Closed-form steady-state solutions for forced vibration of second-order axially moving systems

Second-order axially moving systems are common models in the field of dynamics, such as axially moving strings, cables, and belts. In the traditional research work, it is difficult to obtain closed-form solutions for the forced vibration when the damping effect and the coupling effect of multiple second-order models are considered.In this paper, Green's function method based on the Laplace transform is used to obtain closed-form solutions for the forced vibration of second-order axially moving systems. By taking the axially moving damping string system and multi-string system connected by springs as examples, the detailed solution methods and the analytical Green's functions of these second-order systems are given. The mode functions and frequency equations are also obtained by the obtained Green's functions. The reliability and convenience of the results are verified by several examples. This paper provides a systematic analytical method for the dynamic analysis of second-order axially moving systems, and the obtained Green's functions are applicable to different second-order systems rather than just string systems. In addition, the work of this paper also has positive significance for the study on the forced vibration of high-order systems.

Effect of impeller reflux balance holes on pressure and axial force of centrifugal pump

The size of impeller reflux holes for centrifugal pump has influence on the pressure distribution of front and rear shrouds and rear pump chamber, as well as energy characteristics of whole pump and axial force. Low specific-speed centrifugal pump with Q=12.5 m3/h, H=60 m, n=2950 r/min was selected to be designed with eight axial reflux balance holes with 4.5 mm in diameter. The simulated Q-H curve and net positive suction head(NPSH) were in good agreement with experimental results, which illustrated that centrifugal pump with axial reflux balance holes was superior in the cavitation characteristic; however, it showed to little superiority in head and efficiency. The pressure in rear pump chamber at 0.6 times rate flow is 29.36% of pressure difference between outlet and inlet, which reduces to 29.10% at rate flow and 28.33% at 1.4 times rate flow. As the whole, the pressure distribution on front and rear shrouds from simulation results is not a standard parabola, and axial force decreases as flow rate increases. Radical reflux balance holes chosen to be 5.2 mm and 5.9 mm in diameter were further designed with other hydraulic parts unchanged. With structural grids adopted for total flow field, contrast numerical simulation on internal flow characteristics was conducted based on momentum equations and standard turbulence model(κ-ε). It is found that axial force of pump with radical reflux balance holes of5.2 mm and 5.9 mm in diameter is significantly less than that with radical reflux balance holes of 4.5 mm in diameter. Better axial force balance is obtained as the ratio of area of reflux balance holes and area of sealing ring exceeds 6.

NON-LINEAR FORCED VIBRATION OF AXIALLY MOVING VISCOELASTIC BEAMS

The non-linear forced vibration of axially moving viscoelastic beams excited by the vibration of the supporting foundation is investigated. A non-linear partial-differential equation governing the transverse motion is derived from the dynamical, constitutive equations and geometrical relations. By referring to the quasi-static stretch assumption, the partial-differential non-linearity is reduced to an integro-partial-differential one. The method of multiple scales is directly applied to the governing equations with the two types of non-linearity, respectively. The amplitude of near- and exact-resonant steady state is analyzed by use of the solvability condition of eliminating secular terms. Numerical results are presented to show the contributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity to the response amplitude for the first and the second mode.

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

The parametric dynamic stability of resonant beams with various parameters under periodic axial force is studied. It is assumed that the theoretical formulations are based on Euler-Bernoulli beam theory. The governing equations of motion are derived by using the Rayleigh-Ritz method and transformed into Mathieu equations, which are formed to determine the stability criterion and stability regions for parametricallyexcited linear resonant beams. An improved stability criterion is obtained using periodic Lyapunov functions. The boundary points on the stable regions are determined by using a small parameter perturbation method. Numerical results and discussion are presented to highlight the effects of beam length, axial force and damped coefficient on the stability criterion and stability regions. While some stability rules are easy to anticipate, we draw some conclusions： with the increase of damped coefficient, stable regions arise; with the decrease of beam length, the conditions of the damped coefficient arise instead. These conclusions can provide a reference for the robust design of parametricallyexcited linear resonant sensors.

Horizontal dynamic response of a large-diameter pipe pile considering the second-order effect of axial force

The second-order effect of axial force on horizontal vibrating characteristics of a large-diameter pipe pile is theoretically investigated.Governing equations of the pile-soil system are established based on elastodynamics.Threedimensional wave equations of soil are decoupled through differential transformation and variable separation.Consequently,expressions of soil displacements and horizontal resistances can be obtained.An analytical solution of the pile is derived based on continuity conditions between the pile and soil,subsequently from which expressions of the complex impedances are deduced.Analyses are carried out to examine the second-order effect of axial force on the horizontal vibrating behavior of the pipe pile.Some conclusions can be summarized as follows： stiffness and damping factors are decreased with the application of axial force on the pile head; distributions of the pile horizontal displacement and rotation angle are regenerated due to the second-order effect of the applied axial force; and redistributions of the bending moment and shearing force occur due to the second-order effect of the applied axial force.

Modal and Fatigue Life Analysis on Beam with Multiple Cracks Subjected to Axial Force

Based on the transfer matrix method and Forman equation,a new method is proposed to conduct the modal and fatigue life analysis of a beam with multiple transverse cracks.In the modal analysis,the damping loss factor is introduced by the complex elastic modulus,bending springs without mass are used to replace the transverse cracks,and the characteristic transfer matrix of the whole cracked beam can be derived.In the fatigue life analysis,considering the interaction of the beam vibration and fatigue cracks growth,the fatigue life of the cracked beam is predicted by the timing analysis method.Numerical calculation shows that cracks have a significant influence on the modal and fatigue life of the beam.

Semi-analytic solution of Eringen's two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.

A Combined-Power Model for the Distribution of Axial Force in Shaft Anchor along Its Length

The theoretical results of axial force distribution models differ greatly from tests because of the complication of the rock type material. A three-parameter combined-power model is proposed by curves fitting the test data recorded from the pull tests on anchoring bars used in different engineering projects. Based on the comparison of the mechanical characteristics of shaft anchors and prestressed tendons, a two-parameter combined-power function model for prestressed tendons is proposed. The bounded length derived from the model and the suggested values of the parameters are also proposed. Compared with the Gaussian model, the three-parameter combined-power model is more precise and simple in expression. Results also suggest that the bounded length calculated from the average stress method is not safe enough.

3D FE Analysis of Effect of Ground Subsidence and Piled Spacing on Ultimate Bearing Capacity of Piled Raft and Axial Force of Piles in Piled Raft

The effects of ground subsidence and piled spacing on axial force of piles in squared piled rafts were investigated using numerical analysis. Two cases of piled rafts in soft clay including case 1 (s = 2d) and case 2 (s = 4d) with s and d were piled spacing and piled diameter respectively were considered in this study. Undrained (without ground water pumping) and drained (with ground water pumping) conditions were applied in each case in order to evaluate variations of ultimate bearing capacity of piled raft and axial force of the piles in piled raft. The results showed that ultimate bearing capacity increased about 25% for undrained condition and about 32% for drained condition when piled spacing increased from 2d to 4d. In the same piled spacing, axial force of the piles increased about 9% for piled spacing of 2d and 7% for piled spacing of 4d when drained condition was applied. When piled spacing increased 2 times (2d to 4d), the axial force of piles increased about 7% for undrained condition and about 5% for drained condition.

Axial force measurement for esophageal function testing

The esophagus serves to transport food and fluid from the pharynx to the stomach. Manometry has been the "golden standard" for the diagnosis of esophageal motility diseases for many decades. Hence, esophageal function is normally evaluated by means of manometry even though it reflects the squeeze force (force in radial direction) whereas the bolus moves along the length of esophagus in a distal direction. Force measurements in the longitudinal (axial) direction provide a more direct measure of esophageal transport function. The technique used to record axial force has developed from external force transducers over in-vivo strain gauges of various sizes to electrical impedance based measurements. The amplitude and duration of the axial force has been shown to be as reliable as manometry. Normal, as well as abnormal, manometric recordings occur with normal bolus transit, which have been documented using imaging modalities such as radiography and scintigraphy. This inconsistency using manometry has also been documented by axial force recordings. This underlines the lack of information when diagnostics are based on manometry alone. Increasing the volume of a bag mounted on a probe with combined axial force and manometry recordings showed that axial force amplitude increased by 130% in contrast to an increase of 30% using manometry. Using axial force in combination with manometry provides a more complete picture of esophageal motility, and the current paper outlines the advantages of using this method.

Dynamic Analysis of Axial Magnetic Forces for DVD Spindle Motors

The axial magnetic force, induced by the complicated flux linkage distribution from rotor magnet and stator slotted, is constructed by different relative heights and calculated by 3D finite element method (FEM) to analyze the dynamic characteristics for a DVD spindle motor. The axial magnetic force is designed to provide an axial stiffness, and govern the natural frequency of the dynamic performance. According to the simulation results and experimental measurements, the dynamic behaviors are significantly improved with a variation of relative height of rotor magnet and stator slotted on a DVD spindle motor.

Experimental Investigations of Axial Force Monitoring and Control for Friction Stir Welding Process

Friction stir welding( FSW) is a solid-state welding process that utilizes a rotating tool to induce gross material plastic deformation and join two parts together. A large number of studies have indicated that axial force control can be used to achieve good welding quality. However,in the welding process,due to workpiece's geometry error,improper clamping and other process variations,the axial force can vary significantly and produce welding defects.The control of force in the process of FSW is investigated. At first,the development and evaluation of a closed-loop control system is described,which is equipped with a custom real-time wireless force dynamometer for FSW. Then,an axial force controller is designed based on nonlinear force controllers for FSW. Experimental validations are carried out on an FSW platform. The experimental results demonstrate that the controller maintains the constant axial force and shows desirable dynamic behavior, even when the disturbance is encountered during the welding process.

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

The weakly forced vibration of an axially moving viscoelastic beam is inves- tigated. The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved. The nonlinear equations governing the transverse vibration are derived from the dynamical, constitutive, and geometrical relations. The method of multiple scales is used to determine the steady-state response. The modulation equation is derived from the solvability condition of eliminating secular terms. Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation. The stability of non- trivial steady-state response is examined via the Routh-Hurwitz criterion.

The Tangential Force Distribution on Inner Cylinder of Power Law Fluid Flowing in Eccentric Annuli with the Inner Cylinder Reciprocating Axially

Based on the governing equations of the inner cyinder of the unsteady flow of the power law fluid in eccentric annuli with the inner cylinder reciprocating axially in bipolar coordinate system, the calculation formulae of tangential force were established, and the relevant numerical calculation method was given. Taking the aqueous solution of partially hydrolyzed polyacrylamides (HPAM) for examples, the tangential forces were calculated by using the formulae and numerical calculation method mentioned above;the curves of the tangential force on the wall of the inner cylinder of HPAM aqueous solution were plotted;and the effects on the tangential force of the flow behavior index of the power law fluid, the stroke and the stroke frequency of the inner cylinder were analyzed.

Mobility matrix of a weakly coupled parallel multi-DIM isolator based on axial force solution

A mobility matrix modeling strategy based on axial force solution for a weakly coupled parallel multi-dimentional(multi-DIM)isolator is proposed.Mobility power flow and transmissibility through the isolator are derived from the mobility matrix.Comparison between simulation and experimental results shows the correctness of the proposed modeling strategy.

Viscosity of food boluses affects the axial force in the esophagus

AIM: To study the effect of viscosity on axial force in the esophagus during primary peristalsis using a newly validated impedance-based axial force recording technique. METHODS: A probe able to simultaneously measure both axial force and manometry was positioned above the lower esophageal sphincter. Potable tap water and three thickened fluids were used to create boluses of different viscosities. Water has a viscosity of 1 mPa·s. The three thickened fluids were made with different concentrations of Clinutren Instant thickener. The viscous fluids were in appearance comparable to pudding (2 kPa·s), yogurt (6 kPa·s) and slush ice (10 kPa·s). Six healthy volunteers swallowed 5 and 10 mL of boluses multiple times. RESULTS: The pressure amplitude did not increase with the bolus viscosity nor with the bolus volume whereas the axial force increased marginally with bolus volume (0.1 > P > 0.05). Both techniques showed that contraction duration increased with bolus viscosity (P < 0.01). Association was found between axial force and pressure but the association became weaker withincreasing viscosity. The pressure amplitude did not increase with the viscosity or bolus volume whereas the axial force increased marginally with the bolus size. CONCLUSION: This indicates a discrepancy between the physiological functions that can be recorded with axial force measurements and pressure measurements.

An isolated similarity design method for shaking table tests on reinforced slopes

Physical model test is an effective way to unveil the dynamic response of a slope under seismic condition.The similarity design is the key of physical model test.An isolated similarity design method for shaking table tests was proposed and verified in this work.In this method,the relevant physical quantities were divided into several subsystems and subcharacteristic equations for each subsystem were then established based on the Buckingham similarity theory.Large-scale shaking table tests on a reinforced slope were adopted herein to illustrate the application of the proposed isolated similarity design method.The similarity system for the studied slope was divided into four parts in the process of similarity design.The geometrical dimension L,densityρand gravity g were selected as fundamental quantities for the similarity design,and four subcharacteristic equations were established for each subsystem.The dynamic responses of the recorded acceleration and axis force show that the seismic waves propagate well in the model slope.The proposed isolated similarity design method solves the conflict between the similarity requirement for all relevant physical quantities and the difficulty of test model fabrication to satisfy all similarity relations.

Structural System to Resist Seismic Loads on High Rise Reinforced Concrete Structures

The forces of nature represent the biggest challenge for engineering work in general and perhaps the most prominent of these forces. This generated by earthquake where engineering structure is exposed abnormal loads and stresses which places areal burden on structural engineers to find solutions and structural systems to increase resistance and effectiveness of engineering structure especially high rise concrete structures.

屏蔽式离心泵浮动叶轮轴向力平衡方法

为了实现屏蔽泵离心叶轮轴向力平衡,设计浮动叶轮轴向力自平衡结构,借助全流道三维数值模拟手段,研究浮动叶轮的轴向力平衡作用机理和影响因素.在对数值模拟方法进行试验验证的基础上,开展针对不同工况下叶轮不同轴向位置轴向力的数值计算.数值模拟结果表明,在同一流量工况下,在叶轮浮动范围内,当叶轮向泵入口方向移动时,泵前腔泄漏量减小,扬程和效率增大,且越靠近入口,对泵的性能影响越显著.轴向力呈现先减后增的趋势,在远离入口时方向指向泵入口,在靠近入口时,叶轮轴向力迅速增大,实现反向,指向泵出口,帮助叶轮停止向入口移动,这种轴向力变化趋势可以使叶轮在工作时始终处于浮动状态,实现轴向力自平衡.