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## On a rank factorisation problem arising in gearbox vibration analysis

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Elisa Hubert
• Function : Author
• PersonId : 1050972
Axel Barrau
• Function : Author
• PersonId : 941948
Yacine Bouzidi
• Function : Author
• PersonId : 955292
Roudy Dagher
• Function : Author
• PersonId : 833444

#### Abstract

Given a field $K$, $r$ matrices $D_i \in K^{n \times n}$, a matrix $M \in K^{n \times m}$ of rank at most $r$, in this paper, we study the problem of factoring $M$ as follows $M=\sum_{i=1}^r D_i \, u \, v_i$, where $u \in K^{n \times 1}$ and $v_i \in K^{1 \times m}$ for $i=1, \ldots, r$. This problem arises in modulation-based mechanical models studied in gearbox vibration analysis (e.g., amplitude and phase modulation). We show how linear algebra methods combined with linear system theory ideas can be used to characterize when this polynomial problem is solvable and if so, how to explicitly compute the solutions.

### Dates and versions

hal-03070702 , version 1 (15-12-2020)
hal-03070702 , version 2 (16-12-2020)

### Identifiers

• HAL Id : hal-03070702 , version 2

### Cite

Elisa Hubert, Axel Barrau, Yacine Bouzidi, Roudy Dagher, Alban Quadrat. On a rank factorisation problem arising in gearbox vibration analysis. IFAC 2020 - 21st World Congress, Jul 2020, Berlin / Virtual, Germany. ⟨hal-03070702v2⟩

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