Abstract
We propose a two‐step strategy for the design of passive controllers for the simultaneous confinement and suppression of vibrations (SCSV) in mechanical structures. Once the sensitive and insensitive elements of these structures are identified, the first design step synthesizes an active control law, which is referred to as the reference control law (RCL), for the SCSV. We show that the problem of SCSV can be formulated as an LQR‐optimal control problem through which the maximum amplitudes, associated with the control input and the displacements of the sensitive and insensitive parts, can be regulated. In the second design step, a transformation technique that yields an equivalent passive controller is used. Such a technique uses the square root of sum of squares method to approximate an equivalent passive controller while maximizing the effects of springs and dampers characterizing passive elements that are added to the original structure. The viability of the proposed control design is illustrated using a three‐DOF mechanical system subject to an excitation. It is assumed that all of the masses are sensitive to the excitation, and thus the vibratory energy must be confined in the added passive elements (insensitive parts). We show that the vibration amplitudes associated with the sensitive masses are attenuated at fast rate at the expense of slowing down the convergence of the passive elements to their steady states. It is also demonstrated that a combination of the RCL and the equivalent passive control strategy leads to similar structural performance.
Highlights
In engineering applications, vibrations are likely to excite unwanted resonances in flexible structures
We show that the problem of vibration confinement can be formulated as an LQR-optimal control problem
Motivated by the work of Gluck et al [10], we propose the use of the square root of sum of squares (SRSS) method for the design of a passive controller equivalent to the active controller developed in the preceding section
Summary
Vibrations are likely to excite unwanted resonances in flexible structures. Bendiksen [1] studied the localization phenomenon in engineering structures that include periodic or nearly periodic multi-span beams and trusses, large space structures, and almost periodic structures with circular symmetry, such as bladed disks in turbomachines He discussed both analytical and numerical methods for analyzing and predicting localization in finite- and infinite-dimensional systems. The emphasis is on designing a control strategy that adds passive devices (linear springs and dampers) for confining and suppressing simultaneously the vibrational energy in flexible structures. To this end, we first design an optimal controller using a set of actuators to satisfy desired confinement and suppression specifications. We construct an equivalent passive controller, which produces a similar performance
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