Van der Pol Oscillator. Technical Applications
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der Pol Oscillator. Technical Applications V. Ph. Zhuravlev Ishlinsky Institute for Problems in Mechanics RAS, Moscow, 119526 Russia e-mail: [email protected] Received December 26, 2017; revised December 26, 2017; accepted January 15, 2018
Abstract—Van der Pol equations describing self-oscillations in a quasilinear one-dimensional oscillator are generalized to the case when the generating isotropic oscillator has an arbitrary number of degrees of freedom. Two-dimensional (flat) and three-dimensional (spatial) cases are specifically considered. In contrast to the classical problem, in which a given amplitude of oscillations was stabilized, in the general case it is possible to stabilize not only the oscillation energy, but also the area of a flat elliptical trajectory, its orientation in space, the frequency of the oscillatory process, etc. The technical applications of the respective models are indicated. Keywords: wave solid-state gyroscope, strapdown inertial navigation system DOI: 10.3103/S0025654420010203
1. ONE-DIMENSIONAL VAN DER POL OSCILLATOR Consider the Van der Pol equation in the form given, for example, in [1]
q + q = μ(1 − q 2 )q
(μ > 0),
(1.1)
here μ is a small parameter, which allows us to solve this equation, as is often done, by the averaging method. For the purposes of our further discussion, it is convenient for us to interpret equation (1.1) as the equation of a linear spring oscillator in one-dimensional space (Fig. 1) with feedback superimposed on it by a formalized nonlinear right-hand side of equation (1.1). In [1], a solution to this equation was also given; in particular, it was shown that the equation has two stationary solutions: unstable q = 0, and asymptotically stable
q = 2 cos(t − t0 )
(1.2)
where t0 is an arbitrary constant. The only reason for introducing feedback in the one-dimensional case is the desire to provide a periodic process, despite the inevitable presence of dissipative forces in real systems. Note that the type of feedback
μ(1 − q 2 )q
(1.3)
chosen by Van der Pol and used in various subsequent works, according to nonlinear methods, where this equation is used as an example, is not the best for technical applications. Instead, you should write
μ(1 − q 2 − q2 )q.
(1.4)
q
Fig. 1.
132
VAN DER POL OSCILLATOR
133
q2
T
q1
Fig. 2.
Indeed, for μ = 0 equation (1.1) has the first integral q 2 + q 2 = const and, therefore, for μ ≠ 0 it changes slowly, in contrast to the variable q2, which is rapidly changing in all cases. This not only simplifies the solution, but also improves the quality of management. We will show this. Instead of equation (1.1), we consider the system
q = p, p = −q + μ(1 − q 2 − p2 ) p.
(1.5)
For μ = 0, the solution to system (1.5) is
q = r cos φ,
p = −r sin φ,
φ = t − t0.
(1.6)
Using this solution to move to the new variables (q, p) → (r,φ), we find 2 2 r = μ(1 − r )r sin φ, φ = 1.
(1.7)
We got a system with one slow variable r and one fast variable ϕ. Averaging over the fast variable gives r = μ(1 − r 2 )r /2, when
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