June, 2025 | EN
This article emphasizes the mathematical structure that emerges from my development in the mathematical framework of the Python module for computing the tower crane support reaction. In particular, I use linear algebra for the model and a little functional analysis. Notably I present a bilinear map $\diamond_q$ which maps the product of $\mathbb{R}^q$ with a $q$-direct sum of 3D Eculidean spaces to a 3D Euclidean space, for any $q \in \mathbb{N}$. The map $\diamond_q$ may be equivalent to another established map, and it may be defined via the classical algebraic operation. It is analogous to an inner product, but is different in the traditional sense since the domain consists of different vector spaces. With $\diamond_q$, I can express the application of a load combination in the system as a bounded linear operator on the $q$-direct sum of 3D Euclidean spaces. I also present another map $\overline{\boxtimes}$ which maps the product of a 3D Euclidean space with a $q$-direct sum of 3D Euclidean spaces to the $q$-direct sum via the classical cross product to integrate the moments in the system. The map $\overline{\boxtimes}$ is also bilinear, and it also possesses the antisymmetric property inherited from the cross product in the form of its counterpart $\underline{\boxtimes}$, which is also a bilinear map. I further investigate properties of this linear algebraic model.

July, 2025 | ID
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