The free vibration displacement of an undamped hardening Duffing oscillator is described in exact form by a Jacobi elliptic function. Unlike an undamped linear oscillator, whose displacement is described by a trigonometric function, a Jacobi elliptic function is difficult to interpret by a simple inspection of the function arguments. The displacement of a linear oscillator is often visualised as a rotating vector, which has two characteristics — a constant amplitude and a phase (or frequency). These parameters are readily related to the physical response of the system. In this paper, a similar approach is applied to the free vibration displacement of a Duffing oscillator. However, the rotating vector description of the motion is much more complicated than for a linear system. It still has two characteristics though — an amplitude and a phase, but in general both these quantities are dependent on the position of the vector, i.e., they are frequency modulated. It is shown that there is not a unique rotating vector representation of the cn Jacobi elliptic function. Indeed, there are an infinite number of elliptical loci bounded between an elliptical and a circular locus of the vector. There are two specific cases. One is where the amplitude of the vector is constant and the phase angle is frequency modulated (the circle), and the other is when the amplitude of the vector is frequency modulated and the angular velocity is constant. In all other cases, both the amplitude and the angular velocity of the rotating vector are frequency modulated. To aid in the visualisation of the rotating vectors that represent the free vibration solution of a highly nonlinear hardening Duffing oscillator, two animations are provided.

无阻尼硬化Duffing振荡器的自由振动位移通过Jacobi椭圆函数精确描述。与无阻尼线性振荡器不同，后者的位移由三角函数描述，而雅可比椭圆函数很难通过简单检查函数自变量来解释。线性振荡器的位移通常可视化为旋转矢量，它具有两个特性-恒定振幅和相位（或频率）。这些参数很容易与系统的物理响应有关。在本文中，将类似的方法应用于Duffing振荡器的自由振动位移。但是，运动的旋转矢量描述要比线性系统复杂得多。但是它仍然具有两个特征-幅度和相位，但通常，这两个量都取决于矢量的位置，即它们是频率调制的。结果表明，cn Jacobi椭圆函数没有唯一的旋转矢量表示。实际上，在向量的椭圆和圆形轨迹之间有无限数量的椭圆基因座。有两种具体情况。一个是矢量的振幅是恒定的，并且相位角是经过频率调制的（圆圈），另一种是矢量的振幅是经过频率调制并且角速度是恒定的。在所有其他情况下，旋转矢量的幅度和角速度都经过频率调制。为了有助于可视化表示高度非线性硬化Duffing振荡器的自由振动解的旋转矢量，