Investigation on Mesh and Sideband Vibrations of Helical Planetary Ring Gear Using Structure, Excitation and Deformation Symmetries
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Abstract
Time-variant excitations in planetary gear trains can cause excessive noise and vibration and even damage the system on a permanent basis. This paper focuses on the elastic vibrations of a helical planetary ring gear subjected to mesh and planet-pass excitations. Motivated by the structure, excitation and deformation symmetries, this paper proposes dual-frequency superposition and modulation methods to capture the mesh and sideband vibrations. The transition between ring gear tooth and planet is introduced to address the excitations and vibrations. The phasing effect of ring gear tooth and planet on various deformations is formulated. The inherent connections between the two types of vibrations are identified. The vibrations share identical exciting rules and the wavenumber and modulating signal order both equal the linear combination of tooth and planet counts. The results cover in-plane bending and extensional, out-of-plane bending and torsional deformations. Main findings are verified by numerical calculation and comparisons with the open literature. The analytical expressions can be used to determine whether the sideband is caused by component fault or only by elastic vibration. The methods can be extended to other power-transmission systems because little restriction is imposed during the analysis.
Keywords
Helical planetary ring gear Typical vibration modes Planet phasing SidebandList of Symbols
- N
planet count
- Z_{r}
tooth count of ring gear
- \(\psi_{i}^{\text{c}}\)
spatial phase of the ith (i = 1, 2, 3, …, N) planet
- \(\psi_{i}^{\text{r}}\)
spatial phase of the ith tooth of ring gear
- \(\varphi_{i}^{\text{c}}\)
time phase of the mesh excitation wave at the ith mesh position
- \(\varphi_{j}^{\text{r}}\)
time phase of the planet-pass excitation wave on the jth (j = 1, 2, 3, …, Z_{r}) ring gear tooth
- \(V_{i}^{{l_{\text{m}} }}\)
the l_{m}th harmonic excitation at the ith mesh position
- \(W_{j}^{{l_{\text{s}} }}\)
the l_{s}th harmonic excitation on the jth ring gear tooth
- \(l_{\text{m}}\)
harmonic order of mesh excitation
- \(l_{\text{s}}\)
harmonic order of planet-pass excitation
- \(A_{i}^{{l_{\text{m}} }}\)
amplitude of l_{m}th mesh excitation at the ith mesh position
- \(B_{j}^{{l_{\text{s}} }}\)
amplitude of l_{s}th planet-pass excitation on the jth ring gear tooth
- \(m\)
wavenumber regarding mesh excitation
- \(n\)
wavenumber regarding planet-pass excitation
- \(\theta_{\text{m}}\)
position angle in carrier-fixed frame
- \(\theta_{\text{s}}\)
position angle in ring-fixed frame
- \(\omega_{\text{f}}\)
rotating speed of planet
- \(\gamma_{\uptheta}^{\text{c}}\)
position lag between mesh excitation and response
- \(\gamma_{\text{r}}^{\text{c}}\)
time lag between mesh excitation and response
- \(\gamma_{\theta }^{\text{r}}\)
position lag between planet-pass excitation and response
- \(\gamma_{\text{r}}^{\text{r}}\)
time lag between planet-pass excitation and response
- q_{m}, q_{s}
integers
- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma _{\text{t}}^{{c}}}\)
harmonic excitation phase in carrier-fixed frame
- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma _{\text{t}}^{\text{r}}}\)
harmonic excitation phase in ring-fixed frame
- \(x_{i}^{{v_{\text{m}} }}\)
amplitude-modulating signal at the ith planet
- \(a_{i}^{{v_{\text{m}} }}\)
amplitude of the amplitude-modulating signal
- \(v_{\text{m}}\)
harmonic order of the amplitude-modulating signal
- \(y_{i}^{{l_{\text{m}} }}\)
the l_{m}th signal generated at the ith mesh position
- \(b_{i}^{{l_{\text{m}} }}\)
amplitude of the mesh signal
- \(Q^{\prime}_{\text{m}}\), \(Q^{\prime}_{\text{c}}\)
integers
- C
GCD of the ring gear tooth and planet counts
- f_{m}(θ_{m}, t)
harmonic excitation in the carrier-fixed frame
- f_{s}(θ_{s}, t)
harmonic excitation in the ring-fixed frame
- \(C_{\text{c}}^{{l_{\text{m}} }}\)
the l_{m}th harmonic amplitude
- \(D_{\text{r}}^{{l_{\text{s}} }}\)
the l_{s}th harmonic amplitude
1 Introduction
Noise and vibration reductions are attractive topics in planetary gear trains (PGT), especially for those induced by the mesh and planet-pass excitations. The typical vibrations of a spur ring gear were analytically examined, and the relationships between mesh phase and in-plane elastic vibrations were identified based on symmetry [1]. While similar symmetry is held in helical PGT, the elastic vibration can be more complex to analyze. A question that can possibly be encountered during analysis is how the vibrations change.
Published studies directly or indirectly use the structure and excitation symmetries to capture vibration nature. The studies can be roughly considered in three groups: typical vibration modes [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], planet phasing [1, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], and sideband [35, 36, 37, 38, 39, 40, 41, 42, 43, 44], where the first group relates free vibration, but the last two link forced and parametric ones induced by the mesh or planet-pass excitations. Regardless of excitation patterns, free or forced, rigid or elastic, or even rigid-elastic coupling vibrations, motivations on the problem-solution method or physical explanations on the vibrations can be gained because the symmetric structure and excitation account for the distinctive vibrations.
Existing studies on the distinctive vibrations are generally based on spur PGT with the exception of Refs. [3, 8, 17, 21, 24], etc. The results imply that the vibrations can be classified into rotational, translational, and planet modes for spur PGT [5], but for helical one, it can be categorized as in phase, sequential phase, and counter phase [3], or rotational-axial, translational-tilting, and planet modes [8, 19]. As an extended research, Shi et al. [45] identified the typical vibration in centrifugal pendulum vibration absorbers. Qin et al. [20] examined the vibration modes of a horizontal wind turbine drive train, and they also studied the effect of bearing stiffness on these modes. Bu et al. [21] investigated the herringbone PGT and found the rotational and axial modes, translational modes, planet modes, rotational and axial ring modes, and translational ring modes. The above studies are generally based on rigid body assumption.
The typical vibration modes also arise if the elastic components such as the thin ring gear are considered [11, 12]. Wu and Parker [11] examined the PGT with elastic ring gear and identified the rigid-elastic vibration modes, and later Parker and Wu [12] proved the vibration modes of the PGT with unequally-spaced planets and elastic ring gear. Tanna and Lim [15, 16] examined the ring gear’s free vibration and proved the elastic vibration modes, including in-plane extensional, radial inextensional, out-of-plane bending, and torsional deformations. It seems that similar modes to those in Refs. [11, 12] can be found for the helical planetary ring gear even when considering various deformations owing to the symmetries. However, it could be a challenge to develop an analytical model incorporating many deformations. Even so, the theoretical analysis is another difficulty to be overcome.
Planet phasing is also a typical vibration, which has been well examined based on spur and helical PGT. Kahraman et al. [3, 24] addressed it and obtained general results governing the occurrence or suppression of the forced vibration for arbitrary number of planets. Parker et al. [25] proved the three forced vibration modes induced by the mesh excitation using superposition method, and then they extended it to helical PGT [19]. The phasing behaviors of the spur and helical PGT are in essence the same except for the specific excitations and vibration patterns. Canchi and Parker [28] studied the influence of mesh phase on parametric instability of the planetary ring gear and closed-form results were obtained. Yang and Dai [33] formulated the effect of mesh phase on primary and combination resonances by Multiple Scale method. As a common feature, most of the above studies employ the excitation superposition, through which the resulting response can be obtained and typical vibrations are identified.
Alternatively, Wang et al. [1] used the response instead of the excitation superposition to predict vibration of an elastic spur ring gear and found the phasing effect of basic parameters on typical flexural-extensional vibrations. The results imply that if the wavenumber is zero or unity, the rigid vibration modes are identical with those in Ref. [25]. While the responses of the helical ring gear can be more complex, typical vibrations could remain due to the mechanics similarities in their common symmetries, and the correspondence between the rigid and elastic vibrations could exist in helical PGT. These are the primary concerns of this work.
Another phasing is the mesh sideband associated with the instance where the principal spectrum component is slightly moved from the meshing frequency. Mcfadden and Smith [38] addressed the asymmetry of modulation sideband. To precisely predict the amplitudes of the dominant components, Mcnames [39] generalized their work using continuous-time Fourier series and presented more thorough and intuitive explanations on the observed spectrum. Zhang et al. [41] studied the relationships between tooth count, planet count, and sideband orders. Kahraman [35] asserted that sorting and aligning the planet run-out error in an in-phase configuration during assembly can minimize and in some cases even eliminate the errors effect on the dynamic load sharing. Then Inalpolat and Kahraman [36, 37] proposed a simplified model to describe the modulation sideband, and they developed an experimental PGT set-up to demonstrate the sideband from ring gear radial acceleration measurements. Based on the structure and mesh phase configurations, Vicuña [44] also obtained the analytical expressions governing the sideband. Aiming at the main excitation source, Singh [46, 47] presented a general physical understanding on the basic mechanism causing unequal load sharing. Gu and Velex [48] developed an original lumped parameter model and studied the influence of planet position errors. These researches gained valuable insights into the dynamic behaviors.
Among the aforementioned studies, Inalpolat and Kahraman [36], Mcfadden and Smith [38], Mcnames [39], Keller and Grabill [40], and Vicuña [44] introduced the assumption that the resulting vibration is equal to the sum of those at the ring-planet mesh positions. They found that the mesh frequency cannot be a component in response unless the ring gear tooth count is an integer multiple of planet count. Physically, it is the planet’s motion relative to the transducer that causes the transmitted vibration to vary and thus the mesh vibration to be modulated. Further, it is the mesh phases of the planets that cause the sideband occurrence, asymmetry, or even suppression. Like the vibration modes and planet phasing, the sideband and its asymmetry are both determined by the timing-relation between the inner excitations at different mesh positions. Based on these, more findings can be achieved when fully incorporating the structure and force symmetries of the helical PGT.
Since the two types of phasing phenomena link symmetry, a correspondence can exist between them. As a case in point, they share identical exciting rules [1, 38, 39, 41]. Wang et al. [1] employed the superposition principle to deal with the occurrence of spur planetary ring gear vibrations, and found that they are simply determined by lZ_{r} ± n = qN, where l, Z_{r}, N, and q are the mesh excitation harmonic, ring gear tooth count, planet count, and arbitrary integer. n denotes wavenumber, which is associated with bending and extensional deformations instead of only the former. Mcfedden and Smith [38], Mcnames [39], and Zhang et al. [41] utilized the signal modulation principle to address elastic vibration and obtained the same expression. This expression governs the mesh sideband as the same notation n, but it designates modulating signal order. To gain more physical insights into the two typical vibrations, further work is much needed. This is another focus of this work.
The ring gear vibration is of a classical traveling-load dynamic problem. Such problem has been well studied by many researchers [49]. Compared with those of asymmetric dynamic problems, the vibrations of the helical PGT bear their own distinctions because of the symmetry and timing-relation between the mesh excitations. Based on the prior study [1], the present work analyzes the unique forced vibration with dual-frequency superposition and modulation methods. To facilitate the analysis, the scope is limited to the helical ring gear, though the three-dimensional mesh and sideband vibrations, including the in-plane bending and extensional, out-of-plane bending and torsional deformations, and especially their relationships, are all incorporated. Main results are verified by numerical calculation and comparisons with the results in the open literature.
2 Methods for Mesh and Sideband Vibrations
2.1 Superposition and Modulation Methods
The prediction on the mesh and sideband vibrations can be difficult when using differential equations to describe the physical laws because the three-dimensional excitations create various deformations. Other treatments include the Finite Element and experimental methods, which are sometimes employed as verification tools. Motivated by the previous studies [25, 50, 51], authors of this work uses superposition and modulation methods to address the rigid and elastic vibrations of symmetric power-transmission systems [1, 52, 53, 54]. It is not based on an ordinary or partial differential equation but on the structure, force, and deformation symmetries.
In the previous work [1], the elastic mesh-phasing vibrations of spur planetary ring gears were addressed by the superposition method, which was motivated by the ultrasonic motors [50] and PGT [25], but only in-plane bending and extensional deformations were incorporated. Practically, whatever the deformations are, the various free vibrations are of elastic waves definitely with periodicity due to the ring gear’s closed shape. Since the vibrations at different mesh positions are identical except for a phase lag, the resulting vibration can be obtained via a simple superposition either for bending or extensional deformation. The same can be true for the helical planetary ring gears.
As another point of view, since there exist two types of excitations with mesh and planet-pass frequencies on the ring and planet sides for each and every type of deformation, the resulting response can be explained as a type of modulation. As a result, the same predictions can be made by the superposition and modulation treatments such that insights into the phasing behaviors especially the excitation harmonics can be gained. Authors of this work examined the elastic vibration of permanent magnet motors and obtained complementary results [52]. In another study [54], the rigid-elastic vibrations incorporating the frequency splitting and mode contamination were also analyzed by fully considering mechanical and magnetic symmetries. However, the above two studies are limited to the in-plane vibration. Following the similar procedure, various vibrations of the helical ring gears can be examined analytically.
This work examines the mesh and planet-pass excitations and vibrations of helical ring gears in order to clarify the structure-excitation-vibration relation. To obtain more general results and avoid a specific mathematical model, this work employs the model-free dual-frequency superposition and modulation methods. No over restriction is imposed during the analyses except for the symmetries.
2.2 Periodic Excitations of Gear Meshing
Figure 1(a)‒(c) illustrate the planet P passes over ring teeth T_{1} and T_{2}. During this process, the time-variant mesh excitation occurs. By contrast, Figure 1(d)‒(f) show the process between ring gear tooth T and planets P_{1} and P_{2}, where planet-pass excitation is produced. Symmetry ensures that the excitations on each planet and tooth are the same except for phase lags. The two types of excitations can co-exist even in healthy PGT, and consequently the mesh and planet-pass phasings arise simultaneously. Compared with the mesh-frequency phasing in spur PGT, those in the helical one are more complicated.
2.3 Mesh and Planet-Pass Vibrations
In Figure 2(b), \(\{ O_{\text{r}} - r_{\text{r}} ,\;\theta_{\text{r}} \}\) is a ring-fixed frame, where the polar axis is also directed towards the first tooth center such that the spatial initial phase \(\psi_{1}^{\text{r}}\) is zero. \(W_{j}^{{l_{\text{s}} + }}\) is the l_{s}th harmonic of the traveling wave excited by the excitation on the jth (\(j = 1, \, 2, \, 3, \ldots ,\,\,Z_{\text{r}}\)) ring gear tooth. Similarly, \(\varphi_{j}^{\text{r}}\) and \(B_{j}^{{l_{\text{s}} }}\) are the initial phase and amplitude, respectively. Note that Figure 2 only presents the in-plane bending deformation, but the following analyses also apply to the out-of-plane case.
2.3.1 Mesh-Frequency Vibration
Exciting conditions for elastic mesh-frequency phasing
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 l_{m}th mesh excitation |
\(m \ne \pm q_{\text{m}} N \pm l_{\text{m}} Z_{\text{r}}\) | The mth vibration induced by the l_{m}th mesh excitation is suppressed |
2.3.2 Planet-Pass Frequency Vibration
Similarly, symmetry leads to \(B_{j}^{{l_{\text{s}} }} = B_{1}^{{l_{\text{s}} }}\). The vibration here is different from the previous work [1] in the physical meaning due to the difference in excitations.
Exciting conditions for elastic planet-pass phasing
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 l_{s}th 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 l_{s}th planet-pass-frequency excitation is suppressed |
2.3.3 Transition between Ring and Planets
Exciting conditions for mesh and planet-pass vibrations
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 l_{m}th mesh-frequency and the l_{s}th 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 l_{m}th mesh-frequency and the l_{s}th planet-pass-frequency excitations is suppressed |
Comparison between Tables 1 and 2 implies that \(q_{\text{m}}\) can be the l_{s}th harmonic in the planet-pass excitation, and \(q_{\text{s}}\) can be the l_{m}th harmonic in the mesh excitation. Table 3 unifies the previous results and shows that whatever the tooth and planet counts are, the excited wavenumber is mathematically equal to their linear combination, where the coefficients are the harmonic orders of the excitations on the planet and ring gear teeth, respectively.
3 Mesh Sideband Analysis
The mesh and planet-pass excitations interact with one another leading to sideband. This section addresses it on the basis of symmetries, where the distinction is the treatment using two types of signal collection patterns with the modulation between the mesh and planet-pass signals.
3.1 Signal from Ring-Planet Meshes
Assuming that the l_{m}th signal generated by the ith mesh position is
Exciting conditions for sidebands excited by mesh excitations on planets
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 for sidebands excited by mesh excitations on ring gear teeth
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 for mesh sideband
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 l_{m}Z_{r} ± v_{m} are excited | Sidebands at l_{c}N ± v_{s} are excited |
\(v \ne \pm l_{\text{m}} Z_{\text{r}} \pm l_{\text{s}} N\) | Sidebands at l_{m}Z_{r} ± v_{m} are suppressed | Sidebands at l_{c}N ± v_{s} are suppressed |
4 Unique Vibration of Helical Ring Gear
This work uses analytical methods to deal with various excitations and presents simple expressions governing the occurrence or suppression of the mesh and planet-pass vibrations. The results imply that the two types of apparently isolated vibrations have inherent connections and share the same exciting conditions. The excited wavenumber is the linear combination of the ring gear tooth and planet counts, which is equal to the modulating signal order. The mesh and corresponding planet-pass vibrations are excited simultaneously, and the sideband is the external behavior of the excitations’ modulation.
Exciting conditions for elastic mesh and sideband vibrations
GCDs | Planet counts | Exciting conditions | Wavenumbers | Modulating orders |
---|---|---|---|---|
C = 1 | N = 2, 3 | l_{m}Z_{r} = q_{s}N or l_{s}N = q_{r}Z_{r} | Q _{1} N | Q _{2} N |
l_{m}Z_{r} ± 1 = q_{s}N or l_{s}N ± 1 = q_{r}Z_{r} | Q_{1}N ± 1 | Q_{2}N ± 1 | ||
N ≥ 4 | l_{m}Z_{r} = q_{s}N or l_{s}N = q_{r}Z_{r} | Q _{1} N | Q _{2} N | |
l_{m}Z_{r} ± 1 = q_{s}N or l_{s}N ± 1 = q_{r}Z_{r} | Q_{1}N ± 1 | Q_{2}N ± 1 | ||
Others | Q_{1}N ± S_{1} | Q_{2}N ± S_{1} | ||
1 < C < N | N = 2, 3 | l_{m}Z_{r} = q_{s}N or l_{s}N = q_{r}Z_{r} | Q _{1} N | Q _{2} N |
N ≥ 4 | l_{m}Z_{r} = q_{s}N or l_{s}N = q_{r}Z_{r} | Q _{1} N | Q _{2} N | |
Others | Q_{1}N ± S_{2} | Q_{2}N ± S_{2} | ||
C = N | – | l_{m}Z_{r} = q_{s}N or l_{s}N = q_{r}Z_{r} | Q _{1} N | Q _{2} N |
This classification includes three types of phasing relations: the in-phase, sequential-phase, and counter-phase [3], which corresponds to the cases where phasing factors are zero, one, and others, respectively [25]. The forgoing analyses verify that the sideband can arise even in healthy PGT, in particular when the elasticity is significant for the thin ring gear subjected to heavy load. For this case, the sideband can be even more significant because of the transition effect between the sensor and planets and the traditional planet-pass errors. Whether the specific mesh-frequency, sideband, or its harmonics are excited or not depends on the lead or lag timing-relation between different mesh positions, that is, the combination of the ring gear tooth and planet counts, and to what degree the excitation frequency is close to a natural frequency.
5 Numerical Verification
Specifications of three sample PGT with distinct phases
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/m^{2}) | 206 | ||
Density (kg/m^{3}) | 7.85 × 10^{3} | ||
Poisson ratio | 0.3 |
More specifically, the ring gear tooth and planet counts are different from one case to another. In the three cases, the GCDs are four, two, and one, respectively. The configurations create different mesh phases and lead to distinctive elastic vibrations. Since the tooth count is slightly changed to ensure comparability, the ring gear’s modal properties can be reasonably considered to be unchanged such that the main difference between the forced responses is mainly caused by the mesh phase.
5.1 Free Vibration
Natural frequencies (Hz)
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 | – | – | – | – | – |
5.2 Forced Vibration
The wavenumber in the analytical results governs various deformations. The distinct behaviors of the mesh and sideband vibrations are the excitation or neutralization of the harmonic mesh and planet-pass excitations due to their distinct phases. The fundamental reason behind this is the structure, excitation, and deformation periodicities (symmetries) in the three orthogonal directions. According to the current results, if the transducer is mounted in the axial direction, the same results can be obtained because the symmetries are still held.
6 Comparison and Discussion
The above sections examine the typical vibrations of the helical planetary ring gears. The analytical results regarding mesh vibrations have been verified by numerical method, though the sideband vibration has been not yet. This section makes the following comparisons with the existing results and discusses the mesh and planet-pass vibrations.
6.1 Analysis Method
This work employs the superposition and modulation methods to examine the mesh and sideband vibrations. Different from Refs. [1, 52, 53, 54, 55], this work introduces the dual-frequency excitations to deal with the elastic vibration and signal analysis. The modulation method has been used by the existing literature to address signal modulation between the classical mesh and shaft frequency excitations, though this work uses this idea to examine the excitation interaction between the helical ring gear and planets. Further, the deformations here cover more possibilities, including in-plane bending and extensional, out-of-plane bending and torsional deformations. The superposition and modulation methods have different views toward the mesh and sideband vibrations. The former is focused on the response related to individual component and phase lag needs to be introduced, but the later is concerned with the excitation on concentric components without introducing the phase lag. In spite of the differences, they have been successfully used to deal with the elastic vibration and signal analysis by introducing the dual-frequency excitations and signals.
6.2 Mesh Vibration
Mesh vibrations have been studied by previous researchers with various methods. Wu and Parker [11] and Parker and Wu [12] built up an elastic-discrete model and examined the rigid-elastic vibration of spur PGT, and they proved the rotational, translational, planet, and purely ring modes, each of which has specific ring gear wavenumber. Authors of this work obtained the first three elastic modes by using the superposition method [1, 52]. Since only the ring gear and ring-planet mesh are included there, the purely ring modes in Ref. [11] cannot be found. Similar vibration modes have been found for spur and helical ring gears, though the wavenumber links more deformations. The reason behind this is that the methods focus on the deformation’s period instead of its amplitude such that they have general meaning. This has been verified by the comparison between the results in this work and those in Ref. [1].
6.3 Sideband Vibration
Mcfadden and Smith [38], Mcnames [39], and Zhang et al. [41] examined the sideband vibration and gained much valuable insights. Compared with the previous studies, the distinction here is the dual-signal collection aimed at the two typical excitations. In doing so, the relationships between the sideband vibrations are identified. They are similar in mathematical expressions for different excitations. The sideband order is equal to the modulating signal order. This has practical meaning in fault diagnosis, where special attention needs to be paid to the recognition between the normal elastic vibration and those excited by planet-pass errors in order to identify the actual fault, especially for thin ring gear subjected to heavy load. While this work takes the radial direction as an example, the results are not confined by this. The same results can be obtained by detecting axial deformation without any essential changes to the above analyses.
6.4 Relation between Mesh and Sideband Vibrations
The mesh and planet-pass excitations interact with one another through amplitude modulation leading to mesh sidebands. The present results imply that the elastic vibration and sidebands share the same exciting rules. That is, the excited wavenumber is determined by the mesh and planet-pass excitations, and the same is true for the sideband order. The excited wavenumber and modulating signal order are always mathematically equal to each other. Whether the mesh, sideband, or their harmonics can be excited or not depends on the phase lag, and to what extent the exciting frequency is close to the natural frequency. Although some insights have been gained in this work, only the amplitude modulation is considered. More efforts can be made to identify the source of additional harmonics induced by frequency modulation, and in particular the connections between the mesh and sideband vibrations of the helical ring gears when incorporating the base motions [56], worm gear [57], and even integrated system [58].
7 Conclusions
- (1)
This paper examines the mesh and sideband vibrations of the helical planetary ring gears. Methods and results have been verified by numerical approach and strict comparisons with the existing studies.
- (2)
Dual-frequency superposition and modulation methods are used to address the mesh and sideband vibrations. The relationships between the ring gear tooth count, planet count, and unique vibrations are obtained as simple closed-form expressions.
- (3)
Two types of excitations are combined to examine the mesh and sideband vibrations, which is based on the structure and excitation symmetries and action-reaction relation between the engaged components.
- (4)
The wavenumbers regarding the three distinct mesh phases are QN, QN ± 1, and QN ± (C, 2C, 3C,…, C × INT(N/(2C))), each of which creates distinct sidebands and covers in-plane bending and extensional, out-of-plane bending and torsional deformations.
- (5)
Sidebands can be induced by elastic vibration instead of the planet-pass errors only, where the wavenumber is equal to modulating harmonic order. The mesh and sideband vibrations are determined by the lead or lag timing-relation between the mesh positions and the excitations.
- (6)
Symmetry provides a basis for the superposition and modulation methods, which is also the root cause of the typical vibrations. Since no other restriction is imposed except for symmetry, the methods can find application in the symmetry-related issues in other power-transmission systems.
Notes
Authors’ Contributions
S-YW was in charge of the whole trial; S-YW and CM wrote the manuscript. Both authors read and approved the final manuscript.
Authors’ Information
Shi-Yu Wang, born in 1974, is currently a professor at School of Mechanical Engineering, Tianjin University, China. His primary research interest lies within the dynamics of the high-speed rotary power and transmissions, including the spur/helical/bevel planetary gear trains, permanent magnet motors, rolling ball bearings, and other transmissions having structural symmetry. Recent activities have focused on the dynamic characteristics of the rotary ultrasonic motors and especially the general cyclic symmetric systems having counter-rotating components.
Chanannipat Meesap, born in 1992, is currently a master candidate at School of Mechanical Engineering, Tianjin University, China. He received his bachelor degree from Rajamangala University of Technology Thanyaburi, Thailand, in 2014. His research interests include dynamics analysis of gear transmission system, parametric design based on instability, and the design of rotating components.
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Natural Science Foundation of China (Grant Nos. 51175370, 51675368), Application of Basic Research and Frontier Technology Research Key Projects of Tianjin, China (Grant No. 13JCZDJC34300), and National Basic Research Program of China (973 Program, Grant No. 2013CB035402).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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