An exact analytical method is presented for the analysis of forced vibrations of uniform, open-section channels. The centroid and the shear center of the channel cross-sections considered do not coincide; hence the flexural and the torsional vibrations are coupled. In the context of this study, the type of any existing coupling is defined in terms of the independent motions which are coupled through mass and/or stiffness terms. Hence, if the flexural vibrations in one direction are coupled with the torsional vibrations, the resulting coupling is called double-coupling. On the other hand, if the flexural vibrations in two mutually perpendicular directions and the torsional vibrations are all coupled, the resulting coupling is referred to as triple-coupling. The study also takes the effects of cross-sectional warping into consideration but, since it is derived from torsional characteristics, the warping is not treated as an independent motion. Wherever necessary, the admission of warping is characterized by the inclusion of warping constraint. The current work uses the wave propagation approach in constructing the analytical model. Single-point force excitation has been considered throughout and the channels are assumed to be of Euler-Bernoulli beam type. Both double-and triple-coupling analyses are performed. The coupled wavenumbers, various frequency response curves and the mode shapes are presented for undamped and structurally damped channels. 7 1997 Academic Press Limited 1 2 f 1t 2 = 0, (4)where w is the flexural displacement in the z direction, f is the torsional displacement, EI j is the flexural rigidity in the z direction, GJ is the torsional stiffness, m is the mass per unit length, c y is the eccentricity between the centroid and the shear centre in the y direction,
Purpose-The purpose of this paper is to detail the design of a fractional controller which was developed for the suppression of the flexural vibrations of the first mode of a smart beam. Design/methodology/approach-During the design of the fractional controller, in addition to the classical control parameters such as the controller gain and the bandwidth; the order of the derivative effect was also included as another design parameter. The controller was then designed by considering the closed loop frequency responses of different fractional orders of Continued Fraction Expansion (CFE) method. Findings-The first, second, third and fourth order approximations of CFE method were studied for the performance analysis of the controller. It was determined that the increase in the order resulted in better vibration level suppression at the resonance. The robustness analysis of the developed controllers was also conducted. Practical implications-The experimentally obtained free and forced vibration results indicated that the increase in the order of the approximations yielded better performance around the first flexural resonance region of the smart beam and proved to yield better performance than the classical integer order controllers. Originality/value-Evaluation of the performance of a developed fractional controller was realized by using different approach orders of the CFE method for the suppression of the flexural vibrations of a smart beam.
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