For metal–plastic composites with thin shear-sensitive cores, high damping can be determined. These materials were developed to reduce the structure-borne noise and are therefore used for thin-walled components with bending loads. In addition to the known sensitivity due to the temperature and frequency, these materials show a significant dependency on vibration amplitudes. Within the framework of this study, the nonlinear damping of metal–plastic composites was determined experimentally using free-vibrating cantilever beams. A detailed analysis of the derived velocity–time curves showed a nonlinear damping within the decay. An exponential approach was successfully used to describe the relation between damping and current deflection. Furthermore, a strain energy-based evaluation is introduced to quantify the share of the core contributions. For this purpose, the strain energy of several acoustically sensitive car components as well as the beams with varying supports and vibration lengths were determined numerically with a finite element analysis. The strain energy ratio of the cores was then derived as a comparative measurement and within a finite element-based design of experiments. A cantilever beam setup with a component-specific beam length representing similar core strain energy ratios was retested and showed a similar exponential amplitude dependence but higher damping parameters.