Mathematical and computer modeling vibration protection system with damper

Abstract

The mathematical model of the mechanical system, which is a section of the pipeline with built-in motor pump system and damper for vibration damping, has been developed. The mathematical model is represented by a system of partial differential equations of hydrodynamics for moving fluid in the pipeline and a system of two ordinary differential equations for the damping device. Solution the system of equations is performed using analytical and numerical methods. The numerical solution of the equations of hydrodynamics carried out using the finite element method, implemented in the programming environment of mathematical package freeFEM ++.

Mathematical modeling of the guidance system of satellite antenna

“…The solution of nonlinear differential equations [18][19][20][21][22][23][24][25] can be carried out various approximate analytical methods [26][27][28][29][30][31][32][33][34][35]: the method of Van der Pol, the harmonic balance method, the averaging method, the small parameter method, the method of Krylov-Bogolyubov, method of harmonic linearization, the method of Poincare. We obtained an approximate analytical solution of the modified method of harmonic linearization with Chebyshev polynomials [36][37][38][39][40][41][42] 2 2 4 2 4 2 2 2 2 3 3 3 3 11 11 11 22 3 15 3 1 2 3 3 3 3 3 Figure 15 shows graphs of the vertical oscillations of mobile satellite antenna obtained by analytical method (blue), a numerical method (yellow) and the graph the oscillation without vibration protection devices (green).…”

On the equation of fourth order with quadratic nonlinearity

“…For the construction of solutions of nonlinear differential equations in partial derivatives [6][7][8][9][10][11][12] is used different analytical and numerical methods: the perturbation methods, the small parameter method, the separation of variables method, the linearization method, the averaging method, the method of the stretched coordinates, the method of composite expansions, grid methods -the method of finite differences and the finite element method [13][14][15][16][17][18][19].…”

Non-linear vibrations of rubber membrane

“…For the system described by ordinary differential equations, methods of calculation of amplitude-frequency characteristics are well developed. In this area, many different problems solved [15][16][17][18][19][20][21]. For large deformations of shells and membranes to calculate such characteristics are not simply [22][23][24][25][26].…”