This paper presents a detailed study on the application of a polynomial expansion method to solve inverse problems related to the determination of the displacement and space-dependent forces on a vibrating structure based on observed final displacement data. Beginning with a structured problem formulation, we analyze the challenges of ill-conditioning and solution instability often encountered in the obtained discret systems. To address these issues, we introduce a regularized approach, transforming the original system to enhance numerical stability. Additionally, a preconditioning technique is employed to further improve convergence and computational efficiency. We demonstrate the effectiveness of this method through numerical experiments, analyzing the impact of both regularization and preconditioning on solution accuracy.