The first and second moments of response variables for SDOF system with pseudoelastic material are obtained by a direct linearization procedure. This procedure is an adaptation of well-known statistical linearization methods, and provides concise, model-independent linearization coefficients. The method can be applied to systems that incorporate any SMA hysteresis model having a differential constitutive equation, and can be used for zero and non-zero mean random vibration. This implementation eliminates the effort of deriving linearization coefficients for new SMA hysteresis model. In this paper the complete statistical response of SDOF system containing a mass and a bar made of SMA is obtained via direct linearization procedure. The model considered is modification of phenomenological one-dimensional constitutive model originally proposed by Graesser and Cozzarelli, which provides the capability to model both the martensitic twinning hysteresis and martensitic-austenite pseudoelastic behavior, typical of shape memory alloys. Response statistics for zero mean random vibration are obtained. Furthermore, non-zero mean analysis of the system is carried out and comparisons are made with Monte Carlo simulation

Duval, L.

Noori, Mohammad N.

Seelecke, S.

English

DigitalCommons@CalPoly

DIRECT STOCHASTIC EQUIVALENT LINEARIZATION OF SDOF SMART MECHANICAL SYSTEM L. Duval and M.N. Noori North Carolina State University, Raleigh, NC 27695-7910, USA ldduval@eos.ncsu.edu , mnoori@eos.ncsu.edu S. Seelecke Technical University Berlin, Institute ofThermodynamics Sekr, HF2, Stra{3e des 17, Juni 135, D-10623, Berlin, Germany seelecke@thermo08.pi.tu-berlin.de Abstract The first and second moments of response variables for SDOF system with pseudoelastic material are obtained by a direct linearization procedure. This procedure is an adaptation of well-known statistical linearization methods, and provides concise, model-independent linearization coefficients. The method can be applied to systems that incorporate any SMA hysteresis model having a differential constitutive equation, and can be used for zero and non-zero mean random vibration. This implementation eliminates the effort of deriving linearization coefficients for new SMA hysteresis model. In this paper the complete statistical response of SDOF system containing a mass and a bar made of SMA is obtained via direct linearization procedure. The model considered is modification of phenomenological one-dimensional constitutive model originally proposed by Graesser and Cozzarelli, which provides the capability to model both the martensitic twinning hysteresis and martensitic-austenite pseudoelastic behavior, typical of shape memory alloys. Response statistics for zero mean random vibration are obtained. Furthermore, non-zero mean analysis of the system is carried out and comparisons are made with Monte Carlo simulation. Introduction The analysis of mechanical systems under intense random (i.e., earthquake) excitation, requires the incorporation of nonlinear stress-strain relationships into solution methods for random vibration. Exact solutions for the response statistics of nonlinear systems are very limited, particularly when the material behavior is hysteretic, having a multi-value forced­deformation pattern with non-conservative energy dissipation. The lack of closed form solutions for hysteretic random oscillators necessitates the use of accurate methods for approximate solution. Since the first random vibration study of an inelastic system, performed by Caughey (1960), many researches have applied various approximation methods to nonlinear stochastic analysis. These include amongst others, moment closure techniques (Iyengar et al., 1978; Crandall 1980; Noori and Davoodi, 1990), stochastic averaging (Roberts 1986; Roberts and Spanos, 1986), equivalent nonlinear systems (Nielsel et al., 1990), and an energy dissipation balancing procedure (Cai and Lin, 1990). While these methods offer alternative means for the analysis of hysteretic or generally nonlinear systems, the most widely published of approximation methods has been that of Equivalent Linearization. Formulations of the linearization procedure can be found in the literature (Caughey 1963; Iwan 1973; Spanos 1981). Equivalent Linearization has been used to analyze SDOF systems using several differential mathematical models for hysteresis with approximated solutions comparing favorably to results obtained by pure Monte Carlo Simulation. While the procedure of linearization for each of these models has been identical, following the format presented by Atalik and Utku (1976), each unique constitutive equation has required a unique analytical derivation for the equivalent linear system parameters. Based upon examination of the derivations associated with these models, it is reasonable to conclude that as hysteresis models have evolve into more sophisticated analytical forms to describe progressively more generalized stress-strain behavior, the analytical effort of linearization has become increasingly complex and tedious. The focus of this paper is to present a procedure for obtaining the response statistics (first and second moments) of a SODF system which incorporate a pseudoelastic shape memory alloy, that is generally applicable to any rate-type SMA constitutive model. The expressions for equivalent linear system parameters will be posed in a form that is convenient for numerical solution, so as to allow for a fully automated linearization process, to be accomplished by computer code. A Modified Shape Memory Alloy Constitutive Model Since SMA is intended to be used over a wide range of strain, the Graesser and Cozzarelli model (Graesser and Cozzarelli, 1991) is extended to represent the hardening of the SMA after the transition to martensite is completed as it has been suggested in the literature (Wilde et ai., 2000). As the load increases, the pure martensite follows the elastic response with modulus Em' The modified model is of the form: (Ia,b) + [3a j EE Z + 2azsign(E)EE + ai]ulIJ(E) where the functions Uj(E), Ull(E) and ulll(E)are defined as: IEI~em (2a,b)otherwise _ {I, EE > a and E) < lEI < EmUlll(E ) - . (2c)0, otherWIse As it has been pointed out by Wilde the terms EmEujj(E)in Eq. (Ia) describes the elastic behavior of martensite, which is activated when the strain is higher than Em' Strain, Em' defines the point when the transformation of SMA form austenite to martensite is completed. The smooth transition from the curve of slope by to slope Em is obtained by adding the last term in Eq. (Ia) that is evaluated only during loading and for strain lOt < lEI < Em' The constant aj' az and a3 control the curvature of the transition. Those constants are selected so that the slopes of the function defined by the last term at points E) and Em are consistent with the slopes of SMA "plastic" behavior and martensite elastic response. The smoothness of the transition is governed by the selection of slope at strain Ez. Figure 1 explains the introduced constants in a graphical form. System Modeling Let consider a simpler mechanical system shown in Figure 2. The length and the cross­section area of the bar are denoted by Land A. We assume that the mass m of the bar is negligible. Ifxmeasures the displacement of the mass relative to the base and ymeasures the displacement of the frame, the governing equation of motion of the system is given by Figure 1. Schematic stress-strain relations of the extended hysteretic model of SMA (after Wilde et. al., 2000) mx + ex + kx + AO" =mF(t) (3) where double dots denotes the derivative respect to time. Here a dashpot has been added to account for the viscous damping associated with metals in general which is no accounted for in the constitutive model. Following previous work (Feng and Li 1996) by using the bar length L as the characteristic length and 1/OJ as the characteristic time where OJ is the dimensional natural frequency of the system we can non-dimensionalize Eq. (4). £+2(t:+OJ2E+ 0" =f(t) 0- =g(t:,O") (4)E Since the SMA constitutive model we are considering is rate independent, Eq. (4) is not affected by rescaling. Hereg(E,O") is a general nonlinear constitutive equation that defines the SMA hysteretic behavior. Mass Figure 2. SDOF system containing a shape memory component Statistical Equivalent Linearization The method of equivalent linearization has been proven effective for providing first and second moments of the response for rate-type hysteretic systems. The procedure calls for writing the system ofEqs. (4) in state-space format; letting ql =£, q2 =E and q3 =(5 be non­zero mean random variables, them from Eqs, (4): (5) Taking the expected value of each equation above, with J.li = qi and J.l f =E[/(t)] : . 2 /' J.l3 . [ ( )]J.ll = J.l2 J.l2 =-w J.ll - 2':>J.l2 - E + J.lf J.l3 =E g q2,q3 (6) Subtracting Eqs. (6) from (5) yields zero-mean random variables, Yi =qi - J.li and F =(I - J.lf) such that: · =_w2y - 2/''' - Y3 + FYl =Y2 Y2 1 ~2 E The nonlinear Eq. (7) involving the SMA constitutive hysteresis model g(q2,q3) may be replaced with the linear form: (8) where Ce and K e are unknown equivalent damping and stiffness parameters, respectively. This substitution allows for the generation of a linear system of ODE's whose solution provides response covariance, (9) Equation (6) and (7) govern the means and covariance for the system response. They may be solved simultaneously for a suitable choice for Ce and K e • Excitation of Gaussian White Noise simplifieshlY/'J =hlY3FJ =U, hlY2FJ =nwo , where Wo is Power Spectral Density (PSD). Together, Eqs. (6) and (9) govern the means and covariance of system response. They may be solved numerically if a suitable choice for Ce and Kecan be made. For closure, Ce and Ke are chosen to minimize the error of the linear approximation in Eq. (7), in a statistical sense. Following the results of Atalik and Utkul2 under certain conditions, including the assumption that the state variables E and (5 are jointly Gaussian, the choice which minimizes error is given by the following: C =E[dg(E,(5)] K=E[dg(E,(5)] (10)dE d(5ee These equations for equivalent stiffness and damping coefficients are obtained from an expression that is applicable to MDOF systems with general (not necessarily hysteretic) nonlinearity. It should be noted that the values of these coefficients are system dependent, and must be continually updated as the system of ODE's for means and covariance's is solved numerically. Direct Implementation of Stochastic Linearization The hysteretic system ofEqs. (6) and (7) is an example of a system in which there is only one state variable that is governed by a nonlinear differential equation. This is an advantage to a system with multiple nonlinearities; and leads to the development of linearization coefficients that differs from the definitions given in Eq. (10). Another issue that contributes to this development is the fact that although a linear system is used as an approximation to the equations governing means and covariance's, the response dependence of its linear system prevents its analytical solution. This system dependence of the linearization coefficients influences their new definition. Consider the error of linearization; the difference between the nonlinear system and the approximation ofEq. (8): (11) Only the third variable is non-zero, therefore, the covariances of the error terms are all zero, except to the term: (12) Since Ce and K e are response dependent, and require continual re-evaluation in the numerical solution ofEq. (9), they may be treated as constants at each discrete time step, and brought outside the expected value operator. IfE[g]is assumed to be known at each time step, thenE[YzE[g]] = E[Yz]E[g] = OandE[Y3E[g]] = E[Y3]E[g] = 0, since Yzand Y3 are zero­mean random variables. Nothing that E[gyz] =E[gqz] - pzE[g] and E[gy3] =E[gq3] - P3E[g] , this lead us to the specific definition of linearization coefficients: C = S6E[gE] - SsE[g<J] - S6E[g]pz + SsE[g]P3 e S4S6- S; (13) K = S4E[g<J] - SsE[gE] - S4E[g]P3 + SsE[g]pz e S4S6- S; In essence, both the classical and the currently method of linearization are founded upon the same principle, the mean square minimization of expected error. The difference is in the way the process is implemented, with the current method being better suited for computational methods. The advantage to the above definition over that ofEq. (10) and (14) is that this is posed directly in terms of the desired response statistics and only three expected values, which also do not contain partial derivatives of the SMA hysteresis model. With the excitation being Gaussian white noise, E and <J can be assumed to have a jointly Gaussian probability density function, thereby allowing the evaluation of the three unknown expected values ElgJ' ElgEJ and Elg<JJ, and closure of the system. Using the mean and covariance response solutions of Eqs. (6) and (9), the expected values may be obtained through Gaussian-Quadrature integration. Whatever the choice of SMA hysteresis model, Eqs. (6), (9) and (13) remain unchanged, and the effort of linearization is reduced to simply changing the function of g(E, <J) in the critical expected values. Often, this redefinition, or introduction of a new SMA model, requires the altering of only one line of computer code, and most importantly, completely avoids the type of analytic effort previously associated with equivalent linearization. Conclusions A procedure has been presented for obtaining the first and second moments of response variables for a SDOF system with a pseudoelastic component. The method is an adaptation of classical linearization procedures and eliminates the effort of analytical derivations associated with each SMA hysteresis model. Case of study using a modification to the well­known Graesser and Cozzarelli SMA constitutive model has been presented. The method is accurate when judge by Monte Carlo results, and possibly more accurate then classical linearization routines, due to reduced numerical error. It is possible that with the analytical effort removed from the task of random vibration analysis, this generally applicable procedure may promote the introduction of more sophisticated SMA hysteresis models. Acknowledgements Luis Duval gratefully acknowledges to Profs. Krzysztof Wilde, Yozo Fujino and Mr. Paolo Gardoni for providing their input in the use ofthe extended superelastic model of Shape Memory Alloys References T.K. Caughey, "Random excitation of a system with bilinear hysteresis", J. Applied Mechanics, 12, pp. 649­652,1960. R.N. Iyengar and P.K. Dash, "Study	 of the random vibration of nonlinear systems by the gaussian closure technique", J. Applied Mechanics, 45, pp. 393-397, 1978. S.H. Crandall, "Non-gaussian closure for random vibration	 of nonlinear oscillators", Int. Journal Nonlinear Mechanics, 15, pp. 303-313,1980. M.N. Noori and H. Davoodi, "Comparison between equivalent linearization and gaussian closure for random vibration analysis of several nonlinear systems", Int. Journal Engineering Science, 28, pp. 897-905, 1990. J.B. Roberts, "Application	 of averaging methods to randomly excited hysteretic systems", Proc. IUTAM Symposium on Nonlinear Stochastic Dynamic Engineering Systems, Springer-Verlag, Berlin, pp. 361-374, 1986. J.B. Roberts and P.D. Spanos, "Stochastic averaging: An approximated method	 of solving random vibration problems ", Int. Journal Nonlinear Mechanics, 21, pp. 202-209, 1986. S.R.K Nielsen,	 KJ. M~rk and P. Thoft-Christensen, "Stochastic response of hysteretic systems", Structural Safety, 9, pp. 59-71,1990. G.Q. Cai and Y.K. Lin," On randomly excited hysteretic structures", J. Applied Mechanics ASME, 57, pp. 442­448, 1990. P.D. Spanos, "Stochastic linearization in structural dynamics", Applied Mechanics Review, 34, pp. 1-8, 1981. W.D. Iwan, "A generalization of the concept of equivalent linearization", Int. Journal Nonlinear Mechanics, 8, pp.279-287,1973. T.K. Caughey, "Equivalent linearization techniques", J. Acoust. Soc. Am., 35(N11), pp. 1706-1711, 1963. T.S. Atalik and	 S. Utku, "Stochastic linearization of MDOF nonlinear systems", Earthquake Eng. Struct. Dyn. 4, pp. 411-420, 1976. EJ. Graesser and F.A. Cozzarelli, "Shape memory alloys as a new materials for aseismic isolation", J. Eng. Mechanics, 117(11), pp. 2590-2608, 1991. K. Wilde, P. Gardoni and	 Y. Fujino, "Base isolation system with shape memory alloy device for elevated highway bridges", Engineering Structures, 22, pp. 222-229, 2000. Z.e. Feng and D.Z. Li, "Dynamics of a mechanical system with a shape memory alloy bar", J. Int. Mat. Sys. and Struct., 7, pp. 399-410, 1996. 

Direct Stochastic Equivalent Linearization of SDOF Smart Mechanical System

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Direct Stochastic Equivalent Linearization of SDOF Smart Mechanical System

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