The matrix has the numerical values of the mode vectors for specific position coordinates on the beam. Since this matrix must be square in order to form an inverse, it is convenient to choose the number of degrees of freedom in the system to be equal to the number of modes used in the Galerkin expansion. Then, can contain shape functions evaluated at the nodal degrees of freedom on the beam. In this study the number of mode shapes used in the expansion and the number of degrees of freedom used to model the beam was Nm=N=2. The degrees of freedom were located at the center and tip of the beam. In order to mimic the experimental system, the following parameters were used in the model, which are based on the nominal properties of the experimental hardware: =2700 kg/m3, E=68e9 N/m2, Ab=3.23e-4 m2, I = 4.34e-9 m4, L = 1.016 m. Using these properties with the Ritz-Galerkin method, the two linear natural frequencies of the system are 1/(2)= 9.97 Hz and 2/(2)=62.51 Hz. The transverse stiffness contribution of the spring steel on the experimental beam is approximated in the model as k3=1.4764e9 N/m3. A derivation of this approximation can be found in the appendix. 3.2 Simulated Measurements In order to apply the proposed nonlinear identification, the nonlinear beam must first be driven to respond in a periodic orbit. However, there are many possible periodic orbits that this system may be driven in so it is desirable to consider all of the possible periodic orbits for different forcing configurations. In a companion paper [22], the authors used a numerical continuation technique to calculate the periodic solutions of the beam model for forcing amplitude of A=1 Newton and for forcing frequencies in the band 6-70 Hz. The results of the computation are shown in Figure 4. In (a) and (b), the response curve near the first linear natural frequency is plotted. The red and blue curves in (a) correspond to the displacement initial conditions of the first and second degree of freedom, respectively. The green and black curves in (b) are the analogous velocity initial conditions. This color format is repeated in (c) and (d) for the frequency band near the second linear natural frequency. The curves quantify the resonant responses of the first and second modes of the nonlinear beam (referenced to zero phase of the force), but they also provide the initial state vector for a specific frequency that one can integrate in time to achieve a periodic orbit. A detailed discussion of the dynamics of the frequency response curves is provided in [22]. The important implications of these nonlinear frequency response curves is that the resonance peaks tend to bend towards higher frequencies for this system, which causes regions where multiple periodic orbits are possible for a single forcing frequency, and for this system one of those possible solutions is unstable (unstable solutions are designated in the plots with the dashed lines). The forcing configuration needs to be carefully selected in order to achieve a successful identification of the nonlinear parameters. Guidelines for selecting forcing configurations for the system identification method used here are discussed in detail in [14] for a single degree of freedom system, and those guidelines apply equally well here. 8 10121416182022 10-5 State Displacement Nonlinear FRF, Mode 1, A = 1 Frequency, Hz (a) 8 10121416182022 -0.4 -0.2 0 0.2 0.4 State Velocity Frequency, Hz (b) 59 60 61 62 63 64 65 66 -1 -0.5 0 0.5 1 x 10-3 Nonlinear FRF, Mode 2, A = 1 Frequency, Hz (c) 59 60 61 62 63 64 65 66 -0.4 -0.2 0 0.2 0.4 Frequency, Hz (d) DOF 1 DOF 2 DOF 1 DOF 2 Fig. 4 Initial conditions that result in periodic responses for A=1 plotted versus forcing frequency The nonlinear frequency response curves were used to understand what types of periodic orbits were possible, and a few of the different alternatives were selected for study. It is desirable to drive the system in a stable periodic orbit that is sufficiently nonlinear and isolated from other nearby stable periodic orbits. Sracic and Allen [14] showed that periodic orbits on the larger amplitude branch of the bent resonance peak tend to provide suitable periodic orbit conditions to use for the identification procedure for a hardening resonance such as that of the peak in Figure 4(a). Therefore, a forcing frequency of f1=15.4 Hz was chosen on the first resonance peak of Figure 4(a). Since the beam model has two modes that produce resonance responses, a second forcing frequency of f3=60.9 Hz, which is just below the second resonance peak of Figure 4(c), 110
RkJQdWJsaXNoZXIy MTMzNzEzMQ==