168 V. K. Dertimanis et al. services and regular operation. As an alternative, relatively low-cost continuous monitoring, based on on-board vibration data, collected from revenue-making trains, could be a cost-effective approach to monitor railway tracks [5–9]. It is worth mentioning that on-board monitoring systems using diverse sensor technologies are also investigated in road vehicles [10, 11]. In this work we explore this second option through the adoption of a minimum intervention strategy. This pertains to the use of a single sensor for tracking the train vibration, and the implementation of a dual KF (KF) [12] for the estimation of the unmeasured input force that is developed as the result of train-track interaction [13]. Via estimation of the latter, identification of isolated defects (e.g. squats, turnout frogs, welded joints, etc.), as well as of effects distributed over a certain track length (e.g. superstructure type, ballast condition, soil properties, etc.) can be accomplished. 18.2 Description of the Method Figure 18.1 displays a simplified model for the description of the vertical dynamics of a train vehicle. The chassis is modelled as a lumped mass with two degrees of freedom (DOFs) that correspond to bounce and pitch, while each bogie and wheelset are modelled as lumped masses with a single DOF (e.g. bounce). All vehicle parts are connected through the primary and secondary suspensions and the wheel–rail interaction is modelled with a linear Hertzian spring. A limitation hindering implementation of the proposed method to the state-space model of the original vector structural equation pertains to the input delay that is associated to the rear wheelset. To this end, the dual KF estimation is succeeded by (1) establishing the transfer function that connects the induced wheel-rail interaction force to the acceleration of a measured DOF, which herein assumed to be the bounce of the front wheelset; (2) transforming the latter into a state-space model ; (3) discretizing the state-space model using an appropriate sampling rate; and (4) setting up the dual KF, by adopting a fictitious equation for the unknown forces. In more detail, the differential equation that describes the bounce motion of the front wheelset is mwf ¨xwf(t) +cpf ˙xwf(t) +kwfxwf(t) =fwf(t) +fpf(t) (18.1) inwhichfwf(t) =kwfrf(t) is the wheel contact force andfpf(t) is the interaction force between the wheelset substructure and the rest of the vehicle, given by fpf(t) =cpf ˙xbf(t) − ˙xwf(t) +kpf xbf(t) − ˙xwf(t) (18.2) This latter force carries all uncertainties associated with the parameters of the vehicle (excluding the ones of the wheelset substructure) and especially the mass Mc and inertia Ic of the chassis, which correspond to the varying amount of passengers during operation. The effects of this force to the vibration acceleration of the wheelset mass are expected to be considerably lower, compared to the ones of the wheel contact force, allowing thus a relative accurate estimation of the latter. q Fig. 18.1 Pitch-bounce model of a railway vehicle
RkJQdWJsaXNoZXIy MTMzNzEzMQ==