PSI - Issue 64

Jiantao Li et al. / Procedia Structural Integrity 64 (2024) 500–506 Author name / Structural Integrity Procedia 00 (2019) 000–000

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2.1. Vehicle model As shown in Fig. 1, The train is represented as a simplified vehicle model with four degrees of freedom. The mass and moment of inertia of the vehicle body are represented by m v , I v , respectively. The stiffness and damping coefficients of the vehicle suspension system for the front and rear axles are represented as k i , c i i = 1,2 , respectively. k ti , i = 1,2 are the stiffnesses of the front and rear tyres. The equation of motion for the vehicle model is obtained as M v Z � v + C v Z � v + K v Z v = F v (1) where, M v , C v and K v are the mass, damping and stiffness matrices of the vehicle respectively. Z v = y s ߠ s y 1 y 2 T denotes the states of the vehicle including the bouncing y s and pitching ߠ ݏ of the vehicle body, and the vertical displacements of the front and rear axles y 1 ,y 2 . F v is the force vector applied to the vehicle tyres. The above matrices are given as M v =݉ , C v =ܿ 1 +ܿ 2ܿ 1݀ 1 −ܿ 2݀ 2 −ܿ 1 −ܿ 2ܿ 1݀ 1 −ܿ 2݀ 2ܿ 1݀ 12 +ܿ 2݀ 22ܿ 1݀ 1 −ܿ 2݀ 2 −ܿ 1ܿ 1݀ 1ܿ 1 0 −ܿ 2 −ܿ 2݀ 2 0ܿ 2 , ݕ ݒ ,1 ݕ ݒ ,2 = Py v where ݕ ݒ ,݅ is the contact-point response under the vehicle wheel i, including the road roughness, ݅ݎ and the bridge displacement under the wheel, ܾݓ ,݅ . ݕ ݒ ,݅ = ܾݓ ,݅ + ݅ݎ ,݅ = 1,2 (2) 3. CP displacement response identification based on AKF and BEM 3.1. Augmented state-space model of the vehicle system By introducing state space representation to Eq.(1) and including the CP displacements along with its derivative term into the state, the discrete forms of the augmented vehicle dynamic system are represented as (Xue et al., 2020) x a,k + 1 = A d x a, k +ξ k (3) u k = C d x a, k +υ k (4) where augmented state vector x a = [ x cT ܡ vT ] T = [ x cT y v,1 y � v,1 y v,2 y � v,2 ] T , k is the time step, ξ k =[ w k η k ] and υ k are the process and observation noises of the discrete system. The vector η k represents the process noise corresponding to the road profiles and their derivatives. The relationship between continuous and discrete representations can be represented as A d = e A ca dt , C d = C ca , x a, k = x a kdt , u k = u kdt (5) The covariance matrices of ૆݇ =[ ݇ܟ ࣁ݇ ] and υ k are defined as E w k w kT = Q w,k E η k η kT = Q η ,k (6) E v k v kT = R k 3.2. CP displacement response reconstruction by AKF and BEM After establishing the state-space model specified by Eqs. (3) and (4), the vehicle state and CP responses can be estimated with the integration of AKF and BEM based on the Bayes ’ rule. The proposed method combines the advantages of expectation maximization (EM) algorithms with the principles of Bayesian inference to provide a powerful framework to optimize the performance of the augmented Kalman filter, which can simultaneously estimate the states and unknown inputs of the system along with the tracking of the noise covariance matrices.The ݒ 0 0 0 0 ܫ ݒ 0 0 0 0݉ 1 0 0 0 0݉ 2 K v =݇ 1 +݇ 2݇ 1݀ 1 −݇ 2݀ 2 −݇ 1 −݇ 2݇ 1݀ 1 −݇ 2݀ 2݇ 1݀ 12 +݇ 2݀ 22݇ 1݀ 1 −݇ 2݀ 2 −݇ 1݇ 1݀ 1݇ 1 +݇ ݐ 1 0 −݇ 2 −݇ 2݀ 2 0݇ 2 +݇ ݐ 2 , F v = 0 0 0 0݇ ݐ 1 0݇ 0 ݐ 2