Browsing by Author "Zou, Keguan"
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Item An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation(Elsevier, 2014) Zou, Keguan; Nagarajaiah, SatishA new modification of homotopy analysis method (HAM) is proposed in this paper. The auxiliary differential operator is specifically chosen so that more than one secular term must be eliminated. The proposed method can capture asymmetric and period-2 solutions with satisfactory accuracy and hence can be used to predict symmetry-breaking and period-doubling bifurcation points. The variation of accuracy is investigated when different number of frequencies are considered.Item Study of a piecewise linear dynamic system with negative and positive stiffness(Elsevier, 2014) Zou, Keguan; Nagarajaiah, SatishThe present paper mainly focuses on numerical and analytical study of a piecewise linear dynamic oscillator with negative stiffness followed by positive stiffness which has not been studied to date. The dynamic system of interest stems from a previous analytical and experimental research on adaptive negative stiffness for the purpose of seismic protection. Numerical algorithms meant specifically for simulating piecewise smooth (PWS) systems like this nonlinear system are studied. An appropriate combination of negative stiffness and adequate damping can reduce the peak restoring or transmitted force with a slightly larger peak displacement. Essentially, the negative stiffness system in a dynamic system is very beneficial in reducing the amount of force transmitted. The exact solution is derived for free vibration. A modified Lindstedt–Poincaré method (modified L–P method) is adopted to derive approximate periodic solutions for the forced and damped system and its frequency-response curves are obtained through numerical simulation. The modified L–P solution obtained for the forced and damped case is found to agree well with the numerical results. In the piecewise linear dynamic system with initial negative stiffness followed by positive stiffness, it is found that the response remains bounded in a limit cycle. This system behaves similar to a van der Pol oscillator wherein negative damping is followed by positive damping. Presented herein is a special case as defined by the specified parameter ranges; thus, to make it more general future work is needed.Item Study of Adaptive Passive Stiffness Systems with Nonlinear Vibrations: New Analytical and Computational Techniques(2014-12-08) Zou, Keguan; Nagarajaiah, Satish; Padgett, Jamie E.; Dick, Andrew J.A new class of adaptive passive stiffness systems undergoing nonlinear oscillations are proposed and studied in this Ph.D. thesis. Adaptive passive stiffness systems that have been proposed in this thesis and by other researchers recently allow for new ways of vibration isolation, seismic protection, and tuned mass damping of vibrations. New characteristics which emerge in these adaptive passive stiffness systems, like the coexistence of negative stiffness and positive stiffness, pose the need of analytical methods for studying these systems. Existing analytical and computational techniques that permit such a study are explored and new techniques are developed. A new analytical technique called the multi-frequency homotopy analysis method is proposed to this end. This multi-frequency homotopy analysis method can not only be adopted to solve a wide variety of nonlinear problems with periodic and quasi-periodic steadystate solutions, but can also be utilized to obtain analytical expressions of nonlinear dynamic systems’ transient response. These abilities of the multi-frequency homotopy analysis method make it a powerful tool for studying the new adaptive passive stiffness systems. Besides, the pseudo-force method is adapted to analytically solve a nonlinear system involving negative stiffness. Other analytical and numerical methods like the modified Lindstedt–Poincar´e method, the event-driven simulation technique and the time stepping method are also explored for analyzing such adaptive passive stiffness systems. In this Ph.D. study, a new adaptive passive stiffness device called universal stiffness devices (USD) is proposed. The newly proposed USD can generate a wide range of smooth and piecewise smooth restoring forces with positive as well as negative stiffnesses. The stiffness of the USD can be changed smoothly and continuously by simply adjusting a physical parameter of the USD. The variability of the USD’s stiffness can provide retuning capability of vibration isolation systems and other control systems with such a USD. The proposed USD can serve as a core element in a vibration isolation system to adaptively attenuate the vibration transmitted to the primary structure. Moreover, as an adaptive passive device, the USD brings better adaptability than a purely passive system. An analytical model of the USD has been established and experimental study has been conducted. The force–displacement loops of the USD are measured and compared to the results computed by using the newly formulated analytical model. The USD’s capability of changing its stiffness as it is designed has been tested. The performance of a vibration isolation system with the proposed USD has been verified experimentally under a range of harmonic excitations that cover dominant vibration frequencies and amplitudes. For future research, further application of the proposed analytical method for investigating nonlinear vibration isolation systems of different types could be expected; more practical applications of the proposed USD in vibration control and hazard reduction can be explored.