In this section, the motion behaviors of a single faulty tooth of a planet gear will be discussed. Intuitively, concerning a fixed-sensor, the motion behavior of the planet gear fault-meshing positions should follow some rules. The trouble is that the planet gear faulty tooth may sometimes mesh with the ring gear or sometimes mesh with the sun gear. In this regard, these two conditions will be discussed in the following, respectively.
2.2.1. Initial Fault-Meshing Position at Ring Gear
Assume an initial fault-meshing position that the faulty tooth of a specific planet gear is meshing with the ring gear locating closest in line with the sensor. When the planet gear faulty tooth is again meshing with the ring gear at the same position, a period is completed.
Figure 5 illustrates the schematic of this period.
In
Figure 5, the solid dot represents the faulty tooth of planet gear 1 is meshing with the ring gear. Rotational directions of the sun gear, the analyzed faulty planet gear, and carrier are also annotated. During one rotation cycle of planet gear 1, the faulty tooth will mesh with the sun gear or the ring gear at different positions. Finally, after several rotations of the planet gear 1 or several rotations of the carrier, a period will complete and then the next period will restart. Inspired by [
26], this period is, in essence, similar to the tidal period of the sun gear fault-meshing position. The tidal period of planet gear fault-meshing position should be the analyzed planet gear and carrier both rotating minimal integer cycles, simultaneously. In the following, we will derive this period in terms of rotation angles.
Here, we only consider the fault-meshing position is occurring on the ring gear. In such a scenario, the fault-meshing times only counted once during one complete rotation of the planet gear. Between every two times of the planet gear faulty tooth meshing with the ring gear, we use
to represent the rotation angle of the carrier and
to describe the smallest angle between the fault-meshing point to the sensor, both are shown in
Figure 6.
In
Figure 6, the gears are simplified as circles so that the rotation angles can be conveniently demonstrated. Referring to the fixed sensor, from one fault-meshing point to the next, the rotation angle of planet gear 1,
, must be
radians. The time duration between each fault-meshing,
, can be therefore determined by the rotation angle between two times of fault-meshing divides the angular speed of the planet gear, as is given in Equation (
5):
where
means the rotational frequency of the planet gear measured in Hz. The rotation angle of planet carrier between every two times of fault-meshing of planet gear 1,
, can be determined in the equation below:
where
is measured in radians;
denotes the angular speed of the carrier measured in radians per second and
denotes the rotational frequency of carrier measured in Hz. Along with the operation of planet gear 1 and the carrier, the faulty tooth meshing with ring gear will occur 1st, 2nd ... etc. times. Eventually, at
th fault meshing (
is an integer), the faulty tooth of planet gear 1 will first return to the initial meshing position as shown in
Figure 6a. If we use
to represent the total rotation radians of planet gear 1, and
to represent the total rotation radians of the carrier, both total radians are at the
th fault-meshing:
and
where
and
are both measured in radians.
According to the criteria of the tidal period of the planet gear fault-meshing positions, the planet gear 1 and the carrier must rotate complete cycles, simultaneously. In other words, this means
and
must be an integer number of
, simultaneously. Equation (
7) tells that for any integer value of
,
is integer number of 2
. However,
from Equation (
8), which containing a ratio between
and
, bring difficulty in determining the integer cycles. According to [
22],
and
can be established by the gear-meshing frequency of planetary gearboxes, as is shown in the following equation:
where
and
represent the teeth number of the planet gear and the ring gear, respectively. We substitute Equation (
9) into Equations (
6) and (
8) and then get the expressions of
and
:
and
Equations (
10) and (
11) reveal that between two times of fault-meshing the carrier rotational angle, and total rotational angle of the carrier to be a tidal period, intrinsically depends on the teeth number of planet gear and ring gear. These teeth numbers are necessarily integer number.
Notice that both
and
contains an integer term
, it is reasonable to take a ratio between
and
to cancel the common divisor:
where
and
represent the rotational cycles of the faulty planet gear and carrier to be a tidal period (
). Suppose a greatest common divisor of the numerator and denominator exist in Equation (
12), namely
where
means the greatest common divisor of the whole terms in the bracket.
Equation (
13) tells that, when the planet gear rotates
cycles and simultaneously carrier rotates
cycles, a tidal period is completed. Identically, we can also use the least common multiple to represent the rotation cycles in a tidal period:
and
The total fault meshing times in a tidal period can be therefore determined based on Equation (
14) or Equation (
15):
In this section, with consideration of the initial fault-meshing position that the faulty tooth of a planet gear meshing with the ring gear, the motion period of the fault-meshing positions is derived in terms of the rotation angle, which intrinsically depends on the teeth number of a planetary gearbox. The time duration of this motion period can be determined by the rotational cycles of carrier or planet gear:
where
represents the motion period of the faulty tooth of a planet gear initially meshing at the ring gear. In such a scenario, the period of the fault amplitudes in Equation (
4) should be
. This period should be considered during the signal modeling within faulty planet gear so that some novel fault related sidebands can be revealed. In the next section, the motion period of the faulty tooth initially meshing with the sun gear will be discussed.
2.2.2. Initial Fault-Meshing with Sun Gear
Now assume the initial fault-meshing position that the faulty tooth of a planet gear meshes with the sun gear locating closest in line to the fixed sensor. When the planet gear faulty tooth again meshing with the sun gear at the same position, a period is completed. We give the schematic of this period in
Figure 7.
If each tooth of sun gear is treated identically, the period of the fault-meshing positions equals to
which we have derived in the
Section 2.2.1. Because this period only requires the analyzed planet gear and planet carrier rotating complete cycles, simultaneously. In practice, however, imperfections due to the inevitable manufacturing assembly errors or an uneven load or after a period of operation, different sun gear tooth may suffer from different working conditions. The teeth of sun gear should not be treated identically due to the different tooth will result in different vibrations pattern. In this regard, the period of the faulty tooth of a planet gear meshing with a specific tooth of sun gear will be thoroughly discussed.
Inspired by the derived motion period in
Section 2.2.1, the motion period of a faulty tooth meshing with a specific sun gear tooth requires the sun gear, the analyzed planet gear, and the carrier rotating integer cycles, simultaneously. We symbolize this typical motion period as
and will also derive this period in terms of rotational angles.
Here, we only focus on the fault-meshing point occurring on the sun gear. Correspondingly, this means the fault-meshing times only be counted once with respect to one complete rotation cycle of the faulty planet gear. Between each two times of the faulty tooth of a planet gear meshing with the sun gear, the rotation angle of the carrier,
, and the rotation angle of the specific tooth on sun gear,
, both are shown in
Figure 8.
Between every two times of fault-meshing, the rotation angle of the analyzed planet gear,
, equals to
radians; the rotation angle of carrier has been derived in Equation (
8); the rotation angle of the specific tooth on the sun gear,
, should equal to the angular speed of sun gear shaft,
, multiply by the time duration,
in Equation (
5), as is given in Equation (
18):
where
means the rotational frequency of sun gear shaft which is measured in Hz. The relationship between
and
can be determined through the meshing frequency [
22]:
where
represents the tooth number of the sun gear. According to the machinery handbook [
32], the relationship of the teeth number in a planetary gearbox must subject to the following rule:
Based on Equations (
9), (
19) and (
20),
can be rewritten in the following:
Equation (
21) reveals that
also only depending on the teeth number of the sun gear and planet gear.
Take
Figure 8a as the initial position, assume that after
times of fault-meshing, the faulty tooth on planet gear 1 again meshing with the specific tooth of the sun gear. At the same time, the total rotation radians of planet gear 1, the planet carrier and the sun gear can be expressed as follow:
where
means the total rotation radians of the faulty planet gear;
means the total rotation radians and
means the total rotation radians of the sun gear, all of them are to be a motion period. Based on the requirement of this motion period, all of the rotational radians must be integer multiple of
, simultaneously. A ratio will be taken among Equations (
22)–(
24) to cancel the common divisor:
where
,
and
represent the number of rotations of sun gear, faulty planet gear and carrier in a motion period. Actually, determining the minimum integer rotation cycles of the three components is to find the least common denominator of Equation (
25) and the number of rotating cycles of the carrier can be determined:
where
means calculating the least common denominator of all the terms in the bracket. Correspondingly, the rotation cycles of the sun gear and planet gear to be a motion period can be determined:
The time duration to be a tidal period depend on the number of rotations of the shaft and the input rotational frequency, as is shown:
where
means the time duration of a motion period in which the initial fault-meshing position at sun gear. In such a scenario, the motion period of fault induced amplitudes in Equation (
4) should be
. This period considering the sun gear teeth are different from each other and more complete fault induced information is covered. Therefore, the vibration data measured in this period can be used to diagnostic compound faults, such as the faults occurred on both sun gear and planet gear teeth effectively.
In
Section 2.2.1 and
Section 2.2.2, two typical motion periods for the faulty tooth of the planet gear are discussed. These periods, which intrinsically based on the teeth number of planetary gearboxes, are unique natures to describe the kinematic motion behaviors of the fault-meshing positions. Since the time duration of
covers entire fault induced vibration by all the possible fault meshing positions, we will discuss the application of this period for fault diagnosis by experimental studies.