A recursive model for nonlinear spring-mass-damper estimation of a vehicle localized impact

Transcription

1 A recursive model for nonlinear spring-mass-damper estimation of a vehicle localized impact HAMID REZA KARIMI hamid.r.karimi@uia.no WITOLD PAWLUS witolp9@student.uia.no KJELL G. ROBBERSMYR kjell.g.robbersmyr@uia.no University of Agder, Faculty of Engineering and Science, Department of Engineering Postbox 59, N-4898 Grimstad NORWAY Abstract: In this paper, a recursive autoregressive model is presented to estimate modal parameters of the crash pulse recorded during a full-scale vehicle crash test. Those parameters can be further utilized to characterize a physical model, so called Maxwell model, which is composed of a serial spring-mass-damper model to simulate a vehicle crash event. Results produced by this viscoelastic system closely follow the reference vehicle s kinematics. Thanks to the application of nonlinear model s parameters estimated by RARMAX model it is possible to gain a closer insight into the nature of the vehicle collision. To evaluate the correctness of this methodology, the comparative analysis between the original vehicle s kinematics and behavior of a Maxwell model with the nonlinear parameters estimated by RARMAX model is presented. Key Words: Recursive identification, parameters estimation, nonlinear model, localized impact 1 Introduction Time series data can be successfully predicted by autoregressive models like NAR (nonlinear autoregressive), NARX (nonlinear autoregressive with exogenous input) or NARMAX (nonlinear autoregressive moving average with exogenous input). The following areas of application of autoregressive models were investigated recently: machine diagnostics and vibration data analysis ([1]); time series data forecasting ([2]); modeling of devices for industrial applications ([3]); model s prediction performance assessment ([4]). Physical systems modeling can also be successfully investigated by autoregressive models - nonlinear modal parameters of vehicle crash viscoelastic models were estimated in [5]. FEM (Finite Element Method) and LPM (Lumped Parameter Modeling) application to vehicle crash modeling has been described in [6]-[12]. In [13]-[15] there are discussed results of application of wavelets, neural networks, and fuzzy logic to model a crash event. Those two last approaches possess strong potential for creation of vehicle crash dynamic models and their parameters establishment - in [16] and [17] the parameters of viscoelastic models were obtained thanks to Radial Basis Function (RBF) neural network. Responses of such systems were matching the reference car s behavior well. Artificial Neural Networks (ANNs) are capable of not only reproducing car s kinematics but can be successfully applied to e.g. predict the average speed on highways as it is shown in [18]. On the other hand, [19] presents a method for crash severity assessment based on a number of such inputs, like e.g. driver age, alcohol use, seat belt use, vehicle type, time of the crash, light condition or weather condition. The full-scale crash test which measurements are used in this study is thoroughly described. Subsequently, equations characterizing ARMAX model are presented together with the derivation of recursive ARMAX model. RARMAX model is created and its capabilities to estimate modal parameters of spring-mass-damper model representing a vehicle-topole collision are assessed. Furthermore, kinematics of this model is compared to the original vehicle s behavior. Suitability of the current approach is verified at the end of this work. The main contribution of this paper is establishment of a complete vehicle crash physical model with nonlinear parameters which change in time as well as the evaluation of the modeling results and theoretical considerations with the full-scale experimental data. 2 Crash test overview The detailed description of the analyzed crash test is included in [2]. The initial velocity of the car impacting centrally a rigid pole was 35 km/h, and the mass of the vehicle (together with the measuring equipment ISBN:

2 and dummy) was 873 kg. Particular car s parameters are illustrated in Fig. 1 and listed in Table 1 (please note that all dimensions are in meters[m]). Table 2: Relevant parameters characterizing the real collision Parameter Value Initial impact velocityv [km/h] 35 Rebound velocityv [km/h] 3 Maximum dynamic crushd c [cm] 52 Time when it occurst m [ms] 76 Permanent deformationd p [cm] 5 Figure 1: Car used in the real experiment. Width Height Frontal overhang Wheel track: front axle Length Wheel base Rear overhang Wheel track: rear axle Table 1: Car s relevant dimensions. The plot obtained from the data acquisition analysis (measured car s acceleration together with its velocity and displacement changes) is shown in Fig. 2. For the explanation of the characteristic crash pulse s parameters refer to Table Recursive autoregressive moving average model with exogenous input (RARMAX) analysis 3.1 ARMAX model derivation Analysis of the autoregressive model with moving average and exogenous input (ARMAX) was done according to [21]. ARMAX model is defined as: y(t)+a 1 y(t 1)+ +a na y(t n a ) = (1) b 1 u(t n k )+ +b nb u(t n k n b +1) +c 1 e(t 1)+ +c nc e(t n c )+e(t) where: t - time; y(t) - system s output; a 1,...,a na ; b 1,...,b nb ; c 1,...,c nc - model s parameters; n a - number of model s poles; n b - number of model s zeros + 1; n c - number of model s parameters in C vector; n k - order of delay; y(t 1),...,y(t n a ) - system s output in the previous moment; u(t n k ),...,u(t n k n b + 1) - system s input in the previous moment;e(t 1),...,e(t n c ) - white noise. ARMAX model can also be formulated as: Acceleration [g]; velocity [km/h]; crush [cm] d c V d p q: A(q)y(t) = B(q)u(t n k )+C(q)e(t) (2) where A, B, and C are expressed as functions of A(q) = 1+a 1 q 1 + +a na q na (3) 4 t m Acceleration Velocity Crush B(q) = b 1 +b 2 q 1 + +b nb q nb+1 (4) Figure 2: Real car s kinematics. C(q) = 1+c 1 q 1 + +c nc q nc. (5) ISBN:

3 3.2 RARMAX model derivation By following [22], the recursive identification algorithm in general is given by the equation: ˆΘ(t) = ˆΘ(t 1)+K(t)(y(t) ŷ(t)) (6) where: ˆΘ(t) - the parameter estimate at time t; y(t) - the observed output at time t; ŷ(t) - the prediction of y(t) based on observations up to time t 1; K(t) - the gain which determines how much the current prediction error y(t) ŷ(t) affects the update of the parameter estimate. The gain K(t) is characterized by the equation: K(t) = Q(t)ψ(t) (7) in which the term ψ(t) represents the gradient of the predicted model output ŷ(t Θ) with respect to the parameters Θ. The main purpose of the estimation algorithm is to minimize the prediction error term y(t) ŷ(t). The recursive algorithm presented in this work is called Forgetting Factor Algorithm. It considers models with a linear-regression form: y(t) = ψ T (t)θ (t)+e(t). (8) In this equation,ψ(t) is the regression vector that is computed based on previous values of measured inputs and outputs. Θ (t) represents the true parameters, e(t) is the noise source, which is assumed to be white noise. The particular form of ψ(t) depends on the structure of the polynomial model. In the Forgetting Factor Algorithm which considers linear regression models, the predicted output is given by the following formula: ŷ(t) = ψ T (t)ˆθ(t 1). (9) For the Forgetting Factor Algorithm, the term Q(t) from Eq. 7 is given by: Q(t) = P(t) = P(t 1) λ+ψ(t) T P(t 1)ψ(t) which can be further described as: ( (1) P(t) = λ 1 ) (11). P(t 1) P(t 1)ψ(t)ψ(t)T P(t 1) λ+ψ(t) T P(t 1)ψ(t) To obtain Q(t), it is needed to minimize the following function at timet: t λ t k e 2 (k). (12) k=1 The term λ is referred to as a forgetting factor and typically has a positive value between.9 and 1. When λ = 1, then the recursive estimation algorithm degenerates into simple ARMAX configuration. Recursive identification is efficient in estimating parameters of the systems which are changeable in time. 4 Maxwell model Maxwell model consists of a spring and damper connected in series. This two-parameter model is suitable for component modeling (creep and relaxation) as well as for vehicle localized impact modeling, e.g. localized pole or offset collisions, where the timing at dynamic crush is fairly long [23]. Figure 3: Maxwell model. According to [1], to derive equation of motion of Maxwell model it is proposed to place a small mass m between the spring and damper - as it is shown in Fig. 3. By doing this, the inertia effect which occurs for the spring and damper is neglected and the system becomes third order differential equation which can be solved explicitly [23]. We define d and d as an absolute displacement of massmand absolute displacement of mass m, respectively. Spring stiffness is denoted by k and damping coefficient by c. We establish the following equations of motion (EOM): m d = c( d ḋ ) (13) m d = c( d ḋ ) kd. (14) By differentiating Eq. 13 and Eq. 14 with respect to time and settingm = we obtain: m... d = c( d d ) (15) = c( d d ) kḋ. (16) We sum up both sides of Eq. 15 and Eq. 16 and rearrange: ḋ = m... d. (17) k We substitute Eq. 17 into Eq. 13 and finally obtain such an EOM: ISBN:

4 ... d + k c d+ k m d =. (18) Therefore, the characteristic equation of the Maxwell model is: [ s s 2 + k c s+ k ] =. (19) m By comparing the terms in Eq. 19 to the corresponding terms of the characteristic equation of the 2nd order oscillating element: s 2 +2ζω e s+ω 2 e = (2) the damping factor ζ and natural frequency ω e of the mk Maxwell model are found to be: ζ = andω e = 2c k m. 5 RARMAX model establishment The procedure to estimate modal parameters (i.e. damping factor ζ and natural frequency ω e ) of a Maxwell model shown in Fig. 3 so that it can successfully represent a vehicle-to-pole collision is as follows. As an input to the RARMAX model, the acceleration from Fig. 2 recorded during a full-scale crash test is provided. Subsequently its parameters Θ are estimated and ordersn a andn b as well as forgetting factorλare specified. In the trial and error process it was found that to consider the dynamic nature of a crash event it is required to use high order RARMAX model (n a = n b = 2) with the forgetting factor λ =.92. Furthermore, for each particular value of parameter Θ (since the recorded crash pulse lasts for 175 ms and sampling frequency is 1 khz, in total there are 1751 samples, which at the same time is the length of theθ vector), there are assessed its modal parameters (i.e. damping ζ and natural frequency ω e ). By using formulas k = ω 2 em and c = k 2ζω e spring stiffness k and damping coefficient c of Maxwell model are calculated - see Fig. 4 and Fig. 5. Subsequently the Maxwell model s response is obtained (note that the initial impact velocity is the same as the reference one, i.e. v = 35km/h = 9.7m/s and mass of the model is the same as mass of the vehicle, i.e. m = 873 kg). Spring stiffness K [kn/m] Damping coefficient C [kns/m] x Figure 4: Estimated spring stiffness k Figure 5: Estimated damping coefficientc. ISBN:

5 6 Results comparison Comparative analysis between the model s simulation results and full-scale crash test measurements in terms of acceleration, velocity, and displacement is shown in Fig. 6, Fig. 7, and Fig. 8, respectively. From the comparison it is shown that the Maxwell model with nonlinear parameters estimated by the RARMAX system follows the reference car s kinematics with a high degree of accuracy. Although the acceleration plots are not exactly the same, the overall shape of the estimated acceleration graph is preserved and corresponds to the reference measured acceleration. Similarity between the velocity and crush time histories was also achieved. The obtained satisfactory results prove that the nonlinear Maxwell model s parameters estimated by the RARMAX system are correct. Crush [cm] Figure 8: Crush characteristics. Real RARMAX estimation Real RARMAX estimation 7 Conclusions and future works Acceleration [g] Velocity [km/h] Figure 6: Acceleration signals. Real RARMAX estimation Figure 7: Velocity plots. RARMAX model application to estimate the physical system s parameters which are changing in time is a successful approach to simulate a vehicle collision. Nonlinearity in the system comes from a huge number of elements, joints, and connections involved in a crash event. As it was shown in this paper it is possible to capture and reproduce it by using simple spring-mass-damper model which structural parameters (apart from the mass, which is assumed to be constant as the mass of a vehicle during this impact does not change significantly) change in time, according to the recorded acceleration pulse. RARMAX model orders and forgetting factor play an important role in the parameter estimation process and they need to be selected in a way which takes into consideration the complexity and dynamics of the event being modeled. The higher orders are used, the better insight we get into the phenomenon being modeled because then RARMAX model better captures the behavior of a real system. However, simultaneously, together with the increasing model s orders the number of estimations also increases. Therefore additional attention should be paid to select the proper estimated data set. In the future the performance of another recursive estimation algorithms may be investigated. Apart from the forgetting factor identification methodology introduced in this paper, there exists a variety of approaches which can be utilized for this task. Application of different configurations of springs, dampers, and masses to simulate vehicle collision may be beneficial as well. ISBN:

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