Forschung im Ingenieurwesen

, Volume 76, Issue 1–2, pp 15–31 | Cite as

Theoretical frequency analysis of vibrations from planetary gearboxes

Originalarbeiten/Originals

Abstract

A phenomenological model descriptive of the vibrations generated on a single-stage planetary gearbox when measured with a sensor mounted on the outer part of the ring gear is developed in time domain. First, only the internal meshing processes are considered. The model is analyzed relying on the Fourier theory to provide a detailed theoretical background which explains the different spectral structures observed on planetary gearbox vibrations. It is proposed that planetary gearboxes can be classified in four groups each presenting specific features on the structure of the vibration spectrum. Each group is carefully analyzed and illustrated with examples. Afterwards, the influences on the vibrations of the external gear meshing processes and the transmission path of vibrations from their source to the sensor through the sun gear and carrier plate are studied. It is concluded that the external meshing processes can affect all spectral components of the total vibrations differently, depending on if the number of the teeth of the planet gears is even or odd. Differently, the transmission path through the sun gear and carrier plate affects only the spectral components at the gear mesh frequency and its harmonics. Considering the factors presented in this work, the expected spectral components of the vibrations measured on any non-faulty planetary gearbox can be estimated a-priori, thus representing a valuable aid in view of failure diagnosis on these mechanical systems.

List of

\(a_{i}^{r}(t)\)

Amplitude modulation function of the vibrations \(v_{i}^{r}(t)\)

\(a_{i}^{s}(t)\)

Amplitude modulation function of the vibrations \(v_{i}^{s}(t)\)

\(A_{SC}^{r}\)

Constant with value in [0;1]

\(A_{SC}^{s}\)

Constant with value in [0;1]

f

Frequency variable

\(\mathcal{F}\)

Fourier transform operator

fC

Rotational frequency of the carrier plate [Hz]

\(f_{g}^{p}\)

Gear mesh frequency for a planetary gearbox

i

Index for the planet gear (i=1,2,…,N)

j

\(\sqrt{-1}\)

k

Tooth meshing harmonic index (k∈ℤ)

N

Number of planet gears

q

Sideband harmonic index (q∈ℤ)

t

Measurement time variable

t1

Time shift due to the unknown position of the planet gears at the measurement start [s]

TC

Rotational period of the carrier plate (TC=1/fC)

\(T_{g}^{p}\)

Gear mesh period (\(T_{g}^{p}=1/f_{g}^{p}\))

\(\mathit{TP}_{R}^{r}\)

Direct transmission path of the vibrations generated in the planet/ring gear meshing processes through the ring gear

\(\mathit{TP}_{R}^{s}\)

Direct transmission path of the vibrations generated in the sun/planet gear meshing processes through the ring gear

\(\mathit{TP}_{SC}^{r}\)

Transmission path of the vibrations generated in the planet/ring gear meshing processes through the sun gear and carrier plate

\(\mathit{TP}_{SC}^{s}\)

Transmission path of the vibrations generated in the sun/planet gear meshing processes through the sun gear and carrier plate

\(v_{i}^{r}(t)\)

Vibration generated in the meshing between the i-th planet gear and the ring gear, as experienced by an observer standing in the carrier plate

\(v_{i}^{s}(t)\)

Vibration generated in the meshing between the i-th planet gear and the sun gear, as experienced by an observer standing in the carrier plate

\(v^{r}(t)|_{\mathit{TP}_{R}^{r}}\)

Vibrations generated in the planet/ring gear meshing processes, transmitted to the sensor directly to the ring gear

\(v^{s}(t)|_{\mathit{TP}_{R}^{s}}\)

Vibrations generated in the sun/planet gear meshing processes, transmitted to the sensor directly to the ring gear

\(v^{r}(t)|_{\mathit{TP}_{SC}^{r}}\)

Vibrations generated in the planet/ring gear meshing processes, transmitted through the sun gear and carrier plate

\(v^{s}(t)|_{\mathit{TP}_{SC}^{s}}\)

Vibrations generated in the sun/planet gear meshing processes, transmitted through the sun gear and carrier plate

\(x_{i}^{r}(t)\)

Amplitude-modulated vibration generated in the meshing between the i-th planet gear and the ring gear, as experienced by the sensor

\(x_{i}^{s}(t)\)

Amplitude-modulated vibration generated in the meshing between the i-th planet gear and the sun gear, as experienced by the sensor

xr(t)

Total vibrations experienced by the sensor, considering only the meshing with the ring gear and the transmission through the ring gear

xs(t)

Total vibrations experienced by the sensor, considering only the meshing with the sun gear and the transmission through the ring gear

xr,s(t)

Total vibrations experienced by the sensor, considering the meshing with the ring gear and sun gear and the transmission through the ring gear

x(t)

Total vibrations experienced by the sensor, considering the meshing with the ring gear and sun gear and the transmission through the ring gear and through the sun gear and carrier plate

\(X_{i}^{r}(f)\)

Fourier transform of \(x_{i}^{r}(t)\)

\(X_{i}^{s}(f)\)

Fourier transform of \(x_{i}^{s}(t)\)

Xr(f)

Fourier transform of xr(t)

Xs(f)

Fourier transform of xs(t)

Xr,s(f)

Fourier transform of xr,s(t)

X(f)

Fourier transform of x(t)

ZP

Number of teeth of the planet gear

ZR

Number of teeth of the ring gear

ZS

Number of teeth of the sun gear

\(\alpha_{q}^{r}\)

Fourier coefficients of \(a_{1}^{r}(t)\)

\(\alpha_{q}^{s}\)

Fourier coefficients of \(a_{1}^{s}(t)\)

\(\beta_{k}^{r}\)

Fourier coefficients of \(v_{1}^{r}(t)\)

\(\beta_{k}^{s}\)

Fourier coefficients of \(v_{1}^{s}(t)\)

γri

Phase difference between \(v_{i}^{r}(t)\) and \(v_{1}^{r}(t)\)

γsi

Phase difference between \(v_{i}^{s}(t)\) and \(v_{1}^{s}(t)\)

γsr

Phase difference between \(v_{i}^{r}(t)\) and \(v_{i}^{s}(t)\)

δ(z)

Dirac delta: δ(z)=1 if z=0, and δ(z)=0, if z≠0

θ1

Relative angle between planet 1 and the sensor at the measurement start (t=0)

ψi

Relative angle between the position of the i-th planet gear and the first planet gear (ϕ1=0)

Natural numbers

0

Natural numbers including zero

Real numbers

Integer numbers

R

Angular pitch of the ring gear

S

Angular pitch of the sun gear

Convolution product: a(x)∗b(x)=∫a(xX)b(X)dX

Theoretische Frequenzanalyse für Schwingungen von Planetengetrieben

Zusammenfassung

In diesem Beitrag wird ein phänomenologisches Model zur Beschreibung der in einem einstufigen Planetengetriebe auftretenden Schwingungen vorgeschlagen. Das Model wird für den Fall hergeleitet, wenn die Schwingungen mit einem fest montierten Sensor an der äusseren Oberfläche des Hohlrades gemessen werden. Zuerst wird nur die Innenverzahnungen berücksichtigt. Der Frequenz-Inhalt des Models wird dann mittels der Fourier-Theorie untersucht. Die Ergebnisse lassen den Grund für die im Planetengetriebe beobachteten unterschiedlichen Schwingungsspektrummuster theoretisch erklären. Basierend auf diesen Ergebnissen wird vorgeschlagen, Planetengetriebe in vier Gruppen zu klassieren, wobei jede Gruppe bestimmte Merkmale des Schwingungsspektrums aufweist. Jede Gruppe wird analysiert und durch Beispiele illustriert. Es wird analysiert, wie die Aussenverzahnungen und die Übertragungswege von der Schwingungsquelle bis zum Sensor durch das Sonnenrad und den Planetenträger, die resultierenden Schwingungen beeinflussen. Die Aussenverzahnung beeinflusst alle Schwingungsspektrumslinien in unterschiedlicher Weise, in Abhänhigkeit davon, ob die Zähneanzahl der Planetenräder gerade oder ungerade ist. Andererseits beeinflusst der Schwingungsübertragungsweg durch das Sonnenrad und den Planetenträger nur die Spektrallinien, dessen Frequenz gleich der Zahneingriffsfrequenz und ihrer Vielfachen ist. Unter Berücksichtigung der in diesem Beitrag präsentierten Faktoren, lassen sich die erwarteten Spektrallinien der Schwingungen, die auf einem fehlerfreien einstufigen Planetengetriebe gemessen werden, im Voraus abschätzen. Dies unterstützt die Fehlerdiagnose dieser mechanische Systeme.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Laboratory of Mechanical VibrationsUniversity of ConcepciónConcepciónChile

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