# The Use of Structural Symmetries of a U12 Engine in the Vibration Analysis of a Transmission

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Theorem T1. The Eigenvalues for the System (3) are Eigenvalues for the System (2) As Well

**Proof.**

#### 3.2. If We Consider the Square Polynomial Matrices with Complex Coefficients, of Size n, Noted A, B, C, L, Z = O_{n} and matrix $M=\left(\begin{array}{ccc}A& Z& B\\ Z& A& B\\ L& L& C\end{array}\right)$, then det(M) is Dividable by det(A)

**Proof.**

^{th}order having elements on the first n lines we have [25]:

- For j1 = 1, jn = n we have $\alpha =\mathrm{det}(A)$.
- For j1 = 2n + 1, jn = 3n we have $\alpha =\mathrm{det}(B)$ and $\overline{\alpha}=\mathrm{det}\left(\begin{array}{cc}Z& A\\ L& L\end{array}\right)=-\mathrm{det}(L)\cdot \mathrm{det}(A)$.
- For the rest, we notice that:
- if there is an index $jk\in \left\{n+1,\dots ,2n\right\}$ then the column k from $\alpha $ is null thus $\alpha \beta \gamma =0$.
- $\alpha $ is non-null if $\left\{j1,\dots ,jk\right\}\subset \left\{1,2,\dots ,n\right\}$ and $\left\{jk+1,\dots ,jn\right\}\subset \{2n+1,\dots ,3n\}$ in this case $\alpha =\mathrm{det}(Aj1..AjkBik+1..Bin)$ where $ik+l=jk+l-2n$. For such a fixed $\alpha $ we have three possibilities for $\beta $ namely:
- $\beta $ has a column 0 thus $\beta $ = 0;
- $\beta $ = det(A);
- $\beta $ = $\mathrm{det}(As1..AslBrl+1..Brn)$. In this case, we can determine in a unique way the matrix $\overline{V}$ for each of the two possible versions:
- If there is $t\in \left\{1,\dots ,n\right\}\backslash \left\{j1,\dots jk,s1,\dots ,sl\right\}$ then $\overline{V}$ contains twice the column Lt thus $\alpha \beta \gamma =0$;
- If $\left\{1,\dots ,n\right\}=\left\{j1,\dots jk,s1,\dots ,sl\right\}$ then we consider $\alpha \prime =(\mathrm{det}(As1..AslBrl+1..Brn)$ $\beta \prime =\mathrm{det}(Aj1..AjkBik+1..Bin)$ and $\gamma \prime $ will be a determinant having the same C type columns located in the same position as in $\gamma $ and the L type columns will be the same but permutated as far as the position is concerned. A direct calculation of signs will lead to $\alpha \beta \gamma +\alpha \prime \beta \prime \gamma \prime =0$.

#### 3.3. The Natural Modes of Vibration

**Proof.**

**Proof.**

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Mobile oil drilling plant type TW125 CAA6 mounted on a chassis ROMAN 75,540 MFEG (12 × 8) [24].

No. | Moment of Inertia | Details |
---|---|---|

1 | J_{1} | Cylinder 1 |

2 | J_{2} | Cylinder 2 |

3 | J_{3} | Cylinder 3 |

4 | J_{4} | Cylinder 4 |

5 | J_{5} | Cylinder 5 |

6 | J_{6} | Cylinder 6 |

7 | J_{7}, J_{12} | Gears |

8 | J_{8} | Central Gear |

9 | J_{9} | Flywheel |

10 | J_{10} | Ventilator |

11 | J_{11} | Exit steering wheel |

No. | Rear Clutch Model | Front Clutch Model | ||
---|---|---|---|---|

Moment of Inertia | Values (kg*m^{2}) | Moment of Inertia | Values (kg*m^{2}) | |

1 | J_{1} | 0.1048 | J_{1} | 0.1048 |

2 | J_{2} | 0.0638 | J_{2} | 0.0638 |

3 | J_{3} | 0.1048 | J_{3} | 0.1048 |

4 | J_{4} | 0.1048 | J_{4} | 0.1048 |

5 | J_{5} | 0.0638 | J_{5} | 0.0638 |

6 | J_{6} | 0.1048 | J_{6} | 0.1048 |

7 | J_{7}+J_{8}+J_{12} | 1.81182 | J_{7}+J_{8}+J_{12} | 1.4157 |

8 | J_{9} | 3.41895 | J_{9} | 2.9841 |

9 | J_{11} | 3.70752 | J_{11} | 1.3382 |

Between | Rear Clutch Model | Front Clutch Model | ||
---|---|---|---|---|

Stiffness | Values (Nm/rad) | Stiffness | Values (Nm/rad) | |

1–2 | k_{1} | 2.56 × 10^{6} | k_{1} | 2.56 × 10^{6} |

2–3 | k_{2} | 2.56 × 10^{6} | k_{2} | 2.56 × 10^{6} |

3–4 | k_{3} | 2.53 × 10^{6} | k_{3} | 2.53 × 10^{6} |

4–5 | k_{4} | 2.56 × 10^{6} | k_{4} | 2.56 × 10^{6} |

5–6 | k_{5} | 2.56 × 10^{6} | k_{5} | 2.56 × 10^{6} |

6–7 | k_{6} | 20.87 × 10^{6} | k_{6} | 20.87 × 10^{6} |

7–8 | k_{7} | 12.67 × 10^{6} | k_{7} | 4.683 × 10^{6} |

7–9 | k_{8} | 0.045961 × 10^{6} | k_{8} | 0.030158 × 10^{6} |

No | Rear Clutch Model | Front Clutch Model | Single Engine Model |
---|---|---|---|

Eigenvalues (rpm) | Eigenvalues (rpm) | Eigenvalues (rpm) | |

1 | 0 | 0 | |

2 | 1.338 | 1.596 | |

3 | 14.062 | 13.449 | |

4 | 14.564 | 14.062 | 14.062 |

5 | 30.417 | 22.397 | |

6 | 40.278 | 40.278 | 40.278 |

7 | 41.483 | 41.329 | |

8 | 67.067 | 67.067 | 67.067 |

9 | 67.561 | 67.632 | |

10 | 92.764 | 92.764 | 92.764 |

11 | 93.394 | 93.553 | |

12 | 100.959 | 100.959 | 100.959 |

13 | 101.039 | 101.056 | |

14 | 144.921 | 144.921 | 144.921 |

15 | 151.079 | 152.676 |

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**MDPI and ACS Style**

Mihălcică, M.; Vlase, S.; Păun, M.
The Use of Structural Symmetries of a U12 Engine in the Vibration Analysis of a Transmission. *Symmetry* **2019**, *11*, 1296.
https://doi.org/10.3390/sym11101296

**AMA Style**

Mihălcică M, Vlase S, Păun M.
The Use of Structural Symmetries of a U12 Engine in the Vibration Analysis of a Transmission. *Symmetry*. 2019; 11(10):1296.
https://doi.org/10.3390/sym11101296

**Chicago/Turabian Style**

Mihălcică, Mircea, Sorin Vlase, and Marius Păun.
2019. "The Use of Structural Symmetries of a U12 Engine in the Vibration Analysis of a Transmission" *Symmetry* 11, no. 10: 1296.
https://doi.org/10.3390/sym11101296