The Use of Structural Symmetries of a U12 Engine in the Vibration Analysis of a Transmission
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
3.2. If We Consider the Square Polynomial Matrices with Complex Coefficients, of Size n, Noted A, B, C, L, Z = On and matrix , then det(M) is Dividable by det(A)
- For j1 = 1, jn = n we have .
- For j1 = 2n + 1, jn = 3n we have and .
- For the rest, we notice that:
- if there is an index then the column k from is null thus .
- is non-null if and in this case where . For such a fixed we have three possibilities for namely:
- has a column 0 thus = 0;
- = det(A);
- = . In this case, we can determine in a unique way the matrix for each of the two possible versions:
- If there is then contains twice the column Lt thus ;
- If then we consider and will be a determinant having the same C type columns located in the same position as in 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 .
3.3. The Natural Modes of Vibration
4. Discussion
Author Contributions
Funding
Conflicts of Interest
References
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No. | Moment of Inertia | Details |
---|---|---|
1 | J1 | Cylinder 1 |
2 | J2 | Cylinder 2 |
3 | J3 | Cylinder 3 |
4 | J4 | Cylinder 4 |
5 | J5 | Cylinder 5 |
6 | J6 | Cylinder 6 |
7 | J7, J12 | Gears |
8 | J8 | Central Gear |
9 | J9 | Flywheel |
10 | J10 | Ventilator |
11 | J11 | Exit steering wheel |
No. | Rear Clutch Model | Front Clutch Model | ||
---|---|---|---|---|
Moment of Inertia | Values (kg*m2) | Moment of Inertia | Values (kg*m2) | |
1 | J1 | 0.1048 | J1 | 0.1048 |
2 | J2 | 0.0638 | J2 | 0.0638 |
3 | J3 | 0.1048 | J3 | 0.1048 |
4 | J4 | 0.1048 | J4 | 0.1048 |
5 | J5 | 0.0638 | J5 | 0.0638 |
6 | J6 | 0.1048 | J6 | 0.1048 |
7 | J7+J8+J12 | 1.81182 | J7+J8+J12 | 1.4157 |
8 | J9 | 3.41895 | J9 | 2.9841 |
9 | J11 | 3.70752 | J11 | 1.3382 |
Between | Rear Clutch Model | Front Clutch Model | ||
---|---|---|---|---|
Stiffness | Values (Nm/rad) | Stiffness | Values (Nm/rad) | |
1–2 | k1 | 2.56 × 106 | k1 | 2.56 × 106 |
2–3 | k2 | 2.56 × 106 | k2 | 2.56 × 106 |
3–4 | k3 | 2.53 × 106 | k3 | 2.53 × 106 |
4–5 | k4 | 2.56 × 106 | k4 | 2.56 × 106 |
5–6 | k5 | 2.56 × 106 | k5 | 2.56 × 106 |
6–7 | k6 | 20.87 × 106 | k6 | 20.87 × 106 |
7–8 | k7 | 12.67 × 106 | k7 | 4.683 × 106 |
7–9 | k8 | 0.045961 × 106 | k8 | 0.030158 × 106 |
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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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
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 StyleMihălcică, 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