Available online at ScienceDirect. Procedia Engineering 150 (2016 )

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Procedia Engineering 150 ( 2016 ) 341 – 346

Peer-review under responsibility of the organizing committee of ICIE 2016 doi: 10.1016/j.proeng.2016.06.715

ScienceDirect

International Conference on Industrial Engineering, ICIE 2016

Analysis of Mathieu Equation Stable Solutions in the First Zone of

Stability

A.A. Prikhodko

a,

*, A.V. Nesterov

a

, S.V. Nesterov

a

a _{Kuban State Technological University, 2 Moskovskaya st., Krasnodar, 350072, The Russian Federation}

Abstract

The paper presents the results of a homogeneous Mathieu equation studies. Mathieu equation solutions are oscillations, modulated in amplitude and frequency. In the computational experiments we found dependences of the given oscillations on the ratio of the coefficients. These dependences are shown in graphs that can be used for an approximate estimation of the Mathieu equation solutions without integration.

Peer-review under responsibility ofthe organizing committee of ICIE 2016.

Keywords:Mathieu equation; Ince-Strutt diagram; the spectral characteristics; modulation depth; carrier frequency; modulation frequency.

1. Introduction

Parametric resonance is a well-known phenomenon that can be observed in a variety of mechanical, electrical, optical, and other physical systems [1-5]. In mechanical systems, parametric excitation can result from changes in stiffness [6], the application of the axial load on the cantilever beam [3,7] or drive of the pendulum suspension point [4,8].

One of the equations that describe the parametric oscillations is Mathieu equation: ( 2 cos 2 ) 0.

y a q W y

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* Corresponding author. Tel.: +7-861-251-87-05.

E-mail address: sannic92@gmail.com

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The results of computational experiment, which is to solve the Cauchy problem for this equation with different combinations of coefficients a and q, are presented in this work. Analysis of these solutions y(Ĳ) must answer questions about how they are associated with the named coefficients. This is a problem for dynamic systems, the main condition for the efficiency of which is the requirementy( )W dy_all_.[9, 10]. It is assumed that this requirement is satisfied by appropriate selection of the coefficients a and q. The restrictions also imposed on the spectral characteristics of functions y(Ĳ), for example, the spectral width ǻȦ.

2. Materials and methods

Computer experiment was conducted in the first zone of stability, which is shown in the Ince-Strutt diagram (Fig. 1). The investigated combinations of the Mathieu equation coefficients are shown by the points, uniformly marked on the plane of parameters a and q in the range of the first stability zone.

Fig. 1.Ince-Strutt diagram.

Numerical integration of the Mathieu equation is performed with the function ode23s of computer mathematics system MATLAB 7.0 for all the points shown in Fig. 1. Mathieu equation was integrated in normal form:

1 2 2 1 ; ( 2 cos 2 ) , x x x a q W x ® ¯ (2)

which is equivalent to the equation (1) with y = x1, with the initial conditions:

1 2 (0) 1; (0) 0. x x ® ¯ (3)

Figures 2-4 shows the graphs of three Mathieu equation solutions, which were found in different combinations of

a and q coefficients. These combinations of coefficients are marked by points 1, 2 and 3 on the Ince-Strutt diagram (Fig. 1).

3. Results and discussions

Mathieu equation solutions have a complex oscillatory character, while they are modulated in amplitude and frequency. Analysis of the graphs shows that the ratio of coefficients a and q influences the oscillation type. It enhances or reduces modulation of amplitude and frequency.

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Fig. 2.The function y(Ĳ) (q=0.1, a=0.8)

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Fig. 4.The function y(Ĳ) (q=0.1, a=0.1).

For a comparative analysis of Mathieu equation modulated solutions it is convenient to characterize them by modulation depth m, the carrier frequency Ȧ, and modulation frequency ȍ. The study found that the amplitude modulation depth max min max min A A m A A (4)

lies in the range 0<m<0.4. In particular, the selected solutions 1, 2 and 3 have respectively m1 = 0.15, m2 = 0.04 and

m3§0. The general idea about the relation between the modulation depth and the ratio of the Mathieu equation coefficients is presented in Figure 5.

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In general, low values of m are caused by low values of coefficients a and q, large values are typical for the border area. Minimizing of oscillatory solutions y(Ĳ) by reducing the coefficients a and q is accompanied by the weakening of modulation.

Spectral characteristics Ȧ and ȍ can be determined only approximately, since the investigated solutions y(Ĳ) are quasiperiodic [11]. The following evaluations were obtained excluding the frequency modulation. Moreover, the functions y(Ĳ) are considered as a result of the single tone amplitude modulation. With this assumption, the carrier frequency Ȧ and modulation frequency ȍ are, respectively:

2 ; 2 . M T T S Z S | : | (5)

Periods T and Tm are calculated according to Figure 2. It was found that the carrier frequency is within the range

0<Ȧ<1. For very small q (q<0.1) solution oscillates at a frequency close to the natural Mathieu equation frequency. The second coefficient of equation q also affects the carrier frequency Ȧ. Figure 4 reflects these interconnections. It shows as an example the dependence of the carrier frequency Ȧ from coefficient q at constant coefficient a (a = 0.1

and a = 0.3). It is obvious that the oscillation frequency Ȧ is highest near the stability border.

Fig. 6.Diagram of the oscillation frequency Ȧ.

The modulation frequency ȍ also depends on the ratio of the coefficients a and q. The value ȍdetermines the spectral width of oscillatory solutions y(Ĳ) which is defined as ǻȦ=2ȍ in the case of single-tone amplitude modulation (Fig. 7).

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Fig. 7. Spectrum of oscillatory solutions y(Ĳ).

In contrast to the carrier frequency Ȧit is difficult to estimate the modulation frequency ȍ. The results obtained in the range ȍ<0.1 can be attributed to the preliminary ones, that require further clarification. Close to the stability border, solutions y(Ĳ) are consistent with ideas about the single-tone amplitude modulation, which frequency modulation lies within the range 0.1<ȍ<0.3. Consequently, the maximum spectral width is ǻȦ=0.6, and remains almost constant along the stability border.

4. Conclusion

Thus, it is proposed to consider stable Mathieu equation solutions y(Ĳ) in qualitative terms as the amplitude-modulated oscillations. This allows to use its amplitude A, modulation depth m, carrier frequency Ȧ and modulation frequency ȍ to measure influence of the Mathieu equation coefficients a and q on the character of its solutions. Quantitative estimations of this influence obtained during the computational experiment are presented graphically. The constructed relations omit the numerical integration of the Mathieu equation, since the required spectral characteristics of solutions y(Ĳ) are obtained graphically in any combination of the coefficients a and q.

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