Model-free nonlinear restoring force identification for SMA dampers with double Chebyshev polynomials: approach and validation

Transcription

1 Nonlinear Dyn (215) 82: DOI 1.17/s ORIGINAL PAPER Model-free nonlinear restoring force identification for SMA dampers with double Chebyshev polynomials: approach and validation Bin Xu Jia He Shirley J. Dyke Received: 12 November 214 / Accepted: 1 July 215 / Published online: 14 July 215 Springer Science+Business Media Dordrecht 215 Abstract The initiation and propagation of damage such as cracks in an engineering structure under dynamic loadings is a nonlinear process. Strictly speaking, conventional eigenvalue and eigenvector extraction-based damage identification approaches are suitable for linear systems only. Due to the unique nonlinearities associated with each civil engineering structure, it would be inefficient to attempt to express the nonlinear restoring force (NRF) of an engineering structure such as a reinforced concrete structure in a parametric form. Consequently, it is highly desirable to develop a general nonlinear identification approach to achieve structural damage detection in both qualitative and quantitative ways without the assumption on the parametric model of the hysteretic behavior. In this paper, based on a double Chebyshev polynomial function, a time-domain identification approach is proposed for identifying both the structural NRF and the mass distribution for multi-degree-of-freedom B. Xu (B) College of Civil Engineering, Hunan University, Yuelu Mountain, Changsha 4182, Hunan, People s Republic of China binxu@hnu.edu.cn J. He Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, People s Republic of China S. J. Dyke School of Mechanical Engineering, Purdue University, 585 Stadium Mall Drive, West Lafayette, IN , USA (MDOF) structures under incomplete dynamic loadings. As a typical nonlinear material, shape memory alloy (SMA) is introduced to a MDOF structural model to mimic the nonlinear behavior under dynamic loadings. The feasibility and robustness of the proposed approach is validated via (1) a numerical MDOF structure model equipped with a SMA damper whose restoring force is described by a double-flag-shaped nonlinear model, and (2) an experimental study on a fourstory shear building structure model equipped with an in-house design of a SMA damper used to mimic the structural NRF under incomplete impact loadings. The identified NRF is compared with the theoretical values and the measurements in experiment. Both numerical and experimental results show that the proposed approach is capable of identifying both the mass distribution and structural nonlinearity under dynamic loadings, and could be potentially used for damage initiation and propagation monitoring during vibration of engineering structures under dynamic excitations. Keywords Nonlinear restoring force Identification Double Chebyshev polynomial function Incomplete excitations Double-flag-shaped nonlinear model SMA damper Dynamic experiment 1 Introduction Due to the rapid increase in the number of deteriorating structures and damaged structures under strong dynamic loadings such as earthquakes, it is crucial

2 158 B. Xu et al. to evaluate their current reliability, performance, and condition for the prevention of potential catastrophic events, as well as for the remaining life and dynamic loading spectrum estimation, retrofitting, and strengthening. In the last decades, the development of vibrationbased structural damage detection approaches has been one of the most active research areas in civil and infrastructural engineering as it pertains across a structure s life cycle and considers the condition, safety and performance evaluation, and maintenance. Much progress has been made in this area, and comprehensive literature surveys can be found [1 3]. It has been widely recognized that most of the currently available vibration-based model updating and identification methods for structural damage detection are based on the basic idea of extracting the eigenvalues, mode shapes, and/or mode shape derivatives from structural dynamic response measurements and, strictly speaking, are only suitable for linear systems. However, nonlinearities exist widely in engineering structures, as the occurrence of a fault in an initially linear structure will, in many cases, result in nonlinear behavior especially for engineering structures under strong dynamic excitations such as earthquakes and other natural and manmade hazards. All engineering structures are nonlinear to some extent, and the nonlinearity can be caused by several factors such as crack initiation and development in reinforced concrete (RC) structures, and the looseness and presence of friction characteristics of steel structural joints. RC structure currently is still the most widely employed structures in civil and infrastructural engineering structures. The damage initiation and propagation procedure of RC structures under earthquakes is a typical nonlinear process due to the cracking of concrete materials and the yielding of rebars at beams, columns, and even joints. The damage detection problems for engineering structures under strong dynamic loadings such as earthquakes, impacts, and even blasts are important topics for post-event performance evaluation and should be considered as a nonlinear identification problem. Therefore, the identification of nonlinearities has received increasing attention among civil engineers and is considered of high importance for diagnosis of faults in engineering structures. With the development of powerful computational techniques, great progress has been made on the numerical modeling of the dynamic response analysis of complex nonlinear dynamic systems, especially those with known/assumed nonlinear models. However, the inverse analysis of nonlinear structures is still a challenging task due to the individuality of different nonlinearity of civil engineering systems with different kinds of construction materials and structural types. Early contributions to the identification of nonlinear structural models can be traced back to the 197s [4]. Since then, numerous methods have been developed for highly individualistic nonlinear systems [5]. Worden and Tomlinson [6] described numerous approaches for the detection, identification, and modeling of nonlinear systems in their textbook. Using state variables of nonlinear systems, Masri and Caughey proposed a fruitful approach, named the restoring force surface (RFS) method, to identify and express the nonlinear system characteristics in terms of orthogonal functions, which was first developed for a wide class of single-degreeof-freedom (SDOF) dynamic systems, and then was extended for MDOF systems [7,8]. In the following years, based on the use of power series expansions, a relatively simple nonparametric technique for the identification of nonlinearities of a variety of discrete nonlinear vibrating systems was developed by Yang and Ibrahim [9]. The method was extremely appealing in its simplicity because its starting point was Newton s second law. Masri et al. [1,11] developed a self-starting, multistage, time-domain approach for the nonparametric identification of nonlinear MDOF systems undergoing free oscillations or subjected to arbitrary direct force excitations and/or non-uniform support motions. Kerschen et al. [12] proposed an algorithm based on a Bayesian inference approach for the screening of nonlinear system models. Based on the applied excitation(s) and resulting acceleration, a general procedure was presented for the direct identification of the state equation of complex nonlinear system by Masri et al. [13]. In the following years, Masri et al. [14,15] extended their work to present a general data-based approach by using power series fitting techniques for developing reduced-order, nonparametric models in nonlinear MDOF systems. The development of a general data-based and model-free identification approach is extremely important for civil engineering structures because it is usually hard to know the actual hysteretic behavior of an engineering structure during strong dynamic loadings in prior And to express it in a parametric form. More recently, Tasbihgoo et al. [16] discussed two broad classes of restoring force identification methods. One class relied on the representation of the system restoring forces in a

3 Model-free nonlinear restoring force identification 159 polynomial-basis format, while the other used artificial neural networks (ANNs) as a general mapping tool to describe any arbitrary nonlinear function to map the complex transformations for developing nonparametric models of nonlinear MDOF systems. A computationally efficient ANN model where the need of hidden layers was eliminated by expanding the input pattern by Chebyshev polynomials was presented for the identification of unknown dynamic nonlinear discrete time systems [17]. Based on the usage of the Chebyshev polynomials, the nonparametric restoring force mapping method was employed for identifying the nonlinear behavior of a passive elasto-magnetic suspension system [18]. Rémond et al. [19] presented an improved method for identification of the structural parameters through expansion using a Chebyshev polynomial basis. In the work by Yun et al. [2], model-free identification techniques utilizing nonparametric system identification approaches were used to detect the changes of the nonlinear system, to interpret the physical meaning of the detected changes, and to quantify the uncertainty of the detected system changes. An experimental study was carried out on a test-bed structure to evaluate the effectiveness and reliability of a datadriven nonparametric technique for the identification of nonlinearities in uncertain MDOF chain-like systems [21]. In most of the nonlinear identification approaches above, all excitations (inputs) applied to the degrees of freedom in the nonlinear structural system are assumed to be known and available for the identification. However, in many practical situations, it is either too difficult to excite all of the DOFs of an engineering structure, especially the complex and large-scale civil engineering structures, or it is not easy to obtain complete measurements of the external excitations due to inaccessibility and the limitation of the number of available sensors for excitation measurements. To handle this problem, great efforts have been made by many investigators in recent years. Mohammad et al. [22] proposed a direct parameter estimation method for identifying the physical parameters of linear and nonlinear MDOF structures with only one input excitation. Using a recursive least squares estimation with unknown inputs (RLSE-UI) approach and extended Kalman filter (EKF), Yang et al. [23 25] identified the parameters of a nonlinear structure with known parametric model as well as the unmeasured excitations. More recently, based on the basic idea of equivalent linearization and the symmetry of the identified stiffness matrix, Xu et al. [26,27] proposed a data-based modelfree hysteresis identification approach for nonlinear systems under incomplete excitations. By employing the power series polynomial model (PSPM) to represent the structural nonlinearity, Xu et al. [28] proposed a time-domain data-based approach for identifying the NRFs under spatially incomplete excitations. Moreover, He et al. [29] developed nonlinear identification approach for simultaneously identifying the NRF and the partially unknown excitations by the combination of the PSPM and adaptive iterative least square estimation. In this paper, double Chebyshev polynomials (DCP) involving the instantaneous values of the state variables of a MDOF structural system are proposed to represent system nonlinearities. Based on the incomplete excitations applied to the structure and the corresponding response time series, each coefficient of the DCP is identified by means of a standard least squares algorithm, without any assumptions on the parametric model of the NRF, and the prior knowledge of the mass distribution of the system. Moreover, as compared with the nonparametric identification method proposed by Masri et al. [1,11], there is no eigenvalue problem involved in the proposed approach and the expression of the nonlinear restoring force is directly determined in the time domain by using least square techniques. The feasibility and robustness of the proposed approach is validated firstly via numerical simulation with a 4-DOF structural model incorporating an SMA damper with a double-flag-shaped nonlinear model, which is a typical nonlinear member with hysteretic nonlinearity, and via forced vibration experimental measurements with a four-story steel frame building model equipped with an SMA damper developed by the authors, which was employed to introduce NRF in the model structural system. With the consideration of measurement noise, the structural NRFs identified by the proposed DCP-based approach in the numerical example are compared with those of the previously proposed PSPM-based approach and with the theoretical values from the double-flag-shaped nonlinear model. Results show that the NRFs identified by the proposed DCP-based approach have good agreement with the theoretical ones and provide more accurate identification for SMA damper model as compared with the previously proposed PSPM-based approach. In the dynamic test, the damper force is first identified

4 151 B. Xu et al. by the proposed DCP-based approach and then compared with the corresponding measurements which are obtained from a force transducer in the test setup. The results show that when the constitutive model of the nonlinear structural members with the SMA damper and the structural mass distribution are both assumed to be unknown, the proposed method is still able to identify the NRF with an acceptable accuracy. Thus, this approach provides a promising approach for damage detection where structural nonlinearity needs to be considered. The identified NRF can be used for both quantitative and qualitative evaluation of the nonlinear behavior and the energy consumption of the structural member during vibration, which is useful for post-event condition evaluation of engineering structures. 2 The double Chebyshev polynomials (DCP)-based approach for NRF identification In most of the currently available vibration-based structural damage detection algorithms, the structural damage is identified in the form of the stiffness loss in a structural member or substructural levels by the use of structural parameters identification and model updating algorithms, using eigenvalues or eigenvectors and/or their deviations extracted from structural dynamic response time histories. Strictly speaking, the approaches based on eigenvalue or eigenvector extraction are suitable for linear structures only. Instead of stiffness, restoring forces can describe the linear and nonlinear behavior of structures or structural members under dynamic loadings directly, and moreover, the hysteresis curve (typically found in all structural materials undergoing significant deformations) can be employed to evaluate the energy dissipated during vibration and to identify the damage initiation and development process quantitatively. The identification of restoring forces is more meaningful for nonlinear structures. Unfortunately, the restoring force of a structure under dynamic loadings cannot be measured directly. Also, the true restoring force model of a certain structure such as an RC structure in civil engineering is hard to know a priori and is too difficult to be modeled in a parametric model accurately due to the individuality of different construction materials and structural types. Consequently, efficient and general restoring force identification methodologies using structural dynamic measurements are crucial for damage detection, life cycle performance evaluation, remaining service life forecasting and even retrofitting, and strengthening of engineering structures after strong dynamic loadings. Consider a discrete n-dof lumped-mass chainlike structural system incorporating nonlinear nonconservative dissipative members and subjected to directly applied forces P(t). The motion of this nonlinear system can be governed by the following equation of motion: M ẍ(t) + R[ẋ(t),x(t),g] = P(t) (1) where x(t) is the displacement vector of order n, M is the constant matrix that characterizes the inertia forces, R is the nonlinear non-conservative restoring force vector, g is the vector of system-specific parameters, and P(t) is the directly external forces, respectively. In this study, the NRF of R i,i 1 between two DOFs i and i 1 of the system is assumed to be expressed in a general double Chebyshev polynomial function as shown in the following equation: R i,i 1 [ẋ (t), x (t), g] R i,i 1 [ vi,i 1, s i.i 1, g ] h= j= T (α) = 1 T 1 (α) = α T 2 (α) = 2α 2 1 T 3 (α) = 4α 3 3α T 4 (α) = 8α 4 8α gi,i 1,h, non j T ( ) ( ) h v i,i 1 Tj s i,i 1 (2) (3) where R i,i 1 [ẋ (t), x (t), g] is the NRF between the i-th DOF and the (i 1)-th DOF, gi,i 1,h, non j is the coefficient of the double Chebyshev polynomials, k and q are integers which depend on the nature and extent of the nonlinearity of the system, and T (α) is the first kind of Chebyshev polynomial, v i,i 1 and s i,i 1 are relativevelocity and relative-displacement vectors between the two adjacent DOFs of i and i 1, i.e., for a chain-like lumped-mass system, v i,i 1 and s i,i 1 are the interstory velocity and inter-story displacement vectors and can be defined as v i,i 1 =ẋ i ẋ i 1, s i,i 1 = x i x i 1. Each T n (α) is orthogonal to each other over the interval [ 1, 1]. Because the dynamic response is not exactly within the interval [ 1, 1], the measured data of

5 Model-free nonlinear restoring force identification 1511 dynamic response should be normalized and be mapped onto the appropriate region of [ 1,1] by the linear transformations as follows: v i,i 1 = v i,i 1 ( ) v i,i 1,max + v i,i 1,min /2 ( ) (4) vi,i 1,max v i,i 1,min /2 s i,i 1 = s i,i 1 ( s i,i 1,max + s i,i 1,min ) /2 ( si,i 1,max s i,i 1,min ) /2 (5) Consequently, the equation of motion for each DOF can be rearranged as follows: m 1 ẍ 1 (t) + + h= j= h= j=... m i ẍ i (t) + + h= j= g1,,h, non j T ( ) ( ) h v 1, Tj s 1, g1,2,h, non j T ( ) ( ) h v 1,2 Tj s 1,2 = p1 (t) h= j=... m n ẍ n (t) + = p n (t) gi,i 1,h, non j T ( ) ( ) h v i,i 1 Tj s i,i 1 gi,i+1,h, non j T ( ) ( ) h v i,i+1 Tj s i,i+1 = pi (t) h= j= gn,n 1,h, non j T ( ) ( ) h v n,n 1 Tj s n,n 1 (6) Consider the nonlinear system mentioned above under arbitrary incomplete excitations, the rank of P(t) defined in Eq. (1) will be less than the order n. Consequently, the unknown coefficients including the mass distribution cannot be uniquely determined by implementing least square algorithms directly. Without loss of generality, assume the nth DOF of the structure is not excited and the (n 1)th DOF of the structure is excited, the equation of motion of the nth and the (n 1)th DOF can be written in the following two equations respectively, m n ẍ n (t) = h= j= h= j= g non n,n 1,h, j T h(v n,n 1 )T j (s n,n 1 ) (7) m n 1 ẍ n 1 (t) + gn 1,n 2,h, non j T h(v n 1,n 2 )T j(s n 1,n 2 ) + h= j= g non n 1,n, j,h T h(v n 1,n )T j(s n 1,n ) = p n 1 (t) (8) The algebraic coefficients including the mass of the (n-1)-th DOF and the NRF between the (n-1)-th DOF and the connecting DOFs can be identified with an optimization algorithm such as a least square technique using the applied excitations and the corresponding normalized response time series. For the n-th DOF, which is not excited, its mass can be identified according to the below Eq. (9) usinga linear fitting technique. m n ẍ n (t) = = h= j= h= j= g non n,n 1,h, j T h(v n,n 1 )T j(s n,n 1 ) g non n 1,n,h, j T h(v n 1,n )T j(s n 1,n ) (9) Then, the algebraic coefficients including the mass of the DOFs excited and the corresponding NRF can be identified using least squares techniques in combination with the normalized dynamic measurement and excitation information. The mass corresponding to the DOF with no excitation applied can be identified based on the NRF between it and the DOF with excitation. The process can be carried out in sequence until all mass distribution and NRFs corresponding to all DOFs are identified. 3 Numerical simulation validation with a nonlinear MDOF model equipped with a SMA damper 3.1 Description of a 4-DOF nonlinear numerical model with a SMA damper To illustrate the accuracy of the proposed DCP-model approach, a 4-DOF lumped-mass structure equipped with a SMA damper is considered as a nonlinear numerical simulation example as shown in Fig. 1. Each story of the model is associated with one horizontal DOF. The properties of the linear portions of the structure without the SMA damper are m i = 3 kg, k i = N/m, and c i = 2 N s/m, (i = 1, 2, 3, 4). In this numerical model, to mimic the nonlinear behavior, a SMA

6 1512 B. Xu et al. Fig. 1 4-DOF nonlinear numerical model with a SMA damper SMA Damper M4 M3 K4,C4 k SMA F non = tan( 1 ) SMA 1 α k b SMA 2 = tan( α 2 α c 2 a d ) M2 K3,C3 K2,C2 d c α 1 a b o s a s b s d s c s M1 Fig. 2 Hysteretic model of the SMA damper K1,C1 damper, which is widely used as a typical energy dissipation device in engineering structures for vibration control, is introduced on the 4th floor of the numerical model as shown in Fig. 1. In the numerical validation, a more general case is considered. Only the 3rd floor of the model is excited by a random force, and the corresponding responses of the nonlinear structure are determined by the Newmark- β method. To consider the effects of measurement noise, all structural response measurements are simulated by tures [3,31]. SMAs have distinctive nonlinear characteristics, and there are many available models for SMAs in terms of one-dimension (1-D) or three-dimension (3- D), such as the exponential model, the cosine model, the polynomial model, and so forth [32 34]. Moreover, the technical reviews for the introduction of the development of SMAs can be also found in some literatures [35]. In this study, a double-flag constitutive model which is one of the most widely employed model for SMA dampers as shown in Fig. 2 is adopted for the dynamic response calculation of the 4-DOF numerical model with SMA damper [36 38].The expression of the hysteretic behavior of the SMA damper in a doubleflag form can be given by the following equations: F SMA non = k ( 1 SMA S (oab, ao, o a b, a o) k SMA 1 k SMA ) ( 2 Sb sgn(s) + k2 SMA S (bc, b c ) k SMA 1 k2 SMA ) ( (Sb S c ) sgn(s) + k1 SMA S (cd, c d ) k SMA 1 k2 SMA ) Sa sgn(s) + k2 SMA S (da, d a ) (1) superimposing a white noise with a 3 % noise-to-signal ratio in terms of root-mean-square (RMS) on the theoretically computed responses. The hysteretic performance of the nonlinear system is identified using the proposed DCP-based approach, and the results are discussed in the following sections. 3.2 Identification of NRF provided by SMA damper under incomplete excitations The SMA damper has a unique shape memory effect, super elastic property, high damping, good durability, and corrosion resistance, and can provide large deformations that can be restored, so it has been widely used in the vibration control of engineering struc- where k1 SMA and k2 SMA are the stiffness provided by SMA in different stages as shown in Fig. 2,sgn( ) is the sign function, and S is relative-displacement vectors. The definition of S a, S b, S c, and S d can be found in Fig. 2. In this example, the following values for the constitutive model of the SMA damper are used: S b =.6 m, S a =.4 S b, k1 SMA = N/m, and k2 SMA = N/m Case 1: noise free In order to verify the performance of the proposed approach for NRF identification when the structure are partially excited, the case when only the 3rd floor of the model as shown in Fig. 1 is excited by a set of random excitations is considered. The corresponding responses

7 Model-free nonlinear restoring force identification F1 (KN) F2 (KN) Acc.1 (m/s 2 ) Acc.2 (m/s 2 ) time (s) time (s) time (s) time (s) F3 (KN) F4 (KN) Acc.3 (m/s 2 ) Acc.4 (m/s 2 ) time (s) time (s) time (s) time (s) (a) (b) Vel.1 (m/s) Vel.2 (m/s) Disp.1 (m) Disp.2 (m) time (s) time (s) time (s) time (s) Vel.3 (m/s) Vel.4 (m/s) Disp.3 (m) Disp.4 (m) time (s) time (s) time (s) time (s) (c) (d) Fig. 3 The random excitations applied to each floor of the nonlinear structure with a SMA damper and the corresponding dynamic responses: a the excitation force; b the acceleration response time history; c the velocity response time history; and d the displacement response time history of the system are computed using the Newmark- β method and considered to be noise free in this case. The acceleration, velocity, and displacement responses along with their excitations are shown in Fig. 3. The relative velocity and relative displacement can be obtained and are mapped onto an appropriate region by the linear normalization approach according to Eqs. (4) and (5). Based on the time-domain measurements of the excitation and the normalized response, the expression of the total NRF applied on the 3rd floor, which includes the force from the DOFs 2 and 4, can be determined as shown in the following equation: F 3 = 3. ẍ 3 (t) T (v 3,2 )T (s 3,2 ) T (v 3,2 )T 1(s 3,2 ) T (v 3,2 )T 2(s 3,2 ) T (v 3,2 )T 3(s 3,2 ) T 1 (v 3,2 )T (s 3,2 ) T 1 (v 3,2 )T 1(s 3,2 ) T 1 (v 3,2 )T 2(s 3,2 ) T 1 (v 3,2 )T 3(s 3,2 ) T 2 (v 3,2 )T (s 3,2 )

8 1514 B. Xu et al T 2 (v 3,2 )T 1(s 3,2 ) T 2 (v 3,2 )T 2(s 3,2 ) T 2 (v 3,2 )T 3(s 3,2 ) T 3 (v 3,2 )T (s 3,2 ) T 3 (v 3,2 )T 1(s 3,2 ) T 3 (v 3,2 )T 2(s 3,2 ) T 3 (v 3,2 )T 3(s 3,2 ) T (v 3,4 )T (s 3,4 ) T (v 3,4 )T 1(s 3,4 ).53 T (v 3,4 )T 2(s 3,4 ) T (v 3,4 )T 3(s 3,4 ) T 1(v 3,4 )T (s 3,4 ) T 1 (v 3,4 )T 1(s 3,4 ) T 1 (v 3,4 )T 2(s 3,4 ) T 1 (v 3,4 )T 3(s 3,4 ) 2.68 T 2(v 3,4 )T (s 3,4 ) 67.5 T 2 (v 3,4 )T 1(s 3,4 ) +.5 T 2 (v 3,4 )T 2(s 3,4 ) 21. T 2 (v 3,4 )T 3(s 3,4 ) T 3(v 3,4 )T (s 3,4 ).91 T 3 (v 3,4 )T 1(s 3,4 ) T 3 (v 3,4 )T 2(s 3,4 ) +.62 T 3 (v 3,4 )T 3(s 3,4 ) (11) It is worth noting that the identified coefficient of the acceleration term stands for the mass of the corresponding DOF. From the above equation, the concentrated mass corresponding to the 3rd DOFs is m 3 = 3. kg, which is identical to the exact value set in the numerical model. The NRF applied on the 3rd floor by the 2nd floor can be shown in the following Eq. (12a), RF non 3,2 = T (v 3,2 )T (s 3,2 ) T (v 3,2 )T 1(s 3,2 ) T (v 3,2 )T 2(s 3,2 ) T (v 3,2 )T 3(s 3,2 ) T 1 (v 3,2 )T (s 3,2 ) T 1 (v 3,2 )T 1(s 3,2 ) T 1 (v 3,2 )T 2(s 3,2 ) T 1 (v 3,2 )T 3(s 3,2 ) T 2 (v 3,2 )T (s 3,2 ) T 2 (v 3,2 )T 1(s 3,2 ) T 2 (v 3,2 )T 2(s 3,2 ) T 2 (v 3,2 )T 3(s 3,2 ) T 3 (v 3,2 )T (s 3,2 ) T 3 (v 3,2 )T 1(s 3,2 ) T 3 (v 3,2 )T 2(s 3,2 ) T 3 (v 3,2 )T 3(s 3,2 ) (12a) And the total NRF applied on the 4th floor by the 3rd floor can be obtained and shown in the following Eq. (12b): RF non 4,3 = T (v 4,3 )T (s 4,3 ) T (v 4,3 )T 1(s 4,3 ) +.53 T (v 4,3 )T 2(s 4,3 ) T (v 4,3 )T 3(s 4,3 ) T 1 (v 4,3 )T (s 4,3 ) 5.4 T 1 (v 4,3 )T 1(s 4,3 ) T 1 (v 4,3 )T 2(s 4,3 ) 8.14 T 1 (v 4,3 )T 3(s 4,3 ) T 2 (v 4,3 )T (s 4,3 ) T 2 (v 4,3 )T 1(s 4,3 ).5 T 2 (v 4,3 )T 2(s 4,3 ) T 2 (v 4,3 )T 3(s 4,3 ) 51.9 T 3 (v 4,3 )T (s 4,3 ) +.91 T 3 (v 4,3 )T 1(s 4,3 ) T 3 (v 4,3 )T 2(s 4,3 ).62 T 3 (v 4,3 )T 3(s 4,3 ) (12b) Based on equation of the motion for the 4th floor as shown in Eq. (7), the mass of the 4th floor can be determined using a fitting algorithm. The identified mass corresponding to the 4th floor of the numerical model is m 4 = 3. kg, which is identical to the theoretical value of the model. Since the coefficients of the 3rd DOF are determined before, the parameters of the remaining DOFs can be determined in sequence. The identified mass of the 2nd and 1st floor are m 2 = 3. kg and m 1 = 3. kg, respectively, and the corresponding NRFs applied on the 2nd floor by the 1st floor and the corresponding NRFs between the 1st floor and the foundation could be also determined. It should be noted that the aforementioned identified NRF is the summation of the elastic restoring force, the damping effects of the linear structure itself, and the force provided by the nonlinear member of SMA damper.

9 Model-free nonlinear restoring force identification 1515 Fig. 4 Identified SMA damper restoring force with their theoretical values under incomplete excitations (noise free): a on the 1st floor; b on the 2nd floor; c on the 3rd floor; d on the 4th floor Identified SMA damper force (N) Identified SMA damper force (N) Inter-story shift on the 1st floor (m) (a) Inter-story shift on the 2nd floor (m) (b) Simulated Identified Identified SMA damprforce (N) SMA damper force (N) Inter-story shift on the 3rd floor (m) (c) Inter-story shift on the 4th floor (m) (d) In practical situations, it is almost impossible to directly measure the total NRF between two DOFs of a structure. However, the SMA damper force could be measured with a force transducer. Here, to investigate the performance of the identification approach for the NRF provided by the SMA damper, the SMA damper force is further determined by subtracting the linear elastic restoring force and structural damping force provided by the structure itself from the identified total NRF and then comparing it with the values of the theoretical values determined by its constitutive model. The comparison of the identified SMA damper forces with the corresponding theoretical values is shown in Fig. 4. Note that identical amplitude scales are applied to all the plots. It is easily seen from Fig. 4 that the SMA damper should be located on the 4th floor because the identified SMA damper forces on the other floors are very close to zero. Moreover, it can be found from Fig. 4d that the identified SMA damper restoring force is very close to the theoretical value obtained from its numerical model given by Eq. (1). These findings mean that the identified DCP function can be used to represent the NRFs of the employed SMA damper effectively and that the proposed method

10 1516 B. Xu et al. The difference between the identified value and the simulated value (N) Power series polynomial Double chebyshev polynomial time (s) Fig. 5 Comparison of identification error of the proposed approach with them of the previously proposed PSPM-based approach provides a suitable way to identify the NRF of nonlinear members in the system. To further investigate the effectiveness of the proposed DCP-based nonlinear behavior identification approach, the identification results of the power series polynomial (PSPM)-based nonlinear identification approach proposed by the authors is also employed herein and compared with that of the proposed approach in this study [28,29]. The errors of the identified SMA damper force by the previously proposed PSPM-based approach are plotted in Fig. 5 as a solid line, whereas those by the proposed DCP-based approach in this study are shown as a dashed line. For ease and clarity of comparison, only a short-time segment from.57 sec to.63 s is given in Fig. 5. The identification errors determined by the proposed method in this study are relatively smaller than those of the previous PSPM-based approach. It is because in addition to the nature characteristic of orthogonality, the Chebyshev polynomials have another distinct feature, saying the least squares error is distributed nearly uniformly in the time interval of interest [39]. This desirable feature is well known as equal-error (or equal-ripple) approximation. Similar results can be found in the remaining time segments and are not shown due to space constraints Additional case: 3% noise level in dynamic response measurements In practice, measurement noise is inevitable in dynamic structural response measurements. From this point of view, to investigate the robustness of the proposed approach, the effect of the measurement noise on the nonlinear behavior identification should be considered. Therefore, an identical random excitation is employed to excite the numerical model and all structural response measurements are simulated by superimposing a white noise with a 3 % noise-to-signal ratio in terms of root-mean-square (RMS) on the theoretically computed responses. Similar procedures shown in the previous section are implemented to identify the NRF of the system and SMA damper force. The identified SMA damper forces when noisepolluted measurements are employed are plotted in Fig. 6. It is obvious from Fig. 6 that the SMA damper is located on the 4th floor because the identified SMA damper forces on the 1st, 2nd, and 3rd floor are nearly zero. Moreover, it can be also found from Fig. 6d that the identified SMA force has good agreement with the theoretical one, although the response measurements have been contaminated by 3 % noise. 4 Test validation with a frame structure model equipped with an SMA damper 4.1 Description of the model structure and forced vibration test To illustrate the application of the proposed method with a real structure, a four-story steel frame building model with an SMA damper was constructed in the laboratory and forced vibration tests were carried out. The structure is.3 m.4 m in plane, 1.2 m in height, distributed evenly among the four floors, and has a total mass of kg. The cross section of the columns is 3 mm 5 mm, and the thickness of the floor plates is 1 mm. All the joints are connected using bolts. Figure 7 shows the nonlinear four-story steel frame structure with an SMA damper and vibration test setup. A SMA damper with an in-house design was installed on the 4th floor by four fixtures to induce a nonlinear hysteretic restoring force to the structure during vibration. The SMA damper uses a diameter of 8 mm Ti Ni alloy wire, as shown in Fig. 8. The two ends of the wire are fixed at two supports, and the middle point of the wire is fixed at a support which is connected with a sliding plate. The sliding plate is located in a groove in longitudinal direction of the base of the damper. So the movement of the sliding plate will pro-

11 Model-free nonlinear restoring force identification 1517 Fig. 6 The SMA damper force under incomplete excitations (3 % noise): a on the 1st floor; b on the 2nd floor; c on the 3rd floor; d on the 4th floor Identified SMA damper force (N) Identified SMA damper force (N) Inter-story shift on the 1st floor (m) (a) Inter-story shift on the 2nd floor (m) (b) Simulated Identified Identified SMA damprforce (N) SMA damper force (N) Inter-story shift on the 3rd floor (m) (c) Inter-story shift on the 4th floor (m) (d) duce symmetrical nonlinear restoring force. An impact hammer was employed to excite the structure, and the excitation force was measured directly by a piezoelectric force gauge in the head of the hammer. The corresponding accelerations of the four stories were measured directly by four accelerometers. The excitations and the acceleration responses were recorded simultaneously with a data acquisition system with a sampling frequency of 124 Hz. The velocity and displacement response of the structure was estimated by numerical integration of the measured acceleration time history. In our experiment, a hammer is employed to apply an impact force to the building model structure. The duration of the impact is short and includes high-frequency components. In order to preserve the impact force, a high-pass filter is used. Moreover, in this study, the velocity and displacement response of the structure is obtained from the numerical integration of the measured acceleration time history and the lower-frequency signals will have obvious effects on the integrated displacement and velocity responses. A noise with very low-frequency component (below.2 Hz in this study) would cause a drift when the numerical integration is carried out. Thus, the high filter is set to be.5 Hz

12 1518 B. Xu et al. Accelerometers Force transducer SMA damper Data acquisition system Charge amplifier In this case, our focus is to study the validity of the proposed approach for the identification of the NRF while only the 3rd floor of the structure is excited. The acceleration measurements are all plotted in Fig. 9. Let the sum of the integers k and q of the DCP function given in Eq. (2) be 3. The same procedures shown in the preceding numerical example are implemented to obtain the inter-story NRF as well as the mass distribution of the structure. Based on the impact force, measured acceleration, integrated velocity, and integrated displacement, the expression of the force applied on the 3rd floor can be obtained using a least squares method. Equations (12a) and (12b) show the identified inter-story restoring forces between the 4th and the 3rd stories and between the 3rd and the 2nd stories of the model structure. The identified masses of the 3rd floor and 4th floor are m 3 = kg, m 4 = 16.8 kg, respectively. Hammer with force transducer Fig. 7 Nonlinear multi-story structure with SMA dampers and vibration test setup Fig. 8 Self-developed SMA damper employed in the frame model structure to avoid this drift. The influence of high-frequency noise on the acceleration measurement is not dominant because the high-performance PCB accelerations are used. For comparison, the damping force of the SMA damper was measured by a piezoelectric force gauge and compared with the identification results. 4.2 NRF identification of SMA damper under incomplete excitations Identification of the nonlinear restoring force by SMA damper RF non 4,3 =.8 T (v 4,3 )T (s 4,3 ) T (v 4,3 )T 1(s 4,3 ) T (v 4,3 )T 2(s 4,3 ) T (v 4,3 )T 3(s 4,3 ) 17.4 T 1 (v 4,3 )T (s 4,3 ) T 1 (v 4,3 )T 1(s 4,3 ) T 1 (v 4,3 )T 2(s 4,3 ) 4.54 T 1 (v 4,3 )T 3(s 4,3 ) T 2 (v 4,3 )T (s 4,3 ) T 2 (v 4,3 )T 1(s 4,3 ) T 2 (v 4,3 )T 2(s 4,3 ) T 2 (v 4,3 )T 3(s 4,3 ) T 3 (v 4,3 )T (s 4,3 ) T 3 (v 4,3 )T 1(s 4,3 ) T 3 (v 4,3 )T 2(s 4,3 ) T 3 (v 4,3 )T 3(s 4,3 ) (12a) RF non 3,2 = 6.52 T (v 3,2 )T (s 3,2 ) T (v 3,2 )T 1(s 3,2 ) T (v 3,2 )T 2(s 3,2 ) T (v 3,2 )T 3(s 3,2 ) T 1 (v 3,2 )T (s 3,2 ) T 1 (v 3,2 )T 1(s 3,2 ) 4.43 T 1 (v 3,2 )T 2(s 3,2 ) T 1 (v 3,2 )T 3(s 3,2 ) T 2 (v 3,2 )T (s 3,2 ) 2.53 T 2 (v 3,2 )T 1(s 3,2 ) T 2 (v 3,2 )T 2(s 3,2 ) T 2 (v 3,2 )T 3(s 3,2 ) T 3 (v 3,2 )T (s 3,2 ) T 3 (v 3,2 )T 1(s 3,2 ) T 3 (v 3,2 )T 2(s 3,2 ) T 3 (v 3,2 )T 3(s 3,2 ) (12b)

13 Model-free nonlinear restoring force identification 1519 Fig. 9 The acceleration measurement time history under impact loadings 2 2 Acc.1 (m/s 2 ) Acc.2 (m/s 2 ) time (s) time (s) 2 Acc.3 (m/s 2 ) 3 Acc.4 (m/s 2 ) time (s) time (s) Similar to the numerical example discussed above, the NRFs of the remaining floors as well as the masses are determined in sequence. The identified masses are m 2 = kg and m 1 = kg, respectively. The identified NRFs between the 2nd and the 1st stories and between the 1st story and the ground of the model structure are determined. As mentioned before, the total inter-story NRF identified above consists of the elastic restoring force, the viscous damping force, and the SMA damper force; consequently, it is infeasible to be measured directly. To investigate the accuracy of the identified total NRFs, the SMA damper force is further identified and compared with the corresponding measurement. In order to identify the SMA damper force, the elastic restoring force and the viscous damping force provided by the linear structure itself should be subtracted from the total identified NRFs. Consequently, the above proposed methodology is also employed to identify the parameters of the linear structure without the SMA damper Identification of the linear structure and the SMA damper force In order to identify the elastic stiffness and damping coefficients of the linear structure without SMA damper, the SMA damper was removed from the model structure and similar forced vibration tests were carried out. The above approach is employed to identify the linear system without the SMA damper. Similarly, based on the time-domain vectors, each coefficient of the linear system can be identified by using least square estimation methods. The identified restoring force of the linear model can be expressed as follows: R1, lin s v 1 (13a) R2,1 lin s v 2 (13b) R3,2 lin s v 3 (13c) R4,3 lin s v 4 (13d) The nonlinear member force (i.e., the SMA damper force in this study) between the i-th DOF and the (i-1)- th DOF can be extracted from the total NRFs through the following equation: F non = h= j= k h= j= gi,i 1,h, non j T ( ) ( ) h v i,i 1 Tj s i,i 1 q gi,i 1,h, lin j v i,i 1s i,i 1 (k + q = 1) (14)

14 152 B. Xu et al. Fig. 1 The SMA damper force: a on the 1st floor; b on the 2nd floor; c on the 3rd floor; d on the 4th floor Identified SMA damper force (N) Inter-story shift on the 1st floor (m) x 1-3 (a) Identified SMA damper force (N) Inter-story shift on the 2nd floor (m) x 1-3 (b) Identified SMA damper force (N) Inter-story shift on the 3rd floor (m) x 1-3 (c) SMA damper force (N) Measured Identified Inter-story shift on the 4th floor (m) x 1-3 (d) The SMA damper force in this test study can be identified according to Eq. (14) and displayed in Fig. 1. It is obvious that the SMA damper is not located on the 3rd floor because the identified SMA force on the 3rd floors is very small. Similar results can be found on the 1st and 2nd floors. By comparing the identified SMA damping force with the measured damping force as shown in Fig. 1d, it can be concluded that the proposed method provides a reasonably accurate identification of the NRF, both qualitatively and quantitatively, although no information about the nature of the system s NRFs and mass distribution is utilized. Moreover, no assumption regarding the constitutive model of the damper in the structure is needed for the identification of the nonlinear behavior. 5 Concluding remarks A double Chebyshev polynomial function-based structural nonlinearity identification approach is proposed for the identification of NRF of a chain-like MDOF system under incomplete excitations. The feasibility and robustness of the proposed method is validated via numerical simulation with a 4-DOF system incorporating a parametric SMA model, and via experimental measurements with a four-story steel frame structure equipped with an in-house SMA damper design. The nonlinear devices are employed to provide NRFs in the system with unknown mathematical model. Moreover, the numerical results using the proposed method are compared with that of the previously proposed power series polynomial model-based approach. The following conclusions can be made based on the numerical and experimental validations. 1. The proposed DCP-based approach can be employed to identify the NRF induced by an SMA damper in a partially excited MDOF structure with acceptable accuracy. Though the model of the restoring force and the mass distribution of the model structure are unknown, the proposed approach is still capable of identifying the NRFs. 2. The proposed DCP-based approach is suitable for both linear and nonlinear dynamic systems and is demonstrated to be robust to measurement noise. A distinguishing feature of the proposed nonlinearity identification approach is that, other than the

15 Model-free nonlinear restoring force identification 1521 assumption of chain-like topology, it does not need information about the structure (such as structural characteristics or model class) and only the applied excitations at portions of the structure and the corresponding acceleration response measurements are required. 3. When comparing the capabilities of this method with the previously proposed NRF identification approach, the DCP-based approach proposed in this study identifies the NRF provided by an SMA damper with a higher accuracy. The proposed approach provides a general methodology for the identification of NRFs of engineering structures, which can be used for monitoring of damage initiation and development, and for evaluation of damage severity of engineering structures under dynamic loadings, where significant nonlinearities may be induced. 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