148 V. Yaghoubi and T. Abrahamsson 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -2 -1 0 1 2 3 4 5 6 7 x 10-3 time(sec) Amplitude (m) Ts = 1 µs Ts = 10 µs Ts = 100 µs 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -2 -1 0 1 2 3 4 5 6 7 x 10-3 time(sec) Amplitude (m) Ts = 1 µs Ts = 10 µs Ts = 100 µs Fig. 13.9 Time response of the beam with hard support spring (kNL D0.17 MN/m) and gap nonlinearity with different sampling rates T. (Left) PFSS method, (right) MMRT method Table 13.9 Deviation (in %) of the methods as compared withRK Method Stiffness (N/m) PFSS MMRT RK kNL D1.7 0.01 8.82 – kNL D17 0.39 5.11 – Sampling time step is TD0.1ms Table 13.10 Required time (in seconds) to simulate the beam response with stiff spring for different sample rates Sample rate PFSS MMRT TD0.1ms 3:52 13:65 TD10.0 s 19:98 137:17 TD1.0 s 199:44 1428:7 It is clear from Tables 13.1 and 13.2 for weak nonlinearity the PFSS method worked about 3 times faster than the MMRT method and about 20 times faster than the adaptive step Runge–Kutta method. The result of the PFSS method is about 9 times more accurate than results of the MMRT method. For the case of strong nonlinearity and too low sample rate the PFSS method became unstable but the MMRT yielded results after 0.05 s but large deviation from results of the Runge–Kutta method. Table 13.3 shows that the PFSS method obtained the simulation result for the 3DOF structure with weak nonlinearity and time step 0.01 s after 0.06 s which is about 14 times faster than the MMRT method and about 10 times faster than the Runge–Kutta method. The accuracy of the methods, indicated in Table 13.4, show that the PFSS method is 8 times more accurate in this case. For the more strongly nonlinear case, PFSS works about 2.5 times faster and 3 times more accurate thanMMRT. Simulation results for the second 3DOF structure are shown in Fig. 13.5. The required simulation time and result deviation from results of the Runge–Kutta method has been tabulated in Tables 13.5 and 13.6 respectively. Results indicate that under some conditions MMRT can have big deviation from Runge–Kutta results and also it is significantly slower than the here presented PFSS method. For the beam structure, results have been depicted in Fig. 13.8 show a good match between the results for weak nonlinearity. However, for strong nonlinearity there is deviation between the methods, as is shown Fig. 13.9. Table 13.8 indicates that RK is significantly slower than the other two methods and PFSS is about 12 times faster than MMRT. The main reason for the long simulation time for RK is the fact that stimulus, which is impulse, inserted in very short duration and ode45 could not see it, therefore, a time vector with a very short time step have to send to it. As shown in Table 13.9, PFSS has almost the same accuracy as RK for both cases but MMRT has large deviation from RK in stiff spring case. The convergence of the methods for strongly nonlinear structure, KD170 KN/m, was considered and Fig. 13.9 indicates that the PFSS method has converged for a time step of 0.1 ms and it took 6.43 s to obtain the results (see Table 13.10) but for the MMRT method no convergence could be found even for time steps as small as 1 s with results that took 1,429 s to obtain.
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