应用数学和力学

2023 Vol. 44, No. 5

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Dynamics and Control
Fluid Mechanics
Applied Mathematics

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Dynamics and Control

A Fundamental Surface Theory for Kinetic Analogy of Thin Elastic Shells

XUE Yun, CHEN Liqun

2023, 44(5): 489-498. doi: 10.21656/1000-0887.430222

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Abstract:
The generalization of the Kirchhoff kinetic analogy from thin elastic rods to thin elastic shells, namely the generalized Kirchhoff kinetic analogy, needs a corresponding novel expression of the classical surface theory with its fundamental properties described by means of the concept and method of the rigid body dynamics. A rigid orthogonal-axis system and a curvature-twist vector were defined for the non-orthogonal meshing of a surface, and the Euler angles were used to express the attitude of the system and the partial differential geometric equation of the surface. The curvature-twist vector and the Lamé coefficient were applied to depict the 1st and the 2nd basic quadratic forms of the surface, obtain the normal curvature and calculate the principal curvature and the principal direction. The analysis demonstrates the consistency between the new and the classical expressions of the surface theory. The application example of the proposed method shows that, this method can reasonably express the Rodrigues equation, the Weingarten equation, the Gauss equation and the fundamental equations for the surface, and well describe the differential geometry of the surface. This method has the benefits of conciseness and directness, and lays a mathematical foundation for the generalized Kirchhoff kinetic analogy and its further developments.

Hamiltonian Structures and Stability Analysis for Rigid-Liquid Coupled Spacecraft Systems

YI Zhonggui, YUE Baozeng, LIU Feng, LU Tao, DENG Mingle

2023, 44(5): 499-512. doi: 10.21656/1000-0887.430379

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Abstract:
For the dynamics problems of rigid-liquid coupling spacecraft systems with liquid propellant, a 3D rigid pendulum model was used to simulate the nonlinear sloshing behavior of the propellant. On this basis, the Hamiltonian structure of the rigid-liquid coupling spacecraft system was studied, the $ \mathbb{R}^3$ reduction (corresponding to the translation invariance or the bus momentum invariance of the system) and the S_o(3) reduction (corresponding to the rotation invariance or the total angular momentum invariance of the system) of the system were introduced, with the reduced Poisson brackets of the system in reduced space $ \mathfrak{s}_0^*(3) \times \mathfrak{s}_0^*(3) \times {S_0}(3)$ derived. Then, the spin stability characteristics of the rigid-liquid coupled spacecraft system were studied. Firstly, the relative equilibrium of the rigid-liquid coupled spacecraft system was derived under the principle of symmetric criticality. Based on the energy-momentum method and the block diagonalization technology, the spin stability conditions and the Arnold form stability boundaries of the system were derived. Finally, the spin stability domains illustrated in the form of graph were given according to the specific model parameters.

Research on Constraint Following Control of Flexible Joint Manipulators Based on Singular Perturbation

OU Jingsong, LI Rong, YIN Hui, WANG Huajian

2023, 44(5): 513-524. doi: 10.21656/1000-0887.430024

Abstract(566) HTML (183) PDF(69)

Abstract:
For the control of 2-link flexible joint manipulators, a control method based on the singular perturbation theory and the Udwadia-Kalaba (U-K) method was proposed. The control design was implemented by 2 steps. First, the system order was reduced based on the singular perturbation method and the system was divided into fast and slow sub-systems, to simplify the solution process and overcome the system flexibility. Second, the state feedback constraint following control law for the fast and slow sub-systems was designed with the U-K method, to make the constraint following errors of the fast and slow sub-systems converge to zero, even if the system can't initially satisfy the constraints. The proposed method can deal with holonomic and nonholonomic servo constraints at the same time without the auxiliary variables of the Lagrange multiplier and the pseudo generalized velocity. The method was applied to 2-link flexible joint manipulator systems to solve the flexible oscillation and constraint following problems. Through simulations on MATLAB, and was compared with the traditional PID control, to verify the effectiveness and superiority.

Study on Natural Vibration Characteristics of L-Shaped Cantilever Beams With the Differential Quadrature Method

LI Zhichao, HAO Yuxin

2023, 44(5): 525-534. doi: 10.21656/1000-0887.430382

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Abstract:
The L-shaped cantilever beam structure has many unique advantages such as large flexibility, strong designability, full utilization of space and various deformation modes during vibration, and is widely regarded and studied. A differential quadrature method was proposed to solve the natural frequencies and modes of rectangular-section homogeneous slender L-shaped cantilever beams with additional end masses. In the double coordinate systems, the dynamic equations for the L-shaped cantilever beam based on the Euler-Bernoulli beam theory were established. With selected roots of the Chebyshev polynomial as the node coordinates, the Lagrange interpolation basis function was employed, the weight coefficients of each order were solved, and the boundary conditions were considered, to obtain the natural frequencies and modes of all orders of the structure through resolution of the generalized matrix eigenvalue problem. The theoretical solution of the natural frequencies was verified in comparison with the previous theoretical results and the finite element results. Finally, the effects of the end mass, the length ratio, the width and the thickness of the inner and outer beams on the natural vibration characteristics of all orders were discussed. This method can be further applied to the study of related structural vibrations.

Fluid Mechanics

Simulation and Prediction of the Evaporation Process of Ethanol Droplets Impacting High Temperature Wall

MA Xiaojing, ZHOU Xin, TUSONGJIANG Kari, XU Hanwen

2023, 44(5): 535-542. doi: 10.21656/1000-0887.430139

Abstract(469) HTML (248) PDF(61)

Abstract:
The coupled level set and volume of fluid (CLSVOF) method was used to establish a numerical model for ethanol droplets impacting high temperature wall through introduction of the dynamic contact angle to describe the wetting characteristics of the wall surface. The boiling and evaporation process of ethanol droplets impacting high temperature wall was studied and compared with the experimental data. The results show that, at a fixed droplet temperature, the higher the wall temperature is, the stronger the hydrophilicity will be, and the faster the impacting velocity of ethanol droplet is, the earlier the droplet will boil and the shorter the evaporation time will be. Based on this, a prediction model for droplet evaporation was established with the machine learning algorithm, to study the change of the evaporation residual with time after the ethanol droplet collision with the high temperature wall. The optimal prediction model was selected through comparison of the prediction results of different machine learning algorithms with the simulation results.

Study on Collision Characteristics of Rotating Rod Strings in Annulus Fluid With Wellbores Based on Nested Grids

YUE Qianbei, WANG Xiaoxiao, CAO Wen, LIU Yueqiu, LI Hui, XU Yanlu

2023, 44(5): 543-559. doi: 10.21656/1000-0887.430183

Abstract(513) HTML (194) PDF(47)

Abstract:
To solve the contact problem between the rod string immersed in annulus fluid and the wellbore, a numerical solution method for the collision was established based on the nested grid technology. The annulus fluid domain was divided into 2 sub-domains: the background grid and the component grid. Then, the interpolation calculation formula for the flow field boundary transferred information in each nested domain was derived and the coupling between the annulus fluid domain and the rod solid domain was solved with the subdomain-based method. Through comparison of the frontal and oblique collision experiments on spherical particles and wall surface in stationary fluid, the correctness of the proposed numerical method was verified. The results show that, the force and velocity of the collision between the rod string and the wellbore decrease with the fluid viscosity, i.e., the collision intensity is negatively correlated with the fluid viscosity. Moreover, the force and velocity of the collision between the rod string and the wellbore increase with the rotation speed of the rod, i.e., the collision intensity is positively correlated with the rotation speed.

Dynamic Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation

WANG Heyuan, XIAO Shengzhong, MEI Pengfei, ZHANG Xi

2023, 44(5): 560-572. doi: 10.21656/1000-0887.420336

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Abstract:
To reveal the mechanism of the waterwheel chaotic rotation, the dynamic mechanism and the energy conversion of the waterwheel chaotic rotation were studied with the method of moment analysis. The mathematical model for the Malkus waterwheel rotation was transformed into the Kolmogorov system. Based on the different coupling modes of inertia moments, internal moments, dissipation moments and external moments, the main factors and internal dynamic mechanisms of the Malkus waterwheel chaotic rotation were analyzed and discussed with the method of theoretical analysis and numerical simulation. The conversion among the Hamiltonian energy, the kinetic energy and the potential energy was investigated. The relationship between the energies and the Rayleigh number was discussed. The main factors influencing the chaotic rotation are the external moments and the dissipation moments. The analysis and simulation results show that, the lack-of-moment mode cannot lead to the system chaos, but the full-moment mode can, i.e., the waterwheel chaotic rotation will occur only in the existence of all 4 types of moments and when the dissipation and external forces match well. The Casimir function was introduced to analyze the system dynamics and the energy conversion. The bounds for the chaotic attractor were obtained with the Casimir function. The Casimir function reflects the energy conversion and the distances between the orbits and the equilibria. Numerical simulations depict the relationships among them.

Applied Mathematics

The SAV Scheme Based on the Barycentric Interpolation Collocation Method for the Allen-Cahn Equation

HUANG Rong, DENG Yangfang, WENG Zhifeng

2023, 44(5): 573-582. doi: 10.21656/1000-0887.430149

Abstract(624) HTML (152) PDF(92)

Abstract:
The scalar auxiliary variable (SAV) approach combined with the barycentric interpolation collocation method was proposed to solve the 2D Allen-Cahn equation. Two unconditional energy-stable SAV schemes were constructed based on the Crank-Nicolson scheme and the 2nd-order backward difference scheme for discretization in time, respectively, and the barycentric Lagrange interpolation collocation method for discretization in space. Moreover, the approximation properties of the barycentric Lagrange interpolation were presented. Numerical experiments show that the time-convergence rates of the 2 types of SAV schemes are of the 2nd order and both schemes satisfy the energy decay law. Compared with the finite difference method in space, the barycentric Lagrange interpolation collocation scheme features exponential convergence.

New Class of Difference Schemes With Intrinsic Parallelism for the KdV-Burgers Equation

PAN Yueyue, YANG Xiaozhong

2023, 44(5): 583-594. doi: 10.21656/1000-0887.430128

Abstract(481) HTML (149) PDF(74)

Abstract:
The KdV-Burgers equation as a standard equation for turbulent, has a profound physical background and its fast numerical methods are of great practical application value. A new class of parallel difference schemes were proposed for the KdV-Burgers equation. Based on the alternating segment technology, the mixed alternating segment Crank-Nicolson (MASC-N) difference scheme was constructed with the classic Crank-Nicolson (C-N) scheme, the explicit and implicit schemes. The theoretical analyses indicate that, the MASC-N scheme is uniquely solvable, linearly absolutely stable and 2nd-order convergent. Numerical experiments show that, the MASC-N scheme has higher precision and efficiency than the C-N scheme. Compared with the ASE-I and ASC-N difference schemes, the MASC-N parallel difference scheme has the best performance, and can effectively solve the KdV-Burgers equation.

An Alternating Direction Multiplier Method for 4th-Order Variational Inequalities With Curvature Obstacle

ZHANG Linsen, CHENG Lan, ZHANG Shougui

2023, 44(5): 595-604. doi: 10.21656/1000-0887.430243

Abstract(467) HTML (146) PDF(46)

Abstract:
A self-adaptive alternating direction method of multipliers was proposed for the approximation solution of variational inequalities with biharmonic operators and curvature obstacle. An augmented Lagrange functional was introduced with an auxiliary variable to express the curvature function, and a constrained minimization problem equivalent to a saddle-point one was deduced. Then the alternating direction method of multipliers was applied to solve the saddle-point problem. By means of the balance principle and iterative functions, a self-adaptive rule was obtained to adjust the penalty parameter automatically, and improve the computation efficiency. The convergence of this method was proved and the penalty parameter approximation was given in detail with the iterative functions. The numerical results illustrate the effectiveness of the proposed method.

Preset-Time Consensus of Heterogeneous Fractional-Order Nonlinear Multi-Agent Systems

GONG Ping

2023, 44(5): 605-618. doi: 10.21656/1000-0887.430223

Abstract(563) HTML (233) PDF(64)

Abstract:
The preset-time consensus problem of a class of heterogeneous fractional-order nonlinear multi-agent systems was studied. A type of time-varying function-based preset-time fractional integral controllers were designed, to convert the fractional-order nonlinear multi-agent system into a 1st-order nonlinear multi-agent system. Then, by means of the integer-order Lyapunov function method combined with the preset-time control technology, the accurate bipartite consensus control of multi-agent systems with the connected undirected graph and the directed graph containing spanning trees was realized, respectively. The preset time can be preset with the time-varying function, independent of system initial values and parameters. An example verifies the effectiveness of the theoretical results.