State Space Differential Equations Research Articles

Mesh reduction methods for thermoelasticity of laminated composite structures: Study on the B-spline based State Space Finite Element Method and Physics-Informed Neural Networks

Calculating the mechanical characteristics such as interlaminar stresses of composite laminated structures is a major challenge. To avoid the disadvantages of the traditional finite element method (FEM) of discretizing the whole structure too cumbersome, we implemented a method which only need to discretize on horizontal planes. The state space differential equation is derived from the Hellinger–Reissner variational principle, and both quadratic and cubic B-splines are used to interpolate the in-plane physical quantities, while the precise integration method is carried out along thickness to obtain accurate numerical results efficiently. The method is hence treated as the B-spline based State Space FEM (B-SSFE). To further reduce degree of freedoms, we implemented another method based on physics-informed neural networks (PINNs), which is completely meshless. Deep Learning is combined with differential equations carrying physical information, and neural networks are trained by constraining the loss function. We calculated different boundary conditions for a 3D composite laminated plate under thermal loading. Comparison is conduct among B-SSFE, PINNs and FEM. Numerical results demonstrate that in terms of computational accuracy, B-SSFE can be more accurate than FEM dealing with complex boundary conditions, and PINNs can reflects the stress fields well.

Power loss minimization assessment of a doubly fed induction generator with variable core resistance for wind turbines operation

<p class="Default">This paper assesses the efficiency level and power loss minimization of a doubly fed induction generator (DFIG). A modified DFIG equivalent circuit with multi-core resistance connected in parallel was adopted. State-space differential equations of the DFIG was developed incorporating iron and copper loss components while a minimum flux linkage that aids in the minimization of the overall losses was derived. Simulation results showed that losses were minimized when the equivalent core resistances were connected in parallel with minimum permissible current flow. The results obtained during a transient disturbance showed that at different core resistance values of Rfe = 0.75Ω and 0.25Ω, different efficiency values of 83.45% and 41.21% were realized. An unconstrained optimization test carried out on the DFIG variable parameters showed that the DFIG power loss model was controllable with a positive definite value of 691.9801 and 2.9156〖e〗^(+5) for the leading principal determinants of the Hessian matrix. All simulation processes were achieved in MATLAB/Simulink 2020.</p>

Optimized trajectory tracking control of the clearance manipulator based on improved PSO algorithm

Aiming at the problems of low joint trajectory tracking accuracy and poor anti-interference ability caused by parameter uncertainty, nonlinearity, and external interference in the hydraulic servo system of the clearance manipulator, an active disturbance rejection servo control method optimized based on the improved Particle Swarm Optimization algorithm is proposed. First, through the kinematic and dynamic analysis of the clearance manipulator, the general solutions of its forward and inverse kinematics are derived, and the relationship expression between each joint angle and the extension of the piston rod of the driving cylinder is obtained according to the relationship between the joint space and the driving mechanism space; Second, the state space differential equations of the hydraulic system of the working device are derived using mechanistic modeling, and this is used to design an active disturbance rejection servo controller for each joint, and the parameters of the active disturbance rejection controller are optimized using an improved PSO algorithm to improve the control performance. Simulation and experimental results show that compared with classical PID control and conventional Active Disturbance Rejection Control, the improved ADRC in this paper has higher trajectory tracking control accuracy and the ability to suppress external disturbances.

Three-Dimensional Transient Analysis of FGM Rectangular Sandwich Plate Subjected to Thermal Loading Using State Space Differential Quadrature Method

In this paper, transient response of functionally graded materials (FGM) rectangular sandwich plates subjected to thermal loading based on three-dimensional theory of elasticity is carried out. The facing layers are made of metal and ceramics whereas core layer is FGM with exponential function distribution. For the case of simply supported boundary conditions, obtained state space differential equations are solved analytically by using Fourier series expansions along the in-plane coordinates and Laplace transformation. For other edges, boundary condition semi-analytical method is used by applying differential quadrature method (DQM) along the in-plane direction and state space technique across the transverse coordinate. After carrying out the convergence of the present method, accuracy of this study is validated by comparing the numerical results with those reported in the literature. Finally, influences of the boundary conditions, length-to-thickness ratios, length-to-width ratios, time constant and the FGM graded index on elastic behavior of sandwich plate are examined.

Advanced Proportional-Integral-Derivative Control Compensation Based on a Grey Estimated Model in Dynamic Balance of Single-Wheeled Robot

This paper aims to design a one-wheeled robot as regards its pitch freedom and balance control on the one hand and to assess the application feasibility of the GM (1,1) swing estimation controller on the other. System control focuses mainly on one-wheeled robot stability, body swings in position, and speed control. Mathematical modeling and GM (1,1) prediction control are under investigation. The mathematical modeling is firstly conducted through referencing to the Newtonian mechanics and the Lagrange equation, from which the robot transfer function and state-space differential equation are derived. Next, the linear quadratic regulator is applied as the control rule at the balance point. Applying GM (1,1) to assess the robot gyro signal at a dynamic state is a discussion. Next, model reference estimation control is processed, and a mathematical model of the balance control method is completed. Finally, a simulation is conducted to verify the feasibility of the GM (1,1) estimation reference model. The linear quadratic regulator, which is credited with tenacity, can provide pitch swing and balance control of the one-wheeled robot.

Coupled thermoelasticity analysis of carbon nano tube reinforced composite rectangular plate subjected to thermal shock

In the frame work of Lord-Shulman theory, transient response of a simply supported carbon nanotube reinforced composite (CNTRC) rectangular plate with different pattern of CNT distribution and under thermal shock is studied using generalized coupled thermoelasticity. Applying Fourier series expansion along the in-plane direction and state space technique along the thickness direction to the constitutive equations and governing differential equations of motion results in first state space differential equations which can be solved analytically in Laplace transform. Then response is obtained in time domain using inverse Laplace transform. Moreover, parametric study is conducted to assess the effect of pattern of CNT distribution, CNT volume fraction and pick of thermal shock on transient thermoelastic behavior of FG-CNTRC rectangular plate are examined.

Harmonic Modelling of the Wind Turbine Induction Generator for Dynamic Analysis of Power Quality

Given the increasing integration of wind-based generation systems into the electric grid, efforts have been made to deal with the problem of power quality associated with the intermittent nature of these systems. This paper presents a new modelling approach oriented towards harmonic distortion analysis of the induction machine for wind power applications. The model is developed using companion harmonic circuit modelling, which is a natural approach for analysis of the adverse effects of harmonic distortion in electric power systems, and represents an easier solution method than the well known dynamic harmonic domain, since it solves algebraic equations instead of state-space differential equations. The structure of the companion circuits simplifies both the formulation and solution for power systems with wind-based generation systems. This approach is especially useful for analysis of the harmonic interaction in transient and steady states between the wind power generator and the power system, whose interconnection is made through electronic converters. The proposed model allows us to compute the dynamics of the wind turbine, which are influenced by disturbances such as changes in the wind velocity, voltage fluctuations, electric waveform distortion, and mechanical vibrations, among other factors. Moreover, the cross-coupling between harmonic components at different frequencies is considered. The proposed model represents an integral framework of the electrical and mechanical subsystems of a wind turbine, allowing for analysis of the interactions between them, and understanding power quality degradation behaviour as well as causes and consequences, while also giving useful information on the field of simulation and control. To test the performance of the proposed model, a test power system is used to obtain the behaviour of a wind turbine induction generator in response to typical power quality disturbances, i.e., harmonic distortion, and voltage sags and swells. Then, the dynamics of the variables considering their harmonic interactions are analysed.

Static and vibration analysis of sandwich cylindrical shell with functionally graded core and viscoelastic interface using DQM

Abstract In the present paper, elasticity solution for static and free vibration analysis of sandwich cylindrical shell with functionally graded (FG) core and viscoelastic interface is carried out. Variation of Young's Module and material density of FGM core layer are assumed to obey power-law of radial coordinate with the Poisson's ratio holds to be constant. Imperfect interfaces are modeled according to the Kelvin-Voigt viscoelastic law. State space differential equations are derived using differential equations of motion as well as stress-displacement relations. For sandwich cylindrical shell with simply supported boundary conditions, these equations are solved analytically by using Fourier series expansion along the axial and circumferential direction. Whereas they are solved semi-analytically for the other cases of boundary conditions by using one dimensional differential quadrature method (DQM) along the axial coordinate. Time-dependent behavior is determined by solving first-order differential equation of sliding displacement at the viscoelastic interfaces. Numerical results are computed and compared with the reported results in literature to validate the present approach. In addition, effects of solid, elastic interfaces, different boundary conditions, time and mid radius to thickness ratio on the bending and vibration behavior of the sandwich shell are studied.

Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams

In this paper, the motion differential equations of the bi-directional functionally graded Timoshenko beam are established using Hamilton’s principle. The material properties of the beam change exponentially in both axial and thickness directions. Using the variable substitution method, state space differential equations of the structure are established. First, the dynamic stiffness matrix is formed using the traditional method. Then, the author proposes a new method to directly form an exact dynamic stiffness matrix by using state space differential equations, and this method is compared with the traditional dynamic stiffness matrix method. At the same time, the natural frequency of the structure is computed by combining the Wittrick–William algorithm with a non-iterative algorithm. The influence of gradient parameters α,β on the fundamental frequency, mode shape and frequency response function is analysed through the establishment of the dynamic stiffness matrix of the overall structure. Finally, using the Lagrange equation and the method of modal superposition, structural dynamic differential equations under a harmonic moving load are derived. Using the precise integration method, the dynamic response of the displacement is computed and the influence of gradient parameters α,β on the dynamic response is analysed.

Modeling of Buck Converter Models in MPPT using PID and FLC

PV has become universal for power utility applications in comparison to conventional technologies when it comes to economic competitiveness. As the efficiency of solar PV panel is low, it becomes mandatory to extract maximum power from the PV panel at any given period of time. Maximum Power and efficiency in Photovoltaics can be improved by Maximum Power Point tracking even under distributed temperature and irradiance functions. The paper attempts to compare two different Buck converter models based on predictive control. The two converter models using State space differential equation and direct component in MATLAB/SIMULINK are optimized through PID and FLC to obtain increased gain and desired converter output. A PV system connected with Buck converter using an intelligent controller (FLC) for extracting maximum power at different environmental conditions is proposed and the results are compared with conventional PID controller.

Conservative fourth-order time integration of non-linear dynamic systems

Abstract An energy conserving time integration algorithm with fourth-order accuracy is developed for dynamic systems with nonlinear stiffness. The discrete formulation is derived by integrating the differential state-space equations of motion over the integration time increment, and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts. This process introduces the time derivatives of the state space variables, and these are then substituted from the original state-space differential equations. The resulting discrete form of the state-space equations is a direct fourth-order accurate representation of the original differential equations. This fourth-order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components. Energy conservation for a force potential of general form is obtained by addition of a higher order secant-type correction term. The formulation leads to a consistent representation of the motion within a time increment corresponding to cubic Hermite interpolation in time. This in turn leads to excellent phase representation with only a small fourth-order error, permitting integration of oscillatory systems with only a few integration points per period. Three numerical examples demonstrate the high accuracy of the algorithm.

Static and free vibration analyses of functionally graded sandwich plates using state space differential quadrature method

Abstract This study presents static and free vibration behaviors of two type of sandwich plates based on the three dimensional theory of elasticity. The core layer of one type is functionally graded material (FGM) with the isotropic face sheets whereas in second type, the core layer is isotropic with the face sheets FGM. The effective material properties of FGM layers are estimated to vary continuously through the thickness direction according to a power-law distribution of the volume fractions of the constituents. By using differential equilibrium equations and/or equations of motion as well as constitutive relations, state-space differential equation can be derived. In the case of simply supported condition, applying Fourier series to the quantities along the in-plane coordinates, governing equation can be solved analytically and for the other edges condition, a semi analytical solution can be obtained by using differential quadrature method (DQM) along the in-plane coordinate as well as state spaces technique in the thickness direction. Accuracy and exactness of the present approach is validated by comparing the numerical results with the results of published literature. Moreover, the influences of volume fraction, width-to-thickness ratios and aspect ratio on the vibration and static behaviors of plate are investigated.

Free vibration analysis of functionally graded carbon nanotube-reinforced composite cylindrical panel embedded in piezoelectric layers by using theory of elasticity

Abstract In this paper free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical panel embedded in piezoelectric layers with simply supported boundary conditions is investigated by using three-dimensional theory of elasticity. By using Fourier series expansion along the longitudinal and latitudinal directions and state space technique across the thickness direction, state space differential equations are solved analytically. The traction-free surface conditions then give rise to the characteristic equation for natural frequencies. Accuracy and convergence of the present approach are validated by comparing the numerical results with those found in literature. In addition, the effects of volume fraction of CNT, four cases of FG-CNTRC, piezoelectric layer thickness, mid radius to thickness ration and modes number on the vibration behavior of the hybrid cylindrical panel are also examined.

Global Harmonic Current Rejection of Nonlinear Backstepping Control with Multivariable Adaptive Internal Model Principle for Grid-Connected Inverter under Distorted Grid Voltage

Based on a brief review on current harmonics generation mechanism for grid-connected inverter under distorted grid voltage, the harmonic disturbances and uncertain items are immersed into the original state-space differential equation of grid-connected inverter. A new algorithm of global current harmonic rejection based on nonlinear backstepping control with multivariable internal model principle is proposed for grid-connected inverter with exogenous disturbances and uncertainties. A type of multivariable internal model for a class of nonlinear harmonic disturbances is constructed. Based on application of backstepping control law of the nominal system, a multivariable adaptive state feedback controller combined with multivariable internal model and adaptive control law is designed to guarantee the closed-loop system globally uniformly bounded, which is proved by a constructed Lyapunov function. The presented algorithm extends rejection of nonlinear single-input systems to multivariable globally defined normal form, the correctness and effectiveness of which are verified by the simulation results.

Generalized Frequency Domain State-Space Models for Analyzing Flexible Rotating Spacecraft

The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is described by coupled rigid and flexible body degrees-of-freedom, where the equations of motion are modeled by integro-partial differential equations. Beam-like structures are often useful for analyzing boom-like flexible appendages. The equations of motion are analyzed by introducing generalized Fourier series that transform the governing equations into a system of ordinary differential equations. Though technically straightforward, two problems arise with this approach: (1) the model is frequency-truncated because a finite number of series terms are retained in the model, and (2) computationally intense matrix-valued transfer function calculations are required for understanding the frequency domain behavior of the system. Both of these problems are resolved by: (1) computing the Laplace transform of the governing integro-partial differential equation of motion; and (2) introducing a generalized state space (consisting of the deformational coordinate and three spatial partial derivatives, as well as single and double spatial integrals of the deformational coordinate). The resulting math model is cast in the form of a linear state-space differential equation that is solved in terms of a matrix exponential and convolution integral. The structural boundary conditions defined by Hamilton’s principle are enforced on the closed-form solution for the generalized state space. The generalized state space model is then manipulated to provide analytic scalar transfer function models for original integro-partial differential system dynamics. Symbolic methods are used to obtain closed-form eigen decomposition–based solutions for the matrix exponential/convolution integral algorithm. Numerical results are presented that compare the classical series based approach with the generalized state space approach for computing representative spacecraft transfer function models.

Simulation model for a novel superconducting fault current limiter

A simulation model for a novel superconducting fault current limiter is presented. The components of the novel fault current limiter are melt cast processed BSCCO 2212 tubes with shunt coils wound around them, which are connected in parallel. The simulation model is based on the electrical and thermal equivalent circuits of a component. A corresponding system of state space differential equations describes these equivalent circuits, which are coupled through several variables. MATLAB is used to solve the state space differential equation system, incorporating the material properties of the bulk BSCCO 2212 as well as the copper coil. Through a comparison of the simulation results and measured data the introduced model is validated. With the simulation model, the design of the components for a fault current limiter can be further improved.

Transient Vibration Analysis of Open Circular Cylindrical Shells

A numerical method based on the Rayleigh-Ritz method has been presented for the forced vibration of open cylindrical shells. The equations are derived from the three-dimensional strain-displacement relations in the cylindrical coordinate system. The middle surface of the shell represents the geometry, which is defined by an angle that subtends the curved edges, the length, and the thickness. The displacement fields are generated with a predefined set of grid points on the middle surface using considerably high-order polynomials. Each grid point has five degrees of freedom, viz., three translational components along the cylindrical coordinates and two rotational components of the normal to the middle surface. Then the strain and kinetic energy expressions are obtained in terms of these displacement fields. The differential equation governing the vibration characteristics of the shell is expressed in terms of the mass, stiffness, and the load consistent with the prescribed displacement fields. The transient response of the shell with and without damping is sought by transforming the equation of motion to the state-space model and then the state-space differential equations are solved using the Runge-Kutta algorithm.

Decentralized Adaptive Approach to Control of Chemical Reactors Based on Delta Model Representation

Abstract The contribution presents decentralized adaptive approach to control of multi input-multi output (MIMO) continuous-time dynamic systems. The controlled plant is represented by two serially connected chemical reactors with a recycle. The dynamics of this system is described by fourteen state-space differential equations. The external model of the controlled plant as well as the control design are formulated in a delta operator form. The decentralization of a MIMO plant is performed in the matrix fraction description as a two input - two output system. Control loop consists of local and global control loops. The local controllers are adaptive ones obtained via Diophantine equations in delta polynomials. The global control strategy ensures asymptotic stability of the entire control system. The adaptive property is based on a recursive identification procedure.