We give a detailed derivation of the equation describing the elastic properties of the helical springs in the case of their large deformations. It is shown that the Taylor expansion for the restoring force contains, in addition to a linear term, a term quadratic in the strain. We discuss the physical consequences of this term. A surprising example of this is that for large values of the initial pitch angle during compression of spring, there may be an interval of strain values, for which the differential stiffness becomes negative. The issues outlined in this article will be useful for instructors and advanced undergraduates who deal with modern elasticity theory.

中文翻译：

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螺旋弹簧弹性特性几何非线性的理论研究

我们给出了描述螺旋弹簧在大变形情况下的弹性特性的方程的详细推导。结果表明，除了线性项之外，恢复力的泰勒展开式还包含应变中的二次项。我们讨论这个术语的物理后果。一个令人惊讶的例子是，对于弹簧压缩期间初始螺距角的较大值，可能存在应变值区间，对于该区间，差分刚度变为负值。本文中概述的问题将对处理现代弹性理论的教师和高级本科生有用。