We consider the decay rate of Timoshenko type equations for beams with complementary frictional damping and memory effect in an interval. Under only basic conditions on the memory kernels (without any decaying conditions), we show that the solution energy of Timoshenko systems decays uniformly at the rate $t^{-1}$. Simultaneously, the same rate is also obtained for the corresponding single memory-dissipative wave equation. The proof is based on some appropriate functionals and it improves the previously related results in the literature.

中文翻译：

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具有互补的摩擦阻尼和记忆效应的Timoshenko系统的衰减估计

我们考虑在一定间隔内具有互补摩擦阻尼和记忆效应的梁的Timoshenko型方程的衰减率。仅在内存内核的基本条件下（没有任何衰减条件），我们证明Timoshenko系统的解能以速率均匀衰减${Ť}^{-\mathrm{1个}}$。同时，也为相应的单个记忆耗散波动方程获得了相同的速率。该证明基于一些适当的功能，并且改进了文献中先前相关的结果。