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Semi-active vehicle suspension with magnetorheological dampers is a promising technology for improving the ride comfort of a ground vehicle. However, the magnetorheological damper always exhibits nonlinear hysteresis between its output force and relative velocity, and additional nonlinear stiffness owing to the state transition from liquid to semi-solid or solid, so that the semi-active suspension with magnetorheological dampers features nonlinearity by nature. To control such nonlinear dynamic systems subject to random road roughness, in this paper we present a neural network control, which includes an error back propagation algorithm with quadratic momentum of the multilayer forward neural networks. Both the low frequency of road-induced vibration of the vehicle body and the fast response of the magnetorheological damper enable the neural network control to work effectively on-line. The numerical simulations and an experiment for a quarter-car model indicate that the semi-active suspension with a magnetorheological damper and neural network control is superior to the passive suspensions in a range of low frequency.

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An efficient method of model order reduction is proposed for the dynamic computation of a flexible multibody system undergoing both large overall motions and large deformations. The system is initially modeled by using the nonlinear finite elements of absolute nodal coordinate formulation and then locally linearized at a series of quasi-static equilibrium configurations according to the given accuracy in dynamic computation. By using the Craig-Bampton method, the reduced model is established by projecting the incremental displacements of the locally linearized system onto a set of local modal bases at the quasi-static equilibrium configuration accordingly. Afterwards, the initial conditions for the dynamic computation for the reduced model via the generalized-α integrator can be determined from the modal bases. The analysis of computation complexity is also performed. Hence, the proposed method gives time-varying and dimension-varying modal bases to elaborate the efficient model reduction. Finally, three examples are presented to validate the accuracy and efficiency of the proposed method. KEYWORDSabsolute nodal coordinate formulation, Craig-Bampton method, flexible multibody dynamics, model order reduction Nomenclature: C, vector of constraints; d, vector of augmented external forces; e, vector of generalized coordinates; E, Young's modulus; I, moment of inertia; h, time step; K, global stiffness matrix;K, augmented global stiffness matrix; M, global mass matrix;M, augmented global mass matrix; n, total number of degrees of freedom; n c , number of constraint equations; n e , number of finite elements; n m , number of dynamic equations; n p , number of degrees of freedom per node; n s , revolution per minute; n t , number of truncated modes; n t 1 , initial number of truncated modes; n t max , maximal number of truncated modes; q, vector of augmented modal coordinates; Q, vector of external forces; r, vector of global position; t, time; t N , moment of the Nth local linearization; T, transformation matrix;ũ, vector of augmented coordinates; U, strain energy; XYZ, global frame of coordinates; X i Y i Z i , the ith body frame of coordinates; C , matrix of constrained modes; V , matrix of vibration modes; , vector of modal coordinates of internal part; , vector of Lagrange multipliers; j , the jth eigenvalue; , diagonal matrix of eigenvalues; Ω, configuration region; , vector of augmented acceleration-like variables; s , rotation angle; s , angular velocity; Δ max , threshold of local linearization; , specified value of mode selection; Δ RMS , root of mean squared error. Superscripts and subscripts: () B , symbol related to boundary part; () I , symbol related to internal part; () BI , () IB , symbol related to coupling parts; () e , the Jacobian with respect to vector e; () N , symbol corresponding to the Nth local linearization;() k , symbol corresponding to the kth time step; () d , reserved quantity; () r , residual quantity. Overbar: (^), quantity in the reduced model; (˜), quantity in the non-reduced model; (...