A linear stability analysis is presented of both hydraulically smooth and transitional flows over an erodible bed. The present theory is developed to account for the formation of ripples. It is essentially an extension of the theory of Richards (1980) to include the effect of viscosity upon the bed wave stability. The theory takes into consideration that the formation of ripples does not depend on flow depths, and that only the bed-load transport is involved in the formation of ripples. The effect of gravity is included in the analysis through the local inclination of the wavy bed surface. The results show that the bed is unstable (i.e. ripples exist) when the grain Reynolds number is less than a certain value. The limiting values of the grain Reynolds number for ripple existence obtained through present analysis are found to be in good agreement with observations.

This study firstly proposes some representative simple methods to obtain the suboptimal passive damping and stiffness parameters from the optimal control gain matrix since it is not possible to add the exact optimal damping and stiffness parameters to the structure in practice. It is shown numerically that modifying the structural damping and the stiffness in the proposed suboptimal ways may suppress the uncontrolled vibrations while the performance levels depend on the seismic inputs. Since the proposed approach is intrinsically passive and has no adaptive property against changing dynamic effects, this study secondly proposes a new performance index so that the mechanical energy of the structure, control and the seismic energies are considered simultaneously in the minimization procedure. The implementation of the resulting closedloop control algorithm does not require both a priori knowledge of the seismic excitation and the solution of the nonlinear matrix Riccati equation. The performance of the proposed approach is investigated, e.g., structures subjected to three seismic inputs and compared to the performance of the uncontrolled, the classical linear optimal control, and the passive cases. It is shown by the numerical simulation results that the proposed algorithm is capable of suppressing the uncontrolled seismic structural displacements and the absolute accelerations simultaneously and performs almost as well as the classical linear optimal control in reducing the displacements with comparable control effort and performs better than the classical linear optimal control in reducing the absolute accelerations. The results show that while the proposed active approach has similar performance to the classical linear optimal control, the classical linear optimal control increases the absolute accelerations slightly compared to the proposed active approach in regulating displacements, while the proposed active approach regulates and reduces both displacements and absolute accelerations. The proposed approach is promising in protecting both the structural and non-structural members from the seismic forces since a simultaneous reduction both in the displacements and the absolute accelerations is achieved.C 2012 Computer-Aided Civil and Infrastructure Engineering.

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SUMMARYExact optimal classical closed-open-loop control is not achievable for the buildings under seismic excitations since it requires the whole knowledge of earthquake in the control interval. In this study, a new numerical algorithm for the sub-optimal solution of the optimal closed-open-loop control is proposed based on the prediction of near-future earthquake excitation using the Taylor series method and the Kalman ÿltering technique. It is shown numerically that how the solution is related to the predicted earthquake acceleration values. Simulation results show that the proposed numerical algorithm are better than the closed-loop control and the instantaneous optimal control and proposed numerical solution will approach the exact optimal solution if the more distant future values of the earthquake excitation can be predicted more precisely. E ectiveness of the Kalman ÿltering technique is also conÿrmed by comparing the predicted and the observed time history of NS component of the 1940 El Centro earthquake.

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SUMMARYConsiderable e ort has been devoted to develop optimal control methods for reducing structural response under seismic forces. In this study analytical solution of the linear regulator problem applied widely to the control of earthquake-excited structures is obtained by using the su cient conditions of optimality even though almost all of the optimal controls proposed previously for structural control are based on the necessary conditions of optimality. Since the resulting optimal closed-open-loop control cannot be implemented for civil structures exposed to earthquake forces, the solution of the optimal closed-openloop control is carried out approximately based on the prediction of the seismic acceleration values in the near future. Upon obtaining the relation between the exact optimal solution and future values of seismic accelerations, it is shown numerically that the solution of the optimal closed-open-loop control problem can be performed approximately by using only the ÿrst few predicted seismic acceleration values if a given norm criteria is satisÿed. Calculated performance measures indicate that the suggested approximate solution is better than the closed-loop control and as we predict the future values of the excitation more accurately, it will approach the optimal solution.