A 1/3 Pure Sub-Harmonic Solution and Fractal Characteristic of Transient Process for Duffing’s Equation
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摘要: 通过谐波平衡法和数值积分法研究了杜芬方程的1/3纯亚谐解．提出假设解，找出了亚谐频域，并对参数变化的过渡过程的敏感性和初始值扰动的过渡过程进行了研究．考察了亚谐响应幅值系数对阻尼的敏感性及亚谐振动谐波成分的渐近稳态性．此外，运用广义分形理论对杜芬方程纯亚谐解过渡过程进行了分析．分析表明，广义维数的敏感维数能清楚地描述杜芬方程纯亚谐解过渡过程特征；并对改变初始扰动、阻尼系数、激励幅值情况下，其两个不同频域的杜芬方程纯亚谐解过渡过程的不同分形特性显现出敏感性．Abstract: The 1/3 sub-harmonic solution for the Duffing's with damping equation was investigated by using the methods of harmonic balance and numerical integration. The assumed solution was introduced, and the domain of sub-harmonic frequencies was found. The asympt otical stability of the subharmonic resonances and the sensitivity of the amplitude responses to the variation of damping coefficient were examined. Then, the subharmonic resonances were analyzed by using the techniques from the general fractal theory. The analysis indicates that the sensitive dimensions of the system time-field responses show sensitivity to the conditions of changed initial perturbation, changed damping coefficient or the amplitude of excitation, thus the sensitive dimension can clearly describe the characteristic of the transient process of the subharmonic resonances.
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Key words:
- Duffing’s equation /
- subharmonic /
- transient process /
- fractal characteristic /
- sensitive dimension
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