Study of Eigenvalues and Matrix Eigenvectors Using MATLAB: Vibration Systems of Multi-Purpose Vehicle (MPV)

Abstract

Vehicle vibration is a critical factor influencing both passenger comfort and vehicle performance. In this study, we analyze the multi-degree-of-freedom (MDOF) vibrational behavior of a multi-purpose vehicle (MPV) using matrix eigenvalue and eigenvector methods. The vehicle’s dynamics are modeled by developing a set of equations of motion that account for the forces acting on the front and rear tires, car body, and pitch angle. MATLAB is utilized to numerically compute the system’s eigenvalues and eigenvectors, representing the natural frequencies and vibration modes of the vehicle, respectively. The analysis focuses on the vehicle’s response to a 50 mm displacement at the front tire, simulating the effect of road disturbances. The resulting vibrations in the front and rear tires, car body, and vehicle pitch are illustrated over a 1-second time frame. The findings show that the front tire experiences the largest oscillation amplitude of ±1 mm, while the rear tire exhibits a much smaller displacement of ±0.04 mm. The overall car body displacement reaches a maximum amplitude of ±1.3 mm, indicating partial damping of the front tire vibrations. However, the results reveal that the vehicle’s suspension system lacks effective damping, as the vibrations do not decrease over time. This behavior could negatively impact ride comfort and safety, particularly on uneven roads. The study concludes that improvements to the vehicle’s suspension system are necessary to enhance damping performance. The presented MATLAB-based approach offers a valuable tool for analyzing and optimizing vehicle vibration systems.