Journal of Applied Nonlinear Dynamics

Volume 4 Issue 3 September 2015

ISSN 2164‐6457 (print) ISSN 2164‐6473 (online)

Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics Editors J. A. Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrical Engineering Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto, Portugal Fax:+ 351 22 8321159 Email: [email protected]

Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805 USA Fax: +1 618 650 2555 Email: [email protected]

Associate Editors J. Awrejcewicz Department of Automatics and Biomechanics K-16, The Technical University of Lodz, 1/15 Stefanowski St., 90-924 Lodz, Poland Fax: +48 42 631 2225, Email: [email protected]

Stefano Lenci Dipartimento di Ingegneria Civile Edile e Architettura, Universita' Politecnica delle Marche via Brecce Bianche, 60131 ANCONA, Italy Fax: +39 071 2204576 Email: [email protected]

Miguel A. F. Sanjuan Department of Physics Universidad Rey Juan Carlos Tulipán s/n 28933 Mostoles, Madrid, Spain Fax: +34 916647455 Email : [email protected]

Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University Balgat, 06530, Ankara, Turkey Fax: +90 312 2868962 Email: [email protected]

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Nikolay V. Kuznetsov Mathematics and Mechanics Faculty Saint-Petersburg State University Saint-Petersburg, 198504, Russia Fax:+ 7 812 4286998 Email: [email protected]

C. Nataraj Department of Mechanical Engineering Villanova University, Villanova PA 19085, USA Fax: +1 610 519 7312 Email: [email protected]

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Editorial Board Ahmed Al-Jumaily Institute of Biomedical Technologies Auckland University of Technology Private Bag 92006 Wellesley Campus WD301B Auckland, New Zealand Fax: +64 9 921 9973 Email:[email protected]

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Continued on back materials

Journal of Applied Nonlinear Dynamics Volume 4, Issue 3, September 2015

Editors J. A. Tenreiro Machado Albert Chao-Jun Luo

L&H Scientific Publishing, LLC, USA

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Printed in USA on acid-free paper.

Journal of Applied Nonlinear Dynamics 4(3) (2015) 215–221

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Crises in Chaotic Pendulum with Fuzzy Uncertainty Ling Hong†, Jun Jiang, Jian-Qiao Sun State Key Lab for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, China School of Engineering, University of California at Merced, Merced, CA 95344, USA Submission Info Communicated by Jiazhong Zhang Received 14 December 2014 Accepted 7 March 2015 Available online 1 October 2015 Keywords Nonlinear system Fuzzy uncertainty Chaotic saddle Crisis

Abstract Crises in chaotic pendulum in the presence of fuzzy uncertainty are observed by means of the fuzzy generalized cell mapping method. A fuzzy chaotic attractor is characterized by its topology and membership distribution function. A fuzzy crisis implies a simultaneous sudden change both in the topology of a fuzzy chaotic attractor and in its membership distribution. It happens when a fuzzy chaotic attractor collides with a regular or a chaotic saddle. Two types of fuzzy crises are specified, namely, boundary and interior crises. In the case of a fuzzy boundary crisis, a fuzzy chaotic attractor disappears after a collision with a regular saddle on the basin boundary. In the case of a fuzzy interior crisis, a fuzzy chaotic attractor suddenly changes in its size after a collision with a chaotic saddle in the basin interior. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Physical systems are often subjected to noisy excitations and parametric uncertainties [1–4]. The interplay between noise uncertainty and nonlinearity of dynamical systems can give rise to unexpected global changes in the dynamics, which have no analogue in the deterministic case, even under small noise inputs. For example, noise in nonlinear systems can induce chaos [5, 6], attractor and basin hopping [7, 8], complexity [9], bifurcations [10, 11] and crises [12, 13]. In general, noise is theoretically modeled as a random variable and a fuzzy set leading to the two categories of fuzzy and stochastic dynamics. An important problem is to understand the underlying mechanism for various bifurcations and complicated phenomena in noisy (fuzzy and stochastic) dynamics. Chaos and bifurcation analysis of uncertain nonlinear dynamical systems is in general a difficult subject [14–18], partly because even the definition of chaos and bifurcations is open to discussion. For fuzzy nonlinear dynamics, the subject is even more difficult because the evolution of the membership function of the fuzzy response process can not be readily obtained analytically, especially for fuzzy chaotic response. A master equation has been derived for the evolution of membership functions of fuzzy processes [19, 20]. However, the solution to the equation is rare, particularly for nonlinear dynamical systems. Fuzzy generalized cell mapping [10, 21] is a discrete representation of the master equation. † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6457, eISSN 2164 − 6473/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/JAND.2015.09.001

216

Ling Hong, Jun Jiang, Jian-Qiao Sun /Journal of Applied Nonlinear Dynamics 4(3) (2015) 215–221

More recently, the FGCM method is applied to study bifurcations of fuzzy nonlinear systems [10, 22]. It should be noted that there is little study in the literature on the bifurcation of fuzzy nonlinear dynamical systems. There are studies of bifurcations of fuzzy control systems where the fuzzy control law leads to a nonlinear and deterministic dynamical system. The bifurcation studies are practically the same as that of deterministic systems [23, 24]. The work in [25] deals with bifurcation of fuzzy dynamical systems having a fuzzy response. Numerical simulations are used to simulate the system response with a given parameter and fuzzy membership grade. The eigenvalues and the membership distribution are both used to describe the bifurcation. For a given membership grade, the bifurcation of the system is defined in the same manner as for the deterministic system. In the theory of deterministic nonlinear systems, one often studies bifurcation phenomena as a control parameter is varied. The most dramatic situations are so-called crises, namely the collision of a chaotic attractor with an unstable periodic orbit following the notation of Grebogi et al. [26, 27], in which a chaotic attractor undergoes a sudden discontinuous change. Of special interest are the mechanisms that induce crises. Two different kinds of crises are distinguished. A chaotic attractor suddenly disappears due to a boundary crisis or changes in size due to an interior crisis. This paper applies the fuzzy generalized cell-mapping (FGCM) method to analyze crises in the presence of fuzzy uncertainty. A fuzzy crisis is defined as a simultaneous sudden change both in the topology of a chaotic attractor and in its membership distributions. Two types of a fuzzy chaotic basic set are specified in the crisis analysis, namely, an attractor and a chaotic set in the basin interior. Rigorous set-theoretic definitions of these two chaotic entities for fuzzy dynamics are not available in the literature, to our knowledge, such a crisis induced by fuzzy noise has yet to be addressed.

2 A fuzzy crisis in chaotic pendulum Fuzzy crises occur in chaotic pendulum in the presence of fuzzy uncertainty. dx d2x + κ + sin x = S sin ω t 2 dt dt where S is a fuzzy parameter of the forcing amplitude with a triangular membership function, ⎧ ⎨ [s − (s0 − ε )] /ε , s0 − ε s < s0 μS (s) = − [s − (s0 + ε )] /ε , s0 s < s0 + ε ⎩ 0, otherwise

(1)

(2)

ε > 0 is a parameter characterizing the intensity of fuzziness of S and is called a fuzzy noise intensity. s0 is the nominal value of S with membership grade μS (s0 ) = 1. Here x represents the angle from the vertical of a pendulum subject to an external torque which varies sinusoidally in time with frequency ω and the fuzzy forcing amplitude S. The deterministic counterpart of the chaotic pendulum equation (1) was studied by Rössler et al. [28] when the forcing amplitude S is a deterministic value. In the present work, we are interested in the region of chaotic motions with ω = 0.55, κ = 0.5, s0 = 0.86 and ε = 0. Within the context of the FGCM method, we define a fuzzy attractor as a stable and closed set of self-cycling cells, and a fuzzy saddle as an unstable and transient self-cycling set of cells. There are three types of a chaotic basic set [29], namely, an attractor, a chaotic set in a fractal basin boundary, or a chaotic set in the interior of a basin and disjoint from the attractor. A basic set is a maximal set with a dense trajectory. Maximal means that the basic set does not lie in a strictly bigger set having a dense trajectory. The fuzzy chaotic attractor is characterized by both its global topology in the phase space and steady state membership function. It can undergo sudden and discontinuous changes of the topology and membership function as the system control parameter varies. When this happens, we


217

-0.1

dx/dt

-0.15 -0.2 -0.25 -0.3 -0.35

-3

-2.8

-2.6

x

-2.4

-2.2

-2

Fig. 1 The global topology of the chaotic pendulum (1) with fuzzy noise ε = 0.006. Black symbol “.” denotes fuzzy chaotic attractors, blue symbol “.” saddle, grey symbol “.” boundary.

Fig. 2 The membership distribution function of fuzzy chaotic attractors of the chaotic pendulum (1) with fuzzy noise ε = 0.006.

say that the system goes through a fuzzy crisis according to Grebogi’s definition [26, 27] of crises for deterministic chaotic systems. In the present work, we take the damping κ = 0.5, forcing frequency ω = 0.55 and s0 = 0.86. When applying FGCM method, a cell structure of 141 × 141 cells is used for the region of the state space (−3.14 ≤ x ≤ −2.0) × (−0.35 ≤ dx/dt ≤ −0.06), and 5 × 5 interior sampling points are used within each cell. The membership function is discretized into M = 21 segments. Hence out of each cell, there are 525 trajectories with varying membership grades. These trajectories are then used to compute the

218


-0.1

dx/dt

-0.15 -0.2 -0.25 -0.3 -0.35

-3

-2.8

-2.6

x

-2.4

-2.2

-2

Fig. 3 The global topology of the chaotic pendulum (1) with fuzzy noise ε = 0.007. The legends are the same as that in Fig. 1.


transition membership matrix. A fuzzy boundary crisis varying ε from 0.006 to 0.007 is shown in Figs. 1, 2 and Figs. 3, 4, in which a fuzzy chaotic attractor collides with a period-one saddle in its smooth basin boundary before the crisis, and suddenly disappears, leaving behind a chaotic saddle in the place of the original chaotic attractor in the phase space after the crisis. A fuzzy interior crisis changing ε from 0.007 to 0.008 is shown in Figs. 3, 4 and Figs. 5, 6, in which the attractor collides with a chaotic saddle in its basin interior before the crisis, and suddenly increases


219

-0.1

dx/dt

-0.15 -0.2 -0.25 -0.3 -0.35

-3

-2.8

-2.6

x

-2.4

-2.2

-2

Fig. 5 The global topology of the chaotic pendulum (1) with fuzzy noise ε = 0.008. The legends are the same as that in Fig. 1.


its size after the crisis.

3 Concluding Remarks Chaotic pendulum in the presence of fuzzy uncertainty has been studied by means of the FCMD method. The crisis is characterized by the sudden discontinuous change of the global topology and membership function of a fuzzy chaotic attractor. A fuzzy crisis under the influence of fuzzy noise are

220


investigated. A boundary crisis occurs when a fuzzy chaotic attractor collides with a fuzzy saddle on the basin boundary. An interior crisis is due to the collision of a fuzzy chaotic attractor with a fuzzy chaotic saddle in the basin interior.

Acknowledgements This work was supported by the Natural Science Foundation of China through the grants 11332008, 11172224, 11172223 and 11172197.

References [1] Moss, F. and McClintock, P.V.E. (2007), Noise in Nonlinear Dynamical Systems, Cambridge University Press, Cambridge. [2] Bucolo, M., Fazzino, S., Rosa, M.L., and Fortuna, L.(2003), Small-world networks of fuzzy chaotic oscillators, Chaos Solitons and Fractals, 17, 557–565. [3] Sandler, U. and Tsitolovsky, L.(2001), Fuzzy dynamics of brain activity, Fuzzy Sets and Systems, bf 121, 237–245. [4] Klir,G.J. and Folger,T.A. (1988), Fuzzy Sets, Uncertainty, and Information, Prentice-Hall, Englewood Cliffs, New Jersey. [5] Tung, W.W., Hu, J., Gao,J.B., and Billock,V.A.(2008), Diffusion, intermittency, and noise-sustained metastable chaos in the lorenz equations: Effects of noise on multistability, International Journal of Bifurcation and Chaos,18(6), 1749–1758. [6] Gao, J.B., Hwang, S.K., and Liu,J.M. (1999), When can noise induce chaos?, Physical Review Letters,bf 82(6), 1132–1135. [7] Santitissadeekorn, N. and Bollt, E.M.(2007),Identifying stochastic basin hopping by partitioning with graph modularity, Physica D-Nonlinear Phenomena,231(2), 95–107. [8] Kraut S. and Feudel, U. (2002),Multistability, noise, and attractor hopping: The crucial role of chaotic saddles, Physical Review E,66(1), 015207. [9] Zaks, M.A., Sailer, X., Schimansky-GeierL., and Neiman, A.B. (2005), Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems, Chaos,15(2), 26117. [10] Hong, L. and Sun,J. Q.(2006), Bifurcations of fuzzy nonlinear dynamical systems, Communications in Nonlinear Science and Numerical Simulation,11(1), 1–12. [11] Xu, W., He, Q., Fang, T.,and Rong, H. (2003), Global analysis of stochastic bifurcation in Duffing system, International Journal of Bifurcation and Chaos,13(10), 3115–3123. [12] Sommerer, J. C., Ott, E., and Grebogi, C.(1991),Scaling law for characteristic times of noise-induced crises, Physical Review A,43(4), 1754–1769. [13] Sommerer, J.C., Ditto, W.L., Grebogi, C., Ott, E., Spano, M.L. (1991),Experimental confirmation of the scaling theory for noise-induced crises, Physical Review Letters,66(15), 1947–1950. [14] Xu, W., He, Q., Fang, T., Rong, H. (2004),Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise, International Journal of Non-Linear Mechanics, 39, 1473–1479. [15] Adamy, E. and Kempf, R. (2003), Regularity and chaos in recurrent fuzzy systems, Fuzzy Sets and Systems,140(2), 259–284. [16] Freeman, W. J.(2000), A proposed name for aperiodic brain activity: stochastic chaos, Neural Networks,13(1), 11–13. [17] Doi, S., Inoue, J., Kumagai, S. (1998), Spectral analysis of stochastic phase lockings and stochastic bifurcations in the sinusoidally forced van der Pol oscillator with additive noise, Journal of Statistical Physics, 90(5-6), 1107–1127. [18] Meunier, C. and Verga,A.D. (1988), Noise and bifurcations, Journal of Statistical Physics, 50(1/2), 345–375. [19] Friedman, Y. and Sandler, U.(1999), Fuzzy dynamics as an alternative to statistical mechanics, Fuzzy Sets and Systems, 106, 61–74. [20] Friedman, Y. and Sandler, U. (1996), Evolution of systems under fuzzy dynamic laws, Fuzzy Sets and Systems, 84, 61–74. [21] Sun, J.Q. and Hsu, C.S. (1990), Global analysis of nonlinear dynamical systems with fuzzy uncertainties by the cell mapping method, Computer Methods in Applied Mechanics and Engineering,83(2), 109–120. [22] Hong, L. and Sun, J.Q. (2006) Codimension two bifurcations of nonlinear systems driven by fuzzy noise,


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Physica D-Nonlinear Phenomena,213(2), 181–189. [23] Tomonaga, Y. and Takatsuka, K.(1998), Strange attractors of infinitesimal widths in the bifurcation diagram with an unusual mechanism of onset. Nonlinear dynamics in coupled fuzzy control systems. II, Physica D,111(1-4), 51–80. [24] Cuesta, F., Ponce, E.,and Aracil, J.(2001), Local and global bifurcations in simple Takagi-Sugeno fuzzy systems, IEEE Transactions on Fuzzy Systems,9(2), 355–368. [25] Satpathy, P.K., Das, D., and Gupta, P.B.D.(2004), A fuzzy approach to handle parameter uncertainties in Hopf bifurcation analysis of electric power systems, International Journal of Electrical Power and Energy System, 26(7), 531–538. [26] Grebogi, C., Ott, E., and Yorke, J.A.(1986), Critical exponents of chaotic transients in nonlinear dynamical systems, Physical Review Letter,57, 1284–1287. [27] Grebogi, C., Ott, E., and Yorke, J.A.(1983), Crises, sudden changes in chaotic attractors, and transient chaos, Physica D, 7(1-3), 181–200. [28] Rossler, O.E., Stewart, H. B., and Wiesenfeld, K.(1990), Unfolding a chaotic bifurcation, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences,431(1882), 371–383. [29] Stewart, H.B., Ueda, Y., Grebogi, C., and Yorke, J.A.(1995), Double crises in two-parameter dynamical systems, Physical Review Letters, 75(13), 2478–2481.



On a Class of Generalized Hydrodynamic Type Systems of Equations V.E. Fedorov†, P.N. Davydov Chelyabinsk State University, Chelyabinsk, Russia Submission Info

Abstract By means of the degenerate semigroups theory methods the local existence of a unique solution is proved for initial-boundary value problems to a class of partial differential equations systems of generalized hydrodynamics type. General results are illustrated by examples of a system with the nonlinear viscosity and a weighted system.

Communicated by Jiazhong Zhang Received 21 January 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Partial differential equations system Hydrodynamic type system Initial-boundary value problem Solution existence Solution uniqueness

©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In hydrodynamics equations systems with the incompressibility equation ∇ · v = 0 and vector equations 3

with the sum (v · ∇)v = ∑ vi vxi , v = (v1 , v2 , v3 ), arise often. Such systems will be called as hydrodynamic i=1

type systems. In this work we will research local in the time solvability of initial boundary value problems for a class of generalized hydrodynamic type systems. Denote 1 2 v= (v1x1 , v1x2 , . . . , v3x3 ) , v= (v1x1 x1 , v1x1 x2 , . . . , v3x3 x3 ) , J is an interval in R that containing the point t0 , Ω ⊂ R3 is a bounded region with a smooth boundary ∂ Ω. In a cylinder Ω × J consider a system of equations (1 − χ Δ)vt = ν Δv − r + H(t, x, v) +

3

∑ H i j (t, x, v)vix

i, j=1 3

+

∑

i, j,k=1

H i jk (t, x, v)vix j xk +

3

∑

i, j,k,l=1

j

K i jkl (t, x, v)vix j vkxl

† Corresponding

author. Email address: [email protected], [email protected]


(1)

224

V.E. Fedorov, P.N. Davydov /Journal of Applied Nonlinear Dynamics 4(3) (2015) 223–228 3

∑

+

i, j,k,l,m,s=1

K i jklms (t, x, v)vix j vkxl vmxs

3

1 2

+ ∑ Gi (t, v, v, v)vi + i=1

3

+

∑

3

∑

i, j=1

1 2

Gi j (t, v, v, v)vix j

1 2

i, j,k=1

Gi jk (t, v, v, v)vix j xk , ∇ · v = 0,

(2)

with a boundary condition v(x,t) = 0,

(x,t) ∈ ∂ Ω × J,

(3)

and an initial condition v(x,t0 ) = v0 (x),

x ∈ Ω.

(4)

Systems of the form (1), (2) arise in non-Newtoninan fluids mechanics [1, 2]. As a rule χ ∈ R characterizes elastic properties of a fluid, ν ∈ R describes its viscosity properties. The vector functions of the velocity v = (v1 , v2 , v3 ) and of the pressure gradient r = (r1 , r2 , r3 ) are sought-for. The mappings 1 +32

H, H i j , H i jk , K i jkl , K i jklms : R × Ω × R3 → R3 , Gi , Gi j , Gi jk : R × L1+3 2

→ R3 , i, j, k, l, m, s = 1, 2, 3,

depending on t, v, first and second order partial derivatives of v1 , v2 , v3 with respect to x1 , x2 , x3 , are 3 known. Here Lm 2 is a m-th Cartesian power of the Lebesgue space L2 = (L2 (Ω)) . Let H i j = (H1i j , H2i j , H3i j ), Gi = (Gi1 , Gi2 , Gi3 ), Gi j = (Gi1j , Gi2j , Gi3j ), Gi jk = (Gi1jk , Gi2jk , Gi3jk ), for i, j, k = 1, 2, 3. In the case of Hii j = v j , Hki j = 0 for i = k the system (1), (2) has a hydrodynamic type. Solvability of the problem (1)–(4) in the case of H = H i jk = 0, Hii j = q j (t, x, v), Hki j = 0 for i = k, K i jkl = 0, K i jklms = 0, Gij = 0 for i = j, G11 = G22 = G33 , Gikj = 0 for i = k, G11 j = G22 j = G33 j , Gil jk = 0 for

i = l, G11 jk = G22 jk = G33 jk was studied in [3]. In the present work those results is generalized for the general case. As in [3] investigation is based on the system (1)–(4) considering in the framework of the Showalter problem [4] (5) Pu(0) = u0 for semilinear degenerate evolution equation Lu(t) ˙ = Mu(t) + N(t, u(t)).

(6)

Here U , V are Banach spaces, L : U → V is a linear continuous operator, ker L = {0}, M : U → V is a linear closed operator with dense domain DM in U , U is an open set in R × U , N : U → V is a nonlinear operator. Operator P is a projection on the image of the resolving semigroup identity in the case of (L, σ )-bounded operator M [5, p. 98]. Solvability of the problem (5), (6) with (L, p)-bounded operator was researched in [3]. Similar methods of degenerate operator semigroups theory was used by G. A. Sviridyuk and T. G. Sukacheva for the Oskolkov system, i. e. for (1), (2) with Hii j = v j , Hki j = 0 for k = i, H = H i jk = K i jkl = K i jklms = Gi = Gi j = Gi jk = 0 [6]. In the paper [2] unique solvability was obtained in the sense of weak solutions for the Oskolkov system and several other models of viscoelastic fluids. The second section of the work contains the theorem on solvability of the problem (5), (6) that was proved in [3]. In the third section the problem (1)–(4) was reduced to the form (5), (6) by means of the

V.E. Fedorov, P.N. Davydov /Journal of Applied Nonlinear Dynamics 4(3) (2015) 223–228

225

selection of appropriate functional spaces and operators that acting in them. It was proved that the assumptions of the solvability theorem from the second section are valid for the abstract problem that was obtained by means of reduction. An unique solvability of a system with the nonlinear viscosity and of a loaded system was proved by means of new pesults.

2 Showalter problem for the semilinear equation Let U , V be Banach spaces. Denote by L (U ; V ) the Banach space of linear bounded operators, acting from U to V . If V = U , then denotation will have the form L (U ). The set of linear closed operators with dense domains in U , acting to V , will be denoted as C l(U ; V ). The set C l(U ; U ) will has a denotation C l(U ). Consider a semilinear evolution equation Lu(t) ˙ = Mu(t) + N(t, u(t))

(7)

with operators L ∈ L (U ; V ), M ∈ C l(U ; V ), nonlinear operator N : U → V , where U is an open set in R × U . It is assumed that ker L = {0}. Put ρ L (M) = {μ ∈ C : (μ L − M)−1 ∈ L (V ; U )}. Operator M is called as (L, σ )-bounded [5, p. 89], if ∃a > 0 ∀μ ∈ C

(|μ | > a) ⇒ (μ ∈ ρ L (M)).

Theorem 1. [5, Theorem 4.1.1] Let operator M be (L, σ )-bounded. Then (i) operators ˆ 1 (μ L − M)−1 Leμ t d μ , P= 2π i |μ |=R ˆ 1 L(μ L − M)−1 eμ t d μ , R > a, Q= 2π i |μ |=R are projections ; (ii) L : U k → V k , M : DM ∩ U k → V k , k = 0, 1, where U 0 = ker P, U 1 = imP, V 0 = ker Q, V 1 = imQ ; 1 1 (iii) there exist operators M0−1 ∈ L (V 0 ; U 0 ), L−1 1 ∈ L (V ; U ) ; (iv) M1 ∈ L (U 1 ; V 1 ). Here Lk = L|U k , Mk = M|DMk , DMk = DM ∩ U k , k = 0, 1. Put H = M0−1 L0 ∈ L (U 0 ), N0 = N ∪ {0}. Operator M is called as (L, p)-bounded [5, p. 93] for p ∈ N0 , if it is (L, σ )-bounded, and operator H is nilpotent with a power p. In the case of (L, σ )-bounded operator M consider the generalized Showalter problem Pu(t0 ) = u0 ,

(8)

for degenerate equation (7). For the case of degenerate evolution equation such problem more natural than Cauchy problem u(t0 ) = u0 as it is shown in the following section. It is supposed that u0 ∈ imP = U 1 . A solution of the problem (7), (8) on a segment [t0 ,t1 ] is a function u ∈ C1 ([t0 ,t1 ]; U ), that satisfying condition (8) and for all t ∈ [t0 ,t1 ] inclusions (t, u(t)) ∈ U , u(t) ∈ DM and equality (7) are valid. Theorem 2. Let p ∈ N0 , operator M be (L, p)-bounded, a set U is open in R × U , V = U ∩ (R × U 1 ) is open in R × U 1 , for all (t, u) ∈ U , such that (t, Pu) ∈ U , the equality N(t, u) = N(t, Pu) is valid, N ∈ C p (V ; V ), (I − Q)N ∈ C p+1 (V ; V ), mapping QN : V → V is locally Lipschitzian with respect to u. Then for any pair (t0 , u0 ) ∈ V there exists such t1 > t0 , that the problem (7), (8) has a unique solution u ∈ C1 ([t0 ,t1 ]; U ).

226


3 Generalized hydrodynamic type equations systems Denote L2 = (L2 (Ω))3 , H1 = (W21 (Ω))3 , H2 = (W22 (Ω))3 . The closure of L = {v ∈ (C0∞ (Ω))3 : ∇ · v = 0} in the sense of the L2 norm will be denoted by Hσ , and in the norm of H1 will be denoted as H1σ . Denotations H2σ = H1σ ∩ H2 will be used also. Let Hπ be the orthogonal complement to Hσ in L2 , and Σ : L2 → Hσ , Π = I − Σ be the corresponding orthogonal projections. In the space L2 an operator A = ΣΔ with domain H2σ has a real negative discrete spectrum with finite multiplicities of eigenvalues, condencing only at −∞ [7]. Taking into consideration equation (2), put U = H2σ × Hπ ,

V = L 2 = Hσ × Hπ .

(9)

Therefore u ∈ U has a form u = (v, r), f ∈ V has a form f = (Σ f , Π f ). Then formulas νA 0 1 − χA 0 , M= L= ν ΠΔ −1 −χ ΠΔ 0

(10)

define operators L, M ∈ L (U ; V ). As it is proved in the work [8] the operator M is (L, 0)-bounded. Theorem 3. Let χ = 0, χ −1 ∈ σ (A), ν ∈ R, H, H i j , H i jk , K i jkl , K i jklms ∈ C1 (J × Ω × R3 ; R3 ), for all 3 2 t ∈ J H(t, ·, ·) ∈ C∞ (Ω × R3 ; R3 ), Gi , Gi j , Gi jk ∈ C1 (J × H2σ × L12 2 ; R ), i, j, k, l, m, s = 1, 2, 3, v0 ∈ Hσ , t0 ∈ J. 1 2 1 Then for some t1 ∈ J, t1 > t0 , there exists an unique solution v ∈ C ([t0 ,t1 ]; Hσ ), r ∈ C ([t0 ,t1 ]; Hπ ) of the problem (1)–(4). Proof. An equality 3

(N(t, v, r))(x) = H(t, x, v) +


i, j=1 3

+

∑

i, j,k=1


3

∑

+

i, j,k,l,m,s=1 3

1 2

i=1

3

∑

i, j,k=1

3

∑

i, j,k,l=1



+ ∑ Gi (t, v, v, v)vi + +

j

3

1 2

∑ Gi j (t, v, v, v)vix

i, j=1

j

1 2

Gi jk (t, v, v, v)vix j xk

defines a nonlinear operator N. In according to [9, 10] for all t ∈ J H(t, ·, ·) ∈ C∞ (H2σ × Hπ ; H2 ). Besides, K i jklms (t, x, v)vix j vkxl vmxs L2 ≤ c max K i jklms (t, x, v(x))R3 vix j vkxl vmxs L2 x∈Ω

≤ c max K i jklms (t, x, v(x))R3 v3(W 1 (Ω))3 , x∈Ω

6

because of the continuous enclosures H 2 (Ω) into W61 (Ω) and into C(Ω) in the case of the region dimension n = 3. Analogous reasoning for other terms of the operator N with using of the continuous enclosure H 2 (Ω) into W41 (Ω) for n = 3 shows that N(t, ·, ·) : H2σ × Hπ → L2 for all t ∈ J. As in [3] it can be proved that N ∈ C1 (J × H2σ × Hπ ; L2 ) and N(t, u) = N(t, Pu) for all u = (v, r) ∈ U , because N doesn’t depend on r. Besides, condition (4) is equivalent to the condition Pu(0) = u0 .


227

Thus all conditions of Theorem 2 are satisfied. Consider the case of Gi = 0, Gi j = 0, i, j = 1, 2, 3, ijj 1 Gi (v) =

ˆ

ν1

3

∑

|vkxl (x)|2 dx,

Ω k,l=1

i, j = 1, 2, 3,

where ν1 ∈ R, other Gil jk = 0. Then for the Frechet derivative DGii j j of Gii j j we have 1 1 DGii j j (v) h=

ˆ Ω

ijj 1 1 DG (v) − DGi j j (w) i i

3

∑

2ν1

k,l=1

L (L32 ;R)

vkxl (x)hkxl (x)dx,

≤ c(vH1 + wH1 )v − wH1 .

Hence from Theorem 3 the following assertion is valid. Corollary 4. Let χ = 0, χ −1 ∈ σ (A), ν , ν1 ∈ R, H, H i j , H i jk , K i jkl , K i jklms ∈ C1 (J × Ω × R3 ; R3 ), for all t ∈ J H(t, ·, ·) ∈ C∞ (Ω × R3 ; R3 ), v0 ∈ H2σ , t0 ∈ J. Then for some t1 ∈ J, t1 > t0 , there exists a unique solution v ∈ C1 ([t0 ,t1 ]; H2σ ), r ∈ C1 ([t0 ,t1 ]; Hπ ) of the problem (3), (4) for generalized hydrodynamic type system of equations with nonlinear viscosity ˆ n 2 (1 − χ Δ)vt = [ν + ν1 ∑ vxjm (y) dy]Δv − r Ω

j,m=1

+H(t, x, v) +

3

∑

i, j=1 3

+

∑

i, j,k=1


3

∑

+

i, j,k,l,m,s=1

∇ · v = 0,

H i j (t, x, v)vix j 3

∑

i, j,k,l=1


K i jklms (t, x, v)vix j vkxl vmxs , (x,t) ∈ Ω × J,

(x,t) ∈ Ω × J.

Consider a loaded generalized hydrodynamic type system. Corollary 5. Let χ = 0, χ −1 ∈ σ (A), ν ∈ R, H, H i j , H i jk , K i jkl , K i jklms ∈ C1 (J × Ω × R3 ; R3 ), for all t ∈ J H(t, ·, ·) ∈ C∞ (Ω × R3 ; R3 ), k0 ∈ N, ξ1 , ξ2 , . . . , ξk0 ∈ Ω, gi , gi j , gi jk ∈ C1 (J × R3k0 ; R3 ), i, j, k = 1, 2, 3, v0 ∈ H2σ , t0 ∈ J. Then for some t1 ∈ J, t1 > t0 there exists a unique solution v ∈ C1 ([t0 ,t1 ]; H2σ ), r ∈ C1 ([t0 ,t1 ]; Hπ ) of the problem (3), (4) for a loaded generalized hydrodynamic type system of equations in the cylinder Ω×J (1 − χ Δ)vt = ν Δv − r + H(t, x, v) +

3


i, j=1 3

+

∑

i, j,k=1


3

+

∑

i, j,k,l,m,s=1

3

∑

i, j,k,l=1

j



228

V.E. Fedorov, P.N. Davydov /Journal of Applied Nonlinear Dynamics 4(3) (2015) 223–228 3

+ ∑ gi (t, v(ξ1 ), v(ξ2 ), . . . , v(ξk0 ))vi i=1 3

+

∑ gi j (t, v(ξ1 ), v(ξ2 ), . . . , v(ξk ))vix 0

i, j=1 3

+

∑

i, j,k=1

∇ · v = 0,

j

gi jk (t, v(ξ1 ), v(ξ2 ), . . . , v(ξk0 ))vix j xk ,

(x,t) ∈ Ω × J.

Proof. The continuous differentiability for the mappings of the form G(t, v, r) = g (t, v(ξ1 ), . . . , v (ξk0 )) follows from the continuous enclosure H 2 (Ω) into C(Ω) in the case of region dimension n = 3. Remark. In the case of one-dimensional region Ω Sobolev enclosure theorem allows to consider the weighted system with functions gi , gi j , gi jk , depending not only on values of v in fixed points 1

ξ1 , . . . , ξk0 ∈ Ω, but depending on values of v in fixed points also. Acknowledgements The first author is supported by Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020). The second author is supported by the grant of Russian Foundation for Basic Research.

References [1] Oskolkov, A.P. (1998), Initial-boundary value problems for equations of Kelvin-Voigh fluids and Oldroyd fluids motion. Proceedings of the Steklov Institute of Mathematics, 179, 126–164, 1988 [in Russian]. [2] Zvyagin, V.G. and Turbin, M.V. (2010), The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigh fluids , J. of Math. Sciences, 168 (2), 157–308. [3] Fedorov, V.E., Davydov, P.N.(2013), Semilinear degenerate evolution equations and nonlinear systems of hydrodynamics type, Proceedings of the Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 19 (4), 267–278, 2013 [in Russian]. [4] Showalter, R.E.(1975), Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1), 25–42, 1975. [5] Sviridyuk, G.A. and Fedorov, V.E. (2003), Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston. [6] Sviridyuk, G.A. and Sukacheva, T.G. (1998), On the solvability of a nonstationary problem describing the dynamics of an incompressible viscoelastic fluid, Math. Notes, 63 (3), 388–395. [7] Ladyzhenskaya, O.A. (1969), The Mathematical Theory of Viscous Incompressible Flow, Mathematics and Its Applications 2 (Revised Second ed.), Gordon and Breach, New York–London–Paris–Montreux–Tokyo– Melbourne. [8] Ivanova, N.D., Fedorov, V.E., and Komarova, K.M. (2012), Nonlinear inverse problem for the Oskolkov system, linearized in a stationary solution neighborhood. Bulletin of Chelyabinsk State University. Mathematics. Mechanics. Informatics, 15, 49–70 [in Russian]. [9] Abraham, R. and Robbin, J. (1967), Transversal Mappings and Flows, A. Benjamin Inc., New York. [10] Hassard, B.D., Kazarinoff, N.D., and Wan. Y.H. (1981), Theory and Applications of Hopf Bifurcation, London Math. Society, Lecture Notes, Ser. 41, Cambridge University Press, Cambridge–London–New York– New Rochelle–Melbourne–Sydney.



Influence of Systematic Coupling Stiffness Parameter on Coupling Duffing System Lag Self-synchronization Characteristic Zhao-Hui Ren1†, Yu-Hang Xu1 , Yan-Long Han2 , Nan Zhang1, Bang-Chun Wen1 1 School

of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110004, China of Mechanical Engineering, Chengde Petroleum College, Chengde 067000, China

2 Department

Submission Info Communicated by Jiazhong Zhang Received 7 March 2015 Accepted 21 January 2015 Available online 1 October 2015 Keywords Coupling duffing system Lag self-synchronization Coupling stiffness

Abstract Self-synchronization, compound synchronization and intelligent control synchronization widely exist in the mechanical system engineering, while lag self-synchronization movement is a special form of cooperation movement. Based on coupling Duffing system, this paper studies lag self-synchronization problem, analyses general change law of the system co-rotating synchronization frequency, antisynchronization frequency and lag phase angle by analytic analysis and numerical quantitative analysis, studies coupling parameter influences on systematic lag self-synchronization, and analyses the cause of lag self-synchronization. The results show that the root cause of lag self-synchronization is systematic stiffness namely systematic natural characteristic; that co-rotating synchronization vibration frequency and phase difference depend on coupling stiffness parameter; and that frequency and phase difference of anti-synchronization vibration are independent of coupling stiffness parameter; and when coupling stiffness parameter is larger, phase difference of two oscillators in the two kinds of synchronization is nonzero constant value. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Self-synchronization, compound synchronization and intelligent control synchronization exist in the mechanical system engineering, while lag self-synchronization motion is the special form of cooperation movement. In vibration machinery system, if it was installed with eccentric oscillator felicitously, which made vibration machinery meet synchronization theory, though geometric parameters and initial conditions of vibration machinery were not completely symmetric, finally the oscillator had the same rotational speed. If the phase difference angle was a constant value, namely called realize self-synchronization; if the phase difference angle was a nonzero constant value, namely lag self-synchronization [1,2] was realized. There is a lot of quantitative study about lag synchronization, and many similar theories exist in a number of literatures. Rosenblum et all [3] studied how the oscillator of coupling vibration system was in the transition process from phase synchronization to lag † Corresponding



230

Zhao-Hui Ren et al. /Journal of Applied Nonlinear Dynamics 4(3) (2015) 229–237

synchronization. Pikovsky [4], Kocarev [5], Parlitz [6], Lu [7], Li [8], Njah [9] and others put forward phase synchronization, generalized synchronization and other questions. The paper will study lag self-synchronization based on coupling Duffing system, and analyze general change law of the system co-rotating synchronization frequency, anti-synchronization frequency and lag phase angle by analytic analysis and numerical quantitative analysis, and study coupling parameter influences on systematic lag self-synchronization as well, and then analyze the cause of lag self-synchronization.

2 Study on lag self-synchronization characteristic of dual coupling duffing This study is carried out from the study on lag self-synchronization characteristic of wide representative coupling Duffing oscillators. Dimensionless mathematical model of this system as follows: x¨1 − cx˙1 + k1 x1 + ax˙31 = −b(x˙1 + x˙2 ) − d(x1 + x2 ), x¨2 − cx˙2 + k2 x2 + ax˙32 = −b(x˙1 + x˙2 ) − d(x1 + x2 ).

(1)

Formula (1) is the study in the state equation form, and then x1 , x2 , x˙1 , x˙2 are system state variables; b, d are coupling damping parameter and coupling stiffness parameter respectively; c represents system damping; k1 , k2 represent system stiffness (some natural characteristic of the system k1 = k2 ); a is parameter in measuring system nonlinear degree. The two synchronous parties’ phase remains a fixed nonzero phase difference in the synchronization form, which is lag phase difference. Practical calculation of lag phase difference is obtained by using the method of making two coupling oscillators amplitude normalized. Movement forms of the two oscillators’ lag self-synchronization vibration are: x1 = a1 sin ω t, x2 = a2 sin(ω t − α ).

(2)

Let two oscillators’ movement forms after normalized amplitude be: x01 = sin ω t, x02 = sin(ω t − α ),

(3)

where x01 = ax11 , x02 = ax22 . When two oscillators’ movement forms are co-rotating synchronization, namely x01 and x02 are with same sign. Subtracting: α (4) x01 − x02 = 2 sin sin(ω t + β ), 2 sin α where β = arctan 1−cos α. Namely, α 2 sin = max(x01 − x02 ), 2 (5) max(x01 − x02 ) . α = ±2 arcsin 2 The upper formula is practical calculation form of lag phase angle when two oscillators are in the process of co-rotating synchronization vibration. When two oscillators movement forms are opposite direction, namely x01 and x02 are with opposite signs. Subtracting displacement absolute value: α (6) |x01 | − |x02 | = 2 cos sin(ω t − β ), 2


231

where β with the same formula (4), namely: α cos = max(|x01 | − |x02 |) 2 2 max(|x01 | − |x02 |) . α = ±2 arccos 2

(7)

The upper formula is lag phase difference angle when two oscillators are in the process of antisynchronization vibration. Dual coupling lag synchronization vibration system must vibrate under synchronization frequency form, namely two oscillators have the same frequency which is synchronization frequency. Let the form of lag synchronization solutions be: x1 = e1 cos ω t x2 = e2 cos(ω t + ϕ ).

(8)

In the solutions which is obtained by substituting system equation, let sin ω t, cos ω t, sin(ω t + φ ), cos(ω t + φ ) corresponding coefficients are zero, then obtain analytic solutions, let e2 /e1 = γ have variable substitution, and obtain lag phase expressions: 1 [k1 + d − ω 2 + γ 2 (k2 + d − ω 2 )], 2d γ 1 3ae21 ω 2 3ae22 ω 2 2 +γ b−c+ ]. [b − c + cos ϕ = − 2bγ 4 4

cos ϕ = −

(9)

Natural frequency expressions are: 4b(k1 + k2 γ 2 ) + 4cd(1 + γ 2 ) , 4b(1 + γ 2 ) + 3ad(e21 + e22 γ 2 ) 1 [k1 + d − ω 2 + γ 2 (k2 + d − ω 2 )]. cos ϕ = − 2d γ

ω2 =

(10) (11)

When e1 = e2 = e and φ = 0, the corresponding situation of co-rotating synchronization is:

k1 + k2 + 4d , 2 2c − 4b . eT = 2 3a(k1 + k2 + 4d)

ωT =

(12) (13)

When e1 = e2 = e and φ = π, the corresponding situation of anti-synchronization is:

k1 + k2 , 2 2c . eF = 2 3a(k1 + k2 )

ωF =

(14) (15)

From above, synchronization frequency of the co-rotating synchronization ωT is related to system coupling stiffness parameter d, amplitude is related to coupling parameters b and d; synchronization frequency of anti-synchronization ωF and amplitude is unrelated to coupling parameters b and d.

232


3 Influence of system coupling stiffness parameter on coupling doffing system lag selfsynchronization characteristic Based on system model, program numerical simulation software, this study quantitatively calculates coupling parameter variation and vibration modes of dual coupling oscillator system under the conditions with different initials. If different system of parameters value in formula (1) were: c = 0.1, a = 0.05, b = 0.01, d = 0.01, k1 = 2, k2 = 2.4. In the simulation model, when initial condition of dual coupling oscillator system in formula (1) was [1, 1, 1, 1], system was in co-rotating vibration; when initial condition of state variable was taken [1, 1, −1, −1], system was in inverse vibration. The paper mainly studied co-rotating vibration when initial condition was [1, 1, 1, 1] and inverse vibration when initial condition was [1, 1, −1, −1]. Co-rotating synchronization frequency and anti-synchronization frequency of the system were calculated by using formula (12) and (14). Gained from co-rotating frequency in formula (12), d = 0.5, ωT = 0.285 Hz; d = 1, ωT = 0.326 Hz; d = 2, ωT = 0.396 Hz; d = 5, ωT = 0.556 Hz. Because of anti-synchronization frequency ωF was unrelated to coupling parameters b and d, anti-synchronization frequency ωF = 0.236 Hz was gained from formula (16). The simulation calculation proved accuracy of synchronization frequency in analytic analysis, and then found the influence of coupling parameters on system lag self-synchronization characteristic. 3.1

Influence of system coupling stiffness parameter on lag self-synchronization characteristic when initial condition was [1, 1, 1, 1]

First, if the initial condition of the state variable in dual coupling oscillators system was [1, 1, 1, 1], system was in the process of co-rotating vibration. Respectively changed coupling parameter d, and quantitatively analyzed vibration characteristic of the system. The displacement responses and spectrum graph of dual coupling oscillators system were gained by taking d = 0.01, 0.05, 1, 2 and 5 respectively, fig. 1 ∼ fig. 6. From fig. 1 and fig. 2, when d = 0.01, 0.05, displacement response curve showed displacement was in the same direction, displacement difference was in the beat vibration form, and spectrum graph showed two oscillators’ different frequencies, so under these two conditions, dual coupling vibration systems didn’t have synchronization frequency, two conditions were asynchronous state. With theoretical analyses, when the systematic coupling parameter d = 0.01, 0.05, two oscillators vibrated according to system respective natural frequency. When d = 0.5, 1, 2, 5, with coupling stiffness parameter becoming larger, two oscillators’ vibration amplitude gradually became smaller, movements

Fig. 1 The displacement responses and spectrum graph of co-rotating vibration coupling oscillator when d = 0.01.

Fig. 2 The displacement responses and spectrum graph of co-rotating vibration coupling oscillator when d = 0.05.


233

Fig. 3 The displacement responses and spectrum graph of co-rotating vibration coupling oscillator when d = 0.5

Fig. 4 The displacement responses and spectrum graph of co-rotating vibration coupling oscillator when d=1



gradually tended to coincide, and displacement difference gradually became smaller too. Under each coupling stiffness parameter condition, two oscillators vibrated with the same frequency, which showed that two oscillators made approximate co-rotating synchronization vibration. When d = 0.5, the vibration frequency of two oscillators were both 0.285 Hz, the displacement difference was 0.284, the phase difference was 5.12◦ ; when d = 1, the vibration frequency of two oscillators were both 0.326 Hz, the displacement difference was 0.139, the phase difference was 1.57◦ ; when d = 2, the vibration frequency of two oscillators were both 0.396 Hz, the displacement difference was 0.0579, the phase difference was 0.882◦ ; when d = 5, the vibration frequency of two oscillators were both 0.556 Hz, the displacement difference was 0.0212, the phase difference was 0.327◦ . From the above, when d = 0.5, 1, 2, 5, the dual coupling vibration system had co-rotating synchronization frequency, and its phase difference angle was nonzero constant value. Therefore, the dual coupling vibration system has lag synchronization characteristic. With coupling stiffness parameter increasing, system made approximate co-rotating synchronization vibration with the same natural frequency. While d was increasing, synchronization frequency increased too, the response displacement amplitude of system decreased when dual coupling oscillators

234


were in synchronization vibration; while d was increasing, the displacement difference of dual coupling oscillators gradually decreased, but it was not equal to zero, which showed that a smaller phase difference existed in the response of two oscillators. Then general rules of system were gained when two coupling oscillators co-rotating vibrated in the initial condition [1, 1, 1, 1]: (a) When coupling stiffness parameter d was smaller, two oscillators vibrated according to natural frequency. (b) When coupling stiffness parameter d was larger, two oscillators made approximate co-rotating synchronization vibration according to the same frequency. Because the response difference of two oscillators was not equal to zero, it showed that a smaller phase difference existed in the response of two oscillators, namely the phase difference was nonzero constant value, and system realized lag self-synchronization. (c) The frequency and phase difference of approximate synchronization vibration were closely related to the coupling stiffness parameter. 3.2

Influence of system coupling stiffness parameter on lag self-synchronization characteristic when initial condition was [1, 1, −1, −1]

If the initial condition of the state variable in dual coupling oscillators system was [1, 1, −1, −1], system was in the process of inverse vibration. Respectively changed coupling parameter d, and quantitatively analyzed vibration characteristic of the system. The displacement response and spectrum graph of dual coupling oscillators system were gained by taking d = 0.01, 0.05, 1, 2 and 5 respectively, fig. 7 ∼ fig. 12. From the response and spectrum graph, two oscillators made inverse vibration, with coupling stiffness parameter increasing, the displacement of system tended to coincide, the difference of the displacement absolute value had gradually decrescent trend. When d = 0.01, 0.05, displacement moved in opposite directions, the difference of the displacement absolute value was in the beat vibration form, and the vibration frequencies of the system were different. Under these two conditions, the dual coupling oscillators vibrated according to their own natural frequencies respectively, and they were in asynchronous state. When d = 0.5, 1, 2, 5, with coupling parameter becoming larger, the vibration amplitude of two oscillators’ displacement didn’t have obvious change, the difference of the displacement absolute value gradually became smaller, two oscillators vibrated with the same frequency, it showed that two oscillators made approximate inverse synchronization vibration, and the vibration frequency of two oscillators were both 0.236Hz. When d = 0.5, the difference of the displacement absolute value was 0.409, the phase difference was 174.44◦ ; when d = 1, the difference of the displacement absolute value was 0.218,

Fig. 7 The displacement responses and spectrum graph of inverse vibration coupling oscillator when d = 0.01



235


Fig. 10 The displacement responses and spectrum graph of inverse vibration coupling oscillator when d=1

Fig. 11 The displacement responses and spectrum graph of inverse vibration coupling oscillator when d=2

Fig. 12 The displacement responses and spectrum graph of inverse vibration coupling oscillator when d = 5

the phase difference was 178.34◦ ; when d = 2, the difference of the displacement absolute value was 0.110, the phase difference was 179.56◦ . When d = 5, the difference of the displacement absolute value was 0.0479, the phase difference was 179.89◦ . Through the analysis we could get when d = 0.01, 0.05, two oscillators made approximate inverse vibration with their own natural frequency, when d = 0.5, 1, 2, 5, two oscillators vibrated with system inverse synchronization frequency Therefore, with coupling stiffness parameter increasing, two oscillators made approximate inverse vibration according to the same frequency, while inverse synchronization frequency was unchanged with d increasing. The response displacement amplitude of the system didn’t have obvious change when the system in inverse synchronization vibration, it showed that d had no significant influence on the response displacement amplitude of the system, their difference of the displacement absolute value gradually decreased, but it was not equal to zero, which showed a phase difference of inverse vibration existed in the response of two oscillators. Then it gained general rules of system when dual coupling vibration system inversely vibrated in the initial condition [1, 1, −1, −1]: (a) When the coupling stiffness parameter d was smaller, two oscillators vibrated according to their

236


natural frequency. (b) When coupling stiffness parameter d was larger, two oscillators made approximate inverse synchronization vibration according to the same frequency. Because the difference of the response displacement absolute value of two oscillators was not equal to zero, it showed that a phase difference close to 180◦ existed in the response of two oscillators, namely the phase difference was nonzero constant value, and system realized lag synchronization. (c) The coupling stiffness parameter value had no influence on the frequency and phase difference of approximate synchronization vibration. Two groups calculation both showed: the synchronization frequency and amplitude of co-rotating synchronization or inverse synchronization depended on the coupling parameter value, and the phase difference was nonzero constant value, so both of the two synchronizations were lag self-synchronization. This dual coupling vibration system had some characteristic, when coupling stiffness parameter d was smaller than some value, two oscillators vibrated according to the different natural frequencies; when coupling stiffness parameter was larger than this value, two oscillators made approximate co-rotating or inverse lag self-synchronization vibration according to the same frequency. The paper adopted the numerical simulation analysis method, and quantitatively studied the self-synchronization characteristic of dual coupling Duffing oscillators system

4 Conclusions The paper studies lag self-synchronization of coupling Duffing oscillators which is a representative topic in the field of mechanical vibration etc., and reveals the reason of lag self-synchronization produced by the oscillators, and discusses general rules of the system lag phase angle, co-rotating synchronization frequency and inverse synchronization frequency by analysis and numerical simulation view. The results show that: (1) Systematic stiffness namely systematic natural characteristic is the basic cause of producing lag synchronization. (2) The frequency and phase difference of co-rotating synchronization vibration depends on the coupling stiffness parameter d, and the frequency and phase difference of inverse synchronization vibration is unrelated to coupling parameter d. (3) When coupling stiffness parameter d is larger, whether two oscillators are in co-rotating vibration or inverse vibration, phase difference of two oscillators is nonzero constant value, and system realizes lag synchronization. Acknowledgments The authors gratefully acknowledge the financial support provided by Natural Science Foundation of China (No. 51475084).

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[5] Kocarev, L. and Parlitz, U. (1996), Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems, Physical Review Letters, 76, 1816–1819. [6] Parlitz, U.and Junge, L. (1997), Subharmonic entrainment of unstable period orbits and generalized synchronization, Physical Review Letters, 79, 3158–3161. [7] Lu, J.F. (2008), Communications in Nonlinear Science and Numerical Simulation, 13, 1851–1859. [8] Li, D.M., Wang, Z.D., Zhou, J., Fang, J.A. and Ni, J.J. (2008), A note on chaotic synchronization of timedelay secure communication systems, Chaos, Solitons & Fractals, 38, 1217–1224. [9] Njah, A.N. and Vincent, U.E. (2008), Chaos synchronization between single and double wells Duffing–Van der Pol oscillators using active control, Chaos, Solitons & Fractals, 37, 1356–1361.



Tribo-dynamics Analysis of Satellite-bone Multi-axis Linkage System Jimin Xu1†, Honglun Hong1 , Xiaoyang Yuan1 , Zhiming Zhao2 1 Key

Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi’an Jiaotong University, Xi’an 710049, China 2 College of Mechanical & Electrical Engineering, Shaanxi University of Science Technology, Xi’an 710021, China Submission Info Communicated by Jiazhong Zhang Received 10 February 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Friction torque Multi-axis linkage system Tribo-dynamics Tracking precision

Abstract Friction torque is the most important factor that influences stability and precision of multi-axis linkage system during low-speed running. It is primarily related to shafting structure, external load, motion state and other factors, and presents highly nonlinear characteristics. In this paper, satellite-bone multi-axis linkage system in microgravity environment is taken as the research object to focus on the coupling tribo-dynamics issues of influencing system’s precision. The shafting structural features of two degrees of freedom (2-DOF) vertical-axis turntable, source of friction torque and coating antifriction technology are studied. The corresponding tribo-dynamics model is established. The model shows motion process of multi-axis linkage system is a coupling process of tribology and dynamics. In order to eliminate the effect of friction torque fluctuations, friction compensation based on LuGre friction model is introduced. Then tracking precision of the visual axis is about 40 (large movement range). For the imperfect situations with friction compensation, local actuator is introduced to combine with wide-range basic multi-axis system to realize the accurate movement within small range. Then the tracking precision of visual axis is expected to reach about 2 to 4 (small movement rang). ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Multi-axis linkage systems are widely applied in machine tool, robot, turntable, space telescope, military collimation and other fields [1-3]. High precision, a significant feature of multi-axis system, is the ultimate requirement of system’s functional realization [4]. Two-axis tracking turntables are boarded by different aerospace carriers to form satellite-bone multi-axis linkage system, which is an effective means of space optical communication, space target surveillance, precise observation to ground, scientific target detection and space weapon aiming. It’s also the hosting platform of payloads (such as telescopes, CCD, † Corresponding



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Fig. 1 On-orbit satellite equipped with two-axis tracking system.

etc). Fig. 1 shows a structural diagram of reconnaissance satellite equipped with a sophisticated twoaxis tracking turntable. The turntable needs to constantly adjust the observation accuracy to capture the target effectively. However, the friction force of pivoting friction pairs has a significant impact on the tracking precision. In this sense, the study of multi-axis linkage system’s precision is virtually similar to the study of revolute pairs’ tribo-dynamics issues. Because of the existence of friction, the load is discontinuous function of motion state. Nonlinear systems with such feature can be summarized as “non-smooth dynamical systems”. An effective way to analyze performance of such systems is taking friction and motion into consideration at the same time. With regard to tribological properties of mechanical systems, the literature [5] summarizes the current status and progress of mechanical system’s friction models, analysis tools, compensation controlling and other propositions. Simultaneously, researchers point out precision-growth of machine tools, robots, turntable, space telescopes and other multi-axis linkage systems featured with tracking characteristics has close relationship with the above propositions. Because multi-axis linkage systems generally use ball bearings or roller bearings, their friction and wear have great influence on the movement precision. Friction compensation controlling has obvious effect on precision-growth [6-8]. It is also a kind of precision-growth pathway to introduce the low-friction wear-resistant composite coatings into friction pairs [9, 10]. The characteristics of friction force can be studied through the friction models and experiments. Then the effective self-adaptive compensation controlling method can be realized through the combination of friction and electromechanical dynamics models [11-14], which can reduce the influence of friction on the motion accuracy and realize the precision-growth of multi-axis linkage system. It is worth mentioning that some important research institutions turn the research target into developing new auxiliary function units in recent years. Local actuators featured with small volume, high precision and fast response is installed on the existing multi-axis linkage system to achieve fine controlling within small range. In 2004, Yoshihito et al [15] proposed local actuator applied in electrical discharge machining (EDM) field. In 2008, Zhang et al [16] developed 5-DOF local actuator which has been successfully applied in the actual EDM, and the machining precision can reach 300nm. At the beginning, this paper analyzes the shafting structural features of 2-DOF vertical-axis turntable and the source of friction torque to establish the corresponding tribo-dynamics model. The effect of friction torque on tracking performance under low-speed running is studied. The performance of friction compensation based on Stribeck friction model is analyzed. When the effect of friction compensation is not very well, electromagnetic-fluid local actuator is introduced to collaborate with wide-range basic actuator to realize the high-precision movement. And the tracking accuracy of visual axis is expected to reach 2 to 4 within small movement rang.


Fig. 2 2-DOF horizontal-axis turntable.

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Fig. 3 2-DOF vertical-axis turntable.

2 Structure and tribo-dynamics model of satellite-bone multi-axis linkage system 2.1

Classification on multi-axis linkage system

Taking 2-DOF system as an example, multi-axis linkage system divides into two kinds, namely characterized by horizontal-axis and vertical-axis. System applied in space environment is mainly the latter one, which means its two axes are perpendicular to each other in space. Fig. 2 shows a typical 2DOF horizontal-axis multi-axis linkage system, which is called “manipulator” commonly in engineering field. Its two axes are parallel to each other. This paper uses the terms of satellite-bone instruments: turntable. Fig. 3 shows structural diagram of 2-DOF vertical-axis turntable. It mainly includes the azimuth shafting, elevation shafting, payloads (telescopes, CCD camera, etc), bearings, DC motors for driving shafting and all kinds of sensors. The azimuth axis is in vertical direction and realizes the rotation of main frame. The elevation axis is in horizontal direction and realizes the pitching movement of payloads. The working principle of two axis tracking turntable: when the tracked target is in the perspective of two axis tracking system, turntable performs acquisition command; when the target information is confirmed, the turntable executes tracking command and realizes the accurate tracking for the target. 2.2 2.2.1

Shafting structure of 2-DOF vertical-axis turntable and friction torque Elevation shafting

Fig. 4 shows the elevation shafting structure of 2-DOF vertical-axis turntable. There are two pairs of double rows ball bearings. Designing method of temperature-compensation is adopted for the elevation shafting in consideration of high and low temperature alternation in the space. One end is fixed and the other is freedom in the axial direction, which can avoid precision-decreasing effectively even device failure caused by temperature variation and imbalance force. The left end of shafting is fixed through two angular contact ball bearings installed back to back and a clamp ring is installed on the end face of the inner ring, which can adjust the preload to control friction torque of bearing. The right side adopts a deep groove ball bearing and an elastic collar is added on the end face, which can properly increase the preload to eliminate backlash and ensure rotational accuracy.

242


Fig. 4 Structural layout of elevation shafting.

Fig. 5 Structural layout of azimuth shafting.

2.2.2

Azimuth shafting

Fig. 5 shows the structural arrangement of azimuth shafting. The shafting adopts two angular contact ball bearings installed back to back, which can withstand radial and axial forces and limit the movement of shafting to guarantee its precision. As same as the elevation shafting, a clamp ring is added on the inner ring to adjust the preload and control friction torque. The end face loading approach is applied in the shafting which can ensure precision of shafting effectively. 2.2.3

Source of friction torque and coating antifriction technology

The double or multi rows rolling ball bearings adopted in the elevation and azimuth shafting are the source of friction torque. The value of friction torque is quite large, and can’t be ignored under low-speed running. The existence of friction torque under low-speed running will produce pulsation phenomenon of velocity even crawling phenomenon, namely low-speed instability. Therefore, the instability of visual axis’s pointing influences the tracking accuracy greatly. Besides, the existence of static friction force may also produce dead zone and can’t track normally. In order to reduce the friction torque of rolling bearings, ceramic rolling bearings and low-friction


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Fig. 6 HVO/AF supersonic flame spraying system for preparation of coatings.

wear-resistance coatings is a feasible technological way. KY-HVO/AF supersonic flame spraying device developed by Second Artillery Engineering College of Chinese People’s Liberation Army, as shown in Fig. 6, can be used for the preparation of low-speed wear-resistance coatings. In our future work, we will introduce coating antifriction technology in precision-growth of system with researchers from Second Artillery Engineering College. 2.3

Tribo-dynamics model of multi-axis linkage system

Taking the satellite-bone multi-axis linkage system shown in Fig. 3 as research object, appropriate general coordinates are chosen. Kinetic energy, potential energy, electric energy and magnetic energy of electromechanical coupling system are calculated with coordinate transformation. According to the Lagrange equations, the tribo-dynamics equation is: M θ¨ +Cθ˙ + K θ − N = 0,

(1)

where, ⎡

0 0

0

⎢0 0 0 ⎢ M=⎢ A E 2 E E (0) cos θ sin θ ⎣ 0 0 JA + JA (0) cos θE + Jy (0) sin2 θE − 2Jyz E E ⎡

LA ⎢ 0 ⎢ ⎢ ⎢ 0 C=⎢ ⎢ ⎢ ⎢ ⎣ 0 ⎡

Ra ⎢ 0 ⎢ K=⎢ ⎣ −Ka 0

⎥ ⎥ ,(2) E (0) cos θ − J E (0) sin θ ⎥ −Jxz E E⎦ xy 0

E (0) cos θ − J E (0) sin θ −Jxz E E xy

0 0 0 LB

Ka 0

0

(−2JAE (0) cos θE sin θE + 2JyE (0) sin θE cos θE − E (0) cos2 θ + 2J E (0) sin2 θ )θ˙ 2Jyz E E E yz

0

−(−JAE (0) cos θE sin θE + JyE (0) sin θE cos θE − E (0) cos 2 θ + J E (0) sin2 θ )θ˙ Jyz E E A yz ⎤ 0 0 0 0 0⎥ Re ⎥ ⎥, 0 0 0⎦ −Ke 0 0

⎤

0

JEE 0 Ke

⎤

⎥ ⎥ ⎥ E E ˙ (Jxz (0) sin θE − Jxy (0) cos θE )θE ⎥ ⎥, ⎥ ⎥ ⎥ ⎦ 0

(3)

(4)

244


T N = U1 U2 M fa sgn(θ˙a ) M fe sgn(θ˙e ) T θ = ia ie θA θE .

(5) (6)

The quality characteristics of two-axis turntable are reflected by M matrix when two axes working simultaneously. The coupling property about velocity is reflected through C matrix. The latter two items of N matrix can be replaced by general equations of friction force, such as M fa and M fe (without sign function). According to the established model of tribo-dynamics, friction torque is the most important factor affecting the precision of tracking system when structural parameters and operational parameters are unchanged. In essence, the motion process of multi-axis linkage system is a coupling process of tribology and dynamics. However, sufficient attention hasn’t been given on the practical research of tribo-dynamics. The status of tribo-dynamics in the existing mechanics system isn’t conspicuous. Most researchers pay attention on either tribology or dynamics. The analysis of systematic motion characteristics under the comprehensive effect of tribology and dynamics needs to pay enough attention.

3 Tribo-dynamics analysis of multi-axis system 3.1

Friction torque model under low-speed running

The change trend of friction torque with the variation of speed is nonlinear, as shown in Fig. 7. Friction torque presents different changing laws in different lubricated conditions. In the stationary and quasistationary state, friction torque has a sharp rise process. The value changes from zero to the maximum value and doesn’t depend on the rotational speed. It is mainly the micro-factors of friction pairs that hinder the relative motion at this moment, which can be considered as elastic deformation, namely pre-slip or Dahl effect. In the stage of low rotational speed, friction pairs are in boundary lubrication and the curve slope is negative. The friction torque is actually caused by the shear force between solids. This stage has close relation with crawling phenomenon of system. In the partial lubrication zone, liquid thin film is formed in part of contact surfaces and other surfaces are still contact with solids. This stage is the most difficult to modeling and the phenomenon of friction memory is quite obvious. In the full lubrication stage, liquid thin film is completely formed and the effect of viscous friction is quite significant.

Fig. 7 Relation graph between friction torque and rotational speed.


245

At present, a lot of friction models have been proposed which can be divided into static models and dynamical models. Four models commonly used in engineering filed are static models, namely Column model, viscous friction model, Column + viscous friction model and Stribeck model. Stribeck model is the combination of the former three models. The description of mathematical model: ˙ ˙

2

M f = sgn(θ˙ )Mc + sgn(θ˙ )(Ms − Mc)e−(−θ /θs ) +Cv

(7)

Where, θ˙ (rad/s) is instantaneous angular velocity, θ˙s (rad/s) is Stribeck speed, Mc (N/m) is Column friction torque, Ms (N/m) is static friction torque, Cv (N·m·s−1 ) is viscous friction coefficient and sgn is the sign function. Besides these static models, some dynamical models which can describe dynamic variation process of friction are also proposed, such as Dahl model, Bristle model and LuGre model [17]. In this paper, LuGre model is applied for the analysis. Its mathematical description: ⎧ ⎪ ⎪ ⎨

|θ˙ | z g(θ˙ ) ˙ ˙ 2 ⎪ g(θ˙ ) = α0 + α1 e−(θ /θs ) , ⎪ ⎩ M = σ0 z + σ1 z˙ + α2 θ˙ z˙ = θ˙ −

α0 = Mc /σ0 ,

α1 = (Ms − Mc)/σ0

(8)

Where, z (rad) is deformation of bristles, σ0 (N/m) is stiffness coefficient, σ1 (N·m·s−1 ) is damping coefficient and α2 (N·m·s−1 ) is viscosity damping coefficient. When z˙ equal zero, LuGre model can be simplified as Stribeck model. 3.2

Influence of friction torque on motion characteristics

In order to analyze the influence of friction on motion characteristics easily, a study for 1DOF servo system based on LuGre friction model is conducted. Differential equation of system: J θ¨ = T − M

(9)

Where, J is rotational inertia, θ is rotational angle, T is the driving torque of motor and M is friction torque. According to the above equation, when turntable is in the steady state, T equals M, namely control torque exported by motor is equal to the friction torque of system. Given: J = 1.0, σ0 = 260, σ1 = 2.5, Fc = 0.28, Fs = 0.340, θ¨s = 0.01. PD controlling method is adopted and take k p = 20, kd = 5.0. The control block diagram of system is shown in Figure 8. The input signal is r(t) = 0.1 sin(0.2π t). Tracking performance is obtained through simulation. Tracking performance of angular velocity under 1 Hz and 0.5 Hz are shown in Fig. 9. Dead zone presents when angular velocity tracking is close to zero. System can’t track the given signal accurately. Because of the predictive effect of differential elements, tracking angular velocity doesn’t have obvious distortion except the zero-crossing point with the given PD parameters. If reduce the differential coefficient, the velocity also presents distortion near the maximum value. The distortion phenomenon is more serious when tracking at low speed. From Fig. 9, tracking performance of angular velocity at 0.5Hz is poorer than 1Hz and its variation is more severe. The conclusion that the influence of friction torque on tracking performance at low speed and low frequency is more severe can be obtained from the above analysis. 3.3

Precision based on friction model compensation for multi-axis linkage system

In order to reduce the influence of friction torque on the tracking precision for multi-axis linkage system, feed-forward compensation of friction torque based on LuGre friction model is adopted. The control block diagram based on LuGre friction model compensation is shown in figure 10.

246


ω (rad/s)

ω (rad/s)

Fig. 8 Control block diagram of 1DOF servo system based on LuGre friction model.

t (s)

t (s)

Fig. 9 Tracking performance of angular velocity at 1 Hz and 0.5 Hz.

Fig. 10 Control block diagram based on friction model compensation for multi-axis system.

The given signal is θk+1 = 0.1 sin(0.2π tk )+0.05 sin(0.1π tk )+0.025 sin(0.05π tk ). Fig. 11 shows tracking error graph of angular displacement and angular velocity without friction compensation. Fig. 12 shows


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Fig. 11 Tracking error graph of angular displacement and angular velocity without friction compensation.

Fig. 12 Tracking error graph of angular displacement and angular velocity with friction compensation.

error graph with friction compensation. In Fig. 11, although PID control with high gain is adopted, tracking angular velocity presents serious dithering phenomenon. In Fig. 12, maximum errors of tracking performance are reduced greatly with friction compensation. Tracking error of angular displacement reduces from 0.0015 rad to 0.0007rad. Tracking error of angular velocity reduces from 0.008 rad/s to 0.0007 rad/s.

248


Fig. 13 Three-dimension graph and contour line map of view axis’s precision based on friction compensation.

In a word, friction model compensation is an effective way to improve precision of multi-axis linkage system. The view axis’s precision of three-dimension graph and contour line map is shown in Fig. 12, where, θ1 is azimuth angle and θ2 is elevation angle. Tracking precision can achieve about 40” (large movement range) based on friction compensation.

4 Supplementary introduction about precision-growth principle of local actuator The development trend of satellite-bone multi-axis linkage system is high precision, high reliability and portability. High-precision tracking is an important research direction of researchers. The practical precision-growth technologies applied in space environment are limited to friction compensation technology. However, the effect isn’t ideal. Improving the precision from 40 to second level is the most difficulty needed to be solved right now. Friction compensation is only effective for the fluctuation of friction torque caused by the change of motion state. In view of high-precision controlling can’t achieved by friction compensation, local actuator featured with small volume, high precision and fast response can be developed and applied. Local actuator can be installed on the existing executive component of multi-axis linkage system, as shown in Fig. 14. The large range of motion and adjustment is realized by the original wide range basic multi-axis system. The small range of fine regulation is realized by local actuator. Local actuator can use magnetic-fluid lubrication which the friction torque of solid-liquid friction is smaller than solidsolid interface friction. The local actuator isn’t applied in space environment until now because of the obstacle of controlling system’s reliability and economic index. Therefore, the development of lightweight and high-reliability local actuator is an important research direction in the future. If the local actuator is introduced and the related difficulties are solved, the precision of multi-axis linkage system can reach 2 ∼ 4 (movement range: 60 ∼ 120 ) which can realize high-precision detection and tracking for space targets.

5 Conclusion Taking satellite-bone multi-axis linkage system in microgravity environment as the study object, tribodynamics analysis is conducted in this paper. Main conclusions are as following: 1) Because of the existence of friction, the load is discontinuous function of motion state. The


249

Fig. 14 Working principle and structural diagram of local actuator.

motion precision of multi-axis linkage system is the coupling effect of tribology and dynamics. An effective way to analyze performance of these non-smooth dynamical systems is taking friction and motion into consideration at the same time. 2) Tribo-dynamics model of multi-axis linkage system is established in this paper. Friction torque is the most important factor affecting the tracking precision when structural parameters and operational parameters are unchanged. 3) Multi-axis linkage system is divided into two kinds, namely characterized by horizontal-axis and vertical-axis. The double or multi rows rolling ball bearings adopted in the elevation and azimuth shafting are the source of friction torque. Ceramic rolling bearings, low-friction wear-resistance coatings and friction compensation can realize precision-growth in wide movement range. Local actuator can realize high-precision controlling within small movement rang.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No.51275376). The authors would like to thank Professor Yuefang Wang from Dalian University of Technology for valuable discussions on tribo-dynamics.

References [1] Naozumi Tsuda and Katsutoshi Shimizu (1998), Two-axis pointing mechanism for earth observation system using heterodyne interferometry positioning sensor, SPIE Conference on Current Developments in Optical Design and Engineering, 1998, 169–176. [2] Hilkert, J.M. (2008), Inertially stabilized platform: technology concepts and principles, IEEE control systems magazine, 28(1), 26–46. [3] Michael, K. Masten (2008), Inertially stabilized platforms for optical imaging systems, IEEE control systems magazine, 28(1), 47–64. [4] Hanching Grant Wang and Thomas C. Williams (2008), Strategic inertial navigation systems: high-accuracy inertially stabilized platforms for hostile environments, IEEE control systems magazine, 28(1), 65–85. [5] Brian Armstrong H., Dupont, P., and De Wit, CC (1994), A survey of models, analysis tools and compensation methods for the control of machines with friction, Automatica, 30(7), 1083–1138.

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[6] Xu, J.M., Wang Y., Yuan X.Y. and Zhao, Z.M. (2013), Effects of friction torque and compensation technology for multi-axis tracking system, China National Youth Conference on Ttribology, 2013, H101–H105. [7] Bi, D., Li, Y.F., and Tso, S.K. (2004), Friction modeling and compensation for haptic display based on support vector machine IEEE transactions on industrial electronics, 51(2), 491–500. [8] Devi P, Henk N and Nathan van de W, Analysis of undercompensation and overcompensation of friction in 1DOF mechanical systems (2007), Automatica, 43(8), 1387–1394. [9] Zha, B.L, Qiao S.L., Huang, D.Y., and Yuan, X.Y. (2014), Effect of powder granularity on properties of WC-12Co coatings sprayed by HVOF, Hot Working Technology (China), 4, 138–140. [10] Barletta, M., Bolelli, G., Bonferroni, B., and Lusvarghi, L. (2010), Wear and corrosion behavior of HVOFsprayed WC-CoCr coatings on Al Alloys, Journal of Thermal Spray Technology, 19(1-2), 358–366. [11] Qian, X.S., Zhao Z.M., Guo, Y. and Yuan, X.Y. (2008), The research of self-balanced two-axis system in space thodolite, Journal of Vibration and Shock (China), 27(s), 78–81. [12] Zhao ,Z.M., Yuan, X.Y. Guo, Y., Li, Z.G., and Xu, F. (2010), Modeling and simulation of a two-axis tracking system, Proc. IMechE Part I: J. Systems and Control Engineering, 224, 125–137. [13] Zhao ,Z.M. and Yuan, X.Y. (2010), Backstepping designed sliding mode control for a two-axis tracking system, The 5th IEEE Conference on Industrial Electronics and Applications, 2010, 1593–1598. [14] Xu, F., Gu, Y., Yu, W., Li, Z.G., and Yuan, X.Y. (2008), Simulation and analysis on electromechanical dynamics of optical-electric theodolite, ACTA Photonica Sinica (China), 37(10), 2076–2079. [15] YOSHIHITO Imai, TAKAYUKI Nakagawa, HIDETAKA Miyake, HIROFUMI Hidai and HITOSHI Tokura (2004), Local actuator module for highly accurate micro-EDM, Journal of Materials Processing Technology, 149(1-3), 328–333. [16] Zhang, X., Shinshi, T., and Kajiwara, G. (2008), A 5-DOF controlled maglev local actuator and its application to electrical discharge machining, Precision Engineering, 32(4), 289–300. [17] Canudas de Wit, C., Olsson, H., Astr¨ om, K.J., and Lischinsky, P.(1995), A New Model for Control of Systems with Friction, IEEE Transactions on Automatic Control, 40(3), 419–425.



Stability and Bifurcation of a Nonlinear Aero-thermo-elastic Panel in Supersonic Flow Wei Kang1†, Yang Tang1 , Min Xu1 , Jia-Zhong Zhang2 1. School of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi Province, 710072, P.R. of China 2. School of Energy and Power Engineering, Xi’an Jiaotong University,Shaanxi Province, 710049, P.R. China Submission Info Communicated by Albert C.J. Luo Received 12 January 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Aero-thermo-elasticity Supersonic flow Large deformation Bifurcation

Abstract Stability and bifurcation of a nonlinear supersonic panel under aerothermal loads are analyzed numerically in the present study. In the structural model, von Karman’s large deformation theory is taken into account for the geometric nonlinearity of the panel. In light of Hamilton’s principle, the governing equation of motion of a twodimensional aero-thermo-elastic panel is established. Coupling with the panel vibration, aerodynamic pressure is evaluated by first order supersonic piston theory and aerothermal load is approximated by quasi-steady theory of thermal stress. By transforming the partial differential equation to a series of ordinary differential equations via Galerkin method, fixed points and their stabilities of the system are studied using nonlinear dynamic theory. The complex dynamic responses regions are discussed with temperature loads as a bifurcation parameter. The results show that the thermal stress has a significant influence in the stability of the panel. The panel system undergoes Hopf bifurcation, period doubling, quasi-period and chaos with the increase of the temperature. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Panel flutter is a self-excited aeroelastic oscillation caused by the interaction between the external skin of a high-speed aero-vehicle and aerodynamic loads exerted on it. When the flight speed is supersonic or hypersonic, large thermal stresses caused by aerodynamic heating becomes significant, which even escalates this problem. During the flight, the panel experiences strong coupling of elastic, inertia, acting aerodynamic forces, and thermal stresses, giving rise to a rich variety of nonlinear problems, such as periodic oscillation, periodic-doubling bifurcation, quasi-periodic oscillation, chaotic motion, buckling and so on. These problems may lead to immediate failure or long-term fatigue failure, which is of great † Corresponding



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concern in aerospace engineering. Therefore, to understand the behaviors of the panel system is one of the important issues for the applications of the aerospace vehicles. There have been great interests in the study of nonlinear aeroelastic instability and its control of thin-walled structures [1, 2, 3]. Dowell. [4] and Mei. [5] have studied nonlinear panel flutter problems using Galerkin method, and bifurcation study is utilized for nonlinear complex dynamic behaviors, such as buckling, periodic motion, besides simple harmonic oscillations of the systems. Moreover, the parameter space where flutter occurs is also obtained. An excellent review of studies on nonlinear panel flutter can be found from Mei. [6]. In lights of thermal effects, the studies. [7, 8] indicate that thermal effects play an important part in the static and dynamic behaviors of the panel. [9]. Ventres and Dowell. [10] studies the flutter behavior of clamped plates exposed to transverse pressure loadings or buckled by uniform thermal expansion. In this study, nonlinear behaviors of a supersonic panel under thermal loads are analyzed numerically. The governing equation of motion of a two-dimensional aero-thermo-elastic panel is established according to von Karman’s large deformation theory. Coupling with the panel vibration, aerodynamic pressure is evaluated by first order supersonic piston theory and thermal load is approximated by quasi-steady theory of thermal stress. Galerkin method is used to study the stability of the system are studied. Thermal effects on the stability of the system are discussed.

2 Aero-thermo-elastic model of panel Consider an isotropic infinitely long flat panel model, and the analysis for the panel system will be carried out under the following assumptions: (1) the deformation is elastic; (2) the shearing deformation and inertia effect are ignored; (3) the geometric nonlinearity is considered. Figure 1 shows the panel with the span a, simple supported and the upper surface is exposed to a high supersonic flow and high temperature along the axis x in the Cartesian coordinate system. Due to Von Karman assumption, the nonlinear strain-displacements relations can be then obtained as ∂ u 1 ∂ w 2 ∂ 2w + ( ) − 2, εx = (1) ∂x 2 ∂x ∂x and the stress can be expressed as

σx =

E [εx − (1 + μ )αT T (x, z)], 1 − μ2

(2)

where T (x, z) is the temperature distribution of the panel. E, μ , αT is elastic modulus, Poisson’s ratio and thermal expansion coefficient, respectively.

Fig. 1 Aeroelastic model of supersonic panel.


253

Since the thickness of the panel is relatively small, the temperature is assumed a uniform distribution, i.e. T (x, z)T . Assume that E and αT are influenced by the thermal field linearly, E = E0 + E1 T = E0 (1 + eT )

(3)

αT = α0 + α1 T = α0 (1 + α T )

(4)

where E0 , α0 is free-stress temperature and thermal expansion coefficient. E1 < 0, α1 > 0; e = E1 /E0 , α = α1 /α0 . Accordingly, the relationship between in-plane force, internal moment, strain and curvature can be written below, ˆ h/2 σx dz = σx0 h Nx = ˆ

−h/2 h/2

∂ 2w ∂ x2 −h/2 E ∂u 1 ∂w 2 ) − (1 + μ )αT T (x, z)] σx0 = [ + ( 2 1− μ ∂x 2 ∂x

Mx =

σx zdz = −D

(5)

where ρ is the density of the panel, h the thickness and D = Eh3 /[12(1 − μ 2 )] the bending stiffness. According to Kirchhoff theory, the vibration equation of the panel is written as follows,

∂ 2 Mx ∂ 2w ∂ 2w + N = ρ h − qa , x ∂ x2 ∂ x2 ∂ t2

(6)

where ρ is the transverse displacement, Nx in-plane load, Mx bending momentum and qa aerodynamic loads. Hence, the motion equation of panel under supersonic flow can be presented as the following with one unknown, ˆ 1 a ∂w 2 Eh ∂ 4w ∂ 2w ∂ 2w [ ) ( dx − (1 + μ ) α Ta] − q + ρ h = 0, (7) D 4 − T a ∂x (1 − μ 2 )a 2 0 ∂ x ∂ x2 ∂ t2 where qa is the aerodynamic load. Finally, simply supported boundary conditions are described as w|x=0 = w|x=a = 0 (8)

w |x=0 = w |x=a = 0.

The aerodynamic load over the panel can be obtained by the first-order piston theory of a high supersonic idealized flow, 1 ∂w ρ∞U∞2 ∂ W ( + ) + Δ p, ¯ (9) qa = − M ∂ x U∞ ∂ t where ρ∞ and U∞ are the freestream density and velocity of the flow, while M is the freestream Mach number, and Δ p¯ = p+ − p− is the static pressure difference between the upper and lower surfaces. Define the following dimensionless variables, a¯ =

w , h

x x¯ = , a

D0 (1 − u) Tq = , E0 α0 ha2

t¯= (

D0 1/2 t ) , ρh a2

E0 h3 , D0 = 12(1 − μ 2 )

ΔT =

T , Tq

a4 q¯a = qa , D0 h

(10)


254

and substitute Eq. (9) into Eq. (7), Eq. (7) can be rewritten as follows,

∂ 2 w¯ ∂ 4 w¯ + c − ce [6 e ∂ t¯2 ∂ x¯4

ˆ

1

( 0

∂ w¯ 2 ∂ 2 x¯ ∂ w¯ ρ¯ 1 ∂ w¯ ) d x¯− cα ΔT ] 2 + λ [ +( ] = 0, )2 ∂ x¯ ∂ x¯ ∂ x¯ λ M∞ ∂ t¯

(11)

where ce = 1 + de eΔT Tq = 1 + de eT, cα = 1 + dα α ΔT Tq = 1 + dα α T . de and dα denotes whether the elastic modulus and thermal expansion coefficient of panel material is varying with temperature, respectively, 1 yes; 0 no. The corresponding boundary conditions are ∂ 2 w¯ = 0, = 0. (12) w| ¯ x−0,1 ¯ ∂ x¯2 x−0,1 ¯ 3 Galerkin method The linear operator L of Eq. (11) together with the boundary conditions Eq. (12) is defined as Lw ≡ ∂ 4 w/∂ x4 . Then, {sin nπ x, n = 1, 2, . . . + ∞} constitutes a complete set of eigenfunctions of the operator [3, 11, 12]. These eigenfunctions span an orthogonal basis of the space on which the solution of the governing equation will be projected. The Galerkin procedure is used to approach the solution, that is N

¯ w( ¯ x, ¯ t¯) = ∑ [qi (t¯) sin(iπ x)].

(13)

i=1

Let

∂ 2 w¯ ∂ 4 w¯ G(x, y, w) = 2 + ce 4 − ce [6 ∂ t¯ ∂ x¯

ˆ

1

( 0

∂ w¯ 2 ∂ 2 w¯ ∂ w¯ ρ¯ 1 ∂ w¯ ) d x¯− cα ΔT ] 2 + λ [ +( ]. )2 ∂ x¯ ∂ x¯ ∂ x¯ λ M∞ ∂ t¯

(14)

Following Galerkin procedure, yields, ˆ 0

1

G(x, y, w) sin jπ x = 0

j = 1, 2, 3, · · · , +∞.

(15)

The general form of Eq. (15) can be obtained as, N

q¨ j (t¯) + [ce ( jπ )4 − ( jπ )2 ce cα ΔT ]q j (t¯) + 3ce q j (t¯)( jπ )2 ∑ [(iπ )2 q2i (t¯)] i=1

N

+2λ ∑

i=1

qi

(t¯)i j(1 − (−1)i+ j ) j2 − i2

+(

λ ρ¯ 1 ) 2 q˙ j (t¯) = 0. M∞

(16)

4 Numerical results and analysis The dimensionless parameters of aeroelastic system. [13] are chosen as follows: μ /M = 0.1, the static pressure difference Δ p¯ = 0, T¯q = 0.9784, e = −0.005, α = 0.0039, λ = 160. The number of approximated modes is chosen N = 6. The thermal effects are investigated by studying variation of the elastic modulus and thermal expansion coefficient with temperature. Define a bifurcation parameter Rt = ΔT /π 2 to indicate the thermal degradation. The variables de and dα are chosen as de = dα = 0 and de = dα = 1, respectively. These two numerical cases are studied via Galerkin method and nonlinear behaviors of the two systems are compared and discussed.


255

Figure 2 gives the bifurcation diagram of panel displacement at x¯= 0.75 with bifurcation parameter Rt in two cases. For de = dα = 0 case, as Rt < 2.4334, the panel system has a stable equilibrium. When Rt increases, the equilibrium of the system loses its stability and a Hopf bifurcation occurs at Rt = 2.4334. As Rt ∈ (2.4334, 3.1032], the long term behavior of the system undergoes a limit cycle oscillation, which indicates the beginning of the panel flutter. The motion of the panel becomes simply harmonic. As Rt is increasing from 3.1032 to 4.2828, a series of periodic doubling bifurcation occurs. The behavior of the panel system becomes a multi-periodic motion and goes into quasi-periodic motion as Rt ∈ (4.2828, 4, 3064]. When Rt varies from 4.3064 to 4.891, a period-3 motion is achieved and it goes into chaos state when Rt > 5.0809. For de = dα = 1 case, the behaviors of the system are much different from the other one. When the temperature plays an important effect on the material parameters of the panel (elastic modulus and thermal expansion coefficient in this case), the panel becomes ‘soft’. As Rt < 2.032, the system converges into a stable equilibrium, which loses its stability and a Hopf bifurcation occurs at Rt = 2.032. It indicates that when the thermal effects are considered, the panel flutter can be excited more easily. As Rt increases from 2.032 to 2.9734, the system undergoes a periodic motion. As Rt goes into a narrow region Rt ∈ (2.9734, 3.1723], the system experiences a series of periodic doubling bifurcation and becomes quasi-periodic motion. When Rt goes into the region Rt ∈ (3.5347, 3.6123] ∪ (3.7503, 3.9436], the system regains its stability and becomes periodic motion, which is quite different from the previous case. The system goes into chaos state as 4.0001 < Rt < 6.6926, except a periodic-3 motion in two distinct region Rt ∈ (4.2938, 4.5382] ∪ (4.9332, 5.002]. As Rt > 6.7073, the motion of the system becomes much regular again. However, the amplitude of the panel vibration is increased greatly compared with the ones at Rt < 6.7073. Figure 3 compares typical motions of the panel system with different Rt without and with the thermal effect. It can be seen that thermal effect plays an important role on the stability of the panel system. The amplitudes of the vibration of the panel with thermal effect in the four cases are larger than the one without thermal effect. Moreover, as Rt = 7, the system with thermal effect obtains periodic motion, which is relatively stable than the one without thermal effect. It implies that the system may regain a relatively stable state with suitable control parameter.

Fig. 2 Bifurcation diagrams of displacement at x¯ = 0.75 with bifurcation parameter Rt in two cases.

256


Fig. 3 comparison of motions of the panel system with different Rt .

5 Conclusions In this study, stability and bifurcation of a nonlinear supersonic panel under thermal loads and periodic actuation are analyzed numerically. A model of two-dimensional aero-thermo-elastic panel is established and Galerkin method is used to transform the governing differential equation to a series of ordinary differential equations, and the stability of the system is studied using nonlinear dynamics theory. When the thermal effects are considered, the panel flutter begins from Rt = 2.4334 to Rt = 2.032. It is because the thermal effect reduces in panel stiffness due to softening of the panel material. The coupling system exhibits complex nonlinear behaviors such as stable equilibrium, periodic motion, quasi-periodic motion and chaos. Several types of bifurcations, Hopf bifurcation, period doubling, and chaos are discovered.


257

Acknowledgements The research is supported by the National Natural Science Foundation of China (Grant No. 11402212), the Fundamental Research Funds for the Central Universities, No. 3102014JCQ01002 and the National Fundamental Research Program of China (973 Program), No. 2012CB026002.

References [1] Kang, W., Zhang, J.Z. and Liu, Y., (2010). Numerical simulation and aeroelastic analysis of a local flexible airfoil at low Reynolds numbers, the 8th Asian CFD conference, Hongkong. [2] Kang, W., Zhang, J.Z. and Feng, P.H., (2012). Aerodynamic analysis of a localized flexible airfoil at low Reynolds numbers. Communications in Computational Physics, 11(4), 1300-1310. [3] Kang, W., Zhang, J.Z., Lei, P.F. and Xu, M., (2014). Computation of unsteady viscous flow around a locally flexible airfoil at low Reynolds number. Journal of Fluids and Structures, 46, 42-58. [4] Dowell, E., (1970). Panel flutter-a review of the aeroelastic stability of plates and shells. AIAA Journal, 8(3), 385-399. [5] SHORE, C., Mei, C. and GRAY, C.E., (1991). Finite element method for large-amplitude two-dimensional panel flutter at hypersonic speeds. AIAA journal, 29(2), 290-298. [6] Mei, C., Abdel-Motagaly, K. and Chen, R., (1999). Review of nonlinear panel flutter at supersonic and hypersonic speeds. Applied Mechanics Reviews, 52(10), 321-332. [7] Abbas, L.K., Rui, X., Marzocca, P., Abdalla, M. and De Breuker, R., (2011). A parametric study on supersonic/hypersonic flutter behavior of aero-thermo-elastic geometrically imperfect curved skin panel. Acta mechanica, 222(1-2), 41-57. [8] Librescu, L., Marzocca, P. and Silva, W.A., (2004). Linear/nonlinear supersonic panel flutter in a hightemperature field. Journal of Aircraft, 41(4), 918-924. [9] Gee, D. and Sipcic, S., (1999). Coupled thermal model for nonlinear panel flutter. AIAA journal, 37(5), 642-650. [10] Dowell, E. and Ventres, C., (1970). Comparison of theory and experiment for nonlinear flutter of loaded plates. AIAA Journal, 8(11), 2022-2030. [11] Kang, W., Zhang, J.Z., Ren, S. and Lei, P.F., (2015). Nonlinear galerkin method for low-dimensional modeling of fluid dynamic system using POD modes. Communications in Nonlinear Science and Numerical Simulation, 22, 943-952. [12] Zhang, J.Z., Liu, Y., Lei, P.F. and Sun, X., (2007). Dynamic snap-through buckling analysis of shallow arches under impact load based on approximate inertial manifolds. Dynamics of Continuous Discrete and Impulsive Systems-Series B-Applications & Algorithms, 14, 287-291. [13] Xianhui, Y., (2008). Nonlinear aeroelastic fluter and stability study of panel system. Southwest Jiaotong University. PhD thesis.



Nonlinear Effects of Dusty Plasmas using Homogenous Nonequilibrium Molecular Dynamics Simulations Aamir Shahzad1,2,†, Mao-Gang He1 1 Key

Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education (MOE), Xi’an Jiaotong University, Xi’an 710049, P. R. China 2 Department of Physics, Government College University Faisalabad (GCUF), Allama Iqbal Road, 38000-Faisalabad, Pakistan Submission Info Communicated by Jiazhong Zhang Received 12 January 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Dusty plasma Molecular dynamics Thermal conductivity Non-Newtonian Force field strength Evan’s algorithm

Abstract Three-dimensional strongly coupled complex (dusty) plasma (SCCDP) is modeled using homogenous nonequilibrium molecular dynamics (HNEMD) simulations. The thermal conductivity (λ0 ) and the effects of external force field (F ∗ ) strength on the λ0 of SCCDP are calculated at higher screening strengths (κ ) from generalized Evan’s algorithm. It has been shown that the presented investigations exhibit a non-Newtonian effect that the λ0 (Γ) increases with increasing force field strength that represents interaction contributions in Yukawa conductivity. It is also verified that the results obtained with different external force filed strengths are in satisfactory agreement with earlier numerical results and with reference set of data showed deviations within less than ±10% for most of the present data point. Our very recently computed thermal conductivity at lower κ is validated by comparing the results of λ0 (Γ) at higher κ that also extended the range of force field strength (0.001 ≤ F ∗ ≤ 0.1) which explains the nature of nonlinearity of SCCDP. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The first task of plasma theory or simulation, as of all scientific theories, is to describe the natural world. The transport properties of complex liquids are very different from those of nonionic liquids and thermal properties of strongly coupled complex (dusty) plasmas (SCCDPs) have been very actively investigated in the laboratory and by the computer experiment. Thermal conductivity of complex liquids is an important parameter used in the heat design process for the industries, energy sector and advanced materials [1-7]. The recent developments represent the evolution in plasma technology within the last ten years and mostly such problems were chosen, in which experimental data can be compared with theoretical and simulation results or where the phenomenon has a diagnostic application. Complex † Corresponding

author. Email address: [email protected]; aamirshahzad [email protected]


260

Aamir Shahzad, Mao-Gang He /Journal of Applied Nonlinear Dynamics 4(3) (2015) 259–265

(dusty) plasmas consist of electrons, ions, neutrals, and microparticles with sizes ranging from10nm to some 10μm, which are responsible for the remarkable properties of SCCDPs, such as the structure of liquid or solid phases at strong electrostatic coupling [8]. Complex (dusty) plasma, which is a collection of charged and neutral particles, is the main component of the Universe and is an origin of a vast variety of astrophysical, space and terrestrial phenomena. The dust particles enhance the unique and remarkable fundamental physical properties of dense plasmas and ionized gases can enlighten such phenomena and translate into existing and future industrial applications including nuclear fusion energy, materials synthesis and modification, environmental remediation, aerospace, nano- and biomedical technologies [1, 2, 9]. The dust systems combine the physics of nonideal plasmas and condensed matter and this field has played a significant role in both newly system designs and advance development micro- and nano-technology. Nanotechnology progress in the latest two decades has been disclosed new developments in semiconductor nanowires (NWs) and medical research and their interaction with fluid materials, and thermophysical properties of NWs, including the thermal conductivities, are drastically transformed from the bulk property. Different practical applications and technology improvements have been intended such as thermoelectronic and photoelectronic devices, solar cells, fuel burning and NWs-based thermoelectric [7, 10-13]. A detailed microscopic knowledge and computation of the transport properties of plasma with dust particle interactions in a wide range of plasma parameters is required. The microscopic dynamical origin of heat transport is a basic problem in statistical mechanics with the derivation of modified equations of motion and fully homogenous systems as the ultimate goal [14-17]. In the presented work, we have been employed very recently introduced expressions for the heat energy flux [1, 4, 18, 19] to calculate thermal conductivities of SCCDPs from classical molecular dynamics (MD) simulations. We have been extended our recent published work [19], and parametric effects (external force field) and nonlinear behavior as well as particle lattice structures are calculated and discussed. Homogenous nonequilibrium MD (HNEMD) simulations show that the interaction contributions in the thermal conductivity can be described by shielded Yukawa-type potentials, while the nonlinear behavior is computed under different external force fields.

2 Theory and model for HNEMD In this section, we describe the molecular modelling and implementation parts of our classical MD technique to investigate the thermal properties of SCCDPs. In this paper HNEMD approach is used to determine the influence of external force field on the thermal conductivity. We use the standard well known Green-Kubo relations (GKRs) for the thermal conductivity coefficients of uncharged particles [16, 17, 20, 21]. The usually GKRs expression of simple liquid materials has been employed to the SCCDPs [1-5, 22] and one component Coulomb plasmas (OCCPs) [23] ˆ ∞ V λ= JQ (t) · JQ (0)dt, (1) 3kBV T 2 0 where T is system temperature, V is volume of the system, and kB is Boltzmann’s constant. Here, the microscopic heat energy JQ can be given as N

JQV = ∑ Ei i=1

pi pi 1 − ∑ ri j ( .Fi j ), m 2 i= j m

(2)


261

where ri j = ri − r j is the position vector and Fi j is the interaction force, respectively, on particle i due to j and pi is the particle momentum vector of the ith particle. The further description of microscopic relation for heat energy JQ of the Yukawa liquids is mentioned in [1, 4, 18, 19]. The expression for the energy Ei of particle i, is given as p2 1 (3) Ei = i + ∑ φi j , 2m 2 i= j where φi j is the Yukawa pair potential between particle i and j. The Evan’s generalized the linear response theory (non-Hamiltonian system) that has been employed in our case using the following equations of motion r˙ i = p˙ i =

pi , m

(4)

N

∑ Fi + Di(ri , pi ).Fe(t) − α pi,

(5)

j=1

where the Gaussian thermostat multiplier that used to control the system temperature is α , the total interparticle force acting on particle i is Fi = (−∂ φi j /∂ ri ), and the tensor phase variable that describes the coupling of system to the external force field Fe (t) is Di = Di (ri , pi ). Here the term Ei Fe (t), given in above Eq. (5), is responsible to drive particles at higher potential towards the field and the particles at lower potential against the field which generates heat energy JQ . Since applied force field Fe (t) performs a mechanical work on the system and equilibrium is never maintained. In order to maintain the system temperature a Nose-Hoover thermostat (α ) is applied and it is represented by the expression [15-19] N

α=

∑ [Fi + Di(ri , pi ).Fe (t)].pi

i=1

N

∑

i=1

.

(6)

p2i /mi

Very recently, the presented authors Shahzad and He [1, 4, 18] have reported a detail investigation on the Yukawa conductivity results and the tensor Di (ri , pi ) element in terms of Ewald-Yukawa sums for the case of the microscopic flux of energy current is discussed for a Yukawa system. When external force field Fe (t) = (0, 0, Fz ) is applied parallel to the z-axis, then the thermal conductivity can be written as [19, 24]: ˆ ∞ −JQz (t) V λ= (t)J (0) dt = lim lim , (7) J Q Q z z Fz →0 t→∞ 3kB T 2 0 T Fz where JQz is the z-component of the heat energy vector.

3 Simulation technique and parameters The classical MD technique is used in the present study [7, 25], it involves the solution of the equations of motion of a system of particles that interact with each other through an inter-particle Yukawa potential: φ (r) = (Q2 /4πε0 ) exp(−r/λD )/r, where r is distance between interparticles, λD is the Debye screening length, and Q is a particle charge. It is significant that an accurate model potential can be chosen for the system of interest, MD approach can be used regardless of the phase and thermodynamic conditions of the substances involved. For many physical systems (such as in biology, medicine, physics of polymers and materials for energy production, etc.), this model has also been generally employed for computation of pairwise interaction of repelling particles. Three reduced parameters are used in

262


order to complete description of Yukawa systems: the plasma coupling parameter Γ = Q2 /4πε0 aw skB T , where T is particle kinetic temperature (in energy units) and aw s = (3/4nπ)1/3 is the Wigner-Seitz (WS) radius with n is the dust particle number density, the screening parameter κ ≡ aw s/λD [26]. Additional parameters include the thermal heat energy JQ and the external field force strength Fe (t) = (Fz ), and its normalized value F ∗ = (Fz )(aw s/JQ ) [7, 18, 19, 21]. Take a system of N = 2048-13500 particles enclosed in a cube of volume V interacting through a pairwise potential of Yukawa type. Periodic boundary conditions have been applied to the unit cell in all directions with the minimum image convention of the Yukawa particles. It is noted that this number of particles is sufficient to guarantee a good compromise between numerical cost and minimizing spurious interactions between the repeated images of the dust particles in the directions parallel to the simulation box. A canonical ensemble is used for our simulations that the Gaussian thermostat design is employed in order to keep the system at constant temperature. The simulation time step is dt = 0.001/ω p , where ω p = (nQ2 /ε0 m)1/2 is dust plasma frequency with m is the dust particle mass. A predictor-corrector algorithm is used to solve the numerical equations [7, 25]. The pairwise interaction force acting on the i-th particle Fi = (−∂ φi j /∂ ri ) is calculated by the Yukawa potential between particle i (at ri ) and the particle j (at r j ) and its periodic images. The Ewald sums method has been employed to take care of the long range interaction between the Yukawa dust particles and further details of κ -dependent cut-off radius for Yukawa systems are reported by Shahzad and He [1, 2, 4, 5] and Donko and Hartmann [22]. Production steps are progressed between 2.5 × 105 /ω p and 1.5 × 105 /ω p time units in the series of data recording of λ . In this paper, the HNEMD simulations are performed over wider range of plasma parameters of (2 ≤ Γ ≤ 100) and (1 ≤ κ ≤ 4) for extended F ∗ . 4 Simulation results and discussion In this section we report the results obtained through homogenous nonequilibrium MD algorithm [Eq. (7)] for the thermal conductivity of Yukawa liquids as a function both of the imposed external force field (F ∗ ) and of the number of dust particles, at different screening parameters. We simulate a set of heat (thermal) fluxes by mechanical perturbations using the HNEMD (improved form of Evans’ algorithm) simulations for the SCCDPs and perturb the system with an impulsive force of strength Fz . The thermal conductivities normalized by plasma frequency (ω p ) with applied external field strengths [F ∗ = Fz (JQ /aw s)] are compared and discussed. The normalization employed for the thermal conductivity has been widely used for the earlier studies of the SCCDPs [1-12]. The thermal conductivity λ of 3D Yukawa system can be normalized by the plasma frequency λ0 = λ /nkB ω p aw s2 or by the Einstein √ frequency λ ∗ = λ / 3nkB ωE aw s2 . It has been confirmed that the Einstein frequency decreases with √ √ increasing κ , ωE → ω p / 3 as κ → 0 and λ0 = λ ∗ ( 3ωE /ω p ). First, a separate series of the computer simulations (homogenous NEMD) are carried out for the three values of coupling Γ = 10, 50 and 100, and κ = 2. An intermediary to high rank of order exists throughout the simulation time (ωp t) for higher Γ; on the other hand the Ψ rapidly disappears at the lowest Γ. For the lowest Γ (=10) a system is close disordered liquid state, as shown in Fig. 1. The major thermal conductivity results obtained through an improved HNEMD algorithm are shown in Figs. 2 and 3 for various F ∗ at κ = 1 and 4, respectively. Figure 3(b) suggests that the dominating behavior of potential contribution decreases with further increase of field F ∗ > 0.05 and screening κ strengths. The thermal conductivity is always taken as the limit F ∗ → 0, so just extrapolate the thermal conductivity verses field curve to zero field strength. The homogenous conductivity algorithm makes it possible to study and understand the interaction contributions to the thermal conductivity.


263

As it is expected, at the lowest Coulomb coupling strength Γ the kinetic contribution dominates and the potential contribution at higher Coulomb coupling strength Γ. As the field dependence conductivity results start at low values of F ∗ .

Fig. 1 Particle lattice structure (|Ψ|) as a function of time, ωpt in Yukawa Dusty Plasma Liquids (YDPLs) for Γ = 10, 50 and 100, and N = 2048 at κ = 2. Figure shows the long range order varies with time in the order and disorder states with the application of F ∗ = 0.005. Here, the value of lattice vector is taken as k = 2π/l(1, −1, 1) in our HNEMD simulations

Fig. 2 Comparison of the results obtained λ0 /λREF (Γ) by the present HNEMD at various Γ and F ∗ with EMD of Salin and Caillol [3]: SC (EMD), HPMD of Shahzad and He [4]: SH (HPMD) and InHNEMD of Donkó and Hartmann [22]: DH (NEMD). The dotted lines represent ±10 deviation from the λREF (b) Variation of λ0 (Γ, F ∗ ) as a function of F ∗ and Γ of Yukawa liquid at κ = 1 and N = 13, 500, copyright [19] by American Institute of Physics (AIP) publishing

264


Fig. 3 Comparison of the results obtained λ0 /λREF (Γ) by the present HNEMD at various Γ and F ∗ with EMD of Salin and Caillol [3]: SC (EMD), HPMD of Shahzad and He [4]: SH (HPMD), at κ = 4 and N = 13, 500. The dotted lines represent ±10 deviation from the λREF (b) Reduced thermal conductivity normalized by plasma frequency λ0 as a function of normalized external field strength F ∗ for three selected values of plasma state of screening strength κ = 2, 3 and 4, at fixed Γ = 100

5 Conclusions An improved Evans’ homogenous nonequilibrium MD approach has been used for 3D SCCDPs. Newly improved HNEMD simulations provide more reliable data for 3D Yukawa heat conductivity than previously known simulation data over higher plasma coupling states which are ideal conditions for experimental dusty plasma systems. The presented simulation results are also compared with those obtained with own reference set of data and theoretical approach for complex plasmas. The HNEMD algorithm is a powerful tool for computing thermal properties and this method has satisfactory efficiency than earlier numerical methods. The minimum values of frequency decreases with increasing of κ . This method demonstrates that the measured λ0 of SCCDPs is presented for various plasma states, and it first time provides the application of HNEMD method to the understanding of a significant behavior in non-Newtonian Yukawa liquids. The presented simulation results look forward intensely to new improvements in this plasma field next few years and to an extending range of non-Newtonian technologies, in the fields of both material science and dusty plasmas applications.

Acknowledgments This work was sponsored by the China Postdoctoral Science Foundation (CPSF No. 2013M532042). The authors thank Z. Donkó (Hungarian Academy of Sciences) for providing his thermal conductivity data of Yukawa Liquids for the comparisons of our simulation results, and useful discussions. We are grateful to the National High Performance Computing Centre of Xian Jiaotong University for allocating of computer time to test and run our MD code.

References [1] Shahzad, A. and He, M.-G. (2012), Thermal conductivity of three-dimensional Yukawa liquids (dusty plasmas), Contrib. Plasma Phys., 52, 667–665.


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[2] Shahzad, A. and He, M.-G. (2012), Shear viscosity and diffusion motion of two-dimensional dusty plasma liquids, Phys. Scr., 86, 015502. [3] Salin, G. and Caillol, J.-M. (2003), Equilibrium molecular dynamics simulations of the transport coefficients of the Yukawa one component plasma, Phys. Plasmas, 10, 1220. [4] Shahzad, A. and He, M.-G. (2012), Thermal conductivity calculation of complex (dusty) plasmas, Phys. Plasmas, 19, 083707. [5] Shahzad, A. and He, M.-G. (2012), Thermodynamics characteristics of dusty plasma by using molecular dynamics simulations, Plasma Sci. and Technol., 14, 771–777. [6] Faussurier, G. and Murillo, M.S. (2003), Gibbs-Bogolyubov inequality and transport properties for strongly coupled Yukawa fluids, Phys. Rev. E, 67, 046404. [7] Shahzad, A. and He, M.-G. (2014), Computer Simulation of Complex plasmas: Molecular Modeling and Elementary Processes in complex plasmas, Scholar’s press, Saarbr¨ ucken, Germany. [8] Piel, A. and Melzer, A. (2002), Dynamical Processes in Complex Plasmas, Plasma Phys. Control. Fusion, 44, R1–R26. [9] Thoma, M.H., Kretschmer, M., Rothermel, H. et al., (2005), The plasma crystal, American Journal of Physics, 73, 420–424. [10] Shahzad, A. and He, M.-G. (2014), Homogenous nonequilibrium molecular dynamics evaluations of thermal conductivity 2D Yukawa liquids, Internationl Journal of Thermophysics, pp: 1-12, DOI: 10.1007/s10765-0141671-8. [11] Shahzad, A. and He, M.-G. and Kai, H. (2013), Diffusion Motion of Two-Dimensional Weakly Coupled Complex (Dusty) Plasmas, Phys. Scr., 87, 035501. [12] Shahzad, A., Aslam, A., Sultana, M. and He, M.–G. (2014), Shear Viscosity of Strongly Coupled Complex (Dusty) Liquids, IOP Conf. Ser.: Mater. Sci. Eng., 60 (1), 012014. [13] Wang, Z., Zu, X., Gao, F., Weber, W.J. and Crocombette, J.P. (2007), Atomistic simulation of the size and orientation dependences of thermal conductivity in GaN naowires, Appl. Phys. Lett., 90, 161923. [14] Ciccotti, G., Jacucci G. and McDonald, I.R. (1979), Thought-experiments by molecular dynamics, J. Stat. Phys., 21, 1–22. [15] Hoover, W.G. and Ashurst, W. T. (1975), Nonequilibrium molecular dynamics: Theoretical Chemical Advances in Perspectives, Academic, London. [16] Evans, D.J. and Morriss, G.P. (1990), Statistical Mechanics of Non-equilibrium Liquids, Academic, London. [17] Todd, B.D. and Daivis, P.J. (2007), Homogenous non-equilibrium molecular dynamics simulations of viscous flow: techniques and applications, Mol. Sim, 33, 189. [18] Shahzad A, Maryam S, Arfa A, He, M.-G. (2014), Thermal conductivity measurements of 2D complex liquids using nonequilibrium molecular dynamics simulations, Applied Sciences and Technology (IBCAST), 11th International Bhurban Conference, Jan. 14-18, Proceeding of the IEEE Transaction, 1, 212–217. [19] Shahzad, A. and He, M-G. (2013), Interaction contributions in thermal conductivity of three-dimensional complex liquids, AIP Conf. Proc. 1547, 173. [20] Hansen, J.-P. and McDonald, I.R. (1986), Theory of Simple Liquids, Academic, London. [21] Evans, D.J. (1982), Homogenous NEMD algorithm for thermal conductivity application of non-canonical linear response theory, J. Phys. Lett. A, 91, 457. [22] Donk´ o, Z. and Hartmann, P. (2004), Thermal conductivity of strongly coupled Yukawa liquids, Phys. Rev. E, 69, 016405. [23] Pierleoni, C. Ciccotti, G. and Bernu, B. (1987), Thermal Conductivity of the Classical One-Component Plasma by Nonequilibrium Molecular Dynamics, Europhys. Lett., 4, 1115. [24] Evans, D. J. (1981), Rheological properties of simple fluids by computer simulation, Phys. Rev. A, 23, 1988. [25] Rapaport, D.C. (2004). The Art of Molecular dynamics Simulation, Cambridge University Press, New York. [26] Shahzad A, Arfa A and He M–G, (2014), Equilibrium Molecular Dynamics Simulation of Shear Viscosity of Two Dimensional Complex (Dusty) Plasmas, Radiat. Eff. Defect S., 169, 931–941.



The Adaptive Synchronization of the Stochastic Fractional-order Complex Lorenz System Xiaojun Liu, Ling Hong† State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University Xi’an, 710049, China Submission Info Communicated by J.A.T. Machado Received 8 December 2014 Accepted 7 March 2015 Available online 1 October 2015 Keywords Fractional-order system Random parameter Laguerre polynomial approximation Synchronization

Abstract In this paper, the adaptive synchronization of a stochastic fractionalorder complex Lorenz system is analyzed. Firstly, the Laguerre polynomial approximation method is applied to investigate the fractionalorder system with a random parameter which obeys an exponential distribution. Based on this method, the stochastic system is reduced into the equivalent deterministic one. Besides, based on the stability theory of fractional-order systems, the adaptive synchronization for the deterministic system with unknown parameters is realized by designing appropriate synchronization controllers and estimation laws for uncertain parameters. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Fractional calculus, which has the similar long history with the classic calculus, was proposed in 1695. In the following centuries, the theories of fractional calculus underwent a significant and even heated development, mainly contributed by pure, but not applied, mathematicians. The development of this field is very rapid just in recent several decades due to the application in physics, engineering, secure communications and so on. Meanwhile, it has been found that some fractional-order differential systems can demonstrate chaotic behavior, such as fractional-order Chua circuit [1] , fractional-order Lorenz system [2], fractional-order Chen system [3], and so on. For integer-order systems, there are a lot of references about them with stochastic terms. And more and more researchers begin to realize the importance of investigation of stochastic fractionalorder systems. In [4], for a single stage fractional-order low-pass filter using stochastic and fractional calculus, its noise analysis was investigated. The synchronization of stochastic fractional-order systems with uncertain parameters via Laguerre polynomial was analyzed in [5]. However, the study about the fractional-order complex systems with random parameters is very few. Inspired by the above, in this paper, the fractional-order complex lorenz system with a random parameter is investigated. Firstly, the Laguerre polynomial approximation method is applied to stud† Corresponding



268

Xiaojun Liu, Ling Hong /Journal of Applied Nonlinear Dynamics 4(3) (2015) 267–279

ied the stochastic fractional-order complex system. And the stochastic system is reduced into the equivalent deterministic one by using the method. The dynamics of the system including phase diagrams and time trajectories for different orders are investigated. Besides, based on the stability theory of fractional-order systems, the adaptive synchronization for the deterministic system with fully unknown parameters is realized by designing appropriate synchronization controllers and estimation laws. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme. The paper is organized as follows. In Sect. 2,he stability theory of fractional-order systems is introduced. In Sect. 3, the Laguerre polynomials are introduced as the basis of orthogonal polynomial approximation. In Sect. 4, the transformation of stochastic fractional-order complex Lorenz system into its equivalent deterministic one by Laguerre polynomial approximation is shown. Sect. 5 is devoted to studying the adaptive synchronization of the deterministic system. Finally, we summarize the results in Sect. 6.

2 The stability of the fractional-order chaotic systems Fractional calculus is a generalization of classic ones. There are at least six definitions for the fractional derivative. Caputo derivative is often used in engineering and numerical computation. The main reason is that the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, which is very suitable for practical problems. Therefore, the Caputo definition for the fractional derivatives is employed in this paper. The stability of equilibrium points for a fractional-order chaotic system is very complex and is different from integral chaotic system. The following lemmas can be use to analysis the stability of equilibrium points [6]. Lemma 1. For a commensurate fractional-order system, the equilibrium points of the system are asymptotically stable if all the eigenvalues at the equilibrium E ∗ , satisfy the following condition: | arg(eig(J))| = | arg(λi )| >

π q, i = 1, 2, · · · , n, 2

(1)

where J is the Jacobian matrix of the system evaluated at the equilibria E ∗ . Lemma 2. An incommensurate fractional-order system will asymptotically steady at the equilibrium E ∗ if π (2) | arg(λ )| > r, 2 for all roots λ of the following equation det |diag([λ mq1 λ mq2 · · · λ mqn ]) − J| = 0,

(3)

where m is the least common multiply of the denominators ui s of qi s ,which can be written as qi = vi 1 + ui , vi , ui ∈ Z for i = 1, 2, · · · , n, and r = m . Based on the Lemmas 1 and 2, the minimal commensurate order for a fractional-order system to remain chaotic can be obtained by the following relation q≥

2 min{arg |λi |}, π i

(4)

π − mini {arg|λi |} ≥ 0 which is called the instability and the relation (4) is equivalent to the inequality 2m measure for fractional-order systems (IMFOS). Hence, a necessary condition for fractional order system to remain chaos is IMFOS > 0.


269

3 Exponential probability density function and Laguerre polynomials In this paper, a random parameter which obeys the exponential distribution is described as follows: p(u) =

λ e−λ (u−μ ) , 0,

(x ≥ μ ), (x < μ ),

(5)

where λ = 1, μ = 0. For the expansion of a function of random variables with exponential PDF, Larguerre polynomials are the right choice for its orthogonal basis [4]. The general expansion for Laguerre polynomial is n i n n! k (6) u. Ln (u) = ∑ (−1) i i! i=0 The recurrent formula for Laguerre polynomials is Ln+1 (u) = (2n + 1 − u)Ln (u) − n2 Ln−1 (u).

(7)

The orthogonality of Laguerre polynomial can be described as ˆ

+∞ 0

(i!)2 , e−u Li (u)L j (u)du = 0.

(i = j) (i = j)

(8)

The weighting function is just the same as the exponential PDF, p(u) in (5), the left-hand side of (8) can be regard as the expectation of the product Li (u)L j (u). 4 The stochastic fractional-order complex Lorenz system The Lorenz system is very famous in the field of nonlinear dynamics, which has a chaotic attractor with butterfly shape. Many researchers have studied the dynamics, chaos control and synchronization for the system and its corresponding fractional-order one [7–9]. In [10], the authors proposed the fractional-order complex Lorenz system, and investigated its chaos and synchronization. In this paper, we will study the fractional-order complex Lorenz system with a random parameter with exponential probability density function. The system can be described by the following differential equations: ⎧ q ⎪ ⎨d y1 = a(y2 − y1 ) d q y2 = by1 − y2 − y1 y3 ⎪ ⎩ q d y3 = 12 (y1 y2 + y1 y2 ) − cy3 ,

(9)

vector of state variables. y1 = x + jy, y2 = z + jv are complex variables, y3 = w where y = (y1 , y2 , y3 )T is the √ is a real variable, and j = −1. a, c are system deterministic parameters, b is a random parameter, and it can be expressed as b = b + δ u,

(10)

where b and δ are the mean value and the standard deviation of b respectively. δ is regarded as the intensity of random parameter b. u is a random variable defined on [0, ∞] whose PDF obeys the exponential distribution.

270


Firstly, the complex variables of the system are separated into real and imaginary parts, respectively. Due to the linearity of the Caputo differential operator, the system (9) can be represented as ⎧ ⎪ d q x = a(z − x) ⎪ ⎪ ⎪ q ⎪ ⎪ ⎨d y = a(v − y) d q z = bx − z − xw, ⎪ ⎪ ⎪ d q v = by − v − yw ⎪ ⎪ ⎪ ⎩ q d w = xz + yv − cw

(11)

System (11) is more convenient than (9) for the numerical calculation and discussion for complex space. Therefore, in the rest of the paper, we will use the system (11) as the replacement of the fractional-order complex Lorenz system. It is well known that the stochastic function space, which is constituted with the responses of nonlinear dynamic systems with random parameters, has proved to be a Hilbert space with respect to convergence in the mean square. Therefore the fractional-order system (11) with random parameters can be reduced into the equivalent deterministic system by using of the orthogonal polynomial expansion. It follows from the orthogonal polynomial approximation that the responses of system (11) can be expressed approximately by the following series: ⎧ ⎪ x(t, u) = ∑Ni=0 xi (t)Li (u) ⎪ ⎪ ⎪ N ⎪ ⎪ ⎨y(t, u) = ∑i=0 yi (t)Li (u) z(t, u) = ∑Ni=0 zi (t)Li (u), ⎪ ⎪ ⎪ v(t, u) = ∑Ni=0 vi (t)Li (u) ⎪ ⎪ ⎪ ⎩ w(t, u) = ∑Ni=0 wi (t)Li (u)

(12)

where Li (u) is the Laguerre polynomial. The number N represents the largest order of the polynomials. And when N → ∞, the left of Eqs.(12) is equal to the its right in the sense of mean square convergence. Then substitute the Eqs.(12) into (11), we can get ⎧ ⎪ Dq (∑Ni=0 xi (t)Li (u)) = a(∑Ni=0 zi (t)Li (u) − ∑Ni=0 xi (t)Li (u)) ⎪ ⎪ ⎪ ⎪ ⎪ Dq (∑Ni=0 yi (t)Li (u)) = a(∑Ni=0 vi (t)Li (u) − ∑Ni=0 yi (t)Li (u)) ⎪ ⎪ ⎪ ⎪ ⎪ Dq (∑Ni=0 zi (t)Li (u)) = b(∑Ni=0 xi (t)Li (u)) + δ u(∑Ni=0 xi (t)Li (u)) ⎪ ⎪ ⎪ ⎨−( N z (t)L (u)) − ( N x (t)L (u))( N w (t)L (u)), ∑i=0 i ∑i=0 i ∑i=0 i i i i N N N q ⎪ ( v (t)L (u)) = b( y (t)L (u)) + δ u( D ∑ ∑ ∑ i i ⎪ i=0 i i=0 i i=0 yi (t)Li (u)) ⎪ ⎪ ⎪ N N N ⎪ −(∑i=0 vi (t)Li (u)) − (∑i=0 yi (t)Li (u))(∑i=0 wi (t)Li (u)), ⎪ ⎪ ⎪ ⎪ ⎪ Dq (∑Ni=0 wi (t)Li (u)) = (∑Ni=0 xi (t)Li (u))(∑Ni=0 zi (t)Li (u)) ⎪ ⎪ ⎪ ⎩+(∑N y (t)L (u))(∑N v (t)L (u)) − c(∑N w (t)L (u)) i i i i=0 i i=0 i i=0 i

(13)

All the nonlinear terms in the system (13) can be simplified linear assembly of of Li (u) via the recurrence relationship of Laguerre polynomial, which can be described as follows: ⎧ N N 2N ⎪ ⎪ ⎪(∑i=0 xi (t)Li (u))(∑i=0 wi (t)Li (u)) = ∑i=0 Pi (t)Li (u) ⎪ ⎨( N y (t)L (u))( N w (t)L (u)) = 2N Q (t)L (u) ∑i=0 i ∑i=0 i ∑i=0 i i i i (14) N N 2N ⎪ x (t)L (u))( z (t)L (u)) = F (t)L (u), ( ∑ ∑ ∑ i i i i i i ⎪ i=0 i=0 i=0 ⎪ ⎪ ⎩( N y (t)L (u))( N v (t)L (u)) = 2N G (t)L (u) ∑i=0 i ∑i=0 i ∑i=0 i i i i


271

where Pi (t), Qi (t), Fi (t) and Gi (t) can be get by using the computer software Maple. Meanwhile, the terms δ u(∑Ni=0 xi (t)Li (u)) and δ u(∑Ni=0 yi (t)Li (u)) can be reduced to the following form: N

N

i=0

i=0 N

uδ ( ∑ xi (t)Li (u)) =uδ ( ∑ xi (t)[(2i + 1)Li (u) − i2 Li−1 (u) − Li+1 (u)]) =δ ( ∑ Li (t)[(2i + 1)xi (u) − (i + 1)2 xi−1 (u) − xi+1 (u)] − LN+1 (u)xN (t)), i=0 N

N

(15)

uδ ( ∑ yi (t)Li (u)) =uδ ( ∑ yi (t)[(2i + 1)Li (u) − i2 Li−1 (u) − Li+1 (u)]) i=0

i=0 N

=δ ( ∑ Li (t)[(2i + 1)yi (u) − (i + 1)2 yi−1 (u) − yi+1 (u)] − LN+1 (u)yN (t)). i=0

In order to simplify the calculation, x−1 (t), x4 (t), y−1 (t), and y4 (t) are supposed to zero. Then, substituting the Eqs.(15) and (14) into (13), we have N

N

N

i=0 N

i=0 N

i=0 N

i=0 N

i=0 N

i=0

i=0

i=0

Dq ( ∑ xi (t)Li (u)) =a( ∑ zi (t)Li (u) − ∑ xi (t)Li (u)), Dq ( ∑ yi (t)Li (u)) =a( ∑ vi (t)Li (u) − ∑ yi (t)Li (u)), N

Dq ( ∑ zi (t)Li (u)) =b( ∑ xi (t)Li (u)) + δ ( ∑ Li (t) i=0 2

[(2i + 1)xi (u) − (i + 1) xi−1 (u) − xi+1 (u)] − LN+1 (u) N

2N

i=0

i=0

xN (t)) − ( ∑ zi (t)Li (u)) − ( ∑ Pi (t)Li (u))), N

N

N

i=0

i=0

(16)

Dq ( ∑ vi (t)Li (u)) =b( ∑ yi (t)Li (u)) + δ ( ∑ Li (t) i=0

[(2i + 1)yi (u) − (i + 1)2 yi−1 (u) − yi+1 (u)] − LN+1 (u) N

2N

yN (t)) − ( ∑ vi (t)Li (u)) − ( ∑ Qi (t)Li (u)), i=0

i=0

N

2N

2N

N

i=0

i=0

i=0

i=0

Dq ( ∑ wi (t)Li (u)) =( ∑ Fi (t)Li (u)) + ( ∑ Gi (t)Li (u)) − c( ∑ wi (t)Li (u)). Multiplying both sides of system (16) by Li (u), i = 0, 1, 2, · · · , N in sequence and taking expectation with respect to u, owing to the orthogonality of Laguerre polynomials. We can finally obtain the equivalent deterministic system. Remember that if N → ∞ in (12), then its left is strictly equivalent to the response of stochastic system (11). Otherwise, the (12) is just approximately valid with a minimal residual error. Due to the requirement of computational precision, we take N = 3 in the following numerical analysis. Therefore, we get the equivalent deterministic system approximately as

272


⎧ q D x0 = a(z0 − x0 ) ⎪ ⎪ ⎪ ⎪ ⎪ Dq y0 = a(v0 − y0 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Dq z0 = bx0 + δ (x0 − x1 ) − z0 − P0 ⎪ ⎪ ⎪ ⎪ Dq v0 = by0 + δ (y0 − y1 ) − v0 − Q0 ⎪ ⎪ ⎪ ⎪ ⎪ Dq w0 = F0 + G0 − cw0 ⎪ ⎪ ⎪ ⎪ ⎪ Dq x1 = a(z1 − x1 ) ⎪ ⎪ ⎪ ⎪ ⎪ Dq y1 = a(v1 − y1 ) ⎪ ⎪ ⎪ ⎪ ⎪ Dq z1 = bx1 + δ (3x1 − 4x2 − x0 ) − z1 − P1 ⎪ ⎪ ⎪ ⎪ ⎪ Dq v1 = by1 + δ (3y1 − 4y2 − y0 ) − v1 − Q1 ⎪ ⎪ ⎪ ⎨Dq w = F + G − cw 1 1 1 1 q ⎪ D x2 = a(z2 − x2 ) ⎪ ⎪ ⎪ ⎪ ⎪Dq y2 = a(v2 − y3 ) ⎪ ⎪ ⎪ ⎪ ⎪Dq z2 = bx2 + δ (5x2 − 9x3 − x1 ) − z2 − P2 ⎪ ⎪ ⎪ ⎪ ⎪ Dq v2 = by2 + δ (5y2 − 9y3 − y1 ) − v2 − Q2 ⎪ ⎪ ⎪ ⎪ ⎪ Dq w2 = F2 + G2 − cw2 ⎪ ⎪ ⎪ ⎪ ⎪ Dq x3 = a(z3 − x3 ) ⎪ ⎪ ⎪ ⎪ ⎪ Dq y3 = a(v3 − y3 ) ⎪ ⎪ ⎪ ⎪ ⎪ Dq z3 = bx3 + δ (7x3 − x2 ) − z3 − P3 ⎪ ⎪ ⎪ ⎪ ⎪ Dq v3 = by3 + δ (7y3 − y2 ) − v3 − Q3 ⎪ ⎪ ⎩ q D w3 = F3 + G3 − cw3

(17)

We can get the numerical solutions xi , yi , zi , vi , wi , (i = 0, 1, 2, 3) of system (17) by the improved predictor-corrector algorithm, which is an effective numerical method for fractional-order systems [11]. Therefore, the approximate random responses of the stochastic system (11) can be expressed as ⎧ N N ⎪ ⎨x(t, u) ≈ ∑i=0 xi (t)Li (u), y(t, u) ≈ ∑i=0 yi (t)Li (u) z(t, u) ≈ ∑Ni=0 zi (t)Li (u), v(t, u) ≈ ∑Ni=0 vi (t)Li (u) ⎪ ⎩ w(t, u) ≈ ∑Ni=0 wi (t)Li (u).

(18)

The ensemble mean responses(EMRs) of the stochastic fractional-order complex Lorenz system are described as ⎧ ⎪ E[x(t, u)] ≈ ∑3i=0 xi (t)E[Li (u)] = x0 (t) ⎪ ⎪ ⎪ ⎪ ⎪E[y(t, u)] ≈ ∑3i=0 yi (t)E[Li (u)] = y0 (t) ⎨ (19) E[z(t, u)] ≈ ∑3i=0 zi (t)E[Li (u)] = z0 (t). ⎪ ⎪ 3 ⎪ E[v(t, u)] ≈ ∑i=0 vi (t)E[Li (u)] = v0 (t) ⎪ ⎪ ⎪ ⎩ E[w(t, u)] ≈ ∑3i=0 wi (t)E[Li (u)] = w0 (t) For the equivalent deterministic system (17), when the system parameters are taken as a = 10, b = 28, c = 8/3 and the δ = 0.05, the dynamics of the system (17) with different values of derivative order q can be seen from Fig.1.


5

5

4

4

3

273

3

2 0

x

z0

2 1

1

0

0

−1

−1

−2 −3 −2

−1

0

1

2

x

3

4

−2 0

5

10

20

30

0

(a) The phase diagram for q = 0.97;

40 t

50

60

70

80

(b) the time trajectory for q = 0.97; 6

5

5

4

4

3

3

2

2 x

0

z0

6

1

1

0

0

−1

−1

−2

−2

−3 −4

−2

0

x0

2

4

−3 0

6

(c) The phase diagram for q = 0.98;

10

20

30

40 t

50

60

70

80

(d) the time trajectory for q = 0.98;

20

15

15

10

10 5

x0

z0

5 0

0 −5 −5 −10

−10

−15 −15

−10

−5

0

5

10

x0

(e) The phase diagram for q = 0.995;

15

−15

0

10

20

30

40

50

60

70

80

t

(f) the time trajectory for q = 0.995.

Fig. 1 The phase diagrams and time trajectories of the system with different values of derivative order q.

274


5 Adaptive synchronization In this section, the synchronization of stochastic fractional-order complex Lorenz system with unknown parameters will be investigated. Firstly, we will take the equivalent system (17) as the drive system, and all the system parameters are uncertain which need to be identified. The corresponding response system can be described by the following differential equations ⎧ ⎪ Dq x0 = a1 (z0 − x0 ) + u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq y0 = a1 (v0 − y0 ) + u2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq z0 = b1 x0 + δ (x0 − x1 ) − z0 − P0 + u3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq v0 = b1 y0 + δ (y0 − y1 ) − v0 − Q0 + u4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq w0 = F0 + G0 − c1 w0 + u5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq x1 = a1 (z1 − x1 ) + u6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq y1 = a1 (v1 − y1 ) + u7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq z1 = b1 x1 + δ (3x1 − 4x2 − x0 ) − z1 − P1 + u8 ⎪ ⎪ ⎪ ⎪ q ⎪ ⎪ D v1 = b1 y1 + δ (3y1 − 4y2 − y0 ) − v1 − Q1 + u9 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Dq w = F + G − c1 w + u10 1 1 1 1 (20) ⎪ q ⎪ D x = a (z − x ) + u , 1 2 11 ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ q ⎪ ⎪D y2 = a1 (v2 − y3 ) + u12 ⎪ ⎪ ⎪ ⎪ ⎪ Dq z2 = b1 x2 + δ (5x2 − 9x3 − x1 ) − z2 − P2 + u13 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq v2 = b1 y2 + δ (5y2 − 9y3 − y1 ) − v2 − Q2 + u14 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq w2 = F2 + G2 − c1 w2 + u15 ⎪ ⎪ ⎪ ⎪ q ⎪ ⎪ D x3 = a1 (z3 − x3 ) + u16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq y3 = a1 (v3 − y3 ) + u17 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq z3 = b1 x3 + δ (7x3 − x2 ) − z3 − P3 + u18 ⎪ ⎪ ⎪ ⎪ ⎪ q ⎪ ⎪D v3 = b1 y3 + δ (7y3 − y2 ) − v3 − Q3 + u19 ⎪ ⎪ ⎪ ⎩ q D w3 = F3 + G3 − c1 w3 + u20 where u1 , u2 , · · · , u20 are the adaptive controllers, a1 , b1 , c1 are the estimations of corresponding unknown parameters. Then, the synchronization error variables are defined as ⎧ ⎪ ⎪e1 = x0 − x0 , e2 = y0 − y0 , e3 = z0 − z0 , e4 = v0 − v0 , ⎪ ⎪ ⎪ ⎪ ⎪ e5 = w0 − w0 , e6 = x1 − x1 , e7 = y1 − y1 , e8 = z1 − z1 , ⎪ ⎪ ⎨ (21) e9 = v1 − v1 , e10 = w1 − w1 , e11 = x2 − x2 , e12 = y2 − y2 , ⎪ ⎪ ⎪ ⎪e = z − z , e = v − v , e = w − w , e = x − x , ⎪ 13 2 14 2 15 2 16 3 ⎪ 2 2 2 3 ⎪ ⎪ ⎪ ⎩e = y − y , e = z − z , e = v − v , e = w − w . 17 3 18 3 19 3 20 3 3 3 3 3


275

Meanwhile, ea = a1 − a, eb = b1 − b, ec = c1 − c are the estimation errors of unknown parameters. Subtracting the Eqs.(17) from Eqs.(20), we can get the error dynamical system, which is following ⎧ ⎪ Dq e1 = a1 (e3 − e1 ) + ea (z0 − x0 ) + u1 ⎪ ⎪ ⎪ ⎪ ⎪ q ⎪ D e2 = a1 (e4 − e2 ) + ea (v0 − y0 ) + u2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e3 = b1 e1 + eb x0 + δ (e1 − e6 ) − e3 − P0 + P0 + u3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e4 = b1 e2 + eb y0 + δ (e2 − e7 ) − e4 − Q0 + Q0 + u4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e5 = (F0 − F0 ) + (G0 − G0 ) − c1 e5 − ec w0 + u5 ⎪ ⎪ ⎪ ⎪ ⎪ q ⎪ ⎪ ⎪D e6 = a1 (e8 − e6 ) + ea (z1 − x1 ) + u6 ⎪ ⎪ ⎪ ⎪ ⎪Dq e7 = a1 (e9 − e7 ) + ea (v1 − y1 ) + u7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e8 = b1 e6 + eb x1 + δ (3e6 − 4e11 − e1 ) − e8 − P1 + P1 + u8 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e9 = b1 e7 + eb y1 + δ (3e7 − 4e12 − e2 ) − e9 − Q1 + Q1 + u9 ⎪ ⎪ ⎪ ⎪ ⎪ q ⎨ D e10 = (F1 − F1 ) + (G1 − G1 ) − c1 e10 − ec w1 + u10 (22) q e = a (e − e ) + e (z − x ) + u ⎪ ⎪ D 11 1 13 11 a 2 2 11 ⎪ ⎪ ⎪ ⎪ ⎪ q ⎪ D e12 = a1 (e14 − e12 ) + ea (v2 − y2 ) + u12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e13 = b1 e11 + eb x2 + δ (5e11 − 9e16 − e1 ) − e13 − P2 + P2 + u13 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e14 = b1 e12 + eb y2 + δ (5e12 − 9e17 − e2 ) − e14 − Q2 + Q2 + u14 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e15 = (F2 − F2 ) + (G2 − G2 ) − c1 e15 − ec w2 + u15 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e16 = a1 (e18 − e16 ) + ea (z3 − x3 ) + u16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e17 = a1 (e19 − e17 ) + ea (v3 − y3 ) + u17 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e18 = b1 e16 + eb x3 + δ (7e16 − e11 ) − e18 − P3 + P3 + u18 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dq e19 = b1 e17 + eb y3 + δ (7e17 − e12 ) − e19 − Q3 + Q3 + u19 ⎪ ⎪ ⎪ ⎪ ⎩Dq e = (F − F ) + (G − G ) − c e − e w + u . 20

3

3

3

3

1 20

c 3

20

Our goal is to design suitable controllers and unknown parameter estimation rules such that the synchronization between the drive and response systems can be realized. The following criterion is proposed to ensure the drive system (17) effectively synchronizes the response system (20). Theorem 3. The synchronization between the systems (17) and (20) is realized when the synchronization controllers are designed as following:

276


⎧ u1 = −a1 (e3 − e1 ) − e1 , u2 = −a1 (e4 − e2 ) − e2 ⎪ ⎪ ⎪ ⎪ ⎪ u3 = −b1 e1 − δ (e1 − e6 ) + P0 − P0 ⎪ ⎪ ⎪ ⎪ ⎪ u4 = −b1 e2 − δ (e2 − e7 ) + Q0 − Q0 ⎪ ⎪ ⎪ ⎪ ⎪ u5 = −(F0 − F0 ) − (G0 − G0 ) + c1 e5 − e5 ⎪ ⎪ ⎪ ⎪ ⎪ u6 = −a1 (e8 − e6 ) − e6 , u7 = −a1 (e9 − e7 ) − e7 ⎪ ⎪ ⎪ ⎪ ⎪ u8 = −b1 e6 − δ (3e6 − 4e11 − e1 ) + P1 − P1 ⎪ ⎪ ⎪ ⎪ ⎪ u9 = −b1 e7 − δ (3e7 − 4e12 − e2 ) + Q1 − Q1 ⎪ ⎪ ⎪ ⎨u = −(F − F ) − (G − G ) + c e − e 10 1 1 1 10 10 1 1 ⎪ u11 = −a1 (e13 − e11 ) − e11 , u12 = −a1 (e14 − e12 ) − e12 ⎪ ⎪ ⎪ ⎪ ⎪ u13 = −b1 e11 − δ (5e11 − 9e16 − e1 ) + P2 − P2 ⎪ ⎪ ⎪ ⎪ ⎪ u14 = −b1 e12 − δ (5e12 − 9e17 − e2 ) + Q2 − Q2 ⎪ ⎪ ⎪ ⎪ ⎪u15 = −(F2 − F2 ) − (G2 − G2 ) + c1 e15 − e15 ⎪ ⎪ ⎪ ⎪ ⎪u16 = −a1 (e18 − e16 ) − e16 , u17 = −a1 (e19 − e17 ) − e17 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪u18 = −b1 e16 − δ (7e16 − e11 ) − e18 + P3 − P3 − e18 ⎪ ⎪ ⎪ ⎪ ⎪u19 = −b1 e17 − δ (7e17 − e12 ) − e19 + Q3 − Q3 − e19 ⎪ ⎩ u20 = −(F3 − F3 ) − (G3 − G3 ) + c1 e20 − e20 , and the adaptive laws of the unknown parameter are designed as ⎧ ⎪ Dq ea = −(z0 − x0 )e1 − (v0 − y0 )e2 − (z1 − x1 )e6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪−(v1 − y1 )e7 − (z2 − x2 )e11 − (v2 − y2 )e12 ⎪ ⎪ ⎨−(z − x )e − (v − y )e 3 3 16 3 3 17 q e = −x e − y e − x e − y e − x e − y e ⎪ D 0 3 0 4 1 8 1 9 2 13 2 14 ⎪ b ⎪ ⎪ ⎪ ⎪ −x e − y e 3 18 3 19 ⎪ ⎪ ⎪ ⎩D q e = w e + w e + w e + w e c 0 5 1 10 2 15 3 20

(23)

(24)

proof: By substituting the synchronization controllers (23) into (22), then the error dynamical system can be rewritten as: ⎧ ⎪ Dq e1 = ea (z0 − x0 ) − e1 , Dq e2 = ea (v0 − y0 ) − e2 ⎪ ⎪ ⎪ ⎪ ⎪ Dq e3 = eb x0 − e3 , Dq e4 = −eb y0 − e4 ⎪ ⎪ ⎪ ⎪ ⎪Dq e5 = −ec w0 − e5 , Dq e6 = ea (z1 − x1 ) − e6 ⎪ ⎪ ⎪ ⎪ ⎪Dq e7 = ea (v1 − y1 ) − e7 , Dq e8 = eb x1 − e8 ⎪ ⎪ ⎪ ⎨Dq e = e y − e , Dq e = −e w − e 9 9 10 c 1 10 b 1 (25) q e = e (z − x ) − e , Dq e = e (v − y ) − e ⎪ D 11 a 2 2 11 12 a 2 2 12 ⎪ ⎪ ⎪ ⎪ ⎪ Dq e13 = eb x2 − e13 , Dq e14 = eb y2 − e14 ⎪ ⎪ ⎪ ⎪ ⎪ Dq e15 = −ec w2 − e15 , Dq e16 = ea (z3 − x3 ) − e16 ⎪ ⎪ ⎪ ⎪ ⎪ Dq e17 = ea (v3 − y3 ) − e17 , Dq e18 = eb x3 − e18 ⎪ ⎪ ⎪ ⎩Dq e = e y − e , Dq e = −e w − e . 19 19 20 c 3 20 b 3 Then, we can obtain [Dq e1 , Dq e2 , · · · , Dq e20 , Dq ea , Dq eb , Dq ec ]T = A[e1 , e2 , · · · , ea , eb , ec ]T ,

(26)


277

A1 A2 , A1 = diag[−1, −1, · · · , −1], A2 is the matrix of coefficients for the estimations of

A3 O 20 00 . unknown parameters, A3 = −A2 , and O = 00 Assume that λ is one of the eigenvalues of matrix A and the corresponding non-zero eigenvectors is ζ = (ζ1 , ζ2 , · · · , ζ23 )T , then we can get, (27) Aζ = λ ζ ,

where A =

and the following relation can be easily obtained when conjugate transpose H is taken on both sides of (27) H (Aζ )T = λ ζ . (28) Using (27) multiplied left by (1/2)ζ

H

plus (28) multiplied right by (1/2)ζ , we have

1 1 1 ζ H ( A + AH )ζ = (λ + λ )ζ H ζ . 2 2 2 Then,

H ζ 1 (λ + λ ) = 2

1

1 H 2A + 2A H

ζ ζ

ζ

.

(29)

(30)

By substituting the matrix A into (30), we obtain 1 1 H (λ + λ ) = H ζ Bζ , 2 ζ ζ

(31)

where A1 = diag[−1, −1, · · · , −1, 0, 0, 0] Since λ + λ ≤ 0, that is, any eigenvalue of matrix A satisfies the

23

following relationship:

π π > q, (0 < q < 1). (32) 2 2 According to the stability theory of fractional-order systems (Lemmas 1 and 2), then the equilibrium point of error dynamical system (25) is asymptotically stable. Therefore | arg(λ )| ≥

lim ||e(t)|| = 0,

t→∞

(33)

which means that the adaptive synchronization between the drive system (17) and response system (20) is realized. The proof is completed. In numerical simulations, the real values of the unknown parameters of the drive system (17) are taken as a = 10, b = 28, c = 8/3, the derivative order is taken as q = 0.995, and intensity of random parameter δ = 0.05 . The start point of the parameter estimations is (1, 2, 0.5). The simulation time is 80s and the time step is 0.001. The simulation results are shown in the Figs.2 and 3, from which it can be seen that all the error variables converge to zero and the parameter estimations converge to the real values of the unknown parameters as t → 80s, which implies the drive and response systems are synchronized and the unknown parameters are identified under the controllers and adaptive laws for the unknown parameters. 6 Conclusion In this paper, the adaptive synchronization of the fractional-order complex Lorenz system with a random parameter is analyzed. Firstly, according to the approximation principle of Laguerre orthogonal polynomial method, the equivalent deterministic system is obtained. Besides, based on the stability


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60 50

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theory of fractional-order systems, the synchronization for the deterministic system with unknown parameters is realized by designing appropriate synchronization controllers and estimation laws for uncertain parameters. Numerical simulations are carried out to verify the effectiveness and feasibility of the proposed scheme.


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References [1] Li, C.P., Deng, W.H and Xu, D. (2006) Chaos synchronization of the chua system with a fractional order.Physica A, 360, 171–185. [2] Ge, Z.M. and Ou, C.Y. (2005) Chaos in a fractional order modified duffing system. In: Proc ECCTD. Budapest, 1259–1262. [3] Lu, J.G. and Chen, G.R. (2006) A note on the fractional-order chen system. Chaos Solitons & Fractals, 27, 685-688. [4] Ma, S.J., Xu, W. and Fang, T. (2008) Analysis of period-doubling bifurcation in double-well stochastic duffing system via laguerre polynomial approximation. Nonlinear Dynamics,52, 289–299. [5] Ma, S.J., Shen, Q. and Hou, J. (2013) Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters. Nonlinear Dynamics, 73, 93–100. [6] Tavazoei, M. S. and Haeri, M. (2007) A necessary condition for double scroll attractor existence in fractionalorder systems. Physics Letters A, 367, 102–113. [7] Jia, H.Y. Chen, Z.Q and W. Xue. (2013) Analysis and circuit mplementation for the fractional-order lorenz system., ACTA PHYSICA SINICA, 62, 140503. [8] Moghtadaei, M. and Hashemi Golpayegani, M.R. (2012) Complex dynamic behaviors of the complex lorenz system. Scientia Iranica, 19, 733–738. [9] Wang, X.Y. and Zhang, H. (2013) Bivariate module-phase synchronization of a fractional-order lorenz system in different dimensions. Journal of Computation and Nonlinear Dynamics, 8, 031017. [10] Luo, C. and Wang, X.Y. (2013) Chaos in the fractional-order complex lorenz system and its synchronization. Nonlinear dynamics, 71, 241–257. [11] Diethelm,K., Ford, N.J. and Freed, A.D. (2002), A predictor-corrector approach for the numerical solution of fractional diffrential equations. Nonlinear Dynamics, 29, 3–22.


Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Nonlinear Dynamic Characteristic Analysis of Planetary Gear Transmission System for the Wind Turbine Zhao-Hui Ren1†, Liang Fu1 , Ying-Juan Liu2 , Shi-Hua Zhou1 , Bang-Chun Wen1 1 School 2 Scholl

of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110004, China of Qian’an, Hebei United University, Tangshan, 064400, China Submission Info

Communicated by Jiazhong Zhang Received 22 January 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Planetary gear train Wind turbine Dynamic response Coupled nonlinearity Dynamic model

Abstract A compared with the translational-torsional dynamic model of planetary gear transmission system (PGTS) used in wind turbine is established in order to analyze the dynamic characteristics of the PGTS more accurately. The influences of the meshing stiffness, input and output torques, meshing error, nonlinear characteristics of the support bearing and gravity are considered in the model. Based on previous model, the vibration differential equations of the drive-train are obtained through the Lagrange’s equation. The dynamic response characteristics are investigated using the Runge–Kutta numerical method, and the factors of the above proposed excitation are analyzed. The results show clearly that the vibration responses have different characteristics due to the different speeds of each component in the PGTS. The nonlinear behaviour of support bearing causes the dynamic response more complicated In addition, under the internal and external excitations, more frequency multiplication and frequency combination components appear. The vibration frequency components of the system are mainly concentrated in the frequency range below 200Hz. The results of the study in this paper can provide necessary theoretical basis for natural characteristics study, dynamic response and optimization design method of the MW wind turbine PGTS. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction With the rapid development of rotating machinery, it is well known that the PGTS is effective power transmission system where the high torque to weight ratios, large speed reductions in compact co-axial shaft arrangement, high reliability and superior efficiency are required. As one of the more effective configurations used in wind turbines, automotive aerospace, marine and industrial power trains, in addition, the transmission system can be also found in passenger car, helicopter drive trains. However, due to the structure and the complex working environment of the PGTS, the vibration and noise characteristics are particularly outstanding, which can affect the running accuracy and the transfer † Corresponding



282


efficiency of the devices. To achieve a high reliability model that reduces the vibration for a high performance machine, the further research to the dynamic characteristics of the PGTS is required, and then it also has important theoretical significance and engineering application value [1-2]. At present, a requirement for reliable planetary gear system design calculations is sufficient insight in the dynamics. So the accurate model for practical applications is the main task, which can precisely study the natural characteristic, the nonlinear modal analysis, the dynamic response, the dynamic load, load sharing and nonlinear characteristics of the PGTS. Compared with the general PGTS, the wind power planetary gear has a larger mass, volume and a relatively low speed, which is belonging to low-speed and heavy load gear transmission system, due to the wind turbine PGTS usually works in adverse operation circumstances, which makes the planetary gear bear the varying load, impulse loads and disturbance caused by the mutation of load, and even seriously affects the working performance of the whole system. A number of models were proposed in past to describes the dynamic behavior of the PGTS, and also achieved some remarkable achievements [3-8]. Planetary gear researchers have developed lumped-parameter model (pure torsion model and translational-torsional model) and the finite element model to analyze dynamic characteristics of the PGTS. August [9] established a pure torsional model of PGTS by accounting the effects of input/output, fixed, semi-floating, and fully-floating sun gear conditions, which laid a foundations for further study of the PGTS. Parker [10-12] based on the physical forces acting at the sun-planet and ring-planet meshes, using the well-defined properties of planetary gear vibration modes, the boundaries separating stable and unstable conditions are obtained as simple expressions in terms of mesh parameters. Tooth separation from parametric instability is numerically simulated to show the strong impact of this non-linearity on the response. Kahraman [13-16] set a linear, time-invariant model of double-helical planetary gears, and the torsional, transverse, axial and rotational (rocking) motions of gears and the planet carrier were included in these three-dimensional models. Based on the predictions, the natural modes are classified into three groups: a rigid body mode, asymmetric planet modes and x-symmetric overall modes. Song and Zhang [17-18] established a translational-rotational coupling model of planetary gear train to investigate the natural frequencies and corresponding vibration modes, which are classified into three types, i.e., rotational modes, translational modes, and planetary gear modes. Lin and Parker [19-20] based on gyroscopic effects and time-varying stiffness, established an analytical model of planetary gears and studied the response. The unique characteristics of each type of model were analytically investigated in detail. Yang [21] proposed a dynamic model of the planetary gear system and obtained the differential equation. Taking a typical MW PGTS of wind turbine as an example, the dynamic characteristics were simulated and analyzed by considering the influences of the internal and external incentives on dynamic load and dynamic factor of system. Sun [22] and Qin [23] established a torsional-lateral nonlinear vibration model, and the numerical simulation method was used to solve the dynamic response of PGTS. In addition, the time-varying mesh stiffness, gear meshing errors and gear backlashes on the dynamic characteristics of the system were analyzed. Yang[24] analyzed the helical planetary gear trains including the gyroscopic, time varying stiffness, gear run-out errors, mass eccentricity and gear profile errors, etc. The characterization of helical planetary gear free vibration was studied. Parker and Agashe [25] set a spring-quality model of the planetary gear train and the dynamic response of a helicopter PGTS was examined over a wide range of operating speeds and torques. The calculated response shows classical resonances when a harmonic of mesh frequency coincides with a natural frequency. Jan [26] focused on the gearbox model behavior assessment by means of three more complex modeling techniques of varying complexity: the purely torsional, rigid six degree of freedom with discrete flexibility and flexible multi-body technique. Both simulation and experimental results were discussed. Vijaya [27] and Cheon [28] used two models: a lumped-parameter model and a finite element model to analyze the dynamic response of planetary gear. Mesh stiffness variation excitation, corner contact, and gear tooth contact loss are all intrinsically considered


283

in the finite element analysis, and the dynamics of planetary gears show a rich spectrum of nonlinear phenomena. A model was presented which enabled the simulation of the three-dimensional dynamic behavior of planetary/epicyclical spur and helical gears by Abousleiman [29]. The corresponding equations of motion were solved by combining a time-step integration scheme and a contact algorithm for all simultaneous meshes, and the result illustrated the potential of the proposed hybrid model and the interest of taking into account ring-gear deflections. Chaari [30] and Al-shyyab [31] investigated a plane model of a planetary gear and the influence of gyroscopic in particular was scrutinized by considering time-varying the gear mesh stiffness. Kiracofe [32] extended previous analytical models of simple, single-stage planetary gears to compound, multi-stage planetary gears, which were used to investigate the structured vibration mode and natural frequency properties of compound planetary gears of general description, including those with equally spaced planets and diametrically opposed planet pairs. It can be seen that the references made summaries, comparison, and researches on the dynamic characteristics of the planetary gear in the past. A compared with the translational-torsional dynamic model of the PGTS for wind turbine which considers the external load excitation, meshing stiffness, input and output torques, meshing error, nonlinear characteristics of the support bearing and gravity is rare. In order to analyze the dynamic behaviors of the PGTS more accurately, it is necessary to establish accurate dynamic model of the PGTS. Accurate analytical modeling, including mesh relationships and detailed characterization of the nonlinear dynamics of the PGTS, is needed to estimate planetary gear vibration and predict dynamic forces in industrial applications. Little work has been done to characterize the nonlinear effects of the support bearing and gravity on planetary gear dynamics. The lack of experiment studies to understand the complex dynamics of the PGTS. In this paper, characterize the complex, nonlinear dynamics of planetary gear transmission system using lumped-parameter method propose an analytical model which is systematically studied the vibration characteristics of the PGTS.

2 Modeling of planetary gear dynamics 2.1

The coordinate transformation

Due to exist a number of movable components in the PGTS, the carrier-attached coordinate system in this paper, as shown in Fig. 1, is the reference coordinate system in planetary gear system in order to express easily the relative motion relationships of components [33].

Fig. 1 Carrier-attached coordinate system.

In Fig. 1 the fixed-coordinate system OXY is set up. Oxy represents the motional-coordinate system of the planet carrier. Assuming that the O is the fixed point, and xi -axis points to the center of the planet gear. In the plane, the yi -axis is obtained through the x-axis rotation by 90 degrees

284


counterclockwise. The ideal angular velocity of the carrier is ωc . The θc (θc = ωct) is the rotation angle of the carrier with the motional-coordinate system (Oxy) relative to the X -axis at any t moment. The r i indicates the centroid displacement vector of the planet gear in the PGTS. And the displacement vector can be expressed by the following equation: r i = xi u + yi v ,

(1)

where xi , yi represent the lateral displacement in x-direction and y-direction, u , v are the displacement vector in x-direction and y-direction. By taking the derivative and simplifying the Eq. (1), utilizing the relationship θc = ωct, and finally the equation of the displacement vector can be expressed by: r¨ i = (xï − 2ωc y˙i − ωc2 xi )uu + (yï − 2ωc x˙i − ωc2 yi )vv. 2.2

(2)

Lumped-parameter analytical model of the PGTS

The MW wind turbine PGTS that will be studied in this paper, as shown in Fig. 2, is a lumpedparameter analytical model that is a multi-mesh gear train. Fig. 2 (a) is the structure diagram of planetary gear where the planet carrier is regarded as the input terminal, and the sun gear is the output terminal. The gear mesh is modeled as linear spring acting along the pressure line direction. The other supports (bearings) are modeled as nonlinear springs. The friction forces due to gear teeth contact and other dissipative effects are captured using modal damping. Therefore the translationalrotational model of the planetary gear system is shown in Fig. 2 (b).

Fig. 2 Sketch and dynamics model of PGTS.

In Fig. 2(b), the OXY is the fixed-coordinate system. Oxy represents the motional-coordinate system of the planet carrier. The Oxi points to the theory center of the planet gear, and the ideal angular velocity of the carrier is ωc . s, r, c, pi are the subscripts of the sun, ring, carrier and planet respectively, and the subscripts from 1 to i(i = 1, 2, 3) designate the planets. x j , y j ( j = s, r, c) represent the lateral displacements (the projections in the fixed coordinate system OXY). x pi , y pi (i = 1, 2, 3) indicate the vibration displacement projections of the ith planet gear in o pi x pi y pi coordinate system.


285

rb j ( j = s, r, c, pi ) is the base radius of gears or radius of the carries. θ j ( j = s, r, c, pi ) represents the rotational displacement of the sun, ring, carrier and planet respectively. Besides, it is assumed that the counterclockwise is the positive direction. ϕi is the circumferential position of the ith planet around the sun gear and ϕi = 2π (i − 1)/n, α is the sun-planet and ring-planet operating pressure angles. ks j , cs j ( j = s,t, b) are the bending , torsional of sun gear and the bearing stiffness and damping in x-direction and y-direction respectively, kr j , cr j ( j = s,t) represent the bending of ring and the bearing stiffness and damping in x-direction and y-direction, kc j , cc j ( j = s,t, b) indicate the bending, torsional of the carrier and the bearing stiffness and damping in x-direction and y-direction respectively, kbpi cbpi are the ith bearing stiffness and damping of planet gear in lateral-direction, k pis c pis , k pir c pir are the mesh stiffness and damping of sun-planet and ring-planet in the pressure line direction. 2.3

The rolling bearing analytical model

Figure 3 is the schematic of rolling bearing [34]. Assuming that the bearing outer ring is fixed on bearing seat and the shaft rotates with the bearing inner ring together. What’s more, it is assumed that the balls are uniform distribution between the inner ring and outer ring. So the linear velocities v1 and vo of the contact points between the ball and inner/outer ring can be expressed by: vi = ωi · r,

vo = ωo · R

(3)

where R and r represent the radius of outer and inner rings respectively, ω1 and ωo are the angular velocities, vi and v0 indicate the linear velocities of the contacting point between ball and outer/inner ring respectively.

Fig. 3 The rolling bearing analytical model

Assuming that it is the pure rolling between the balls and bearing inner/outer ring and the revolving angular velocity of cage is equal to the revolving angular velocity of the ball. So: vb = (vo + vi )/2 = (ωo R + ωir)/2.

(4)

By taking the derivative and simplifying the Eq. (4), utilizing the relationships ω0 = 0, ωi = ω , and finally, the angular velocity of cage can be expressed as follows:

ωb = 2vb /(R + r) = ωi · r/(R + r).

(5)

θi is the circumferential angle of the ith planet gear. So: θi = ωb · t + 2π(i − 1)/Nb ,

(i = 1, 2, 3, . . . , Nb ),

(6)

286


where Nb represents the number of rolling ball. It is assumed that the x and y represent linear vibration displacements of ring in X −direction andY direction, and γ0 is the bearing clearance. The normal contacting deformation between ith rolling ball and bearing rings can be expressed by:

δi = x cos θi + y sin θi − γ0 .

(7)

According to nonlinear hertz contact theory, the contact pressure between the ith rolling ball and bearing inner and outer rings is fi , and considering the normal stress can only be generated between rolling ball and bearing rings, so when δi > 0, the force can be expressed: fi = kb (δi )3/2 = kb (x cos θi + y sin θi − γ0 )3/2 ·H(x cos θi + y sin θi − γ0 ),

(8)

where kb is the hertz contact stiffness, H(x) represents Heaviside function. Therefore, the non-linear bearing force of the rolling ball bearing Fbx and Fby in x-direction and y-direction can be described as follows: F=

Nb

Nb

i=1

i=1

∑ fi = ∑ [kb (x cos θi + y sin θi − γ0)3/2

·H(x cos θi + y sin θi − γ0 ) cos θi , Fx =

Nb

∑ fix = F cos θi ,

i=1

2.4

(9)

Nb

Fy = ∑ fiy = F sin θi . i=1

The relative displacement analysis of the components in the PGTS

The force analysis of each component for the PGTS is shown in Fig. 4

Fig. 4 Force analysis of planetary gear train.

(10)


287

1. The elastic deformation of sun-planet in the pressure line direction. xs , ys are the linear displacements of the sun gear in x-direction and y-direction, and the θs rbs is the circumferential angular displacement. xs sin ϕsi , −ys cos ϕsi and −θs rbs are the projections of sun-planet along with the pressure line. x pi , y pi are the linear displacement of the planet gear in x-direction and y-direction, and the θ pi rbpi is the circumferential angular displacement. x pi sin α , −y p cos α and −θ pi rbpi are the projections of sun-planet along with the pressure line. So the compressive deflections in the sun-planet mesh spring can be expressed as follows:

δ pis = xs sin ϕsi − ys cos ϕsi − rbs θs + x pi sin α + y pi cos α − rbpi θ pi , ϕsi = ϕi − α .

(11)

2. The elastic deformation of ring-planet in the pressure line direction. xr , yr are the linear displacement of the ring in x-direction and y-direction, and the θr rbr is the circumferential angular displacement. xr sin ϕri , −yr cos ϕri and −θr rbr are the projections of ring-planet along with the pressure line. x pi , y pi are the linear displacement of the planet gear in x-direction and y-direction, and the θ pi rbpi is the circumferential angular displacement. −x pi sin α , y pi cos α and θ pi rbpi are the projections of ring-planet along with the pressure line. So the compressive deflections in the ring-planet mesh spring can be expressed as follows:

δ pir = xr sin ϕri − yr cos ϕri − rbr θr − x pi sin α + y pi cos α + rbpi θ pi , ϕri = ϕi + α .

(12)

3. The relative elastic deformation between planet gear and carrier. δ pix and δ piy are the relative displacement between planet gear and carrier in the oxi yi coordinate system, the elastic deformations are:

δ pix = x pi − xc cos ϕi − yc sin ϕi , δ piy = y pi + xc sin ϕi − yc cos ϕi − rbc θc . 2.5

(13)

Vibration differential equations of the planetary gear transmission system

It is assumed that the meshing stiffness of the sun-planet and the ring-planet are constant, and ignore the meshing backlash, friction and the change at meshing position. The meshing force along with the pressure line acts on the centre of tooth width. Based on the dynamic model of the PGTS in Fig.2, a generalized displacement vector of the planetary gear train can be defined with respect to the global co-ordinate system by: X = [xc yc θc xbc ybc xr yr θr x p1 y p1 θ p1 x p2 y p2 θ p2 x p3 y p3 θ p3 xs ys θs xbs ybs ]T . (14) The vibration differential equation of the carrier: 3

mc x¨c + 2ccs (x˙c − x˙bc ) − ∑ c pi (δ˙pix cos ϕi − δ˙ pix cos ϕi ) i=1

3

+2kcs (xc − xbc ) − ∑ k pi (δ pix cos ϕi − δ pix cos ϕi ) = Fbcx , i=1

3

mc y¨c + 2ccs (y˙c − y˙bc ) − ∑ c pi (δ˙pix sin ϕi + δ˙ pix cos ϕi ) i=1

3

+2kcs (yc − ybc ) − ∑ (k pi δ pix sin ϕi + k pi δ pix cos ϕi ) = Fbcy − mc g, i=1 3

3

2 ˙ 2 θc − ∑ c piy rbc δ˙piy + kct rbc θc − ∑ k piy rbc δ piy = −Td . Jc θ¨c + cct rbc i=1

i=1

(15)


288

The vibration differential equation of the ring gear: 3

3

i=1 3

i=1 3

mr x¨r + cr x˙r − ∑ c pir δ˙pir sin(ϕi + α ) + kr xr − ∑ k pir δ pir sin(ϕi + α ) = 0, mr y¨r + cr y˙r + ∑ c pir δ˙pir cos(ϕi + α ) + kr yr + ∑ k pir δ pir cos(ϕi + α ) = −mr g, i=1

(16)

i=1

3

3

i=1

i=1

2 ˙ 2 θr − ∑ c pir rbr δ˙pir + krt rbr θr − ∑ k pir rbr δ pir = 0. Jr θ¨r + crt rbr

The vibration differential equation of the ith planet gear: m pi (x¨pi − 2ωc y˙pi − ωc2 x pi ) + c pi δ˙pix + (c pis δ˙pis − c pir δ˙pir ) sin α + k pi δ pix + (k pis δ pis − k pir δ pir ) sin α = 0, ms (y¨pi + 2ωc x˙pi − ωc2 y pi ) + c pi δ˙piy + (c pis δ˙pis + c pir δ˙pir ) cos α + k pi δ piy + (k pis δ pis + k pir δ pir ) cos α = −ms g, (17) Jpi θ¨ pi − c pis rbpi δ˙pis + c pir rbpi δ˙pir − k pis rbpi δ pis + k pir rbpi δ pir = 0. The vibration differential equation of the sun gear: 3

3

i=1 3

i=1 3

ms x¨s − ∑ c pis δ˙pis sin(ϕi − α ) − ∑ k pis δ pis sin(ϕi − α ) = Fbsx , ms y¨s + ∑ c pis δ˙pis cos(ϕi − α ) + ∑ k pis δ pis cos(ϕi − α ) = Fbsy − ms g, i=1

(18)

i=1

3

3

i=1

i=1

2 ˙ 2 θs − ∑ c pis rbs δ˙pis + kst rbs θs − ∑ k pis rbs δ pis = Tl . Js θ¨s + cst rbs

3 Nonlinear dynamic response of planetary gear system From the previous conclusions and results, it can be seen that the vibration characteristics of nonlinear PGTS, supported by rolling bearing, become more complicated due to the gear meshing, the coupled translational-rotational vibration and the adverse operation circumstances. Therefore it is necessary to give a detailed analysis of the PGTS. The dynamic behaviors of system are investigated by RungeKutta numerical simulation method. On this basis, the planetary gear geometry parameters of the system analyzed in this paper are given in table 1. Table 1 Parameters of planetary gear drive train system Parameters

Carrier

Ring gear

Planet gear

Sun gear

mc

2125.56

mr

5236.78

m pi

157.24

ms

538.36

Jc

146.12

Jr

1422.04

Jpi

5.65

Js

19.72

rc

0.3256

rr

0.5375

r pi

0.2119

rs

0.1137

Number of teeth

zr

104

z pi

41

zs

22

Bearing stiff.(N/m)

kc

3.35 × 1010

kr

2.62 × 1010

k pi

4.54 × 109

ks

5.36 × 109

Torsional stiff.(N/m)

kct

8.55 × 108

krt

5.42 × 108

k pit

6.2 × 108

kst

8.4 × 108

Module (mm)

m

12 6.87 × 109

k pis

6.08 × 109

Mass (kg) Inertia

(kg.m2 )

Base circle radius (mm)

k pir

Mesh stiff.(N/m) Pressure angle (deg)

22.5


3.1

289

Dynamic response of the planetary gear

By applying the coupled translational-rotational vibration model of a PGTS, the vibration analysis and calculations are carried out. In this paper, an example of the MW wind turbine planetary gear transmission is considered here to study the accuracy of the vibration response by Runge-Kutta numerical simulation of the model to conclude the key influence factors including the external load excitation, meshing stiffness, input and output torque, meshing error, nonlinear characteristics of the support bearing and gravity. The coupled analyses of the PGTS were performed so as to include all the components of influence factors. Fig. 5 ∼ Fig. 8 and Fig. 11 ∼ Fig. 13 show the vibration response of the each component in x, y, θ directions, in which the carrier driver, having a rated speed of 12r.p.m, drives the planetary to rotate. The response curves (vibration waveform and spectrum) of the component are made appropriate interception in order to analyze the dynamic characteristics of the wind turbine PGTS more accurately. Fig. 5 ∼ Fig. 8 show the time-domain waveform of each component in lateral-direction and torsionaldirection. The negative value in figure indicates the opposite direction with respect to the predetermined positive direction in the model. It can be seen from Fig. 5 ∼ Fig. 8 that the vibration waveform of each component in the train system presents a fluctuated changing trend. Fig. 5 shows the time domain response of the carrier in x, y, θ directions. It can be seen that the fluctuation in x-direction is significantly higher than that in y-direction, nevertheless, the amplitude of vibration displacement in y direction far higher than that in x-direction. In addition, the high-harmonic components appear in x-direction and y-direction. The time domain responses of the ring gear in x, y, θ directions are shown in Fig.6, which can be seen that the fluctuation in x-direction is smoother to some extent than that in y-direction. Besides, the clap-vibration phenomenon slightly appears in y-direction due to the influence of the internal and external excitations. Fig. 7 shows the vibration response of the one of planet gears due to the three planets displaying a symmetric distribution. It can be seen that the fluctuation is significant lower than others, but the amplitude of vibration displacement are almost 2 times of the carrier. Compared Fig. 7 ∼ Fig. 8 with Fig. 5 ∼ Fig. 6, due to the effects of input and output torques the translational amplitudes of the vibration displacements are lower than the rotational amplitudes in the transmission system, which is affected slightly by external excitation. With the increasing of the

Fig. 5

Fig. 6 Vibration waveform of the ring gear in x, y, θ directions.

290


Fig. 7 Vibration waveform of the planet gear in x, y, θ directions.

Fig. 8 Vibration waveform of the sun gear in x, y, θ directions.

rotating speed and the effect of the mesh gears, the high-harmonic components of the planet gear and sun gear become significant increasing, and then the transverse displacement fluctuation trends of the sun gear and planet gear in x, y directions are significant larger than others, in addition, the vibration amplitudes are also obviously increasing and the magnitude of amplitude reach the 0.1 mm. It shows that due to the effects of the internal and external excitations, the support bearings take dramatic changes in their work to bear the moving load, which makes the dynamic characteristics of the PGTS become more complex. Fig. 5(c)-Fig. 8(c) show the time-domain waveform of each component in torsional-direction. Through comparing with the waveform, it can be found that the vibration displacement of ring gear reaches the minimum, and the vibration amplitude of the sun gear takes second place (it is consistent with previous analysis results). It is caused by the support bearing and the coupled translational-rotational vibration. In addition to the influence of gear meshes, the support bearings also have a significant effect on the dynamic behavior of the PGTS. In fact, bearing is one of main factor controlling vibration in a power transmission system because they play a key role in determining the system’s natural frequencies, and they can also be responsible for vibration amplification and transmission to other parts. Hence, the effects of bearing must be included, especially in a complicated power transmission system. Fig. 9 shows the axis orbits of support bearings at the carrier, planet and sun gearlocations. It can be seen from Fig.9 that the axis orbit show very complex characteristics, and have no rules of the each component. 3.2

Frequency response characteristics of the planetary gear

Because the PGTS is a continuous transmission and the mesh between the gears obtains the same period. At the same time, it can be found that the planet-ring and planet-sun have the same meshing frequency by applying these transmission relations in the PGTS. So the meshing frequency can be expressed by the follows [36]:

fm =

fs zs zr = fc zr , zs + zr

(19)


291

Fig. 9 Axis orbit of support bearing.

where fm , fs and fc are the meshing frequency, rotational frequency of the sun gear and the planet gear respectively; zr and zs are the number of teeth of the planet gear and sun gear. Corresponding to the waveform of each component in planetary gear train system, Fig. 10 ∼ Fig. 13 show the frequency domain response of the each component in lateral-direction and torsional-direction. It can be seen that the frequency domain response appear a lot of frequency multiplication and frequency combination components. For example, the frequency combination of the 53.25 Hz is composed of 2 fm (the meshing frequency fm = fc *zr = 20.8 Hz) and the bearing variable stiffness frequency f ps ( f ps = 11.6 Hz). The frequency combination of the 69.73Hz is composed of 2 fm , a half-times the bearing variable stiffness frequency of the carrier f pc ( f pc = 11.7 Hz) and a half-times the bearing variable stiffness frequency of the planet f pp ( f pp = 11.6 Hz). The frequency domain response of the sun gear and planet gear are shown in Fig. 11 ∼ Fig. 12. Some response frequencies of the system are concentrated distribution at both sides of certain frequencies which show the nonlinear characteristics. For instance, the frequency components of 70.15 Hz, 78.18 Hz, 86.21 Hz, 105 Hz, 106 Hz, 131.4 Hz, 167 Hz, and 189 Hz are intensive at both sides of the 127 Hz. The results indicate that the vibration frequency of the wind power planetary gear transmission system has a very obvious probability distribution, which is mainly concentrated in the frequency range below 200 Hz. This phenomenon is caused by the nonlinear characteristics of the support bearing.

Fig. 10 Spectrogram of the carrier in x, y, θ directions.

Fig. 12 Spectrogram of the planet gear in x, y, θ directions.

292


Fig. 13 Spectrogram of the sun gear in x, y, θ directions.

It can be seen from the previous analysis of the planetary gear transmission that the each component shows various characteristics due to the influences on the characteristics of assembly, the gear geometry parameters and the non-linear characteristics of the support bearing. Because the wind turbine PGTS is an overloaded gear transmission structure, the vibration feature of each component becomes more complicated by the effects of internal and external loads. In this paper, the dynamic response analysis of the PGTS for wind power can reveal the actual project and causes to damage of the support bearing. The results of the study in this paper can prove necessary theoretical of natural characteristics, dynamic response and optimization design method of the MW wind power planetary gear transmission with favorable dynamic behaviors

4 Conclusions In this paper, a lumped-parameter mathematical model is used to more precisely analyze the nonlinear dynamic model of coupled vibration with translation and torsion for the PGTS of wind turbine. The main conclusions are as follows: (1) Based on Lagrange equation, using the lumped-parameter method establishes a planetary gear dynamic model by considering the effects of internal and external excitation. The Runge-Kutta numerical method is used to solve the dynamic response of each component and get the frequency response characteristics, which has a certain foundation of the reducing vibration and structural optimization for the wind power planetary gear system. (2) Under the effects of internal and external excitation, the low-frequency, high-frequency and frequency combination components appear significantly in the vibration response. In addition, with the changing of rotational speed, the coupling between the lateral and torsional displacements has different vibration amplitudes. (3) The vibration frequency of the wind PGTS has a very obvious probability distribution, which is mainly concentrated in the frequency range below 200Hz. This phenomenon is caused by the nonlinear characteristics of the support bearing, which make the features of planet gear, sun gear and other components become more complicated. Hence, the effects of support bearing must be included, especially in a complicated power transmission system.

Acknowledgment The authors gratefully acknowledge the financial support provided by Natural Science Foundation of China (No. 51475084).


293

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Real-time 2D Concentration Measurement of CH4 in Oscillating Flames Using CT Tunable Diode Laser Absorption Spectroscopy Takahiro Kamimoto1 , Yoshihiro Deguchi1†, Ning Zhang1,2 , Ryosuke Nakao1, Taku Takagi1 , Jia-Zhong Zhang2 1 Graduate

School of Advanced Technology and Science, The University of Tokushima, Tokushima, 770-8506, Japan 2 School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China Submission Info Communicated by J.A.T. Machado Received 12 January 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Detection systems Diagnosis Gas turbines Transport properties Combustion oscillation Computed tomography Tunable diode laser absorption spectroscopy

Abstract One of the major problems of gas turbine combustors is the combustion oscillation. The combustion oscillation of gas turbines has many complex causes such as pressure fluctuations, combustion instabilities, and mechanical designs of the combustion chamber. Although the significant research efforts have been dedicated to this topic, the combustion oscillation problems have not yet been solved because of its complexity and nonlinearity. In this study, the theoretical and experimental research has been conducted in order to develop the noncontact and fast response 2D CH4 distribution measurement method to elucidate nonlinear combustion oscillation problems. The method is based on a computed tomography (CT) method using tunable diode laser absorption spectroscopy (TDLAS). The CT-TDLAS method was applied to oscillating flames and the time resolved 2D CH4 concentration distributions were successfully measured using 16 path CT-TDLAS measurement cell. CT-TDLAS has the kHz response time and the method enables the real-time 2D species concentration measurement to be applicable to the nonlinear phenomena of combustion oscillation problems in gas turbines. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction It is necessary to make efforts to protect natural ecosystems and effectively utilize fossil fuels in various fields. Combustion is widely used for energy conversion technique in the world. Measurements of combustion parameters such as temperature and species concentration are important to understand the combustion structure, improve combustion efficiency and reduce the production of pollutants such as NOx, CO, and particles. Especially, 2D temperature and concentration distributions play an important role for the combustion structure and the combustor efficiency in engines, burners, gas turbine combustors and so on. Among them gas turbine combustors have played a more important role since † Corresponding



Takahiro Kamimoto et al. /Journal of Applied Nonlinear Dynamics 4(3) (2015) 295–303

296

the Fukushima Daiichi nuclear disaster on 11 March 2011, because they have better energy conversion efficiency by employing natural gas combined cycles. One of the major problems of gas turbine combustors is the combustion oscillation. The combustion oscillation of gas turbines has many complex causes such as pressure fluctuations, combustion instabilities, and mechanical designs of the combustion chamber [1]. Elimination of combustion oscillations is often time-consuming and costly because there is no single approach to solve this problem. Although the significant research efforts have been dedicated to this topic, the combustion oscillation problems have not yet been solved because of its complexity and nonlinearity. It is also difficult to predict this phenomenon using CFD except for some specific problems. The complexity and nonlinearity of this phenomenon occur partly due to the complicated interaction between flame chemistries, turbulent flow structures and acoustic modes of combustors. Recently, laser diagnostics such as tunable diode laser absorption spectroscopy (TDLAS) [2-4] and laser induced fluorescence (LIF) [5, 6] has been developed and intensively applied to the gas turbine combustors [7]. With these engineering developments, transient phenomena of combustion phenomena in gas turbines have been gradually elucidated in various conditions. However, time resolved 2D temperature and concentration measurements in combustors have still been a challenging problem. In this study, the theoretical and experimental research has been conducted in order to develop the noncontact and fast response 2D temperature and concentration distribution measurement method. The method is based on a computed tomography (CT) method using tunable diode laser absorption spectroscopy[9-17]. The CT-TDLAS method was applied to oscillating flames to measure time resolved 2D CH4 concentration distributions. The temporal and spatial resolutions of this method have also been discussed to demonstrate its applicability to the combustion oscillation problem of gas turbine combustors.

2 Theory 2.1

Absorption spectroscopy

Gas temperature and species concentration can be determined by measuring molecular absorbance at multiple wavelengths. Tunable diode laser absorption spectroscopy was used in this research. It is possible to continuously scan laser wavelengths and measure absorption spectra. The principle of TDLAS is based on Lambert Beer’s law. When light permeates an absorption medium, the strength of the permeated light is related to absorber concentration according to Lambert Beer’s law. TDLAS uses this basic law to measure temperature and species concentration. The number density of the measured species n is related to the amount of light absorbed as in the following formula [7, 8]. Iλ /Iλ 0 = exp{−Aλ } = exp{− ∑(ni L ∑ Si, j (T )GVi, j )}. i

(1)

j

Here, Iλ 0 is the incident light intensity, Iλ is the transmitted light intensity, Aλ is the absorbance, ni is the number density of species i, L is the path length, Si, j is the temperature dependent absorption line strength of the absorption line j, and Gvi, j is the line broadening function. A CH4 absorption spectrum at 1635.34 nm is shown in Fig. 1(a). According to Eq. (1), the absorbance bear a proportionate relationship to the number density of species (CH4 ) has a linear relation and it has been also confirmed experimentally as shown in Fig. 1(b). This relation is used to measure 2D concentration distribution using a CT algorism. 2.2

CT-TDLAS

Absorption of transmitted light through absorption medium occurs on the optical path. The absorption signal strength becomes an integrated value of the optical path. In this study, several optical paths


297

Fig. 1 Proportionate relationship between the methane concentration and the absorption intensity.

Fig. 2 CT grid and laser path.

are intersected to each other to form analysis grids, reconstructing the 2D temperature distribution by a computed tomography method. Concept of analysis grids and laser beam paths is shown in Fig. 2. The integrated absorbance in the path p is given by [9-11]. Aλ ,p = ∑ nq L p,q αλ ,q .

(2)

q

CH4 concentration at each analysis grid was determined using a multifunction minimization method to minimize the spectral fitting error. Error = ∑{(Aλ ,q )theory − (Aλ ,q )experimet }2 .

(3)

The HITRAN database [18] was used to calculate theoretical values. A polynomial noise reduction technique [2] was also used to reduce noises.

3 Experimental setup Fig. 3 shows the outline of a CT-TDLAS measurement system used in this study. A DFB laser (NTT Electronics Co., NLK1U5EAAA) at 1635 mm was used to measure CH4 concentration. The laser wavelength was scanned at 4 kHz and absorption spectra of CH4 were measured. 10 spectra (2.5 ms time resolution) were averaged to calculate the instant 2D CH4 concentration. The diameter of 16 path measurement cell was 70 mm. A laser beam was separated by fiber splitter (OPNETI CO., SMF-28e SWBC 1 × 16) and the separated laser beams were irradiated into target gas by 16 collimators (THORLABS Co., 50-1310-APC). The transmitted light intensities were detected by photodiodes (Hamamatsu

298


Fig. 3 CT-TDLAS measurement system.

Fig. 4 Experimental setups.

Photonics and G8370-01), and taken into the analyser (HIOKI E.E. Co., 8861 Memory Highcoda HD Analog16). The experiment was performed using two types of experimental setups. One was an experiment system for the accuracy evaluation of 2D CH4 concentration measured by CT-TDLAS. The experimental setup is shown in Fig. 4(a). A jet flow of 1% CH4 with a buffer gas of N2 was introduced into the 16 path CT-TDLAS measurement call at the flow rate of 1.7 × 10−5 m3 /s. The inner diameter of the jet pipe was 8 mm and the N2 guard flow from an outer pipe with the inner diameter of 65 mm was formed at the flow rate of 3.3 × 10−4 m3 /s. The CH4 concentration distribution at 3 mm above the CH4 jet pipe was measured by CT-TDLAS. The CH4 concentration distribution at Y = 0 mm was also measured by sampling the gas and measuring the CH4 concentration using a TDLAS method. The other experiment system was aimed to measure time resolved 2D CH4 concentrations in an oscillating CH4 -Air Bunsen-type flame using CT-TDLAS. The experimental setup is shown in Fig. 4(b). The flow rate of CH4 was oscillated at 50 Hz and 100 Hz using a direct drive servovalve (Moog Inc., D633). The air and average CH4 flow rates were 6.0 × 10−5 m3 /s and 9.0 × 10−6 m3 /s, respectively. The inner diameter of the burner was 8 mm and the air guard flow from an outer pipe with the inner diameter of 65 mm was formed at the flow rate of 3×.3 10−4 m3 /s. A stainless steel mesh was placed at 10 mm above the burner to simulate a lifted flame and CH4 concentration distribution at 3 mm above the burner was measured by CT-TDLAS. The oscillating flame was also measured by a CCD camera (Panasonic, HX-WA30-K) at 480fps to confirm the flame oscillation characteristics.


299

4 Results and Discussion 4.1

Accuracy evaluation of 2D CH4 concentration measured by CT-TDLAS

Fig. 5 shows CH4 concentration measurement results by CT-TDLAS and gas sampling methods shown in Fig. 4(b). Measured CH4 concentration distributions show almost the same structure between the two methods. The high concentration of CH4 was measured at X = Y = 0 mm and CH4 was distributed from X = −6 mm to 6 mm. Because of the limited path number of CT-TDLAS, the sharp edge of CH4 concentration change was not measured by CT-TDLAS at X = −4 mm and 4 mm. This is because of the achievable spatial resolution using the 16 path CT-TDLAS measurement cell (3-4 mm) and the spatial resolution can be modified by increasing the number of laser paths.

Fig. 5 CH4 concentration measurement results.

4.2

Time resolved 2D CH4 concentration measurement in an oscillating CH4 -Air flame

Fig. 6 and Fig. 7 show absorbance time-histories of laser path 3, 4, and 14. The laser path 3 was located at the center of the burner, the laser path 4 at 10 mm from the center of the burner, and the laser path 14 at 3.5 mm from the center of the burner. The absorbance of laser path 3 and 4 shows the periodical oscillation about ±25% (50 Hz) and ±5% (100 Hz) were changed using the set of 16 path absorbance time-histories, the time resolved 2D CH4 concentration distributions were calculated using the CT algorism.

Fig. 6 Absorbance time-histories in 50 Hz oscillating flame.

300


Fig. 7 Absorbance time-histories in 100 Hz oscillating flame.

Fig. 8 Photographs of 50 Hz oscillating flame.

Fig. 8 and Fig. 9 show the direct photographs of the oscillating flame using the experimental setup shown in Fig. 4(b). The flame was oscillating periodically at 50 Hz and 100 Hz, and the scattered flame structure was formed above the mesh because of the periodical equivalence ratio change by the direct drive servovalve. The oscillation of the flame was periodical and shows the agreement with the absorbance time-histories shown in Fig. 6 and Fig. 7. Fig. 10 and Fig. 11 show the time-history of 2D CH4 concentration distribution in oscillating flames measured by CT-TDLAS. The high concentration of CH4 was measured at X = Y = 0 mm and CH4 was distributed from X = −6 mm to 6 mm. The 2D CH4 concentration was successfully reconstructed by the set of 16 path absorbance. The precise structure of 50 Hz and 100 Hz oscillation of 2D CH4


301

Fig. 9 Photographs of 100 Hz oscillating flame.

Fig. 10 Time-history of 2D CH4 concentrations distribution in 50 Hz oscillating flame.

concentration distribution was also detected showing the agreement with the flame structure measured by the CCD camera (Fig. 8 and Fig. 9).


302

Fig. 11 Time-history of 2D CH4 concentrations distribution in 100 Hz oscillating flame

5 Conclusions The 2D methane concentration measurement method using CT-TDLAS was investigated. Based on the results of this study, the conclusions are summarized as follows. 1) Relation between the methane concentration and the absorption intensity were evaluated. The results show a good agreement with the absorption theory. 2) 2D methane concentrations in oscillating flames were measured by the CT-TDLAS method. It was demonstrated that the measurement results of CT-TDLAS were matched with methane flow rates through the nozzle with 50 Hz and 100 Hz oscillation and it enables the real-time 2D concentration measurement to be applicable to the practical combustion processes such as gas turbines.

References [1] Huang, Y. and Yang, V. (2005), Effect of swirl on combustion dynamics in a lean-premixed swirl-stabilized combustor, Proceedings of the Combustion Institute, 30, 1775–1782. [2] Yamakage, M., Muta, K., Deguchi, Y., Fukada, S., Iwase, T., and Yoshida, T. (2008), Development of direct and fast response exhaust gas measurement, SAE Paper 20081298. [3] Zaatar, Y., Bechara, J., Khoury, A., Zaouk, D., and Charles, J.P. (2000), Diode laser sensor for process control and environmental monitoring, Applied Energy, 65, 107–113. [4] Liu, X., Jeffries, J.B., Hanson, R.K., Hinckley, K.M., and Woodmansee, M.A. (2006), Development of a tunable diode laser sensor for measurements of gas turbine exhaust temperature, Applied Physics B, 82(3), 469–478. [5] Ax, H., Stopper, U., Meier, W., Aigner, M., and G¨ uthe, F. (2010), Experimental analysis of the combustion behavior of a gas turbine burner by laser measurement techniques, Journal of Engineering for Gas Turbines and Power, 132(5), p. 051503/1-051503/9. [6] Deguchi, Y., Noda, M., Fukuda, Y., Ichinose, Y., Endo, Y., Inada, M., Abe, Y., and Iwasaki, S. (2002), Industrial applications of temperature and species concentration monitoring using laser diagnostics, Measurement Science and Technology, 13(10), R103–R115. [7] Deguchi, Y. (2011), Industrial applications of Laser Diagnostics, CRS Press, Taylor & Francis. [8] Wang, F., Cen, K.F., Li, N., Jeffries, J.B., Huang, Q.X., Yan, J.H., and Chi, Y. (2010), Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy, Measurement Science and Technology, 21, 045301.


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[9] Wright, P., Terzijaa, N., Davidsona, J.L., Garcia-Castillo, S., Garcia-Stewart, C., Pegrumb, S., Colbourneb, S., Turnerb, P., Crossleyc, S. D., Litt, T., Murrayc, S., Ozanyana, K.B., and McCanna, H. (2010), High-speed chemical species tomography in a multi-cylinder automotive engine, Chemical Engineering Journal, 158(1), 2–10. [10] Ma, L. and Cai, W. (2008), Numerical investigation of hyperspectral tomography for simultaneous temperature and concentration imaging, Applied Optics, 47(21), 3751–3759. [11] Deguchi, Y., Yasui, D., and Adachi, A. (2012), Development of 2D temperature and concentration measurement method using tunable diode laser absorption spectroscopy, Journal of Mechanics Engineering and Automation, 2(9), 543–549. [12] Deguchi, Y., Kamimoto, T., Wang, Z.Z., Yan, J.J., Liu, J.P., Watanabe, H., and Kurose, R. (2014), Applications of laser diagnostics to thermal power plants and engines, Applied Thermal Engineering, 73, 1453–1464 [13] Pal, S. and McCann, H. (2011), Auto-digital gain balancing: a new detection scheme for high-speed chemical species tomography of minor constituents, Measurement Science and Technology, 22(11), 115304. [14] Terzija, N., Davidson, J. L., Garcia-Stewart, C.A., Wright, P., Ozanyan, K.B., Pegrum, S., Litt, T.J., and McCann, H. (2008), Image optimization for chemical species tomography with an irregular and sparse beam array, Measurement Science and Technology, 19(9), 094007. [15] Ma, L., Li, X.,1 Sanders, S.T., Caswell Andrew, W., Roy, S., Plemmons, D.H., and Gord, J.R. (2013), 50-kHz-rate 2D imaging of temperature and H2O concentration at the exhaust plane of a J85 engine using hyperspectral tomography, Optics Express, 21(1), 1152–1162. [16] Cai, W. and Kaminski, C.F. (2014), A tomographic technique for the simultaneous imaging of temperature, chemical species, and pressure in reactive flows using absorption spectroscopy with frequency-agile lasers, Applied Physics Letter, 104, 034101. [17] Cai, W. and Kaminski, C.F. (2014), Multiplexed absorption tomography with calibration-free wavelength modulation spectroscopy, Applied Physics Letter, 104, 154106. [18] Rothman, L.S., Gordon, I.E., Barbe, A., ChrisBenner, D., Bernath, P.F., Birk, M., Boudon, V., Brown, L.R., Campargue, A., Champion, J.P., Chance, K., Coudert, L.H., Dana, V., Devi, V.M., Fally, S., Flaud, J.M., Gamache, R.R., Goldman, A., Jacquemart, D., Kleiner, I., Lacome, N., Lafferty, W.J., Mandin, J.-Y., Massie, S.T., Mikhailenko, S.N., Miller, C.E., Moazzen-Ahmadi, N., Naumenko, O.V., Nikitin, A.V., Orphal, J., Perevalov, V.I., Perrin, A., Predoi-Cross, A., Rinsland, C.P., Rotger, M., Simeckova, M., Smith, M.A.H., Sung, K., Tashkun, S.A., Tennyson, J., Toth, R.A., Vandaele, A.C. and VanderAuwera, J. (2009), The HITRAN2008 molecular spectroscopic database, Journal of Quantitative Spectroscopy & Radiative Transfer, 110, 533–572.


Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Fluid-Structure Coupling Effects on the Aerodynamic Performance of Airfoil with a Local Flexible Structure at Low Reynolds Number Wei Kang1†, Min Xu1 , Jia-Zhong Zhang2 1 School

of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi Province, 710072, P.R. of China 2 School of Energy and Power Engineering, Xi’an Jiaotong University, Shaanxi Province, 710049, P.R. China Submission Info Communicated by Albert C.J. Luo Received 12 January 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Fluid structure interaction Aerodynamic performance Performance enhancement Low Reynolds number flow

Abstract A fluid structure interaction method for an airfoil with a local flexible structure is presented for flow control of Micro Air Vehicles. An improved ALE-CBS scheme is developed for unsteady viscous flow coupling in combination with the theory of shallow arch with large deformation. After the verification of the presented algorithm, the method is used to study the interaction between flow and airfoil with a local flexible structure with different elastic stiffness. The momentum and energy exchange are investigated to reveal the unsteady coupling effects on aerodynamic performance. The results show that the coupling between fluid and structure enhances the momentum and energy exchange from main flow into the boundary layer. It induces the separation bubble moving downstream, and decreases the negative pressure in the separation zone on the upper surface, which leads to lift enhancement. The utilization of local flexible structure can be considered as an effective flow control technique for enhancement of the aerodynamic performance of Micro Air Vehicles. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The efficient flight capabilities of insects and birds have drawn much attention involving in the investigation of low Reynolds number aerodynamics. In the flights of these animals, flexible wings are coupling with unsteady low Reynolds number flow to obtain better aerodynamic performance. Inspired by biological flight, flexible structure is designed as lift surface for control of micro air vehicles (MAVs). It is expected to gain the potential of the shape adaptation of flexible structure, and to result in lift enhancement and drag reduction. To utilize such flow control technique addresses a challenge of multidisciplinary computational techniques and fundamental understanding of unsteady aerodynamics and fluid-structure interaction. † Corresponding



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Wei Kang, Min Xu, Jia-Zhong Zhang/Journal of Applied Nonlinear Dynamics 4(3) (2015) 305–312

An extensive study has been conducted on the coupling effects between the fluid flow and flexible structures on aerodynamic performances of MAVs at low Reynolds numbers. In 1995, Smith and Shyy [1] established a model of membrane airfoil coupling with unsteady flow at Reynolds number 4000. Since then, Shyy’s group focused on the aerodynamics and aeroelasticity of MAVs. Excellent reviews on flexible and flapping wings of MAV can be referred to as [2, 3]. The results from existing literatures [4, 5, 6, 7] show that flexibility of structure gains the potential of shape adaptation for severe flow condition, and to improve aerodynamic performance. Moreover, the stall is delayed due to the vibration of flexible structure. In these works, the whole lift surface of wings was modelled as flexible structure. Such design demands high quality of structure materials, and may result in irreversible damage during strong fluid-structure coupling. For the normal-sized wing, researchers also attempted to utilize the coupling effect to obtain a better flight capability. In 2006, Sinha and Ravande [8] proposed a laminar flow control technique, flexible composite surface deturbulator to stabilize large extents of separated flow over aerodynamic surfaces through passive interactions with a boundary layer flow. They studied the aerodynamic performance of NLF-0414 wings using this technology experimentally. The results show a drag reduction of 60-80%, and lift enhancement of 12% due to the coupling effect of the flexible surface. In the study of rotary-wing aircraft, Sahin et al [9] investigated dynamic stall of an airfoil with a dynamically deformed leading-edge shape (DDLE) at Mach number 0.3 and 0.4. They found that lift, drag, and pitching-moment hysteresis loops were milder for the DDLE airfoil compared to the baseline airfoil. In light of the study of DDLE airfoil, a model of airfoil with a local flexible structure (LFS airfoil) is presented for flow control of MAVs in this study. The flexible surface is located on the upper surface near the leading edge, which could be a better choice for airfoil design with minor demand of material in comparison with the design of whole flexible lift surface of MAV wing. The coupling effect of LFS airfoil on the aerodynamic performance is studied compared with rigid one, in order to check whether LFS airfoil could be used as flow control to improve the lift of the airfoil. The paper is organized as follows. In Section 2 and 3, an aeroelastic model for the flow around the airfoil with a local flexible structure is proposed and an ALE-CBS algorithm is developed coupling with a structure solver for the shallow arch model. Then, a benchmark case for fluid-structure interaction is simulated using the presented algorithm for the verification in Section 4. Subsequently, the interaction between the viscous flow and LFS airfoil is studied. The momentum and energy exchange between the boundary layer and main flow are analyzed to investigate the effect of coupling between the fluid and structure on the lift. Finally, some conclusions are drawn in Section 5.

2 Aerodynamic solver 2.1

Flow equations

When Mach number for low Reynolds number flow is less than 0.3, the flow can be considered as incompressible. Let the airfoil chord c be the characteristic scale, freestream velocity U∞ be the characteristic velocity, and dimensionless variables are defined as x x∗ = , c

y y∗ = , c

t∗ =

U∞t , c

p∗ =

p , ρ∞U∞2

u∗ =

u , U∞

v∗ =

v , U∞

(1)

For the sake of simplicity, the “*” is dropped here, and the standard summation conventions are used hereafter. The ALE framework is introduced for fluid-structure interaction. Hence, the governing equation for two-dimensional unsteady incompressible flow in ALE configuration can be written as follows,

∂ ui =0 ∂ xi

(2)


307

1 ∂ 2 ui ∂ ui ∂ ui ∂p + (u j − uˆ j ) =− + ∂t ∂xj ∂ xi Re ∂ x j ∂ x j

(3)

where uˆ j is the velocity of the grid, Re = ρ U∞ c/μ . 2.2

ALE-CBS algorithm The ALE-CBS algorithm is summed up below [10, 11]. Step 1. obtain intermediate velocities. Ui∗ −Uin = Δt[−c j

1 ∂ 2Uin Δt 2 1 ∂ 2Uin ∂ ∂ ∂ (Uin ) + ]+ ( (c jUin ) + ). ck ∂xj Re ∂ x j ∂ x j 2 ∂ xk ∂ x j Re ∂ x j ∂ x j

(4)

where c j = u j − uˆ j . Step 2. solve the continuity equation implicitly using pre-conditioning conjugate gradient method. (

1 ∂ 2 pn+1 ∂U ∗ ∂U n ) = (θ1 i + (1 − θ1 ) i ). ∂ xi ∂ xi Δt ∂ xi ∂ xi

(5)

Step 3. correct the velocities with obtained pressure. Uin+1 −Ui∗ = −Δt

∂ pn+1 ∂ xi

(6)

Equations (4-6) are temporal discretization form of Navier-Stokes equations with ALE-CBS scheme and these equations can be approached by the standard finite element method. Linear shape functions are used for the velocities and pressure for the spatial discretization.

3 LFS airfoil solver 3.1

Model of LFS airfoil

The locally flexible part of the airfoil is modeled using the theory of shallow arch with large deformation, since the airfoil surface with small curvature is analogous to the shallow arch. The model of LSF airfoil is shown in Figure 1. The shallow arch with pin-pin boundaries on the surface of the airfoil, subjected to normal aerodynamic pressure. It is located at the front of the surface so that the perturbation caused by the arch can propagate downstream. Following the Hamilton’s principle, the governing equation for shallow arch can be derived as the following, ∂ 2 ws Es h3s ∂ ws ws +Vs (ys0 − ws ) + ds ρs hs 2 + = Δp f , (7) ∂ ts 12 ∂ ts where ρs is the density of the shallow arch, hs the thickness of the shallow arch, ls the span of the shallow arch, Es the elastic stiffness of the shallow arch, ws the displacement of shallow arch, ys0 the ∂ ws , differential with respect to xs , Δp f , pressure initial positions of the middle axis of shallow arch, ws ∂ x s ˆ 1 ls Es hs 2 (ys − y2 [ variation comes from the aerodynamic solver, Vs = s0 )]dxs . ls 0 2 The chord of the airfoil c and the freestream density ρ∞ and velocity U∞ of the fluid are referred to as characteristic quantities, and the dimensionless variables are defined as x=

xs , c

y=

ys , c

w=

ws , c

t=

U∞ts , c

E=

Es , ρ∞U∞2

Δp =

Δp f , ρ∞U∞2

(8)

308


Fig. 1 Configuration of LFS airfoil

the governing equation in dimensionless form for the structure can be presented as the following,

ρh

∂ 2 w Eh3 ∂w w +V (y0 − w ) + d = Δp, + 2 ∂t 12 ∂t

(9)

ˆ 1 1 Eh 2 [ (y − y2 where V = 0 )]dx. l 0 2 The pin-pin boundary conditions for Eq (9) are described as

w|x=0 = w |x=0 = w|x=l = w |x=l = 0.

(10)

The initial shape or configuration of the shallow arch, i.e., the surface of the airfoil is approximated by fifth order polynomial as the following, 5

y0 (x) = ∑ ai xi ,

(11)

i=1

where x = 0, y0 (0) = 0; x = l, y0 (l) = 0. 3.2

Structure solver

For deformable shallow arch, Galerkin method is used to solve its governing equation [10, 11, 12, Eh3 w , 13, 14]. A complete functional space is spanned by the eigenfunctions of the operator Lw = 12 nπ i.e., {sin x, n = 1, 2, . . . + ∞}, and the solution of the governing equation will be projected into the l functional space. The displacement of the structure is written as w(x,t) =

∞

∑ wn(t) sin

n=1

nπ x. l

(12)

Let

∂ 2w ∂w Eh3 w +V (y0 − w ) + ρ h 2 + d − Δp. 12 ∂t ∂t Following Galerkin procedure, yields, ˆ 1 mπ x = 0 m = 1, 2, 3, · · · + ∞. R(x, y, w) sin l 0 R(x, y, w) =

(13)

(14)

Equation (14) can be solved by Fourth order Runge-Kutta algorithm by coupling with the aerodynamic solver and then dynamic behaviors of the flexible surface can be obtained.


309

4 Numerical results and discussions 4.1

validation for fluid-structure interaction

A benchmark case for fluid structure interaction of incompressible flow is simulated for the validation. The problem is driven open-cavity with flexible bottom, which is described in Fig. 2. In Bathe’s work [15], the flexible part is considered as plate and dimensionless structural parameters is listed in Table 1. The Reynolds number is 100, and the driven velocity is varying periodically. The time interval for the computation for this case is 7 × 10−4 . Table 1 Dimensionless structural parameters for benchmark case density

thickness

elastic stiffness

damping

500

0.002

2.5 × 104

0.0

The transient response of the flexible bottom is presented in Fig. 3, compared with the one from [15]. Excellent agreement between the presented results and Bathe’s results is achieved both in the transient response and stable response, indicating that the algorithm presented is feasible for fluid-structure interaction.

Fig. 2 description of the benchmark problem.

4.2

Fig. 3 transient response of the flexible bottom at middle point.

Coupling effect on lift

Figure 4 gives time-average lift coefficients of LFS airfoil with different elastic stiffness, with a dash line for the lift coefficient of rigid airfoil. It is seen that the elastic stiffness has a great influence in the lift of the airfoil. When elastic stiffness is between the distribution [5.0 × 104 , 2.0 × 105 ], the ratio of the lift between LFS airfoil and rigid airfoil is greater than 1.4, and reaches its maximum 1.6173 at E = 5 × 104 , which means the lift is enhanced 61.73% by the coupling effect. When elastic stiffness is between the distribution [4.0 × 105 , 6.0 × 105 ], the ratio of the lift between LFS airfoil and rigid airfoil is less than 1.0, and reaches its minimum 0.9568 at E = 5 × 105 . When elastic stiffness goes up to 7.0 × 105 , the ratio of lift is approximately 1.0, indicating the coupling effect on lift becomes less significant. In the subsequent section, the case for E = 5 × 104 with maximum lift will be discussed.

310


Fig. 4 Time averaged lift coefficients of locally flexible airfoil with different elastic stiffness.

4.3

Coupling effect on momentum and energy exchange between boundary layer and main flow

Ten points along x direction upon the airfoil surface are chosen for the comparison of the momentum and energy transfer of flow in LFS airfoil case and rigid airfoil case. Table 2 lists different positions √ and flow characteristics at 10 given points. The fluctuating kinetic energy can be calculated by ex = ∑Ni (ui − u¯i )2 /2N.

Table 2 Characteristics of given points near the airfoil Point number

Position

Flow characteristics

5

(x = 0.1 ∼ 0.5, 0.1)

Shear layer

3

(x = 0.6 ∼ 0.9, 0.1)

Separation bubble

2

(x = 0.9 ∼ 1.0, 0.1)

Vortex shedding in the wake

Figure 5 shows the time-average flow velocities at 10 given points with different elastic stiffness. As is shown in Fig 5, time-average velocities are increased while the elastic stiffness becomes greater. The time-average velocities for rigid airfoil are higher than other cases in x ∈ (0.1, 0.5). The streamwise velocity for rigid airfoil at (x = 0.1, y = 0.1) is 0.99272, which is the thickness of the boundary layer. At the same spatial position, the streamwise velocity for E = 5.0 × 104 is 0.66514, far less than 0.99 with U∞ = 1. It indicates that the accelerating motion of the fluid past the leading edge on the upper surface is actuated by the vibration of the structure, which increases the thickness of boundary layer in x ∈ (0.1, 0.5). In x ∈ (0.5, 0.9), the streamwise velocities for E = 5.0 × 104 and E = 2.0 × 105 become dominant, compared with the one for rigid airfoil, while the streamwise velocity for E = 3.0 × 105 , E = 5.0 × 105 and E = 7.0 × 105 is less than the one for rigid airfoil. The comparison of fluctuating kinetic energy in x ∈ (0.5, 0.9) is shown in Fig. 6. It is seen from Fig 6 that the fluctuating kinetic energy for E = 5.0 × 104 and E = 2.0 × 105 is greater than the one in other cases. It is seen that the perturbations caused by the vibration of the flexible structure for E = 5.0 × 104 and E = 2.0 × 105


Fig. 5 Time-average velocities at 10 points in streamwise direction with different elastic stiffness.

311

Fig. 6 Fluctuating kinetic energy at 10 points in streamwise direction with different elastic stiffness.

promote the energy transfer between the boundary layer and main flow. The fluid in the separation zone gains extra energy from the upstream due to the vibration of the flexible structure. It increases the momentum and kinetic energy of the fluid to make the separation bubble moving downstream. Accordingly, the negative pressure in the separation zone on the upper surface is improved due to the increase of the kinetic energy, which plays an important role in the lift enhancement. In the region of x ∈ (0.9, 1.0), there is no significant difference of streamwise velocities between the rigid airfoil and LFS airfoil. However, it is seen from Fig 6 the fluctuating kinetic energy for E = 5.0 × 104 and E = 2.0 × 105 , which are related to the cases of lift enhancement, is much greater than the ones for other elastic stiffness. As the perturbations caused by the vibration of the flexible structure for E = 5.0 × 104 and E = 2.0 × 105 are propagating downstream, the fluctuating amplitude of the fluid velocity is enhanced, indicating the acceleration of the energy exchange between the main flow and the wake vortex shedding. It leads to the pressure drop at the trailing edge of the airfoil.

5 Conclusions In this paper, an ALE-CBS algorithm for unsteady viscous flow around LFS airfoil is developed coupling with a nonlinear structural solver for shallow arch. The results show that the coupling between the fluid and structure has important effects on the momentum and energy exchange between boundary layer and main flow with different elastic stiffness. The unsteady coupling for E = 5.0 × 104 and E = 2.0 × 105 increase the momentum and kinetic energy of the fluid to make the separation bubble moving downstream, and the negative pressure in the separation zone on the upper surface is improved accordingly, which plays an important role in the lift enhancement. Further study will be carried out to examine such aeroelastic system with a thoroughly parametric investigation.

Acknowledgements The research is supported by the National Natural Science Foundation of China (Grant No. 11402212), the Fundamental Research Funds for the Central Universities, No. 3102014JCQ01002 and the National High Technology Research Program of China (863 Program), No. S2012AA052303.

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References [1] Smith, R. and Shyy, W., (1995). Computation of unsteady laminar flow over a flexible two-dimensional membrane wing. Physics of Fluids, 7(9), 2175–2185. [2] Shyy, W., (2008). Aerodynamics of low reynolds number flyers. Cambridge Univ Press. [3] Shyy, W., Berg, M. and Ljungqvist, D., (1999). Flapping and flexible wings for biological and micro air vehicles. Progress in Aerospace Sciences, 35(5), 455–505. [4] Gordnier, R.E., (2009). High fidelity computational simulation of a membrane wing airfoil. Journal of fluids and structures, 25(5), 897–917. [5] Lian, Y., Shyy, W., Viieru, D. and Zhang, B., (2003). Membrane wing aerodynamics for micro air vehicles. Progress in Aerospace Sciences, 39(6-7), 425–465. [6] Mueller, T.J. and DeLaurier, J.D., (2003). Aerodynamics of small vehicles. Annual Review of Fluid Mechanics, 35(1), 89–111. [7] Rojratsirikul, P., Wang, Z. and Gursul, I., (2009). Unsteady fluid-structure interactions of membrane airfoils at low Reynolds numbers. Exp Fluids, 46(5), 859–872. [8] Sinha, S.K. and Ravande, S.V., (2006). Drag reduction of natural laminar flow airfoils with a flexible surface deturbulator. AIAA Paper, 3030, 5–8. [9] Sahin, M., Sankar, L.N., Chandrasekhara, M. and Tung, C., (2003). Dynamic stall alleviation using a deformable leading edge concept-a numerical study. Journal of aircraft, 40(1), 77–85. [10] Kang, W., Zhang, J.Z., Lei, P.F. and Xu, M., (2014). Computation of unsteady viscous flow around a locally flexible airfoil at low Reynolds number. Journal of Fluids and Structures, 46, 42–58. [11] Kang, W., Zhang, J.Z. and Liu, Y., (2010). Numerical simulation and aeroelastic analysis of a local flexible airfoil at low Reynolds numbers, the 8th Asian CFD conference, Hongkong. [12] Kang, W., Zhang, J.Z. and Feng, P.H., (2012). Aerodynamic analysis of a localized flexible airfoil at low reynolds numbers. Communications in Computational Physics, 11(4), 1300–1310. [13] Zhang, J.Z., Liu, Y., Lei, P.F. and Sun, X., (2007). Dynamic snap-through buckling analysis of shallow arches under impact load based on approximate inertial manifolds. Dynamics of Continuous Discrete and Impulsive Systems-Series B-Applications & Algorithms, 14, 287–291. [14] Kang, W., Zhang, J.Z., Ren, S. and Lei, P.F., Nonlinear galerkin method for low-dimensional modeling of fluid dynamic system using pod modes. Communications in Nonlinear Science and Numerical Simulation, 22, 943–952. [15] Bathe, K.J. and Zhang, H., (2009). A mesh adaptivity procedure for cfd and fluid-structure interactions. Computers and Structures, 87, 604–617.



Modal Analyses of a Thin Shell with Constrained Layer Damping (CLD) Based on Rayleigh-Ritz Method Xu-Yuan Song, Hong-Jun Ren, Jing-Yu Zhai, Qing-Kai Han† School of Mechanical Engineering, Dalian University of Technology, Liaoning, 116024, PR China Submission Info Communicated by Jiazhong Zhang Received 12 January 2015 Accepted 7 March 2015 Available online 1 October 2015 Keywords Thin shell CLD Modal analyses Rayleigh-Ritz method

Abstract This paper presents analytical results of natural frequencies and loss factors of node-diameter and node-circumferential modes of a thin shell treated with constrained layer damping (CLD) based on Rayleigh-Ritz method. General differential equations of motion of the thin CLD shell are derived firstly by following the Donnell-Mushtari shell theory. By taking the beam characteristic functions as the admissible functions, the Rayleigh-Ritz method is employed to deduce the higher degree equations of natural frequencies of the thin CLD shell under different boundary conditions. It is confirmed that the present method is accurate and convenient so that it is applicable to the thin CLD shell compared with classic analytic method or transfer matrix method. Several examples are achieved and compared to illustrate the effects of the viscoelastic material (VEM) and constrain layer’s thickness ratios on the natural frequencies and modal loss factors, besides the effect of boundary conditions. The results show that the thickness ratio of VEM affects sensitively the modal frequencies and the total damping capacity of the thin CLD shell. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Thin shell structures are widely used in many engineering applications due to their good stiffness and low mass, such as pressure vessel, airplane and rocket. These shell-type structures often undertake seismic loads, aerodynamic loads, wave loads, wind loads and many other kinds of dynamic loads that lead to higher stress or higher vibration level. The special damping treatment by using of viscoelastic materials (VEM) which has high polymeric molecular property is quickly developed to reduce the shell structure vibrations and so that compensate its lifetime under seriously operating condition [1-2]. It is of great importance to understand the dynamic characteristics and vibration abatement of thin shell, especially when it is treated by constrained layer damping (CLD) in which the core layer is made of VEM. The modes or free vibrations of thin shell have been studied extensively for more than one hundred years. As Leissa [3] and Soedel [4] summarized systematically in their monographs, various mechanical † Corresponding



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theories were derived to solve the vibration of the thin shell depending on variously simplified assumptions of the strain-displacement relation, stress-strain relation and the equilibrium equation. Among them the most common used shell theories include Love’s [5], Donnell’s [6], Naghdi’s [7], Sanders’[8] and Flugge’s [9] ones. Based on those thin shell theories, the mechanical performances of thin shell treated by CLD have also been studied by many researchers, including dynamic modeling, vibration analysis and vibration reduction [10]. Until now, the dynamic analyses for many kinds of CLD structures are mostly achieved based on finite element method. Based on the finite element method, Rameshand and Ganesan [11-13] developed a finite element based on a discrete layer theory assuming the variables of in-plane displacements along thickness and giving a constant transverse displacement. They discussed the effects of the boundary condition, shear modulus and thickness ratio of the VEM core on the natural frequencies and the loss factors of CLD shells. Sainsbury [14] used the finite element method and a curved shell element to formulate the add-on damping material. Wang and Chen [15] analyzed a cylindrical shell with a partially constrained damping layer and get the natural characteristics of it. In fact, the common used FEM based modal analysis for thin shells with CLD often is neither highly efficient due to a great deal of degrees of freedom of the total structure, nor good accuracy of high-order modes because of the improper shape functions adopted for damping material. The reason for the finite element method being widely used and well-developed is largely due to its convenience in modeling structures which are geometrically complicated. Besides, some exact or semi-analytical methods are also valuable to be as a benchmark for the finite element method. Recently, the transfer matrix method, a kind of semi-analytical one, is developed to solve the natural modes of the CLD structures in a higher precision. Li et al [16] developed a new transfer matrix method for solving the natural frequencies of CLD beams and cylindrical shells. This method can also be applied to solve dynamic problem of the sandwich shells with various boundary conditions and with partially constrained damping layers. Although various methods have been used in dynamic analyses of the CLD structures, the RayleighRitz method is another available method for the goal of improving the analyzing precision and convenience. This method, also known as one kind of energy method, has received intensive attention for plates and shells of a variety of specifications, configurations and boundary conditions [17-19]. During the procedure of Rayleigh-Ritz method, admissible functions should be determined according to different boundary conditions. Many researchers use simple algebraic polynomials as admissible functions in the Rayleigh-Ritz method to solve vibration characteristics of many composite structures effectively [20-22]. The Rayleigh-Ritz method is now widely acceptable due to its simplicity in implementation and capability to provide satisfactory results. In this paper, the modal analyses of thin shell with CLD in which the damping material layer is made of VEM are achieved firstly based on the Rayleigh-Ritz method. The energy equations describing the vibration of the CLD shell are derived by considering the energy dissipation due to the shear deformation of viscoelastic layer. Then, the Rayleigh-Ritz method is used to solve the natural characteristics of the bare shell and the CLD shell respectively. The results of natural frequencies of a bare shell under simply supported boundary condition are used to validate the present method’s accuracy. The effects of CLD parameters on the modal characteristics of CLD shells are calculated and compared thoroughly. The obtained numerical results indicate that the viscoelastic damping layer can affect the vibration character of the thin shell greatly under different boundary conditions.

2 Problem Description and Basic Equations Consider a thin shell treated with constrained damping layer, shown in Fig. 1. The length of CLD shell is L. Ri , Hi and ρi (i = s, v, c) are the radius, thickness, and density of each layer, where s, v, and c, respectively, are short notes for the bare shell, the viscoelastic damping layer and the constrain layer. The axial, circumferential and radial coordinates are represented by x, θ and z respectively, and their


315

corresponding displacements on the middle surface of the CLD shell are denoted by ui , vi (i = s, v, c). In addition, for a thin shell, the simplification of usz = uvz = ucz = w is acceptable.

Fig. 1 Geometric configuration of thin shell with CLD.

According to the Donnell thin shell theory, the strains at an arbitrary point of bare shell and the constrain layer are given by as follows ⎫ ⎧ (0)i ⎫ ⎫ ⎧ ⎧ i i ⎪ ⎪ ε ε κ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎬ ⎪ ⎨ ⎬ ⎬ ⎨ x ⎪ ⎨ x⎪ (0)i εθi = εθ . (i = s, c) (1) + z κθi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎭ ⎪ ⎩ ⎩ i ⎪ i ⎩ (0)i ⎭ γxθ χxθ γxθ (0)

(0)

where the middle surface strain εx , εθ defined as follows ⎤ ⎡ ∂ ⎫ ⎧ ⎧ 0 0 (0)i ⎥ ⎪ ui ⎪ ∂x ⎪ ⎢ ⎪ εx ⎪ ⎥ ⎪ ⎪ ⎢ ⎪ ⎬ ⎢ ⎨ ⎥⎨ i ∂ 1 1 (0)i ⎥ v =⎢ εθ ⎢ 0 ⎪ ⎪ Ri ∂ θ Ri ⎥ ⎥⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎭ ⎣ 1 ∂ ⎩ (0)i ⎪ ⎦⎩ w ∂ γxθ 0 Ri ∂ θ ∂x

(0)

and γxθ , the middle surface curvature κx , κθ and χxθ are

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

⎡ ⎧ ⎫ 0 κi ⎪ ⎢ ⎪ ⎪ ⎨ x ⎪ ⎬ ⎢ ⎢ κθi =⎢ ⎢0 ⎪ ⎪ ⎪ ⎪ ⎩ i ⎭ ⎢ ⎣ χxθ 0

0 1 ∂ R2i ∂ θ 1 ∂ Ri ∂ x

∂2 − 2 ∂x 1 ∂2 − 2 2 Ri ∂ θ 2 ∂2 − Ri ∂ x∂ θ

⎤

⎧ ⎫ ⎥⎪ ui ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎬ ⎨ ⎥ i ⎥ v . (i = s, c) ⎥⎪ ⎪ ⎪ ⎥⎪ ⎭ ⎩ wi ⎪ ⎦⎪

(2) Substituting Eq. (2) into Eq. (1), the strain at an arbitrary point of bare shell and the constrain layer can be written as follows

∂ ui ∂ 2 wi −z 2 ∂x ∂x i 1 ∂v 1 ∂ 2 wi 1 ∂ vi εθi = ( + wi ) − z 2 + z . (i = s, c) Ri ∂ θ Ri ∂ θ 2 R2i ∂ θ εxi =

γxiθ =

(3)

2 ∂ 2 wi 1 ∂ vi ∂ vi 1 ∂ ui + −z +z ∂ x Ri ∂ θ Ri ∂ x∂ θ Ri ∂ x

The bare shell and the constrain layer are assumed to be thin, isotropic, and homogeneous, the


316

stress-strain relationship of the bare shell and the ⎫ ⎡ ⎧ σxi ⎪ Q12 Q ⎪ ⎪ ⎪ ⎬ ⎢ 11 ⎨ σθi =⎢ ⎣ Q21 Q22 ⎪ ⎪ ⎪ ⎭ ⎩ i ⎪ τxθ Q31 Q32 where Q11 = Q22 =

Ei , 1 − μi2

Q12 =

constrain layer can be written as ⎤ Q13 ⎧ i ⎫ ⎥ ⎨ εxi ⎬ Q23 ⎥ ⎦ ⎩ εiθ ⎭ . (i = s, c) γxθ Q33

μi Ei , 1 − μi2

Gi =

Ei 2(1 + μi )

(i = s, c)

(4)

(5)

where Ei is the Young’s modulus, μi is Possion’s ratio and Gi is the shear modulus. The damping material layer deforms linearly with respect in z-axis and experiences axial and transverse displacements. Shear deformation occurs in the xz -plane and yz -plane only. Axial, compressive, and tensile constraints in this layer can be neglected due to the fact that the Young’s modulus of the viscoelastic layer is very small as compared to that of the base shell and the constrain layer [2]. Thus the in-plain shear stress in xz - and yz - direction are given as follows respectively

∂w ∂x vv 1 ∂w γθv z = βθv − + Rv Rv ∂ θ v = βv + γxz x

(6)

where, ui , vi , w(i = s, c) are the radial displacement components of the shell along x-axis y-axis and z-axis. βxv and βθv denote the rotations in x- and θ - directions of the damping material layer. According to Kirchhoff’s hypothesis and the thin shell assumption, the rotations of bare shell and constrain layer in x- and θ - directions are expressed as follows

βxi = −

∂w , ∂x

βθi =

1 i ∂w (v − ), Ri ∂θ

i = s, c

(7)

Applying the Hook’s law, the amplitudes of shear stress in x- and θ - directions can be expressed as v = G γv τxz v xz

τxvθ = Gv γθv z

(8)

where, Gv is the complex shear modulus of the damping material. The displacement distribution among the shell-damping material layer and constrain layer is illustrated in Fig. 2.

Fig. 2 Deformation distribution and continuity of the shell with CLD


317

Assuming the damping material undergoes only shear strain, the following relationship can be obtained as follows Hc c Hv βi (x, θ ) = uvi (x, θ ) + βiv (x, θ )(i = x, θ ) 2 2 H H s v C − D : usi (x, θ ) + βis (x, θ ) = uvi (x, θ ) − βiv (x, θ )(i = x, θ ) 2 2 A − B : uci (x, θ ) −

(9)

Based on Eq. (9), the displacement of the damping material layer uvi (i = x, θ ) and angular displacement βiv (i = x, θ ) are obtained as 1 Hc Hs uvi (x, θ ) = [uci (x, θ ) − βic (x, θ ) + usi (x, θ ) + βis (x, θ )](i = x, θ ) 2 2 2 1 c Hc Hs βiv (x, θ ) = [ui (x, θ ) − βic (x, θ ) − usi (x, θ ) − βis (x, θ )](i = x, θ ) Hv 2 2

(10)

Substituting Eq. (10) into Eq. (6) yields 1 Hc Hs [(uc − βxc ) − (us + βxs )] Hv 2 2 v = 1 (uc − us ) + 1 (H + H + 2H ) ∂ w γxz c s v Hv 2Hv ∂x

βxv =

(11)

The shear strain γθv z in damping material layer can be written by

γθv z = βθv −

vv 1 ∂w + Rv Rv ∂ θ

(12)

where βθv and displacement vv can be expressed as

βθv =

1 c Hc 1 c ∂ w 1 Hs s ∂ w [(v − (v − )) − (vs + (v − ))] Hv 2 Rc ∂θ 2 Rs ∂θ

(13)

The displacement of the damping material layer can be expressed as follows 1 Hc Hs vv = [(vc − βθc ) + (vs + βθs )] 2 2 2 1 Hs s ∂ w 1 c Hc 1 c ∂ w (v − ) + vs + (v − )] = [v − 2 2 Rc ∂θ 2 Rs ∂θ 1 Hc Hs uv = [(uc − βxc ) + (us + βxs )] 2 2 2 Hs ∂ w 1 c Hc ∂ w s +u + ] = [u + 2 2 ∂x 2 ∂x

(14)

(15)

where wv = w. Substituting Eq. (13) and Eq. (14) into Eq. (12), the shear strain of the damping layer can be written as follows 1 Hc 1 Hc 1 Hs 1 Hs − − + )vc − ( + + + )vs Hv 2Hv Rc 2Hv 4Rv Rc Hv 2Hv Rs 2Rv 4Rs Rv Hs Hc Hs 1 ∂w Hc + − + + ) +( 2Hv Rc 2Hv Rs 4Rv Rc 4Rs Rv Rv ∂ θ

γθv z = (

(16)

318


Thus the amplitudes of shear strain in x- and θ - direction for damping layer can be expressed as 1 c 1 ∂w , (u − us ) + (Hc + Hs + 2Hv ) Hv 2Hv ∂x 1 Hc 1 Hc 1 Hs 1 Hs γθv z = ( − − + )vc − ( + + + )vs Hv 2Hv Rc 2Rv 4Rv Rc Hv 2Hv Rs 2Rv 4Rs Rv Hs Hc Hs 1 ∂w Hc + − + + ) . +( 2Hv Rc 2Hv Rs 4Rv Rc 4Rs Rv Rv ∂ θ v γxz =

(17)

The kinetic energy of the bare shell, damping material layer and constrain layer are expressed as follows ˆ 2π ˆ L 1 (u˙2i + v˙2i + w˙ 2 )Ri dxdθ (i = s, v, c). (18) Ti = Hi ρi 2 0 0 The strain energy of the bare shell and constrain layer is Ri Ui = 2

ˆ

−Hi /2 ˆ L ˆ 2π

−Hi /2

0

0

{

Ei 2μi Ei i i [(εxi )2 + (εθi )2 ] + εx εθ + Gi(γxiθ )2 }dθ dxdz, 2 2 1 − μi 1 − μi

(19)

where, Gi = Ei /2(1 + μi )(i = s, c) denote to the shear modulus of bare shell and constrain layer. Because the damping layer undergoes only shear strain, the strain energy of the damping material layer is ˆ ˆ ˆ Rv −Hs /2 L 2π v v (τxz γxz + τθv zγθv z )dθ dxdz. (20) Uv = 2 −Hs /2 0 0 The strain energy of the aforementioned shell can be expressed in the form U = Us +Uv +Uc .

(21)

3 General solutions The general relation for the mode displacement of a CLD shell can be expressed for any circumferential wave number n as follows, where x is replaced by ξ . ⎧ u (ξ , θ ,t) = Ui (ξ ) cos(nθ )e− jω t , ⎪ ⎪ ⎨ i (i = s, v, c) (22) vi (ξ , θ ,t) = Vi (ξ ) sin(nθ )e− jω t , ⎪ ⎪ ⎩ w(ξ , θ ,t) = W (ξ ) cos(nθ )e− jω t , where, ω is the natural frequency of CLD shell, Uim (ξ ), Vim (ξ ), Wm (ξ ) are the axial modal functions describing the vibrational mode in longitudinal directions. These axial modal functions Uim (ξ ), Vim (ξ ), Wm (ξ ) can be approximated by using of the linear combination of the admissible functions that satisfy the geometric boundary condition applied on the shell, they are ⎧ ⎪ Us (ξ ) = am ϕsu (ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ Vs (ξ ) = bm ϕ v (ξ ), ⎪ s ⎪ ⎨ (23) W (ξ ) = cm ϕ w (ξ ), ⎪ ⎪ ⎪ ⎪ Uc (ξ ) = dm ϕcu (ξ ) ⎪ ⎪ ⎪ ⎪ ⎩ Vc (ξ ) = em ϕcv (ξ )


319

where, ϕmu (ξ ), ϕmv (ξ ) and ϕmw (ξ ) are the admissible functions, and each of which is multiplied by an unknown constants am , bm , cm , dm and em respectively. The basis functions of ϕmu (ξ ), ϕmv (ξ ) and ϕmw (ξ ) are as follow ⎧ ∂ φ (ξ ) ⎪ ⎪ ϕ u (ξ ) = , ⎪ ⎨ m L∂ ξ (24) ϕmv (ξ ) = φ (ξ ), ⎪ ⎪ ⎪ ⎩ ϕmw (ξ ) = φ (ξ ), In present analysis, the axial modal function φ (ξ ) is chosen as the same as a beam function and is expressed in a general form as follows [22]

φ (ξ ) = a1 cosh(λm ξ ) + a2 cos(λm ξ ) − σm [a3 sinh(λm ξ ) + a4 sin(λm ξ )],

(25)

where ai (i = 1, 2, 3, 4) are to be determined from the boundary conditions. The eigvalue σm also is defined by boundary conditions. Substituting the expressions of the displacement field of Eq. (22) into Eq. (18), the kinetic energy of the CLD shell is written as ˆ 1 1 πω 2 −2 jω t L 2 2 e {ρs Hs Rs (Usm +Vsm +W 2 ) + ρvHv Rv ( Us2 + Uc2 (26) T =− 2 4 4 0 Hc2 Hs2 Hs Hc 2 n2 Hs Hc n2 Hc2 n2 Hs2 2 1 U + − )W + + )W + U + ( +(1 − c s 8Rs Rc 16R2c 16R2s 2 16 16 8 nH 2 nHs Hc nHc nHs nHs nHc nHc2 nHs Hc +( 2s − − + )VsW + ( − + − )VcW 8Rs 8Rs Rc 4Rc 4rs 4Rs 4Rc 8R2c 8Rs Rc H2 H2 Hs 2 1 Hc 2 Hc Hs 1 )Vs + ( + c 2 − )Vc + ( − )UsW +( + s 2 + 4 16Rs 4Rs 4 16Rc 4Rc 4 4 1 Hs Hc Hc Hs Hc Hs 2 2 +( − − + )VcVs + ( − )UcW ) + ρc hc rc (Ucm +Vcm +W 2 )}dξ . 2 8Rs Rc 4Rc 4Rs 4 4 After substituting the expression of the displacement field of Eq. (22) into Eq. (19), the strain energy of the bare shell Us and constrain layer Uc can be expressed in the following forms of Ui , (i = s, c) ˆ L Ei hi π H 2 n4 H2 H 2 n3 −2 jω t e (( i 2 + 1)W 2 + n2 ( s 2 + 1)Vi2 + ( i 2 + 2n)ViW (27) Ui = 2 2Ri (1 − μi ) 12Ri 12Ri 6Ri 0

μi n2 Hi2 μi nHi2 Hi2 R2i 2 W + R2i Ui2 + 2nRi μiUiVi + 2μi RiUiW − W W− W Vi 12 6 6 (1 − μi ) 2 2 nHi2 H2 n2 Hi2 2 (n Ui + W Vi + (R2i + i )Vi2 + W − 2nRiVsUi ))dξ . + 2 3 12 3

+

Substituting the expression of the displacement field of Eq. (23) into Eq. (20), and considering the relationship Eq. (17), the strain energy of the viscoelastic damping layer can be written as follows ˆ LGv π Rv − jω t 1 1 1 Hv2 1 Hs2 Hs Hv Hv Hs 2 e { ( + 1 + + + + )V (28) Uv = Hv 2 4 R2v 4 R2s Rs Rv Rv Rs s 0 1 1 + (Hc + Hs + 2Hv)UcW − (Hc + Hs + 2Hv )UsW 2L 2L Hc2 Hv2 1 Hc Hv Hc Hv 2 1 2 1 2 + (4 + 2 + 2 − 4 − 4 + 4 )V + U + U 8 Rc Rv Rc Rv Rc Rv c 2 c 2 s 1 H2 H2 −UcUs + 2 ( c + s + Hv2 + HcHv + HcHs + HsHv )W 2 2L 4 4


320

1 1 Hc2 1 Hs2 Hv2 Hc Hv HvHs Hc Hs 2 + n2 ( + + 2 +2 +2 + )W 4 2 R2c 2 R2s R2v Rc Rv Rv Rs Rc Rs n 1 Hc2 Hv2 HvHc 1 Hc Hs Hs Hv Hc + ( + 2 +2 + − − 2 − )VcW 2 2 2 Rc Rv Rv Rc 2 Rc Rs Rs Rv Rc 2 2 n 1 Hs Hv Hv Hs 1 Hc Hs Hs Hv Hc + ( + 22 + + + 2 + )VsW 2 2 2 Rs Rv Rv Rs 2 Rc Rs Rs Rv Rc 2 1 Hv2 1 1 Hc Hs Hs Hc 1 Hc Hv − + − − 2 + )VcVs }dξ + ( 2 2 Rc Rs Rs Rc 4 Rc R2v 2 R2v Thus, the total strain energy U of the CLD shell is obtained as follows U = Us +Uv +Uc

(29)

Considering the condition Tmax = Umax , leads to the Rayleigh’s quotient as

ω2 =

Umax Tnum

(30)

where Umax = (Us )max + (Uv )max + (Uc )max

(31)

By minimizing Rayleigh’s quotient with respect to the constant coefficients

∂ ω2 ∂ ω2 ∂ ω2 ∂ ω2 ∂ ω2 = = = = =0 ∂ am ∂ bm ∂ cm ∂ dm ∂ em

(32)

The set of equations in matrix form is assembled as follow X =0 K − ω 2 M ]X [K where, X = {a given by

b c d ⎡

Kaa ⎢ Kba ⎢ K =⎢ ⎢ Kca ⎣ Kda Kea

(33)

e}T is the Ritz basis vector. The 5 × 5 complex stiffness matrix K and M are

Kab Kbb Kcb Kdb Keb

Kac Kbc Kcc Kdc Kec

Kad Kbd Kcd Kdd Ked

⎤ Kae Kbe ⎥ ⎥ Kce ⎥ ⎥ , Kde ⎦ Kee 5×5

⎡ ⎢ ⎢ M =⎢ ⎢ ⎣

⎤

Maa

⎥ ⎥ ⎥ ⎥ ⎦

Mbb Mcc Mdd Mee

(34)

5×5

The elements of matrix K and M are listed in Appendix A and B. Eq. (33) can be solved by imposing the condition of non-trivial solutions and equating the characK − ω 2 M ] to zero. Expanding the characters determinant, a polynomial equation teristic determinant [K about ω is obtained as follows ω 6 + ω 4 β1 + ω 2 β2 + β3 = 0 (35) where βi (i = 1, 2, 3) are constant and Eq. (35) is solved to yield natural frequencies. For the thin shell with CLD treatment, the natural frequency ω can be expressed as follows

ω = ω0 (1 + iη )

(36)

where η is corresponding to the modal loss factor and ω0 is the natural frequency. The lowest frequency is corresponding to a radial motion mode. The other frequencies refer to either longitudinal or circumferential motions.


321

4 Confirm of the present method In order to validate the convergence of the present analysis method, the computed natural frequencies ω of a bare shell are compared with those of published studies available in literature, including by using of the transfer matrix method [23] and a classically analytical method. The geometrical parameter and material parameter of the shell is given in Tab. 1, which boundary condition is a simple-simple supported one. Table 1 Properties and geometrical parameter of the thin shell Property

Symbol

Value

Unit

Young’s modulus

E

1.1 × 1011

Pa

Poisson’s ratio

μ

0.3

Density

ρ

4480

Length

L

0.256

m

Radius

R

0.16

m

Thickness

H

0.0025

m

kg/m3

The obtained results are shown in Fig. 3, where the orders of the axial mode for the thin shell is denoted by m = 1, 2 and 3, and the circumferential wave number n increases from 1 to 10. It can be seen from the values’ comparison that the three methods are excellent agreement with each other in such a wide range.

Fig. 3 Comparison of the modal frequencies of the bare shell by using three different methods.

5 Numerical Examples The CLD shell discussed here is also shown in Fig. 1. The geometrical parameter and material parameter of the bare steel shell is the same as those listed in Tab. 1. The damping layer are applied on the total out-surface of the shell made of Zn33, Hv = 1 mm, Gv = 0.896 (1+0.9683i) MPa, ρv = 999

322


kg/m3 . The constrain layer is made of Aluminum, which parameter values are Hc = 1 mm, Ec = 70 Gpa, μc = 0.3, ρc = 2700 kg/m3, respectively. 5.1

The effect of CLD on the natural frequencies

Fig. 4 shows the natural frequencies of the CLD shell compared with the bare shell under simple supported boundary condition. By checking the obtained modes of the bare shell and the CLD shell with viscoelastic damping material, it is observed that the natural frequencies change greatly. When the circular wave number n < 8, the VEM increases the natural frequencies of the whole shell obviously. The mode frequencies from n = 8 to n = 15, i.e. the higher modes, are quite different from the bare shell to a CLD one.

Fig. 4 Vibration model of the shell with two edges with simply supported boundary condition.

5.2

The effect of the boundary conditions on the natural frequencies and modal loss factors

Fig. 5 shows the effect of the boundary conditions on the natural frequencies and the modal loss factors the CLD shell, where the half wave number m = 1 and circumferential wave number increase from 1 to 15. From Fig. 5(a), the boundary has little effect on the natural when the circumferential wave number n < 5. When n = 9 the lowest natural frequency appears. The differences of the first order natural frequency among these three boundary conditions are the largest. With the circumferential wave number n increases, the differences of natural frequencies are all small. The clamp to clamp boundary has the largest modal frequency obviously. Fig. 4(b) shows that the largest modal loss factor appears at the circumferential wave number n = 11 for the C-C boundary condition. 5.3

The effect of thickness ratio of VEM on the natural frequencies and modal loss factors

In order to compare the thickness ratio of the damping material on the natural characteristic of the CLD shell, three different thickness ratio (R1 = hv /hs ) of damping material layer are chosen to indicate the effectiveness. The obtained results are shown in Fig. 6, where the boundary condition of the CLD shell is clamp to clamp. Fig. 6(a) shows the effect of the thickness ratio on the natural frequencies of the CLD shell. These curves are at the same trend. With the thickness ratio increasing, the natural frequency decreases.


323

Fig. 5 Vibration model of the shell with different boundary condition.

Fig. 6 Modal analyses of thin CLD shell with different thickness ratios R1 .

The lowest natural frequency appears at n = 9. Fig. 6(b) shows the effect of the thickness ratio on the modal loss factors of the CLD shell. The largest loss factor appears at the circumferential wave number n = 12 and the thickness ratio is 0.2. 5.4

The effect of thickness ratio of constrain layer on the natural frequencies and modal loss factors

Fig. 7 shows the natural frequencies and modal loss factors the CLD shell with different cover thickness ratio (R2 = hc /hv ) of the constrain layer, where the boundary condition is clamp to clamp supported at two ends. From Fig. 6(a), the thickness ratio of constrain layer affects weekly on the natural frequencies of the CLD shell. Figure 5(b) shows the effect of the thickness ratio of the constrain layer on the modal loss factor of the CLD shell. With the increment of the circumferential wave number, the different of the modal loss factors become larger. The largest modal loss factor appears at the circumferential wave number n = 12 and the thickness ratio is 0.5.

324


Fig. 7 Modal analyses of thin CLD shell with different thickness ratios R2 .

6 Conclusions The proposed modal analysis based on Rayleigh-Ritz method for a thin shell with constrained layer damping (CLD) treatment is accuracy enough and convenience to deal with different boundary conditions. The governing differential equations of the CLD shell is deduced based on the Donnell-Mushtari thin shell theory and viscoelastic material theory, following energy expression of different layers where the energy dissipation is attributable to the shear deformation of the viscoelastic layer. The Rayleigh-Ritz method is introduced to solve the modal characteristics of the CLD shell. Comparison and convergence of a simple supported bare shell show that the result by using of the present method is agreement with those of classic analytic method and transfer matrix method. The effect of the geometric parameters of the VEM and constrain layer are discussed on the natural frequencies of the CLD shell. The numerical results indicated that the viscoelastic damping material affects the natural frequencies and the modal loss factors of the CLD shell greatly. Otherwise, the thickness ratio of constrain layer only affects the modal loss factors. Different boundary conditions also influence the modes of the CLD shell.

Acknowledgments This work is supported by Natural Science Foundation of China (Grant No. 51175070, 11472068), and National Basic Research Program of China (No. 2012CB026000-05, 2013CB035402-2), and the Fundamental Research Funds for the Central Universities (DUT14RC(3)095).

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Appendix A

LEs Hs π Kaa = 1 − μs2

ˆ ˆ

1

0

Rv Gv π Rs u u n2 (1 − μs ) u u ϕs ϕs + ϕs ϕs )dξ + 2 L 2Rs Hv2

1

ˆ 0

1

ϕsu ϕsu dξ

nEs Hs π ϕsu ϕsv dξ 2 1 − μs 0 ˆ ˆ Es Hs π 1 u w Rv Gv π (2Hv + Hs + Hc ) 1 u w Kac = ϕ ϕ d ξ − ϕs ϕ dξ 1 − μs2 0 s 2Hv2 L 0 ˆ Rv Gv π 1 u u Kad = − ϕs ϕc dξ Hv2 0 Kab =

Kae = 0 Kba =

nπ Es Hs 1 − μs2

ˆ

1 0

ϕsv ϕsu dξ


326

ˆ 1 n2 v v Rs v v 2Rv Gv π 3 Hs 9 ( ϕs ϕs + (1 − μs ) 2 ϕs ϕs )dξ + ( + ) ϕsv ϕsv dξ 2 R 2L H 4 R 8 s s 0 0 v ˆ 1 ˆ 1 LEs Hs π n n(1 − μs ) v w nRv Gv π Hc 6Hs Kbc = ( ϕsv ϕ w − ϕs ϕ )dξ + ( − ) ϕ v ϕ wdξ 2 1 − μs 0 Rs 2L 8Hv Rc Rs 0 s

LEs Hs π Kbb = 1 − μs2

ˆ

1

Kbd = 0

ˆ 1 2Rv Gv π Hs Hc (3 + −3 ) ϕ v ϕ v dξ Kbe = − 4Hv2 Rs Rc 0 s c ˆ ˆ Es Hs π 1 w u Rv Gv π (2Hv + Hs + Hc ) 1 w u Kca = ϕ ϕs dξ − ϕ ϕs dξ 1 − μs2 0 2Hv2 L 0 ˆ 1 ˆ LEs Hs π 1 n w v n(1 − μs ) w v nRv π Gv Hc 6Hs Kcb = ( ϕ ϕs − ϕ ϕs )dξ + ( − ) ϕ wϕsv dξ 1 − μs2 0 Rs 2L 8Hv Rc Rs 0 ˆ LEs Hs π 1 Hs2 n4 w w Hs2 Rs w w 1 w w n2 Hs2 w w Kcc = ( ϕ ϕ + ϕ ϕ + ϕ ϕ − ϕ ϕ 1 − μs2 0 12R3s 12L4 rs 6Rs L2 ˆ n2 Hs2 (1 − μs ) w w 2Rv Gv π 1 (Hc + Hs + 2Hv )2 w w Hs Hc + ϕ ϕ )dξ + ( ϕ ϕ + n2 ( + )2 ϕ w ϕ w )dξ 2 2 2 6Rs L 8Hv L Rs Rc 0 ˆ 1 2 4 2LEc Hc π Hc n w w Hc2 Rc w w 1 w w n2 Hc2 w w Hc2 n2 (1 − μc ) w w + ( ϕ ϕ + ϕ ϕ + ϕ ϕ − ϕ ϕ + ϕ ϕ )dξ 2(1 − μc2 ) 0 12R3c 12L4 Rc 6Rc L2 6Rc L2 ˆ ˆ Ec Hc π 1 w u π Rv Gv (2Hv + Hs + Hc) 1 w u Kcd = ϕ ϕc dξ + ϕ ϕc dξ 1 − μc2 0 2LHv2 0 ˆ 1 ˆ LEc Hc π 1 n w v n(1 − μc ) v w nRv Gv π Hc Hs Kce = ( ϕ ϕc − ϕc ϕ )dξ − ( − ) ϕ w ϕcv dξ 1 − μc2 0 Rc 2L 4Hv2 Rc Rs 0 ˆ Rv Gv π 1 u u Kda = − ϕc ϕs dξ Hv2 0 Kdb = 0 Kdc Kdd Kde

Ec Hc π = 1 − μc2

ˆ

1 0

ϕcu ϕ w dξ

ˆ

1

π Rv Gv (2Hv + Hs + Hc) + 2LHv2

2π Ec Rc Hc = ϕcu ϕcu dξ + 2 2L(1 − μc ) 0 ˆ nπ Ec Hc 1 u v = ϕ ϕ dξ 1 − μc2 0 c c

Kea = 0

2n2 π LE

ˆ

c Hc (1 + μc )

4Rc

1

0

ˆ

ϕcu ϕ w dξ 1

0

ϕcu ϕcu dξ

2Rv Gv π + 2Hv2

ˆ

1 0

ϕcu ϕcu dξ

ˆ 1 Rv Gv π Hs Hc (3 + − 3 ) ϕ v ϕ v dξ 4Hv2 Rs Rc 0 c s ˆ 1 ˆ LEc Hc π 1 n w v n(1 − μc ) v w nRv Gv π Hc Hs Kec = ( ϕ ϕc − ϕc ϕ )dξ − ( − ) ϕ w ϕcv dξ 1 − μc2 0 Rc 2L 4Hv2 Rc Rs 0 ˆ nπ Ec Hc 1 v u Ked = ϕ ϕ dξ 1 − μc2 0 c c ˆ ˆ ˆ 1 2n4 π LEc Hc 1 v v 2π Ec Hc Rc 1 v v 2Rv Gv π 2Hc Kee = ϕ ϕ dξ + ϕ ϕ dξ + (1 − ) ϕcv ϕcv dξ 2Rc (1 − μc2 ) 0 c c 4L(1 + μc ) 0 c c 8Hv2 Rc 0 Keb = −


Appendix B ˆ Maa = πρs Hs Rs

0

ˆ Mbb = πρs Hs Rs

L

0

L

ϕ u ϕ u dξ ϕ v ϕ v dξ ˆ

Mcc = π L(ρs Hs Rs + ρv Hv Rv + ρc Hc Rc ) ˆ 1 Mdd = π Lρc Hc Hc ϕ u ϕ u dξ 0 ˆ 1 Mee = π Lρc Hc Rc ϕ v ϕ v dξ 0

1 0

ϕ wϕ w dξ

327



The Method of High Order Fatigue Test of Thin Plate Composite Structure With Hard Coating Hui Li†, Wei Sun, Zhong Luo, Bengchun Wen School of Mechanical Engineering & Automation, Northeastern University, TX 024-8368-4500, China Submission Info Communicated by Jiazhong Zhang Received 8 December 2014 Accepted 7 March 2015 Available online 1 October 2015 Keywords High order fatigue Test method Hard coating Dynamic strain Strain softening

Abstract In order to solve the high order fatigue test problem of thin plate composite structure with hard coating, a new test method and its technological process is proposed based on the summarizing the experience of massive experiments. Besides, by considering technical difficulties of the high order fatigue test of the hard coating composite structure, several key techniques are described in details, such as how to measure dynamic strain without damaging the hard coating, how to predict excitation amplitude required by the high order fatigue test, and how to avoid the interference resulted from the strain softening of hard coating and to measure high order nature frequencies accurately. Finally, an experimental test is done to verify the practicability and reliability of this method. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The hard coating is a kind of coating materials prepared by the metal substrate, ceramic substrate or their mixtures. The coatings with high hardness and high resistance of high temperature and corrosion are mainly used in barrier coatings, anti-friction and anti-erosion coatings. Recently, as a representative coating material, the ceramic hard coating have been applied to the blade of a gas compressor, which is aimed at solving the problem of high order fatigue failures by reducing vibration and achieving anti-scour performance simultaneously [1-4]. In addition, the high order fatigue characteristics of hard coating itself are also very crucial in the application. At present, the high order fatigue characteristics of the hard coating composite structure have drawn some research interest. Take the blade or blade-like structure for an example, many experimental investigations have been done. Black well et al.[5] experimentally verified that the increased damping, the decreased vibration stress and extended fatigue life can be achieved after the titanium plate was coated by MgO+Al2 O3 coating. Ivancic et al. [6] compared the damping and the fatigue characteristics of a coating structure with or without MgO+Al2 O3 coating. They found that the anti-fatigue properties of a hard coating titanium plate enhanced remarkably, e.g., under the same load, its life can prolong five times compared with the plate without coating.Besides, some research efforts have been focused † Corresponding

author.

Email address: ISSN 2164 − 6457, eISSN 2164 − 6473/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/JAND.2015.09.012

330

Hui Li et al. /Journal of Applied Nonlinear Dynamics 4(3) (2015) 329–337

on how to measure fatigue property of the blade. For example, Shokrieh et al. [7] tested the vibration fatigue strength of turbine blades. Cowles [8] presented a vibration fatigue test method of aircraft gas turbines. However, most of researches only considered the fatigue tests under low order vibration conditions. Until now, few research works are carried out in the field of high order fatigue tests and the related test methods are still scarce, especially on the hard coating composite structure. This study uses a hard coating composite plate as an example and proposes a test method and its technological process to solve the high order fatigue test problem. Besides, by considering technical difficulties of the high order fatigue test of the hard coating plate, several key techniques are described in details, such as how to measure dynamic strain without damaging the hard coating, how to predict excitation amplitude which is required by the high order fatigue test, and how to avoid the interference resulted from the strain softening of the hard coating and to measure high order nature frequencies accurately. Finally, the application of this method is verified by a titanium plate specimen coated with MgO+Al2 O3 . The result shows it can not only be used to objectively evaluate the high order anti-fatigue capability of hard coating materials, but also to check the effect of the coating technology.

2 Technical difficulties in high order fatigue test of hard coating composite structure The high order fatigue test on the hard coating composite structure is different from that of the traditional test on single metal structure, because the dynamic characteristics of composite structure have been affected by coating material and their high order fatigue tests require the new experimental techniques. The main technical difficulties are listed in the following aspects: (1) Precise measurement of nature frequency A structure without coating material is generally regarded as a linear structure, and its nature frequencies will not change as the external excitation level increase. However, the strain softening phenomenon is significant when the hard coating material is applied in the raw structure[9-10], and the nature frequency of the composite structure will decrease as the excitation amplitude increase. For example, Fig. 1 shows the fifth nature frequencies of hard coating composite plate under different excitation level obtained by experiment, from which we can observe that nature frequencies and damping parameters keep changing when the different excitation levels are applied on such plate. Thus, it is necessary to determine nature frequencies with considering strains oftening of hard coating. Besides, precise test of nature frequency is not easy to complete due to sweep direction and sweep rate, we will discuss them in section 4.

Fig. 1 The fifth nature frequencies of hard coating composite plate under different excitation levels obtained by experiment.


331

(2) Non-permit of contact strain measurement Generally, the fatigue test needs to continuously monitor and acquire dynamic strain of the maximum stress point of the structure, which is an important parameter in judging whether the structure is fatigue or not. However, the traditional strain measurement method needs to attach strain gauges to the tested surface, which is not suitable for the structure coated with hard coating materials, because it will destroy the coating in the polishing procedure. Besides, this method will add an extra mass, which will result in experimental errors on frequency and damping. (3) Precise measurement of high order vibration response High order vibration of hard coating composite structure always contains multiple nodes and nodal lines, and their complexity will further rise as the vibration order increases. The physical dimension of conventional accelerometer is larger than laser point, so it is very likely to be placed at nodes or nodal lines, which will lead to the reduction of response test accuracy. Another problem is that the amplitude of the high order vibration is small, e.g., the response amplitude of 7 th mode of titanium plate with MgO+Al2 O3 coating is only 17 um under the excitation amplitude of 1 g, the accelerometer often fails to recognize such a small vibration level.

3 Test method of high order fatigue Considering the above technical difficulties in section 2, a high order fatigue test method of thin plate composite structure with hard coating is proposed, which includes the following 7 key steps: (1) Select the fatigue order and the maximum stress point Before carrying out the high order fatigue test of the hard coating composite structure, the most important task is to determine the tested fatigue order. It should match with the order where the real structure’s high order fatigue occurs. In this study, the fifth fatigue is employed as an example to discuss and the fifth double strips strain modal shape of the thin plate with hard coating is obtained by finite element method, as seen in Fig. 2, the maximum stress point (the danger point of the fatigue) can be determined from the result. (2) Test high order nature frequency accurately The accuracy nature frequency result is very important to the high order vibration fatigue test of the hard coating composite structure. It is recommended that using hammer excitation method to roughly get frequency value of each order, and then employing vibration shaker to conduct sine sweep test by selecting sweep range on the basis of 75%∼125% of this frequency. Because of the strain softening phenomenon, the measurement should be carried out from high frequency to low frequency. Fig. 3 gives the 3D waterfall of the sweep test of the 3rd mode, from which we can accurately identify the related nature frequency of the hard coating composite structure.

Fig. 2 The fifth double strips strain modal shape of the thin plate with hard coating.

332


Fig. 3 The 3D waterfall of sweep test containing the 3rd nature frequency of the hard coating composite structure.

(3) Determine high order fatigue cycle index and the ultimate stress The hard coating material is mainly used to suppress and prevent the high order fatigue of the structure, so the corresponding cycle index should be over 106 . After the fatigue life parameter is determined, the ultimate stress can also be approximately obtained by the S − N curve of the substrate material [11]. Because the S − N curve of hard coating composite structure is hardly obtained and its high order fatigue life is closely related to the substrate material, we can use the S − N curve of the substrate material as a reference. Assuming σmax is the ultimate stress of the substrate material, and if it is within the level of elastic deformation and the relation of strain and the stress obeys the Hooke’s law. Thus, the ultimate strain εmax can be obtained by the following: εmax = σmax /E, (1) where, E is the young’s modulus of the substrate material. (4) Determine excitation time of high order fatigue failure Assuming the substrate material is undergone a cycle process with the ultimate stress, and its failure time of the fatigue, t0 can be calculated as: t0 =

1 × 10n , fh

n ≥ 6,

(2)

where, fh is the high order nature frequency chosen for high order fatigue test. After hard coating material is applied on the structure, its fatigue life should be prolonged theoretically, so when testing the high order fatigue characteristic of such composite structure, the excitation time t1 should be longer than t0 , which can be chosen by referencing the following: t1 = (1.5 ∼ 2) × t0 .

(3)

(5) Determine excitation amplitude required for fatigue failure Generally, electromagnetic vibration shaker is widely used in the vibration fatigue test, as it can provide excitation energy for the test specimen with large amplitude continuously and stably. It is necessary to determine base excitation level, which can ensure ultimate stress is recurrently applied on thin plate structure with or without hard coating. It can be determined by dynamic strain test using traditional strain gauges. The details will be discussed in section 4. (6) Establish the strain-displacement curve of the maximum stress point


333

As the hard coating is quite thin, the traditional strain method like pasting strain gauges is not suitable, for the surface needs to be polished when using this method, which could destroy the coating. Thus, indirect strain measurement is adopted by establishing the strain-displacement curve of the maximum stress point of thin plate structure. We will discuss it in details in section 4. (7) Determine the failure criterion Set up the fatigue excitation time as well as the required excitation amplitude for the vibration shaker, and carry out the high order fatigue test of the hard coating composite structure. Measure its vibration response by the laser Doppler vibrometer, if the nature frequency of coating-structure or steady-state vibration response changes under the constant resonance excitation, it means the fatigue failure happens. Stop the excitation and observe the fatigue failure of the hard coating composite structure, e.g., check whether or not the coating-structure will crack or the coating peel off.

4 Key techniques of high order fatigue test Based on the summarizing the experience of massive experiments on the traditional vibration fatigue test [12-13], this section presents 3 key techniques to solve the technical problems of high order fatigue test of hard coating composite structure. (1) Measure dynamic strain indirectly Usually dynamic strain is a critical parameter during the fatigue characteristic test, but the coating thickness is thin, e.g., the thickness of MgO+Al2 O3 coating is between 20 um and 30 um. Thus, the traditional strain method like pasting strain gauges is not suitable, for the surface needs to be polished when using this method, which could destroy the coating. Besides, lots of experiments indicate that after a long time excitation with ultimate stress, the strain gauge will often fall off firstly before the coating. Therefore, an indirect strain measurement should be used to overcome the limitations of strain gauges. To this end, two types of response signal are required on the uncoated structure. On the one hand, according to the result of the finite element analysis, strain signal still needs to be measured by pasting strain gauges onto its maximum stress position; On the other hand,response signal is also required by using the laser Doppler vibrometer, which can be expressed in the form of displacement or velocity. In this way, we can establish the strain-displacement curve of the maximum stress point. Because dynamic stress of hard coating composite structure is basically same with the uncoated structure, we can indirectly measure dynamic stress of the coating-structure via the displacement signal.However, it should be noticed that the laser point should not be put on structural nodes or nodal lines,otherwise the preciseness of strain-displacement curve will be heavily affected. (2) Predict the base excitation amplitude The emphasis of the high order fatigue test is to ensure that the structure is measured under the ultimate stress condition. If fatigue cycle of 106 is selected, refer to the S − N curve of the uncoated substrate material, the corresponding ultimate stress of the hard coating structure can be obtained as the ultimate stress point remain unchanged before and after the structure is coated, and dynamic stress of hard coating composite structure is basically same with the uncoated structure. In actual test, the ultimate stress can be reached to by increasing the base excitation amplitude repeatedly until the max response amplitude is experimentally measured by laser Doppler vibrometer. Then, through the strain-displacement curve as well as Hooke’s law, we can predict the base excitation amplitude required by the high order fatigue test. (3) Measure nature frequency accurately The strain softening phenomenon makes it inconvenient to test the high order fatigue of the hard coating composite structure. Good practice in our test dictates that two test methods can be employed to overcome this difficulty. 1) Sweep from high frequency to low frequency


334

Measure the nature frequency of the hard coating composite structure by sweeping from high frequency to low frequency. If the sine sweep test is done from a low frequency to a high frequency, as seen in Fig. 4, the obtained real response amplitude will increase along the curve fromA to B, and then jump from B to D directly before it will decrease along the curve from D to E. In this way, it is imposable to measure the nature frequency and the maximum amplitude of the system. However, if the frequency can be swept from high to low, the response amplitude can rise along the curve from E to D, then increase continuously from D from C and jump from C to A directly before it will decrease along the curve from A to the low frequency. Consequently, the accurate nature frequency can be obtained by identifying C.

Fig. 4 The frequency response of hard coating composite structure.

2) Determine the reasonable sweep rate It is important to determine the reasonable sweep rate so that the tested structure has enough time to reach the resonance peak. According to the reference [14], the sweep rate S (Hz/s) can be determined by the following inequality: (4) S < ξ 2 fh2 , where, ξ is the damping ratio of the coating-structure, which can be obtained by measurement. 5 A case study 5.1

The test specimen and test system

Use titanium plate as an example, as seen in Fig.5(a), it is in clamped-free boundary condition and its length, width and thickness are 150 mm, 110 mm and 1 mm, respectively. Hard coating material such as MgO+Al2 O3 is coated on the one side of the TI-6AL-4V plate with the coating thickness of 20 um, as seen in Fig. 5(b). The material parameters of the composite plate are listed in Tab. 1, and the instruments used in the high order fatigue test are listed in Tab. 2. Table 1 The material parameters of the titanium plate and the hard coating Name

Young’s modulus (GPa)

Poisson’s ratio

Density (kg/m3 )

TI-6AL-4Vplate

111.5

0.31

4420

MgO+Al2 O3 hard coating

70.36

0.31

2841


335

Fig. 5 The fifth fatigue test of hard coating titanium plate. Table 2 The instruments used in the high order fatigue test Serial No.

Name

1

LMS SCADAS data acquisition front-end and

2

Dell notebook computer

3

PCB 086C01 Hammer

4

EM-1000F electromagnetic vibration shaker system

5

Polytec PDV-100 laser Doppler vibrometer

6

Coinv INV1861 dynamic strain indicator

In the test set-up, the excitation is provided by EM-1000F electromagnetic vibration shaker system, and Polytec PDV-100, mounted to a rigid support, is used to measure the plate velocity at a single point along the excitation direction of vibration table, while strain signal is simultaneously acquired by Coinv INV1861 dynamic strain indicator. Besides, PCB 086C01 Hammer is used to give pulse excitation on the composite plate to roughly get its nature frequencies. All the excitation and response signals are recorded by LMS SCADAS data acquisition front-end and Dell notebook computer (with Intel Core i7 2.93 GHz processor and 4G RAM) is used to operate LMS Test.Lab 10B software and store measured data. 5.2

Test results and discuss

Firstly, select the laser point B as the test point, as seen in Fig. 5, and measure nature frequencies of the uncoated and the coated plate by sine sweep test under 1g excitation amplitude, the related test results are listed in Tab. 3. Then, establish the relationship of strain-displacement between the maximum stress point and laser point B. The fifth nature frequency is employed as the excitation frequency, and the excitation amplitude is repeatedly adjusted to obtain the strain-displacement curve, as shown in Fig. 6. It should be noted that the fifth nature frequencies obtained by experiment under different excitation levels are already given in Fig. 1. Table 3 Nature frequencies of the uncoated and the coated plate (Hz) Modal order

1

2

3

4

5

6

Uncoated plate

64.3

157.0

388.3

534.5

569.8

1008.0

Coated plate

64

162.5

400.8

559.8

592.8

1020.8

336


Fig. 6 The strain-displacement curve

Fig. 7 The fifth fatigue failure of hard coating titanium plate.

Finally, complete fifth fatigue test of titanium plate specimen coated with MgO+Al2 O3 . By referring to the titanium S − N curve in the reference [15], the ultimate stress σmax of the thin plate should reach to 500 Mpa for completing fatigue cycle of 106 . And calculating by using (1) and (2),the fatigue failure time t1 of the coated titanium plate is approximately 45∼60min and the related ultimate strain εmax is about 4.5 muε which requires 36 g excitation amplitude provided by vibration shaker. Using this excitation level to conduct fatigue test for 60 min, but the phenomenon of fatigue does not occur which is probably due to inaccurate of S − N curve as well as the strain-displacement curve. Therefore, larger base excitation amplitude such as 40 g is used to excite hard coating titanium plate, the corresponding excitation frequency is set to 589.0 Hz and response amplitude of the laser point is keeping at about 0.16 mm. By 47: 20, the response signal has suddenly dropped at about 0.036 mm, the phenomenon of fatigue occurs with the exfoliated coating in the maximum stress, as shown in Fig. 7. This result has proved the practicability and efficiency of the high order fatigue test method proposed in this research.

6 Conclusions This study proposes high order fatigue test method and its practicability and reliability have been verified by using the hard coating thin plate as an object. (1) Consider the strain softening effect of the hard coating and propose key techniques of high order fatigue test via lots of experimental research; (2) Establish the strain-displacement curve of an uncoated plate with strain gauges and laser Doppler vibrometer, and dynamic strain of the hard coating plate can be indirectly obtained through this curve;


337

(3) Propose technological process to test the higher order fatigue of thin plate composite structure with hard coating.

Acknowledgments This work was supported by the National Science Foundation of China [grant numbers 51105064]; the National Program on Key Basic Research Project [grant numbers 2012CB026000]; and the Natural Science Foundation of Liaoning Province [grant numbers 201202056].

References [1] Patsias, S., Saxton, C., and Shipton, M. (2004). Hard damping coatings: an experimental procedure for extraction of damping characteristics and modulus of elasticity. Materials Science and Engineering: A, 370(1), 412–416. [2] Reed, S.A., Palazotto, A.N., and Baker, W.P. (2008).An experimental technique for the evaluation of strain dependent material properties of hard coatings. Shock and Vibration, 15(6), 697–712. [3] Xu, Q.Z., Liang, C.H., Sun, G.H., et al. (2008). Development of thermal barrier coating for foreign turbofan engine turbine blade. Aeroengine, 34(3), 52–56. [4] Zhai, J. Y., Li, H., and Han, Q. K. (2011).Damage Process Simulation of Damping Coating of Thin Plate. In Advanced Engineering Forum, 2, 739-742. [5] Blackwell, C., Palazotto, A., George, T.J., et al. (2007). The evaluation of the damping characteristics of a hard coating on titanium. Shock and Vibration, 14(1), 37–51. [6] Ivancic, F.T., Palazotto, A., and Cross, C. (2003). The effect of a hard coating on the damping and fatigue life of titanium. Defense Technical Information Center, Wright-Patterson. [7] Shokrieh, M.M. and Rafiee, R. (2006).Simulation of fatigue failure in a full composite wind turbine blade. Composite Structures, 74(3), 332–342. [8] Cowles, B.A. (1989).High cycle fatigue in aircraft gas turbines—an industry perspective. International Journal of Fracture, 80(2-3), 147–163. [9] Herman Shen, M.-H. (2002).Development of a Free Layer Damper using Hard Coatings, 7th National Turbine Engine High Cycle Fatigue Conference. May 14-17, 2002, Palm Beach Gardens, Florida. [10] Amabili, M. (2008). Nonlinear vibrations and stability of shells and plates. Cambridge University Press, Cambridge. [11] Sonsino, C.M. (2007). Course of SN-curves especially in the high-cycle fatigue regime with regard to component design and safety. International Journal of Fatigue, 29(12), 2246–2258. [12] Wozney, G.P. (1962).Resonant-vibration fatigue testing.Experimental Mechanics, 2(1), 1–8. [13] George, T.J., Seidt, J., Herman,Shen, M.H., et al. (2004).Development of a novel vibration-based fatigue testing methodology. International Journal of Fatigue, 26(5), 477–486. [14] Torvik, P.J. (2011).On estimating system damping from frequency response bandwidths. Journal of Sound and Vibration, 330(25), 6088–6097. [15] Monteiro, S.N. and Reed, H.R.E. (1973).An empirical analysis of titanium stress-strain curves. Metallurgical Transactions, 4(4), 1011–1015.

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Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University St-Petersburg, 198504, Russia Email: [email protected]

S.C. Sinha Department of Mechanical Engineering Auburn University Auburn, Alabama 36849, USA Fax: +1 334 844 3307 Email: [email protected]

Jiazhong Zhang School of Energy and Power Engineering Xi’an Jiaotong University Shaanxi Province 710049, China Email: [email protected]

Shijun Liao Department of Naval Architecture and Ocean Engineering Shanghai Jiaotong University Shanghai 200030, China Fax: +86 21 6485 0629 Email: [email protected]

Jian-Qiao Sun School of Engineering University of California Merced, 5200 N. Lake Road Merced, CA 95344, USA Fax: +1 209 228 4047 Email: [email protected]

Yufeng Zhang College of Sciences China University of Mining and Technology Xuzhou 221116, China Email: [email protected]

Journal of Applied Nonlinear Dynamics Volume 4, Issue 3

September 2015

Contents Crises in Chaotic Pendulum with Fuzzy Uncertainty Ling Hong, Jun Jiang, Jian-Qiao Sun……………………………….……...………......

215－221

On a Class of Generalized Hydrodynamic Type Systems of Equations V.E. Fedorov, P.N. Davydov………………………...……………..….…………………

223－228

Influence of Systematic Coupling Stiffness Parameter on Coupling Duffing System Lag Self-synchronization Characteristic Zhao-Hui Ren, Yu-Hang Xu, Yan-Long Han, Nan Zhang, Bang-Chun Wen…..……….....

229－237

Tribo-dynamics Analysis of Satellite-bone Multi-axis Linkage System Jimin Xu, Honglun Hong, Xiaoyang Yuan and Zhiming Zhao…..….….…………….....…

239－250

Stability and Bifurcation of a Nonlinear Aero-thermo-elastic Panel in Supersonic Flow Wei Kang, Yang Tang, Min Xu, Jia-Zhong Zhang…..….……..……………………..……

251－257

Nonlinear Effects of Dusty Plasmas using Homogenous Nonequilibrium Molecular Dynamics Simulations Aamir Shahzad, and Mao-Gang He………...………………..…..…………..................…

259－265

The Adaptive Synchronization of the Stochastic Fractional-order Complex Lorenz System Xiaojun Liu, Ling Hong………………...…………………..…….………………...……..

267－279

Nonlinear Dynamic Characteristic Analysis of Planetary Gear Transmission System for the Wind Turbine Zhao-Hui Ren, Liang Fu, Ying-Juan Liu, Shi-Hua Zhou, Bang-Chun Wen……...…...…...

281－294

Real-time 2D Concentration Measurement of CH4 in Oscillating Flames Using CT Tunable Diode Laser Absorption Spectroscopy Takhiro Kamimoto, Yoshihiro Deguchi, Ning Zhang, Ryosuke Nakao, Taku Takagi, JiaZhong Zhang…………………..……………………………………………………….….

295－303

Fluid-Structure Coupling Effects on the Aerodynamic Performance of Airfoil with a Local Flexible Structure at Low Reynolds Number Wei Kang, Min Xu, Jia-Zhong Zhang………...……………………..………….…….….

305－312

Modal Analyses of a Thin Shell with Constrained Layer Damping (CLD) Based on Rayleigh-Ritz Method Xu-Yuan Song, Hong-Jun Ren, Jing-Yu Zhai, Qing-Kai Han……………………….…….

313－327

The Method of High Order Fatigue Test of Thin Plate Composite Structure With Hard Coating Hui Li, Wei Sun, Zhong Luo, Bengchun Wen…………………………………..………….

329－337

Available online at https://lhscientificpublishing.com/Journals/JAND-Download.aspx

Printed in USA

Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics

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