114 M. Kwarta and M. S. Allen As shown in Eq. (1), the ∗-indicator expresses the relative difference between the coefficient that was enforced to be found as real and the real part of the complex one. Hence, ∗ will be small when the two algorithms produce consistent results. In contrast, ∗∗ is defined as the relative difference between the real and imaginary parts of the complex solution and gives a measure of how large the imaginary part of the solution is. Hence, the nonlinear term is considered to be dominant when its ∗ value is low enough and ∗∗ is close to 1. Note that these two requirements must be satisfied simultaneously. The accuracy thresholds could be specified with a parameter , as presented in Eq. (1c). The value of should be a small number, say =0.05. With the two -indicators defined, we are ready to illustrate the black-box capabilities of the NIXO algorithms. The next sections present a successful black-box identification performed on a numerical model of a curved beam. 2 Simulated Black-Box Identification of a Curved Beam The numerical test is performed on an ICE-ROM of a clamped–clamped curved beam subjected to a uniformly distributed swept cosine forcing signal. The beam has a length of 304.8 mm, a width of 12.7 mm, a thickness of 0.508 mm and a radius of curvature of 11.43 m. It is made of steel with a Young’s modulus of 207.4334×10 11 GPa, a density of 7850 kg/m3 and a Poisson’s ratio of 0.29. The ICE-ROM consists of the first three symmetric modes, i.e., modes 1, 3, and 5. Their linear natural frequencies and damping ratios are {65.181, 158.636, 385.882} Hz and {0.035, 0.0262, 0.0174}, respectively. The nonlinear equation of motion of the system, including every possible nonlinear term, is presented in (2). Since (1) the nonlinear part consists of the quadratic and cubic parts, and (2) there are three modes present in the ROM, the number of nonlinear terms that can occur in the EOM is at most 16. In each case study run, we assume the most general form of the NLEOM, see Eq. (2); hence, NIXO can point out the terms dominant in the system’s response out of the most general set of 16 terms. This brief publication focuses on identification of the nonlinear mode 1, and thus we presented the nonlinear equation of motion of this mode only; the equations for the two remaining modes are analogous. Note that the subscripts of the nonlinear coefficients correspond to the product of polynomial terms they multiply; e.g., β111 multiplies termq 3 1, while β123 multiplies termq1q2q3. ¨q1 +2ζ1ω1 ˙q1 +ω 2 1q1 linear part +α 1 11q 2 1 +α 1 12q1q2 +· · · quadratic stiffness part +β 1 111q 3 1 +β 1 112q 2 1q2 +· · · cubic stiffness part = T 1 f(t). (2) 3 Identification of Mode 1 The beam is excited with swept cosine signals of various magnitudes, such that it oscillates at different response levels in every test. These input/output signals are later provided to the NIXO algorithms, which use them to estimate the underlying linear and the nonlinear parts of the system. Since the first mode occurs at approximately 65 Hz, the authors decided to excite the system with 300-second-long (up and/or down) sweeps, with frequencies ranging from 1 to 115 Hz. The output signals obtained in these numerical tests are illustrated in Fig. 1 and can be grouped into two sets: down- and up-sweeps, which correspond to the force amplitudes of F0 ∈-1×10−7 , . . . , 8×10−4. and F0 ∈ {2.4, . . . , 4.75}×10−3 newtons, respectively. Note that some of the responses shown are not symmetrical with respect to the equilibrium position. This result was expected, since the beam is curved and thesnap-througheffect causes this asymmetry. Moreover, this observation explains the importance of including the quadratic stiffness terms in the nonlinear equation of motion. 4 Sample Black-Box ID Outcomes from High-Amplitude Vibration Tests The results from a black-box system identification attempt are presented in this section. The input signals provided to the NIXO algorithms are the sweep cosines with magnitudes of 2.4×10−3 (up-sweep) and 5.0×10−4 newtons (down-sweep). The corresponding output signals are already shown in Fig. 1. Note that the signals used have very different amplitudes; the proposed algorithm seemed to work best when this is the case.
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