlinear energy sink. Due to a cubic nonlinearity of the VdPD oscillator, a frequency of oscillations near the unstable origin strongly differs from the frequency of limit cycle oscillations (LCO). The paper demonstrates that, despite the strong nonlinearity of the model system, one can efficiently describe the dynamics with a combination of averaging and multiple scales methods. Global structure of possible response regimes is revealed. It is also demonstrated that the nonlinear energy sink can efficiently control and mitigate the undesired LCOs in this system. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Systems exhibiting limit cycle oscillations (LCO) abound in science and nature [1–5]. For some structures, the LCOs can become extremely dangerous. An example for such dangerous effects can be seen in fighter aircrafts′ wings—at high subsonic and transonic speeds a flutter can appear and cause severe structural damage. One of means recently developed to reduce such dangerous self-excitation is an attachment of a strongly nonlinear oscillator with relatively small mass to a primary structure [6–11]. Such attachment under certain conditions can facilitate a targeted energy transfer (TET) from the main body to the small mass. Such essentially nonlinear attachment is generally referred to as a nonlinear energy sink (NES); it has been studied widely as a tool for suppressing undesired oscillations in externally excited and self-excited systems [6–11]. Van der Pol (VdP) oscillator [1] is a paradigmatic model for description of self-excited vibrations, although historically it was preceded by mathematically equivalent Rayleigh model [2]. The VdP model is widely used for approximate description of limit-cycle oscillations (LCOs) in many mechanical and physical systems [2–5]. The mitigation of the LCOs in the VdP system with attached NES was investigated in [12,13]. However, a common VdP oscillator due to its simplicity simulates faithfully only very limited class of the self-excited systems. In particular, if a nonlinear term in the VdP oscillator is small, then a frequency of oscillations is very close to a natural frequency of a linear component both near an unstable origin and at the limit cycle. Such situation is by no means generic. In real physical systems, where essential nonlinearities are involved, the frequency of the LCOs may be very different from eigenfrequencies near the unstable states of equilibrium. This is the case in simple 2dof model of aeroelastic instability in an aircraft wing; mitigation of the LCOs in this system by means of the NES was described in papers [11,12,14]. Analytic approach in these works was based on the averaging of the dynamical flow over a fast frequency of the primary system unstable mode. Therefore, this analysis was able to address the changes in stability of the system caused by attachment of the NES and to predict some of its low-amplitude response regimes. n
Corresponding author. Tel.: +97 248 293 877; fax: +97 248 295 711. E-mail addresses: [email protected] (E. Domany), [email protected] (O.V. Gendelman).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.05.001
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Transitions to fully developed LCO remained unattended due to large variations of the basic frequency, both in stationary LCOs and in quasiperiodic responses. These large frequency variations preclude use of common averaging over some constant basic frequency. Consequently, the methods used previously for analysis of dynamic responses in this sort of systems, at least require essential modification. The latter challenge is a main motivation for this paper. In order to pursue the analytic approach, we adopt relatively simple model of the single-dof primary system with the same peculiarity—large variations of the basic frequency. The hope is that the general behavior of such systems will not depend on “fine details” of the model. To obtain such simple model with large frequency variations, it is enough to add an essential cubic nonlinearity to the linear term in the VdP model; in a modified system, the frequency of the LCOs will be significantly higher than that of the linear component. This modified system is referred to as Van der Pol–Duffing (VdPD) oscillator [15]. More specifically, in this research we describe different response regimes possible in the system comprising the VdPD oscillator coupled with common purely cubic NES. The secondary goal is to verify whether the addition of the NES enables one to suppress or to mitigate an undesired limit-cycle oscillations. As it was already mentioned, both problems require development of new analytic tools, since for some response regimes peculiar for this sort of system (for instance, for so-called strong modulated responses (SMRs) [16–19]) the instantaneous frequency is not constant. We are going to suggest an alternative analytic approach that allows explicit account of relatively slow frequency variations. 2. Description of the model and expression for an instantaneous frequency Let us consider the VdPD oscillator with an attached ungrounded purely cubic NES. The coupling is realized through linear damping and cubic stiffness. Equations of motion for this system can be written as follows: 2 d d d d x1 − x2 þ Kðx1 −x2 Þ3 ¼ 0 m1 2 x1 þ c x1 ðx21 −A2 Þ þ ql x1 þ qnl x31 þ γ dt dt dt dt 2 d d d x2 − x1 þ Kðx2 −x1 Þ3 ¼ 0 (1) m2 2 x 2 þ γ dt dt dt Here mi,xi(i¼ 1,2) are masses and coordinates of the primary oscillator and the NES respectively, coefficients c and A define a common VdP term, ql and qnl are linear and nonlinear stiffness coefficients, γ is the linear damping coefficient and K is the cubic stiffness coefficient. Relative orders of magnitude of the system parameters will be specified in an asymptotic analysis presented below. After rescaling System (1), one obtains the following non-dimensional system: 4 u€ 1 þ εαu_ 1 ðu21 −1Þ þ u1 þ 43 βu31 þ ελðu_ 1 −u_ 2 Þ þ εkðu1 −u2 Þ3 ¼ 0 3 4 u€ 2 þ λðu_ 2 −u_ 1 Þ þ kðu2 −u1 Þ3 ¼ 0 3 Non-dimensional parameters in Eq. (2) are determined as pffiffiffiffiffiffiffiffiffiffiffiffiffi τ ¼ ωn t; ωn ¼ ql =m1 ; ui ¼ xi =A; α ¼ cA2 =m2 ωn ;
λ ¼ γ=m2 ωn ;
i ¼ 1; 2;
k ¼ 3KA2 =4m2 ω2n ;
(2)
ε ¼ m2 =m1 ; β ¼ 3qnl A2 =4ql
A dot denotes a differentiation with respect to a new time variable τ. Further simplification can be achieved by introducing a new variable: r ¼ u1 −u2
(3)
Substituting (3) into (2), one obtains: 4 u€ 1 þ εαu_ 1 ðu21 −1Þ þ u1 þ 43 βu31 þ ελr_ þ εkr 3 ¼ 0 3 4 2 3 4 r€ þ εαu_ 1 ðu1 −1Þ þ u1 þ 3 βu1 þ ð1 þ εÞλr_ þ ð1 þ εÞkr 3 ¼ 0 3
(4)
In this non-dimensional system, all parameters are considered to be of order unity, besides the parameter ε. The latter is equal to a ratio between the masses of the attachment and of the primary system. From a viewpoint of possible applications, this ratio should be kept relatively small. In theoretical and experimental works devoted to the systems with the NES, similar parameters are adopted to stay in a range 0.01–0.1 and this convention will be followed in current work. As it will be demonstrated below, relative smallness of ε is crucial, if one develops the analytic approach to the problem. We also adopt that the same small parameter determines the scales of the VdP term and of the linear damping and the nonlinear stiffness of the NES. As it was mentioned above, the instantaneous frequency in the primary system depends on its amplitude. In order to perform averaging over varying “fast” frequency of the primary VdPD oscillator, we first formally introduce this frequency as ω¼ω(τ). Then, an additional change of variables is performed in order to apply a complexification-averaging approach [20,21]: iωðτÞu1 þ u_ 1 ¼ R exp ðiδ1 ðτÞÞ
(5)
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Here R is the amplitude of the primary oscillator, δ1 is its phase. The instantaneous frequency and the “fast” phase are related in an obvious way: δ_ 1 ¼ ωðτÞ
(6)
For the relative displacement of the NES, the following ansatz is used: iωðτÞr þ r_ ¼ ξ exp ðiδ1 Þ
(7)
Here ξ(τ) is a complex variable. We consider the simplest 1:1 resonance between the primary VdPD oscillator and the NES. As it will be clear later, this approximation is sufficient for analytic description of essential global features of the dynamical responses. More detailed studies of the frequency content in this kind of essentially nonlinear systems resort on numeric methods [11,12] and demonstrate that high-order resonances may be very significant for adequate understanding of details of transient dynamics in this class of systems. These details are beyond the scope of current paper, since we are more interested in the qualitative description of possible response regimes. If only 1:1 resonance is considered, the variables R(τ) and ξ(τ) may be treated as “slow” variables of the system. After substituting (5)–(7) into (4), we average out the “fast” phase δ1, thus obtaining the following “slow-flow” equations: " # " # 2 2 _ _R ¼ − εαR R −1 − ελ ξ þ iεk jξj2 ξ þ iR 1−ω2 þ βR þ ωR 2ω 2 4ω2 2 2ω 2ω3 ω2 " # 2 3 _ _ξ ¼ i ðR−ω2 ξÞ− ð1 þ εÞλ ξ þ ð1 þ εÞ ik jξj2 ξ− εαR R −1 þ iβR þ ω ξ (8) 2ω 2 2 4ω2 2ω3 2ω3 2ω Extracting imaginary terms of order O(1) from the first equation of (8), we obtain: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7 1 þ 4βR2 β 1−ω2 þ 2 R2 ¼ 0⇒ω2 ¼ 2 ω
(9)
In the limit β-0 the primary system turns into a common VdP oscillator and the instantaneous frequency should tend to unity in this limit [13]. Consequently, in Eq. (9) one has to choose a positive sign and the final expression for the instantaneous frequency will be as follows (up to order O(1)): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t1 þ 1 þ 4βRðτÞ2 (10) ωðτÞ ¼ 2 By differentiating Eq. (9) and after simple algebraic manipulations, one arrives to the following expression for the derivative of the instantaneous frequency: _¼ ω
βRR_ ωð2ω2 −1Þ
(11)
Substituting this expression and Eq. (9) into the first equation of System (8), one can reshape the latter equation in the following way: ðεαR=2Þ½ðR2 =4ω2 Þ−1−ðελ=2Þξ þ ðiεk=2ω3 Þjξj2 ξ R_ ¼ − þ Oðε2 Þ 1−ðβR2 =ð2ω2 ð2ω2 −1ÞÞÞ " # ! εð2ω2 −1Þ R2 iεk 2 αR −1 −λξ þ jξj ξ þ Oðε2 Þ ¼− 3ω2 −1 2ω3 4ω2
(12)
_ From Eq. (12) it is clear that ROðεÞ for all diapason of ω≥1. Although ω(R) can be now computed from Eq. (10), below in some instances the variable ω will be preserved for the sake of brevity. The next step of treatment is an asymptotic analysis based on the small parameter ε (the ratio of masses of the attachment and of the primary system). At the first stage, only terms having order O(1) will be considered. The slow variable ξ(τ) is presented in a polar form: ξ ¼ P exp ðiΔÞ
(13)
Then, substituting expressions (10), (11) and (13) into the second equation of System (8) and keeping only the leadingorder terms in an explicit form, one obtains: λ ω P_ ¼ − P þ R sin Δ þ OðεÞ 2 2 _Δ ¼ ω 1− R cos Δ − k P 2 þ OðεÞ 2 P 2ω3
(14)
System (12)–(14) is already simple enough to yield to an asymptotic analysis which will be described in the next section.
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3. Asymptotic analysis For the sake of the asymptotic analysis, the following formal assumptions are made: ε⪢1;
α; k; β; λOð1Þ
(15)
Physical meaning of these assumptions has been discussed above in Section 2. Phase flow defined by System (12)–(14) belongs to a state space ℝ2 ⊗S1 , but the derivatives of the variables have different orders of magnitude with respect to the small parameter ε. Thus, further asymptotic decomposition of the system is possible with the help of a method of multiple scales [2,11,21]. Variables P(τ) and Δ(τ) evolve at the time scale determined by averaging of the complete phase flow (4); this time scale is further referred to as “slow” one. Variable R(τ) evolves at even slower time scale determined by the small parameter ε; this time scale is further referred to as “super-slow”. A term “fast” denotes the time scale determined by fast oscillations of the primary system with frequency ω(τ). This time scale is “averaged out” when Eq. (8) are computed. So, three different time scales are used for analysis of this problem. The multiple-scale analysis is performed with the help of a common decomposition [2,11,21]: τj ¼ εj τ⇒
∂ ∂ ∂ ∂ ¼ εj ¼ þε þ⋯ ∂τ ∂τj ∂τ0 ∂τ1
(16)
where j¼ 0 corresponds to the “slow” time scale and j ¼1–to the “super-slow” time scale. For the slow time scale, system (12)–(14) is reduced to the following form: ∂R ¼0 ∂τ0 ∂P λ ω ¼ − P þ R sin Δ ∂τ0 2 2 ∂Δ ω R k 1− cos Δ − 3 P 2 ¼ ∂τ0 2 P 2ω
(17)
It is clear from (17) that R¼R(τ1), and two other equations form a planar system. It is possible to prove, with the help of the Bendixson criterion [15] that LCOs do not exist in the averaged slow system (17), and that all trajectories will be attracted to stable fixed points. These fixed points are denoted as follows: P 0n ðτ1 Þ ¼ lim Pðτ0 Þ; τ0 -∞
Δ0n ðτ1 Þ ¼ lim Δðτ0 Þ; τ0 -∞
R0n ðτ1 Þ
¼ lim Rðτ0 Þ
(18)
τ0 -∞
Isolating trigonometric terms from (17), we obtain: λ2 P 20n ω60n þ P 20n ðkP 20n −ω40n Þ2 −R20n ω80n ¼ 0;
ω0n ¼ ωðR0n Þ
(19)
For simplicity we define: Y ¼ R20n ;
Z ¼ P 20n
(20)
Then, the following equation is obtained: λ2 ω60n Z þ ZðkZ−ω40n Þ2 −Yω80n ¼ 0
(21)
Eq. (21) defines the slow invariant manifold (SIM) of System (14), which, in general, consists of two stable and one unstable branches, as can be seen in Fig. 1. The instability of the middle branch is easy to prove by common methods from System (17). The stable branches of the SIM are dynamical attractors for the slow flow, and since ∂R/∂τ0 ¼0 it is clear that for every initial condition (IC) on the Y–Z plane the flow will be attracted to one of these branches. Thus, the phase point on the plane Y–Z will move “horizontally” towards one of the stable branches—namely, the one which can be approached without crossing the unstable branch. The super-slow time scale will govern further evolution of the phase flow. If the trajectory will reach either of “fold points” Z1,2, it will “jump” horizontally towards “landing points” Zu,d respectively. Such structure of the SIM provides a possibility for relaxation oscillations [22–24]. This relaxation motion is very similar to that exhibited in the VDP oscillator coupled to the NES [13,16]. Further analysis of the response regimes requires consideration of the super-slow time scale in System (12)–(14). It is possible to circumvent explicit exploration of this time scale by observing that the super-slow flow in System (12)–(14) in main approximation should be taken into account only for variable R(τ); consequently, the equation determining the evolution of the system on this time scale will be of the first order. Therefore, the dynamics on one-dimensional SIM (19) is completely determined by positions of the global fixed points of flow (8). To determine these fixed points, the instantaneous energy of the system is defined as sum of its kinetic and potential energies, while the damping and VdP terms are disregarded. The fixed points under consideration correspond to stationary limit cycles of initial system and satisfy obvious necessary condition: over single cycle, input and output of energy in the system should be exactly balanced. With account of the damping and VdP terms, the instantaneous energy is not constant,
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and a rate of the energy variation in initial System (4) may be expressed as follows: " # 2 βu41 dE d u_ 1 þ u21 εu_ 22 εk 4 ¼ þ þ ðu1 −u2 Þ þ dτ dτ 2 2 2 3 ¼ −ε½αu_ 21 ðu21 −1Þ þ λðu_ 1 −u_ 2 Þ2
(22)
The right-hand side of (22) describes the non-conservative part of the energy flow. Fixed points of the averaged flow (8) will obey the condition 〈dE=dτ〉δ1 ¼ 0. Performing the averaging according to (5)–(11), one obtains: ! R20n −1 (23) −λP 20n ¼ αR20n 4ω20n With account of (20), we finally obtain an additional condition for the fixed points at the SIM: " # Y α 1− 2 Y−λZ ¼ 0 4ω0n
(24)
Eqs. (10), (21) and (24) are sufficient to uncover the structure of the response regimes in the system under consideration. This can be done with the help of geometric approach presented in the next section.
Fig. 1. Slow invariant manifold (SIM) of the system. The unstable branch is dashed. Z1,2 correspond to the fold points, Zu,d correspond to the “landing” points of the relaxation oscillations. Arrows depict the trajectories which the system follows while switching from one stable branch to the other.
Fig. 2. Y–Z plot for the case of complete elimination of the LCO. The SIM curve (21) (thick line) and the “energy conservation” parabola (24) (thin line) intersect only in the trivial fixed point.
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4. Numerical verifications and specific bifurcation scenarios Further analysis of particular response regimes relies on geometry of intersections of the SIM (21) with “energy conservation” parabola (24) on Z–Y plane. If not stated otherwise, values of β¼1 and ε¼ 0.05 were used in the simulations.
3
2
u1
1
0
−1
−2
−3
0
50
100 τ
150
200
0
50
100 τ
150
200
5 4 3 2
u1−u2
1 0 −1 −2 −3 −4 −5
Fig. 3. Numeric time series for the case of complete LCO suppression for set of parameters (25) and initial conditions u1(0)¼ 2, u_ 1 ð0Þ ¼ 0, u2(0) ¼0, u_ 2 ð0Þ ¼ 0. (a) Primary mass displacement u1 and (b) relative displacement of the NES u1−u2.
Fig. 4. The case of partial LCO elimination. The intersection occurs in both the trivial fixed point and in the nontrivial fixed point.
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4.1. Complete suppression of LCO For certain range of parameters, curves (21) and (24) will intersect only in the origin (Fig. 2). We consider such a case for the parameter set: α ¼ 0:05;
k ¼ 0:36;
λ ¼ 0:15
(25)
In this case, the origin is the only stable attractor of the system and, consequently, the LCOs will be eliminated. This conclusion is confirmed numerically by simulation presented in Fig. 3.
1 0.8 0.6 0.4
u1
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
50
100
0
50
100
τ
150
200
250
150
200
250
1 0.8 0.6 0.4
u1−u2
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
τ
Fig. 5. Numeric time series for the case of partial LCO suppression for set of parameters (28) and ICs u1(0)¼ 0.6, u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0. (a) Primary mass displacement u1 and (b) relative displacement of the NES u1−u2.
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This special case is very interesting, and one can reveal for what range of parameters it can occur. It is easy to see that the complete elimination will occur when near the origin the slope of the “energy conservation” parabola is larger than the slope of the SIM curve. This condition reads as λ 4 λ2 þ 1 α
(26)
This relationship does not contain any parameters related to the system nonlinearities. It is quite natural, since the regime of complete LCO elimination is related to a linear stability of the origin. From (26) it follows that the complete elimination occurs in the following range of the system parameters: 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1−4α ; þ 1−4α (27) λelimination ∈ − 2 2α 2 2α One can see that the latter will only exist for α≤0.5, i.e. for relatively small “intensity” of the VdPD oscillator. 4.2. Partial suppression of LCO Let us increase the “intensity parameter” α slightly above the maximum value for complete suppression and choose the parameter set as follows: α ¼ 0:6;
k ¼ 0:4;
λ ¼ 0:3
(28)
Mutual positions of curves (21) and (24) in this case are presented in Fig. 4. In this case, one can see that two fixed points appear. Since the system does not grow without bound, the fixed point with the higher excitation level (i.e. the nontrivial fixed point) must be stable, and the origin will be unstable. Thus, all dynamical trajectories will be attracted to this fixed point. It means that the amplitude of the LCO will be nonzero but small compared to full-scale LCOs. Fig. 5 presents the numerical time series for this case. A regime of partial suppression is obtained as predicted. Values of both the amplitudes of the primary mass displacement, and the relative displacement between the primary mass and the NES are rather close to the analytic predictions. 4.3. Canard explosion and the appearance of SMR If the “intensity parameter” α will grow further, the upper fixed point will cross the fold and will appear on the unstable branch of the SIM. For instance, such situation is realized for the following set of parameters (Fig. 6): α ¼ 1;
k ¼ 0:4;
λ ¼ 0:3
(29)
One can see that the nontrivial fixed point belongs to the unstable SIM branch. Consequently, the relaxation oscillations demonstrated in Fig. 6 turn out to be the only stable attractor of the system. This dynamical response is referred to as Strongly Modulated Response (SMR) [17,18]. Time series for this response regime are presented in Fig. 7a,b. It is interesting to compare this case with the previous one. As one can notice the transition from the previous case was caused by increasing the intensity of the self-excitation α. In our asymptotic consideration the transition from the stable fixed point to the full-scale relaxation oscillations is abrupt. However, the initial system is smooth and such abrupt transition is not possible. Instead, normal Hopf bifurcation will take place and the transition to the relaxation oscillations will occur at exponentially small interval of the bifurcation parameter. This phenomenon is well-known in systems with the relaxation oscillations and is commonly referred to as a “canard explosion” [25]. This transition is illustrated in Fig. 8a–c.
Fig. 6. The case of strong modulated response (SMR). The intersection occurs on the unstable branch.
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1 0.8 0.6 0.4
u1
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
50
100
0
50
100
τ
150
200
250
150
200
250
2 1.5 1
u1−u2
0.5 0 −0.5 −1 −1.5 −2
τ
Fig. 7. Numeric time series for the case of the SMR. Parameters are chosen according to (29) and the ICs are u1(0)¼ 0.6, u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0. (a) Primary mass displacement u1 and (b) relative displacement of the NES u1−u2.
The nontrivial fixed point coalesces with the fold of the SIM curve at α≈0.8. From the numeric time series, one can observe the “explosion” at α≈0.86. It is instructive to observe that the frequency of oscillations grows in Fig. 7, as the amplitude increases. It occurs due to the Duffing term, which makes the frequency essentially amplitude-dependent. Time series obtained for the SMR allow us to verify to some extent expression (10) for the relationship between the instantaneous frequency and the amplitude of the primary mass oscillations. The frequency content of the response is computed with the help of a common wavelet transform, and the predicted value of the instantaneous frequency—from the
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1.5
1
u1−u2
0.5
0
−0.5
−1
−1.5
0
100
200
0
100
200
0
100
200
τ
300
400
500
300
400
500
300
400
500
1.5
1
u1−u2
0.5
0
−0.5
−1
−1.5
τ
2 1.5 1
u1−u2
0.5 0
−0.5 −1 −1.5 −2
τ
Fig. 8. Numerical illustration (time series of the NES relative displacement) for the canard explosion. Initial conditions are u1(0)¼ 0.6, u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0 and the bifurcation parameter is (a) α¼ 0.79, (b) α ¼ 0.83, and (c) α¼ 0.86.
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time series. Comparison between the theoretical predictions and the numeric findings appears in Fig. 9. One can see that the coincidence is rather satisfactory, and thus the analytic approach developed above is justified a posteriori.
4.4. Shi’lnikov homoclinic bifurcation of the SMR and the co-existence of a LCO and a SMR Further increasing the self-excitation intensity, one obtains new topological structure of the fixed points. This structure is illustrated in Fig. 10 for the following set of parameters: α ¼ 1:8;
k ¼ 0:8;
λ ¼ 0:3
(30)
Fig. 9. Wavelet analysis of a single SMR cycle for the case depicted in Fig. 7: (a) time series of the primary mass response and (b) wavelet transform of the signal. Thick line corresponds to theoretical prediction of the frequency value calculated from (9).
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As one can see in Fig. 10, there are four intersections between SIM curve (21) and the “energy conservation” parabola (23). One fixed point (the highest one) should be stable, as the system has no divergent responses. In this case, the system exhibits two stable attractors for different sets of initial conditions—the stable LCO for the initial points in the zone of “High IC”, and the SMR for the initial points in the zone of “Low IC”. Numeric verifications of this conclusion appear in Fig. 11. The SMR disappears with increase of the parameter α by global bifurcation scenario known as Shi’lnikov homoclinic bifurcation [26]. This bifurcation occurs when the upper saddle point coincides with the “landing point” Zu, and thus the SMR turns into the homoclinic trajectory for the upper saddle point. As one can see in Fig. 12, this bifurcation occurs for α¼ 2.335. For higher values of α, the SMR will not exist and all phase trajectories will be attracted to the stable LCO.
(a) 10
9 8 7
Y
6 5 4 3 2 1 0
0
5
10
Z
15
20
25
(b)
1
Y
0.8
0.6
0.4
0.2
0
1
2
3 Z
4
5
6
Fig. 10. Co-existence of the stable SMR and stable LCO. A boundary (horizontal dashed line) passing through the unstable fixed point separates the domains of attraction in the space of ICs. The high ICs lead to the LCO and the low ICs lead to the SMR. (a) Complete figure and (b) close-up of the SMR area.
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(a)
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0.8 0.6 0.4 0.2
u1
0 −0.2 −0.4 −0.6 −0.8
(b)
0
50
100
τ
150
200
3
2
u1
1
0
−1
−2
−3
0
50
100 τ
150
200
Fig. 11. Numeric time series of the primary mass displacements u1 for the case of co-existence of the LCO and the SMR. Parameters are chosen according to (34) and the ICs are u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0. (a) Low IC u1(0)¼ 0.6 and (b) high IC u1(0) ¼2.
To illustrate this scenario numerically one should note that as the SMR turns into the homoclinic trajectory, its period should grow. Theoretically, the period of the homoclinic trajectory is infinite, but due to approximate character of the treatment one cannot expect to hit the Shilnikov bifurcation point in the original four-dimensional state space. However, significant growth of the SMR period with extremely small variations of α is observed numerically, as illustrated in Fig. 13a–c. Such behavior is clear “signature” of dynamics close to the Shilnikov bifurcation.
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Fig. 12. Homoclinic connection leading to Shil'nikov bifurcation.
4.5. Heteroclinic bifurcation of the SMR Another global bifurcation scenario is described in Fig. 14a,b. As one of the system parameters varies, the saddle-node bifurcation occurs at the stable branch of the SIM between points Z2 and Zu. As this happens, the SMR turns into a heteroclinic connection between the node and the saddle. This particular bifurcation scenario is illustrated for the following set of parameters: α ¼ 0:3;
k ¼ 0:22;
λ∈ð0:01; 0:03Þ
(31)
In this example, the damping coefficient is varied and all other parameters are constant. The bifurcation should theoretically occur where λ ¼0.0162 as seen in Fig. 14, and simulations presented in Fig. 15 show that the critical value is λ≈0.025. The heteroclinic bifurcation is also detected through essential growth of the SMR period with tiny variations of the damping coefficient. 4.6. Co-existence of two LCOs The last case we wish to discuss here is the co-existence of two different limit cycles. The parameters examined here are: α ¼ 5;
k ¼ 0:185;
λ ¼ 0:2
(32)
This scenario is observed as the self-excitation coefficient is relatively large and the damping and the nonlinear stiffness are relatively low. The SIM curve (21) and the “energy conservation” parabola (24) for this case appear in Fig. 16. It is clear from Fig. 16 that we obtain two different stable fixed points, each representing its own limit cycle in the state space of the initial system. The unstable SIM branch and dashed black line represent the boundary between domains of attraction of two limit cycles in the space of initial conditions of the averaged system. Numeric verification of the existence of two different LCOs is presented in Fig. 17. 5. Basins of attraction—the case of two LCOs In this section, we would like to verify numerically the predictions for basins of attraction for the case of two LCOs presented in Fig. 16. Specifically, we would like to check whether the phase trajectory of the original system with initial conditions corresponding to the ICs of the averaged flow (8) in each zone in Fig. (16) would be attracted to the LCO predicted by the analytic procedure. The numerical results appear in Fig. 18. One can that in general the analysis provides some clue on the shape of the basins of attraction; however, the boundaries are far from exact. This result reveals the limitations of proposed analytic approach—the qualitative predictions regarding the structure of the response regimes ant their bifurcations are predicted correctly. However, the accuracy of the quantitative predictions is limited. 6. Concluding remarks In the paper, it was demonstrated that one can mitigate the LCOs in the self-excited system with strongly varying frequency by means of the NES. This result is in some sense expectable, since the NES differs from common means of the vibration suppression (like a tuned mass damper) by absence of a resonant frequency [11]. Therefore, it is reasonable to
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2 1.5 1
u1−u2
0.5 0 −0.5 −1 −1.5 −2 5000
5500
6000
6500
7000
6500
7000
6500
7000
τ 2 1.5 1
u1−u2
0.5 0 −0.5 −1 −1.5 −2 5000
5500
6000
τ 3
2
u1−u2
1
0
−1
−2
−3 5000
5500
6000 τ
Fig. 13. Numeric illustration of Shilnikov bifurcation of the SMR for parameters k¼ 0.8, λ¼ 0.3, ε ¼0.005 and (a) α ¼2.0155, (b) α ¼2.0155146085, (c) α ¼ 2.01552.
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Fig. 14. Heteroclinic bifurcation scenario for α¼ 0.3, k ¼0.22 and (a) λ¼ 0.03, (b) λ ¼ 0.0162.
expect that the NES might be efficient for the mitigation of broadband excitations. Results presented above confirm this conjecture for important particular case of the self-excited systems with essential nonlinearity. Moreover, it was demonstrated that one could suggest the approximate analytic procedure allowing description of a multitude of response regimes in this system. Global qualitative picture of possible response regimes obtained by this analysis seems complete—at least, the numeric simulations have not revealed any response regimes different from the predicted ones. Of course, this statement will remain true only if the system parameters will remain in the range suitable for the simplifying assumptions made. The quantitative coincidence is very good for some of these regimes and less profound for the others. It seems that better quantitative coincidence may be achieved if higher-order approximations would be taken into account. This problem will be a subject of future studies.
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2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2
0
500
1000
1500
2000 τ
2500
3000
3500
4000
0
500
1000
1500
2000 τ
2500
3000
3500
4000
2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2
2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2
0
1000
2000
3000 τ
4000
5000
6000
Fig. 15. Numerical illustrations of the heteroclinic bifurcation for α ¼0.3, k ¼ 0.22 and (a) λ ¼0.03, (b) λ¼ 0.025, (c) λ ¼0.02.
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Fig. 16. The case of co-existence of two LCOs. The ICs in zone I will lead to the higher amplitude LCO and the ICs in zone II will lead to the lower amplitude cycle (computed for parameter set (32)).
2.5 2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2 −2.5
0
20
40
60
80
100
60
80
100
τ 2.5 2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2 −2.5
0
20
40
τ Fig. 17. The case of co-existence of two LCOs for parameters (32) and ICs u_ 1 ð0Þ ¼ 0, u2(0) ¼0, u_ 2 ð0Þ ¼ 0 and (a) u1(0) ¼0.6, (b) u1(0)¼ 2.
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5507
25
20
Y
15
10
5
0
0
20
40
Z
60
80
100
Fig. 18. Basins of attraction map, corresponding to the case of co-existence of two LCOs. Blue asterisks (*) represents ICs leading to the LCO in zone I, and green crosses (x) represent ICs leading to the LCO in zone II. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Reviews 1 (1920) 701–710. 754–762. A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley Interscience, New York, 1995. E.H. Dowell, Non-linear oscillator models in bluff body aeroelasticity, Journal of Sound and Vibration 75 (1981) 251–264. B. van der Pol, J. van der Mark, Frequency demultiplication, Nature 120 (1927) 363–364. O.M. Griffin, R.A. Skop, The vortex-excited resonant vibrations of circular cylinders, Journal of Sound and Vibration 27 (1973) 235–249. O.V. Gendelman, Transition of energy to nonlinear localized mode in highly asymmetric system of nonlinear oscillators, Nonlinear Dynamics 25 (2001) 237–253. O.V. Gendelman, A.F. Vakakis, L.I. Manevitch, R. McCloskey, Energy pumping in nonlinear mechanical oscillators I: dynamics of the underlying Hamiltonian system, Transactions of ASME, Journal of Applied Mechanics 68 (2001) 34–41. A.F. Vakakis, O.V. Gendelman, Energy pumping in nonlinear mechanical oscillators II: resonance capture, Transactions of ASME, Journal of Applied Mechanics 68 (2001) 42–48. O.V. Gendelman, Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment, Nonlinear Dynamics 37 (2004) 115–128. L.I. Manevitch, E. Gourdon, C.H. Lamarque, Towards the design of an optimal energetic sink in a strongly inhomogeneous two-degree-of-freedom system, Transactions of ASME, Journal of Applied Mechanics 74 (2007) 1078–1086. A.F. Vakakis, O.V. Gendelman, G. Kerschen, L.A. Bergman, M.D. McFarland, Y.S. Lee, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Springer, 2009. Y.S. Lee, A.F Vakakis, L.A. Bergman, D.M. McFarland, Suppression of limit cycle oscillations in the Van der Pol oscillator by means of passive nonlinear energy sinks (NESs), Structural Control and Health Monitoring 13 (2006) 41–75. O.V. Gendelman, T. Bar, Bifurcations of self-excitation regimes in a Van der Pol oscillator with a nonlinear energy sink, Physica D 239 (2010) 220–229. O.V. Gendelman, A.F. Vakakis, L.A. Bergman, D.M. McFarland, Asymptotic analysis of passive nonlinear suppression of aeroelastic instabilities of a rigid wing in subsonic flow, SIAM Journal of Applied Mathematics 70 (2010) 1655–1677. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, Berlin, New York, 1983. O.V Gendelman, Targeted energy transfer in systems with external and self-excitation, Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 225 (2011) 2007–2043. O.V. Gendelman, Yu. Starosvetsky, Quasiperiodic response regimes of linear oscillator coupled to nonlinear energy sink under periodic forcing, Transactions of ASME, Journal of Applied Mechanics 74 (2007) 325–331. O.V. Gendelman, Y. Starosvetsky, M. Feldman, Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: description of response regimes, Nonlinear Dynamics 51 (2008) 31–46. Y. Starosvetsky, O.V. Gendelman, Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry, Physica D 237 (2008) 1719–1733. L.I. Manevitch, The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables, Nonlinear Dynamics 25 (2001) 95–109. L.I. Manevitch, O.V Gendelman., Tractable Models in Solid Mechanics, Springer, Berlin, 2011. V.I. Arnold, V.S. Afrajmovich, Yu.S. Il'yashenko, L.P. Shil'nikov, Dynamical Systems V. Encyclopedia of Mathematical Sciences, Springer Verlag, Berlin, New York, 1994. P. Szmolyan, M. Wechselberger, Relaxation oscillations in R3, Journal of Differential Equations 200 (2004) 69–104. J. Guckenheimer, K. Hoffman, W. Weckesser, Bifurcations of relaxation oscillations near folded saddles, International Journal of Bifurcations and Chaos 15 (2005) 3411–3421. E. Benoit, J.F. Callot, F. Diener, M. Diener, Chasse au canard, Collection Mathematique 31–32 (1981) 37–119. L.P. Shil'nikov, On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Matematicheskij Sbornik, USSR 6 (1968) 427–438.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Dynamic responses and mitigation of limit cycle oscillations in Van der Pol–Duffing oscillator with nonlinear energy sink E. Domany, O.V. Gendelman n Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e i n f o
abstract
Article history: Received 24 September 2012 Received in revised form 28 April 2013 Accepted 1 May 2013 Handling Editor: L.N. Virgin Available online 15 June 2013
The paper considers dynamics of Van der Pol–Duffing (VdPD) oscillator with attached nonlinear energy sink. Due to a cubic nonlinearity of the VdPD oscillator, a frequency of oscillations near the unstable origin strongly differs from the frequency of limit cycle oscillations (LCO). The paper demonstrates that, despite the strong nonlinearity of the model system, one can efficiently describe the dynamics with a combination of averaging and multiple scales methods. Global structure of possible response regimes is revealed. It is also demonstrated that the nonlinear energy sink can efficiently control and mitigate the undesired LCOs in this system. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Systems exhibiting limit cycle oscillations (LCO) abound in science and nature [1–5]. For some structures, the LCOs can become extremely dangerous. An example for such dangerous effects can be seen in fighter aircrafts′ wings—at high subsonic and transonic speeds a flutter can appear and cause severe structural damage. One of means recently developed to reduce such dangerous self-excitation is an attachment of a strongly nonlinear oscillator with relatively small mass to a primary structure [6–11]. Such attachment under certain conditions can facilitate a targeted energy transfer (TET) from the main body to the small mass. Such essentially nonlinear attachment is generally referred to as a nonlinear energy sink (NES); it has been studied widely as a tool for suppressing undesired oscillations in externally excited and self-excited systems [6–11]. Van der Pol (VdP) oscillator [1] is a paradigmatic model for description of self-excited vibrations, although historically it was preceded by mathematically equivalent Rayleigh model [2]. The VdP model is widely used for approximate description of limit-cycle oscillations (LCOs) in many mechanical and physical systems [2–5]. The mitigation of the LCOs in the VdP system with attached NES was investigated in [12,13]. However, a common VdP oscillator due to its simplicity simulates faithfully only very limited class of the self-excited systems. In particular, if a nonlinear term in the VdP oscillator is small, then a frequency of oscillations is very close to a natural frequency of a linear component both near an unstable origin and at the limit cycle. Such situation is by no means generic. In real physical systems, where essential nonlinearities are involved, the frequency of the LCOs may be very different from eigenfrequencies near the unstable states of equilibrium. This is the case in simple 2dof model of aeroelastic instability in an aircraft wing; mitigation of the LCOs in this system by means of the NES was described in papers [11,12,14]. Analytic approach in these works was based on the averaging of the dynamical flow over a fast frequency of the primary system unstable mode. Therefore, this analysis was able to address the changes in stability of the system caused by attachment of the NES and to predict some of its low-amplitude response regimes. n
Corresponding author. Tel.: +97 248 293 877; fax: +97 248 295 711. E-mail addresses: [email protected] (E. Domany), [email protected] (O.V. Gendelman).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.05.001
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Transitions to fully developed LCO remained unattended due to large variations of the basic frequency, both in stationary LCOs and in quasiperiodic responses. These large frequency variations preclude use of common averaging over some constant basic frequency. Consequently, the methods used previously for analysis of dynamic responses in this sort of systems, at least require essential modification. The latter challenge is a main motivation for this paper. In order to pursue the analytic approach, we adopt relatively simple model of the single-dof primary system with the same peculiarity—large variations of the basic frequency. The hope is that the general behavior of such systems will not depend on “fine details” of the model. To obtain such simple model with large frequency variations, it is enough to add an essential cubic nonlinearity to the linear term in the VdP model; in a modified system, the frequency of the LCOs will be significantly higher than that of the linear component. This modified system is referred to as Van der Pol–Duffing (VdPD) oscillator [15]. More specifically, in this research we describe different response regimes possible in the system comprising the VdPD oscillator coupled with common purely cubic NES. The secondary goal is to verify whether the addition of the NES enables one to suppress or to mitigate an undesired limit-cycle oscillations. As it was already mentioned, both problems require development of new analytic tools, since for some response regimes peculiar for this sort of system (for instance, for so-called strong modulated responses (SMRs) [16–19]) the instantaneous frequency is not constant. We are going to suggest an alternative analytic approach that allows explicit account of relatively slow frequency variations. 2. Description of the model and expression for an instantaneous frequency Let us consider the VdPD oscillator with an attached ungrounded purely cubic NES. The coupling is realized through linear damping and cubic stiffness. Equations of motion for this system can be written as follows: 2 d d d d x1 − x2 þ Kðx1 −x2 Þ3 ¼ 0 m1 2 x1 þ c x1 ðx21 −A2 Þ þ ql x1 þ qnl x31 þ γ dt dt dt dt 2 d d d x2 − x1 þ Kðx2 −x1 Þ3 ¼ 0 (1) m2 2 x 2 þ γ dt dt dt Here mi,xi(i¼ 1,2) are masses and coordinates of the primary oscillator and the NES respectively, coefficients c and A define a common VdP term, ql and qnl are linear and nonlinear stiffness coefficients, γ is the linear damping coefficient and K is the cubic stiffness coefficient. Relative orders of magnitude of the system parameters will be specified in an asymptotic analysis presented below. After rescaling System (1), one obtains the following non-dimensional system: 4 u€ 1 þ εαu_ 1 ðu21 −1Þ þ u1 þ 43 βu31 þ ελðu_ 1 −u_ 2 Þ þ εkðu1 −u2 Þ3 ¼ 0 3 4 u€ 2 þ λðu_ 2 −u_ 1 Þ þ kðu2 −u1 Þ3 ¼ 0 3 Non-dimensional parameters in Eq. (2) are determined as pffiffiffiffiffiffiffiffiffiffiffiffiffi τ ¼ ωn t; ωn ¼ ql =m1 ; ui ¼ xi =A; α ¼ cA2 =m2 ωn ;
λ ¼ γ=m2 ωn ;
i ¼ 1; 2;
k ¼ 3KA2 =4m2 ω2n ;
(2)
ε ¼ m2 =m1 ; β ¼ 3qnl A2 =4ql
A dot denotes a differentiation with respect to a new time variable τ. Further simplification can be achieved by introducing a new variable: r ¼ u1 −u2
(3)
Substituting (3) into (2), one obtains: 4 u€ 1 þ εαu_ 1 ðu21 −1Þ þ u1 þ 43 βu31 þ ελr_ þ εkr 3 ¼ 0 3 4 2 3 4 r€ þ εαu_ 1 ðu1 −1Þ þ u1 þ 3 βu1 þ ð1 þ εÞλr_ þ ð1 þ εÞkr 3 ¼ 0 3
(4)
In this non-dimensional system, all parameters are considered to be of order unity, besides the parameter ε. The latter is equal to a ratio between the masses of the attachment and of the primary system. From a viewpoint of possible applications, this ratio should be kept relatively small. In theoretical and experimental works devoted to the systems with the NES, similar parameters are adopted to stay in a range 0.01–0.1 and this convention will be followed in current work. As it will be demonstrated below, relative smallness of ε is crucial, if one develops the analytic approach to the problem. We also adopt that the same small parameter determines the scales of the VdP term and of the linear damping and the nonlinear stiffness of the NES. As it was mentioned above, the instantaneous frequency in the primary system depends on its amplitude. In order to perform averaging over varying “fast” frequency of the primary VdPD oscillator, we first formally introduce this frequency as ω¼ω(τ). Then, an additional change of variables is performed in order to apply a complexification-averaging approach [20,21]: iωðτÞu1 þ u_ 1 ¼ R exp ðiδ1 ðτÞÞ
(5)
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Here R is the amplitude of the primary oscillator, δ1 is its phase. The instantaneous frequency and the “fast” phase are related in an obvious way: δ_ 1 ¼ ωðτÞ
(6)
For the relative displacement of the NES, the following ansatz is used: iωðτÞr þ r_ ¼ ξ exp ðiδ1 Þ
(7)
Here ξ(τ) is a complex variable. We consider the simplest 1:1 resonance between the primary VdPD oscillator and the NES. As it will be clear later, this approximation is sufficient for analytic description of essential global features of the dynamical responses. More detailed studies of the frequency content in this kind of essentially nonlinear systems resort on numeric methods [11,12] and demonstrate that high-order resonances may be very significant for adequate understanding of details of transient dynamics in this class of systems. These details are beyond the scope of current paper, since we are more interested in the qualitative description of possible response regimes. If only 1:1 resonance is considered, the variables R(τ) and ξ(τ) may be treated as “slow” variables of the system. After substituting (5)–(7) into (4), we average out the “fast” phase δ1, thus obtaining the following “slow-flow” equations: " # " # 2 2 _ _R ¼ − εαR R −1 − ελ ξ þ iεk jξj2 ξ þ iR 1−ω2 þ βR þ ωR 2ω 2 4ω2 2 2ω 2ω3 ω2 " # 2 3 _ _ξ ¼ i ðR−ω2 ξÞ− ð1 þ εÞλ ξ þ ð1 þ εÞ ik jξj2 ξ− εαR R −1 þ iβR þ ω ξ (8) 2ω 2 2 4ω2 2ω3 2ω3 2ω Extracting imaginary terms of order O(1) from the first equation of (8), we obtain: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7 1 þ 4βR2 β 1−ω2 þ 2 R2 ¼ 0⇒ω2 ¼ 2 ω
(9)
In the limit β-0 the primary system turns into a common VdP oscillator and the instantaneous frequency should tend to unity in this limit [13]. Consequently, in Eq. (9) one has to choose a positive sign and the final expression for the instantaneous frequency will be as follows (up to order O(1)): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t1 þ 1 þ 4βRðτÞ2 (10) ωðτÞ ¼ 2 By differentiating Eq. (9) and after simple algebraic manipulations, one arrives to the following expression for the derivative of the instantaneous frequency: _¼ ω
βRR_ ωð2ω2 −1Þ
(11)
Substituting this expression and Eq. (9) into the first equation of System (8), one can reshape the latter equation in the following way: ðεαR=2Þ½ðR2 =4ω2 Þ−1−ðελ=2Þξ þ ðiεk=2ω3 Þjξj2 ξ R_ ¼ − þ Oðε2 Þ 1−ðβR2 =ð2ω2 ð2ω2 −1ÞÞÞ " # ! εð2ω2 −1Þ R2 iεk 2 αR −1 −λξ þ jξj ξ þ Oðε2 Þ ¼− 3ω2 −1 2ω3 4ω2
(12)
_ From Eq. (12) it is clear that ROðεÞ for all diapason of ω≥1. Although ω(R) can be now computed from Eq. (10), below in some instances the variable ω will be preserved for the sake of brevity. The next step of treatment is an asymptotic analysis based on the small parameter ε (the ratio of masses of the attachment and of the primary system). At the first stage, only terms having order O(1) will be considered. The slow variable ξ(τ) is presented in a polar form: ξ ¼ P exp ðiΔÞ
(13)
Then, substituting expressions (10), (11) and (13) into the second equation of System (8) and keeping only the leadingorder terms in an explicit form, one obtains: λ ω P_ ¼ − P þ R sin Δ þ OðεÞ 2 2 _Δ ¼ ω 1− R cos Δ − k P 2 þ OðεÞ 2 P 2ω3
(14)
System (12)–(14) is already simple enough to yield to an asymptotic analysis which will be described in the next section.
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3. Asymptotic analysis For the sake of the asymptotic analysis, the following formal assumptions are made: ε⪢1;
α; k; β; λOð1Þ
(15)
Physical meaning of these assumptions has been discussed above in Section 2. Phase flow defined by System (12)–(14) belongs to a state space ℝ2 ⊗S1 , but the derivatives of the variables have different orders of magnitude with respect to the small parameter ε. Thus, further asymptotic decomposition of the system is possible with the help of a method of multiple scales [2,11,21]. Variables P(τ) and Δ(τ) evolve at the time scale determined by averaging of the complete phase flow (4); this time scale is further referred to as “slow” one. Variable R(τ) evolves at even slower time scale determined by the small parameter ε; this time scale is further referred to as “super-slow”. A term “fast” denotes the time scale determined by fast oscillations of the primary system with frequency ω(τ). This time scale is “averaged out” when Eq. (8) are computed. So, three different time scales are used for analysis of this problem. The multiple-scale analysis is performed with the help of a common decomposition [2,11,21]: τj ¼ εj τ⇒
∂ ∂ ∂ ∂ ¼ εj ¼ þε þ⋯ ∂τ ∂τj ∂τ0 ∂τ1
(16)
where j¼ 0 corresponds to the “slow” time scale and j ¼1–to the “super-slow” time scale. For the slow time scale, system (12)–(14) is reduced to the following form: ∂R ¼0 ∂τ0 ∂P λ ω ¼ − P þ R sin Δ ∂τ0 2 2 ∂Δ ω R k 1− cos Δ − 3 P 2 ¼ ∂τ0 2 P 2ω
(17)
It is clear from (17) that R¼R(τ1), and two other equations form a planar system. It is possible to prove, with the help of the Bendixson criterion [15] that LCOs do not exist in the averaged slow system (17), and that all trajectories will be attracted to stable fixed points. These fixed points are denoted as follows: P 0n ðτ1 Þ ¼ lim Pðτ0 Þ; τ0 -∞
Δ0n ðτ1 Þ ¼ lim Δðτ0 Þ; τ0 -∞
R0n ðτ1 Þ
¼ lim Rðτ0 Þ
(18)
τ0 -∞
Isolating trigonometric terms from (17), we obtain: λ2 P 20n ω60n þ P 20n ðkP 20n −ω40n Þ2 −R20n ω80n ¼ 0;
ω0n ¼ ωðR0n Þ
(19)
For simplicity we define: Y ¼ R20n ;
Z ¼ P 20n
(20)
Then, the following equation is obtained: λ2 ω60n Z þ ZðkZ−ω40n Þ2 −Yω80n ¼ 0
(21)
Eq. (21) defines the slow invariant manifold (SIM) of System (14), which, in general, consists of two stable and one unstable branches, as can be seen in Fig. 1. The instability of the middle branch is easy to prove by common methods from System (17). The stable branches of the SIM are dynamical attractors for the slow flow, and since ∂R/∂τ0 ¼0 it is clear that for every initial condition (IC) on the Y–Z plane the flow will be attracted to one of these branches. Thus, the phase point on the plane Y–Z will move “horizontally” towards one of the stable branches—namely, the one which can be approached without crossing the unstable branch. The super-slow time scale will govern further evolution of the phase flow. If the trajectory will reach either of “fold points” Z1,2, it will “jump” horizontally towards “landing points” Zu,d respectively. Such structure of the SIM provides a possibility for relaxation oscillations [22–24]. This relaxation motion is very similar to that exhibited in the VDP oscillator coupled to the NES [13,16]. Further analysis of the response regimes requires consideration of the super-slow time scale in System (12)–(14). It is possible to circumvent explicit exploration of this time scale by observing that the super-slow flow in System (12)–(14) in main approximation should be taken into account only for variable R(τ); consequently, the equation determining the evolution of the system on this time scale will be of the first order. Therefore, the dynamics on one-dimensional SIM (19) is completely determined by positions of the global fixed points of flow (8). To determine these fixed points, the instantaneous energy of the system is defined as sum of its kinetic and potential energies, while the damping and VdP terms are disregarded. The fixed points under consideration correspond to stationary limit cycles of initial system and satisfy obvious necessary condition: over single cycle, input and output of energy in the system should be exactly balanced. With account of the damping and VdP terms, the instantaneous energy is not constant,
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and a rate of the energy variation in initial System (4) may be expressed as follows: " # 2 βu41 dE d u_ 1 þ u21 εu_ 22 εk 4 ¼ þ þ ðu1 −u2 Þ þ dτ dτ 2 2 2 3 ¼ −ε½αu_ 21 ðu21 −1Þ þ λðu_ 1 −u_ 2 Þ2
(22)
The right-hand side of (22) describes the non-conservative part of the energy flow. Fixed points of the averaged flow (8) will obey the condition 〈dE=dτ〉δ1 ¼ 0. Performing the averaging according to (5)–(11), one obtains: ! R20n −1 (23) −λP 20n ¼ αR20n 4ω20n With account of (20), we finally obtain an additional condition for the fixed points at the SIM: " # Y α 1− 2 Y−λZ ¼ 0 4ω0n
(24)
Eqs. (10), (21) and (24) are sufficient to uncover the structure of the response regimes in the system under consideration. This can be done with the help of geometric approach presented in the next section.
Fig. 1. Slow invariant manifold (SIM) of the system. The unstable branch is dashed. Z1,2 correspond to the fold points, Zu,d correspond to the “landing” points of the relaxation oscillations. Arrows depict the trajectories which the system follows while switching from one stable branch to the other.
Fig. 2. Y–Z plot for the case of complete elimination of the LCO. The SIM curve (21) (thick line) and the “energy conservation” parabola (24) (thin line) intersect only in the trivial fixed point.
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4. Numerical verifications and specific bifurcation scenarios Further analysis of particular response regimes relies on geometry of intersections of the SIM (21) with “energy conservation” parabola (24) on Z–Y plane. If not stated otherwise, values of β¼1 and ε¼ 0.05 were used in the simulations.
3
2
u1
1
0
−1
−2
−3
0
50
100 τ
150
200
0
50
100 τ
150
200
5 4 3 2
u1−u2
1 0 −1 −2 −3 −4 −5
Fig. 3. Numeric time series for the case of complete LCO suppression for set of parameters (25) and initial conditions u1(0)¼ 2, u_ 1 ð0Þ ¼ 0, u2(0) ¼0, u_ 2 ð0Þ ¼ 0. (a) Primary mass displacement u1 and (b) relative displacement of the NES u1−u2.
Fig. 4. The case of partial LCO elimination. The intersection occurs in both the trivial fixed point and in the nontrivial fixed point.
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4.1. Complete suppression of LCO For certain range of parameters, curves (21) and (24) will intersect only in the origin (Fig. 2). We consider such a case for the parameter set: α ¼ 0:05;
k ¼ 0:36;
λ ¼ 0:15
(25)
In this case, the origin is the only stable attractor of the system and, consequently, the LCOs will be eliminated. This conclusion is confirmed numerically by simulation presented in Fig. 3.
1 0.8 0.6 0.4
u1
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
50
100
0
50
100
τ
150
200
250
150
200
250
1 0.8 0.6 0.4
u1−u2
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
τ
Fig. 5. Numeric time series for the case of partial LCO suppression for set of parameters (28) and ICs u1(0)¼ 0.6, u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0. (a) Primary mass displacement u1 and (b) relative displacement of the NES u1−u2.
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This special case is very interesting, and one can reveal for what range of parameters it can occur. It is easy to see that the complete elimination will occur when near the origin the slope of the “energy conservation” parabola is larger than the slope of the SIM curve. This condition reads as λ 4 λ2 þ 1 α
(26)
This relationship does not contain any parameters related to the system nonlinearities. It is quite natural, since the regime of complete LCO elimination is related to a linear stability of the origin. From (26) it follows that the complete elimination occurs in the following range of the system parameters: 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1−4α ; þ 1−4α (27) λelimination ∈ − 2 2α 2 2α One can see that the latter will only exist for α≤0.5, i.e. for relatively small “intensity” of the VdPD oscillator. 4.2. Partial suppression of LCO Let us increase the “intensity parameter” α slightly above the maximum value for complete suppression and choose the parameter set as follows: α ¼ 0:6;
k ¼ 0:4;
λ ¼ 0:3
(28)
Mutual positions of curves (21) and (24) in this case are presented in Fig. 4. In this case, one can see that two fixed points appear. Since the system does not grow without bound, the fixed point with the higher excitation level (i.e. the nontrivial fixed point) must be stable, and the origin will be unstable. Thus, all dynamical trajectories will be attracted to this fixed point. It means that the amplitude of the LCO will be nonzero but small compared to full-scale LCOs. Fig. 5 presents the numerical time series for this case. A regime of partial suppression is obtained as predicted. Values of both the amplitudes of the primary mass displacement, and the relative displacement between the primary mass and the NES are rather close to the analytic predictions. 4.3. Canard explosion and the appearance of SMR If the “intensity parameter” α will grow further, the upper fixed point will cross the fold and will appear on the unstable branch of the SIM. For instance, such situation is realized for the following set of parameters (Fig. 6): α ¼ 1;
k ¼ 0:4;
λ ¼ 0:3
(29)
One can see that the nontrivial fixed point belongs to the unstable SIM branch. Consequently, the relaxation oscillations demonstrated in Fig. 6 turn out to be the only stable attractor of the system. This dynamical response is referred to as Strongly Modulated Response (SMR) [17,18]. Time series for this response regime are presented in Fig. 7a,b. It is interesting to compare this case with the previous one. As one can notice the transition from the previous case was caused by increasing the intensity of the self-excitation α. In our asymptotic consideration the transition from the stable fixed point to the full-scale relaxation oscillations is abrupt. However, the initial system is smooth and such abrupt transition is not possible. Instead, normal Hopf bifurcation will take place and the transition to the relaxation oscillations will occur at exponentially small interval of the bifurcation parameter. This phenomenon is well-known in systems with the relaxation oscillations and is commonly referred to as a “canard explosion” [25]. This transition is illustrated in Fig. 8a–c.
Fig. 6. The case of strong modulated response (SMR). The intersection occurs on the unstable branch.
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1 0.8 0.6 0.4
u1
0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
50
100
0
50
100
τ
150
200
250
150
200
250
2 1.5 1
u1−u2
0.5 0 −0.5 −1 −1.5 −2
τ
Fig. 7. Numeric time series for the case of the SMR. Parameters are chosen according to (29) and the ICs are u1(0)¼ 0.6, u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0. (a) Primary mass displacement u1 and (b) relative displacement of the NES u1−u2.
The nontrivial fixed point coalesces with the fold of the SIM curve at α≈0.8. From the numeric time series, one can observe the “explosion” at α≈0.86. It is instructive to observe that the frequency of oscillations grows in Fig. 7, as the amplitude increases. It occurs due to the Duffing term, which makes the frequency essentially amplitude-dependent. Time series obtained for the SMR allow us to verify to some extent expression (10) for the relationship between the instantaneous frequency and the amplitude of the primary mass oscillations. The frequency content of the response is computed with the help of a common wavelet transform, and the predicted value of the instantaneous frequency—from the
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1.5
1
u1−u2
0.5
0
−0.5
−1
−1.5
0
100
200
0
100
200
0
100
200
τ
300
400
500
300
400
500
300
400
500
1.5
1
u1−u2
0.5
0
−0.5
−1
−1.5
τ
2 1.5 1
u1−u2
0.5 0
−0.5 −1 −1.5 −2
τ
Fig. 8. Numerical illustration (time series of the NES relative displacement) for the canard explosion. Initial conditions are u1(0)¼ 0.6, u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0 and the bifurcation parameter is (a) α¼ 0.79, (b) α ¼ 0.83, and (c) α¼ 0.86.
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time series. Comparison between the theoretical predictions and the numeric findings appears in Fig. 9. One can see that the coincidence is rather satisfactory, and thus the analytic approach developed above is justified a posteriori.
4.4. Shi’lnikov homoclinic bifurcation of the SMR and the co-existence of a LCO and a SMR Further increasing the self-excitation intensity, one obtains new topological structure of the fixed points. This structure is illustrated in Fig. 10 for the following set of parameters: α ¼ 1:8;
k ¼ 0:8;
λ ¼ 0:3
(30)
Fig. 9. Wavelet analysis of a single SMR cycle for the case depicted in Fig. 7: (a) time series of the primary mass response and (b) wavelet transform of the signal. Thick line corresponds to theoretical prediction of the frequency value calculated from (9).
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As one can see in Fig. 10, there are four intersections between SIM curve (21) and the “energy conservation” parabola (23). One fixed point (the highest one) should be stable, as the system has no divergent responses. In this case, the system exhibits two stable attractors for different sets of initial conditions—the stable LCO for the initial points in the zone of “High IC”, and the SMR for the initial points in the zone of “Low IC”. Numeric verifications of this conclusion appear in Fig. 11. The SMR disappears with increase of the parameter α by global bifurcation scenario known as Shi’lnikov homoclinic bifurcation [26]. This bifurcation occurs when the upper saddle point coincides with the “landing point” Zu, and thus the SMR turns into the homoclinic trajectory for the upper saddle point. As one can see in Fig. 12, this bifurcation occurs for α¼ 2.335. For higher values of α, the SMR will not exist and all phase trajectories will be attracted to the stable LCO.
(a) 10
9 8 7
Y
6 5 4 3 2 1 0
0
5
10
Z
15
20
25
(b)
1
Y
0.8
0.6
0.4
0.2
0
1
2
3 Z
4
5
6
Fig. 10. Co-existence of the stable SMR and stable LCO. A boundary (horizontal dashed line) passing through the unstable fixed point separates the domains of attraction in the space of ICs. The high ICs lead to the LCO and the low ICs lead to the SMR. (a) Complete figure and (b) close-up of the SMR area.
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(a)
5501
0.8 0.6 0.4 0.2
u1
0 −0.2 −0.4 −0.6 −0.8
(b)
0
50
100
τ
150
200
3
2
u1
1
0
−1
−2
−3
0
50
100 τ
150
200
Fig. 11. Numeric time series of the primary mass displacements u1 for the case of co-existence of the LCO and the SMR. Parameters are chosen according to (34) and the ICs are u_ 1 ð0Þ ¼ 0, u2(0)¼ 0, u_ 2 ð0Þ ¼ 0. (a) Low IC u1(0)¼ 0.6 and (b) high IC u1(0) ¼2.
To illustrate this scenario numerically one should note that as the SMR turns into the homoclinic trajectory, its period should grow. Theoretically, the period of the homoclinic trajectory is infinite, but due to approximate character of the treatment one cannot expect to hit the Shilnikov bifurcation point in the original four-dimensional state space. However, significant growth of the SMR period with extremely small variations of α is observed numerically, as illustrated in Fig. 13a–c. Such behavior is clear “signature” of dynamics close to the Shilnikov bifurcation.
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Fig. 12. Homoclinic connection leading to Shil'nikov bifurcation.
4.5. Heteroclinic bifurcation of the SMR Another global bifurcation scenario is described in Fig. 14a,b. As one of the system parameters varies, the saddle-node bifurcation occurs at the stable branch of the SIM between points Z2 and Zu. As this happens, the SMR turns into a heteroclinic connection between the node and the saddle. This particular bifurcation scenario is illustrated for the following set of parameters: α ¼ 0:3;
k ¼ 0:22;
λ∈ð0:01; 0:03Þ
(31)
In this example, the damping coefficient is varied and all other parameters are constant. The bifurcation should theoretically occur where λ ¼0.0162 as seen in Fig. 14, and simulations presented in Fig. 15 show that the critical value is λ≈0.025. The heteroclinic bifurcation is also detected through essential growth of the SMR period with tiny variations of the damping coefficient. 4.6. Co-existence of two LCOs The last case we wish to discuss here is the co-existence of two different limit cycles. The parameters examined here are: α ¼ 5;
k ¼ 0:185;
λ ¼ 0:2
(32)
This scenario is observed as the self-excitation coefficient is relatively large and the damping and the nonlinear stiffness are relatively low. The SIM curve (21) and the “energy conservation” parabola (24) for this case appear in Fig. 16. It is clear from Fig. 16 that we obtain two different stable fixed points, each representing its own limit cycle in the state space of the initial system. The unstable SIM branch and dashed black line represent the boundary between domains of attraction of two limit cycles in the space of initial conditions of the averaged system. Numeric verification of the existence of two different LCOs is presented in Fig. 17. 5. Basins of attraction—the case of two LCOs In this section, we would like to verify numerically the predictions for basins of attraction for the case of two LCOs presented in Fig. 16. Specifically, we would like to check whether the phase trajectory of the original system with initial conditions corresponding to the ICs of the averaged flow (8) in each zone in Fig. (16) would be attracted to the LCO predicted by the analytic procedure. The numerical results appear in Fig. 18. One can that in general the analysis provides some clue on the shape of the basins of attraction; however, the boundaries are far from exact. This result reveals the limitations of proposed analytic approach—the qualitative predictions regarding the structure of the response regimes ant their bifurcations are predicted correctly. However, the accuracy of the quantitative predictions is limited. 6. Concluding remarks In the paper, it was demonstrated that one can mitigate the LCOs in the self-excited system with strongly varying frequency by means of the NES. This result is in some sense expectable, since the NES differs from common means of the vibration suppression (like a tuned mass damper) by absence of a resonant frequency [11]. Therefore, it is reasonable to
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5503
2 1.5 1
u1−u2
0.5 0 −0.5 −1 −1.5 −2 5000
5500
6000
6500
7000
6500
7000
6500
7000
τ 2 1.5 1
u1−u2
0.5 0 −0.5 −1 −1.5 −2 5000
5500
6000
τ 3
2
u1−u2
1
0
−1
−2
−3 5000
5500
6000 τ
Fig. 13. Numeric illustration of Shilnikov bifurcation of the SMR for parameters k¼ 0.8, λ¼ 0.3, ε ¼0.005 and (a) α ¼2.0155, (b) α ¼2.0155146085, (c) α ¼ 2.01552.
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Fig. 14. Heteroclinic bifurcation scenario for α¼ 0.3, k ¼0.22 and (a) λ¼ 0.03, (b) λ ¼ 0.0162.
expect that the NES might be efficient for the mitigation of broadband excitations. Results presented above confirm this conjecture for important particular case of the self-excited systems with essential nonlinearity. Moreover, it was demonstrated that one could suggest the approximate analytic procedure allowing description of a multitude of response regimes in this system. Global qualitative picture of possible response regimes obtained by this analysis seems complete—at least, the numeric simulations have not revealed any response regimes different from the predicted ones. Of course, this statement will remain true only if the system parameters will remain in the range suitable for the simplifying assumptions made. The quantitative coincidence is very good for some of these regimes and less profound for the others. It seems that better quantitative coincidence may be achieved if higher-order approximations would be taken into account. This problem will be a subject of future studies.
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2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2
0
500
1000
1500
2000 τ
2500
3000
3500
4000
0
500
1000
1500
2000 τ
2500
3000
3500
4000
2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2
2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2
0
1000
2000
3000 τ
4000
5000
6000
Fig. 15. Numerical illustrations of the heteroclinic bifurcation for α ¼0.3, k ¼ 0.22 and (a) λ ¼0.03, (b) λ¼ 0.025, (c) λ ¼0.02.
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Fig. 16. The case of co-existence of two LCOs. The ICs in zone I will lead to the higher amplitude LCO and the ICs in zone II will lead to the lower amplitude cycle (computed for parameter set (32)).
2.5 2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2 −2.5
0
20
40
60
80
100
60
80
100
τ 2.5 2 1.5 1
u1
0.5 0 −0.5 −1 −1.5 −2 −2.5
0
20
40
τ Fig. 17. The case of co-existence of two LCOs for parameters (32) and ICs u_ 1 ð0Þ ¼ 0, u2(0) ¼0, u_ 2 ð0Þ ¼ 0 and (a) u1(0) ¼0.6, (b) u1(0)¼ 2.
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5507
25
20
Y
15
10
5
0
0
20
40
Z
60
80
100
Fig. 18. Basins of attraction map, corresponding to the case of co-existence of two LCOs. Blue asterisks (*) represents ICs leading to the LCO in zone I, and green crosses (x) represent ICs leading to the LCO in zone II. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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