One of the most important things to note about Amass is that each one of its structure-element computations is independent of the structure as a whole. It generates the added mass matrix (M_ij) of a structure one piece at a time, without taking into consideration interactions between the different components, or even if the components overlap. For instance, if the user inputs two concentric spheres, both with the same radius, the M_11 = M_22 = M_33 components for the final structure will be off by a factor of 2. It is important to be aware of overlapping when creating structures, as Amass has no capability to deal with it.

The M_ij of individual components is computed for each component in turn, and then summed to get the total M_ij of the entire structure. This is done using strip theory for the cylinders, and a direct equation for the spheres. Initially, the cylinders are lined up with the X axis such that one end is at the origin. With the cylinder in this position it is possible to derive an analytical solution for the generalized cylinders Amass uses. (i.e. R_1 >= 0, R_2 >= 0, R_1 may or may not = R_2). This is done using particular solutions of strip theory. Once this initial M_ij is calculated, it is possible to rotate and translate the M_ij matrix such that M_ij' is M_ij for the location in which the cylinder ultimately ends up in the structure. This is even easier for spheres since there is no rotation component, only strict translation.

I should also comment on how M_11 is initially computed for cylinders. As you know, the slender body theory tells us that M_11 is not readily computable. However, Amass makes the assumption that the along the initial axis of the cylinder (the 1 = X-axis), before the rotation and translation of M_ij to the cylinder's final position, M_11 is simply the sum of the M_11 components of two half spheres, each with the radius of R_1 or R_2, the radii of the bases of the cylinder. All other M_1j = M_i1 components are zero.

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