A regularization method for solving dynamic problems with singular configuration

In the simulation process for multi-body systems, the generated redundant constraints will result in ill-conditioned dynamic equations, which are not good for stable simulations when the system motion proceeds near a singular configuration. In order to overcome the singularity problems, the paper presents a regularization method with an explicit expression based on Gauss principle, which does not need to eliminate the constraint violation after each iteration step compared with the traditional methods. Then the effectiveness and stability are demonstrated through two numerical examples, a slider-crank mechanism and a planar four-bar linkage. Simulation results obtained with the proposed method are analyzed and compared with augmented Lagrangian formulation and the null space formulation in terms of constraints violation, drift mechanical energy and computational efficiency, which shows that the proposed method is suitable to perform efficient and stable dynamic simulations for multi-body systems with singular configurations.

Application of Gauss principle of least constraint in multibody systems with redundant constraints

Redundancy in the constrained mechanical systems often occurs in complex multibody mechanic systems in the existence of excessive constraints and singular positions due to system motion. In this work, Gauss principle of least constraint (GPLC) is applied to solve the dynamic motion of system with redundant constraints without changing the physics of system. Furthermore, the particle swarm optimization method is used to handle the minimization optimization problem. Eventually, the effectiveness of GPLC is validated through the dynamic modelling and simulation of two numerical examples (a planar four-bar mechanism and a spatial parallelogram mechanism). The simulation results are analyzed and compared with those obtained from Udwaia-Phohomsiri formulation and augmented Lagrangian formulation, in terms of constraint violation, computational efficiency and variation of the mechanical energy. From the viewpoint of computational efficiency and accuracy, GPLC can be regarded as a practical real-time simulation method for multibody systems with redundant constraints.

DUAS ABORDAGENS PARA A SOLUÇÃO DE PROBLEMAS: SOFTWARE E HISTÓRIA DA CIÊNCIA.

ResumoNosso objetivo é fornecer aos estudantes de física e engenharia dois caminhos - formal e conceitual - para resolução de problemas, e aos estudantes de cursos de outras áreas uma estratégia mais intuitiva e mais fácil de memorizar no longo prazo. Para atingir este objetivo o mesmo problema - a máquina de Atwood composta com quatro corpos - é abordado de duas formas diferentes. A abordagem formal enfoca a obtenção das equações de movimento usando a segunda lei de Newton, as restrições geométricas ou vínculos e um software de computação algébrica. A abordagem conceitual consiste em uma transferência de conhecimento da máquina de Atwood convencional. Neste caso, o raciocínio desenvolvido para resolver o problema é o mesmo que o usado para a máquina simples, só que aplicado mais de uma vez. O mesmo vale para o uso do princípio de Gauss, tal como era compreendido durante o século XIX. Um exemplo numérico mostrará que todas as abordagens levam ao mesmo resultado. Palavras-chave: máquina de Atwood; Newton; Gauss; física conceitual; habilidades formais. Abstract The aim of the present study is to provide Physics and Engineering students with two paths regarding problem solving and non-physics students with a more intuitive solving strategy and easier to memorize in the long term. To achieve this goal, the same problem is approached in two different ways: one, formal and the other, conceptual. The problem is the compound Atwood machine with four bodies. The formal approach focuses on obtaining the equations of motion using Newton's second law, stresses the role of geometric constraints and uses an algebraic computation software. The conceptual approach consists of a knowledge transfer from the simple Atwood machine problem. In this case, the thinking process for solving the compound machine problem is the same as for the simple one but it is applied more than once. The same goes for the use made of the Gauss principle, as it was understood in the 19th century. A numerical example will show that all approaches lead to the same result. Keywords: Atwood machine; Newton; Gauss; conceptual Physics; formal skills