The exact periodic solution of an archetypal non-natural oscillator with quadratic inertia and cubic nonlinear static stiffness has been derived in terms of the elliptic integral of the third kind. The periodic solution was derived based on a new form of elliptic integral that arose naturally from the model of the archetypal non-natural oscillator. It was shown that this new form of elliptic integral is an alternate form of the elliptic integral of the third kind and it has several advantages over the classical Legendre form. Hence, some new properties of elliptic integrals were derived based on the new elliptic integral. A bifurcation analysis of the present oscillator revealed the possible combinations of static stiffness parameters that can produce periodic solutions and the corresponding initial amplitude range where these periodic solutions exist. The bifurcation analysis informed investigations on the frequency–amplitude response which included the effect of the different static stiffness parameters. It was observed that the frequency increased for an increase in the positive cubic nonlinear stiffness parameter and vice versa. Additionally, the large-amplitude displacement history showed the same profile notwithstanding the combination and magnitudes of the stiffness parameters.

根据第三类椭圆积分，推导了具有二次惯性和三次非线性静态刚度的原型非自然振荡器的精确周期解。周期解是根据原型非自然振荡器模型自然产生的一种新形式的椭圆积分得出的。结果表明，这种新的椭圆积分形式是第三种椭圆积分的一种替代形式，它比经典的勒让德形式具有许多优点。因此，基于新的椭圆积分导出了椭圆积分的一些新性质。本振荡器的分叉分析揭示了可能产生周期解的静态刚度参数的可能组合，以及存在这些周期解的相应初始振幅范围。分叉分析为研究频率-振幅响应提供了依据，其中包括不同静态刚度参数的影响。观察到频率随着正立方非线性刚度参数的增加而增加，反之亦然。另外，尽管刚度参数的组合和大小，大振幅位移历史也显示出相同的轮廓。观察到频率随着正立方非线性刚度参数的增加而增加，反之亦然。此外，尽管刚度参数的组合和大小，大振幅位移历史也显示出相同的轮廓。观察到频率随着正立方非线性刚度参数的增加而增加，反之亦然。另外，尽管刚度参数的组合和大小，大振幅位移历史也显示出相同的轮廓。