Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients

The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.

Time-dependent Diffusion Coefficient and Conventional Diffusion Constant of Nanoparticles in Polymer Melts by Mode-coupling Theory

Time-dependent diffusion coefficient and conventional diffusion constant are calculated and analyzed to study diffusion of nanoparticles in polymer melts. A generalized Langevin equa- tion is adopted to describe the diffusion dynamics. Mode-coupling theory is employed to calculate the memory kernel of friction. For simplicity, only microscopic terms arising from binary collision and coupling to the solvent density fluctuation are included in the formalism. The equilibrium structural information functions of the polymer nanocomposites required by mode-coupling theory are calculated on the basis of polymer reference interaction site model with Percus-Yevick closure. The effect of nanoparticle size and that of the polymer size are clarified explicitly. The structural functions, the friction kernel, as well as the diffusion coefficient show a rich variety with varying nanoparticle radius and polymer chain length. We find that for small nanoparticles or short chain polymers, the characteristic short time non-Markov diffusion dynamics becomes more prominent, and the diffusion coefficient takes longer time to approach asymptotically the conventional diffusion constant. This constant due to the microscopic contributions will decrease with the increase of nanoparticle size, while increase with polymer size. Furthermore, our result of diffusion constant from mode- coupling theory is compared with the value predicted from the Stokes-Einstein relation. It shows that the microscopic contributions to the diffusion constant are dominant for small nanoparticles or long chain polymers. Inversely, when nanonparticle is big, or polymer chain is short, the hydrodynamic contribution might play a significant role.

Propagator for a Time-Dependent Damped Harmonic Oscillator with a Force Quadratic in Velocity

The propagator for a time-dependent damped harmonic oscillator with a force quadratic in velocity is obtained by making a specific coordinate transformation and by using the method of time-dependent invariant.

Variable Coefficient KdV Equation for Amplitude of Nonlinear Solitary Rossby Waves in a Sort of Time-Dependent Zonal Flow

On the basis of the quasi-geostrophic vorticity equation,theoretical research has been down upon the evolution of the amplitude of solitary Rossby waves employing the perturbation method,and come to the conclusion that the evolution of the amplitude satisfies the variable coefficient Korteweg-de Vries(KdV) equation.

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schrodinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.

Auto-Backlund Transformation and Extended Tanh-Function Methods to Solve the Time-Dependent Coefficients Calogero-Degasperis Equation

In this paper, the Auto-B?cklund transformation connected with the homogeneous balance method (HB) and the extended tanh-function method are used to construct new exact solutions for the time-dependent coefficients Calogero-Degasperis (VCCD) equation. New soliton and periodic solutions of many types are obtained. Furthermore, the soliton propagation is discussed under the effect of the variable coefficients.

Numerical Investigation of Mooring Line Damping and the Drag Coefficients of Studless Chain Links

The chain/wire rope/chain combination is a common choice for mooring offshore floating platforms. However, data of the drag coefficients of chain links are rather limited, resulting in uncertainties with the calculations of the drag force, and hence the damping of the mooring system. In this paper, the importance of the selection of the drag coefficient is first investigated. The computational fluid dynamics(CFD) method is then used to determine the drag coefficients of a studless chain under steady flows. Numerical model validation is first completed by simulating a smooth circular cylinder under steady flows. In particular, the performance of different turbulence models is assessed through the comparisons between the calculations and the experimental results. The large eddy simulation(LES) model is finally selected for the simulation of steady flows past a chain. The effects of the Reynolds number on the drag coefficient of a stud-less chain is also studied. The results show that the calculated drag coefficients of a stud-less chain are fairly consistent with the available experimental data.

Numerical investigation on damping coefficient of the integral squeeze film damper

The elimination of rotor vibration is usually achieved by applying additional damping to the system.Squeeze film dampers are widely used in various aerospace and turbine equipments.The research is carried out on flow characteristics in the integral squeeze film dampers(ISFDs).The dynamic response to the operation condition is investigated through the computational fluid dynamics(CFD) model of ISFD.Due to the large pressure loss at the oil inlet,the oil film force only changes slightly with the increase of oil supply pressure,and the damping increases slightly.The vibration amplitude only affects the film force,but has no effect on the damping.The oil film force and damping show an upward tendency with the decrease of thickness of the end seal clearance.

L^(2)-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

In this paper,we are concerned with the asymptotic behavior of L^(∞) weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping-m/(1+t)^(λ).As λ∈(0,l/7],we prove tht the L^(∞) weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation(GPME)in L^(2)(R).As λ∈(1/7,1),we prove that the L^(∞) weak-entropy solution converges to an expansion around the nonlinear diffusion wave in L^(2)(R),which is the best asymptotic profile.The proof is based on intensive entropy analysis and an energy method.

Blowup of Solutions to the Non-Isentropic Compressible Euler Equations with Time-Dependent Damping and Vacuum

This paper mainly studies the blowup phenomenon of solutions to the compressible Euler equations with general time-dependent damping for non-isentropic fluids in two and three space dimensions. When the initial data is assumed to be radially symmetric and the initial density contains vacuum, we obtain that classical solution, especially the density, will blow up on finite time. The results also reveal that damping can really delay the singularity formation.

Study on Hydrodynamic Coefficients of Double Submerged Inclined Plates

Added mass and damping coefficients are very important in hydrodynamic analysis of naval structures. In this paper,a double submerged inclined plates with ‘/\’ configuration is firstly considered. By use of the boundary element method(BEM) based on Green function with the wave term, the radiation problem of this special type structure is investigated. The added mass and damping coefficients due to different plate lengths and inclined angles are obtained. The results show that: the added mass and damping coefficients for sway are the largest. Heave is the most sensitive mode to inclined angles. The wave frequencies of the maximal added mass and damping coefficients for sway and roll are the same.

A method for stochastic seismic response analysis of non-classically damped structures

A method to calculate the stationary random response of a non-classically damped structure is proposed that features clearly-defined physical meaning and simple expression. The method is developed in the frequency domain. The expression of the proposed method consists of three terms, i.e., modal velocity response, modal displacement response, and coupled (between modal velocity and modal displacement response). Numerical results from the parametric study and three example structures reveal that the modal velocity response term and the coupled term are important to structural response estimates only for a dynamic system with a tuned mass damper. In typical cases, the modal displacement term can provide response estimates with satisfactory accuracy by itself, so that the modal velocity term and coupled term may be ignored without loss of accuracy. This is used to simplify the response computation of non-classically damped structures. For the white noise excitation, three modal correlation coefficients in closed form are derived. To consider the modal velocity response term and the coupled term, a simplified approximation based on white noise excitation is developed for the case when the modal velocity response is important to the structural responses. Numerical results show that the approximate expression based on white noise excitation can provide structural responses with satisfactory accuracy.

The study on the bottom friction and the breaking coefficient for typhoon waves in radial sand ridges—the Lanshayang Channel as an example

Owing to the interactions among the complex terrain, bottom materials, and the complicate hydrodynam-ics, typhoon waves show special characteristics as big waves appeared at the high water level （HWL） and small waves emerged at low and middle water levels （LWL and MWL） in radial sand ridges （RSR）. It is as-sumed that the mud damping, sandy bed friction and wave breaking effects have a great influence on the typhoon wave propagation in this area. Under the low wave energy, a mud layer will form and transport into the shallow area, thus the mud damping effects dominate at the LWL and the MWL. And high Collins coef-ficient （c around 1） can be applied to computing the damping effects at the LWL and the MWL. But under the high wave energy, the bottom sediment will be stirred and suspended, and then the damping effects disappear at the HWL. Thus the varying Collins coefficient with the water level method （VCWL） is imple-mented into the SWAN to model the typhoon wave process in the Lanshayang Channel （LSYC） of the RSR, the observed wave data under “Winnie” （“9711”） typhoon was used as validation. The results show that the typhoon wave in the RSR area is able to be simulated by the VCWL method concisely, and a constant wave breaking coefficient （γ） equaling 0.78 is better for the RSR where wide tidal flats and gentle bed slopes exist.

3D Effects on Vortex-Shedding Flow and Hydrodynamic Coefficients of A Vertically Oscillating Cylinder with A Bottom-Attached Disk

Vortex-shedding flow induced by the vertical oscillation of a cylinder with bottom-attached disks of different diameter ratio Dd/Dc and thickness ratio td/Dc is studied by a 3D （three-dimensional） numerical model developed in this paper, and compared with the results obtained through 2D （two-dimensional） numerical model. The high-order upwind scheme is applied to stabilize the computation, and convergence is accelerated by the multi-grid method. Qualitative and quantitative analyses of the differences between the 2D and 3D simulation results reveal the 3D effect on the flow field characteristics and hydrodynamic coefficients of the vertically oscillating cylinder with a bottom-attached disk. The 3D effect on the fluid field is mainly reflected in the significance of three vortex-shedding patterns： ωx has a greater effect on the flow fields around the sharp edges relative to the vortices generated in the 2D simulation. In the slice along the axial orientation, the vortex effect of ωy along the radial axis is smaller than that of ωx along the circumferential direction, indicating the radial effect on the velocity more pronounced than the circumferential effect around the sharp edges of the disk. The rotational interaction ωz of the fluid in the horizontal plane during the heave motion is insignificant. Based on the 2D and 3D simulation results, the turning points that separate the increasing regimes of the added mass coefficient and damping ratio are identified. The dependence of the turning point on the diameter ratio Dd/Dc and thickness ratio td/Dc are discussed in detail.

Complex complete quadratic combination method for damped system with repeated eigenvalues

A new response-spectrum mode superposition method, entirely in real value form, is developed to analyze the maximum structural response under earthquake ground motion for generally damped linear systems with repeated eigenvalues and defective eigenvectors. This algorithm has clear physical concepts and is similar to the complex complete quadratic combination （CCQC） method previously established. Since it can consider the effect of repeated eigenvalues, it is called the CCQC-R method, in which the correlation coefficients of high-order modal responses are enclosed in addition to the correlation coefficients in the normal CCQC method. As a result, the formulas for calculating the correlation coefficients of high-order modal responses are deduced in this study, including displacement, velocity and velocity-displacement correlation coefficients. Furthermore, the relationship between high-order displacement and velocity covariance is derived to make the CCQC-R algorithm only relevant to the high-order displacement response spectrum. Finally, a practical step-by-step integration procedure for calculating high-order displacement response spectrum is obtained by changing the earthquake ground motion input, which is evaluated by comparing it to the theory solution under the sine-wave input. The method derived here is suitable for generally linear systems with classical or non-classical damping.

GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION

In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation（δ2 gij）/δt2＋μ/（（1 ＋ t）λ）（δ gij）/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ ＋ 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R（t, x） of the solution metric gij remains uniformly bounded.

Dynamic and Seismic Property Experiments of High Damping Concrete and Its Frame Models

The loss tangent and storage modulus of cement mortar containing treated silica fume and styrene-acrylic are measured. Shaking table tests on three groups of 1/4 models of high damping concrete frames are conducted to compare the effects of reducing vibration. A relationship between the viscosity coefficient of high damping concrete and the damping coefficient of structure is established by analyzing dynamic parameters like damping ratio and frequency, and comparing responses of relative displacement and absolute acceleration. Surface treatment of silica fume and styrene-acrylic emulsion prior to incorporation in a cement mortar increases the loss tangent of the resulting composite by 40-400% as compared to the cement mortar without any admixtures, and the damping ratio of the reinforced concrete is raised by 7%-9%. In addition, a theoretical method of determining the viscosity coefficient of the material on the basis of shaking table test results is put forward.

Optimum placement and characteristics of velocity-dependent dampers under seismic excitation

In this study, through novel drift-based equations of motion in the frequency domain, optimum placement and characteristics of linear velocity-dependent dampers are investigated. In this study, the sum of the square of the absolute values of transfer matrix elements for interstory drifts is considered as the optimization index. Optimum placement and characteristics of dampers are simultaneously obtained by minimizing the optimization index through an incremental procedure. In each step of the procedure, a predefined value is considered as the damper characteristic. The optimum story for this increment is selected such that it leads to a minimum value for the optimization index. The procedure is repeated for the next increments until the optimization index meets its target value, which is obtained according to the desired damping ratio for the overall structure. In other words, the desired overall damping ratio is the input to the proposed procedure, and the optimal placement and characteristics of the dampers are its output. It is observed that the optimal placement of a velocitydependent damper depends on the damping coefficient of the added damper, frequency of the excitation, and distribution of the mass, stiffness, and inherent damping of the main structure.

The roll damping assessment via decay model testing (new ideas about an old subject)

The methodology to obtain the non-linear roll damping from decay tests is very old. It has been proposed by Froude in the 19th century and used from then on. Behind it there is a quadratic model [θ｜θ｜] for the damping and a subsequent equivalent linearization. Probably all model basin in the world follows this approach to assess the damping from a methods to get the P1-P2 coefficients. This is very applied to any kind of hull. However, it has become decay test. This is well documented and so is the general in the sense that in principle, it could be clear that for hull with a flat bottom such as a very large crude carrier （VLCC）, this approach may lead to confusing results such as negative P2. Faced with this, the work presents a completely new idea. Avoiding the polynomial approximation, the basic attitude is to devise two regions from the decaying test response. The first, called the large amplitude response region yields a larger damping, probably due to the large bilge keel vortices that are attracted to the hull flat bottom. The second is the small amplitude response region where the vortices are not attracted to the bottom but travels approximately 45° sidewise. These observations has led to a new approach called the bi-linear approach as discussed in the work after analyzing several （many） model test results. In fact, a new modified bi-linear approach is ultimately proposed after the understanding of a transition region instead of a transition angle.

A Novel Shear Mode MR Damper and Its Mechanical Performance

This paper presents a linear shear mod magneto-rheological(MR) damper which can be applied to vibration control system.The proposed MR damper featured by a small amount of MR fluid,absence of a gas chamber or diaphragm and piston with helix slotted.Because of the absence of a gas chamber or diaphragm,unnecessary damping force caused by gas compression is not generated.Magnitude and damping coefficient of damping force are two important indexes to evaluate performance of MR damper.The piston with helix slotted is developed based on mechanical analysis on rheological characteristics of MR damper,and the damping performance of MR damper with helix slotted piston is investigated through performance experiments and comparison with analytical simulation.The results indicate that helix slot may increase friction coefficient on surface of the piston,and improve the maximum damping force without reducing damping coefficient of the damper.The reflux of MR fluid may be increased by adjusting helix angle suitably,which avoids the settlement of MR fluid.