轴向受力梁强非线性超谐波与次谐波共振的能量迭代法

研究了在横向动载荷作用下存在轴向力的一类梁的强非线性振动;建立了梁振动的二阶强非线性非自治微分方程,并对求解强非线性自治系统的能量迭代法加以改进,用于求解梁的强非线性非自治系统,其方法是:由能量法得到主共振、超谐共振和次谐共振的一次近似解的表达式;引入牛顿迭代的思想和最小二乘法,得到高次近似解的表达式.研究结果表明:用改进后的能量迭代法求解强非线性非自治系统精度较高;分析这种非线性梁的振动时,除了要考虑其主共振外,还要考虑超谐共振和次谐共振.

Energy-iterative method for solution of super-harmonic and sub-harmonic resonance of strongly non-linear vibrations with an axially forcing beam

Strongly non-linear vibrations of an axially forcing beam under transverse dynamic loads were investigated. The Dulling equation, the equation of strongly non-linear and non-autonomous oscillation system of the beam were set up. The energy-iteration method for strongly non-linear autonomous system was modified to deal with strongly non-linear non-autonomous system, and the modified new method was developed, the analytic expressions of solution of primary resonance and superharmonic and subharmonic resonance by the energy method were obtained, and the higher order analytic expression of solution was gotten by means of Newton iteration idea and least square technique principle. The results show that the modified energy iterative method for strongly non-linear and non-autonomous system is both effective and accurate, and that not only the primary resonance but also the superharmonic and subharmonic resonance must be considered in this sort of strongly non-linear beams.