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This paper deals with limit cycles in one degree of freedom systems. The van der Pol equation is an example of an equation describing systems with clear limit cycles in the phase space (displacement-velocity 2 dimensional plane). In this paper, it is shown that a system with nonlinear loading, representing the drag load acting on structures in an o...
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... the van der Pol Equation, the damping term will, for some time, work as negative damping, and pumping energy into the system. The phase plane diagram for the van der Pol equation with μ=1 is shown in Figure 1. The starting point, decided by the initial conditions (ICs) of each trajectory is marked by a square on the figure, with an arrow indicating the direction of the trajectories. ...
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... order to achieve this, the values of a and b in Equation (6) are set to 1,0 and 0,002 respectively. Figure 10 shows the phase plane diagram for four different trajectories, which all spiral towards a limit cycle. ...
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... different trajectories are shown, all spiralling towards the limit cycle. Figure 11: Position vs time plot for the system at resonance subjected to loading from Equation (6). Note that all the trajectories overlap at approximately t = 50 s. ...
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... that all the trajectories overlap at approximately t = 50 s. Figure 11 shows the position vs time plot for the system at resonance with forcing term from Equation (6). The trajectories overlap after approximately 50s, reaching the limit cycle. ...
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... trajectories overlap after approximately 50s, reaching the limit cycle. Figure 12 shows the limit cycles for different values of b. As b increases, the velocity and position amplitudes of the limit cycle increases, as the structure becomes less stiff, resulting in more structural movements. ...
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... results in larger amplitude limit cycles. Figure 13 shows the limit cycles for the system subjected to Equation (6) at resonance (red), at second order resonance (blue), at third order resonance (magenta) and out of resonance (black). Figure 14 shows the position vs time plot for the same trajectories. ...
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... 13 shows the limit cycles for the system subjected to Equation (6) at resonance (red), at second order resonance (blue), at third order resonance (magenta) and out of resonance (black). Figure 14 shows the position vs time plot for the same trajectories. Position vs time plot for systems with different degrees of resonance, subjected to loading from Equation (6). Figure 15 shows the phase plane diagram for systems at varying degrees of resonance, with F0 adjusted to a realistic value of the nonlinear forcing. ...
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... vs time plot for systems with different degrees of resonance, subjected to loading from Equation (6). Figure 15 shows the phase plane diagram for systems at varying degrees of resonance, with F0 adjusted to a realistic value of the nonlinear forcing. The position vs time plot for the same systems are shown in Figure 16. ...
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... vs time plot for systems with different degrees of resonance, subjected to loading from Equation (6). Figure 15 shows the phase plane diagram for systems at varying degrees of resonance, with F0 adjusted to a realistic value of the nonlinear forcing. The position vs time plot for the same systems are shown in Figure 16. It should be noted that the adjustment of F0 reflects the estimation of the relative loading from waves with the associated periods (at resonance the wave period is 2π and at second order resonance the wave period is 6π). ...
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Citations
... The member's velocity dx dt appears in the forcing function and could then represent a damping term. This situation has been considered by the author, see [2], and the solutions are discussed in Sect. 2 below. ...
... Systems with a stable limit cycle will, Fig. 1 Phase plane diagram for a system with nonlinear drag force at resonance, β = 1, Four different trajectories (with different initial conditions) are shown, which all spiral towards the limit cycle marked in red. Notice that systems started with initial conditions inside the limit cycle, evolve outwards towards the limit cycle (Copied from [2]) independent of the starting conditions, settle into a steady trajectory in the phase plane, where there is a balance between generation and dissipation of energy. ...
... As shown in [2], for forcing functions given by (6) and (8), trajectories starting at different initial conditions, all are spiraling towards the limit cycle, see Fig. 1. ...
Abstract A single degree of freedom oscillator system subject to regular sinusoidal loading does exhibit a well-known linear solution. In real situations, we shall add damping as there in any physical system will be a damping mechanism that will limit
the oscillations. In case of forced oscillations in a fluid, the loading and the damping may be nonlinear and the response may then exhibit nonlinear characteristics. A simple drag loading term proportional to the square of the oscillator’s velocity is studied as an example of a system exhibiting limit cycles: When the external loading and the oscillator are in higher order resonances, where the natural period of the loading is an integer of the natural period of the system, resonances will occur. The resulting response of the system is a limit cycle oscillation. Furthermore, for oscillators with non-negligible motions, the motion will act as a nonlinear damping term, that for some situations may be interpreted as “negative damping.” In case of irregular wave loading, the system can get into higher order resonances occasionally (when the periods of the main waves are an integer of the oscillator’s periods), triggering large temporary limit cycle motions. A review of other nonlinear damping models is also given with a brief discussion of the performance of systems exposed to these damping models.
... The method of phase plane presentation will be used to identify limit cycles for certain combinations of initial values. A detailed study of the effects of the parameters of the nonlinear forcing equation is presented in [13]. The main contribution presented in the present paper is to link the theoretical findings to a case example: the strong vibrations of a onetower concrete platform located in the Norwegian Sea, when hit by a very large wave [14]. ...
... In paper [13], Equation (2) is considered, and the results of solutions are given for linear, constant damping, c = 0.5. ...
... The drag force damping term, ⁄ , is proportional to the velocity of the structural displacement, and the drag loading term is proportional to the wave water velocity squared. In [13], we investigated whether the drag forcing can act as a damping term in the system, and also whether it could act as negative damping, enhancing the motions of the structure. It is also of importance in this regard that the term sin |sin | can be linearized using Fourier series transformation, whereby the wave forcing has terms with frequencies at integers of the wave frequency, , potentially at higher order resonance frequencies: ...
In marine engineering, the dynamics of fixed offshore structures (for oil and gas production or for wind turbines) are normally found by modelling of the motion by a classical mass-spring damped system. On slender offshore structures, the loading due to waves is normally calculated by applying a force which consists of two parts: a linear “inertia/ mass force” and a non-linear “drag force” that is proportional to the square of the velocity of the particles in the wave, multiplied by the direction of the wave particle motion. This is the so-called Morison load model. The loading function can be expanded in a Fourier series, and the drag force contribution exhibits higher order harmonic loading terms, potentially in resonance with the natural frequencies of the system. Currents are implemented as constant velocity terms in the loading function. The paper highlights the motion of structures due to non-linear resonant motion in an offshore environment with high wave intensity. It is shown that “burst”/“ringing” type motions could be triggered by the drag force during resonance situations.
... Previous research has been conducted by a number of authors relating to the van der Pol equation [5][6][7][8] in physics [9][10], biology [11], economics [12], etc. [13][14][15]. Amongst them, Verhulst [5] provided a theorem about the order of accuracy of the formal expansion solution with respect to the perturbation factor in the damping term. ...
We investigate the validity of the formal expansion method for solving a second order ordinary differential equation raised from an electrical circuit problem. The formal expansion method approximates the exact solution using a series of solutions. An approximate formal expansion solution is a truncated version of this series. In this paper, we confirm using simulations that the approximate formal expansion solution is valid for a specific interval of domain of the free variable. The accuracy of the formal expansion approximation is guaranteed on the time-scale 1.
Bipolar disorder is a common psychiatric dysfunction characterized by recurrent episodes of mania and depression. The dynamics of the mood of a bipolar person can be seen from the fluctuations in mood change. This study uses the Van Der Pol equation approach to the mood mathematical model of a bipolar type II person. The mathematical model for the mood is in the form of a nonlinear differential equation. By analyzing stability through linierization, a stable solution is obtained towards x = 0 and towards the limit cycle. Furthermore, the existence of the limit cycle is tested using Lienard's theorem. When looking for the amplitude of the limit cycle analytically using the Average Method, the results of this study obtained several interpretations, namely that the solution to the limit cycle with normal amplitude in bipolar with treatment must be greater than dysregulation minus the levels of mood stabilizer nerve cells in the body so that the mood reaches normal levels. Another interpretation is that the greater the abnormality in sleep patterns, the faster the mood swings occur. By knowing the stability of the mathematical model of mood, it is hoped that the appropriate therapy for bipolar disorder can be determined.
Mood swings in patients with bipolar disorder (BD) are difficult to control and can lead to self-harm and suicide. The interaction between the therapist and BD will determine the success of therapy. The interaction model between the therapist and BD begins by reviewing the models that were previously developed using the Systematic Literature Review and Bibliometric methods. The limit of articles used was sourced from the Science Direct, Google Scholar, and Dimensions databases from 2009 to 2022. The results obtained were 67 articles out of a total of 382 articles, which were then re-selected. The results of the selection of the last articles reviewed were 52 articles. Using VOSviewer version 1.6.16, a visualization of the relationship between the quotes “model”, “therapy”, “emotions”, and “bipolar disorder” can be seen. This study also discusses the types of therapy that can be used by BD, as well as treatment innovations and the mathematical model of the therapy itself. The results of this study are expected to help further researchers to develop an interaction model between therapists and BD to improve the quality of life of BD.
Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua.
Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two.
