Non-reciprocal interactions of discrete or continuous systems may induce surprising responses such as flutter instabilities. It is shown in this paper that a finite one-dimensional lattice under non-symmetrical elastic interactions may flutter for sufficiently strong unsymmetrical interactions. An exact solution is presented for the vibration of such one-dimensional lattices with direct and non-symmetrical elastic interactions. An internal force controlling the interactions is included in the model as an additional force for each mass, which acts proportionally to the elongation of a spring at its position. This non-conservative problem due to this circulatory interaction is solved from the resolution of a linear difference equation for this unsymmetrical repetitive lattice. It is possible to derive the exact eigenfrequency dependence with respect to the unsymmetrical interaction parameter, which plays the role of a bifurcation parameter. Divergence and flutter instabilities of this fixed–fixed non-conservative axial lattice under non-Hermitian interactions are theoretically predicted, from a direct approach or by solving the difference equation whatever the number of masses of the lattice. It is shown that the system may flutter for sufficiently strong unsymmetrical interactions, whatever the size of the system, for even or odd number of masses. However, divergence instability may arise in such a system only for even number of masses. The drastic change of response of the present system for odd or even number of particles is specific of the discrete nature of the dynamic system.

离散或连续系统的非互易相互作用可能会引起令人惊讶的响应，例如颤振不稳定性。本文表明，非对称弹性相互作用下的有限一维晶格可能会因足够强的非对称相互作用而颤振。针对这种具有直接和非对称弹性相互作用的一维晶格的振动提出了精确的解。控制相互作用的内力作为每个质量的附加力包含在模型中，其作用与弹簧在其位置的伸长成比例。由于这种循环相互作用而产生的非保守问题可以通过解决这种不对称重复晶格的线性差分方程来解决。可以导出关于不对称相互作用参数的精确特征频率依赖性，该参数起着分岔参数的作用。这种固定-固定非保守轴向晶格在非厄米相互作用下的发散和颤振不稳定性可以从理论上通过直接方法或通过求解差分方程来预测，无论晶格的质量数是多少。结果表明，对于偶数或奇数质量，无论系统大小如何，系统都可能因足够强的不对称相互作用而颤振。然而，在这样的系统中，仅当质量数为偶数时才可能出现发散不稳定性。本系统对奇数或偶数个粒子的响应的剧烈变化是动态系统的离散性质所特有的。