A multi-vibration mode framework for impact force reconstruction in dynamical systems

Impact events on structures, ranging from minor collisions to catastrophic accidents, often trigger structural vibrations. The accurate reconstruction of impact forces is critical for analyzing dynamic interactions and designing structures that can withstand such events. The nature of these impacts depends on factors such as relative velocity, material properties, and structural characteristics. Traditional force reconstruction techniques face challenges due to the ill-posed nature of inverse problems, sensitivity to noise, and dependence on assumptions about material properties and system geometry. This paper presents an accurate and robust modal-based impact force reconstruction framework, minimizing prior assumptions about the applied force. It exploits the multi-mode vibrational response of the impacted body, accounting for its deformation. Each mode represents an independent system whose oscillations are ruled by the shape, magnitude and duration of the impact force. The impact force is approximated as a finite harmonic series, enabling reconstruction of a wide variety of transient force shapes. A dedicated experiment setup involving a glass plate suspended as a pendulum and impacted by an impact hammer validates the technique. The plate response is measured by a laser interferometer, making it possible to resolve even small vibrations. It is shown that the accuracy of the force reconstruction improves when more vibration modes are detected. The technique however is effective even when few modes are considered, allowing a reliable estimation of the impact duration and exchanged momentum. It is adaptable to different boundary conditions and demonstrates robustness through mode redundancy. The versatile framework is particularly suited for applications where minimal prior knowledge of the excitation is available, including structural health monitoring, and nondestructive evaluation.