Discrete Dynamic Equations Research Articles

A formulation for fluid–structure interaction problems with immersed flexible solids: Application to splitters subjected to flow past cylinders with different cross-sections

In the finite element method framework, a fluid–structure formulation is developed by coupling an Eulerian fixed-mesh fluid approach with a Lagrangian deforming-mesh description for a flexible solid. The coupled formulation is solved using a staggered scheme during time. For the fluid solution stage, the solid walls are considered as a time-variable internal boundary. The velocity and pressure fields are obtained by solving the weak form of the fluid dynamic equations in which the solid velocity is imposed on the internal boundary via a penalization term. For the solid solution stage, the displacement field is obtained by solving the discrete solid dynamic equations which consider traction forces computed by integrating pressures and viscous stresses on the nodes belonging to the solid walls. This novel technique is firstly applied to analyze a flexible splitter under the shedding of a flow past square cylinder due to this problem is considered as a benchmark in the literature. The present solutions agree with those computed using body-fitted techniques, thus validating the proposal. Secondly, flexible splitter motions under the shedding of flow past cylinders with different cross-sections and splitter lengths are comprehensively studied. Overall, the computed results confirmed that the hydrodynamic coefficients on the cylinders were reduced because of the presence of the splitter.

Coupled vibration analysis of the spacecraft with the flexible shaft and solar panels assembly

Dynamical characteristics of the spacecraft with flexible components are investigated in this paper. The component consists of a flexible shaft and solar panels, where solar panels are completely fixed on the shaft. A novel power series multiplier polynomial method is proposed to describe the connecting condition between the shaft and solar panels. The deformation and force matching conditions between the one dimensional shaft and two dimensional panels are established, which is the major contribution about the multidimensional combined structure of this study. According to constraining boundary conditions of the shaft and solar panels, the bending-torsion coupled deformation of the component is expressed. The attitude maneuvering of the central platform and the elastic vibration of the component cause the rigid-flexible coupled effect. Then the discrete dynamical equation of the spacecraft is obtained by using the rigid-flexible coupled modal shapes. The correctness of the proposed method is verified by comparing the natural characteristics with that of the finite element model. The relative error of the higher order frequency is not more than 4%. Numerical results show that the degree of the bending angle polynomial should be at least quadratic to ensure the accuracy of the calculation. The proposed power series multiplier polynomial breaks through the barriers that the traditional Lagrange multiplier cannot express the continuous constraint. It is an effective method to fit the constrained boundary by the polynomial rather than by the discrete multipoint method.

Random vibration analysis of composite laminated conical–cylindrical cabin structures of air vehicles considering thermal load

This study proposes a novel semi-analytical random dynamic model using the meshless method to precisely predict and quickly analyze the frequency-domain random response properties of composite laminated conical–cylindrical cabin structures (CL-CCCSs) of air vehicles under stationary random loads. In addition, thermal load is considered in the analysis. The first-order shear deformation theory (FSDT) is used to develop a random dynamic theoretical model for CL-CCCSs based on the spectro-geometric method (SGM) and pseudo-excitation method (PEM). Variable stiffness springs are employed to handle diverse boundary cases and compatibility conditions in the current model. Also, the displacement admissible functions for laminated conical shell and cylindrical shell are uniformly depicted with SGM, which depends on sine and cosine functions. After that, the PEM is employed to transform stationary random loads into pseudo harmonic loads. Subsequently, the energy variation principle is used to derive theoretical discrete random dynamic equations for vibration model of CL-CCCSs considering the thermal effect. The results of numerical vibration analyses are compared to confirm the reliability of the established model. Finally, the effect of certain parameters, such as the lamination scheme, layer number, and temperature, on the random vibration response of a CL-CCCS is thoroughly explained.

Applications of Differential-Difference Algebra in Discrete Calculus

Discrete calculus deals with developing the concepts and techniques of differential and integralcalculus in a discrete setting, often using difference equations and discrete function spaces. This paperexplores how differential-difference algebra can provide an algebraic framework for advancing discretecalculus. Differential-difference algebra studies algebraic structures equipped with both differential anddifference operators. These hybrid algebraic systems unify continuous and discrete analogues ofderivatives and shifts. This allows the development of general theorems and properties that cover bothsettings. In particular, we construct differential-difference polynomial rings and fields over discretefunction spaces. We define discrete derivatives and shifts algebraically using these operators. We thenstudy integration, summation formulas, fundamental theorems, and discrete analogues of multivariatecalculus concepts from an algebraic perspective. A key benefit is being able to state unified theorems indifferential-difference algebra that simultaneously yield results for both the continuous and discretecases. This provides new tools and insights for discrete calculus using modern algebraic techniques.We also discuss applications of representing discrete calculus problems in differential-differencealgebras. This allows bringing to bear algebraic methods and software tools for their solution. Specificexamples are provided in areas such as numerical analysis of discrete dynamical systems definedthrough difference equations. The paper aims to demonstrate the capabilities of differential-differencealgebra as a unifying framework for further developing the foundations and applications of discretecalculus. Broader connections to algebraic modeling of discrete physical systems are also discussed.

Size-Dependent Rigid–Flexible Coupling Dynamics of Functionally Graded Rotating Moderately Thick Microplates

Micro air vehicles, which are typical small-sized rotating-motion systems, have seen major advancements in recent years. To provide some theoretical basis for developing and producing micro air vehicles, this study establishes a novel rigid–flexible coupling dynamic model for functionally graded (FG) moderately thick rectangular microplates attached to a central rotating rigid hub based on the modified couple stress theory and first-order shear deformation theory. The proposed model incorporates nonlinear coupling term of in-plane deformation to reflect the dynamic stiffening effect caused by rotational motion. Material characteristics of the FG microplate have a linear power-law distribution along the thickness axis. Further, the discrete form dimensionless coupling dynamic equations and their numerical solutions are obtained by combining the Euler–Lagrange equation and the Chebyshev–Ritz method. Convergence and comparative studies are carried out to demonstrate the accuracy and validity of the proposed model. Thereafter, the influence of material length scale parameter, rotational speed, gradient index, and aspect ratio on the frequency of the microplates is investigated. Numerical results reveal that couple stress and dynamic stiffening effects both enhance the rigidity of the microplates, whereas the gradient index decreases the rigidity. Nonlinear coupling term which leads to significant differences in frequency value and trace line can’t be ignored for rotative structure. In-plane motion and its coupling terms play a significant function for the moderately thick or thick microplates. The increase of rotational speed and gradient index will reduce the size dependency of the microplate. Furthermore, the frequency trajectory steering and corresponding mode transition phenomenon are graphically represented.

Generating discrete dynamical system equations from input–output data using neural network identification models

This research presents a novel framework for generating equations describing discrete dynamical systems from only input–output data. The framework operates in two steps, creating a system identification model from input–output data using neural networks and then performing sensitivity analysis on the model. The sensitivity analysis is driven by a uniquely constrained functional decomposition of the identification model that breaks a complex identification problem into a group of small curve fitting problems. The resultant system equation represents the neural network identification model and by proxy the original system from which the input–output data belongs. The analysis allows for system equations to be generated from both black box systems and identification models, which can then be used for transparent and interpretable replacement of opaque system models. Transparent models can be better understood, leading to increased trustworthiness, safety, and reliability. An open source code implementation of the framework is created and made publicly available.

Numerical and experimental study on the performance of an electromagnetic energy harvester distributed along drill-string

Real-time measurement of the downhole information from sensors distributed along a slender drill-string plays an important role in oil and gas drilling. However, an urgent problem is how to provide power to the downhole sensor network. In this study, combining with the centrifugal softening effect and an oscillator with clearance, a bi-stable electromagnetic energy harvester (EMEH) is proposed to enhance the energy harvesting performance in the low-frequency rotation of the drill-string. The discrete dynamic equations of the coupled EMEHs & drill-string system are established by using the finite difference method and Lagrange's equations. The accuracy of the numerical simulation is validated through model test, and the effects of damping, excitation amplitude, speed and gravity on the performance of a single EMEH are analyzed through numerical simulation. Then, the coupled dynamic model is used to simulate and analyze the performance of the EMEHs in the drilling process of vertical wells and deviated wells. The numerical results show that the resonant frequency of EMEH is related to the excitation amplitude and rotating speed. It first decreases and then increases with the increase of the excitation amplitude, which can cover most frequencies below the resonant frequency of the linear system with δi0 = 0 mm. The variation range of the resonant frequency decreases with the increase of rotating speed. In vertical wells, the output power of EMEH increases with the increase of drill-string depth, and the maximum output power at the well bottom can reach 2.0 W. In deviated well, the output power is mainly generated by the continuous rotation of the drill-string and gravity, which increases with the increase of the well inclination. The results show that the output power of a single EMEH can meet the power demand of most downhole sensors.

Local regularity estimates for general discrete dynamic programming equations

We obtain an analytic proof for asymptotic Hölder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.

Dynamics Analysis of Rotating Cantilever Beams with Free End Mass

An improved modeling method for the dynamics analysis of rotating cantilever beams with free end mass is introduced. The proposed improved modeling method is based on the nonlinear Green strain theory in this study. By using the Hamilton principle and the Galerkin method, the discrete dynamic equations of the axial and chordwise motions are obtained. Meanwhile, the equations for free vibrations analysis of the rotating cantilever beam with free end mass are derived by introducing the equilibrium axial deformation into both axial and chordwise motions. Numerical simulations are performed to validate the proposed model, and the results are compared and discussed with the other modeling methods. The comparison shows that even a small free end mass has an important influence on the dynamic characteristics of the rotating beam. Furthermore, the influences of various parameters on the vibration frequencies and axial deformations of the rotating cantilever beams are analyzed, such as the angular speed, the free end mass to beam mass ratio, and the center body radius to beam length ratio. Under the same conditions, the influence of the free end mass on the variation in the first- and high-order frequencies with the increasing angular speed are different.

Meshless numerical approach to flutter analysis of rotating pre-twisted nanocomposite blades subjected to supersonic airflow

In this study, the flutter analysis of a rotating pre-twisted functionally graded graphene nanoplatelets reinforced composite (FG GPLRC) blade under supersonic airflow is investigated. The pre-twisted blade is multi-layered and reinforced with graphene nanoplatelets (GPLs) evenly distributed in each layer while the GPL weight fraction changes from layer to layer through the thickness direction. The effective Young's modulus is determined by Halpin-Tsai micromechanical model while the Poisson's ratio and mass density are predicted by Voigt's rule for GPLRC layers. According to assumptions of the first-order shear deformation theory (FSDT), the first-order piston theory and shell theory, the dynamic model of rotating pre-twisted GPLRC blades subjected to supersonic flow is developed. Meshless the improved moving least-square Ritz method (IMLS-Ritz) is used to derive the discrete dynamic equations of rotating pre-twisted GPLRC blades under aerodynamic load. The accuracy of the IMLS-Ritz method for this problem is validated by comprehensive convergence studies and careful comparison studies. A detailed parameter investigation of the effects of GPL distribution configuration, rotation velocity and geometrical parameters on flutter behavior characteristics of GPLRC blades is systematically conducted.

Dynamic modeling and orbit maneuvering response analysis for a three-axis attitude stabilized large scale flexible spacecraft installed with hinged solar arrays

An analytical dynamic model for a three-axis attitude stabilized large scale flexible spacecraft (LSFS) installed with hinged solar arrays is established. The natural property of the LSFS is investigated by simplifying it as a rigid central body hinged with two sets of multi-panel structures in this paper. The Gram-Schmidt process is used to construct a set of characteristic orthogonal polynomials as the displacement field of the solar panel. Lagrange multipliers are introduced to describe the constraints at hinges. For the model of LSFS, the Rayleigh-Ritz method is employed to obtain analytical global modes which are explicitly functions of space coordinates and effective for the modeling of lower-order discrete dynamic equations and design of active vibration controllers for the system. Moreover, the method is applied to investigate the parametric influence on natural characteristics for the LSFS. The paper develops a novel method to obtain rigid-flexible coupling global modes and construct the low-dimensional dynamical model for LSFS and study the rigid-flexible coupling dynamic response of in-orbit LSFS with complicated external excitation.

МОДЕЛЮВАННЯ ЛІНІЙНИМИ РІЗНИЦЕВИМИ РІВНЯННЯМИ ПЕРШОГО ПОРЯДКУ

The article considers economic and mathematical models and studies the socio-economic processes that develop over time, as well as mathematical models that describe them. These are dynamic models. All variables in dynamic models generally depend on the time that acts as an independent variable. In economic research, there are often problems in which variables acquire discrete numerical values. For example, at the end of the month, quarter, year, etc., production results are optimized; accrual of interest on the bank deposit at the end of the month, six months, at the end of the year. In addition, because computers operate only with numbers, so when using computer technology, all continuous processes are reduced to discrete. In this case, from differential equations that describe certain economic processes, we move to difference equations. There are dynamic models with continuous and discrete time, ie continuous and discrete models. Therefore, depending on the type of dynamics of the system under study, dynamic models can be divided into discrete and continuous. In discrete dynamic models, difference equations or systems of difference equations are used; differential equations or systems of differential equations are used in continuous dynamic models. In addition, in some cases there may be systems with mixed dynamics, then differential-equation equations are used to describe them. Difference equations and systems of equations are used successfully in modeling dynamic processes (in economics, banking, etc.). It is when the change of process occurs abruptly, or discretely, that it is convenient and expedient to apply difference equations and systems of equations. The theory of dynamical systems with discrete time, which arose as a result of building mathematical models of real economic and physical processes at the junction of the theory of difference equations and discrete random processes, is currently experiencing a period of rapid development and widespread use in various spheres of human life. In this paper, we investigate the following equations, as well as show their application to solve economic problems. In particular, discrete models described by first-order difference equations are considered. Considerable attention is paid to the analysis of specific models that are meaningful and widely used in economic theory, banking, etc.

Study on analytical global modes for a multi-panel structure connected with flexible hinges

A dynamic modeling method for a multi-panel structure connected with flexible hinges is proposed in this paper. For the solar array composed of several flexible rectangular panels, the global explicit modal functions of spatial coordinates are obtained by employing the Rayleigh-Ritz method, which are in favor of deriving the lower-dimension discrete dynamic equations and the design of controllers for the system in the further research. The Gram-Schmidt characteristic orthogonal polynomials are adopted to construct the flexible deformation expressions of the solar panel. The Lagrange multipliers are introduced to describe the constraint at flexible hinges. The eigen equation of the multi-panel structure is derived to obtain the natural frequencies and corresponding global modes of the system through the Rayleigh-Ritz procedure. Considering the natural frequency calculated from the finite element method as a reference value, the correctness of the modeling process is validated by comparing results with those obtained from the finite element method. Moreover, a high accuracy of the proposed method are confirmed by a very good agreement between results from ANSYS and the theoretical method proposed here. The proposed method derives analytical global modes of the whole system which can be conveniently used to design the controller and work out the nonlinear dynamic model and with a few degree-of-freedoms directly. Finally, the interesting mode interchanged phenomenon is observed with the variation of the stiffness of the rotation spring at flexible hinges.

Trajectory optimisation of satellite launch vehicles using network flow-based algorithm

ABSTRACTA trajectory optimisation procedure is addressed to generate a reference trajectory for Satellite Launch Vehicles (SLVs). Using a grid-based discrete scheme, a Modified Minimum Cost Network Flow (MCNF)-based algorithm over a large-scale network is proposed. By using the network grid around the Earth and the discrete dynamic equations of motion, the optimum trajectory from a launch point to the desired orbit is obtained exactly by minimisation of a cost functional subject to the nonlinear dynamics and mission constraints of the SLV. Several objectives such as the flight time and terminal conditions may be assigned to each arc in the network. Simulation results demonstrate the capability of the proposed algorithm to generate an admissible trajectory in the minimum possible time compared with previous works.

Investigation on global analytic modes for a three-axis attitude stabilized spacecraft with jointed panels

The natural property of a three-axis attitude stabilized spacecraft is investigated by simplifying it as a rigid central body jointed with 4 solar panels in this paper. For the spacecraft composed of a rigid central body and flexible panels, the Rayleigh-Ritz method is employed to obtain global modes which are explicitly functions of space coordinates and are useful for the modeling of lower-order discrete dynamic equations and design of active vibration controllers for the system. The Gram-Schmidt process is used to construct a set of characteristic orthogonal polynomials as the displacement field of the solar panel. Lagrange multipliers are introduced to describe the constraint at joints. The characteristic equation of the whole system is derived through the Rayleigh-Ritz procedure. Then, natural frequencies and corresponding global modes of the spacecraft with jointed panels are obtained. Taking the natural frequency obtained from the finite element method as a reference value, the process of getting natural frequencies is validated by comparing results obtained with those from the finite element method. Moreover, an excellent convergence and high accuracy of the present method is demonstrated by a very good agreement between results from ANSYS and theoretical method proposed here. Finally, parametric studies on characteristics of the spacecraft are conducted. The interesting mode shift phenomenon is observed when parameters of the spacecraft are changed.

Flutter analysis of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method

This paper investigates the flutter characteristics of graphene nanoplatelet (GPL) reinforced laminated composite quadrilateral plates using the element-free IMLS-Ritz method. The modified Halpin-Tsai model and rule of mixture are employed to predict the effective material properties including Young's modulus, mass density and Poisson's ratio. The energy functions of GPL reinforced composite (GPLRC) quadrilateral plates are obtained by the first-order shear deformation theory (FSDT) and first order piston theory. Based on the IMLS-Ritz approximation, the discrete dynamic equation of GPLRC quadrilateral plates is derived. The accuracy of the IMLS-Ritz results is examined by comparing the natural frequencies with those obtained from published values. A comprehensive parametric study is carried out, with a particular focus on the effects of weight fraction, distribution pattern, total number of layers, geometry and size of GPL reinforcements and geometric parameters of quadrilateral plates on the flutter boundary of GPLRC quadrilateral plates.

A Semi-Continuous PWA Model Based Optimal Control Method for Nonlinear Systems

To alleviate the mode mismatch of multiple model methods for nonlinear systems when completely discrete dynamical equations are adopted, a semi-continuous piecewise affine (SCPWA) model based optimal control method is proposed. Firstly, a SCPWA model is constructed where modes evolve in continuous time and continuous states evolve in discrete time. Thanks to this model, a piecewise affine (PWA) system can switch at any time instant whereas mode switching only occurs at sample instants when a completely discrete PWA model is adopted, which improves the prediction accuracy of multi-models. Secondly, the switching condition is relaxed such that operating subspaces have overlaps and switching condition parameters are introduced. As a consequence, an optimal control problem with fixed mode switching sequence is established. Finally, a SCPWA model based model predictive control (MPC) policy is designed for nonlinear systems. The convergence of the MPC algorithm is proved. Compared with widely used mixed logical dynamic (MLD) model based methods, the proposed method not only alleviates mode mismatch, but also lightens the computing burden, hence improves the control performance and reduces the computation time. Some numerical examples are provided as well to show the efficiency of the method.

Dynamic modeling and vibration control of a rotating space flexible arm with enhanced active constrained layer damping treatment

In this paper, an active-passive control technology called as enhanced active constrained layer damping (EACLD) is applied to structural vibration suppression of a rigid-flexible coupled hub-beam system. The longitudinal shortening of the beam caused by the transverse deformation is considered so that the high-order geometrically nonlinear terms are obtained. By using the Euler-Bernoulli beam theory and describing the VEM with a complex shear modulus, and using the assumed mode method together with Lagrange’s equations, the discrete high-order dynamic equations of the system with EACLD treatment are derived in the closed-loop case with a PD controller. Based on the new dynamic model, the dynamic behavior of the rotating EACLD beam is described in the time domain. The effects of coverage ratio of the control patch, mass of edge element, control gain and edge element location on the dynamic responses of the system are investigated. The results obtained show that the influences of edge element stiffness and mass can be helpful to designs of structural vibration control.

Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales

AbstractWe establish theHyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, which include the continuous (step size zero) and the discrete (step size constant and nonzero) dynamic equations as important special cases. In particular, for certain parameter values in relation to the graininess of the time scale, we find the minimum HUS constants. A few nontrivial examples are provided. Moreover, an application to a perturbed linear dynamic equation is also included.

Discrete dynamical laser equation for the critical onset of bistability, entanglement and disappearance

We transform the semi-classical laser equation for single mode homogeneously broadened lasers to a one-dimensional nonlinear map by using the discrete dynamical approach. The obtained mapping, referred to as laser logistic mapping (LLM), characteristically exhibits convergent, cyclic and chaotic behavior depending on the control parameter. Thus, the so obtained LLM explains stable, bistable, multi-stable, and chaotic solutions for output field intensity. The onset of bistability takes place at a critical value of the effective gain coefficient. The obtained analytical results are confirmed through numerical calculations.