Natural vibration and critical velocity of translating Timoshenko beam with non-homogeneous boundaries

In most practical engineering applications,the translating belt wraps around two fixed wheels.The boundary conditions of the dynamic model are typically specified as simply supported or fixed boundaries.In this paper,non-homogeneous boundaries are introduced by the support wheels.Utilizing the translating belt as the mechanical prototype,the vibration characteristics of translating Timoshenko beam models with nonhomogeneous boundaries are investigated for the first time.The governing equations of Timoshenko beam are deduced by employing the generalized Hamilton's principle.The effects of parameters such as the radius of wheel and the length of belt on vibration characteristics including the equilibrium deformations,critical velocities,natural frequencies,and modes,are numerically calculated and analyzed.The numerical results indicate that the beam experiences deformation characterized by varying curvatures near the wheels.The radii of the wheels play a pivotal role in determining the change in trend of the relative difference between two beam models.Comparing the results unearths that the relative difference in equilibrium deformations between the two beam models is more pronounced with smaller-sized wheels.When the two wheels are of equal size,the critical velocities of both beam models reach their respective minima.In addition,the relative difference in natural frequencies between the two beam models exhibits nonlinear variation and can easily exceed 50%.Furthermore,as the axial velocities increase,the impact of non-homogeneous boundaries on modal shape of translating beam becomes more significant.Although dealing with non-homogeneous boundaries is challenging,beam models with non-homogeneous boundaries are more sensitive to parameters,and the differences between the two types of beams undergo some interesting variations under the influence of non-homogeneous boundaries.

The Outer-Performance Distortion of the Suspension Shock Absorber and Its Critical Velocity

On the basis of analysing the outer performance degradation of shock absorber on suspenson and from the relationship between outer and inner performances of the shock absorber, an internal relationship between the structure design and degradation of the shock absorber is discussed in the paper. From dynamic property, analysed the dynamic cause for degradation, the paper proposes a technical method of improving outer performance and a concept of critical velocity, and discusses what effects the critical velocity and the outer performance mance degradation has.

Numerical Experiments on Critical Ventilation Velocity and Back-layer in Tunnel Fire

Full-scale numerical experiments were carried out on the vehicular fire in a long tunnel to study the critical ventilation velocity and back-layer distance with heat release rate of 5, 20 and 100 MW respectively. A computational fluid dynamics (CFD) model of fire-driven fluid flow FDS(Fire Dynamics Simulator) was used to solve numerically a form of the Navier-Stokes equations for fire. The results were compared with the expressions proposed in the literature. A modified equation for the critical ventilation velocity was given to better fit the experimental results. A bi-exponential model that well fitted the numerical experimental results was proposed to describe the relationship between back-layer distance and ventilation velocity.

A Novel Pre-control Method of Vehicle Dynamics Stability Based on Critical Stable Velocity during Transient Steering Maneuvering

The current research of direct yaw moment control（DYC） system focus on the design of target yaw moment and the distribution of wheel brake force. The differential braking intervention can effectively improve the lateral stability of the vehicle, however, the effect of DYC can be improved a step further by applying the control of vehicle longitudinal velocity. In this paper, the relationship between the vehicle longitudinal velocity and lateral stability is studied, and the simulation results show that a decrease of 5 km/h of longitudinal velocity at a particular situation can bring 100° increasing of stable steering upper limit. A critical stable velocity considering the effect of steering and yaw rate measurement is defined to evaluate the risk of losing steer-ability or stability. A novel velocity pre-control method is proposed by using a hierarchical pre-control logic and is integrated with the traditional DYC system. The control algorithm is verified through a hardware in-the-loop simulation system. Double lane change（DLC） test results on both high friction coefficient（μ） and low μ roads show that by using the pre-control method, the steering effort in DLC test can be reduced by 38% and 51% and the peak value of brake pressure control can be reduced by 20% and 12% respectively on high μ and low μ roads, the lateral stability is also improved. This research proposes a novel DYC system with lighter control effort and better control effect.

The Influence of the Imperfectness of Contact Conditions on the Critical Velocity of the Moving Load Acting in the Interior of the Cylinder Surrounded with Elastic Medium

The dynamics of the moving-with-constant-velocity internal pressure acting on the inner surface of the hollow circular cylinder surrounded by an infinite elastic medium is studied within the scope of the piecewise homogeneous body model by employing the exact field equations of the linear theory of elastodynamics.It is assumed that the internal pressure is point-located with respect to the cylinder axis and is axisymmetric in the circumferential direction.Moreover,it is assumed that shear-spring type imperfect contact conditions on the interface between the cylinder and surrounding elastic medium are satisfied.The focus is on the influence of the mentioned imperfectness on the critical velocity of the moving load and this is the main contribution and difference of the present paper the related other ones.The other difference of the present work from the related other ones is the study of the response of the interface stresses to the load moving velocity,distribution of these stresses with respect to the axial coordinates and to the time.At the same time,the present work contains detail analyses of the influence of problem parameters such as the ratio of modulus of elasticity,the ratio of the cylinder thickness to the cylinder radius,and the shear-spring type parameter which characterizes the degree of the contact imperfection on the values of the critical velocity and stress distribution.Corresponding numerical results are presented and discussed.In particular,it is established that the values of the critical velocity of the moving pressure decrease with the external radius of the cylinder under constant thickness of that.

Experimental study of the critical sand starting velocity of gas-watersand flow in an inclined pipe

The purpose of this paper is to study the critical sand starting velocity and transformation law of flow pattern based on gas-water-sand three-phase flow in an inclined pipe.Firstly,the indoor simulation experiment system of gas-water-sand three-phase flow was used to test the conversion law of flow pattern based upon the different gas void fraction.Secondly,the influence of slug bubbles on sand migration was investigated according to distinctive hole deviation angles,gas void fraction and sand concentration.Finally,the critical sand starting velocity was tested based on dissimilar hole deviation angles,gas void fraction,sand concentration and sand particle size,and then the influence of the abovementioned key parameters on the sand starting velocity was debated based on the force analysis of the sand particles.The experimental results illustrated that when the gas void fraction was less than 5%,it was bubbly flow.When it increased from 5%to 30%,the bubbly flow and slug flow coexisted.When it was between 30%and 50%,the slug flow and agitated flow coexisted.When it reached 50%,it was agitated flow.Providing that the hole deviation angle was 90°,the phenomenon of overall migration and wavelike migration on the surface of sand bed was observed.On the contrary,the phenomenon of rolling and jumping migration was recognized.The critical sand starting velocity was positively correlated with the hole deviation angle and sand particle size,but negatively associated with the gas void fraction and sand concentration.This research can provide a certain reference for sand-starting production in the field of petroleum engineering.

Critical velocity of sandwich cylindrical shell under moving internal pressure

Critical velocity of an infinite long sandwich shell under moving internal pressure is studied using the sandwich shell theory and elastodynamics theory. Propagation of axisymmetric free harmonic waves in the sandwich shell is studied using the sandwich shell theory by considering compressibility and transverse shear deformation of the core, and transverse shear deformation of face sheets. Based on the elastodynamics theory, displacement components expanded by Legendre polynomials, and position-dependent elastic constants and densities are introduced into the equations of motion. Critical velocity is the minimum phase velocity on the desperation relation curve obtained by using the two methods. Numerical examples and the finite element （FE） simulations are presented. The results show that the two critical velocities agree well with each other, and two desperation relation curves agree well with each other when the wave number k is relatively small. However, two limit phase velocities approach to the shear wave velocities of the face sheet and the core respectively when k limits to infinite. The two methods are efficient in the investigation of wave propagation in a sandwich cylindrical shell when k is relatively small. The critical velocity predicted in the FE simulations agrees with theoretical prediction.

Critical velocity and dynamic respondency of pipe conveying fluid

Presents the calculation of critical velocity, natural frequency and dynamic respondency of fluid conveying pipe are calculated under different boundary conditions using finite element method, and the use of calculation results to design and research rocket pipes feeding fuel and watery turbine pipes conveying water etc.

On the energy conservation and critical velocities for the propagation of a "steady-shock" wave in a bar made of cellular material

The propagation of shock waves in a cellular bar is systematically studied in the framework of continuum solids by adopting two idealized material models, viz. the dynamic rigid, perfectly plastic, locking （D-R-PP-L） model and the dynamic rigid, linear hardening plastic, locking （D-R-LHP-L） model, both considering the effects of strain-rate on the material properties. The shock wave speed relevant to these two models is derived. Consider the case of a bar made of one of such material with initial length L 0 and initial velocity v i impinging onto a rigid target. The variations of the stress, strain, particle velocity, specific internal energy across the shock wave and the cease distance of shock wave are all determined analytically. In particular the ＂energy conservation condition＂ and the ＂kinematic existence condition＂ as proposed by Tan et al. （2005） is re-examined, showing that the ＂energy conservation condition＂ and the consequent ＂critical velocity＂, i.e. the shock can only be generated and sustained in R-PP-L bars when the impact velocity is above this critical velocity, is incorrect. Instead, with elastic deformation, strain-hardening and strain-rate sensitivity of the cellular materials being considered, it is appropriate to redefine a first and a second critical impact velocity for the existence and propagation of shock waves in cellular solids. Starting from the basic relations for shock wave propagating in D-R-LHP-L cellular materials, a new method for inversely determining the dynamic stress-strain curve for cellular materials is proposed. By using e.g. a combination of Taylor bar and Hopkinson pressure bar impact experimental technique, the dynamic stress-strain curve of aluminum foam could bedetermined. Finally, it is demonstrated that this new formulation of shock theory in this one-dimensional stress state can be generalized to shocks in a one-dimensional strain state, i.e. for the case of plate impact on cellular materials, by simply making proper replacements of the elastic and plastic constants.

Numerical Simulation of Fire-Smoke Diffusion Caused by Vehicles in a Tunnel

Urban tunnels are generally narrow and fire smoke can hardly diffuse.In the present study,numerical simulation is used to analyze the diffusion of high temperature smoke produced by fire inside a specific tunnel(the Kaiyuan tunnel).The results are compared with similar data relating to other tests to determine the validity of the numerical method.Moreover,the critical velocity obtained by numerical simulation of 5 MW,20 MW,and 50 MW fires in curved and linear sections of the considered tunnel is compared with the values obtained using empirical formulas.The results show that,for the tunnel ventilation design,it is necessary to consider the fan pressurization at different sections and the fan pressurization should be higher at curved sections than that at linear sections.The safety of personnel escaping under different critical velocity values in the linear section has also been considered.On the basis of our findings,if only relying on natural ventilation,people can escape safely for the case of small fires,whereas for medium and large fires,it is necessary to turn on mechanical ventilation in time(and in order to avoid the danger caused by rapid diffusion of smoke,the timing of mechanical ventilation should be carefully tuned).

Aerodynamic unstable critical wind velocity for three-dimensionalopen cable-membrane structures

The aerodynamic unstable critical wind velocity for three-dimensional open cable-membrane structures is investigated. The geometric nonlinearity is introduced into the dynamic equilibrium equations of structures. The disturbances on the structural surface caused by the air flow are simulated by a vortex layer with infinite thickness in the structures. The unsteady Bernoulli equation and the circulation theorem are applied in order to express the aerodynamic pressure as the function of the vortex density. The vortex density is then obtained with the vortex lattice method considering the coupling boundary condition. From the analytical expressions of the unstable critical wind velocities, numerical results and some useful conclusions are obtained. It is found that the initial curvature of open cable-membrane structures has clear influence on the critical wind velocities of the structures.

A new model for predicting the critical liquid-carrying velocity in inclined gas wells

Based on the assumption of gas-liquid stratified flow pattern in inclined gas wells,considering the influence of wettability and surface tension on the circumferential distribution of liquid film along the wellbore wall,the influence of the change of the gas-liquid interface configuration on the potential energy,kinetic energy and surface free energy of the two-phase system per unit length of the tube is investigated,and a new model for calculating the gas-liquid distribution at critical conditions is developed by using the principle of minimum energy.Considering the influence of the inclination angle,the calculation model of interfacial friction factor is established,and finally closed the governing equations.The interface shape is more vulnerable to wettability and surface tension at a low liquid holdup,resulting in a curved interface configuration.The interface is more curved when the smaller is the pipe diameter,or the smaller the liquid holdup,or the smaller the deviation angle,or the greater gas velocity,or the greater the gas density.The critical liquid-carrying velocity increases nonlinearly and then decreases with the increase of inclination angle.The inclination corresponding to the maximum critical liquid-carrying velocity increases with the increase of the diameter of the wellbore,and it is also affected by the fluid properties of the gas phase and liquid phase.The mean relative errors for critical liquid-carrying velocity and critical pressure gradient are 1.19%and 3.02%,respectively,and the misclassification rate is 2.38%in the field trial,implying the new model can provide a valid judgement on the liquid loading in inclined gas wells.

Numerical study on critical flow velocity of ice slurry in the pipe of ice source heat pump system

The traditional evaporator of water source heat pump system is easy to freeze when the water source temperature is too low in winter.A novel ice source heat pump system is proposed,which can use low temperature surface water to supply heat to buildings.One of the key problems for safe transportation is velocity range of ice slurry in ice outlet pipeline.In order to study the critical flow velocity of ice slurry in the pipe,an Euler model of horizontal straight pipe and 90°elbow pipe were established by Fluent software.The influence of five factors on the critical flow velocity of two kinds of pipes,including initial ice packing fraction,ice particle size,pipe diameter,length of straight pipe and radius curvature of elbow was studied.It found that the initial ice content should be controlled from 10%to 15%.The inlet flow rate was 20%higher than the critical flow rate under the most unfavorable condition.Ice particle size was less than 0.2 mm.The total length of ice discharge pipeline was less than 50 m.Curvature radius of the bend should be met the minimum requirements of the specification of the pipe used.

Hydraulic calculation of gravity transportation pipeline system for backfill slurry

Taking cemented coal gangue pipeline transportation system in Suncun Coal Mine, Xinwen Mining Group, Shandong Province, China, as an example, the hydraulic calculation approaches and process about gravity pipeline transportation of backfill slurry were investigated. The results show that the backfill capability of the backfill system should be higher than 74.4 m3/h according to the mining production and backfill times in the mine; the minimum velocity (critical velocity) and practical working velocity of the backfill slurry are 1.44 and 3.82 m/s, respectively. Various formulae give the maximum ratio of total length to vertical height of pipeline (L/H ratio) of the backfill system of 5.4, and then the reliability and capability of the system can be evaluated.

Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach

In this article, transverse free vibrations of axially moving nanobeams subjected to axial tension are studied based on nonlocal stress elasticity theory. A new higher-order differential equation of motion is derived from the variational principle with corresponding higher-order, non-classical boundary conditions. Two supporting conditions are investigated, i.e. simple supports and clamped supports. Effects of nonlocal nanoscale, dimensionless axial velocity, density and axial tension on natural frequencies are presented and discussed through numerical examples. It is found that these factors have great influence on the dynamic behaviour of an axially moving nanobeam. In particular, the nonlocal effect tends to induce higher vibration frequencies as compared to the results obtained from classical vibration theory. Analytical solutions for critical velocity of these nanobeams when the frequency vanishes are also derived and the influences of nonlocal nanoscale and axial tension on the critical velocity are discussed.

VIBRATION AND STABILITY OF VERTICAL UPWARD-FLUID-CONVEYING PIPE IMMERSED IN RIGID CYLINDRICAL CHANNEL

A theoretical model is developed for the vibration and stability of a vertical pipe subjected concurrently to two dependent axial flows. The external fluid, after exiting the outer annular region between the pipe and a rigid cylindrical channel, is conveyed upwards inside the pipe. This configuration thus resembles of a pipe that aspirating fluid. The equation of planar mo- tion is solved by means of the differential quadrature method （DQM）. Calculations are conducted for a slender drill-string-like and a bench-top-size system, for different confinement conditions of the outer annular channel. It is shown that the vibrations of these two systems are closely related to the degree of confinement of the outer annular channel. For a drill-string-like system with narrow annuli, buckling instability may occur in the second and third modes. For a bench-top-size system, however, both buckling and flutter may occur in the lowest three modes. The form of instability depends on the annuli size.

New insight into the stability and dynamics of fluid-conveying supported pipes with small geometric imperfections

In several previous studies,it was reported that a supported pipe with small geometric imperfections would lose stability when the internal flow velocity became sufficiently high.Recently,however,it has become clear that this conclusion may be at best incomplete.A reevaluation of the problem is undertaken here by essentially considering the flow-induced static deformation of a pipe.With the aid of the absolute nodal coordinate formulation(ANCF)and the extended Lagrange equations for dynamical systems containing non-material volumes,the nonlinear governing equations of a pipe with three different geometric imperfections are introduced and formulated.Based on extensive numerical calculations,the static equilibrium configuration,the stability,and the nonlinear dynamics of the considered pipe system are determined and analyzed.The results show that for a supported pipe with the geometric imperfection of a half sinusoidal wave,the dynamical system could not lose stability even if the flow velocity reaches an extremely high value of 40.However,for a supported pipe with the geometric imperfection of one or one and a half sinusoidal waves,the first-mode buckling instability would take place at high flow velocity.Moreover,based on a further parametric analysis,the effects of the amplitude of the geometric imperfection and the aspect ratio of the pipe on the static deformation,the critical flow velocity for buckling instability,and the nonlinear responses of the supported pipes with geometric imperfections are analyzed.

Effects of storm waves on rapid deposition of sediment in the Yangtze Estuary channel

Recent research on short-term topographic change in the Yangtze Estuary channel under storm surge conditions is briefly summarized. The mild-slope, Boussinesq and action balance equations are compared and analyzed. The action balance equation, SWAN, was used as a wave numerical model to forecast strong storm waves in the Yangtze Estuary. The spherical coordinate system and source terms used in the equation are described in this paper. The significant wave height and the wave orbital motion velocity near the bottom of the channel during 20 m/s winds in the EES direction were simulated, and the model was calibrated with observation data of winds and waves generated by Tropical Cyclone 9912. The distribution of critical velocity for incipient motion along the bottom was computed according to the threshold velocity formula for bottom sediment. The mechanism of rapid deposition is analyzed based on the difference between the root-mean-square value of the near-bottom wave orbital motion velocity and the bottom critical tractive velocity. The results show that a large amount of bottom sediments from Hengsha Shoal and Jiuduan Shoal are lifted into the water body when 20 m/s wind is blowing in the EES direction. Some of the sediments may enter the channel with the cross-channel current, causing serious rapid deposition. Finally, the tendency of the storm to induce rapid deposition in the Yangtze Estuary channel zone is analyzed.

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half- space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of （sub-） hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of （sub-） soft beam and hard half-space but are similar to each other for the system of （sub-） hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity （the shear wave velocity of the beam for the system of soft beam and hard half-space）. The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave racliation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.

High Velocity Impact Experiment on Ti/CFRP/Ti Sandwich Structure

Aircraft laminated composite components often suffer a variety of high velocity impacts with large quantity of energy,which usually affects aircraft behavior and would incur component damages,even disastrous consequences.Therefore,one investigates the impact resistance of a new type of composite material,Ti/CFRP/Ti sandwich structure,and launches impact tests by using an air gun test system.Then one acquires the critical breakthrough rate of the structure and analyzes the damages.The results show that the main failure mode of the front titanium sheet is shear plugging and brittle fracture of adhesive layer with fiber breakage,while the back titanium sheet is severely ripped.The rear damage is worse than the front one.Compared with traditional CFRP laminates,the critical breakthrough rate of Ti/CFRP/Ti sandwich structure is improved by 69.9% when suffered the impact of a bearing ball with 2mm radius.