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In this paper, we try to use modeling based on singular perturbation theory, in order to control satellite attitude during the wide rolling angle maneuvering through nonlinear H∞ control strategy. Differential equations describing dynamics of the satellite are presented first, and by choosing the appropriate dynamic model for actuators and based on the standard singular perturbation model, the closed-loop system is created. Next, this model is put into the appropriate form to solve H∞ problem. Then, after solving the HJI equation, the control law is determined. Simulation results for a nominal satellite control based on our approach are finally presented.

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The main objective of this research is to study the nonlinear vibrations of a single walled carbon nanotube. For this purpose, the lattice structure of carbon nanotube is replaced with a continuum structure using nanoscale continuum mechanics. Firstly, each carbon-carbon bond is replaced with an equivalent beam element and then the whole discrete structure of carbon nanotube is replaced with a virtual continuum medium representing hollow cylinder. Then, governing equations for vibrations is obtained taking into account geometric nonlinearity arisen from stretching of a mid-plane due to bending. Perturbation technique is used to analyze the nonlinear vibrations of carbon nanotubes. Frequency responses of carbon nanotubes for free vibrations and force vibrations in both primary and secondary resonance cases are studied. Obtained results are in a very good agreement with numerical integration technique. The results imply on hardening behavior of carbon nanotube. Moreover, nonlinear bifurcation and nonlinear jump phenomena are observed.

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Cavitation is changing liquid phase to gas phase due to decreasing local pressure of flow induced by increasing local velocity. In situation of maximum velocity, some bubbles that contain air and vapor are produced and traveled from point of high pressure to lower pressure, so bubbles are destroyed rapidly and produce acoustic noise. Providing sufficient numerical model for simulation of acoustic waves induced by cavitation or supercavitation is so important for monitoring and controlling of these phenomena. For analyzing propagation of acoustic waves in fluid, sound is part of fluid dynamics, so momentum, energy and mass conservation equations like fluid dynamics are basics equation for identification of supercavitation. In this paper, to provide a numerical model contains hydrodynamic and acoustic parts of fluid dynamics, first by using scaled analysis, non dimensional forms of conservation equations are generated. Then by using perturbation method and considering acoustic term as a term in lower order than hydrodynamic term, conservation equations can be separated to two group equations with different orders. Leading order is hydrodynamic equations and first order is acoustic form of conservation equations. Results in first order equation show coupling of acoustic terms with hydrodynamic terms of fluid flow.

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In this paper, the governing equations for a vibratory beam with moderately large deflection are derived using the first order shear deformation theory. These equations which are a system of nonlinear partial differential equations with constant coefficients are solved analytically with the perturbation technique and the natural frequencies and the buckling load of the system are determined. A parametric study is performed and the effects of the geometrical and material properties on the natural frequency and buckling load are investigated and the effect of normal transverse strain and axial load on natural frequency are examined. Some results based on the first order shear deformation theory are consistent with classic theories of beams and some yield different results. Formulation presented to calculate the transverse frequency, determines the axial frequency too. Also, the natural frequencies and buckling load are calculated with the finite elements method by applying one and three-dimensional elements and the results are compared with the analytical solution.

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In this study, static/dynamic instability and nonlinear vibrations of FG plates resting on elastic foundation under parametric forcing excitation, are investigated. Based on CPT, applying the von-Karman nonlinear strain–displacement relation and the Hamilton’s principle, the governing nonlinear coupled partial differential equations are derived. By considering six vibration modes, the Galerkin’s procedure is used to reduce the equations of motion to nonlinear Mathieu equations. In the absence of elastic foundation, the validity of the formulation for analyzing the static buckling, dynamic instability and nonlinear deflection is accomplished by comparing the results with those of the literature. Then in the presence of the foundation and by deriving the regions of dynamic instability, it is shown that as the parameters of the foundation increases, the natural frequency and the critical buckling load increase and the dynamic instability occurs at higher excitation frequencies. The frequency response equations in the steady-state condition are derived by applying the multiple scales method, and the parametric resonance is analyzed. Then the conditions of existence and stability of nontrivial solutions are discussed. Moreover, the effects of the system parameters, including excitation frequency, amplitude of excitation, foundation parameters and damping, on the nonlinear dynamics of the FG plate are investigated. Also it is shown that the presence of the foundation has a considerable influence on the resonance characteristic curves.

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In this paper, steady creeping motion of non-Newtonian falling drop through a viscous fluid is investigated analytically. Here, the Upper Convected Maxwell model (UCM) is used for drop phase and Newtonian model is considered for external fluid. The perturbation technique is used to solve both exterior and interior flows and Deborah number that indicated the elastic effect is considered as the perturbation parameter. The present solution is derived up to second order of perturbation parameter so the present solution has a suitable accuracy for drops that made from dilute polymeric solutions. We found that the Newtonian drop has a spherical shape during the creeping motion but the non-Newtonian drop loses this shape and takes an oblate form. By increasing the elastic effect, a dimple at the rear end of the drop is created and developed. Here, it is shown that the present results have more agreement with experimental data than the previous analytical studies. The origin of drop deformation is also considered and it is proofed that the elastic property of drop phase creates a concentrated normal stress at the rear end of the drop that causes the dimple shape in this region.

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Galloping of cables is a kind of self-excited vibration and characterized with high amplitude and low frequency vibration. In this paper for investigating the nonlinear galloping of an inclined cable, considering flexural and torsional stiffness, a cable-beam model is used. The iced cable is formulated under the effects of combined wind flow and support motion. Assuming low sag to span ratio and using physical parameter values of the cable, the governing equation of motion is obtained as a classical equations of the perfectly flexible cable, plus a further equation governing the twist motion. These two degrees of freedom system is discretized via the Galerkin method, by taking in-plane and out-of-plane modes as trial function. Two resulting non-homogeneous ordinary differential equations are coupled and contain quadratic and cubic nonlinearities in both velocity and displacement terms. By using multiple scale method for 1:1 internal resonance, a first order amplitude-phase modulation equation, governing the slow dynamic of the cable, is obtained. In this paper the wind speed and the eccentricity of the iced section are set as the control parameters. Without consideration the eccentricity, the value of amplitude is increased as the wind speed is increased. But considering the eccentricity is reduced to firstly increasing and then decreasing the amplitude.

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For formation flying of two satellites in a satellite constellation, the relative motion and attitude determination algorithms are the key components that affect the quality of flight and mission efficiency. In this paper orbital relative motion of two satellites with arbitrary Keplerian elliptic orbit and in large distance will be analyzed and also exact and efficient solution for relative motion of the satellite with j2 perturbation which is one of the important perturbation in Low Earth Orbit (LEO) using spherical geometry is proposed. Direct geometric method using spherical coordinates are utilized to achieve this solution. In this method relative position and relative velocity of two satellites are calculated in the satellite constellation based on orbital elements. The obtained results from simulation with STK software, comparison of results with extracted results from equations for satellite with different eccentricity and analysis of the proposed method’s accuracy and fault show that the solution obtained from the geometric method presents the relative motion of the satellite with high accuracy. Thus, the proposed solution will be applicable and effective for relative motion of constellation satellites in space missions.

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Motion and deformation of the drop falling in an immiscible fluid has become a benchmark problem in fluid mechanics and has a wide range of application in petroleum, medicine processing, metals extraction, power plant and heat exchanger. In this paper, an exact analytical solution of a falling viscous drop at low Reynolds number is investigated. Analytical solution for both internal and external flows is obtained using the perturbation method. The Reynolds numbers and capillary are considered as the perturbation parameters. Drop’s shape remains spherical for sufficient small ones. The falling drop’s shape at Newtonian phase, deforms from its spherical shape as its volume increases. Inertial forces, surface tension, normal components stresses have the most influence on the falling drop’s shape. Drop’s deformation is due to the forces at the interfaces acting between two fluids. By volume increase of the falling drop, normal components stresses overcome to the surface tension and cause a dimple at the bottom drops in addition to the inertial force enhancement. For small non-dimensional parameters (Reynolds number and capillary) drop’s deformation is exactly similar to a sphere and then by increase in Reynolds number and capillary, the drop’s shape alters and cause a dimple at the bottom drops. Analytical solution show suitable agreement in terminal velocity and drop shape estimation with experimental results.

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In this paper, steady motion of non-Newtonian falling drop through a Newtonian fluid at low Reynolds number is investigated analytically. Here, the Upper Convected Maxwell model (UCM) is used for drop phase and Newtonian model is considered for external fluid. During the past few decades, studies relating to non-Newtonian instabilities especially those involving free surfaces are amongst the most striking. These types of studies can be used to optimize design processes in, for example, the petroleum and medicine related processes, metal extraction, and paint and power-plant related fields. Analytical solution is obtained using the perturbation method. Reynolds and Deborah numbers are used to linearize the equations governing the problem in analytical method. Deborah number indicates the elastic effect of drop. The drag force increases by the growth of the elastic effect of non-Newtonian Drop’s. The non-Newtonian drop loses its shape and exchanges to an oblate form. Increment in Deborah number enhances the dimple at the bottom of the drop and results in an increment in its drag force and as a consequence its terminal velocity decreases. A hole is created at the rear of the drop due to the presence of inertia force and focus of normal component of stress at the rear of the drop. The novelty of this study is to consider the convection (non-linear) term of the momentum equations which was neglected in the previous studies due to the creeping flow.

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In this study, free vibrations and resonances analysis of a nanocomposite beam with internal damping is investigated. For this purpose the various distributions of carbon nanotubes with arbitrary average volume fractions are considered. System includes the geometry and inertia nonlinearities. With the aid of Hamilton principle the equations of motion are derived and using the Galerkin method are reduced to ordinary ones. To analyze the system the multiple scales method is utilized. In free analysis the analytical expressions for amplitude, phase and nonlinear natural frequency are obtained. Also, the effect of system parameters such as damping coefficients, kind of the carbon nanotube distribution, average volume fraction of nanotubes in them are probed. In free analysis, it is observed that by increasing the external damping the amplitude is decreased. Also, by increasing the average volume fraction, the nonlinear natural frequency is increased. In resonance analysis, by depicting the frequency response curves, it is observed that by increasing internal damping coefficient the amplitude is decreased and the loci of the bifurcations is changed. Also carbon nanotube distribution and average volume fractions of them on the solution and bifurcations have an important effect. Also, it is seen that by decreasing the external force, the amplitude of the system is decreased and bifurcations occur in higher internal damping coefficients. An isotropic beam in the highest and a nano-composite beam in the lowest values of internal damping coefficients become completely stable.

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In this study, control design of a T shaped mass connected to the clamped-clamped microbeam excited by electrostatic actuation is investigated. The actuation force is generated by applying an electric voltage between the horizontal part of T shaped mass and an opposite electrode plate. In this model, the micro-beam is modeled by Euler-Bernoulli theory as a continuous beam. The T-shaped assembly connected to the the microbeam is assumed as a rigid body and nonlinear effect of electrostatic force is considered. Equations of motion and associated boundary condition are derived using the Lagrange’s principle. The differential equation of nonlinear vibration around the static position is discretized using Galerkin method.. The discretized equations are solved by the perturbation theory. To improve the dynamics behavior of systems, nonlinear control feedback has been presented. The controller regulates the pass band of microcantilever and analytically approximate the nonlinear resonance frequency and amplitudes of the periodic solutions when the microcantilever is subjected to one point and fully distributed feedback forces. The results of paper may be used for improving the design of mass sensors based on nonlinear jump phenomena.

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In the present research, higher resonance frequencies are employed to improve the performance of the atomic force microscopy in the non-contact mode. Conventional models already used in the literature to study AFM microcantilever dynamics such as point-mass approach are not only incapable of modeling higher vibrational modes but also fail to predict microcantilever complicated dynamics with a sufficient accuracy. In this paper, the Hamilton’s extended principle is used to obtain equations governing the nonlinear oscillations of the AFM probe. Euler-Bernoulli beam assumptions and small deflection theory are assumed. The resulting partial differential equation is often converted to a set of ordinary differential equations and then this set is solved either numerically or based on perturbation methods. In the present research, however, the partial differential equation is attacked directly by a special perturbation technique. The accuracy of the present method is then verified by a combination of the Galerkin discretization scheme and a Rung - Kutta numerical solution. Finally, different behaviors of the AFM probe including static behavior, linear mode shapes and frequency response curves are investigated through several numerical simulations. It is found out that higher vibrational modes have smaller frequency shift. It is also found out that higher modes are faster in gathering surface information and also more sensitive to the excitation.

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In the present study, the instability in the interface of two semi-infinite fluid layers with applying a shock is studied. To this end, the effect of various parameters such as fluid densities, velocities of fluids, and magnetic field on the instability is explored. By using the magneto-hydrodynamic equations, a general equation is developed for the evolution of perturbation amplitude near the interface. Analytical and graphical results indicate that the time dependent part of perturbation amplitude is the same for both the constant and varying density cases and the instability depends on the growth rate. Remarkably, the growth rate depends on the characteristics of the fluids and magnetic field and can be real or imaginary; hence, the stability condition is determined with respect to this parameter. Furthermore, it is shown that the spatial part of the perturbation amplitude in the constant density case, even with different densities, is symmetric and independent from the layer densities and damps exponentially in the two sides of the interface. On the other hand, it is shown that in the varying density case, the function of the spatial part of the perturbation amplitude depends on the parameters of the environment and the fluid; so the spatial part of the perturbation amplitude in the two fluid damps asymmetrically. Moreover, the results attained in the constant density case match the findings of the previous studies.