Summary

We study the equilibrium of a mechanical system composed by two rods that bend under the action of a pressure difference; they have one fixed endpoint and are partially in contact. This system can be viewed as a bi-valve made by two smooth leaflets that lean on each other. We obtain the balance equations of the mechanical system exploiting the principle of virtual work and the contact point is identified by a jump condition. The problem can be simplified exploiting a first integral. In the case of quadratic energy, another first integral exists: its peculiarity is discussed and a further reduction of the equations is carried out. Numerical integration of the differential system shows how the shape of the beams and the position of the contact point depend on the applied pressure. For small pressure, an asymptotic expansion in a small parameter allows us to find an approximate solutions of polynomial form which is in surprisingly good agreement with the solution of the original system of equations, even beyond the expected range of validity. Finally, the asymptotics predicts a value of the pressure that separates the contact from the no-contact regime of the beams that compares very well with the one numerically evaluated.

中文翻译：

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两杆在压力下接触的平衡

概要

我们研究了由两个在压力差作用下弯曲的杆组成的机械系统的平衡。它们具有一个固定的端点，并且部分处于接触状态。该系统可以看作是由两个彼此靠在一起的光滑小叶制成的双阀。我们利用虚功原理获得了机械系统的平衡方程，并通过跳跃条件确定了接触点。利用第一积分可以简化该问题。在二次能量的情况下，存在另一个第一个积分：讨论了它的特殊性，并进一步简化了方程。微分系统的数值积分显示了梁的形状和接触点的位置如何取决于所施加的压力。对于小压力，小参数的渐近展开使我们能够找到多项式形式的近似解，这与原始方程组的解出乎意料的良好一致性，甚至超出了预期的有效性范围。最后，渐近线会预测将接触与梁的非接触状态分开的压力值，该值与数值评估的结果非常好。