Increasing the damping properties of a structure while limiting the addition of mass is a challenging task. Originally developed for their acoustic absorption capabilities, microperforated plates (MPP) are simple devices that can feature additional viscous damping at low frequencies. Dissipation arises in the boundary layers of the microperforations through viscous friction mechanisms induced by the relative motion between the solid and the fluid in which the MPP is immersed. This leads to sound absorption in an acoustic context and to an added viscous damping in a solid dynamics context. The added viscous damping exhibited by the MPP can be captured by considering adapted fluid-solid interaction terms in the governing equations. A model was experimentally validated in a linear context. However, when the fluid velocity in the microperforations becomes sufficiently large, the inertial effects occurring in the microperforations can also have a significant impact. These are captured by Forchheimer's correction. Introducing this correction into the MPP linear vibration model leads to a nonlinear system with an additional antisymmetric quadratic damping term. The resulting equations of motion are solved numerically for a large amplitude forcing and quantitative analyses are performed. It is shown that the influence of antisymmetric quadratic nonlinear damping on the added viscous damping at the MPP depends on the magnitude of the external mechanical force. The model suggests the existence of maximum added viscous damping as a function of the external mechanical force magnitude.