Interval type-2 fuzzy α-plane minimax co-design of multi-tuned mass dampers for broadband vibration control under bounded uncertainty

Abstract

Passive tuned mass dampers are effective near their design frequencies but can lose performance or exceed travel limits when structural properties are imprecise. This paper develops an interval type-2 fuzzy α-plane minimax method for the simultaneous allocation, tuning, and damping design of a three-absorber bank on a six-degree-of-freedom coupled machine–foundation system. Primary mass, stiffness, and damping are represented by lower and upper triangular membership functions, so uncertainty in both parameter values and membership widths is retained. At every α-plane, outer and inner vertex responses are propagated through the complex frequency-response matrix. A membership-weighted objective combines worst peak acceleration, broadband root-mean-square acceleration, footprint width, and a 16 mm relative-stroke constraint. Differential evolution determines nine absorber variables for a fixed 6% auxiliary-mass budget. The frequency-domain solver reproduces classical equal-peak tuning ratios to machine precision and absorber damping ratios within 0.20%. In 2,048 independent outer-support Sobol scenarios, the proposed design decreases the mean peak acceleration from 6.058 g to 1.262 g and the worst peak acceleration from 9.005 g to 1.682 g. The deterministic optimum attains a smaller mean peak of 1.112 g but violates the stroke constraint in 55.42% of the scenarios; the type-1 robust design violates it in 2.05%, whereas the proposed design has no violations and a maximum stroke of 15.992 mm. The results show that explicitly preserving membership-function uncertainty changes the mass allocation and first-mode detuning sufficiently to obtain a feasible broadband design without active control.