The present paper proposes the use of discrete time V olterra series expanded with Kautz ﬁlters for predicting the experimental response of a nonlinear beam, taking into account uncertainties that are inherent to the system. Nonlinear behavior is simulated considering large amplitude of vibration in a clamped-free beam, with a magnet near the free end to cause cubic stiﬀness eﬀect. The V olterra kernels and Kautz functions are estimated in a stochastic way, where the limits of conﬁdence are established to the system response and used as a reference model to detect damage. The detection approach is based on a nonlinear stochastic index and hypothesis test, considering diﬀerent levels of simulated damage, associated with loss of mass. The results have shown that considering the uncertainties when identifying the model is very important in the process of damage detection and the nonlinear metric proved to be more appropriated to describe the system response with better results mainly in the nonlinear regime of motion. c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: Nonlinear dynamics; V olterra series; Uncertainties quantiﬁcation; SHM. 1. Introduction The structural health monitoring (SHM) problem is more complicated in systems with intrinsic nonlinear behavior, because the nonlinearities can be confused with damage [1]. In addition, many mechanical systems can operate with strong nonlinear behavior that make them exhibit complex responses containing harmonic distortion, jumps, modalinteractions, bifurcation, and possible chaos [2]. In this situation, it is necessary to predict the nonlinear behavior inthe healthy condition and there is not a general approach to be applied in all nonlinear systems, once the nonlinearitieshave a lot of particularities. Furthermore, another source of errors in SHM is the presence of uncertainties, since any real system is uncertain with respect to the project nominal values, due to material imperfections, irregularities on the manufacturing process, noise in the measurements, etc. and the natural variability of systems can be confused withdamage [3,4]. To overcome these problems, the V olterra series expanded in the Kautz orthonormal basis have beenused to predict the dynamic responses of systems with nonlinear behavior [5,6]. However, in the previous papers, ∗Corresponding author. Tel.: +55-018-3743-1930. E-mail address: luisgustavovillani@gmail.com; samuel@dem.feis.unesp.br; americo@ime.uerj.br 1877-7058 c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017.Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000 www.elsevier.com/locate/procedia X International Conference on Structural Dynamics, EURODYN 2017 Damage detection in an uncertain nonlinear beam Luis Gustavo Giacon Villania, Samuel da Silvaa, Americo Cunha Jrb aUNESP - Universidade Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de Engenharia Mecˆ anica, Av. Brasil, 56, Ilha Solteira, 15385-000, SP , Brasil bUERJ - Universidade do Estado do Rio de Janeiro, NUMERICO – Nucleus of Modeling and Experimentation with Computers, R. S˜ ao Francisco Xavier, 524, Rio de Janeiro, 20550-900, RJ, Brasil Abstract The present paper proposes the use of discrete time V olterra series expanded with Kautz ﬁlters for predicting the experimental response of a nonlinear beam, taking into account uncertainties that are inherent to the system. Nonlinear behavior is simulated considering large amplitude of vibration in a clamped-free beam, with a magnet near the free end to cause cubic stiﬀness eﬀect. The V olterra kernels and Kautz functions are estimated in a stochastic way, where the limits of conﬁdence are established to the system response and used as a reference model to detect damage. The detection approach is based on a nonlinear stochastic index and hypothesis test, considering diﬀerent levels of simulated damage, associated with loss of mass. The results have shown that considering the uncertainties when identifying the model is very important in the process of damage detection and the nonlinear metric proved to be more appropriated to describe the system response with better results mainly in the nonlinear regime of motion. c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: Nonlinear dynamics; V olterra series; Uncertainties quantiﬁcation; SHM. 1. Introduction The structural health monitoring (SHM) problem is more complicated in systems with intrinsic nonlinear behavior, because the nonlinearities can be confused with damage [1]. In addition, many mechanical systems can operate with strong nonlinear behavior that make them exhibit complex responses containing harmonic distortion, jumps, modalinteractions, bifurcation, and possible chaos [2]. In this situation, it is necessary to predict the nonlinear behavior inthe healthy condition and there is not a general approach to be applied in all nonlinear systems, once the nonlinearitieshave a lot of particularities. Furthermore, another source of errors in SHM is the presence of uncertainties, since any real system is uncertain with respect to the project nominal values, due to material imperfections, irregularities on the manufacturing process, noise in the measurements, etc. and the natural variability of systems can be confused withdamage [3,4]. To overcome these problems, the V olterra series expanded in the Kautz orthonormal basis have beenused to predict the dynamic responses of systems with nonlinear behavior [5,6]. However, in the previous papers, ∗Corresponding author. Tel.: +55-018-3743-1930. E-mail address: luisgustavovillani@gmail.com; samuel@dem.feis.unesp.br; americo@ime.uerj.br 1877-7058 c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017.2 Villani et al. /Procedia Engineering 00 (2017) 000–000 the nonlinear model considered was deterministic and here this approach is extended considering the uncertainties in data acquisition and the variabilities of the system over diﬀerent days. The mainly advantage of this nonlinear robustmodel is the capability to diﬀer the presence of damage for the nonlinear behavior and uncertainties. The system instudy is a clamped-free beam with nonlinear behavior induced by the presence of a magnet near the free end, excitedin two diﬀerent levels, and the V olterra kernels are identiﬁed considering the data variation as stochastic referencemodel. The damage detection approach is based on the model limits and hypothesis test, considering diﬀerent levels of damage simulated with loss of mass. The linear and nonlinear analysis are compared since the V olterra series allow the separation of linear and nonlinear contributions in the response. The nonlinear approach proved to be moreadequate with a higher rate of damage detection. 2. The Stochastic Volterra Series In this paper, a stochastic version of the V olterra series is used to identify the nonlinear system of interest. The details to the deterministic version of the V olterra series theory can be found in [6–9]. Since only the data uncertainties are considered in this work, a parametric probabilistic approach is employed [10,11]. So, the model parameters subjected to uncertainties, the experimental output signal and consequently the V olterra kernels, are described as random processes, deﬁned on the probability space (Θ, Σ,P), where Θis a sample space, Σis aσ-algebra over Θ, and Pis a probability measure. It is assumed that any random variable θ∈Θ/mapsto→Y(θ)∈Rin this probabilistic setting, with probability distribution P Y(dy) onR, admits a probability density function (PDF) y/mapsto→pY(y) with respect to dy. The discrete time V olterra series can be used to represent diﬀerent types of nonlinear systems using the convolutionconcept [7]. Considering the presence of uncertainties, the random output signal y(θ,k) of a nonlinear system can be described by y(θ,k)=∞/summationdisplay η=1N1−1/summationdisplay n1=0...Nη−1/summationdisplay nη=0Hη(θ,n1,..., nη)η/productdisplay i=1u(k−ni)=y1(θ,k)+y2(θ,k)+y3(θ,k)+···, (1) where u(k) is the deterministic input signal, the random process y(θ,k) is the uncertain response of the system, (θ,n1, ..,nη)∈Θ×Zη/mapsto→Hη(θ,n1,..., nη) represents the random version of the η-order V olterra kernel, y1(θ,k) is the linear contribution and y2(θ,k)ey 3(θ,k) are the quadratic and cubic contributions of y(θ,k). The V olterra series’ inconvenience is the diﬃculty in its convergence when a high number of terms is used N1, ...,Nη. So, the kernels can be expanded in a base of orthonormal functions to minimize the problem, in this case, with the use of Kautz functions [12–14]. Now, the V olterra kernels can be deﬁned by Hη(θ,n1,..., nη)≈J1/summationdisplay i1=1...Jη/summationdisplay iη=1Bη/parenleftBig θ,i1,..., iη/parenrightBigη/productdisplay j=1ψij(θ,nj), (2) where J1,..., Jηare the number of samples in each orthonormal projections of the V olterra kernels, the random processes (θ, i1,..., iη)∈Θ×Zη/mapsto→Bη(θ,i1,..., iη) represent the random V olterra kernels expanded in the orthonormal basis. The deﬁnition of Kautz functions is related to the dynamic response of the system y(θ,k) and it depends on the physical parameters of the system (damping ratio and natural frequency). Then, the Kautz functions are assumed asrandom processes (θ, n)∈Θ×Z/mapsto→ψ(θ,n). In this context, the V olterra series can be rewritten as the multidimensional convolution between the random orthonormal kernels B η(θ,i1,..., iη) and the deterministic input signal ﬁltered by the random Kautz functions y(θ,k)≈∞/summationdisplay η=1J1/summationdisplay i1=1...Jη/summationdisplay iη=1Bη/parenleftBig θ,i1,..., iη/parenrightBigη/productdisplay j=1lij(θ,k), (3) where the random process lij(θ,k) is a simple ﬁltering of the deterministic input signal u(k) by the random Kautz function ψij(θ,nj). Then, for each realization θ, the terms of the orthonormal kernels Bη/parenleftBig θ,i1,..., iη/parenrightBig can be grouped into a vector Φand estimated by the least squares method Φ=(ΓTΓ)−1ΓTY, (4) where the matrix Γhas the ﬁltered input signal lij(θ,k) and Y=[y(θ,1)···y(θ,k)]for each realization θ, where kis the number of samples used. More information about the method can be found in [5,6,14]. Luis Gustavo Giacon Villania et al. / Procedia Engineering 199 (2017) 2090 –2095 2091 Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000 www.elsevier.com/locate/procedia X International Conference on Structural Dynamics, EURODYN 2017 Damage detection in an uncertain nonlinear beam Luis Gustavo Giacon Villania, Samuel da Silvaa, Americo Cunha Jrb aUNESP - Universidade Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de Engenharia Mecˆ anica, Av. Brasil, 56, Ilha Solteira, 15385-000, SP , Brasil bUERJ - Universidade do Estado do Rio de Janeiro, NUMERICO – Nucleus of Modeling and Experimentation with Computers, R. S˜ ao Francisco Xavier, 524, Rio de Janeiro, 20550-900, RJ, Brasil Abstract The present paper proposes the use of discrete time V olterra series expanded with Kautz ﬁlters for predicting the experimental response of a nonlinear beam, taking into account uncertainties that are inherent to the system. Nonlinear behavior is simulated considering large amplitude of vibration in a clamped-free beam, with a magnet near the free end to cause cubic stiﬀness eﬀect. The V olterra kernels and Kautz functions are estimated in a stochastic way, where the limits of conﬁdence are established to the system response and used as a reference model to detect damage. The detection approach is based on a nonlinear stochastic index and hypothesis test, considering diﬀerent levels of simulated damage, associated with loss of mass. The results have shown that considering the uncertainties when identifying the model is very important in the process of damage detection and the nonlinear metric proved to be more appropriated to describe the system response with better results mainly in the nonlinear regime of motion. c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: Nonlinear dynamics; V olterra series; Uncertainties quantiﬁcation; SHM. 1. Introduction The structural health monitoring (SHM) problem is more complicated in systems with intrinsic nonlinear behavior, because the nonlinearities can be confused with damage [1]. In addition, many mechanical systems can operate with strong nonlinear behavior that make them exhibit complex responses containing harmonic distortion, jumps, modalinteractions, bifurcation, and possible chaos [2]. In this situation, it is necessary to predict the nonlinear behavior inthe healthy condition and there is not a general approach to be applied in all nonlinear systems, once the nonlinearitieshave a lot of particularities. Furthermore, another source of errors in SHM is the presence of uncertainties, since any real system is uncertain with respect to the project nominal values, due to material imperfections, irregularities on the manufacturing process, noise in the measurements, etc. and the natural variability of systems can be confused withdamage [3,4]. To overcome these problems, the V olterra series expanded in the Kautz orthonormal basis have beenused to predict the dynamic responses of systems with nonlinear behavior [5,6]. However, in the previous papers, ∗Corresponding author. Tel.: +55-018-3743-1930. E-mail address: luisgustavovillani@gmail.com; samuel@dem.feis.unesp.br; americo@ime.uerj.br 1877-7058 c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017.Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000 www.elsevier.com/locate/procedia X International Conference on Structural Dynamics, EURODYN 2017 Damage detection in an uncertain nonlinear beam Luis Gustavo Giacon Villania, Samuel da Silvaa, Americo Cunha Jrb aUNESP - Universidade Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de Engenharia Mecˆ anica, Av. Brasil, 56, Ilha Solteira, 15385-000, SP , Brasil bUERJ - Universidade do Estado do Rio de Janeiro, NUMERICO – Nucleus of Modeling and Experimentation with Computers, R. S˜ ao Francisco Xavier, 524, Rio de Janeiro, 20550-900, RJ, Brasil Abstract The present paper proposes the use of discrete time V olterra series expanded with Kautz ﬁlters for predicting the experimental response of a nonlinear beam, taking into account uncertainties that are inherent to the system. Nonlinear behavior is simulated considering large amplitude of vibration in a clamped-free beam, with a magnet near the free end to cause cubic stiﬀness eﬀect. The V olterra kernels and Kautz functions are estimated in a stochastic way, where the limits of conﬁdence are established to the system response and used as a reference model to detect damage. The detection approach is based on a nonlinear stochastic index and hypothesis test, considering diﬀerent levels of simulated damage, associated with loss of mass. The results have shown that considering the uncertainties when identifying the model is very important in the process of damage detection and the nonlinear metric proved to be more appropriated to describe the system response with better results mainly in the nonlinear regime of motion. c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: Nonlinear dynamics; V olterra series; Uncertainties quantiﬁcation; SHM. 1. Introduction The structural health monitoring (SHM) problem is more complicated in systems with intrinsic nonlinear behavior, because the nonlinearities can be confused with damage [1]. In addition, many mechanical systems can operate with strong nonlinear behavior that make them exhibit complex responses containing harmonic distortion, jumps, modalinteractions, bifurcation, and possible chaos [2]. In this situation, it is necessary to predict the nonlinear behavior inthe healthy condition and there is not a general approach to be applied in all nonlinear systems, once the nonlinearitieshave a lot of particularities. Furthermore, another source of errors in SHM is the presence of uncertainties, since any real system is uncertain with respect to the project nominal values, due to material imperfections, irregularities on the manufacturing process, noise in the measurements, etc. and the natural variability of systems can be confused withdamage [3,4]. To overcome these problems, the V olterra series expanded in the Kautz orthonormal basis have beenused to predict the dynamic responses of systems with nonlinear behavior [5,6]. However, in the previous papers, ∗Corresponding author. Tel.: +55-018-3743-1930. E-mail address: luisgustavovillani@gmail.com; samuel@dem.feis.unesp.br; americo@ime.uerj.br 1877-7058 c/circlecopyrt2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017.2 Villani et al. /Procedia Engineering 00 (2017) 000–000 the nonlinear model considered was deterministic and here this approach is extended considering the uncertainties in data acquisition and the variabilities of the system over diﬀerent days. The mainly advantage of this nonlinear robustmodel is the capability to diﬀer the presence of damage for the nonlinear behavior and uncertainties. The system instudy is a clamped-free beam with nonlinear behavior induced by the presence of a magnet near the free end, excitedin two diﬀerent levels, and the V olterra kernels are identiﬁed considering the data variation as stochastic referencemodel. The damage detection approach is based on the model limits and hypothesis test, considering diﬀerent levels of damage simulated with loss of mass. The linear and nonlinear analysis are compared since the V olterra series allow the separation of linear and nonlinear contributions in the response. The nonlinear approach proved to be moreadequate with a higher rate of damage detection. 2. The Stochastic Volterra Series In this paper, a stochastic version of the V olterra series is used to identify the nonlinear system of interest. The details to the deterministic version of the V olterra series theory can be found in [6–9]. Since only the data uncertainties are considered in this work, a parametric probabilistic approach is employed [10,11]. So, the model parameters subjected to uncertainties, the experimental output signal and consequently the V olterra kernels, are described as random processes, deﬁned on the probability space (Θ, Σ,P), where Θis a sample space, Σis aσ-algebra over Θ, and Pis a probability measure. It is assumed that any random variable θ∈Θ/mapsto→Y(θ)∈Rin this probabilistic setting, with probability distribution P Y(dy) onR, admits a probability density function (PDF) y/mapsto→pY(y) with respect to dy. The discrete time V olterra series can be used to represent diﬀerent types of nonlinear systems using the convolutionconcept [7]. Considering the presence of uncertainties, the random output signal y(θ,k) of a nonlinear system can be described by y(θ,k)=∞/summationdisplay η=1N1−1/summationdisplay n1=0...Nη−1/summationdisplay nη=0Hη(θ,n1,..., nη)η/productdisplay i=1u(k−ni)=y1(θ,k)+y2(θ,k)+y3(θ,k)+···, (1) where u(k) is the deterministic input signal, the random process y(θ,k) is the uncertain response of the system, (θ,n1, ..,nη)∈Θ×Zη/mapsto→Hη(θ,n1,..., nη) represents the random version of the η-order V olterra kernel, y1(θ,k) is the linear contribution and y2(θ,k)ey 3(θ,k) are the quadratic and cubic contributions of y(θ,k). The V olterra series’ inconvenience is the diﬃculty in its convergence when a high number of terms is used N1, ...,Nη. So, the kernels can be expanded in a base of orthonormal functions to minimize the problem, in this case, with the use of Kautz functions [12–14]. Now, the V olterra kernels can be deﬁned by Hη(θ,n1,..., nη)≈J1/summationdisplay i1=1...Jη/summationdisplay iη=1Bη/parenleftBig θ,i1,..., iη/parenrightBigη/productdisplay j=1ψij(θ,nj), (2) where J1,..., Jηare the number of samples in each orthonormal projections of the V olterra kernels, the random processes (θ, i1,..., iη)∈Θ×Zη/mapsto→Bη(θ,i1,..., iη) represent the random V olterra kernels expanded in the orthonormal basis. The deﬁnition of Kautz functions is related to the dynamic response of the system y(θ,k) and it depends on the physical parameters of the system (damping ratio and natural frequency). Then, the Kautz functions are assumed asrandom processes (θ, n)∈Θ×Z/mapsto→ψ(θ,n). In this context, the V olterra series can be rewritten as the multidimensional convolution between the random orthonormal kernels B η(θ,i1,..., iη) and the deterministic input signal ﬁltered by the random Kautz functions y(θ,k)≈∞/summationdisplay η=1J1/summationdisplay i1=1...Jη/summationdisplay iη=1Bη/parenleftBig θ,i1,..., iη/parenrightBigη/productdisplay j=1lij(θ,k), (3) where the random process lij(θ,k) is a simple ﬁltering of the deterministic input signal u(k) by the random Kautz function ψij(θ,nj). Then, for each realization θ, the terms of the orthonormal kernels Bη/parenleftBig θ,i1,..., iη/parenrightBig can be grouped into a vector Φand estimated by the least squares method Φ=(ΓTΓ)−1ΓTY, (4) where the matrix Γhas the ﬁltered input signal lij(θ,k) and Y=[y(θ,1)···y(θ,k)]for each realization θ, where kis the number of samples used. More information about the method can be found in [5,6,14]. 2092 Luis Gustavo Giacon Villania et al. / Procedia Engineering 199 (2017) 2090 –2095 Villani et al. /Procedia Engineering 00 (2017) 000–000 3 2.1. Damage detection approach The identiﬁed Volterra kernels can be used as the reference model for the system. The stochastic response of the system can be obtained through convolution between the input signal and the Volterra kernels given by (Eq. 3) and the envelope of reliability, with probability Pc=99%, can be calculated based on the percentiles. The reliability interval at discrete time kis deﬁned by the set I99(k)=y(k)|r−(k)<=y(k)<=r+(k), (5) where y(k) is the model response in each discrete time k,r−(k) and r+(k) are the inferior and superior limits of the interval, respectively. If only the ﬁrst kernel is used y(k)=ylin(k)=y1(k), it means linear analysis; and if the ﬁrst three kernels are used y(k)=ynlin(k)=y1(k)+y2(k)+y3(k), nonlinear analysis. For each discrete time kthe distance between the experimental signal yexp(k) and the set I99(k) can be calculated by dlin(k)=inf/braceleftBig |yexp(k)−ylin(k)|for all y lin(k)∈I 99(k)/bracerightBig , (6) dnlin(k)=inf/braceleftBig |yexp(k)−ynlin(k)|for all y nlin(k)∈I 99(k)/bracerightBig . (7) So, by considering the calculated distances, a damage index can be computed, assuming the linear and nonlinear analysis λlin=||dlin|| A, (8) λnlin=||dnlin|| A, (9) where ||.||is the Euclidean norm operator and the excitation amplitude Ais used to normalize the index considering high and low levels of amplitude, making the comparison between the indexes in diﬀerent conditions easier. If the structure is in reference condition it is expected that the index has low value because the distance between the exper-imental signal and the response conﬁdence bands is zero or close to zero. The damage detection is done considering the hypothesis test /braceleftBiggH 0:λ≤Λ, H1:λ>Λ ,(10) where Λis the threshold value that is determined based on the reference data. The reference data has unknown distribution, thus the threshold is determined considering the interquartile range Λ= Q3+1.5(Q3−Q1), (11) where Q1andQ3are the ﬁrst and third quartiles and Λis the upper quartile of the Box plot theory [15]. By applying this equation, every quantity above the Λvalue is considered an outlier. In order to estimate the threshold value the index is applied to the reference data considering linear and nonlinear behavior. The next section presents an experimental example of this approach. 3. Application in a Nonlinear Beam The experimental setup is the same used in [6]. It is composed by an aluminum beam with dimensions of 300 ×18×3 [mm] with a steel mass connected to the free end to cause a magnetic interaction. To simulate structural variations and study the proposed approach, a bolted connection is placed 150 [mm] from the free end with four nuts with 1 [g]each one. The healthy state considers the four nuts in the system and the damage is simulated with the loss of mass. Figures 1 and 2 show the experimental setup used. A MODAL SHOP shaker (Model Number: K2004E01) is attached 50 mm from the clamped and used to excite the structure considering diﬀerent levels of voltage amplitude 0.01 V (low level), 0.10 V (medium level) and 0.15 V (high level). A vibrometer laser Polytec (Model: OFV-525/-5000-S) and a Dytran load cell (Model: 1022V) are used to measure the velocity at the free end of the beam and the force excitation, respectively. All signals are measured considering a sampling rate of 1024 Hz and 4096 samples saved using a m+p4 Villani et al. /Procedia Engineering 00 (2017) 000–000 Vib Pilot data acquisition system. The magnet interaction with the beam generates a nonlinear hardening behavior. This eﬀect is illustrated on the beam velocity curves obtained during the stepped sine test shown in Fig. 3. When the input signal is high, the beam presents a jump phenomenon that conﬁrms the nonlinear behavior of the system in this condition of excitation. More information about the hardening behavior and jump phenomenon can be found in [16]. Fig. 1. Experimental setup (photo) [6]. Fig. 2. Experimental setup (scheme) [6].10-1100101102 15 20 25 30 35Amplitude [(ms-1)/V] Frequency [Hz] Fig. 3. Beam velocity during a stepped sine test, where /trianglerepresents the low level of input signal (0.01 V), ◦the medium level (0.10 V) and/squarethe high level (0.15 V).3.1. Volterra kernels identiﬁcation This section describes the model identiﬁcation/validation through the Volterra kernels extraction. All tests used a sampling frequency of 1024 Hz and 4096 samples. The tests were repeated 156 times in diﬀerent days during 2 weeks to obtain large variation of data. The estimation of Volterra kernels was done in two steps. First, a chirp signal varying the frequency from 10 to 50 Hz with rate of 10 Hz/s, considering low level of amplitude (0.01 V), was applied to estimate the ﬁrst kernel (linear contribution). Then, the same chirp signal considering high level of amplitude (0.15 V) was applied to estimate the second and third kernels (nonlinear contribution). With the Volterra kernels estimated the system response can be obtained considering the convolution of the input signal with the kernels identiﬁed. Figure 4shows the 99% conﬁdence bands of the response in comparison with new experimental data to a chirp input varying the frequency from 10 to 50 Hz with rate of 10 Hz/s, considering two levels of amplitude (low - 0.01 V and high - 0.15 V). In both cases, linear and nonlinear behavior, the model is able to describe the responses, considering the data variations. The dispersion of the responses is probably caused by noise inthe measurements, sensor position, actuator position, etc. since the tests were carried out in diﬀerent days, precisely to consider a great dispersion of data, diﬀerently from what was done in [6], where only the statistical analysis of the damage index was enough to describe the dispersion of the responses without the uncertainties analysis in Volterraseries. -0.06-0.04-0.0200.020.040.06 1 1.2 1.4 1.6 1.8 2 2.2 2.4Velocity [m/s] Time [s] (a) Low input.-0.8-0.6-0.4-0.200.20.40.60.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4Velocity [m/s] Time [s] (b) High input. Fig. 4. Conﬁdence bands for the system responses obtained through the Volterra model with 99% of conﬁdence. The mean is represented by the blue line and the experimental data by the reddots. As a great advantage, Volterra series allow the separation of linear and nonlinear contributions of the response through ﬁltering by the kernels. Figure 5shows the contributions of the ﬁrst, second and third kernels with 99% of conﬁdence, considering low and high input. When the system has linear behavior (low input - 0.01 V) only the ﬁrst kernel contributes to the response. Now, with nonlinear behavior (high input - 0.15 V) the response is a sum of the ﬁrst and third kernel contributions. The second kernel has no contribution because the response is approximately Luis Gustavo Giacon Villania et al. / Procedia Engineering 199 (2017) 2090 –2095 2093 Villani et al. /Procedia Engineering 00 (2017) 000–000 3 2.1. Damage detection approach The identiﬁed Volterra kernels can be used as the reference model for the system. The stochastic response of the system can be obtained through convolution between the input signal and the Volterra kernels given by (Eq. 3) and the envelope of reliability, with probability Pc=99%, can be calculated based on the percentiles. The reliability interval at discrete time kis deﬁned by the set I99(k)=y(k)|r−(k)<=y(k)<=r+(k), (5) where y(k) is the model response in each discrete time k,r−(k) and r+(k) are the inferior and superior limits of the interval, respectively. If only the ﬁrst kernel is used y(k)=ylin(k)=y1(k), it means linear analysis; and if the ﬁrst three kernels are used y(k)=ynlin(k)=y1(k)+y2(k)+y3(k), nonlinear analysis. For each discrete time kthe distance between the experimental signal yexp(k) and the set I99(k) can be calculated by dlin(k)=inf/braceleftBig |yexp(k)−ylin(k)|for all y lin(k)∈I 99(k)/bracerightBig , (6) dnlin(k)=inf/braceleftBig |yexp(k)−ynlin(k)|for all y nlin(k)∈I 99(k)/bracerightBig . (7) So, by considering the calculated distances, a damage index can be computed, assuming the linear and nonlinear analysis λlin=||dlin|| A, (8) λnlin=||dnlin|| A, (9) where ||.||is the Euclidean norm operator and the excitation amplitude Ais used to normalize the index considering high and low levels of amplitude, making the comparison between the indexes in diﬀerent conditions easier. If the structure is in reference condition it is expected that the index has low value because the distance between the exper- imental signal and the response conﬁdence bands is zero or close to zero. The damage detection is done considering the hypothesis test /braceleftBiggH0:λ≤Λ, H1:λ>Λ ,(10) where Λis the threshold value that is determined based on the reference data. The reference data has unknown distribution, thus the threshold is determined considering the interquartile range Λ= Q3+1.5(Q3−Q1), (11) where Q1andQ3are the ﬁrst and third quartiles and Λis the upper quartile of the Box plot theory [15]. By applying this equation, every quantity above the Λvalue is considered an outlier. In order to estimate the threshold value the index is applied to the reference data considering linear and nonlinear behavior. The next section presents an experimental example of this approach. 3. Application in a Nonlinear Beam The experimental setup is the same used in [6]. It is composed by an aluminum beam with dimensions of 300 ×18×3 [mm] with a steel mass connected to the free end to cause a magnetic interaction. To simulate structural variations and study the proposed approach, a bolted connection is placed 150 [mm] from the free end with four nuts with 1 [g]each one. The healthy state considers the four nuts in the system and the damage is simulated with the loss of mass. Figures 1 and 2 show the experimental setup used. A MODAL SHOP shaker (Model Number: K2004E01) is attached 50 mm from the clamped and used to excite the structure considering diﬀerent levels of voltage amplitude 0.01 V (low level), 0.10 V (medium level) and 0.15 V (high level). A vibrometer laser Polytec (Model: OFV-525/-5000-S) and a Dytran load cell (Model: 1022V) are used to measure the velocity at the free end of the beam and the force excitation, respectively. All signals are measured considering a sampling rate of 1024 Hz and 4096 samples saved using a m+p4 Villani et al. /Procedia Engineering 00 (2017) 000–000 Vib Pilot data acquisition system. The magnet interaction with the beam generates a nonlinear hardening behavior. This eﬀect is illustrated on the beam velocity curves obtained during the stepped sine test shown in Fig. 3. When the input signal is high, the beam presents a jump phenomenon that conﬁrms the nonlinear behavior of the system in thiscondition of excitation. More information about the hardening behavior and jump phenomenon can be found in [16]. Fig. 1. Experimental setup (photo) [6]. Fig. 2. Experimental setup (scheme) [6].10-1100101102 15 20 25 30 35Amplitude [(ms-1)/V] Frequency [Hz] Fig. 3. Beam velocity during a stepped sine test, where /trianglerepresents the low level of input signal (0.01 V), ◦the medium level (0.10 V) and/squarethe high level (0.15 V).3.1. Volterra kernels identiﬁcation This section describes the model identiﬁcation/validation through the Volterra kernels extraction. All tests used a sampling frequency of 1024 Hz and 4096 samples. The tests were repeated 156 times in diﬀerent days during 2 weeks to obtain large variation of data. The estimation of Volterra kernels was done in two steps. First, a chirp signal varying the frequency from 10 to 50 Hz with rate of 10 Hz/s, considering low level of amplitude (0.01 V), was applied to estimate the ﬁrst kernel (linear contribution). Then, the same chirp signal considering high level of amplitude (0.15 V) was applied to estimate the second and third kernels (nonlinear contribution). With the Volterra kernels estimated the system response can be obtained considering the convolution of the input signal with the kernels identiﬁed. Figure 4shows the 99% conﬁdence bands of the response in comparison with new experimental data to a chirp input varying the frequency from 10 to 50 Hz with rate of 10 Hz/s, considering two levels of amplitude (low - 0.01 V and high - 0.15 V). In both cases, linear and nonlinear behavior, the model is able to describe the responses, considering the data variations. The dispersion of the responses is probably caused by noise inthe measurements, sensor position, actuator position, etc. since the tests were carried out in diﬀerent days, precisely to consider a great dispersion of data, diﬀerently from what was done in [6], where only the statistical analysis of the damage index was enough to describe the dispersion of the responses without the uncertainties analysis in Volterraseries. -0.06-0.04-0.0200.020.040.06 1 1.2 1.4 1.6 1.8 2 2.2 2.4Velocity [m/s] Time [s] (a) Low input.-0.8-0.6-0.4-0.200.20.40.60.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4Velocity [m/s] Time [s] (b) High input. Fig. 4. Conﬁdence bands for the system responses obtained through the Volterra model with 99% of conﬁdence. The mean is represented by the blue line and the experimental data by the reddots. As a great advantage, Volterra series allow the separation of linear and nonlinear contributions of the response through ﬁltering by the kernels. Figure 5shows the contributions of the ﬁrst, second and third kernels with 99% of conﬁdence, considering low and high input. When the system has linear behavior (low input - 0.01 V) only the ﬁrst kernel contributes to the response. Now, with nonlinear behavior (high input - 0.15 V) the response is a sum of the ﬁrst and third kernel contributions. The second kernel has no contribution because the response is approximately 2094 Luis Gustavo Giacon Villania et al. / Procedia Engineering 199 (2017) 2090 –2095 Villani et al. /Procedia Engineering 00 (2017) 000–000 5 symmetric. The model validation was done considering a simple sine input in the region of the ﬁrst mode shape (≈23 Hz) with high level of input (0.15 V), considering new experimental data. Figure 6shows the response in time and frequency domains with 99% of conﬁdence bands. Multiples harmonics in the system response can be seen(nonlinear behavior) and the model is able to describe the multiple components in frequency domain. The frequencyband in which the model is not able to describe the system behavior (≈ 50 to≈68 Hz) has low amplitude and little contribution in the response predicted. -0.06-0.04-0.0200.020.040.06 1 1.2 1.4 1.6 1.8 2 2.2 2.4Velocity [m/s] Time [s] (a) Low input.-0.8-0.6-0.4-0.200.20.40.60.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4Velocity [m/s] Time [s] (b) High input. Fig. 5. Conﬁdence bands for the kernels contribution to the output signals obtained through the Volterra model with 99% of conﬁdence. B represents the linear contribution, Brepresents the quadratic contribution and Brepresents the cubic contribution. -0.8-0.6-0.4-0.200.20.40.60.8 2.5 2.6 2.7 2.8 2.9 3Velocity [m/s] Time [s] (a) Time domain.10-1410-1210-1010-810-610-410-2100 0 10 20 30 40 50 60 70Amplitude [(m/s)2/Hz] Frequency [Hz] (b) Frequency domain. Fig. 6. Conﬁdence bands for the system responses obtained through the Volterra model with 99% of conﬁdence. The mean is represented by the blue line and the experimental data are represented by the redline. 3.2. Damage detection In order to include structural changes and to apply the stochastic Volterra series model in the damage detection, some mass reductions were analyzed by removing some nuts of the structure (Figs. 1 and 2). The conditions used are shown in the Tab. 1. To establish the threshold value, the method described in section 2.1was applied considering 312 samples in reference condition with diﬀerent excitation amplitudes. The threshold value was determined as Λ=1.14. In each damage and repair condition, 156 samples were measured in diﬀerent days to study the sensitivity of themethod in presence of uncertainties. The results in the Tab. 2represent the percentages of damage detection through the hypothesis test applied in the two damage indexes (λ linand λnlin) considering two diﬀerent levels of input applied (0.0 1 V - linear and 0.15 V - nonlinear behavior). The false alarm refers to the situation where the hypothesis test indicates damage and the systemis in healthy/repair condition (states 0 and R). It can be seen that when it was used the linear index to detect damage and the structure was operating in nonlinear regime of motion the hypothesis test has shown that in every state the structure was damaged, i. e., the model can not diﬀer the damage from the nonlinear behavior. Additionally, when thestructure operated in linear regime of motion the eﬀectiveness of the linear and nonlinear indexes has been the same,with no false alarms and 50% of true detection in the situation where just one mass was removed. The nonlinear index has presented better results when the structure operated with nonlinear behavior with 100% of true detection and no false alarms showing that the nonlinear index is more sensitive to detect damage.6 Villani et al. /Procedia Engineering 00 (2017) 000–000 Table 1. Structure variations. State Condition State Condition 0 4 masses (reference) III 1 mass I 3 masses R 4 masses (repair) II 2 masses – – Table 2. Damage detection obtained through the hypothesis test applied to damage indexes considering linear and nonlinear regime of motion. Condition Linear regime (input - 0.01 V) Nonlinear regime (input - 0.15 V) Linear index (λ lin) [%] Nonlinear index (λ nlin) [% ] Linear index ( λlin) [%] Nonlinear index ( λnlin)[ %] 00 /axisshort/axisshort/arrowaxisrightfalse alarm 0/axisshort/axisshort/arrowaxisrightfalse alarm 100/axisshort/axisshort/arrowaxisrightfalse alarm 0/axisshort/axisshort/arrowaxisrightfalse alarm I 50/axisshort/axisshort/arrowaxisrighttrue detection 50 /axisshort/axisshort/arrowaxisrighttrue detection 100/axisshort/axisshort/arrowaxisrighttrue detection 100 /axisshort/axisshort/arrowaxisrighttrue detection II 100/axisshort/axisshort/arrowaxisrighttrue detection 100 /axisshort/axisshort/arrowaxisrighttrue detection 100/axisshort/axisshort/arrowaxisrighttrue detection 100 /axisshort/axisshort/arrowaxisrighttrue detection III 100/axisshort/axisshort/arrowaxisrighttrue detection 100 /axisshort/axisshort/arrowaxisrighttrue detection 100/axisshort/axisshort/arrowaxisrighttrue detection 100 /axisshort/axisshort/arrowaxisrighttrue detection R0 /axisshort/axisshort/arrowaxisrightfalse alarm 0/axisshort/axisshort/arrowaxisrightfalse alarm 100/axisshort/axisshort/arrowaxisrightfalse alarm 0/axisshort/axisshort/arrowaxisrightfalse alarm 4. Final remarks The proposed methodology based on stochastic V olterra series had a satisfactory result considering the large data variation along the samples, mainly in the nonlinear regime of motion. The nonlinear approach has proved to be more eﬃcient with better results principally in nonlinear regime of motion, when it was able to tell the diﬀerence between the nonlinear behavior compared to damage. Other indexes and hypothesis tests have to be studied to improve the approach whereas the estimation of the threshold based on reference samples is always subject to errors. Acknowledgements The authors are thankful for the ﬁnancial support provided by S ˜ao Paulo Research Foundation (FAPESP), grant number 2012/09135-3, 2014/02971-6 and 2015/25676-2, by Research Support Foundation of the State of Rio de Janeiro (FAPERJ), grant number E-26/010.002178/2015, and CNPq grant number 303403/2013-6.