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Van der Pol–Duffing Oscillator Equation
Published in Snehashish Chakraverty, Susmita Mall, Artificial Neural Networks for Engineers and Scientists, 2017
Snehashish Chakraverty, Susmita Mall
The Van der Pol–Duffing oscillator equation is a classical nonlinear oscillator, which is a very useful mathematical model for understanding different engineering problems. This equation is widely used to model various physical problems, namely, electrical circuits, electronics, mechanics, etc. [1]. The Van der Pol oscillator equation was proposed by a Dutch scientist Balthazar Van der Pol [2], which describes triode oscillations in electrical circuits. The Van der Pol–Duffing oscillator is a classical example of a self-oscillatory system and is now considered as a very important model to describe a variety of physical systems. Also this equation describes self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. The nonlinear Duffing oscillator equation and the Van der Pol–Duffing oscillator equations are difficult to solve analytically. In recent years, various types of numerical and perturbation techniques, such as Euler, Runge–Kutta, homotopy perturbation, linearization, and variational iteration methods, have been used to solve the nonlinear equation.
Marine propulsion shafting: A study of whirling vibrations
Published in C. Guedes Soares, T.A. Santos, Progress in Maritime Technology and Engineering, 2018
S. Busquier, S. Martinez, M.J. Legaz
The Van der Pol oscillator is a dynamical system which includes positive feedback and a nonlinear damping term. In its original application, at the turn of the 20th century, it was used in the electrical field as an electrical oscillator with a nonlinear element was the forerunner of the early commercial radios. A circuit of this type helps small oscillations and causes the large ones to damp.
Oscillators
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
In electronics, to develop the calculations and get insights on the system working principle, the behavior of the active component is often assumed locally linear, but globally active components are always nonlinear. The van der Pol oscillator is one of the examples where the nonlinearity is fundamental to achieve an emergent behavior.
Nonlinear dynamic characteristics analysis of the surge motion of a tender-assisted drilling operation system
Published in Ships and Offshore Structures, 2023
Zhuang Kang, Rui Chang, Shangmao Ai
The phase differences between two signals are captured by employing the Hilbert transform (Pikovsky et al., 2003). Pikovsky et al. (2003) defined three synchronisation states of a Van der Pol oscillator by using phase dynamics. The phase differences between motion responses and excitations corresponding to the synchronisation states are divided into: (i) phase locking, (ii) phase trapping, and (iii) phase drifting and slipping, as shown in Figure 20. The yellow line of the phase difference is locked to a constant value, which is referred to as phase locking. The phase dynamics of the blue line, presenting an oscillatory trend around a constant value, are characterised by phase trapping. The behaviour in which the phase difference keeps a slow drift for a period of time and then shows a fast slip is defined as phase drifting and slipping, which can be understood by viewing the green line. The last state of phase difference, represented by the black line, is near a slash and shows no phase slip (Ren et al., 2019), which is classified as the other state in the following discussion.
Data-driven feedback stabilisation of nonlinear systems: Koopman-based model predictive control
Published in International Journal of Control, 2023
Abhinav Narasingam, Sang Hwan Son, Joseph Sang-Il Kwon
In our first example, we consider the Van der Pol oscillator which is described by the following equations: At u = 0, the unforced dynamics of the Van der Pol oscillator are characterised by a limit cycle with an unstable equilibrium point at the origin. We will see whether the proposed Koopman LMPC is able to stabilise the system at the origin. First, the data required to build the Koopman bilinear model is generated by simulating the unforced system of (42). The simulations were initialised uniformly over a circle around the origin, and a number of trajectories for 10 s were collected with a sampling time of s, i.e. time-series samples per trajectory. In the next step, the states were lifted to the high-dimensional space by using monomials of degree 5 as the dictionary functions , i.e.. This results in a lifted system of dimension , and the system matrix Λ was constructed using the algorithm described in Section 2.3. To determine the B matrix in the controlled setting, the relation between the Koopman eigenfunctions and dictionary functions was used as shown in (21). The derivatives of the eigenfunctions were computed using the symbolic toolbox in MATLAB. This completes the identification of the Koopman bilinear model of (14).
Filtering adaptive tracking control for uncertain switched multivariable nonlinear systems
Published in International Journal of Control, 2022
The control problem of Van der Pol oscillator that can represent various types of electrical circuits consisting of resisters, inductance coils a capacitor, and a triode with two DC power sources. The balance equation of the switched nonlinear system is described as below: In the simulation, the actuator faults are chosen as during and during , during , during . The tracking command is . In the implementation of the filtering adaptive tracking controller, the sampling time step is chosen as . The low-pass filters are chosen as and .