On dynamics of elastic rod based on exact Cosserat model

The analysis of kinematics and dynamics of an elastic rod with circular cross section is studied on the basis of exact Cosserat model under consideration of the tension and shear deformation of the rod. The dynamical equations of a rod with arbitrary initial shape are established in general form. The dynamics of a straight rod under axial tension and torsion is discussed as an example. In discussion of static stability in the space domain the Greenhill criteria of stability and the Euler load are corrected by the influence of tension and shear strain. In analysis of dynamical stability in the time domain it is shown that the Lyapunov and Euler stability conditions of the rod in space domain are the necessary conditions of Lyapunov＇s stability in the time domain. The longitudinal, torsional and lateral vibrations of a straight rod based on exact model are discussed, and an exact formula of free frequency of lateral vibration is obtained. The free frequency formulas of various simplified models, such as the Rayleigh beam, the Kirchhoff rod, and the Timoshenko beam, can be seen as special cases of the exact formula under different conditions of simplification.

Dynamic Response Study of Steel Catenary Riser Based on Slender Rod Model

A numerical model of the steel catenary riser(SCR) is built based on the slender rod model. The slender rod model,which describes the behavior of the slender riser in terms of the center line position, can solve the geometrical nonlinearity effectively. In a marine environment, the SCR is under the combined internal flow and external loads,such as wave and current. A general analysis considers only the inertial force and the drag force caused by the wave and current. However, the internal flow has an effect on the SCR; it is essential to explore the dynamic response of the SCR with the internal flow. The SCR also suffers the lift force and the fluctuating drag force because of the current. Finite element method is utilized to solve the motion equations. The effects of the internal flow, wave and current on the dynamic response of the SCR are considered. The results indicate that the increase of the internal flow density leads to the decrease of the displacement of the SCR, while the internal flow velocity has little effect on the SCR. The displacement of the SCR increases with the increase of the wave height and period. And the increasing wave period results in an increase in the vibration period of the SCR. The current velocity changes the displacements of the SCR in x-and z-directions. The vibration frequency of the SCR in y-direction increases with the increase of the current velocity.

Analytical models for the penetration of semi-infinite targets by rigid,deformable and erosive long rods

A theoretical study is presented herein on the pen- etration of a semi-infinite target by a spherical-headed long rod for Yp 〉 S, where Yp is the penetrator strength and S is the static target resistance. For Yp 〉 S, depending upon initial impact velocity, there exist three types of penetration, namely, penetration by a rigid long rod, penetration by a deforming non-erosive long rod and penetration by an erosive long rod. If the impact velocity of the penetrator is higher than the hydrodynamic velocity （VH）, it will penetrate the target in an erosive mode; if the impact velocity lies between the hydrodynamic velocity （VH） and the rigid body velocity （VR）, it will penetrate the target in a deformable mode; if the impact velocity is less than the rigid body velocity （VR）, it will penetrate the target in a rigid mode. The critical conditions for the transition among these three penetration modes are proposed. It is demonstrated that the present model predictions correlate well with the experimental observations in terms of depth of penetration （DOP） and the critical transition conditions.

An Axisymmetric Numerical Model for Simulating Kinetically-Limited Growth of a Cylindrical Rod in 3D Laser-induced Chemical Vapor Deposition

Laser-induced chemical vapor deposition (LCVD) is an important process for freeform microfabrication of high aspect ratio prototypes. The system consists of a laser beam focused onto a movable substrate in a vacuum chamber. Heat from the laser at or near the focal spot of the beam causes gas in the chamber to react. As a result, solid-phase reaction products are deposited on the substrate to form the microstructure. In this paper, we develop a numerical model for simulating growth of an axisymmetric cylindrical rod by pre-specifying the surface temperatures required for growing the rod and then by solving for the laser power that satisfies the pre-specified temperatures. The solution using least squares is obtained by minimizing the sum of square deviations between the pre-specified surface temperatures and the calculated temperatures from the heat equation with a given laser power as a heat source. Model predictions of the laser power over growth time helped in optimizing the growth process. Rods grown based on the predicted laser power from the numerical model were very close to being cylindrical in shape. Ways to further improve the model are being investigated.

General Kinetostatic Modeling and Deformation Analysis of a Two-Module Rod-Driven Continuum Robot with Friction Considered

Continuum robots actuated by flexible rods have large potential applications,such as detection and operation tasks in confined environments,since the push and pull actuation of flexible rods withstand tension and compressive force,and increase the structure's rigidity.In this paper,a generalized kinetostatics model for multi-module and multi-segment continuum robots considering the effect of friction based on the Cosserat rod theory is established.Then,the model is applied to a two-module rod-driven continuum robot with winding ropes to analyze its deformation and load characteristics.Four different in-plane configurations under the external load term as S1,S2,C1,and C2 are defined.Taking a bending plane as an example,the tip deformation along thex-axis of these shapes is simulated and compared,which shows that the load capacity of C1 and C2 is generally larger than that of S1 and S2.Furthermore,the deformation experiments and simulations show that the maximum error ratio without external loads relative to the total length is no more than 3%,and it is no more than 4.7%under the external load.The established kinetostatics model is proven sufficient to accurately analyze the rod-driven continuum robot with the consideration of internal friction.

Stability analysis of helical rod based on exact Cosserat model

Helical equilibrium of a thin elastic rod has practical backgrounds, such as DNA, fiber, sub-ocean cable, and oil-well drill string. Kirchhoff＇s kinetic analogy is an effective approach to the stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff＇s theory without the extension of the centerline and the shear deformation of the cross section are not adoptable to real soft materials of biological fibers. In this paper, the dynamic equations of a rod with a circular cross section are established on the basis of the exact Cosserat model by considering the tension and the shear deformations. Euler＇s angles are applied as the attitude representation of the cross section. The deviation of the normal axis of the cross section from the tangent of the centerline is considered as the result of the shear deformation. Lyapunov＇s stability of the helical equilibrium is discussed in static category. Euler＇s critical values of axial force and torque are obtained. Lyapunov＇s and Euler＇s stability conditions in the space domain are the necessary conditions of Lyapunov＇s stability of the helical rod in the time domain.

Approximate solutions of the Alekseevskii–Tate model of long-rod penetration

The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.

Qualitative Knowledge Based Discrete Rod Model with Simulation Case

Constructing a qualitative model for discrete rods warhead,the kinematics analysis and dynamics analysis of the rods are completed.On the basis of the qualitative model of discrete rods,a simulation case is provided.The qualitative simulation result shows that hexagon cross section shape of rod will receive the lowest air drag while by means of better preplaced dip angle,controlling the rotation velocity of the rod and keeping every rod with identical dip angle to the axis of warhead will make the rod get a lateral direction velocity so as to have the rod gain identical casting initial velocity in its length direction.Simulation experiment shows that when dip angle of preplaced discrete rod is 6°,the damage cycle reaches the optimal value.

Simulation analysis of wear characteristics of connecting rod bearing bush based on improved mixed lubrication model

The failure rate of crankpin bearing bush of diesel engine under complex working conditions such as high temperature,dynamic load and variable speed is high.After serious wear,it is easy to deteriorate the stress state of connecting rod body and connecting rod bolt,resulting in serious accidents such as connecting rod fracture and body damage.Based on the mixed lubrication characteristics of connecting rod big endbearing shell of diesel engine under high explosion pressure impact load,an improved mixed lubrication mechanism model is established,which considers the influence of viscoelastic micro deformation of bearing bush material,integrates the full film lubrication model and dry friction model,couples dynamic equation of connecting rod.Then the actual lubrication state of big end bearing shell is simulated numerically.Further,the correctness of the theoretical research results is verified by fault simulation experiments.The results show that the high-frequency impact signal with fixed angle domain characteristics will be generated after the serious wear of bearing bush and the deterioration of lubrication state.The fault feature capture and alarm can be realized through the condition monitoring system,which can be applied to the fault monitoring of connecting rod bearing bush of diesel engine in the future.

Numerical Analysis of Steady Smoldering of Biomass Rods

Understanding the steady mechanism of biomass smoldering plays a great role in the utilization of smoldering technology.In this study numerical analysis of steady smoldering of biomass rods was performed.A two-dimensional(2D)steady model taking into account both char oxidation and pyrolysis was developed on the basis of a calculated propagation velocity according to empirical correlation.The model was validated against the smoldering experiment of biomass rods under natural conditions,and the maximum error was smaller than 31%.Parameter sensitivity analysis found that propagation velocity decreases significantly while oxidation area and pyrolysis zone increase significantly with the increasing diameter of rod fuel.

Variable-based Ramberg-Osgood constitutive model of power spinning bushing

The influences of power spinning process parameters on the mechanical properties of spinning parts were analyzed with an SXD100/3-CNC numerical control power spinning machine.The unidirectional tensile tests were carried out.Based on the experimental data,a ternary quadratic regression equation was established by orthogonal experiment.The Ramberg-Osgood constitutive model of tin-bronze connecting rod bushing was obtained.Referred to the constitutive relation of macroscopic incremental,the incremental elastoplastic constitutive relation of spinning parts was deduced based on the Mises yield criterion and kinematic hardening model.The results can be applied to the elastoplastic analysis in finite element numerical simulation.

Rod-pinch二极管理论及数值模拟

介绍了Rod-pinch二极管的基本结构和工作原理,采用Laminar模型分析了Rod-pinch二极管中的粒子运动过程及其阻抗特性。考虑背景空间离子电荷的影响,用1维Laminar方程分析Rod-pinch二极管中电子的自箍缩过程,并且利用Magic程序对其中的粒子运动进行数值模拟,求解二极管中的电压和电流,最终得出二极管的阻抗特性,在较低电压下,负极性RPD的性能明显不如正极性RPD。根据临界电流经验公式,初步验证Laminar理论模型的可行性。

Physics-based Modeling and Simulation for Motional Cable Harness Design

The design work of motional cable in products is vital due to the difficulty in estimating the potential issues in current researches.In this paper,a physics-based modeling and simulation method for the motional cable harness design is presented.The model,based on continuum mechanics,is established by analyzing the force of microelement in equilibrium.During the analysis procedure,three coordinate systems：inertial,Frenet and main-axis coordinate systems are used.By variable substitution and dimensionless processing,the equation set is discretized by differential quadrature method and subsequently becomes an overdetermined nonlinear equation set with boundary conditions solved by Levenberg-Marquardt method.With the profile of motional cable harness obtained from the integral of arithmetic solution,a motion simulation system based on＂path＂and＂profile＂as well as the experimental equipments is built.Using the same parameters as input for the simulation and the real cable harness correspondingly,the issue in designing,such as collision,can be easily found by the simulation system.This research obtains a better result which has no potential collisions by redesign,and the proposed method can be used as an accurate and efficient way in motional cable harness design work.

Surface effect and non-local elasticity in wave propagation of functionally graded piezoelectric nano-rod excited to applied voltage

A non-local solution for a functionally graded piezoelectric nano-rod is pre- sented by accounting the surface effect. This solution is used to evaluate the charac- teristics of the wave propagation in the rod structure. The model is loaded under a two-dimensional （2D） electric potential and an initially applied voltage at the top of the rod. The mechanical and electrical properties are assumed to be variable along the thick- ness direction of the rod according to the power law. The Hamilton principle is used to derive the governing differential equations of the electromechanical system. The effects of some important parameters such as the applied voltage and gradation of the material properties on the wave characteristics of the rod are studied.

The effects of compressibility and strength on penetration of long rod and jet

The approximate compressible model is adopted to study the effects of strength and compressibility on the penetration by WHA long rod and copper jet into semi-infinite target in detail. For WHA rod penetrating PMMA at 2 km/s <V <5 km/s, the compressibility has a significant effect on the penetration efficiency. We clarify how compressibility affects the penetration efficiency by changing the stagnation pressures of the rod and target. For WHA rod penetrating 4340 Steel and 6061-T6 Al at 2 km/s < V < 10 km/s, the effect of strength is strong and the effect of compressibility is negligible at lower impact velocity, whilst the effect of strength is weak and the effect of compressibility becomes stronger at higher impact velocity. For the copper jet penetrating 4030 Steel, 6061-T6 Al and PMMA. the virtual origin model is adopted, and the compressibility and strength are implicitly considered by the linear relation between the penetration velocity and impact velocity. The effects of compressibility and target resistance on penetration efficiency are studied. The results show that the target resistance has a significant effect on the penetration efficiency. Howver PMMA is much more compressible than copper and the huge difference of compressibility has a significant effect on the penetration by hypervelocity copper jet into PMMA.

Flow stress equation in range of intermediate strain rates and high temperatures to predict roll force in four-pass continuous rod rolling

A flow stress equation was proposed to compute the roll force in the finishing stands of an actual rod mill where the strain rate and the temperature of the material range from 100 to 400 s-1 and from 900 to 1050 ℃,respectively.The underlying idea is to modify the Shida model and Misaka model,which provide flow stress equations（constitutive equations） frequently used to depict deformation behavior of high temperature material at different strain rates.The modified model was coupled with finite element method to compute the roll force during four-pass continuous rod rolling,where strain rates are in the range of 100-400 s-1 at high temperatures（900-1050 ℃）.The roll forces and the surface temperatures of the material at each stand were measured,and the measured data were compared with the computed values.Results reveal that the Misaka model is better than the Shida model for high temperatures and intermediate strain rates.The roll force error was-5.7 % when the Misaka model was used at 900 ℃.However,the error increased by-15.2% at 1050 ℃.When the modified Misaka model was used,the error was reduced to 1.8% on average.It can consequently be deduced that the modified Misaka model can be used to depict the deformation resistance behavior in intermediate ranges of strain rate and high temperature ranges in continuous rod rolling process.

ANALYSIS OF THERMAL POST-BUCKLING OF HEATED ELASTIC RODS

Based on the nonlinear geometric theory of extensible rods, an exact mathematical model of thermal post_buckling behavior of uniformly heated elastic rods with axially immovable ends is developed, in which the arc length s(x) of axial line and the longitudinal displacement u(x) are taken as the basic unknown functions. This is a two point boundary value problem of first order ordinary differential equations with strong non_linearity. By using shooting method and analytical continuation, the nonlinear boundary value problems are numerically solved. The thermal post_buckled states of the rods with transversely simply supported and clamped ends are obtained respectively and the corresponding numerical data tables and characteristic curves are also given.

Single-phase Computational Fluid Dynamics Applicability for the Study of Three-dimensional Flow in 5×5 Rod Bundles with Spacer Grids

Spacer grids play an important role in pressurized water reactor（PWR） fuel assembly in that they have significant influence on the thermal-hydraulic characteristics of the reactor core.But so far,the numerical studies are performed without regarding dimple and spring of spacer grids,just considering mixing vane.Moreover,these studies use k-ε turbulence model without considering the suitability of the other turbulence models upon the different spacer grids flow.A study is carried out to understand the 3-D single-phase flow in AFA-2G 5×5 rod bundles with spacer grids based on numerical method.In order to investigate the suitability of different turbulence models,k-ε model and k-ω model,the influence of different parts of spacer grid on the fluid flow is also predicted.By using second-order upwind scheme,hybrid grids technique,and improved SIMPLEC algorithm,the Reynolds averaged mass conservation and momentum conservation equations are solved,and the pressure and velocity field of flow are obtained.The numerical simulation results are compared with experiment results and the agreement is satisfactory.The simulation results show the influences of the spring,dimple and mixing vane,and the different characteristics of the k-ε model and k-ω model.Comparing with the experiment results,the simulation results suggest that the k-ω model is suitable for the simulation of the rod bundle flow with spacer grids;the spring and dimple are the main causes of the pressure loss in the spacer grid channel.The friction coefficient of the channel with spring and dimple is 1.5 times the coefficient of the channel with the vane.These results are beneficial to enhance the simulation ability of spacer grids flow and optimization design ability of spaces grid.

Effects of rod radius and voltage on streamer discharge in a short air gap

Streamer discharge is the inaugural stage of gas discharge,and the average electron energy directly determines the electron collision reaction rate,which is a key parameter for studying streamer discharge.Therefore,taking into account the average electron energy,this work establishes a fluid chemical reaction model to simulate and study the course of evolution of a streamer discharge in a 5 mm rod–plate gap,considering 12 particles and 27 chemical reactions.It introduces the electron energy drift diffusion equation into the control equation,and analyzes the temporal and spatial changes of average electron energy,electric field intensity and electron density with change in rod radius and voltage.The effects of voltage and rod radius on the course of streamer discharge can be reflected more comprehensively by combining the average electron energies.Three different values of 0.3 mm,0.4 mm and 0.5 mm are set for the rod radius,and three different values of 5 k V,6 k V and 7 k V are set for the voltage.The influence of an excitation reaction on the streamer discharge is studied.The findings indicate that,as voltage raises,the streamer head’s electron density,electric field and average electron energy all rise,and the streamer develops more quickly.When the rod radius increases,the electron density,electric field and average electron energy of the streamer head all decrease,and the streamer’s evolution slows down.When an excitation reaction is added to the model,the average electron energy,the magnitude of the electric field and the density of electrons decrease,and the evolution of the streamer slows down.An increase in average electron energy will lead to an increase in electric field strength and electron density,and the development of the streamer will be faster.

Long-rod penetration:the transition zone between rigid and hydrodynamic penetration modes

Long-rod penetration in a wide range ol" velocity means that the initial impact velocity varies in a range from tens of meters per second to several kilometers per second.The long rods maintain rigid state when the impact velocity is low,the nose of rod deforms and even is blunted when the velocity gets higher,and the nose erodes and fails to lead to the consumption of long projectile when the velocity is very high clue to instantaneous high pressure.That is,from low velocity to high velocity,the projectile undergoes rigid rods,deforming non-erosive rods,and erosive rods.Because of the complicated changes of the projectile,no well-established theoretical model and numerical simulation have been used to study the transition zone.Based on the analysis of penetration behavior in the transition zone,a phenomenological model to describe target resistance and a formula to calculate penetration depth in transition zone are proposed,and a method to obtain the boundary velocity of transition zone is determined.A combined theoretical analysis model for three response regions is built by analyzing the characteristics in these regions.The penetration depth predicted by this combined model is in good agreement with experimental result.