Study On Symmetry Theories And Numerical Calculation Of Vehicle Body Vibration Systems

Symmetry theories are described by infinitesimal transformations leaving structural relations unchangeable and play an important role in mathematics, physics, mechanics and other fields. It is closely contacted between symmetries and conserved quantities of dynamical systems.To study symmetries of the system can be used to reduce the order of the differential equations of motion and get the conserved laws. In this paper, the inverse problem of Lagrangian mechanics of the vehicle body vibration system is researched to provide a new symmetry solution to the vehicle body vibration problem. The calculation formula and iterative method on Runge-Kutta via Matlab for numerical simulation, and many significant results are obtained.Firstly, The vertical translation vibration and the pitching vibration around the mass center are addressed by means of Lagrange function to construct a truck body vibration, and the motion equations of this system are obtained. The relative theory and methods of the inverse research of the vehicle body vibration system are obtained, and the Lagrangian of the vehicle body vibration coupled two-dimensional system is also obtained.Secondly, we studied the Noether symmetry and conserved quantities of vehicle body vibration coupled two-dimensional system. Start with the variation of the Hamilton action of the vehicle body vibration system, we study the symmetrical transformations and the quasi-symmetrical transformations of given system through the infinitesimal transformations. And the corresponding Noether identity、Noether theory and Killing equations are obtained. Finally, we deduce the conservation laws of momentum by Noether theorems.Thirdly, we studied the Lie symmetry and conserved quantities of vehicle body vibration coupled two-dimensional system. Start with the differential equations of motion of the vehicle body vibration, the Lie symmetrical transformations, the determining equations of Lie symmetries, structural equation of Lie symmetries and the existent conservation laws of given system through the infinitesimal transformations are studied for the invariance of the differential equations of motion. Finally, we deduce the conservation laws of the vehicle body vibration coupled two-dimensional system by Lie theorems.At last, the numerical simulation is conducted on Runge-Kutta of the vehicle body vibration coupled two-dimensional system by Matlab. First, the motion equation of the two order vehicle body vibration system is transformed into the first order form. Then the calculation formula and iterative method of the given system are given by Runge-Kutta. Ultimately, we simulate by MATLAB, and get the graph of the vertical translation vibration displacement, the graph of the vertical translation vibration velocity, the graph of the angular displacement of the pitching vibration around the mass center, the graph of the angular velocity of the pitching vibration around the mass center.