In fact in mathematics Duhamel's integral is presented rather as a technique to find a solution of a non-homogeneous differential equation based on the known solution of homogeneous equation. Maybe in this abstract form is more easier to understand why is useful for a complex linear system. You have a good technique to calculate particular oscillations which are not specifically linked with a perturbation (let's say modal analyses) and you need a technique to calculate the system response under various perturbations as forces. The perturbations are the terms responsible for "non-homogeneous". Duhamel's integral makes this step in solving the problem.

For the case of oscillations for one degree linear system subject to various forcing forces, Duhamel's integral can be presented with a physical meaning as in the book I've mentioned.

In the same book the Duhamel's solution has a more general form which is "a perfectly general expression for the response of an undamped, linearly elastic one-degree system subjected to any load function and/or initial conditions."

In the form you've written, M is the mass of the system, ω is the natural circular frequency and depends on both k and M (with k the "spring constant"). So k, M and ω are specific to the system.

The article you've downloaded informs that the displacements of a system due to any arbitrary load can be calculated through the application of Duhamel's principle and integral.

"CAESAR II provides the user with a Pulse Table/DLF Spectrum Generator, which performs the automatic integration of Duhamel's integral. This module takes a user supplied, segmented pulse, and creates the appropriate equation for each segment. Displacements are calculated at each terminus of the segment, and the equation is differentiated in order to locate any displacement minima or maxima occurring within the interior of the segment (this assures that the maximum displacement is found, without using a hit-ormiss approach). The absolute maximum dynamic displacement is then selected from the largest of the segmental values, and the DLF calculated from that. This process is repeated for the number of natural frequencies specified by the user, which, when plotted, create the response spectrum."