7 Identification of a Time-Varying Beam Using Hilbert Vibration Decomposition 77 x(t) Source separation x(t) → s(t) Analytic signal z(t) = s1(t)+ iH(s1(t)) Phase extraction φ(t) = Ðz(t) Trend extraction φ(t) → φ(k)(t) VKF x(k)(t), Vk(t) Sifting process x(t) := x(t) − x(k)(t) Fig. 7.4 Flow chart of the method 7.6 Numerical Application The numerical application considered in this study consists of a beam on which a moving mass is traveling. The beam is 2 m-long and is simply supported at both ends; two random external force excitations are applied in the vertical and lateral directions at a quarter of the total length of the beam. Five equally spaced measurement points are selected to record the time response of the beam in the vertical and lateral directions. A mass of 3 kg travels the whole beam in 60 s during which the external forces are applied. The response of the system is computed using the LMS-Samcef Mecano [9] software in which a slider element is used to make the connection between the beam and the lumped mass. The measured time responses are sampled at a rate of 1,000 Hz and to better simulate a real measurement process, a normally distributed noise is added on each signal with a signal-to-noise ratio of 1 %. The simulated system and the measurement set-up are illustrated in Fig. 7.5. The properties of the system are the following: • Beam length: l D2 m • Beam cross section80mm(width) 20mm (height) • Density: D2;700kg/m3 • Young’s modulus: ED70;000MPa • Poisson’s ration: 0.33 • Moving mass: 3kg • Pinned connections: x.0/ Dy.0/ Dz.0/ Dx.l/ Dy.l/ Dz.l/ D0 The eigenfrequencies of the beam subsystem are listed in Table 7.1 with their corresponding mode-shapes. Using cross correlation between time signals, it can be verified that the responses in the lateral and vertical directions are completely separated, so that the two sets of measurements can be treated separately. In Fig. 7.6 wavelet spectra of the response of the first node in both lateral Fig. 7.6a and vertical Fig. 7.6b directions are shown. In this figure, the white dashed lines correspond to the natural frequencies of the beam subsystem (when the mass is located on one support, before beginning its motion). It can be observed that, for a given mode, the natural frequency decreases as the mass is moving and comes back to its initial value every time the mass passes upon a vibration node. This is due to the fact that the additional inertia force
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