150 F. Beltrán-Carbajal et al. Table 14.1 Mechanical system parameters for two DOF m1 D1.2678 kg m2 D1.3317 kg k3 D360N/m c1 D2.9Ns/m c2 D1.7Ns/m – k1 D178N/m k2 D360N/m – 0 -0.01 -0.005 0 0.005 0.01 0.015 0.012 0.001 0.008 0.006 0.004 0.002 0 0 20 40 60 1 2 3 t [s] x1 [m] X1(w) [m] w [rad/s] 4 Numerical model Experimental 5 Fig. 14.2 Free vibration response of the first mass carriage x1 and experimental FRF computed from a sine sweep from 0.1 to 10 Hz and constant force of 1.0 N Moreover, one could replace xi(s) into Eq. 14.6 by some Frequency Response FunctionHi(!): Hi .!/ D n X iD1 i .j!/ Ai j! i C A i j! i (14.17) where i are the system poles, Ai are the residues and i are polynomial functions representing the influence of the system initial conditions, and thus synthesize from Eq. 14.6 an algebraic identification scheme for on-line estimation of the modal parameters using some experimental Frequency Response Function. 14.4 An Illustrative Case: Simulation and Experimental Results The experimental setup is a rectilinear mechanical plant (Model 210a) provided by Educational Control Products®. The mechanical system consists of two mass carriages, interconnected by bidirectional cylindrical helical springs. Each mass carriage suspension has anti-friction ball bearing systems and, therefore, the linear dashpots are included only to describe small viscous dampings. Each mass carriage has a (rotary) high resolution optical encoder to measure its actual positions via cable-pulley systems (with effective resolutions of 2,266 pulses/cm or 4.413 10 3 mm/pulse). The signal and algebraic identification are obtained through a high-speed DSP board into a standard PC running under Windows XP® and Matlab®/Simulink®. The performance of the proposed on-line algebraic modal parameter identification approach was numerically and experimentally verified on a two-DOF mechanical system with the set of physical system parameters given in Table 14.1. Therefore, the actual values of the characteristic polynomial coefficients can be easily computed as: a0 D1.5267 10 5, a1 D1,778.44, a2 D967.94, a3 D3.564, with corresponding modal parameters !n1 D14.1244 rad/s, 1 D0.06679, !n2 D27.6636 rad/s and 2 D0.03032. Here we only employ measurements of the position of the first mass carriage x1. The application of the algebraic identification scheme in Eqs. 14.13, 14.14 and 14.15 was performed in the numerical case using Runge-Kutta 4/5 methods with fixed step time of 9 ms and for the real-time (experimental) algebraic estimation case using cumulative trapezoidal numerical integration with fixed sampling time of 9 ms. In both cases the initial conditions were x1(0)D0.01275m, Px1.0/ D0 m=s, x2(0)D0mand Px0.0/ D0 m=s. The numerical and experimental transient response for the displacement x1 and FRF computed only for the experimental data are shown in Fig. 14.2. The algebraic estimation of the coefficients of the fourth order characteristic polynomial of the two DOF system is described in Fig. 14.3. Note the satisfactory performance of the algebraic identifiers (14.13), (14.14) and (14.15), using only measurements of the position output variable x1. The effective and fast estimation of the coefficients of the characteristic
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