After 67 iterations (CPU time is about 8 seconds), the inverse solutions can be obtained and plotted in Figure 6 for the exact and estimated external forces.

From the above two test cases we learned that an inverse nonlinear damped force convection problem in estimating simultaneously the unknown external forces is now completed. Reliable estimations can be obtained when using either exact or error measurements.

　

7. CONCLUSIONS

　

The Conjugate Gradient Method (CGM) was successfully applied for the solution of the inverse nonlinear force vibration problem in a multiple degree-of-freedom system to determine simultaneously the unknown transient external forces by utilizing simulated displacement measurements obtained from sensors. Several test cases involving different system parameters, measurement errors and external forces were considered. The results show that the inverse solutions obtained by CGM remain stable and regular as the measurement errors are large. Moreover the CPU time needed in the inverse calculations is very short and the initial guesses for external forces can be arbitrarily chosen as zero.

　

Figure 1. A multiple-degree of-freedom nonlinear force vibration system considered in the present study.

　

Figure 2. The exact (f₁ and f₂) and estimated (f₁ and f₂) external forces using displacement measurements with σ₁ = σ₂ = 0.0 in numerical test case 1.

　

ACKNOWLEDGMENT

　

This work was supported in part through the National Science Council, R. O. C., Grant number, NSC-89-2611-E-006-027.

　

RBFERENCES

　

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