In the multi-valued region, according to the stability criteria of Krylov and Bogoliubov, there are two stable solutions and one unsteady solution (the dotted line in the figure) for each determined Ω value. For example, f0=2N/kg, when Ω<1.75rad/s or Ω>3.21rad/s, the amplitude solution is single-valued; and when 1.75rad/s<Ω<3.21rad/s, there are 3 different The A value, the solid line corresponds to the stable solution, and the dotted line corresponds to the unsteady solution. At the same time, when f and γ0 remain unchanged, when the parameter Ω slowly increases from a relatively small value, the system has a significant amplitude jump phenomenon; vice versa.
In order to determine the fixed frequency, amplitude A produces a range of f0 for multivalued solutions, 2 gives ε0=2姨N/(kgm3), γ0=0.11Ns/(mkg), and Ω=2rad/s, amplitude A with f0 The curve of change. It can be seen from 2 that under the above conditions, the amplitude-frequency curve can be in the multi-valued region only when f0∈(0.8,3)N/kg.
The bifurcation of the system and the bifurcation of the chaotic nonlinear system are of great significance for investigating the chaotic motion of the system. The bifurcation diagram and the Lyapunov exponent spectrum are good tools for studying the topology of the system with the parameters. Therefore, this section takes f0 as the control parameter, and uses the bifurcation diagram and Lyapunov exponent spectrum to study the bifurcation process of equation (2), and analyzes its path to chaos. Let ε0=2姨N/(kgm3), Ω=1rad/s, γ0=0.006Ns/(mkg), initial value (x(0), y(0))=(1.2m, 0.58m/s), The fourth-order Runge-Kutta method [5] is used for numerical calculation and a bifurcation diagram is drawn.
The bifurcation parameter f0 starts from 0.5N/kg, and the system first appears a long single-cycle motion; when f0=1.06N/kg, the system has 3 times periodic bifurcation; continue to increase f0, the system goes through multiple times The bifurcation and inversion bifurcation process, but always in periodic motion; until f0=2.3N/kg, the chaotic motion appears in the system. Compared with the previous periodic motion, the chaotic interval is very short, indicating the periodic motion comparison of the system. Stable; further increase f0, the system occurs backwards, when f0 = 2.6N / kg, the system returns to single-cycle motion.
The Lyapunov exponent is an important index for judging chaotic motion. The Lyapunov exponent curve of the system with the control parameter f0 is calculated by the algorithm proposed in the literature, as shown in Fig. 5. The positive maximum Lyapunov exponent indicates that the system performs chaotic motion, and the negative maximum Lyapunov exponent indicates that the system performs periodic motion. When the maximum Lya-punov exponent is 0, the system bifurcates. Comparing the results of the two is consistent, thus confirming the above analysis results of system bifurcation.
After adding the next λk+1, k+1i=1Σλi becomes a negative number, then the Lyapunov dimension of the system is Dky=k+1-λk+1ki=1Σλi. Figure 5 gives the Lyapunov dimension of the formula calculated by the formula. Spectrum. It can be seen that the Lya-punov dimension of the system corresponding to equation (2) is a fraction when doing chaotic motion, and the Lyapunov dimension is equal to 2 when doing periodic motion.
The differential equation of forced transverse vibration of symmetric double spring oscillator is nonlinear. Applying the progressive method to solve the periodic solution and verifying with computer numerical solution can help us better understand the multivalued and amplitude jump phenomena of the periodic solution of nonlinear systems. The bifurcation diagram and Lyapunov exponent spectrum of the system can study the variation of the system topology with the parameters, and then analyze the system into chaos. These can be used as a reference for understanding the problem of general nonlinear forced vibration, and also have some guiding significance for dealing with some practical engineering problems.
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