(ii) Backlash in the real substructure Real-time dynamic substructuring is .... C.A. Taylor, âDynamic testing of the Second Severn Crossing during construction,â.
RA Z: Understanding the theoretical basis behind making real-time dynamic substructuring work (All Investigators) - Engineering Collaborators David Stoten (Mech Eng), Nick Lieven (Aero Eng) and Colin Taylor (Civil Eng). The overall aim of this part of the Programme is that we will have mathematical confidence in realtime dynamic substructuring. This theoretical study will be carried out in parallel with experimental substructuring tests within the BLADE facility. Whilst we have a clear idea of the content of the first year (see (i) below), the precise course that this part of the Programme will take depends on the results from all the other four themes (including workshops and associated visitor programmes). Therefore, the Programme Scientific Steering Committee will select specific objectives from the three main challenges below. We will apply our results to two test-bed problems; an auto-parametric resonant system and cable-deck interaction in cable-stayed bridges. Background In many branches of mechanical, aerospace and civil engineering, it is of fundamental importance that structures are tested under dynamic loads before deployment. In most cases the properties of the use materials mean that a scale model will behave differently than its full size counterpart. Unfortunately, the need to test at full scale means expensive or even uneconomic prototyping. Real-time dynamic substructuring [11] circumvents this problem by dividing the structure under consideration into two parts. One part is the real substructure, which is the full-size nonlinear component of the structure under test, whose dynamics are often poorly understood. The other part is the virtual substructure, which is the rest of the structure represented by a computer model. Typically this represents a large but essentially understood part of the system. The two parts are connected by a control loop consisting of a control signal from the computer model, which sets off actuators that move the real substructure. The return loop is from sensors on the real substructure, which give the computer model its input. In this project we focus on three main challenges: delay and backlash in the real substructure, and the computer model used in the virtual substructure. (i) Delay As we showed in our presentation in London, finite delay in the actuator makes real-time dynamic substructuring unstable [3, 4]; a direct link with RA A. Another source of delay is the finite computational time of the virtual substructure. However, it is also well known [7] that some systems can actually be stabilised by introducing delay. Building on the work done on delay in the first two years of RA A (together with its workshop and visitor programme) we will, in the first year of RA Z, produce a delay compensation scheme that is superior to the linear [3] or cubic [4] extrapolation techniques currently available, which are both restricted to structures with low natural frequencies. We will exploit the fact that delay can be incorporated into the combined system in a number of ways. We will seek strategies to combine the real and virtual substructures such that it is stable to delay but whose dynamics qualitatively matches that of a system that is nearby in parameter space to the original structure in the absence of delay. Here we shall exploit ideas of shadowing in dynamical systems and the use of invariant manifold theory to track the error dynamics between the original system and the substructured one. An outstanding problem in this area is that the delay of the actuator varies with both amplitude and frequency of oscillation. There is no theory for this problem. Stepan, Hale and Verdyn Lunel will provide the necessary theoretical input as the project develops. (ii) Backlash in the real substructure Real-time dynamic substructuring is destabilised by backlash as identified in practice in [1]. There are no delay compensation schemes for such problems, because both existing schemes [3, 4] assume smoothness of the system vector fields. We will propose a compensation scheme for PWS systems and then refine it for use at higher frequencies. Both RA A and RA B (and their associated workshops and visitor programmes) will conclude before this stage starts. Hence this part of RA Z begins in year 4 of the Programme. In particular the compensation scheme 1
will need to be closely analysed for C-bifurcations (which will lead to loss of stability). The numerical algorithms developed in RA B will be of central importance to this stage of RA Z. (iii) Computational and modelling issues in the virtual substructure Speed of computation is absolutely essential in the implementation of real-time dynamic substructuring. However, accuracy in describing the dynamics is just as important. Results and experience from RA C and RA D will be used to determine what level of modelling is necessary to preserve the relevant dynamics. Step size selection is an issue of equal importance. In particular, all delay-compensation schemes to date use not only a fixed step size but also require that the step size is a rational fraction of the actuator delay time. Clearly this situation is untenable in practice because the delay varies. In addition (and more seriously) fixed-time-step numerical integration schemes are not well suited to stiff or PWS systems, such as arise in real structures. Therefore, we will seek a compensation scheme based on variable step methods. There are also issues surrounding the dynamics of the controller used to transfer the output of the numerical model to the actuators. The MCS algorithm developed at Bristol by Stoten, with di Bernardo, is able to adaptively control nonlinear systems numerically [8] using the virtual substructure as a reference model. However, there are potential problems with the digital implementation of this type of algorithm, especially in the presence of significant nonlinearity [10]. Stepsize selection and the complexity of the model are thus crucial here too.
Test-bed problems Over the course of RA Z, the Programme Scientific Steering Committee will choose from the above challenges and we will apply our results to two test-bed problems: 1. Auto-parametric resonant system We will study a mass-spring-pendulum system where the pendulum’s natural frequency is half that of the much heavier mass-spring [9, Chapter 4]. This is known to exhibit large amplitude motion of the pendulum that can be periodic or chaotic and which quenches the motion of the primary system. This represents a severe test for real-time dynamic substructuring since information flows from the (light) pendulum real substructure to the virtual substructure of the (heavy) spring-mass system. We will compare our results with experiments to be conducted in the BLADE Automatic Control and Transmission Laboratory. Auto-parametric resonant interaction has also been observed between the light cables and heavy deck of a cable-stayed bridge [5]. 2. Cable-deck interaction in cable-stayed bridges Cable-deck interaction is an ideal application for the development of real-time dynamic substructuring, since the bridge deck is essentially linear while the cable dynamics can be highly nonlinear and there is just a single point of contact between the two. There is considerable expertise at Bristol (Taylor and MacDonald) in testing and modelling cable-stayed bridges [2, 5, 6] and it is proposed to use this as a major test-bed problem for BLADE, using the BLADE Earthquake Laboratory and real-time dynamic substructuring. Here a single cable is the physical substructure, and the rest of the bridge the virtual one. Substructuring would allow much greater control of the physical parameters and loads, and would enable us to isolate the dynamics of the cable, which can undergo complicated nonlinear motion including backlash and saturation (directly linked to RA B). Building on results and experience from RA D we will determine the level of modelling required of the numerical substructure to reproduce faithfully the results of a far more complex numerical model of the entire structure (for example, by homogenising over many cables). We will use realtime dynamic substructuring MCS to understand under what conditions of cable sag, initial extension and cable damping can the cable dynamics cause an undesirable response in the deck. The methods developed by RA C will be relevant here, because (secondary) oscillatory instabilities are expected to be prominent in the system when many modes are excited.
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References [1] A. Blakeborough, M.S. Williams, A.P. Darby, and D.M. Williams, “The development of real-time substructure testing,” Phil. Trans. 359: 1869–1891, 2001. [2] C.T. Georgakis, J.H.G. Macdonald, and C.A. Taylor, “Nonlinear analysis of wind-induced cable-deck interaction,” IABSE Conf. Cable-supported Bridges, Paper no. 330, Seoul, South Korea, 12-14 June 2001. [3] T. Horiuchi, M. Inoue, T. Konno, and Y. Namita, “Real-time hybrid experimental system with actuator delay compensation and its application to a piping system with energy absorber,” Earthquake Eng. Struct. Dyn. 28: 1121–1141, 1999. [4] T. Horiuchi and T. Konno, “A new method for compensating actuator delay in real-time hybrid experiments,” Phil. Trans. 359: 1893–1909, 2001. [5] J.H.G. Macdonald, E.L. Dagless, B.T. Thomas, and C.A. Taylor, “Dynamic measurements of the Second Severn Crossing,” Proc. ICE: Transport Vol. 123(4), pp 241–248, 1997. [6] J.H.G. Macdonald and C.A. Taylor, “Dynamic testing of the Second Severn Crossing during construction,” 2nd Int. Conf. Struct. Dynamics Modelling, pp 449–460, Windermere, Cumbria, 1996. [7] G. Stepan, Retarded Dynamical Systems: Stability and Characteristic functions, (Longman Scientific and Technical, 1989). [8] D.P. Stoten and M. di Bernardo, “Application of the minimal control synthesis algorithm to the control and synchronization of chaotic systems,” Int. J. Control 65(6): 925–938, 1996. [9] A. Todl, T. Ruijgrok, F. Verhulst, and R. Nabergoj, Autoparametric resonance in mechanical systems, (Cambridge University Press, 2000). [10] D.J. Wagg and D.P. Stoten, “Substructuring of dynamical systems via the adaptive minimal control synthesis algorithm,” Earthquake Engineering and Structural Dynamics 30: 865–877, 2001. [11] M. S. Williams (Ed.), Dynamic Testing of Structures, Special Issue of Phil. Trans. 359(1786), 2001.
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