An analytical approach to integral resonant control of second-order systems

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Structural modelling and testing of failed high energy pipe runs: 2D and 3D pipe whip

Copyright @ 2011 ElsevierThe sudden rupture of a high energy piping system is a safety-related issue and has been the subject of extensive study and discussed in several industrial reports (e.g. [2], [3] and [4]). The dynamic plastic response of the deforming pipe segment under the blow-down force of the escaping liquid is termed pipe whip. Because of the potential damage that such an event could cause, various geometric and kinematic features of this phenomenon have been modelled from the point of view of dynamic structural plasticity. After a comprehensive summary of the behaviour of in-plane deformation of pipe runs [9] and [10] that deform in 2D in a plane, the more complicated case of 3D out-of-plane deformation is discussed. Both experimental studies and modelling using analytical and FE methods have been carried out and they show that, for a good estimate of the “hazard zone” when unconstrained pipe whip motion could occur, a large displacement analysis is essential. The classical, rigid plastic, small deflection analysis (e.g. see [2] and [8]), is valid for estimating the initial failure mechanisms, however it is insufficient for describing the details and consequences of large deflection behaviour

Reproducing kernel collocation method applied to the non-linear dynamics of pipe whip in a plane

A windowed collocation method, based on a moving least squares reproducing kernel particle approximation of functions, is explored for spatial discretization of the strongly non-linear system of partial differential equations governing large, planar whipping motion of a cantilever pipe subjected to a follower force pulse (the blow-down force) normal to the deflected centreline at its tip. This problem was discussed by Reid et al. [An elastic–plastic hardening–softening cantilever beam subjected to a force pulse at its tip: a model for pipe whip. Proc R Soc London A1998;454:997–1029] where a space–time finite difference discretization was employed to solve the governing partial differential equation of motion. It was shown that, despite the deflected shape predictions being accurate, numerical solutions of these equations might exhibit problematic (possibly spurious) steep localized gradients. The resolution of this problem in the context of structural mechanics is novel and is the subject of this paper. In particular, it is demonstrated that it is possible to reduce significantly such spurious and localized numerical instabilities through a windowed collocation approach with a suitable choice of the window size. The collocation procedure presently adopted is based on the moving least squares reproducing kernel particle method. Material and structural non-linearity in the beam (pipe) model is incorporated via an elastic– plastic hardening– softening moment–curvature relationship. The projected ordinary differential equations are then integrated in time through a fifth order, explicit Runge–Kutta method with adaptive step sizes. The whipping motion of a pipe is often associated with the formation of one or more so-called moving plastic ‘hinges’ (regions). Depending on the magnitude and duration of the follower load pulse, plastic hinges may form at intermediate positions along the pipe. In such cases numerical solutions, especially those for curvatures and their derivatives, obtained through central difference or classical finite element methods may exhibit considerable fluctuations in space and time and may even ‘blow up’ following the formation of a kink at the location of the plastic hinge. However, it is shown that the presently adopted windowed collocation strategy has a far superior numerical performance, arresting these fluctuations to a great extent. Indeed, as demonstrated in this paper, there is a band of window sizes for which such fluctuations may nearly vanish

Hybrid modelling of machine tool axis drives.

NoThe x-axis dynamics of a milling machine where the workpiece and saddle are mounted on supporting slides is considered. A permanent magnet motor, lead screw, ball nut and bearings are employed as the machine, traverse actuator mechanism. Hybrid, distributed¿lumped parameter methods are used to model the machine tool x-axis drive system. Inclusion of the spatial configuration of the drive generates the incident, travelling and reflected vibration signature of the system. Lead screw interactive torsion and tension loading, which is excited by cutting and input disturbance conditions, is incorporated in the modelling process. Measured and results from simulation exercises are presented in comparative studies enabling the dynamic characteristics of the machine to be identified under, no load and with the application of cyclic, cutting force disturbances. The effect of the lead screw length, cutting speed and hence the load disturbance frequency are examined and the resulting performance accuracy is commented upon