Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique

A non-linear attitude control method for a satellite with magnetic torque rods using the state-dependent Riccati equation (SDRE) technique has been developed. The magnetic torque caused by the interaction with the Earth's magnetic field and the magnetic moment of torque rods plays a role of the control torque. The detailed equations of motion for this system are presented using angular velocity and quaternions. The SDRE controller is developed for the non-linear systems which can be formed in pseudo-linear representations using the state-dependent coefficient (SDC) method without linearization procedure. The aim of this control system is to achieve a stable attitude within 5°, and minimize the control effort. The stability regions for the SDRE controlled satellite system are estimated through the investigation of the stability conditions developed for pseudo-linear systems and the application of Lyapunov's theorem. For comparisons, the Linear Quadratic Regulator (LQR) method using the solution of the algebraic Riccati equation (ARE) is also applied to this non-linear system. The performance of the SDRE non-linear control system demonstrates more robustness and stability than the LQR control system when subjected to a wide range of initial conditions.     

Non-linear free periodic vibrations of variable stiffness composite laminated plates

Steady-state free vibrations, with large amplitude displacements, of variable stiffness composite laminated plates (VSCL) are analysed. The intentions of this research are: (1)?to find out how the natural frequencies and (mode) shapes evolve with the displacement amplitude in this new type of laminated composite material; (2)?to describe modal interactions in VSCL due to energy interchanges under the coupling induced by non-linearity; (3)?to compare the VSCL with traditional, constant stiffness, laminated plates. The VSCL of interest here have curvilinear fibres and the numerical analysis carried out is based on a recently developed p-version finite element with hierarchic basis functions. The element follows first-order shear deformation theory and considers Von Kármán??s non-linear terms. The time domain equations of motion are first reduced using the linear modes of vibration and then transformed to the frequency domain via the harmonic balance method. These frequency domain equations are solved by an arc-length continuation method.     

An analytical approximate technique for a class of strongly non-linear oscillators

An analytical approximate technique for large amplitude oscillations of a class of conservative single degree-of-freedom systems with odd non-linearity is proposed. The method incorporates salient features of both Newton's method and the harmonic balance method. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton's method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of non-linear algebraic equations without analytical solution. With carefully constructed iterations, only a few iterations can provide very accurate analytical approximate solutions for the whole range of oscillation amplitude beyond the domain of possible solution by the conventional perturbation methods or harmonic balance method. Three examples including cubic-quintic Duffing oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique.     

Non-linear vibrations of transversely isotropic rectangular'' plates

The large amplitude free flexural vibration of transversely isotropic rectangular plate, incorporating the effects of transverse shear and rotatory inertia, is studied using the von Karman field equations. A mode shape, consisting of three generalised-coordinates together with the Galerkin technique, results in a system of three non-linear simultaneous ordinary differential equations which govern the motion of the plate. These equations are integrated using a fourth-order Runge-Kutta method to obtain the period for each amplitude of vibration. The non-linear period vs amplitude behaviour is of the hardening type and it is also found that transverse shear and rotary inertia effects increase the period and that this increase is quite significant even for thin transversely isotropic plates. The results are compared with earlier results which were based on a one-term or one generalised coordinate solution and using the Berger approximation or the von Karman field equations.     

A perturbation solution to a plate postbuckling problem

The modal coupling behavior of a compressed, imperfect plate is examined as a six degree of freedom structural system. A direct variational solution procedure is developed by the use of the functional for the total potential energy of the system. The non-linear algebraic equations generated are solved by use of a continued perturbation technique. The segmented deflection history is then developed for a variety of imperfection compositions and aspect ratios.     

Non-linear analysis of structures using two-point method

Non-linear algebraic equations must be solved by an iterative method, the non-linear equations being linearized by evaluating the non-linear terms with the known solution from the preceding iteration. The Newton-Raphson method, which is based on the Taylor series expansion and uses the tangent stiffness matrix, has been extensively used to solve non-linear problems. In this paper, a new Newton-Raphson algorithm is developed for analyses involving non-linear behavior. Our method, here named as a two-point method, is constructed as a predictor-corrector one, most frequently taking Newton's method in the first iteration. It should be noted that our concern in this research ignores the problem of passing limit points. The presented method incorporates the known information at each stage of the loading process to determine the subsequent unknown variables. Compared with the classic Newton-Raphson algorithm, it offers a strategy that can be deployed to reduce both the number of the iterations and the computing time involved in non-linear analysis of structures.     

Non-linear vibration of a single link viscoelastic Cartesian manipulator

In this present work, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied. The temporal equation of motion with complex coefficients of the system is obtained by using D’Alembert's principle and generalized Galarkin method. The temporal equation of motion contains non-linear geometric and inertia terms with forced and non-linear parametric excitations. It may also be found that linear and non-linear damping terms originated from the geometry of the large deformation of the system exist in this equation of motion. Method of multiple scales is used to determine the approximate solution of the complex temporal equation of motion and to study the stability and bifurcation of the system. The response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. The response curves obtained using viscoelastic beams are compared with those obtained from a linear Kelvin-Voigt model and also with an equivalent elastic beam. The effect of the material loss factor, amplitude of base excitation, and mass ratio on the steady state responses for both simple and subharmonic resonance conditions are investigated.     

计算圆板大振幅非线性振动频率的平均刚度法

本文用平均刚度法研究圆板大振幅非线性振动的频率问题，导出了相应的非线性广义特征值方程，构造了一种避免发散并能加速收敛的加权平均迭代法，计算结果与Ｋａｎｔｏｒｏｖｉｃｈ时间平均法的解十分吻合。     

A Newton's method scheme for solving free-surface flow problems

A Newton's method scheme is described for solving the system of non-linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy-driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.     

A finite element study on the large amplitude flexural vibration characteristics of FGM plates under aerodynamic load

The large amplitude flexural vibration characteristics of functionally graded material (FGM) plates are investigated here using a shear flexible finite element approach. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The effective material properties are then evaluated based on the rule of mixture. The FGM plate is modeled using the first-order shear deformation theory based on exact neutral surface position and von Kármán’s assumptions for large displacement. The third-order piston theory is employed to evaluate the aerodynamic pressure. The governing equations of motion are solved by harmonic balance method to study the vibration amplitude of FGM plates under supersonic air flow. Thereafter, the non-linear equations of motion are solved using Newmark’s time integration technique to understand the flexural vibration behavior of FGM plates in time domain (simple harmonic or periodic or quasi-periodic). This work is new in the sense that it deals with the non-linear flutter characteristics of FGM plates under high supersonic airflow accounting for both the geometric and aerodynamic non-linearities. Some parametric study is conducted to understand the influence of these non-linearities on the flutter characteristics of FGM plates.     

Dynamics of a Flexible Cantilever Beam Carrying a Moving Mass

The motion of a flexible cantilever beam carrying a moving spring-mass system is investigated. The beam is assumed to be an Euler–Bernouli beam. The motion of the system is described by a set of two nonlinear coupled partial differential equations where the coupling terms have to be evaluated at the position of the mass. The nonlinearities arise due to the coupling between the mass and the beam. Due to the nonlinearities the system exhibits internal resonance which is investigated in this work. The equations of motion are solved numerically using the Rayleigh–Ritz method and an automatic ODE solver. An approximate solution using the perturbation method of multiple scales is also obtained.     

Wiener-Hopf analysis of an acoustic plane wave in a trifurcated waveguide

In this paper, we have made Wiener-Hopf analysis of an acoustic plane wave by a semi-infinite hard duct that is placed symmetrically inside an infinite soft/hard duct. The method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 × 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which is solved by using the pole removal technique. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. These systems of linear algebraic equations are solved numerically. The graphs are plotted for sundry parameters of interest. Kernel functions are also factorized.     

Nonlinear flexural–torsional dynamic analysis of beams of variable doubly symmetric cross section—application to wind turbine towers

In this paper, a boundary element solution is developed for the nonlinear flexural–torsional dynamic analysis of beams of arbitrary doubly symmetric variable cross section, undergoing moderate large displacements, and twisting rotations under general boundary conditions, taking into account the effect of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions and to twisting and/or axial loading. Four boundary-value problems are formulated with respect to the transverse displacements, to the axial displacement, and to the angle of twist and solved using the Analog Equation Method, a Boundary Element Method (BEM) based technique. Application of the boundary element technique yields a system of nonlinear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled with algebraic equations. Numerical examples of great practical interest including wind turbine towers are worked out, while the influence of the nonlinear effects to the response of beams of variable cross section is investigated.     

Finite element analysis of incompressible laminar boundary layer flows

A numerical procedure was developed to solve the two-dimensional and axisymmetric incompressible laminar boundary layer equations using the semi-discrete Galerkin finite element method. Linear Lagrangian, quadratic Lagrangian, and cubic Hermite interpolating polynomials were used for the finite element discretization; the first-order, the second-order backward difference approximation, and the Crank-Nicolson method were used for the system of non-linear ordinary differential equations; the Picard iteration and the Newton-Raphson technique were used to solve the resulting non-linear algebraic system of equations. Conservation of mass is treated as a constraint condition in the procedure; hence, it is integrated numerically along the solution line while marching along the time-like co-ordinate. Among the numerical schemes tested, the Picard iteration technique used with the quadratic Lagrangian polynomials and the second-order backward difference approximation case turned out to be the most efficient to achieve the same accuracy. The advantages of the method developed lie in its coarse grid accuracy, global computational efficiency, and wide applicability to most situations that may arise in incompressible laminar boundary layer flows.     

Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM

In this paper, a boundary element method is developed for the non-linear flexural–torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderately large deflections and twisting rotations under general boundary conditions, taking into account the effects of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions as well as to twisting and/or axial loading. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique leads to a system of non-linear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled to algebraic equations. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the non-linear effects to the response of the beam.     

On the identification of non-linear mechanical systems using orthogonal functions

Real world mechanical systems present non-linear behavior and in many cases simple linearization in modeling the system would not lead to satisfactory results. Coulomb damping and cubic stiffness are typical examples of system parameters currently used in non-linear models of mechanical systems. This paper uses orthogonal functions to represent input and output signals. These functions are easily integrated by using a so-called operational matrix of integration. Consequently, it is possible to transform the non-linear differential equations of motion into algebraic equations. After mathematical manipulation the unknown linear and non-linear parameters are determined. Numerical simulations, involving single and two degree-of-freedom mechanical systems, confirm the efficiency of the above methodology.     

Off-center impact of an elastic column by a rigid mass

Based on the Timoshenko beam model the equations of motion are obtained for large deflection of off-center impact of a column by a rigid mass via Hamilton's principle. These are a set of coupled nonlinear partial differential equations. The Newmark time integration scheme and differential quadrature method are employed to convert the equations into a set of nonlinear algebraic equations for displacement components. The equations are solved numerically and the effects of weight and velocity of the rigid mass and also off-center distance on deformation of the column are studied.     

A NUMERICAL STUDY OF THE FLOW OVER ELLIPSOIDAL OBJECTS INSIDE A CYLINDRICAL TUBE

A numerical scheme is developed to obtain the flow field around one, two and five ellipsoidal objects inside a cylindrical tube. The scheme uses the Galerkin finite element technique and the primitive variable(u–v–p) formulation. The two-dimensional incompressible Navier–Stokes equations are solved numerically by using the direct mixed interpolation method. A Picard iteration scheme is used for the solution of the resulting system of non-linear algebraic equations. The computer code is verified by checking with known analytical solutions for the flow past a sphere. Results for the shear stress distributions along the ellipsoids, forces and drag coefficients are obtained for different geometric ratios and Reynolds numbers. Some of the intermediate computational results on the velocity fields developed are also reported.     

Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells

A general approach, based on shearable shell theory, to predict the influence of geometric non-linearities on the natural frequencies of an elastic anisotropic laminated cylindrical shell incorporating large displacements and rotations is presented in this paper. The effects of shear deformations and rotary inertia are taken into account in the equations of motion. The hybrid finite element approach and shearable shell theory are used to determine the shape function matrix. The analytical solution is divided into two parts. In part one, the displacement functions are obtained by the exact solution of the equilibrium equations of a cylindrical shell based on shearable shell theory instead of the usually used and more arbitrary interpolating polynomials. The mass and linear stiffness matrices are derived by exact analytical integration. In part two, the modal coefficients are obtained, using Green's exact strain-displacement relations, for these displacement functions. The second- and third-order non-linear stiffness matrices are then calculated by precise analytical integration and superimposed on the linear part of equations to establish the non-linear modal equations. Comparison with available results is satisfactorily good.     

Non-linear shear vibrations of piezoceramic actuators

A wide range of non-linear effects are observed in piezoceramic materials. For small stresses and weak electric fields, piezoceramics are usually described by linearized constitutive equations around an operating point. However, typical non-linear vibration behavior is observed at weak electric fields near resonance frequency excitations of the piezoceramics. This non-linear behavior is observed in terms of a softening behavior and the decrease of normalized amplitude response with increase in excitation voltage. In this paper the authors have attempted to model this behavior using higher order cubic conservative and non-conservative terms in the constitutive equations. Two-dimensional kinematic relations are assumed, which satisfy the considered reduced set of constitutive relations. Hamilton's principle for the piezoelectric material is applied to obtain the non-linear equation of motion of the piezoceramic rectangular parallelepiped specimen, and the Ritz method is used to discretize it. The resulting equation of motion is solved using a perturbation technique. Linear and non-linear parameters for the model are identified. The results from the theoretical model and the experiments are compared. The non-linear effects described in this paper may have strong influence on the design of the devices, e.g. ultrasonic motors, which utilize the piezoceramics near the resonance frequency excitation.