The modal damping characteristics of beams partially treated with magnetorheological (MR) fluid elements are studied using the modal strain energy approach and the finite element method. Different configurations of a sandwich beam partially treated with MR fluid are considered, including a beam with a cluster of MR fluid segments and a beam with arbitrarily located MR fluid segments. The significance of the location of the MR fluid segments on the modal damping factor is investigated under different end conditions. An optimization problem is formulated by combining finite element analysis with optimization algorithms based on sequential quadratic programming (SQP) and the genetic algorithm (GA) to identify optimal locations for MR fluid treatment to achieve maximum modal damping corresponding to the first five modes of flexural vibration, individually and simultaneously. The solutions of the optimization problem revealed that the GA converges to the global solutions rapidly compared to the SQP method, which in some modal configurations usually entraps in the local optimum. The results suggest that the optimal location of the MR fluid treatment is strongly related to the end conditions and also the mode of vibration. Furthermore, partial treatments with MR fluid can significantly alter the deflection modes of the beam. It has also been demonstrated that optimal locations of the MR fluid segments based on linear combination of the modal damping factors of the first five modes are identical to those obtained based on the first mode, irrespective of the end conditions. However, the optimal locations of the MR fluid segments, identified based on the logarithmic summation of the modal damping factors of the first five modes, would yield a more uniform shear energy distribution compared to that attained by considering individual modes or a linear summation of the individual modes.
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