74 M.V. van der Seijs et al. and assembled to other measured or simulated IRFs, as was already demonstrated for a relatively simple 1D bar problem [9]. From the paper by Rixen it followed that one of the biggest issues is in fact the imperfection of the impulse. In this paper some of the solutions proposed for handling this imperfect impulse are discussed in more detail. Section 6.2 gives a summary of the IRF-based dynamic simulations and the procedures to couple substructures in the time-domain. Also the representation of displacement, velocity and acceleration IRFs is discussed. In Sect. 6.3 the time-domain IRF determination is presented and illustrated for a 1D POM bar problem. Thereafter some observations and practical challenges are discussed in Sect. 6.4. 6.2 Recap of Impulse Based Substructuring This section will give of brief recap of the theory of using impulse response functions for solving dynamic substructuring problems. First, the theory of using impulse response functions for determining dynamic structural responses is recalled. In addition, this idea is expanded in order to handle partitioned systems by applying Lagrange multipliers for enforcing compatibility between the different substructures. As one can choose to measure accelerations, velocities and or displacements, we will show for all cases how the method is to be implemented and which of those IRFs is used in this work. 6.2.1 Using Impulse Response Functions for Time Integration Let us call Y.t/ the matrix of the displacement responses due to a unit impulse at t D0for a linear system that is initially at rest. In other words a coefficient ŒY.t/ ij of the impulse response matrix represents the response of degree of freedom (DoF) i to a unit impulse on DoFj at time t. The response of the linear system, initially at rest, to an applied force f.t/ can then be evaluated by the convolution product (Duhamel’s integral) between the impulse response function matrix and the applied forces: u.t/ DY.t/ f.t/ DZ t 0 Y.t /f. /d (6.1a) This is a classical result of time analysis of linear systems, usually obtained using Laplace transforms (see for instance [6]). Similarly, we can define this as well for the velocities and accelerations of the system. By calling PY.t/ and RY.t/, respectively the matrices of the velocity and acceleration responses due to a unit impulse at t D0for a linear system initially at rest, such that: Pu.t/ DZ t 0 PY.t /f. /d (6.1b) Ru.t/ DZ t 0 RY.t /f. /d (6.1c) These convolution products can be interpreted as follows: the response at time t is an infinite sum of the responses to the infinitesimal impulses f. /d before time t (see Fig. 6.1). Each impulse at time gives a contribution through the impulse response from tot, that is Y.t /. τ Δτ f(τ) Fig. 6.1 Forcing function as a series of impulses [10]
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