SUPER-HARMONIC AND SUB-HARMONIC SIMULTANEOUS RESONANCES OF FRACTIONAL-ORDER DUFFING OSCILLATOR

The super-harmonic and sub-harmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied in this paper. The first-order approximate analytical solution is obtained by averaging method. The definitions of equivalent linear damping coefficient and equivalent linear stiffness efficient for super-harmonic and subharmonic simultaneous resonance are presented. The analytical amplitude-frequency equation for steady-state solution of simultaneous resonance is established. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness and higher-order precision of the approximately analytical results. Then, a further comparison between the fractional-order and traditional integer-order Duffing oscillator is fulfilled through the definitions of equivalent linear damping coefficient and equivalent linear stiffness coefficient, and the results prove that the fractional-order parameters has the effects of both damping and stiffness, which is similar in other fractionalorder system. At last the numerical simulation is used to analyze the effects of different fractional-order parameters on multi-value characteristics and jumping phenomena of the amplitude-frequency curve under simultaneous resonance, and the differences between the super-harmonic and sub-harmonic resonances under single-frequency excitation are analyzed in detail. It could be found that the fractional-order parameters not only affect the response amplitude and resonant frequency of the system, but also has significant influence on the number, existing area and the occurrence order of periodic solutions. Moreover, single super-harmonic resonance, single sub-harmonic resonance and both existing of these two resonances could be respectively found under different basic parameters, which is important to study the dynamic characteristics of the similar system.