11 Extending the Fixed-Points Technique for Optimum Design of Rotational Inertial Tuned Mass Dampers 85 0 2 4 6 8 10 12 Dynamic Magnification Factor (DMF) Tuned Damping Infinity Damping De-tuned Damping 1 De-tuned Damping 3 De-tuned Damping 4 De-tuned Damping 2 Zero Damping 0.6 0.8 1.2 Frequency Ratio Q1 P2 P1 Q2 1.4 1.6 1.8 1 Fig. 11.4 Fixed-points in the RITMD’s frequency response curves Frequency Ratio Dynamics Magnification Factor (DMF) Proposed Fixed Points Method Nonlinear Programming Optimization 1 0.5 0 1 2 3 4 1.5 Fig. 11.5 Extended Fixed-Point Method vs Numerical Optimization; m1 D10% ms (rotational mass D15%m1) which is a forth degree polynomial with four real positive roots. Extending the fixed-points method, we assumed the optimum frequencies condition occurs in the case of a pair equality of the primary structure response magnitude in the fixedpoints (kHP1kDkHQ1k; kHp2kDkHQ2k). This assumption leads to two high-order nonlinear equations which are solved numerically to find the optimum frequency ratios. In the final step, the optimum damping can be found by putting the maximum response magnitude of one of two frequency ratios equal to response magnitude of the other optimum frequency ratio. 11.3 Results and Discussion To examine the proposed extended fixed-points technique accuracy, the frequency response of the primary system with optimum design values from the proposed method is compare with the optimum H-infinity design utilizing an numerical nonlinear programming optimization method [10] (Fig. 11.5). The propose method is an approximate method, thus there are small difference in comparison to the exact H-infinity optimization utilizing nonlinear programming; however, the results from the proposed method are close, which demonstrates the accuracy of the assumptions in the extended proposed method. In addition, the response of the primary structure of both RITMD and TMD systems with the same secondary mass ratio (10%) and optimized with the fixed point method is shown in Fig. 11.6. It can be observe that the RITMD exhibits superior performance in reducing the vibration amplitude of the primary system in compare to the TMD.
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