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Today, base isolation of buildings is a conventional approach to earthquake resistance. The prominent goal is to reduce displacement of structure by movement of elastomeric bearings installed on the base of structures on the ground. Considering widespread construction of asymmetric buildings well as the intensity of damages to such types of structures resulting from earthquake, the present research covers study of interaction mechanics of asymmetric base-isolated structures, where motion equations are presented in two coordinates, one fixed on the building base (global coordinate) and the other on the torsional isolation level (local coordinate). In this conventional approach, the motion equations are calculated on linear form in the initial coordinate system, whereas in the new approach proposed in this research, motion mechanics analysis in the secondary coordinate system will lead to non-linear equations. Three types of structures are proposed with ratio of torsional-lateral correlated natural frequency on asymmetric natural frequency. Responses of both linear and nonlinear methods for the three types of structures under harmonic effects and earthquake are compared while analyzing time history and frequency. Some differences are observed between the linear and nonlinear methods. Then, some non-linear phenomena such as saturation, energy transfer between modes, and rigid displacement in such structures are also analyzed.

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In this paper, a modified linear-quadratic-Gaussian (MLQG) optimal control algorithm is proposed for controlling the seismic response of frame structures. Environmental loads (e.g., earthquakes) at the moment of calculation and exertion of control forces to structures, can not be measured. So these loads are not included in the conventional control algorithms, such as linear quadratic regulator and linear-quadratic-Gaussian control. Therefore the command of LQG optimal controller is merely a proportional feedback of estimated state of structure at the moment of exertion. This state approximation is performed by optimal state stimator or Kalman filter. In the proposed control algorithm, using a new variable, including control force andearthquake force, acceleration of gound motion, which is non-measurable duting exertion of control force, is considered in the state space equation of motion and also in both of Kalman Filter estimator and the optimal regulator. According to the proposed control algorithm, two ways are selected. So first command control are sum of the control force and ratios of the estimated state and measurement output of sensors, which are obtained and used in previous time step. The estimated state of system, used in the first command control, is calculated by the conventional and knownKalman Filter. but in second strategy of control, First, the Kalman Filter estimator is modified based on new state space equations, and then the estimated state of structure obtained from it, is used for calculation of command control. Numerical simulation of a seven-storey structure with active control system under several far-fault and near-fault earthquakes are performed to show effectiveness of two proposed controls on mitigation of structural responses and compare to those of a uncontrolled structure and a structure controlled with conventional control. Also sensitivity of some perforemance measures for controllers are investigated against changes of some controlling and perturbation parameters of systems or uncertainties. The alalysis results demonstrate that control performance of the proposed controllers, specially the second one, are better and also stable and robust under variations of uncertainties. So that the greatest reduction in maximum displacement (even up to 80 percent) compared to uncontrolled displacement of structure and meanwhile, very low energy consumption are attained by the second proposed control strategy.but in second strategy of control, First, the Kalman Filter estimator is modified based on new state space equations, and then the estimated state of structure obtained from it, is used for calculation of command control. Numerical simulation of a seven-storey structure with active control system under several far-fault and near-fault earthquakes are performed to show effectiveness of two proposed controls on mitigation of structural responses and compare to those of a uncontrolled structure and a structure controlled with conventional control. Also sensitivity of some perforemance measures for controllers are investigated against changes of some controlling and perturbation parameters of systems or uncertainties. The alalysis results demonstrate that control performance of the proposed controllers, specially the second one, are better and also stable and robust under variations of uncertainties. So that the greatest reduction in maximum displacement (even up to 80 percent) compared to uncontrolled displacement of structure and meanwhile, very low energy consumption are attained by the second proposed control strategy.

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The use of members with non-uniform cross-sections due to the reduction of the number of materials and the weight of the structure is widely used in industrial structures and metal bridges. Buckling is one of the major problems engineers face in the design of axial compression members (columns). For this reason, several researches have been conducted by researchers in the field of column buckling. Most of the previous research is limited to investigating stability and buckling in Non-prismatic elastic columns in the static state. During an earthquake, the structure is subjected to vertical and lateral earthquake loads. To evaluate the dynamic behavior of the structure during an earthquake, the stability and dynamic buckling of the column must be evaluated. The effect of the earthquake's vertical load and the dynamic axial load has an effect on the dynamic stability of the member in the form of the second-order effect of buckling. In this article, the dynamic buckling of a column with a variable section and viscous damper under alternating axial load is investigated in a comprehensive model. The alternating axial load effect is assumed as a cosine function and the viscous damping effect at the end of the member is assumed as a Dirac delta function. The changes in the moment of inertia along the length of the column are considered in three modes: linear, cubic, and fourth-order changes. The constituent differential equation includes column strain energy, second order effect of alternating axial load, inertia per unit length of the column, and damping of a viscous damper. To solve the constitutive equation, first the weak form of the governing differential equation is written. Lagrange interpolation functions are used as the shape function and the Fourier function (proposed by Bolotin) as the dynamic response of the equation. In the next step, the matrices of material hardness, geometric hardness, and mass are extracted. After extracting the above matrices, the eigenvalues (Buckling load factor, natural frequency) of the equation are checked. Muller root finding technique is used by coding in MATLAB software to calculate eigenvalues. For accuracy in calculations, the function of the form of the equation is checked by the Lagrange method with the number of thirty terms. Also, finding the roots of the equation to calculate the eigenvalue is done with a step of 0.05 using Mueller's method. The buckling load coefficient of the column is evaluated for different values of the expansion coefficient and the damping percentage of the viscous damper in different boundary conditions. The results show that the mentioned values have a significant effect on the changes in the buckling load factor in terms of excitation frequency and resonance frequency. Depending on the boundary conditions, increasing the opening factor causes the diagram to move to the right or left side of the dimensionless excitation frequency axis. Also, increasing the damping coefficient of the viscous damper causes the diagram to move to the left side of the dimensionless excitation frequency axis. Dimensionless parameters such as bar coefficient, excitation frequency, and opening coefficient have been used to report the dynamic behavior of the set in all the tables and figures. The results of this research can be generalized for the design of columns under periodic axial load. The results of this article are verified and compared with previous research. There is an acceptable agreement between the results of the present article and previous research.