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首页> 外文期刊>Journal of aerospace engineering >Asymptotic Distribution of Eigenvalues for Damped String Equation: Numerical Approach
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Asymptotic Distribution of Eigenvalues for Damped String Equation: Numerical Approach

机译：阻尼弦方程特征值的渐近分布：数值方法

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In the present paper, we consider a one-parameter family of the nonself-adjoint operators, which are the dynamics generators for systems governed by the wave equations containing dissipative terms. The equations contain viscous damping terms and are equipped with the boundary conditions involving an arbitrary complex parameter. In the current engineering literature, this type of boundary condition is used to model the action of smart materials (self-sensing/self-straining actuators). In the previous research of the first writer, the aforementioned dynamics generators have been studied analytically and precise asymptotic formulas for the eigenvalues have been derived (the asymptotic when the number of the eigenvalues tends to infinity). The goal of the present paper is to demonstrate that the analytic formulas are not only important theoretically, but also extremely efficient practically. Namely, we show that the leading terms in the asymptotic formulas approximate the actual eigenvalues with excellent accuracy. To justify the results, we use two methods, i.e., the Newton method and the Tchebychev method. First, Newton's method is applied to the characteristic equation using asymptotic formulas as initial guesses to find the eigenvalues. The convergence of Newton's method is improved by modifying the asymptotic formula. Second, we ase Tchebychev discretization to circumvent the nonlinear characteristic equation and to obtain a finite-dimensional generalized eigenvalue problem that approximates the infinite-dimensional one. Finally, to solve the generalized eigenvalue problem, we use the QT algorithm.
机译：在本文中，我们考虑一类非自伴算子，这是由包含耗散项的波动方程控制的系统的动力学生成器。这些方程式包含粘性阻尼项，并配备有涉及任意复数参数的边界条件。在当前的工程文献中，这种边界条件用于对智能材料（自感应/自拉伸执行器）的作用进行建模。在第一作者的先前研究中，已对上述动力学生成器进行了分析研究，并得出了特征值的精确渐近公式（特征值的数量趋于无穷大时的渐近式）。本文的目的是证明解析公式不仅在理论上很重要，而且在实践中也非常有效。即，我们表明渐近公式中的前导项以极好的精度逼近实际特征值。为了证明结果的正确性，我们使用两种方法，即牛顿法和Tchebychev法。首先，使用渐近公式作为初始猜测，将牛顿法应用于特征方程，以找到特征值。通过修改渐近公式，牛顿法的收敛性得到了改善。其次，我们利用Tchebychev离散化来规避非线性特征方程，并获得逼近无限维的有限维广义特征值问题。最后，为了解决广义特征值问题，我们使用了QT算法。

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