应用数学和力学

2018 Vol. 39, No. 7

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An Efficient Numerical Method for Computing Dynamic Responses of Periodic Piecewise Linear Systems

HE Dongdong, GAO Qiang, ZHONG Wanxie

2018, 39(7): 737-749. doi: 10.21656/1000-0887.390055

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Abstract:
An efficient method based on the parametric variational principle (PVP) was proposed for computing the dynamic responses of periodic piecewise linear systems with multiple gap-activated springs. Through description of gap-activated springs with the PVP, the complex nonlinear dynamic problem was transformed to a standard linear complementary problem. This method can avoid iterations and updating the stiffness matrix in the computing process and can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method (PIM), an efficient numerical time-integration method was developed to obtain the dynamic responses of the system. This method indicates that there are a large number of identical elements and zero elements in the matrix exponents of a periodic structure, and saves computation load and computer storage by avoiding repeated calculation and storage of these elements. Numerical results validate the proposed method. The dynamic behaviors of a 5-DOF piecewise linear system under harmonic excitations were analyzed, including the stable periodic motion, the quasi-periodic motion and the chaotic motion. In comparison with the Runge-Kutta method, the proposed method has satisfactory correctness and efficiency.

Effective Elastic Properties of Transversely Isotropic Matrix Based Composites

ZHANG Chunchun, WANG Yanchao, HUANG Zhengming

2018, 39(7): 750-765. doi: 10.21656/1000-0887.380267

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Abstract:
One of the main objectives of micromechanics is to predict the effective elastic properties of composites. Most existent explicit micromechanics models are based on an assumption of isotropic matrices and on that only 2-phase constituent materials are involved. In reality, a composite may possess a 3rd interphase between the fiber and the matrix, which is generally transversely isotropic. Accordingly, the prediction of the elastic properties of a 3-phase composite can be achieved through the combination of 2 kinds of 2-phase composites, to which a micromechanics model with transversely isotropic matrix should be applicable. The explicit bridging tensor elements to correlate the internal stresses of a transversely isotropic matrix with those of a reinforcing fiber in a concentric cylinder assemblage (CCA) model were derived firstly. Then this obtained bridging tensor was used to deduce analytical formulae for all the 5 effective elastic moduli of the composite made with the transversely isotropic matrix. An extension of the bridging model applicable to fiber reinforced transversely isotropic matrix composites was achieved as well. With properly chosen bridging parameters, the predicted elastic moduli of the composite with the 2 models are quite close to each other.

T-Stress in a Centrally Cracked Brazilian Disk Under Nonuniform Pressure Load

PENG Fan, DONG Shiming

2018, 39(7): 766-775. doi: 10.21656/1000-0887.380136

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Abstract:
In an engineering structure, the rock is often subjected to nonuniform load; in addition, there is a small distribution angle under a concentrated force in actual experimental study, and the load distribution also should be nonuniform. Based on this case, a class of nonuniform loads were proposed in the trigonometric function form. Then the maximum loading pressure appeared in the middle of the distribution angle and gradually became small to both sides until 0. By means of the centrally cracked Brazilian disk (CCBD) T-stress formula under a radial concentrated load, the analytical T-stress solution under this kind of nonuniform load was acquired via the integral in the range of the distribution angle, meanwhile the finite element simulation was conducted to obtain the numerical solution. There is a very good consistency between the numerical results and the analytical results, which verifies the accuracy of each other. Compared with those under the uniform load, the dimensionless T-stress values under this nonuniform load are closer to the values under the concentrated load when the other condition is the same, and the discrepancy between the stresses under the nonuniform load and the concentrated load is very small. The further calculated results show that it is accurate and reasonable to use the T-stress formula for the CCBD subjected to the concentrated forces in actual testing.

The Noether Theorem for Nonlinear Optimal Control Problems of Mechanical Multibody System Dynamics

ZHENG Mingliang

2018, 39(7): 776-784. doi: 10.21656/1000-0887.380295

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Abstract:
A Noether-type conservation law for the nonlinear optimal control problems of mechanical multibody system dynamics was proposed based on the group invariance principle. The controlled mechanical multi-rigid-body systems under ideal holonomic constraints were studied, and the dynamic Euler-Lagrange equations were expressed in the form of the state space with the augmented vector method. The state equations, adjoint equations and governing equations for the optimal solution to the optimal control problem were obtained with the variational method. The Noether symmetric infinitesimal transformation with time, state variables, covariate variables and control variables was applied to the system performance index functional, then the conservation laws of the optimal solution equations were obtained, and the optimal solution relation was expressed in the form of a set of algebraic equations, which lays a solid foundation for the integral method and various numerical algorithms of the optimal solution. Finally, an example about the optimal energy control of the nonlinear dynamics of the mechanical arm under the basic vibration was given to illustrate the correctness of the proposed symmetry method.

Linear Theory of Velocity Redistribution for Flow in Compound Open Channels Under Inflow With Small Oblique Angles

XU Dong, HUANG Xionghe, JI Chunning, BAI Yuchuan

2018, 39(7): 785-797. doi: 10.21656/1000-0887.380183

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Abstract:
The directions of flow in natural compound river channels are frequently influenced by the discharge and floodplainmain channel configuration, which may lead to a small intersection angle between the flow and the channel. Hence, lots of existent theories for floodplain flows under the assumption of uniform straight channels may be inapplicable. To investigate the influence of oblique inflow, shallow water equations were used to describe flow motion in compound open channels. The perturbation method was used with a small parameter of the oblique angle to derive the analytical solution for flow in compound open channels under inflow with small oblique angles. The solutions were verified with numerical simulation results and the flow velocity distributions agreed well. Theoretical results show that the obliquity of inlet flow lead to asymmetrical velocity distribution across the cross section of a compound channel, which is featured by an increase of velocity in the upstream side and a decrease in the downstream side of the main channel. Under an oblique angle of 13° and a floodplainmain channel water depth ratio of 3∶8, the flow velocity deviation may reach 21.8% of the value in the straight flow case. Such a deviation increases with the decreasing of the floodplainmain channel water depth ratio. The proposed modification to flow velocity distribution may improve the accuracy of hydrodynamic conditions for further research on sediment transport and evolutions of compound channel rivers.

Research of Wind Speed Time Series Based on the Hurst Exponent

YUAN Quanyong, YANG Yang, LI Chun, KAN Wei, YE Kehua

2018, 39(7): 798-810. doi: 10.21656/1000-0887.380154

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Abstract:
In order to investigate self-similarity and long-rang dependence of wind speed time series, the rescaled range analysis and the detrended fluctuation analysis (DFA) were carried out to calculate the Hurst exponent of the wind speed time series. The power spectral density was analyzed and the spectral indexes were calculated. The results show that, the Hurst exponents calculated with the both methods are close to 1, which means that the wind speed time series has strong self-similarity and long-range positive dependence. However, the Hurst indexes from the 2 methods are different, which means that the DFA method could reflect the power-law feature of the non-stationary wind speed time series. Furthermore, the results of the Hurst exponents and the spectral indexes show that the wind speed fluctuation belongs to the “1/f fluctuation”. The conclusions offer some references for further study on fractal chaos and short-time wind speed prediction.

Research on Influential Factors of Initiation Pressure in Fracture Formation for Perforated Boreholes

DING Yi, LIU Xiangjun, LUO Pingya

2018, 39(7): 811-820. doi: 10.21656/1000-0887.380209

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Abstract:
Hydraulic fracturing is the most common and effective stimulation treatment. To better the fracturing design and improve the stimulation treatment, the mechanism of fracture initiation and propagation was investigated. In fracture formation, lots of natural fractures exist, having strong anisotropy and causing extreme complication and uncertainty of fracture initiation and propagation. Therefore, the mechanism of fracture initiation was stressed with the tensile criterion to establish the initiation pressure model in view of the natural fracture number, the occurrence and the intersection between perforation and fracture. The results indicate that, the natural fracture tends to cause initiation along the fracture plane and decrease the initiation pressure. In particular, with an increasing number of fracture planes, this decline will be more significant. Besides, the distribution of initiation pressure is related to several engineering parameters. For a high perforation azimuth and a low wellbore deviation, the initiation pressure will be relatively larger. The findings can offer reference for hydraulic fracturing in application.

State Estimation for Delayed Neural Networks With Stochastic SampledData Control

ZENG Deqiang, WU Kaiteng, SONG Qiankun, ZHANG Ruimei, ZHONG Shouming

2018, 39(7): 821-832. doi: 10.21656/1000-0887.380273

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Abstract:
The problem of the state estimation for delayed neural networks with stochastic sampleddata control was studied. First, a unified probability framework involving the stochastic sampling interval and the sampling input delay was proposed. Second, based on this unified probability framework, a new LyapunovKrasovskii functional (LKF) with some new terms was constructed. Third, with this LKF and some inequality technologies, a less conservative criterion was established, which can ensure the stochastic stability of the error system. The desired state estimator was designed. Finally, numerical simulation results show the effectiveness and advantages of the proposed method.

A Spectral Element Method for Transmission Eigenvalue Problems of the Helmholtz Equation

DAI Hai, PAN Wenfeng

2018, 39(7): 833-840. doi: 10.21656/1000-0887.380327

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Abstract:
A Chebyshev spectral element method for the transmission eigenvalue problems of the Helmholtz equation was proposed, which combined the flexibility of the finite element method to deal with the boundary and region and the fast convergence of the spectral method. By means of the principle of weighted residuals, the basic theory and mathematical formulae of the Chebyshev spectral element method for transmission eigenvalue problems were obtained. The original problem was transformed into quadratic eigenvalue problems. Furthermore, several numerical examples were given to illustrate the effectiveness of the proposed method.

Non-Equidistant GM(1,1) Models Based on Fractional-Order Reverse Accumulation and the Application

ZENG Liang

2018, 39(7): 841-854. doi: 10.21656/1000-0887.380252

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Abstract:
For the prediction of non-equidistant decreasing series, a non-equidistant GM(1,1) model based on the 1st-order reverse accumulation was constructed, and the least square solutions of the model parameters and the discrete time response functions applicable to prediction were given. In order to further improve the prediction accuracy, a fractional-order reverse accumulation non-equidistant GM(1,1) model was proposed. With the objective of minimizing the average relative error of simulation, a nonlinear programming model was established to obtain the optimal order. Finally, numerical simulation and an example of the prediction of the fatigue strength of Ti alloy were given to verify the validity and practicability of the proposed model.