General Dynamic Equation Research Articles

1∶2内共振情况下点阵夹芯板动力学的奇异性分析

Inner resonance is a typical nonlinear dynamic behavior, and the symmetric crossply composite sandwich plates have been widely used in aerospace. The studies about inner resonance of such sandwich plates have both theoretical and engineering significances. Based on the dynamic equations for the sandwich-plates, of which the boundary conditions were simply supported on 4 sides, the transverse and inplane excitations were both considered. The average equations in the polar form were obtained with the multiscale method, and the algebraic equations in the steady state form were derived through the average equations. The singularity theory was utilized to investigate 1∶2 resonant bifurcations of the symmetric crossply sandwich plates. Based on the algebraic equations in the steady state form, the restricted tangent space was obtained for the bifurcation equations with 2 tuning parameters, an inplane excitation and a transverse excitation. Then the algebraic equations were simplified under strong equivalence, and the normal form of the algebraic equations were obtained in nondegenerate cases. The singularity theory were generalized for the general nonlinear dynamic equations with 2 state variables and 4 bifurcation parameters, and the 18 universal unfoldings of bifurcation equations with codimension 4 were obtained in the case of 1∶2 internal resonance. The transition sets in the parameter plane and the bifurcation diagrams were depicted. The relationships between the tuning parameters and the exciting parameters were determined when bifurcation, hysteresis, and double limit points happened. The numerical results indicate that the vibration modes in different bifurcation regions are different.

A Specific Sliding Mode Control for Autopilot Design of Bank-to-Turn Missiles

This paper studies the BTT missile's sliding mode flight controller. Firstly, the control model is established according to dynamic inverse method and the general dynamic equation of the missile flying in the atmosphere. Then, the sliding mode controller is designed and it adopts the time-varying sliding mode surface, the self-defined reaching law and the adaptive control parameters. Finally, three sets of simulation are carried out to prove the flight controller designed in this paper for BTT missile can follow the guidance instructions and meet operational needs.

Manipulating Twisted Electron Beams.

A theoretical description of twisted (vortex) electrons interacting with electric and magnetic fields is presented, based on Lorentz transformations. The general dynamical equations of motion of a twisted electron with an intrinsic orbital angular momentum in an external field are derived. Methods for the extraction of an electron vortex beam with a given orbital polarization and for the manipulation of such a beam are developed.

ON A PENDULUM MOTION IN MULTI-DIMENSIONAL SPACE. PART 1. DYNAMICAL SYSTEMS

In the proposed cycle of work, we study the equations of the motion of dynamically symmetric fixed n-dimensional rigid bodiespendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free n-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. In thit work, we derive the general multi-dimensional dynamic equations of the systems under study.

Dynamic super Efimov effect

Super Efimov effect is a recently proposed three-body effect characterized by a double-exponential scaling, which has not been observed experimentally yet. Here, we present the general dynamic equations determining the cloud size of a scale invariant quantum gas in a time dependent harmonic trap. We show that a double-log periodicity as the hallmark of the super Efimov effect emerges when the trap frequency is decreased with a specially designed time-dependence. We also demonstrate that this dynamic super Efimov effect can be realized with realistic choices of parameters in current experiments.

Estimation of atmospheric particle formation rates through an analytical formula: validation and application in Hyytiälä and Puijo, Finland

Abstract. The formation rates of 3 nm particles were estimated at SMEAR IV, Puijo (Finland), where the continuous measurements extend only down to 7 nm in diameter. We extrapolated the formation rates at 7 nm (J7) down to 3 nm (J3) based on an approximate solution to the aerosol general dynamic equation, assuming a constant condensational growth rate, a power-law size-dependent scavenging rate, and negligible self-coagulation rate for the nucleation mode particles. To evaluate our method, we first applied it to new particle formation (NPF) events in Hyytiälä (Finland), which extend down to 3 nm, and, therefore, J3 and J7 can be determined directly from the measured size distribution evolution. The Hyytiälä results show that the estimated daily mean J3 values slightly overestimate the observed mean J3, but a promising 91 % of the estimated J3 values are within a factor of 2 from the measured ones. However, when considering detailed daily time evolution, the agreement is not as good due to fluctuations in data as well as uncertainties in estimated growth rates, which are required in order to calculate the time lag between formation of 3 and 7 nm particles. At Puijo, the mean J7 for clear NPF days during April 2007–December 2015 was 0.44 cm−3 s−1, while the extrapolated mean J3 was 0.61 cm−3 s−1.

A new balance formula to estimate new particle formation rate: reevaluating the effect of coagulation scavenging

Abstract. A new balance formula to estimate new particle formation rate is proposed. It is derived from the aerosol general dynamic equation in the discrete form and then converted into an approximately continuous form for analyzing data from new particle formation (NPF) field campaigns. The new formula corrects the underestimation of the coagulation scavenging effect that occurred in the previously used formulae. It also clarifies the criteria for determining the upper size bound in measured aerosol size distributions for estimating new particle formation rate. An NPF field campaign was carried out from 7 March to 7 April 2016 in urban Beijing, and a diethylene glycol scanning mobility particle spectrometer equipped with a miniature cylindrical differential mobility analyzer was used to measure aerosol size distributions down to ∼ 1 nm. Eleven typical NPF events were observed during this period. Measured aerosol size distributions from 1 nm to 10 µm were used to test the new formula and the formulae widely used in the literature. The previously used formulae that perform well in a relatively clean atmosphere in which nucleation intensity is not strong were found to underestimate the comparatively high new particle formation rate in urban Beijing because of their underestimation or neglect of the coagulation scavenging effect. The coagulation sink term is the governing component of the estimated formation rate in the observed NPF events in Beijing, and coagulation among newly formed particles contributes a large fraction to the coagulation sink term. Previously reported formation rates in Beijing and in other locations with intense NPF events might be underestimated because the coagulation scavenging effect was not fully considered; e.g., estimated formation rates of 1.5 nm particles in this campaign using the new formula are 1.3–4.3 times those estimated using the formula neglecting coagulation among particles in the nucleation mode.

Density perturbations inf(R,ϕ)gravity with an application to the varying-power-law model

Density perturbations in the cosmic microwave background within general $f(R,\phi)$ models of gravity are investigated. The general dynamical equations for the tensor and scalar modes in any $f(R,\phi)$ gravity model are derived. An application of the equations to the (varying power)-law modified gravity toy-model is then made. Formulas and numerical values for the tensor-to-scalar ratio, the scalar tilt and the tensor tilt are all obtained within this specific model. While the model cannot provide a theoretical reason for the value of the energy scale at which inflation should occur, it is found, based on the latest observations of the density perturbations in the sky, that the model requires inflation to occur at an energy scale less than the GUT-scale; namely, $\sim10^{14}\,{\rm GeV}$. The different energy intervals examined here show that the density perturbations recently obtained from observations are recovered naturally, with very high precision, and without fine tuning the model's parameters.

Convective Heat Transfer and Resistance Characteristics of Nanofluids with Cylindrical Particles

ABSTRACTNumerical research on convective heat transfer and resistance characteristics of TiO2/water nanofluids with cylindrical particles in laminar channel flow are performed by solving the governing equations of fluid flow with the additional term of cylindrical nanoparticles, the equation of probability density functions for cylindrical nanoparticle orientation, and general dynamics equation for nanoparticle volume concentration. The nonuniformity of nanoparticle distribution is considered and the effects of both particle volume concentration and Reynolds number on friction factor and local Nusselt number are mainly analyzed. The results show that the friction factor of nanofluid flow increases with an increase in particle volume concentration. And the friction factor decreases with increasing Reynolds number and is not dependent on the volume concentration at high Reynolds numbers. The Nusselt number declines when the Reynolds number decreases, and finally approaches an asymptotic value after the Reynolds number falls to a certain value. The Nusselt number is higher in the entrance region than at the downstream locations, and will become steady at somewhere downstream when the flow is thermally and hydraulically developed.

On the asymptotic and oscillatory behavior of solutions of third-order neutral dynamic equations on time scales

The oscillatory and asymptotic behavior results for a class of third-order nonlinear neutral dynamic equations on time scales are presented. The results obtained can be extended to more general third-order neutral dynamic equations of the type considered here. Examples are provided to illustrate the applicability of the results.

Building dynamic equations of manipulator with flexible links


 
 
 This paper presents the research of general model and building dynamic equations of two flexible links manipulator motion in the horizontal plane. Dynamic modeling is considered with adding factors which are payload, elastic friction, mass and initial moment of rotors. So it is closed to reality. Dynamic equations are derived through finite element method based on Lagrange approach. Dynamic behaviors of the system with payload were simulated like a specific example. The results can be used to building the control system which increases accuracy position of manipulators under influence of elastic displacements of links. 
 
 

Free vibration characteristics of delaminated composite pretwisted stiffened cylindrical shell

Effects of delamination on free vibration characteristics of laminated stiffened cylindrical shells with pretwist are analyzed by finite element method. The investigation is carried out using an eight-noded quadratic isoparametric shell element, which incorporates the transverse shear deformation and rotary inertia along with a three-noded beam element for the stiffener. The multipoint constraint algorithm has been included to guarantee the compatibility of deformation, equilibrium of resultant forces, and moments at delamination crack tip. The general dynamic equilibrium equation is derived from Lagrange’s equation of motion for moderate rotational speeds for which the Coriolis effect is neglected. The standard eigenvalue problem is solved utilizing QR iteration algorithm. The accuracy of the present formulation is validated with benchmark solutions is available in the literature. The present work concerns about the effects of delamination, fiber orientation, twist angle, stiffener depth-to-shell thickness ratio, and rotational speed on the fundamental frequency of shallow cylindrical shells with stiffener. Representative mode shapes for some typical case of the stiffened shell for different twist angles and rotational speeds are also presented.

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be written in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression for the anisotropic damping tensor which is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chiral, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises furthermore both electronic interband and intraband transitions. We also show that when the externally applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torque exerted on the spin moment.

Kinetic studies of mercury adsorption in activated carbon modified by iodine steam vapor deposition method

The objective of this study was to develop a high efficiency activated carbon (AC) sorbent for mercury removal. The vapor deposition method was used to modify the raw AC. This method is relatively easy to carry out and uniformly modifies the AC. The results of this study indicate that AC modified by iodine steam vapor deposition has much better Hg capturing capacity than AC modified by bromide deposition under the same experimental conditions. London dispersion forces and exposure of the sorbent surfaces play important roles in elemental mercury (Hg0) adsorption. Adsorption kinetic behavior was also studied under different experiment conditions. Results indicated that the pseudo-second model was the best fit to the actual mercury adsorption behavior onto AC-I2 sorbent. The effect of adsorbent mass and gas flow rate on adsorption behavior were examined. Results indicated that increasing the sorbent mass led to an increase in Hg adsorption capacity, while increasing the gas flow rate resulted in a significant decrease in Hg adsorption capacity. Decreases in Hg adsorption capacity were likely due to a shorter contact time and lower partial pressure reducing the number of contact opportunities between the Hg atoms and the sorbent surface. Finally, a general dynamic equation was determined that can be used to help predict Hg adsorption curve of AC-I2 sorbent without completing a real fixed-bed Hg adsorption experiment.

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 distribution 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 number 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.

Phase-field modeling of isothermal quasi-incompressible multicomponent liquids.

In this paper general dynamic equations describing the time evolution of isothermal quasi-incompressible multicomponent liquids are derived in the framework of the classical Ginzburg-Landau theory of first order phase transformations. Based on the fundamental equations of continuum mechanics, a general convection-diffusion dynamics is set up first for compressible liquids. The constitutive relations for the diffusion fluxes and the capillary stress are determined in the framework of gradient theories. Next the general definition of incompressibility is given, which is taken into account in the derivation by using the Lagrange multiplier method. To validate the theory, the dynamic equations are solved numerically for the quaternary quasi-incompressible Cahn-Hilliard system. It is demonstrated that variable density (i) has no effect on equilibrium (in case of a suitably constructed free energy functional) and (ii) can influence nonequilibrium pattern formation significantly.

Wave propagation through an inhomogeneous slab sandwiched by the piezoelectric and the piezomagnetic half spaces

Wave propagation through a gradient slab sandwiched by the piezoelectric and the piezomagnetic half spaces are studied in this paper. First, the secular equations in the transverse isotropic piezoelectric/piezomagnetic half spaces are derived from the general dynamic equation. Then, the state vectors at piezoelectric and piezomagnetic half spaces are related to the amplitudes of various possible waves. The state transfer equation of the functionally graded slab is derived from the equations of motion by the reduction of order, and the transfer matrix of the functionally gradient slab is obtained by solving the state transfer equation with the spatial-varying coefficient. Finally, the continuous interface conditions are used to lead to the resultant algebraic equations. The algebraic equations are solved to obtain the amplitude ratios of various waves which are further used to obtain the energy reflection and transmission coefficients of various waves. The numerical results are shown graphically and are validated by the energy conservation law. Based on the numerical results on the fives of gradient profiles, the influences of the graded slab on the wave propagation are discussed. It is found that the reflection and transmission coefficients are obviously dependent upon the gradient profile. The various surface waves are more sensitive to the gradient profile than the bulk waves.

A general method of fractional dynamics, i.e., fractional Jacobi last multiplier method, and its applications

In this paper, we present a general method of finding conserved quantities, i.e., fractional Jacobi last multiplier method, and explore its applications to fractional dynamical system. Based on the definition of Riesz–Riemann–Liouville fractional derivative, we study general fractional dynamical equations, construct its fractional Jacobi last multiplier, and, respectively, give the determining equation and three important properties of the multiplier. And then, we present the fractional Jacobi last multiplier method, which includes three theorems of finding conserved quantities of fractional dynamical systems. Further, in fractional framework, we explore the relation between Lie symmetry and Jacobi last multiplier. Furthermore, the fractional Jacobi last multiplier method is applied to the fractional generalized Hamiltonian system, the complete fractional Hamiltonian system, the standard fractional Hamiltonian system, the fractional Nambu system and the fractional Birkhoffian system, and five corresponding propositions are given. Also by using the fractional Jacobi last multiplier method, we respectively find the conserved quantities of a fractional relativistic Buchdahl model, a fractional Euler–Poinsot model and a fractional Duffing oscillator model. This work constructs a basic theoretical framework of fractional Jacobi last multiplier method, and provides a general method of fractional dynamics.

Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron

We study general dynamical equations describing homogeneous isotropic cosmologies coupled to a scalaron $\psi$. For flat cosmologies ($k=0$), we analyze in detail the gauge-independent equation describing the differential, $\chi(\alpha)\equiv\psi^\prime(\alpha)$, of the map of the metric $\alpha$ to the scalaron field $\psi$, which is the main mathematical characteristic locally defining a `portrait' of a cosmology in `$\alpha$-version'. In the `$\psi$-version', a similar equation for the differential of the inverse map, $\bar{\chi}(\psi)\equiv \chi^{-1}(\alpha)$, can be solved asymptotically or for some `integrable' scalaron potentials $v(\psi)$. In the flat case, $\bar{\chi}(\psi)$ and $\chi(\alpha)$ satisfy the first-order differential equations depending only on the logarithmic derivative of the potential. Once we know a general analytic solution for one of these $\chi$-functions, we can explicitly derive all characteristics of the cosmological model. In the $\alpha$-version, the whole dynamical system is integrable for $k\neq 0$ and with any `$\alpha$-potential', $\bar{v}(\alpha)\equiv v[\psi(\alpha)]$, replacing $v(\psi)$. There is no a priori relation between the two potentials before deriving $\chi$ or $\bar{\chi}$, which implicitly depend on the potential itself, but relations between the two pictures can be found by asymptotic expansions or by inflationary perturbation theory. Explicit applications of the results to a more rigorous treatment of the chaotic inflation models and to their comparison with the ekpyrotic-bouncing ones are outlined in the frame of our `$\alpha$-formulation' of isotropic scalaron cosmologies. In particular, we establish an inflationary perturbation expansion for $\chi$. When all the conditions for inflation are satisfied and $\chi$ obeys a certain boundary (initial) condition, we get the standard inflationary parameters, with higher-order corrections.

Mathematical modeling of di-ethyl-hexyl-sebacate nanoparticle formation in a free turbulent jet under high nucleation rate conditions

Mathematical modeling of di-ethyl-hexyl-sebacate (DEHS) nanoparticle formation due to homogeneous nucleation, condensation and coagulation in a hot free turbulent jet issuing into a colder environment is performed. The Reynolds-averaged Navier–Stokes equations are used to describe the fluid flow. The aerosol dynamics is treated by transforming the general dynamic equation into moment equations which are closed by assuming a lognormal shape of the particle size distribution. The contribution of condensation and coagulation to particle formation in the case of high saturation ratios and nucleation rates is studied. The sensitivity of the model to existing approximations of the vapor saturation pressure curve and the ambient temperatures is analyzed. The influence of the main control parameters of the system, such as vapor saturation temperature and nozzle exit velocity, on the aerosol characteristics formed in the turbulent jet is investigated.