830. Nonlinear dynamic characteristics of SMA simply supported beam in axial stochastic excitation

Transcription

1 8. Noninear dynamic characteristics of SMA simpy supported beam in axia stochastic excitation Zhi-Wen Zhu 1, Wen-Ya Xie, Jia Xu 1 Schoo of Mechanica Engineering, Tianjin University, 9 Weijin Road, Tianjin 7, P. R. China Institute of Eectrica and Mechanica Engineering, Hainan University 58 Renmin Road, Haikou 578, P. R. China Tianjin Key Laboratory of Noninear Dynamics and Chaos Contro 9 Weijin Road, Tianjin 7, P. R. China E-mai: 1 zhuzhiwen@tju.edu.cn, xiewenya199@yahoo.cn, xujia_d@16.com (Received 1 Juy 1; accepted 4 September 1) Abstract. In this paper, noninear dynamic characteristics of shape memory aoy (SMA) simpy supported beam in axia stochastic excitation were studied. Von de Po noninear difference item was introduced to interpret the hysteresis phenomenon of the strain-stress curve of SMA, and the hysteretic noninear dynamic mode of SMA simpy supported beam in axia stochastic excitation was deveoped. The oca stochastic stabiity of the system was anayzed according to the argest Lyapunov exponent, and the goba stochastic stabiity of the system was discussed in singuar boundary theory. The steady-state probabiity density function and the joint probabiity density function of the system were obtained in quasi-nonintegrabe Hamitonian system theory. The resut of simuation shows that the stabiity of the trivia soution varies with bifurcation parameter, and stochastic Hopf bifurcation appears in the process. The resut is hepfu to stochastic bifurcation contro to SMA simpy supported beam. Keywords: shape memory aoy (SMA), hysteretic noninearity, stochastic bifurcation. Introduction Shape Memory Aoy (SMA) is a kind of smart materias and appied in engineering fied widey. It has many specia characteristics such as shape memory effect, arge damping and super-easticity, based on which the SMA smart structure can be designed to reduce engineering vibration. SMA simpy supported beam is a kind of basic smart structure, which was appied in vibration reduction fied widey and has compex noninear dynamica characteristics. Lau anayzed vibration characteristics of SMA beams with different boundary conditions [1]. Liew studied the pseudoeastic behavior of a SMA beam by the eement-free Gaerkin method []. Zbiciak discussed dynamic characteristics of pseudoeastic SMA beam []. Scarpa deveoped spectra eement formuation for SMA beams under random vibration excitation [4]. Hashemi deveoped the dynamic mode of SMA beam [5]. Coet anayzed vibration behavior of SMA beam under dynamica oading [6]. This paper aims to offer a kind of anaysis method to noninear dynamica characteristic of SMA simpy supported beam in axia stochastic excitation in theoreticay. Von de Po noninear difference item was introduced to interpret the hysteresis phenomenon of strain-stress curve of SMA, and the hysteretic noninear dynamic mode of SMA simpy supported beam in axia stochastic excitation was deveoped. The oca stochastic stabiity of the system was anayzed according to the argest Lyapunov exponent, and the goba stochastic stabiity of the system was discussed in singuar boundary theory. The steady-state probabiity density function and the joint probabiity density function were obtained in quasi-nonintegrabe Hamitonian system theory. Finay, the theoretic resut was proved by simuation. VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN

2 Hysteresis Noninear Mode of SMA Simpy Supported Beam in Axia Stochastic Excitation The strain-stress curve of SMA was shown in Fig. 1. Obviousy, there is hysteretic noninearity in the strain-stress curve of SMA. Most of SMA modes were based on thermodynamics theory and micromechanics theory, where the percentage content of martensite was taken as main variabe of stress-strain equation. As resuts, those SMA modes were mosty shown as equations with subsection function or doube integra function, and hard to be anayzed in theory [7-1]. Usuay, research resuts to those modes can ony be obtained by numerica method or experiment method [14-18]. In this paper, Von de Po hysteretic cyce mode was introduced to describe the hysteretic noninear characteristic of SMA. Fig. 1. The strain-stress curve of SMA The initia Von de Po hysteretic mode describes hysteretic oop which is symmetrica about the initia point (, ). It can be shown as foows: x y= f ( x) = f ( x) + a 1 xɺ (1) b where f ( ) x is skeeton curve of hysteretic oop and usuay expressed in poynomia function, a and b are coefficients which determine the difference between the skeeton curve and the rea curve. Supposing the strain-stress curve of SMA is symmetrica about the point G ( ε, σ ), the strain-stress curve of SMA can be shown as foows: ε ε σ σ = b1 ( ε ε) + b ( ε ε) + b 1 ɺ ε () b 4 where σ is stress, ε is strain, b i ( i= 1,,, 4) are coefficients, skeeton curve is chosen as: f( x) = b1 x+ b x. b4 = ε since the oading curve has the same vaue as the unoading curve when ε =, and σ b1ε bε = because the initia stress of SMA must be avoided for industry appication, so Eq. can be rewritten as foows: 16 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN

3 σ = aε + a ε + aε + ( a ε aε ) ɺ ε () b where: b a1 = b1 + bε, a = bε, a = b, a4 =, a5 =. b4 b4 Mode of SMA simpy supported beam in axia stochastic excitation was shown in Fig., where w( x, t ) is dispacements of points of SMA beam, N is axia excitation, N = N c ξ ( t), N is initia excitation, c N is coefficient, ξ ( t) is Gauss white noise whose mean is zero and intensity is D, D >. N Fig.. The mode of SMA simpy supported beam in axia stochastic excitation In this paper, tension and compression were assumed as symmetrica, so the neutra axis was ocated in the geometrica center. A rectanguar cross-section under bending moment M was shown in Fig.. Fig.. The cross-section and curvature of the assumed beam The boundary conditions of SMA simpy supported beam can be written as foows: x= : w=, w = x xx = : w=, w xx = In main mode, w can be shown as foows: π w( t, x) = u( t)sin x (4) where u( t ) is ampitude of the fundamenta mode. The geometrica deformation condition is: w ε = y x VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN (5) 161

4 According to the reationship between stress and strain shown in Eq., the bending moment M can be shown as foows: h / σ [ / 1ε ε ε ( 4ε 5ε ) ɺ ε ] (6) h M = yda= b y a + a + a + a a dy Considering Eq. 4 and Eq. 5, we obtained: π 6 π 6 π M = I1a1π usin x Iaπ u sin x + Ia5π u sin x u ɺ (7) 5 bh bh where I1 =, I = The dynamica equation of SMA simpy supported beam can be shown as foows: M w w w x x t t + N + c + ρ A = (8) where c is inear damping coefficient, ρ is density of the SMA and A is area of the crosssection of SMA beam. Considering Eq. 4 and Eq. 7, we obtained differentia equation of vibration ampitude with parametric excitation as foows: a1 I1π Fπ c a5 Iπ a Iπ Fπ uɺɺ + u+ u u u u ( t) ɺ + = ξ (9) ρa ρ A 4ρ A 4ρ A ρa Introducing non-dimensiona transformation: 1 1 * a1 I1 * a t = t, 1I1 Ω = Ω ρa ρa and ignoring the symbo *, we obtained the non-dimensiona motion equation as foows: ɺɺ ( ) ɺ ( ) (1) u+ ku+ α u + µ γ u u= euξ t where: k N π 4 = π, p a1 1 Iπ a1i1 8 8 a =, Iπ c α =, µ =, a5i π γ =, 4a I ρ Aa I 4 ρ Aa I c N π e=. a I 1 1 Stochastic Stabiity Anaysis of System Let u= q, uɺ = p, Eq. 1 can aso be shown as foows: qɺ = p pɺ = kq α q ( µ γ q ) p + eqξ ( t) (11) Considering that the items αq, ( µ γ q ) p and eqξ ( t) are a sma, the Hamitonian function of Eq. 11 can be shown as foows: 16 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN

5 1 ( ) H = p + kq (1) According to the quasi-nonintegrabe Hamitonian system theory, the Hamitonian function H(t) converges weaky in probabiity to an one-dimensiona Ito diffusion process. The averaged Ito equation about the Hamitonian function can be shown as foows: dh = m( H ) dt+ σ ( H ) db( t) (1) where B( t ) is standard Wiener process, m( H ) and σ ( H ) are drift and diffusion coefficients of Ito stochastic process, which can be obtained in stochastic averaging method: µ k a γ = + (14) / m( H ) H H H k k π k σ ( H ) = H (15) k Based on quasi-nonintegrabe Hamitonian system theory [19], the argest Lyapunov exponent of a inearized system is defined as foows: 1 λ = im n Z ( t, z ) (16) t t The inearized Ito differentia equation can be shown as foows after the system was inearized in the trivia soution H = : dh = m () Hdt+ σ () HdB( t) (17) Then the associated argest Lyapunov exponent is: { m [ σ ] } 1 1/ µ λ= im n H = () () / / = (18) t t 4k Now the oca stochastic stabiity of the system can be discussed as foows: 1) The trivia soution H = is ocay asymptotic stabe if and ony if λ <, which means µ > ; k ) The trivia soution H = is ocay asymptotic unstabe if and ony if λ >, which means µ < ; k ) Bifurcation shoud appear near the trivia soution H = if and ony if λ =, which means µ =. k The argest Lyapunov exponent can ony estimate the oca stabiity. In this paper, the boundary cassification method was used to anayze the goba stabiity of the trivia soution of the system. Generay, the boundaries of diffusion process are singuar, and the boundary VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN

6 cassification is often determined by diffusion exponent, drift exponent and character vaue []. When H : k a µ γ m( H ) = H H + H / k k π k σ ( H ) = H H k k So: µ k α =, β = 1, c = 1 where µ k H k α is diffusion exponent, β is drift exponent, c is character vaue, is eft boundary. Thus, the eft boundary H = beongs to the first kind of singuar boundary. According to the cassification for singuar boundary [], we obtained: c > 1; 1) The eft boundary H = is repusivey natura if ) The eft boundary H = is stricty natura if c = 1; ) The eft boundary H = is attractivey natura if c < 1. Simiary, the right boundary H = beongs to the second kind of singuar boundary. When H : µ k a γ m( H) = H H + H / / k k π k a γ H H k π + k σ ( H ) = H H k k So: kπγ 4a α r =, β r =, cr = kπ where r is the right boundary. Thus, the right boundary H = is an entrance boundary. The necessary and sufficient conditions for gobay asymptotic stabiity of the trivia soution require that the eft boundary be attractivey natura and the right boundary be entrance. Thus, the trivia soution H = is gobay asymptoticay stabe ony c < 1, which means µ >. The infuence of the character vaue to the stabiity was shown in Fig. 4. k Stochastic Bifurcation and Simuation Fig. 4. The infuence of the character vaue to the stabiity The averaged FPK equation of Eq. 11 is: 164 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN

7 f σ = [ m( H) f ] + t H H 1 [ ( H) f ] (19) where f is probabiity density. Thus, the stationary probabiity density function of the system is: dσ f ( H ) = A exp m( t) / dt dt a γ AH H H k π H η σ = exp 1/ () µ k where A is a normaization constant, η =. The joint probabiity density function of the system is: η 1 1 4a π kγ 1 1 f ( p, q) = A p + kq exp p kq + π k (1) The resuts of numerica simuation were shown in Fig. 5 Fig. 8, where k =. 5, D =. 5, c =.5, = 1, M = 4, E = 1, A = 8 1, I = f H Fig. 5. The steady-state probabiity density when µ = Fig. 6. The joint probabiity density when µ = From Fig. 5 Fig. 8, we can see that: 1 1) p = and q = when H = since H = ( p + kq ), so the trivia soution H = corresponds to the origin (, ) in the figure of joint probabiity density; 165 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN

8 ) The steady-state probabiity density of H = is the max when µ =, which is ocay asymptotic unstabe; ) The steady-state probabiity density of H = decreases when µ increases, and its stabiity varies from unstabe to stabe; 4) Stochastic Hopf bifurcation appears when the bifurcation parameter µ varies. We can obviousy see that there is a imit cyce in the figures of joint probabiity density, which is accord with the resut of stochastic stabiity. f H Fig. 7. The steady-state probabiity density when µ =.1 Fig. 8. The joint probabiity density when µ =.1 Concusions In this paper, noninear dynamic characteristics of shape memory aoy (SMA) simpy supported beam in axia stochastic excitation were studied. Von de Po noninear difference item was introduced to interpret the hysteresis phenomenon of the strain-stress curve of SMA, and the hysteretic noninear dynamic mode of SMA simpy supported beam in axia stochastic excitation was deveoped. The oca stochastic stabiity of the system was anayzed according to the argest Lyapunov exponent, and the goba stochastic stabiity of the system was discussed in singuar boundary theory. The steady-state probabiity density function and the joint probabiity density function of the system were obtained in quasi-nonintegrabe Hamitonian system theory. The resut of simuation shows that the stabiity of the trivia soution varies from unstabe to stabe when the bifurcation parameter µ varies, and stochastic Hopf bifurcation appears in the process. The resut is hepfu to stochastic bifurcation contro to SMA simpy supported beam. 166 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN

9 Acknowedgements The authors gratefuy acknowedge the support of Natura Science Foundation of China (NSFC) through Grant No. 11 and the Ph. D. Programs Foundation of Ministry of Education of China through Grant No References [1] Lau K. T. Vibration characteristics of SMA composite beams with different boundary conditions. Materias and sign, Vo., Issue 8,, p [] Liew K. M., Ren J., Kitipornchai S. Anaysis of the pseudoeastic behavior of a SMA beam by the eement-free Gaerkin method. Engineering Anaysis with Boundary Eements, Vo. 8, Issue 5, 4, p [] Zbiciak A. Dynamic anaysis of pseudoeastic SMA beam. Internationa Journa of Mechanica Sciences, Vo. 5, Issue 1, 1, p [4] Scarpa F., Ruzzene M., Hassan M. R. Spectra eement formuation for SMA beams under random vibration excitation. Smart Structures and Materias, Vo., Issue 8, 4, p [5] Hashemi S. M. T., Khadem S. E. Modeing and anaysis of the vibration behavior of a shape memory aoy beam. Internationa Journa of Mechanica Sciences, Vo. 48, Issue 1, 6, p [6] Coet M., Fotete E., Lexceent C. Anaysis of the behavior of a shape memory aoy beam under dynamica oading. European Journa Mechanics A-Soids, Vo., Issue 4, 1, p [7] Tanaka K. A thermomechanica sketch of shape memory effect: one-dimensiona tensie behavior. Res. Mechanics, Vo. 18, Issue, 1986, p [8] Boyd J. G., Lagoudas D. C. Thermodynamica constitutive mode for shape memory materias. Internationa Journa of Pasticity, Vo. 1, Issue 6, 1996, p [9] Brinson L. C. One-dimensiona constitutive behavior of shape memory aoys: thermomechanica derivation with nonconstant materia functions and redefined martensite interna variabe. Journa of Inteigent Materia Systems and Structures, Vo. 4, Issue, 199, p [1] Graesser E. J., Cozzarei F. A. A proposed three-dimensiona constitutive mode for shape memory aoys. Journa of Inteigent Materia Systems and Structures, Vo. 5, Issue 1, 1994, p [11] Ivshin Y., Pence T. J. Thermomechanica mode for a one variant shape memory materia. Journa of Inteigent Materia Systems and Structures, Vo. 5, Issue 4, 1994, p [1] Auricchio F., Lubiner J. Uniaxia mode for shape-memory aoys. Internationa Journa of Soids and Structures, Vo. 4, Issue 7, 1997, p [1] Zhu Z. W., Wang J., Xu J. Modeing of shape memory aoy based on hysteretic noninear theory. Appied Mechanics and Materias, Vo. 44, Issue, 1, p [14] Savi M. A., Pacheeo P. M., Braga M. B. Chaos in shape memory two-bar truss. Internationa Journa of Non-Linear Mechanics, Vo. 7, Issue 8,, p [15] Sohn J. W., Han Y. M., Choi S. B. Vibration and position tracking contro of a fexibe beam using SMA wire actuators. Journa of Vibration and Contro, Vo. 15, Issue, 9, p [16] Li H., Liu Z. Q., Ou J. P. Experimenta study of a simpe reinforced concrete beam temporariy strengthened by SMA wires foowed by permanent strengthening with CFRP pates. Engineering Structures, Vo., Issue, 8, p [17] Wang C. G., Tan H. F. Experimenta and numerica studies on wrinking contro of an infated beam using SMA wires. Smart Structures and Materias, Vo. 19, Issue 1, 1, p [18] Speicher M. S., s Roches R., Leon R. T. Experimenta resuts of a NiTi shape memory aoy (SMA)-based recentering beam-coumn connection. Engineering Structures, Vo., Issue 9, 11, p [19] Zhu W. Q., Huang Z. L. Stochastic stabiity of quasi non-integrabe Hamitonian systems. Journa of Sound and Vibration, Vo. 18, Issue 5, 1998, p [] Zhu W. Q., Huang Z. L. Stochastic Hopf bifurcation of quasi non-integrabe Hamitonian systems. Internationa Journa of Non-Linear Mechanics, Vo. 4, Issue, 1999, p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 1. VOLUME 14, ISSUE. ISSN