1 Introduction
2 Methods for Mesh and Sideband Vibrations
2.1 Superposition and Modulation Methods
2.2 Periodic Excitations of Gear Meshing
2.3 Mesh and Planet-Pass Vibrations
2.3.1 Mesh-Frequency Vibration
Exciting conditions | Mesh-frequency vibrations |
---|---|
\(m = \pm q_{\text{m}} N \pm l_{\text{m}} Z_{\text{r}}\)
| The mth vibration is excited by the lmth mesh excitation |
\(m \ne \pm q_{\text{m}} N \pm l_{\text{m}} Z_{\text{r}}\)
| The mth vibration induced by the lmth mesh excitation is suppressed |
2.3.2 Planet-Pass Frequency Vibration
Exciting conditions | Planet-pass frequency vibrations |
---|---|
\(n = \pm\,q_{\text{s}} Z_{\text{r}} \pm l_{\text{s}} N\)
| The nth elastic vibration is excited by the lsth planet-pass-frequency excitation |
\(n \ne \pm\,q_{\text{s}} Z_{\text{r}} \pm l_{\text{s}} N\)
| The nth elastic vibration induced by the lsth planet-pass-frequency excitation is suppressed |
2.3.3 Transition between Ring and Planets
Exciting conditions | Mesh and planet-pass frequency vibrations |
---|---|
\(w = \pm l_{\text{m}} Z_{\text{r}} \pm l_{\text{s}} N\)
| The wth elastic vibration is excited by the lmth mesh-frequency and the lsth planet-pass-frequency excitations |
\(w \ne \pm l_{\text{m}} Z_{\text{r}} \pm l_{\text{s}} N\)
| The wth elastic vibration related to the lmth mesh-frequency and the lsth planet-pass-frequency excitations is suppressed |
3 Mesh Sideband Analysis
3.1 Signal from Ring-Planet Meshes
Exciting conditions | Sidebands |
---|---|
\(v_{\text{m}} = \pm l_{\text{m}} Z_{\text{r}} \pm Q^{\prime}_{\text{m}} N\)
| Sidebands at \(l_{\text{m}} Z_{\text{r}} \pm v_{\text{m}}\) are excited |
\(v_{\text{m}} \ne \pm l_{\text{m}} Z_{\text{r}} \pm Q^{\prime}_{\text{m}} N\)
| Sidebands at \(l_{\text{m}} Z_{\text{r}} \pm v_{\text{m}}\) are suppressed |
3.2 Signal from Ring Gear Tooth
Exciting conditions | Sidebands |
---|---|
\(v_{\text{s}} = \pm l_{\text{s}} N \pm Q^{\prime}_{\text{c}} Z_{\text{r}}\)
| Sidebands at \(l_{\text{s}} N \pm v_{\text{s}}\) are excited |
\(v_{\text{s}} \ne \pm l_{\text{s}} N \pm Q^{\prime}_{\text{c}} Z_{\text{r}}\)
| Sidebands at \(l_{\text{s}} N \pm v_{\text{s}}\) are suppressed |
3.3 Comparison between Signal Collections
Exciting conditions | Sidebands collected from planets | Sidebands collected from ring teeth |
---|---|---|
\(v = \pm l_{\text{m}} Z_{\text{r}} \pm l_{\text{s}} N\)
| Sidebands at lmZr ± vm are excited | Sidebands at lcN ± vs are excited |
\(v \ne \pm l_{\text{m}} Z_{\text{r}} \pm l_{\text{s}} N\)
| Sidebands at lmZr ± vm are suppressed | Sidebands at lcN ± vs are suppressed |
4 Unique Vibration of Helical Ring Gear
GCDs | Planet counts | Exciting conditions | Wavenumbers | Modulating orders |
---|---|---|---|---|
C = 1 | N = 2, 3 | lmZr = qsN or lsN = qrZr |
Q
1
N
|
Q
2
N
|
lmZr ± 1 = qsN or lsN ± 1 = qrZr | Q1N ± 1 | Q2N ± 1 | ||
N ≥ 4 | lmZr = qsN or lsN = qrZr |
Q
1
N
|
Q
2
N
| |
lmZr ± 1 = qsN or lsN ± 1 = qrZr | Q1N ± 1 | Q2N ± 1 | ||
Others | Q1N ± S1 | Q2N ± S1 | ||
1 < C < N | N = 2, 3 | lmZr = qsN or lsN = qrZr |
Q
1
N
|
Q
2
N
|
N ≥ 4 | lmZr = qsN or lsN = qrZr |
Q
1
N
|
Q
2
N
| |
Others | Q1N ± S2 | Q2N ± S2 | ||
C = N | – | lmZr = qsN or lsN = qrZr |
Q
1
N
|
Q
2
N
|
5 Numerical Verification
Items | Case I | Case II | Case III |
---|---|---|---|
Sun gear tooth count | 40 | 38 | 39 |
Ring gear tooth count | 64 | 66 | 65 |
Planet tooth count | 12 | 14 | 13 |
Planet count | 4 | ||
Ring gear | |||
Inner radius (m) | 0.116 | ||
Outer radius (m) | 0.128 | ||
Axial length (m) | 0.020 | ||
Elastic modulus (GN/m2) | 206 | ||
Density (kg/m3) | 7.85 × 103 | ||
Poisson ratio | 0.3 |
5.1 Free Vibration
Modes | Wavenumber w | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
RIN | – | – | 508 | 1428 | 2711 | 4328 | 6254 | 8462 | 10924 | 13616 |
OPB | – | – | 715 | 2123 | 4086 | 6501 | 9284 | 12362 | 15677 | – |
EXT | 6710 | 9466 | 14946 | – | – | – | – | – | – | – |
TOR | 5753 | 6679 | 8937 | 11801 | 14929 | – | – | – | – | – |