The present model describes the vibrations of a composite beam consisting of two interconnected beams. One component is a material that exhibits only structural dissipation, while the other is a material subjected to additional frictional dissipation. It is well known that the solutions for laminated Timoshenko beams with one component converge to the equilibrium point at rates that depend on the wave propagation velocity. Here, we show that the solution of the complete system decays with exponential or rational rates when the components share a transmission condition. Proofs of the main results are provided using the multiplier method in the frequency domain. Technically, the main ingredients are inequalities of observability type that serve as “propagators” of asymptotic properties from one component to another.

本模型描述了由两个互连梁组成的复合梁的振动。一种成分是仅表现出结构耗散的材料，而另一种成分是承受额外摩擦耗散的材料。众所周知，具有一个分量的叠层铁木辛柯梁的解会以取决于波传播速度的速率收敛到平衡点。在这里，我们表明，当组件共享传输条件时，整个系统的解会以指数或有理速率衰减。使用频域乘法器方法提供了主要结果的证明。从技术上讲，主要成分是可观测性类型的不等式，它们充当从一个分量到另一个分量的渐近属性的“传播器”。