THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

Desired Dynamic Equation for Primary Frequency Modulation Control of Gas Turbines

Gas turbines play core roles in clean energy supply and the construction of comprehensive energy systems.The control performance of primary frequency modulation of gas turbines has a great impact on the frequency control of the power grid.However,there are some control difficulties in the primary frequency modulation control of gas turbines,such as the coupling effect of the fuel control loop and speed control loop,slow tracking speed,and so on.To relieve the abovementioned difficulties,a control strategy based on the desired dynamic equation proportional integral(DDE-PI)is proposed in this paper.Based on the parameter stability region,a parameter tuning procedure is summarized.Simulation is carried out to address the ease of use and simplicity of the proposed tuning method.Finally,DDE-PI is applied to the primary frequency modulation system of an MS6001B heavy-duty gas turbine.The simulation results indicate that the gas turbine with the proposed strategy can obtain the best control performance with a strong ability to deal with system uncertainties.The proposed method shows good engineering application potential.

SOME OSCILLATION CRITERIA FOR A CLASS OF HIGHER ORDER NONLINEAR DYNAMIC EQUATIONS WITH A DELAY ARGUMENT ON TIME SCALES

In this paper,we establish some oscillation criteria for higher order nonlinear delay dynamic equations of the form[rnφ(⋯r2(r1x^(Δ))^(Δ)⋯)^(Δ)]^(Δ)(t)+h(t)f(x(τ(t)))=0 on an arbitrary time scale T with supT=∞,where n≥2,φ(u)=|u|^(γ)sgn(u)forγ>0,ri(1≤i≤n)are positive rd-continuous functions and h∈C_(rd)(T,(0,∞)).The functionτ∈C_(rd)(T,T)satisfiesτ(t)≤t and lim_(t→∞)τ(t)=∞and f∈C(R,R).By using a generalized Riccati transformation,we give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.The obtained results are new for the corresponding higher order differential equations and difference equations.In the end,some applications and examples are provided to illustrate the importance of the main results.

Oscillation of Second-order Nonlinear Dynamic Equation on Time Scales

The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many known results for second order dynamic equations. Some examples are given to illustrate the main results of this article.

Solution of general dynamic equation for nanoparticles in turbulent flow considering fluctuating coagulation

A new averaged general dynamic equation （GDE） for nanoparticles in the turbulent flow is derived by considering the combined effect of convection, Brownian diffusion, turbulent diffusion, turbulent coagulation, and fluctuating coagulation. The equation is solved with the Taylor-series expansion moment method in a turbulent pipe flow. The experiments are performed. The numerical results of particle size distribu- tion correlate well with the experimental data. The results show that, for a turbulent nanoparticulate flow, a fluctuating coagulation term should be included in the averaged particle GDE. The larger the Schmidt number is and the lower the Reynolds number is, the smaller the value of ratio of particle diameter at the outlet to that at the inlet is. At the outlet, the particle number concentration increases from the near-wall region to the near-center region. The larger the Schmidt number is and the higher the Reynolds num- ber is, the larger the difference in particle number concentration between the near-wall region and near-center region is. Particle polydispersity increases from the near-center region to the near-wall region. The particles with a smaller Schmidt number and the flow with a higher Reynolds number show a higher polydispersity. The degree of particle polydispersity is higher considering fluctuating coagulation than that without considering fluctuating coagulation.

INVESTIGATION ON KANE DYNAMIC EQUATIONSBASED ON SCREW THEORY FOR OPEN-CHAIN MANIPULATORS

First,screw theory,product of exponential formulas and Jacobian matrix are introduced.Then definitions are given about active force wrench,inertial force wrench,partial velocity twist,generalized active force,and generalized inertial force according to screw theory.After that Kane dynamic equations based on screw theory for open-chain manipulators have been derived. Later on how to compute the partial velocity twist by geometrical method is illustrated. Finally the correctness of conclusions is verified by example.

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DYNAMIC EQUATIONS WITH p-LAPLACIAN AND DAMPING

This paper concerns the oscillation of solutions of the second order nonlinear dynamic equation with p-Laplacian and damping（r（t）φ（x^△（t））^△＋p（t）φα（x^△α（t）＋q（t）f（xδ（t））=0on a time scale T which is unbounded above. Sign changes are allowed for the coefficient functions r, p and q. Several examples are given to illustrate the main results.

Hyers-Ulam Stability of First Order Nonhomogeneous Linear Dynamic Equations on Time Scales

This paper deals with the Hyers-Ulam stability of the nonhomogeneous linear dynamic equation x~?(t)-ax(t) = f(t), where a ∈ R^+. The main results can be regarded as a supplement of the stability results of the corresponding homogeneous linear dynamic equation obtained by Anderson and Onitsuka(Anderson D R, Onitsuka M. Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales. Demonstratio Math., 2018, 51: 198–210).

New Oscillation Criteria for Second Order Half-Linear Neutral Type Dynamic Equations on Time Scales

In this paper, we will study the oscillatory properties of the second order half-linear dynamic equations with distributed deviating arguments on time scales. We obtain several new sufficient conditions for the oscillation of all solutions of this equation. Our results not only unify the oscillation of second order nonlinear differential and difference equations but also can be applied to different types of time scales with sup T = ∞. Our results improve and extend some known results in the literature. Examples which dwell upon the importance of our results are also included.

COORDINATION OF THE MOTION PARAMETERS WITHIN STEP-BY-STEP INTEGRATION FOR DYNAMIC EQUATION

A method is presented that coordinates the calculation of the displacement, velocity and acceleration of structures within the time-steps of different types of step-by-step integration. The dynamic equation is solved using an energy equation and the calculating data of the original method. The method presented is better than the original method in terms of calculating postulations and is in better conformity with the system's movement. Take the Wilson-θ method as an example. By using the coordination process, the calculation precision has been greatly im proved (reducing the errors by approximately 90% ), and the greater part of overshooting of the calculation result has been eliminated. The study suggests that the mal-coordination of the motion parameters within the time-step is the major factor that contributes to the result errors of step-by-step integration for the dynamic equation.

Research and Application of Dynamic Equation for Full Frontal Impact

Full frontal impact theory needs researching and exploring to satisfy the primary safety design of occupant restraint system,avoiding the increasingly "engineering"trend in order to develop and design safety vehicle. After occupant restraint system is simulated by using linear elastic stiffness k,the occupant-vehicle frontal rigid barrier impact model is established. Dynamic equation of dummy chest coupling vehicle is built for full frontal impact based on ordinary vehicle deceleration by Hooke law,and the equation is solved by comparing coefficient and satisfying boundary qualifications. While relative vehicle characteristic parameters are kept unchanging,the actual vehicle deceleration is fitted to the simplified equivalent square wave( ESW),tipped equivalent square wave( TESW) and equivalent dual trapezoids wave( EDTW). Phase angle  and amplitude A of dynamic equations based on ESW,TESW and EDTW are calculated and deduced. The results show that: the dynamic equation of dummy chest coupling vehicle can be well utilized to instruct the primary safety design of full frontal impact for objective vehicle to satisfy chest deceleration demands and the equation based on TESW is best for this design.

POSITIVE SOLUTIONS FOR p-LAPLACIAN DYNAMIC EQUATIONS ON TIME SCALES

The three-point boundary value problems of p-Laplacian dynamic equations on time scales are investigated. By using Krasnosel＇skii＇s fixed-point theorem and fixed-point index theorem, criteria are achieved for the existence of at least one, two or 2n positive solutions. Furthermore, some examples are included to illustrate the main theorems.

Dynamic equations for curved submerged floating tunnel

In virtue of reference Cartesian coordinates, geometrical relations of spatial curved structure are presented in orthogonal curvilinear coordinates. Dynamic equations for helical girder are derived by Hamilton principle. These equations indicate that four generalized displacements are coupled with each other. When spatial structure degenerates into planar curvilinear structure, two generalized displacements in two perpendicular planes are coupled with each other. Dynamic equations for arbitrary curvilinear structure may be obtained by the method used in this paper.

GENERAL DYNAMIC EQUATION AND DYNAMICAL CHARACTERISTICS OF VISCOELASTIC TIMOSHENKO BEAMS

In this paper, a governing differential equation of viscoelastic Timoshenko beam including both extension and shear viscosity is developed in the time domain by direct method. To measure the complex moduli and three parameters of standard linear solid, the forced vibration technique of beam is successfully used for PCL and PMMA specimens. The dynamical characteristics of viscoelastic Timoshenko beams, especially the damping properties, are derived from a considerable number of numerical computations. The analyses show that the viscosity of materials has great influence on dynamical characteristics of structures, especially on damping, and the standard linear solid model is the better one for describing the dynamic behavior of high viscous materials.

On the Oscillation of Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Based on Riccati transformation and the inequality technique, we establish some new sufficient conditions for oscillation of the second-order neutral delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales. At the end of this paper, we give an example to illustrate the main results.

Oscillation of Third-order Delay Dynamic Equations on Time Scales

This paper is concerned with the oscillatory behavior of a class of third-order nonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequality technique, we establish some new oscillation criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of third-order nonlinear variable delay functional differential equations and functional difference equations with a nonlinear neutral term. Some examples are given to illustrate the importance of our results.

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper,Runge-Kutta Discontinuous Galerkin（RKDG） finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ？ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ？ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.

Oscillation Theorems for Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

By employing the generalized Riccati transformation technique,we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation [r(t)[y(t)+p(t)y(■(t))]~Δ]~Δ+q(t)f(y((δ(t)))=0 on a time scale■.The results improve some oscillation results for neutral delay dynamic equations and in the special case when■our results cover and improve the oscillation results for second- order neutral delay differential equations established by Li and Liu[Canad.J.Math.,48(1996), 871 886].When■,our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh[Comp.Math.Appl.,36(1998),123-132].When ■ ■our results are essentially new.Some examples illustrating our main results are given.

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE DYNAMIC EQUATIONS FOR KOITER SHELLS

In this paper, the dynamic equations for Koiter shells have been studied by Galerkin method, the existence and uniqueness to the solutions are proved.

A Well-Balanced Kinetic Scheme for Gas Dynamic Equations under Gravitational Field

In this paper,a well-balanced kinetic scheme for the gas dynamic equations under gravitational field is developed.In order to construct such a scheme,the physical process of particles transport through a potential barrier at a cell interface is considered,where the amount of particle penetration and reflection is evaluated according to the incident particle velocity.This work extends the approach of Perthame and Simeoni for the shallow water equations[Calcolo,38(2001),pp.201-231]to the Euler equations under gravitational field.For an isolated system,this scheme is probably the only well-balanced method which can precisely preserve an isothermal steady state solution under time-independent gravitational potential.A few numerical examples are used to validate the above approach.