应用数学和力学

2014 Vol. 35, No. 4

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Iterative Symplectic Perturbation Method for the Dynamic Analysis of Rigid-Flexible Bodies Equations

WU Feng, GAO Qiang, ZHONG Wan-xie

2014, 35(4): 341-352. doi: 10.3879/j.issn.1000-0887.2014.04.001

Abstract(1240) PDF(1047)

Abstract:
The iterative symplectic perturbation method was proposed for the dynamic analysis of rigid-flexible bodies equations. With the proposed method, the low-frequency motion and high-frequency motion were treated separately. The symplectic perturbation method was applied to the coupling terms jointly caused by the low- and high-frequency motions. The proposed method could give correct numerical results with relatively larger time steps. It overcomes the difficult stiff integral problem. Numerical examples show that the proposed method is valid for the dynamic analysis of rigid-flexible bodies equations.

Dynamic Behavior Analysis of Rotational Flexible Blades Based on Time-Domain Finite Element Method

WANG Xin-dong, DENG Zi-chen, WANG Yan, FENG Guo-chun

2014, 35(4): 353-363. doi: 10.3879/j.issn.1000-0887.2014.04.002

Abstract(1075) PDF(853)

Abstract:
The time-domain finite element method was introduced to investigate the dynamic responses of rotational flexible blades. Firstly, the rotational flexible blades were modeled as a classic rigid hub-flexible beam system. Based on the first-order approximate coupling (FOAC) model, the Lagrangian function for the rotational flexible blades system was derived. Then, with the assumed mode method (AMM), the time-domain finite element scheme was constructed. Finally, the dynamic behavior of the rotational flexible blades was analyzed with the time-domain finite element method through numerical simulation. Constructed directly without derivation of the kinetic equations, the proposed discrete scheme is naturally endowed with symplectic conservation, high computational accuracy and good stability. Numerical results show that the time-domain finite element method can effectively solve the rigid-flexible coupling problem, in which the low-frequency large motion and the high-frequency elastic vibration of the blades are interactive.

Capillary Force Analysis of Constant-Volume Liquid Bridges in Atomic Force Microscopes

WEI Zheng, CHEN Shao-yong, ZHAO Shuang, SUN Yan

2014, 35(4): 364-376. doi: 10.3879/j.issn.1000-0887.2014.04.003

Abstract(1475) PDF(1098)

Abstract:
The capillary force and rupture energy of the constant-volume liquid bridge in the atomic force microscope (AFM) were investigated with the surface and interface thermodynamic and mechanical methods respectively. Comparison between results of these two methods was given. Validity of the circular-arc shape approximation of the liquid bridge was discussed. Rupture energy of the liquid bridge dominates the energy dissipation in the tapping mode of the AFM and makes a key factor effecting phase shift. Moreover, the contact angle hysteresis effect on the capillary force and rupture energy was presented. The work may have some reference for understanding of the imaging mechanism of the tapping mode of AFMs and analysis of the force-distance curves in AFMs.

Direct Numerical Simulation of Head-on Binary Collision of Aluminum Oxide Droplets

XIA Sheng-yong, HU Chun-bo

2014, 35(4): 377-388. doi: 10.3879/j.issn.1000-0887.2014.04.004

Abstract(1355) PDF(890)

Abstract:
Direct numerical simulations of head-on binary collisions between equal-sized aluminum oxide droplets were conducted to investigate the collision physics and mechanics of aluminum oxide droplets in solid rocket motors. Trial simulations of head-on collisions of tetradecane droplets in nitrogen medium were performed firstly to give results in good agreement with those of the previous experiments. After the positive validation of the numerical method, the head-on binary collisions of equal-sized aluminum oxide droplets were numerically computed with various Weber numbers under 6 MPa ambient pressure and 3 387 K ambient temperature. The Weber number ranged from 10 to 200, and the Ohnesorge number kept at 0.036 4, which covered three different types of outcomes: bouncing, coalescence and reflexive separation. The results show that the critical Weber number between bouncing and coalescence after substantial deformation is 26, and the one between coalescence after substantial deformation and reflexive separation is 44. The collision model for aluminum oxide droplets can be obtained through modification of the collision model for other fluid droplets with critical Weber numbers.

Hydrodynamic Interactions Between Multiple Ships Advancing Parallel in Close Proximity in Waves

XU Yong, DONG Wen-cai

2014, 35(4): 389-400. doi: 10.3879/j.issn.1000-0887.2014.04.005

Abstract(1354) PDF(1457)

Abstract:
A three-dimensional frequency-domain calculation model based on the potential flow theory and multi-body dynamics theory was developed to investigate wave loads and free motions of multiple closely-spaced, hydrodynamically interacting ships advancing parallel in waves. In order to consider the speed effects on free surface in the gaps between these ship bodies, the translating-pulsating source Green’s function was chosen to simulate the radiation and diffraction waves. The model was validated through solving of the hydrodynamic terms and free motions of two parallel closely-arranged ship models advancing at zero or positive forward speeds in regular waves. Numerical results were also presented in the case of three parallel ships. Comparison of the dynamic behaviors between the two-ship case and three-ship case shows that in the two cases hydrodynamic interaction effects are quite different. The presented model is useful to predict wave loads and free motions of multiple vessels in underway replenishment on the sea.

Pressure Variation in Sealed Cavity and Force Analysis on Vanes in Variable Displacement Vane Pumps

ZHANG Guo-tao, YIN Yan-guo

2014, 35(4): 401-411. doi: 10.3879/j.issn.1000-0887.2014.04.006

Abstract(1141) PDF(853)

Abstract:
Aimed at the characteristics of large pressure variation in sealed cavity of transition zones and difficulty in vanes’protraction from the rotor slot of the variable displacement vane pump, the pressure distribution around the damping groove and its kinetic effects on the vane tip were investigated through MATLAB simulation. The results show that introduction of the damping grooves can effectively reduce the hydraulic shock and pressure jump caused by large pressure difference in the pump; it also can lower the peak of pressure gradient. Consideration of the leakage at friction pairs does not reduce the efficiency of pre-pressurizing and pre-depressurizing processes, but lowers the peak of pressure gradient and smooths the pressure distribution in transition zones. Compared with the traditional spired vane, the round vane has no difficulty in protraction in the transition zones under high working pressure, and has smoother contact force change at the vane tip.

Convergence of Finite Element Method in Rheology

HOU Lei, SUN Xian-yan, ZHAO Jun-jie, LI Han-ling

2014, 35(4): 412-422. doi: 10.3879/j.issn.1000-0887.2014.04.007

Abstract(1279) PDF(769)

Abstract:
Convergence of the first-order mixed-type hyperbolic parabola partial differential equations in non-Newtonian fluid problems was studied. The coupling partial differential equations (Cauchy fluid equation, P-T/T stress equation) were used to simulate the flow zone generated by the free surface elements or excessively tensile elements. The semi-discrete finite element method was applied to solve these equations coupling with time. The finite element method was used in space. The trilinear functional was employed to solve the nonlinear problems of partial differential equations. In the time domain the Euler scheme was adopted. The convergence order of the equation set reached O(h2+Δt). Numerical results of the equations were obtained through priori and posteriori error estimation of high performance computation. And the deformed sizes of the grids were presented.

A New Model and Calculating Method for Pump Dynamograph Diagnosis of Sucker Rod Pumping Systems

SUN Ren-yuan, ZHANG Yun-fei, FAN Kun-kun, WANG Jian, LIU Zhuo, LU Xue-jiao

2014, 35(4): 423-431. doi: 10.3879/j.issn.1000-0887.2014.04.008

Abstract(1317) PDF(1427)

Abstract:
The existing Gibbs model was improved through analysis of the dynamic characteristics of the micro units out of hybrid sucker rods. A new pump dynamograph diagnosis mathematical model was established for sucker rod pumping systems. The model was solved with the methods of implicit difference scheme and variable step size finite difference scheme. The displacement and load expressions at the interface of the box coupling and at any position of the uniform sucker rod segments were obtained. The formula of equivalent damping coefficient was derived and calculated through the corresponding iterative algorithm. With a well surface dynamometer card as an example, the pump dynamograph was calculated by means of the new model, and the results were compared with those from the Gibbs method. The comparison indicates that the new model and algorithm is stable and reliable, and the resulting pump dynamograph fits the actual situation better than that from the Gibbs model.

Analytic Solution and Experimental Study on Shear Lag Effect of Steel-Concrete Composite Beams With Wide Flanges

HU Shao-wei, YU Jiang, XIE Jian-feng

2014, 35(4): 432-443. doi: 10.3879/j.issn.1000-0887.2014.04.009

Abstract(1039) PDF(770)

Abstract:
The sheer lag effect of the steel-concrete composite beam leads to nonuniform distribution of longitudinal stress along the width of flange, thus probably producing disadvantageous transverse cracks. And so, it is necessary to further study the mechanism of shear lag effect of the steel-concrete composite beams with wide flanges, in order to prevent those cracks. The cross-section normal stress differential equation was established for the thin-walled double-room box composite beams, based on the combination of flange micro-unit deformation compatibility condition and equilibrium differential equation; and the analytic stress solution was obtained with the simple-support boundary condition at both ends of the composite beam considered. Finally, the analytic solution was compared with the results of model tests, to prove the accuracy and applicability of the presented method.

Deflection Analysis of Load Trajectories for General Hoisting Mechanisms

QI Zhao-hui, SONG Hui-tao

2014, 35(4): 444-458. doi: 10.3879/j.issn.1000-0887.2014.04.010

Abstract(934) PDF(752)

Abstract:
A systematic method for analyzing load trajectory deflections of general hoisting mechanisms was presented. In this method, the first order differential equations for generalized degrees of freedom and wire rope tensions were established based on the governing equations of equilibrium and compatibility conditions. Properties and formulations of the variables involved in these equations were studied in detail, in which the displacements and velocities of tangent points between pulleys and the tension directions of ropes as well as the change rates of tension directions were derived. The compatibility condition for drum parameters, as well as for rope lengths in the system was provided. A method to solve the angular speeds of spatial pulleys was proposed. In the process of solving the differential equations, the pulley’s angular speed which determined the relation of tensions at both ends of the pulley, was derived at each step. The systematic method is applicable to hoisting mechanisms with single reeving branches, and solves the difficult problem of determining the rotating directions of pulleys. To a large extent the proposed method is universal and makes a reference for designing and analyzing general hoisting mechanisms.

Singularity Analysis for Notches in Orthotropic Composite Plates With the Interpolating Matrix Method

GE Ren-yu, CHENG Chang-zheng, YANG Zhi-yong, NIU Zhong-rong

2014, 35(4): 459-470. doi: 10.3879/j.issn.1000-0887.2014.04.011

Abstract(1363) PDF(862)

Abstract:
Based on asymptotic expansion of generalized displacement field at the V-notch tip, a new method for analyzing the stress singularity exponents of the notches in orthotropic composite plates was proposed. Through introduction of the typical terms in asymptotic expansion of the generalized displacement functions into the basic elastic equations of the plate, the eigenvalue problem of a set of nonlinear ordinary differential equations(ODEs) about the stress singularity exponents of the notch was obtained, then the nonlinear eigenvalue problem was transformed into a linear one by means of variable substitution, and the interpolating matrix method was employed to solve the problem to determine the stress singularity exponents and associated characteristic functions at the notch tip in the orthotropic bi-material plate. With the present method, both the stress singularity exponents and the associated characteristic angle functions can be acquired simultaneously, and the stress singularity exponents can be easily distinguished between plane and anti-plane singularities according to the corresponding characteristic angle functions. Validity of the present method is confirmed in comparison with the existing results through numerical calculation.