Abstract

The differential eigenvalue problem governing eigenvibrations of an elastic bar with fixed first end and mechanical resonator attached to second end is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We introduce limit differential eigenvalue problems and derive the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as a resonator parameter tending to infinity. The original differential eigenvalue problem is approximated by the finite element method on a uniform mesh. Error estimates for approximate eigenvalues and eigenfunctions are established.

中文翻译：

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带机械谐振器的棒材本征振动问题的研究

摘要

研究了具有固定第一端和连接到第二端的机械谐振器的弹性杆的特征振动的微分特征值问题。这个问题有一个递增的正简单特征值序列，极限点在无穷远处。对于特征值序列，对应一个完整的特征函数标准正交系统。我们引入极限微分特征值问题，并推导出初始问题的特征值和特征函数收敛到极限问题的相应特征值和特征函数作为趋于无穷大的谐振器参数。原始微分特征值问题通过有限元方法在均匀网格上近似。建立近似特征值和特征函数的误差估计。