应用数学和力学

2018 Vol. 39, No. 1

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Global Sensitivity Analysis of Structural Dynamic Characteristics Considering Metamodel Uncertainty

WAN Huaping, ZHONG Jian, REN Weixin

2018, 39(1): 1-10. doi: 10.21656/1000-0887.380018

Abstract(1512) PDF(859)

Abstract:
Uncertainty of structural parameters will unavoidably lead to uncertainty of structural natural frequencies. The global sensitvity analysis (GSA) is an effective approach to quantify the contributions of individual parameters to the induced uncertainty of dynamic characteristics. However, the GSA has the issue of high computational cost that needs to be addressed. The fast-running Gaussian process model (GPM) was used as a surrogate for the costly computer models, to reduce the computational burden of the GSA. Moreover, the influence of the metamodel uncertainty associated with the GPM was taken into account. The effectiveness of the presented GPM-based method for the GSA was verified with a test function. Finally the GPM-based approach was applied to the GSA of structural dynamic characteristics of the Anqing Yangtze River Railway Bridge.

Application of the Bi-Potential Integration Algorithm to Non-Associated Materials

ZHOU Yangjing, FENG Zhiqiang, PENG Lei

2018, 39(1): 11-28. doi: 10.21656/1000-0887.380139

Abstract(1479) PDF(1543)

Abstract:
Given the formulation of material free energy, the bi-potential theory allows one to divide standard materials into 2 main categories: explicit or implicit. The Drucker-Prager (D-P) model was taken as an example, which typically describes non-associated materials through the constitutive cones. With a new description of the orthogonal law, the dual constitutive cones were proposed, which not only satisfy the constitutive law of the D-P model, but also meet the requirements of the implicit flow rules. On the basis of the D-P model, and according to the bi-potential theory, 5 forms of bi-potential functions were established: the elastic stage in rate form, the plastic stage in rate form, the elastic stage in incremental form, the plastic stage in incremental form and the elasto-plastic stage in incremental form. The bi-potential integration algorithm was then obtained. A numerical simulation example was given to verify the accuracy and stability of the bi-potential integration algorithm.

Dynamics Modeling and Motion Planning for Snakeboard Systems Based on Differential Geometry

YAO Qijia, GE Xinsheng

2018, 39(1): 29-40. doi: 10.21656/1000-0887.380107

Abstract(1187) PDF(884)

Abstract:
Dynamics modeling and motion planning for snakeboard systems were investigated, and a hybrid optimization strategy based on the genetic algorithm (GA) and the Gauss pseudospectral method (GPM) was presented. Firstly, the Euler-Lagrange equations for the snakeboard system were derived based on the Riemannian manifold and the affine connection theory in differential geometry. The configuration space of the snakeboard corresponds to the manifold space, the velocity space corresponds to the tangent space, the torque space corresponds to the cotangent space, and the inertia matrix provides a Riemannian measure on the manifold. The set of admissible velocities were represented by the appropriate bases to simplify the kinematics equations. Then the optimal motion planning problem was transformed into a nonlinear programming problem with the GPM. The optimal trajectory and the optimal control inputs were obtained with the sequential quadratic programming (SQP) algorithm. The GA was applied to generate the initial values of the GPM. Finally, through numerical simulation, the optimal trajectory agrees well with actual conditions, and the control inputs match the various constraints closely. The results demonstrate the effectiveness of the proposed method.

A Concurrent Microstructured Model for Complex Solids and Existence of Solitary Waves

NARANMANDULA

2018, 39(1): 41-49. doi: 10.21656/1000-0887.380074

Abstract(839) PDF(634)

Abstract:
A concurrent microstructured nonlinear model involving 2 kinds of microscale nonlinear effects was established to describe the motion of complex solids with 2 microstructures of different properties. The existence of asymmetric solitary waves was proved according to the qualitative analysis theory and the bifurcation theory for dynamic systems under certain conditions in concurrent microstructured solids, and the existence conditions for the asymmetric solitary waves were given. The results indicate that the symmetry properties of solitary waves were influenced by the 2 kinds of microscale nonlinear effects simultaneously. The asymmetric properties of solitary waves are more obvious when the microscale nonlinear effects become stronger. Finally, the results of qualitative analysis were validated further through numerical simulation.

Random Sound Radiation of Thin Plates Under Turbulent Boundary Layer Excitations With a Symplectic Method

PAN Chenge, LI Yuyin, ZHANG Yahui

2018, 39(1): 50-63. doi: 10.21656/1000-0887.380151

Abstract(1248) PDF(702)

Abstract:
The random sound radiation of thin plates subjected to turbulent boundary layer (TBL) excitations was studied in the symplectic duality system. Firstly, the cross power spectral density of the TBL was represented by a Fourier series, and the problem of the random sound radiation of structures excited by a random field was reduced to solve the deterministic response function, i.e. the structural response to a spatial and temporal harmonic pressure of unit magnitude. Secondly, the free vibration analysis of thin plates was introduced to the symplectic duality system, then a symplectic eigenproblem was formed with the method of separation of variables. Finally, the decoupled governing equations were derived through expansion of the response and excitation vectors in the symplectic space, to reduce the difficulty of solving the equations, and the symplectic analytical solution was obtained. In contrast to the modal decomposition method (MDM), the presented method is formulated in the symplectic duality system and does not need modal truncation, hence the computations are of high precision. In the numerical examples, the harmonic response functions for the thin plate were studied, and a comparison was made with the MDM to verify the effectiveness of the presented method. Thereafter, the sound pressure levels (SPL) of the power spectral density of the sound pressure response to the TBL were obtained, the convergence induced by the Fourier series expansions was examined, and the directivity functions of the radiation sound field were extensively investigated.

Dynamic Mode Decomposition of Horseshoe Vortex Flow Structures Around Square PrismPlate Junctions

WANG Jianming, MING Xiaojie, WANG Han, MA Yang, WANG Chengjun

2018, 39(1): 64-76. doi: 10.21656/1000-0887.380125

Abstract(1037) PDF(847)

Abstract:
The multi-frequency phenomenon exists with the horseshoe vortex flow structure in junction flow. In order to demonstrate the oscillatory characteristics and the basic dynamics, the periodical oscillatory horseshoe vortex system was simulated near the square prism-plate junction. The periodical oscillatory flow of the horseshoe vortex makes a multi-frequency phenomenon. Then the velocity field in the plane of symmetry was decomposed with the dynamic mode decomposition (DMD) method. The 1st 3 modes were reconstructed and superposed on the mean flow mode respectively, and their progressive evolutions were analyzed. The results show that all the modes reveal the merging of horseshoe vortices in different sizes, and develop in different styles.

Soil Infiltration Rates and Hydrology Model Classifications Based on the Hausdorff Fractal Derivative Richards Equation

CHEN Wen, LIANG Yingjie, YANG Xu

2018, 39(1): 77-82. doi: 10.21656/1000-0887.380101

Abstract(1025) PDF(744)

Abstract:
The time-dependent soil infiltration rate was derived based on the Hausdorff fractal derivative Richards equation. This model requires only 2 parameters, among which the Hausdorff derivative order characterizes the underlying water transport environment property in heterogeneous soil, while the pore size distribution index categorizes different hydrological models. Two applications show that a fractal order α≠1 of the Hausdorff derivative indicates the history-dependent process. Namely, a lower α exhibits slower decay of the infiltration rate with time evolution, reflecting stronger memory and further departure from the classical integer-order models. It is also observed that a smaller pore size distribution index indicates slower decay of the infiltration rate, making a fundamental index of soil infiltration.

Research on Grouting Infiltration Mechanism for Time-Dependent Viscous Slurry Considering Effective Void Ratios in Saturated Clay

KOU Lei, XU Jianguo, WANG Bo

2018, 39(1): 83-91. doi: 10.21656/1000-0887.380050

Abstract(1181) PDF(665)

Abstract:
The calculation formula of effective void ratios was established according to the characteristic parameters of soil in view of the influence of the strong bound water layer on the permeability characteristics of clay. The empirical formula of permeability coefficients of clay was obtained through modification of the empirical formula of the Kozeny-Carman permeability coefficient. Based on the permeability coefficient of clay, the formulas for calculating the cylindrical and spherical diffusion radii and grouting pressures of time-dependent viscous slurry in saturated clay were derived. The empirical formula of permeability coefficients of clay was validated, then the diffusion radii with different grouting pressures and grouting times were compared between cases with or without the effective void ratio, and between cases with or without the time-dependent slurry viscosity. The results show that the diffusion radius with the natural void ratio is over 1.5 times larger than that with the effective void ratio, and the difference between them increases with the grouting pressure and the grouting time, so the permeability of clay is under the significant effects of the strong bound water layer.

A Theoretical Model for MagnetoMechanical Coupling Behaviors of Magnetorheological Elastomers

LI Xu, WAN Qiang, SHI Ping’an

2018, 39(1): 92-103. doi: 10.21656/1000-0887.380021

Abstract(1189) PDF(1415)

Abstract:
Based on the magnetic dipole interaction theory, a theoretical model was proposed to describe the magneto-mechanical coupling behaviors of magnetorheological elastomers with the principle of minimum potential energy. In this model, the fully coupled interaction among all particles and chains was considered according to the micro-structure of magnetorheological elastomers. The energy equations of magnetic interaction and the elastic potential energy equations based on the Mooney-Rivlin model were derived respectively. Then a theoretical model was developed to describe the stress-strain relationship of magnetorheological elastomers under uniaxial load. This model agrees well with existing experimental data and can be used to explain the micro-mechanism of magneto-induced stress. The results show that the mechanism of magneto-induced stress is closely related to the inner micro-structure, and the nonlinear property of magneto-induced stress mainly depends on the interaction among both particles and chains.

An Operational Matrix Method for Fractional Advection-Diffusion Equations With Variable Coefficients

ZHU Xiaogang, NIE Yufeng

2018, 39(1): 104-112. doi: 10.21656/1000-0887.380041

Abstract(1178) PDF(763)

Abstract:
A numerical method for the Caputo-fractional advection-diffusion equations with variable coefficients was investigated. Based on Chebyshev cardinal functions, an effective operational matrix was derived for Riemann-Liouville fractional integral, and with it, a new operational matrix method was proposed for the fractional advection-diffusion equations with variable coefficients. This method reduces the equation to an algebraic system and is characterized by small computing cost and easy programming. The numerical results and the comparisons with some existing methods illustrate that it is convergent and possesses advantages in accuracy.

Singular Perturbation Solutions to 1D Stochastic Burgers Equations Under Weak Noises

BAO Liping, HONG Wenzhen

2018, 39(1): 113-122. doi: 10.21656/1000-0887.380068

Abstract(1354) PDF(670)

Abstract:
The singular perturbation solutions to a class of bounded stochastic Burgers equations under colored noises were discussed, of which the volatility followed the weak noise Ornstein-Uhlenbeck (O-U) process. With the Kolmogorov equation satisfied by the probability density function of wave motion, the Kolmogorov equation satisfied by the expectation of the random Burgers equation was obtained. Since the initial boundary conditions for the Kolmogorov equation relate to a class of deterministic solutions to the Burgers equation, this problem is actually a simultaneous form of the Burgers equation and the Kolmogorov equation. Firstly, the regular asymptotic expansion of a class of deterministic Burgers equations was given. Based on the Schauder estimates and the Ascoli-Arzela theorem, boundedness and existence of the asymptotic solutions to the nonlinear parabolic equations were proved; moreover, according to the Lax-Milgram theorem, boundedness and existence of the asymptotic solutions to the linear parabolic equations were proved. The formal asymptotic solution of wave expectation was obtained. Secondly, with the singular perturbation theory, the asymptotic expansion of singular perturbation and the boundary layer correction of a class of expected equations were got. The existence and boundedness of the asymptotic solutions to the boundary layer functions were obtained according to the theory of linear partial differential equations. By means of the extremum principle and the De-Giorgi iterative techniques, the boundedness of the remainder terms of the asymtotic solutions of wave velocity and wave expectation was proved respectively, and the uniformly valid estimate for the asymptotic solution of the system was obtained.