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1 Acta Phys. Sin. Vol. 6, No. 5 () 553 (II) * (, 543 ) ( 3 ; 5 ),,,,,,,, :,,, PACS: a, 45..Hj 3,, 5., /,,, 3 3 :,,, ;, (memory hereditary),,, ( ) 6 9 ( ) 3 ( ) 6 3.,, Deng 4,5,,,,, * ( : 758,936), ( : E47), ( : IRT97) shenyongjun@6.com c Chinese Physical Society

2 Acta Phys. Sin. Vol. 6, No. 5 () 553,, 9 ( p ), 3 Duffing 9,3,,,, : mẍ(t) + kx(t) + cẋ(t) + K D p x(t) + K D p x(t) = F cos(ωt), () D p x(t) D p x(t) x(t) t ((n ) p n ) (n p n, n ), K K K >, K >.,, Caputo ( n < p < n) D p x(t) = Γ (n p) t x (n) (u) du, () (t u) p n+ Γ (z) Gamma, Γ (z + ) = zγ (z). k ω = m, εµ = c m, εk = K m, εk = K m, εf = F m, (3) () ẍ(t) + ω x(t) + εµẋ(t) + εk D p x(t) + εk D p x(t) = εf cos(ωt). (4), ω ω, ω ω, ω = ω + εσ (σ ), (4) ẍ(t) + ω x(t) =ε { f cos(ωt) + σx(t) µẋ(t) (5) k D p x(t) k D p x(t) }. (5) x = a cos ϕ, ẋ = aω sin ϕ, (6) ϕ = ωt + θ, ȧ = ω P (a, θ) + P (a, θ) + P 3 (a, θ) sin ϕ, (7a) a θ = ω P (a, θ) + P (a, θ) + P 3 (a, θ) cos ϕ, (7b) P (a, θ) =εf cos(ϕ θ) + σa cos ϕ + µaω sin ϕ, P (a, θ) = εk D p a cos ϕ, P 3 (a, θ) = εk D p a cos ϕ. 3, (7) T a θ ȧ = T ω P (a, θ) + P (a, θ) + P 3 (a, θ) sin ϕdϕ, (8a) a θ = T ω P (a, θ) + P (a, θ) + P 3 (a, θ) cos ϕdϕ, (8b) T T = π( P i (a, θ) (i =,, 3) ), T = ( P i (a, θ) (i =,, 3) ). (8) π ȧ = P (a, θ) sin ϕdϕ πω = εf sin θ εµa, ω (9a) a θ = π P (a, θ) cos ϕdϕ πω 553-

3 Acta Phys. Sin. Vol. 6, No. 5 () 553 = εf εσa cos θ ω ω. (9b) (8), B = T sin(ωt) t p = ω p Γ ( p) cos B = T cos(ωt) t p = ω p Γ ( p) sin ( pπ ( pπ ), ). () 9,3 (8a) ȧ = T εk = T T ω T ω εk ȧ = T T ω P (a, θ) sin ϕdϕ D p a cos ϕ sin ϕdϕ, () sin(ωt + θ) } { D p a cos(ωt + θ) = ( )n εak ω n Γ (n p ) T T { T t sin(ωu + θ) (t u) p n+ du } sin(ωt + θ) s = t u ds = du,. () ȧ = ( )n εak ω n Γ (n p ) T T { T t sin(ωt + θ ωs) ds sin(ωt + θ) } = ( )n εak ω n Γ (n p ) { t s p n+ T T cos(ωs) s p n+ ds sin(ωt + θ) sin(ωt + θ) } + ( )n εak ω n Γ (n p ) { t T T sin(ωs) s p n+ ds cos(ωt + θ) sin(ωt + θ) }. (3) A A = ( )n εak ω n 3 4Γ (n p ) T { ωt sin(ωt + θ) T t } cos(ωs) s p n+ ds ( )n εak ω n 3 4Γ (n p ) T T ωt sin(ωt + θ) cos(ωt). T t p n+ (4) () T, A = ( )n εak ω p (p n + )π sin, (5), (3) T, ȧ =A = ( )n εak ω p (p n + )π sin. (6a), a θ = ( )n εak ω p (p n + )π cos. (6b) (8) ȧ 3 = ( )n εak ω p (p n + )π cos, (7a) a θ 3 = ( )n εak ω p (p n + )π sin. (7b) 553-3

4 Acta Phys. Sin. Vol. 6, No. 5 () 553 (9), (6) (7), ȧ = εf ω sin θ εµa ( )n εak ω p (p n + )π sin ( )n εak ω p (p n + )π cos, (8a) a θ = εf εσa cos θ ω ω + ( )n εak ω p (p n + )π cos ( )n εak ω p (p n + )π sin. (8b) ȧ = F mω sin θ a m C(p), a θ = F ωa cos θ mω + a mω K(p), (9a) (9b) C(p) =c + ( ) n K ω p (p n + )π sin + ( ) n K ω p (p n + )π cos, (a) K(p) =k + ( ) n K ω p (p n + )π cos ( ) n K ω p (p n + )π sin, (b) (),. () : (),, ; K K, p p., p (n ),, p n, ;, n = ( p ), 9 p n,, p n, ;, n = ( p ),,,,, p, ( ).,,,,, 3 ( ). ȧ = θ =, āωc(p) = F sin θ, āk(p) mω ā = F cos θ. θ, (a) (b) ā { ω C (p) + K(p) mω } = F, () ā = F ω C (p) + K(p) mω, (3) ωc(p) θ = arctan K(p) mω. (4), a = ā + a, θ = θ + θ, (9), d a d θ = C(p) m = a F cos θ mω θ, F cos θ F sin θ mā a + θ. (5) ω māω (), θ C(p) det m λ ωā ā mω K(p) ω ā + =. mωā K(p) C(p) m λ (6) 553-4

5 Acta Phys. Sin. Vol. 6, No. 5 () 553 λ + C(p) m λ + C (p) 4m + λ, = C(p) m ± i mω K(p) ω =. (7) mω K(p) ω. (8) () (8) 9. K > K >, λ,, (3) K = K =, (), C(p) K(p) c k, ā = F ω c + k mω, (9) 3,33, 4 () p p, n =, p p. 4. (), m = 5, k =, c =.3, F =, K =, p =.6, K =, p =.4, (3), ω/ω, a/a, a F/k.,,,3 (), l D p x(t l ) h p C p j x(t l j), (3) j= t l = lh, h, C p j, : ( C p =, Cp j = + p ) C p j j. (3) h =.4, 3 s, s s,,,, 4. 9, K.3,., 3., ( (a) K =, (b) ). K,, ( ) ;,, ( ),, K :, p.,.4,.6,.8, 3 ( 3(a) K =, 3(b) ). p,, ( ) ;,, ( ),, p :, 553-5

6 Acta Phys. Sin. Vol. 6, No. 5 () 553 9,, ( ) p (a) K = ; (b) K = K (a) K = ; (b) K = 4.3 (b), K,, ( k + K ω p p π ) cos K (p ω p )π, (3) sin K(p),, K, ω, ω, ω =., K, p, p (3) 4 6, (3), K, p, p, ω =.. p (, K ),, ( K ω p p π ) cos K (p ω p )π, (33) sin K(p) k,, ω.,, K, ω, ω = ω, K, p, p (33)

7 Acta Phys. Sin. Vol. 6, No. 5 () p 4 K (a) ; (b) 7 K 8 p 5 p (a) ; (b) 7 9, (33), K, p, p, ω,, K, p, p, K ( 7), p ( 9)., p 553-7

8 Acta Phys. Sin. Vol. 6, No. 5 () 553, p p, p.5. 9 p 5,,,, :,,,, Oldham K B, Spanier J 974 The Fractional Calculus-Theory and Applications of Differentiation and Integration to Arbitrary Order (New York: Academic Press) p Podlubny I 999 Fractional Differential Equations (London: Academic) p 3 Petras I Fractional-Order Nonlinear System (China: Higher Education Press) p9 4 Rossikhin Y A, Shitikova M V Applied Mechanics Reviews Yang S P, Shen Y J 9 Chao, Solitons and Fractals Wang Z Z, Hu H Y Science China: Physics, Mechanics & Astronomy Wang Z Z, Du M L Shock and Vibration Rossikhin Y A, Shitikova M V 997 Acta Mechanica 9 9 Li G G, Zhu Z Y, Cheng C J Applied Mathematics and Mechanics 94 Cao J Y, Ma C B, Xie H, Jiang Z D ASME Journal of Computational and Nonlinear Dynamics 5 4 Wu X J, Lu H T, Shen S L 9 Physics Letters A Chen J H, Chen W C 8 Chaos, Solitons and Fractals Lu J G 6 Physics Letters A Zhang C F, Gao J F, Xu L 7 Acta Phys. Sin (in Chinese),, Liu C X 7 Acta Phys. Sin (in Chinese) Chen X R, Liu C X, Wang F Q, Li Q X 8 Acta Phys. Sin (in Chinese),,, Zhang R X, Yang Y, Yang S P 9 Acta Phys. Sin (in Chinese),, Hu J B, Xiao J, Zhao L D Acta Phys. Sin (in Chinese),, Li Q D, Chen S, Zhou P Chin. Phys. B 5 Zhang R X, Yang S P 9 Chin. Phys. B Qi D L, Wang Q, Yang J Chin. Phys. B 55 Wu Z M, Xie J Y 7 Chin. Phys Zhou P 7 Chin. Phys Deng W H, Li C P 8 Phys. Lett. A Deng W H. 7 Journal of Computational Physics Chen L C, Zhu W Q 9 Journal of Vibration and Control Wahi P, Chatterjee A 4 Nonlinear Dynamics Huang Z L, Jin X L 9 Journal of Sound and Vibration 39 9 Shen Y J, Yang S P, Xing H J Acta Phys. Sin (in Chinese),, Shen Y J, Yang S P, Xing H J, Gao G S Commun. Nonlinear Sci. Numer Simulat Sanders J A, Verhulst F, Murdock J 7 Averaging methods in nonlinear dynamical systems (New York: Springer) p5 3 Ni Z H 988 Vibration Mechanics p79 (Xi an: Xi an Jiaotong University Press) (in Chinese) 988 ( : ) Liu Y Z, Chen W L, Chen L Q 998 Vibration Mechanics p36 (Beijing: Higher Education Press) (in Chinese),, 998 ( : )

9 Acta Phys. Sin. Vol. 6, No. 5 () 553 Dynamical analysis of linear SDOF oscillator with fractional-order derivative (II) Shen Yong-Jun Yang Shao-Pu Xing Hai-Jun ( Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 543, China ) ( Received December ; revised manuscript received 5 January ) Abstract A linear single degree-of-freedom (SDOF) oscillator with two kinds of fractional-order derivatives is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical properties, including the fractional coefficients and the fractional orders in the two kinds of fractional-order derivatives, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and the results is entirely different from the results given in the existing literature. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The following analysis of the effects of the fractional parameters on the amplitude-frequency is presented, and it is found that the fractional coefficients and the fractional orders can affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness. Finally, the effects of the fractional coefficient in the second fractional-order derivative on resonance frequency are analyzed, and the design rule for the fractional coefficient in the second fractional-order derivative to meet the satisfactory vibration control performance is pointed out. Keywords: fractional-order derivative, averaging method, approximately analytical solution, vibration control PACS: a, 45..Hj * Project supported by the National Natural Science Foundation of China (Grant Nos. 758, 936), the Natural Science Funds for Distinguished Young Scholar of Hebei Province (Grant No. E47), the Program for New Century Excellent Talents in University and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT97). shenyongjun@6.com 553-9

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