The fundamental subject of a single degree of freedom system's free vibration is essential in the field of mechanical vibrations, with applications in a wide range of engineering fields. This paper presents a novel numerical method for solving this problem based on Bernoulli polynomials in matrix form. The method is simple to implement and requires only basic linear algebra operations. The method is also very efficient; and can be used to solve problems with the single degree of freedom system. The proposed method is illustrated by a numerical example, and the results are compared with those of the exact solution. The results show that the proposed method is highly accurate and efficient.
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