Lili Xia

Modeling of dynamic systems with nonholonomic constraints usually involves constraint multipliers. Consequently, the dynamic equations in the laboratory coordinate system have a complex form, and as a result, the corresponding numerical algorithms need to be improved in terms of both efficiency and accuracy. This paper addresses establishing the mathematical model of the hydrodynamic sleigh in the Hamel framework. Firstly, the Lie symmetry and the Noether theorem conserved quantities of classic Chaplygin sleigh in which the inertial frame is reviewed. Based on the symmetries and the nonholonomic constraints, the frame of the sleigh can be directly realized in the algebraic space. Based on the mutual coupling mechanism between the fluid and the sleigh in a potential flow environment, the reduced equations in the moving frame are proposed in nonintegrable constraint distributions. The corresponding Hamel integrator is constructed based on the discrete variational principle. For the sleigh model in potential flow, the Hamel integrator is used to verify the feasibility of parameter control based on rotation angles and mass distribution, and to obtain the dynamic characteristics of the sleigh blade with both a rotational offset and translational offset. Numerical results indicate that the modeling method in the Hamel framework provides a more concise and efficient approach for exploring the dynamic behavior of the hydrodynamic sleigh.

The optimization of main beam and side lobe performance in array signal processing can be achieved by adjusting the weighting coefficient of each array element. In this paper, the influence of the variation law of the weighting coefficient on the beam pattern is researched. The uniform linear array is taken as an example and the simulation of the uniform weighting, discrete prolate spheroidal sequences (DPSS), Kaiser weighting is conducted. The DPSS and Kaiser weighting are the form of first increasing then decreasing. Then a kind of quadratic function weighting with opposite variation trend is proposed, which decreases first and then increases. The factors affecting the beam pattern performance are simulated and analyzed. Finally, the simulation results of the above weighting methods are compared and comprehensively analyzed. The simulation results show that the trend of first increasing and then decreasing can better suppress the side lobe, but the directivity becomes worse. Additionally, due to the influence of the number of array elements and the direction of incoming waves, the DPSS weighting does not always meet the trend of increasing first and then decreasing, which will also affect the ability of suppressing the side lobe. The quadratic function weighting can achieve a narrower main beam, but there will be higher side lobe at the same time. A new method to generate narrow main beam can be acquired by this way.

Perturbations to Mei symmetry and the numerical algorithm of disturbed nonholonomic systems are studied under total variational discretization. The discrete equations on regular lattices of nonholonomic systems in the undisturbed and the disturbed cases are presented. The determining equations of Mei symmetry are established for undisturbed and disturbed systems. The exact invariants of Noether type led by Mei symmetry for undisturbed nonholonomic systems are given under infinitesimal transformations of Lie groups. For discrete disturbed nonholonomic systems, the condition of existence of adiabatic invariants led by perturbation to Mei symmetry and their forms are presented. The numerical simulations demonstrate that the geometric numerical algorithm has a higher precision and longer time stability than the standard numerical method.

Geometry modeling methods can conserve the geometry characters of a system, which helps the dynamic equations more concisely and is good for long simulations. Reduced attitude, Lie group and Lie algebra are three different expressions of geometry. Models for the dynamics of a planer pendulum and a 3D pendulum were built with these three geometry expressions. According to the variation method, the dynamics models as ordinary differential equations were transformed into nonlinear equations which are solved by Newton iteration. The simulation results show that Lie group and Lie algebra calculations can conserve the geometric structure, but have different long-time behavior. The complete Lie group expression has the best long simulation behavior and has the lowest sensitivity to the time step in both planer and 3D pendulum simulations, because it saves the complete geometry of the system in the dynamics model.

Symmetry preserving difference schemes approximating equations of Hamiltonian systems are presented in this paper. For holonomic systems in the Hamiltonian framework, the symmetrical operators are obtained by solving the determining equations of Lie symmetry with the Maple procedure. The difference type of symmetry preserving invariants are constructed based on the three points of the lattice and the characteristic equations. The difference scheme is constructed by using these discrete invariants. An example is presented to illustrate the applications of the results. The solutions of the invariant numerical schemes are compared to the noninvariant ones, the standard and the exact solutions.