CONSIDERATION ABOUT THE USAGE OF THE DUHAMEL INTEGRAL IN VIBRATION MECHANICS

The Duhamel integral, also called convolution integral, usually satisfies the commutative law. It is used to derive the general solution of a linear dynamic system subjected to arbitrary loads. Through mathematical and physical analysis, we have uncovered a notable exception: when a system is subjected to segmented loads, the Duhamel integral ceases to uphold the commutative law. Furthermore, even in the context of continuous loads, the commutative property may not hold when computations are performed in segmented manner. In light of this discovery, this paper elucidates the correct methodology for segmented calculations. The validity of these conclusions is further supported by illustrative examples.