Sessions 2022-2023

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Man is a slow, sloppy, and brilliant thinker;
computers are fast, accurate, and stupid.

October 10, 2023

A. S. Smirnov, S. A. Kravchinskiy (Peter the Great SPbPU, IPME RAS)
Oscillations of a double pendulum with a weak nonlinearity
The report considers nonlinear oscillations of a double mathematical pendulum with identical parameters of links and end weights. Exact nonlinear equations of motion of the system are derived, from which a classical linear model of small oscillations is obtained, as well as a weakly nonlinear oscillation model that takes into account cubic nonlinearity. A well-known solution of the problem of small oscillations of a double pendulum is given, which serves as a basis for further research. With the help of asymptotic methods of nonlinear mechanics, an approximate solution is constructed for a weakly nonlinear model under arbitrary initial conditions of motion. It is shown that the found solution has a nontrivial structure and represents polyharmonic oscillations on eight different frequencies. The resulting approximate formulas are accompanied by graphical illustrations that compare the behavior of the angles of deviation of the links of a double pendulum from the vertical when using linear and weakly nonlinear models, as well as the original nonlinear model, which is calculated using numerical integration. The conclusions drawn are of interest for analytical mechanics and theory of oscillations, and they can also be used in practice in applied problems of robotics and biomechatronics.
Smirnov Alexey S. – Assistant Professor at the Higher School of Mechanics and Control Processes (Peter the Great St. Petersburg Polytechnic University), Junior Researcher at the Laboratory of Mechatronics (Institute of Problems of Mechanical Engineering of the Russian Academy of Sciences). Research interests: analytical mechanics, theory of mechanical oscillations, rigid body dynamics, stability of equilibrium and motion, control of mechanical systems, optimization in mechanics.
Kravchinskiy Sergey A. – Master's Student at the Higher School of Mechanics and Control Processes (Peter the Great St. Petersburg Polytechnic University). Research interests: analytical mechanics, theory of mechanical oscillations, computational mechanics, computer engineering. Scientific supervisor - Assistant Professor A. S. Smirnov.
Smirnov Alexey S. – Assistant Professor at the Higher School of Mechanics and Control Processes (Peter the Great St. Petersburg Polytechnic University), Junior Researcher at the Laboratory of Mechatronics (Institute of Problems of Mechanical Engineering of the Russian Academy of Sciences). Research interests: analytical mechanics, theory of mechanical oscillations, rigid body dynamics, stability of equilibrium and motion, control of mechanical systems, optimization in mechanics.