The high-speed impact of a droplet onto a flexible substrate is a highly non-linear process of practical importance, which poses formidable modelling challenges in the context of fluid–structure interaction. We present two approaches aimed at investigating the canonical system of a droplet impacting onto a rigid plate supported by a spring and a dashpot: matched asymptotic expansions and direct numerical simulation (DNS). In the former, we derive a generalisation of inviscid Wagner theory to approximate the flow behaviour during the early stages of the impact. In the latter, we perform detailed DNS designed to validate the analytical framework, as well as provide insight into later times beyond the reach of the proposed analytical model. Drawing from both methods, we observe the strong influence that the mass of the plate, resistance of the dashpot, and stiffness of the spring have on the motion of the solid, which undergo forced damped oscillations. Furthermore, we examine how the plate motion affects the dynamics of the droplet, predominantly through altering its internal hydrodynamic pressure distribution. We build on the interplay between these techniques, demonstrating that a hybrid approach leads to improved model and computational development, as well as result interpretation, across multiple length and time scales.

液滴对柔性基底的高速撞击是具有实际重要性的高度非线性过程，这在流体与结构相互作用的背景下提出了巨大的建模挑战。我们提出了两种方法来研究液滴撞击到由弹簧和减震器支撑的刚性板上的规范系统：匹配渐近展开法和直接数值模拟（DNS）。在前者中，我们推导了无粘性瓦格纳理论的推广，以近似于冲击早期的流动行为。在后者中，我们执行详细的DNS，旨在验证分析框架，并提供对超出提议的分析模型影响范围的更深入了解。从这两种方法中，我们观察到板块质量的强烈影响，减震器的阻力以及弹簧的刚度对固体的运动产生影响，固体会经历强制的阻尼振荡。此外，我们研究了板运动如何主要通过改变其内部流体动压分布来影响液滴的动力学。我们基于这些技术之间的相互作用，证明了混合方法可以在多个长度和时间范围内改善模型和计算开发以及结果解释。