应用数学和力学

The Basic Equation of Transverse Vibration for Rotor Dynamics

WU Feng, XU Xiao-ming, LI Ming-wu, GAO Qiang, ZHONG Wan-xie

2015, 36(10): 1011-1018. doi: 10.3879/j.issn.1000-0887.2015.10.001

Abstract(1307) PDF(1271)

Abstract:
The basic equation of transverse vibration for rotor systems was investigated. First the previous precise nonlinear mathematical model for rotor dynamics, which had been built through the introduction of the Euler angle representation to describe the nonlinear motions of rotors, was discussed. Then a linearized model for rotor dynamics based on the precise nonlinear mathematical model was developed. Results of the numerical tests verify the correctness and validity of the proposed linear basic equation.

The Reciprocal Theorem for the Finite Displacement Theory and Its Applications

FU Bao-lian

2015, 36(10): 1019-1034. doi: 10.3879/j.issn.1000-0887.2015.10.002

Abstract(1504) PDF(823)

Abstract:
The reciprocal theorem of 3D linear elasticity for the finite displacement theory was proposed. On the basis of the theorem, the reciprocal theorem of rectangular plates in large defection was derived. Meanwhile, the reciprocal theorem of plate strips in large deflection was directly obtained through simplification of the theorem of the rectangular plates. For applications, the bending of a plate strip in large deflection with 2 ends fixed under uniformly distributed load and the bending of a rectangular plate in large deflection with 4 edges fixed under uniformly distributed load were calculated. The calculation shows, on the basis of the reciprocal theorem of bending thin plates in large deflection, the bending rectangular plates in large deflection can easily be sovled with the aid of the basic solution corresponding to the smalldeflection case.

The Singular Boundary Method for Obliquely Incident Water Wave Passing a Submerged Breakwater

LI Jun-pu, FU Zhuo-jia, CHEN Wen

2015, 36(10): 1035-1044. doi: 10.3879/j.issn.1000-0887.2015.10.003

Abstract(1926) PDF(2373)

Abstract:
The singular boundary method (SBM) was implemented to solve the obliquely incident water wave passing a submerged breakwater. The SBM was a recently emerging boundary-type collocation method with the merits of being meshless, integration-free, mathematically simple and easy-to-program. The accuracy and efficiency of the SBM was first investigated through the benchmark examples in comparison with the boundary element method. Then the effects of the position, size and geometry of the breakwater on the water wave propagation were analyzed through extensive numerical experiments. The numerical results verify that the present SBM provides accurate solutions in good agreement with those of the boundary element method. Then the numerical investigations show that the size of the breakwater has a great effect on the water wave propagation. The larger the dimensionless height of the breakwater is, the more obvious the observed shield effect is on the water wave propagation passing the submerged breakwater. With the increasing dimensionless width of the breakwater, the shield effect of the breakwater first rises and then falls. The slope of the breakwater has no obvious shield effect on the water wave propagation. The absorbing submerged breakwater has smaller transmission coefficient T and larger reflection coefficient R than the rigid submerged breakwater, which means a more obvious shield effect.

An Artificial Viscosity Based on the Subcell-Edged Approximate Riemann Solver

ZHAI Chuan-lei, YONG Heng

2015, 36(10): 1045-1057. doi: 10.3879/j.issn.1000-0887.2015.10.004

Abstract(1398) PDF(887)

Abstract:
The artificial viscosity method is generally used to capture shock waves in the Lagrangian hydrodynamics algorithms, and the properties of the artificial viscosity influence the simulation results essentially. A new artificial viscosity based on the subcell-edged approximate Riemann solver was presented. This new method was prove to have the merits of momentum conservation and satisfaction of entropy inequality. With the introduced limiters for the differences of velocities on the subcell edges, the presented artificial viscosity is able to distinguish the shock wave from the isoentropic compression and satisfy the wave front invariance in the spherical symmetric problems. Various numerical examples demonstrate the robustness and effectiveness of the new artificial viscosity.

Analysis of Non-Newtonian Blood Flow in Stenotic Carotid Artery Under Fluid-Structure Interaction

LIU Ying, YIN Yan-fei, ZHANG De-fa, ZHANG Zhi-liang

2015, 36(10): 1058-1066. doi: 10.3879/j.issn.1000-0887.2015.10.005

Abstract(1307) PDF(919)

Abstract:
The non-Newtonian transient blood flow with fluid-structure interaction was numerically simulated for 6 stenosis ratios of carotid arteries with the computational fluid dynamics method. The effects of the carotid artery stenosis ratio on the hemodynamic performance were investigated to clarify the relationship between the stenosis ratio and the atherosclerotic plaque formation and development in the carotid artery. The results show that, different stenosis ratios of the carotid artery result in obviously dissimilar hemodynamic characteristic distributions. Compared with the stenosis ratios of 0.05, 0.1, 0.2, 0.3 and 0.4, the stenosis ratio of 0.5 corresponds to strikingly larger blood stagnant vortex flow zones around the stenotic section. Under the action of the complex bood flow field, lower wall pressure, abnormal wall shear stress distribution, larger total mesh displacement and higher von Mises stress will occur around this section, where the lipid and fibrin macromolecules may easily deposit due to low-speed blood flow. Meanwhile, low wall pressure may cause obvious‘negative pressure’effects, and in turn insufficient blood supply for brain. Furthermore, the atherosclerotic plaques are liable to rupture and fall off under abnormal wall shear stress distribution, and consequently block the blood vessel in brain. Large vascular von Mises stress may cause stress concentration and rupture of blood vessel, providing favorable conditions for the occurrence of stroke. Therefore, the larger stenosis ratio the carotid artery has, the greater the influence is on the formation and development of atherosclerotic plaques，and the higher the possibility of cerebral ischemic stroke occurs.

Traveling Wave Solutions to the Davey-Stewartson Equation With the Riccati-Bernoulli Sub-ODE Method

YANG Xiao-feng, DENG Zi-chen, WEI Yi

2015, 36(10): 1067-1075. doi: 10.3879/j.issn.1000-0887.2015.10.006

Abstract(1464) PDF(861)

Abstract:
The Riccati-Bernoulli subsidiary ordinary differential equation (sub-ODE) method was proposed to construct the exact traveling wave solutions to the nonlinear partial differential equations (NLPDEs). Through traveling wave transformation, the NLPDE was reduced to a nonlinear ODE. With the aid of the Riccati-Bernoulli sub-ODE, the nonlinear ODE was converted into a set of nonlinear algebraic equations. The exact traveling wave solutions to the NLPDE were obtained as soon as this set of nonlinear algebraic equations were solved. Application of this method to the Davey-Stewartson equation directly gave the exact traveling wave solutions. The Bcklund transformation of the Davey-Stewartson equation was also given. The results were compared with those of the first-integral method. The proposed method is effective and easy to be generalized to deal with other types of nonlinear partial differential equations.

The Soliton Solutions to a Class of Nonlinear Hyperbolic Evolution Equations

FENG Yi-hu, CHEN Xian-feng, MO Jia-qi

2015, 36(10): 1076-1084. doi: 10.3879/j.issn.1000-0887.2015.10.007

Abstract(1244) PDF(932)

Abstract:
A class of nonlinear evolution equations were investigated. With the undetermined functions and functional homotopic mapping methods, the exact solitary solution to the non-disturbed evolution equation and the arbitrary order approximate travelling wave solitary solution to the disturbed evolution equation were obtained. A homotopic mapping was introduced, and an initial approximate function was chosen to find out successively the arbitrary order solitary approximate analytic solutions to the nonlinear hyperbolic evolution equation based on the homotopic mapping theory. With the perturbation method, the examples illustrated the validity and approximation degree of the arbitrary order approximate solutions. A discussion shows the practicability and high accuracy of the approximate solutions obtained with the proposed homotopic mapping method.

Periodic Solutions to the Stochastic-Dynamic Climate Model With Sea-Air Interaction

CHEN Li-juan, LU Shi-ping, XU Jing

2015, 36(10): 1085-1094. doi: 10.3879/j.issn.1000-0887.2015.10.008

Abstract(1034) PDF(738)

Abstract:
Usually, most of the stochastic-dynamic climate models were addressed under the assumption that the stochastic forcing terms were white noises. However, many fast climate variables are expressed as nonlinear stochastic processes other than white noises. The stochastic forcing terms in the sea-air interaction model were improved, and a reasonable model was built accordingly. The Mawhin’s continuation theorem as a very effective and general method to study the existence of periodic solutions to dynamic systems, was applied to the problem of periodic solutions to the proposed stochastic-dynamic climate model with sea-air interaction, in which the stochastic forcing terms were some stochastic processes differring from white noises. The existence of periodic solutions to the model under certain conditions was proved, and the potential application value of the results was discussed.

Some Notes on the Scalarization of Function Δ for Vector Optimization Problems

TANG Li-ping, LI Fei, ZHAO Ke-quan, YANG Xin-min

2015, 36(10): 1095-1106. doi: 10.3879/j.issn.1000-0887.2015.10.009

Abstract(1680) PDF(919)

Abstract:
Recently，Xia Yuan-mei,et al.(Journal of Chongqing Normal University(Natural Science), 2015,32(1): 12-15) studied the ?-properly efficient solutions to vector optimization problems via scalar function Δ in terms of the nonlinear scalarization method, and gave some examples to illustrate their results. It was point out here that theorem 1 established by Xia Yuan-mei,et al. was a special case of Theorem 4.6(i) obtained by Gao, et al.(Journal of Industrial and Management Optimization,2011,7（2）: 483-496), and the proof of Theorem 2 given by Xia Yuan-mei,et al. had some deficiency. Through investigation the nonlinear scalarization of function Δ for the (C, ε)-properly efficient solutions, theorem 2 obtained by Xia Yuan-mei, et al. was proved again rigorously. In the end, some examples in which ?-properly efficient solutions did exist, were given to illustrate the main results.

Global Stability of a Class of Delayed Epidemic Models With Nonlinear Incidence Rates

XIE Ying-chao, CHENG Yan, HE Tian-yu

2015, 36(10): 1107-1116. doi: 10.3879/j.issn.1000-0887.2015.10.010

Abstract(1375) PDF(804)

Abstract:
In view of the demographic effects, the latent period and the complexity of disease spread, the dynamic behavior of a class of delayed SIRS epidemic models with nonlinear incidence rates was investigated. The characteristic equation of the corresponding linearized approximation system was analyzed to prove the local stability of the disease-free equilibrium. By means of the Lyapunov-LaSalle invariant set principle, it was proved that the disease-free equilibrium was globally asymptotically stable when the basic reproduction number was less than 1; and the sufficient conditions were obtained for the global asymptotic stability of the endemic equilibrium when the basic reproduction number was greater than 1. Consequently, the conclusions provide a theoretical reference for the effective prevention and control of the spread of communicable diseases.

2015 Vol. 36, No. 10