Planet Modes Research Articles

Nonlinear dynamics of lumped-parameter planetary gears with general mesh phasing

This work derives analytical solutions for resonances and parametric instabilities of all rotational, translational, and planet modes, regardless of the degeneracy/multiplicity of the natural frequencies, for planetary gears with time-varying mesh stiffness excitation and tooth separation nonlinearity. Degenerate modes are the focus, but the results apply to all modes. For a degenerate mode resonance, use of mesh phasing results and modal properties from the literature reduces a problem with multiple, unknown, coupled modal amplitudes to one with a single unknown modal amplitude. This reduction allows derivation of a closed-form amplitude–frequency relation that is algebraically prohibitive otherwise. The analytical solution reveals important features such as the peak resonant amplitude and tooth contact loss occurrence. An instability suppression rule governs the occurrence or suppression of a potential parametric instability between any two natural frequencies, regardless of their multiplicity. Closed-form instability boundaries that bound the range of mesh frequencies where an unsuppressed parametric instability occurs are derived with and without damping. Counter-intuitively, damping can destabilize a parametric instability between two different natural frequencies by enlarging the instability bandwidth. The critical damping at which a parametric instability is eliminated is determined analytically. For parametric instabilities of a common type, the amplitude of nonlinear response with tooth separation is derived in closed-form. Numerical integration of a planetary gear model verifies all the analytical results. These numerical results also show rich nonlinear behaviors near parametric instabilities such as jump phenomena, period-doubling/halving, and quasi-periodic response.

Modal properties and parametrically excited vibrations of spinning epicyclic/planetary gears with a deformable ring

This work investigates the modal properties and parametric instabilities of high-speed spur epicyclic and planetary gears with an elastically deformable ring. Coriolis (i.e., gyroscopic) and centripetal acceleration effects are modeled. The gyroscopic effects lead to complex-valued (i.e., traveling-wave) modes. These modes fall into three categories: rotational, translational, and planet modes. Each category has highly structured modal deflections. The modal properties are proved analytically by discretizing the elastic ring such that the system falls into the class of general cyclically symmetric systems with proven modal properties. Such a proof readily extends to helical planetary gears with any or all of the sun, carrier, and ring modeled as elastic bodies. The time-varying sun-planet and ring-planet mesh stiffnesses are the sources of parametric excitation. The conditions where combinations of the mesh frequency and mesh stiffness amplitudes cause parametric instabilities are determined in closed form with the method of multiple scales. The analytical results agree with numerical results from Floquet theory. Many potential parametric instabilities are suppressed. A rule to determine whether a given potential parametric instability is suppressed or not is given. The rule is closely linked to the modal properties noted above and the planet mesh phasing parameters (sun and ring tooth numbers and number of planets).

Modal Properties of Contra-rotating Encased Differential Gear Train Used in Coaxial Helicopter

Planetary gear train is widely used in the aerospace and automotive industries, but the vibration problems caused by the dynamic meshing forces and bearing forces of the system affect its reliability and longevity. Modal analysis is the basis for optimal design of structural dynamic characteristics. In this paper, the characteristics of natural frequencies and mode shapes of an encased differential gear train for the main reducer of a coaxial helicopter are analyzed. Considered with the coupling between stages and gyroscopic effect, a translational–rotational dynamic model for the planetary gear train which can transmit a single input to two contra-rotating outputs with equal speeds is established by the lumped-parameter method. Based on this model, the modal properties of the system are studied by numerical simulation and mathematically proved. The results indicate that all vibration modes of the planetary gear train can be categorized as four classes: rotational mode, translational mode, stepped planet mode and planet mode, and the typical mode shapes are graphically displayed. The magnitude of the natural frequencies corresponding to the planet mode and the stepped planet mode is not affected by the number of planets.

Modal properties of a multi-stage power-split planetary gear set

Planetary gears are recognized as one of the most effective forms of power transmission. A dynamic model of an N-stage power-split planetary gear set is proposed in this article. Three-dimensional translations and axial rotation of all members, mesh stiffness, bearing stiffness, and the interstage coupling stiffness are considered. Subsequently, equations of motion of the whole system are derived. Natural frequencies and vibration modes are obtained by solving the eigenvalue problem governed by corresponding equations. According to the modal properties, all vibration modes of the N-stage system can be categorized as N + 2 groups: the overall axial translational–rotational mode, the overall radical translational mode, and the ith-stage planet mode ( Ni > 3). Furthermore, the typical vibration modes of a two-stage system are graphed. It is also found that the radical interstage coupling stiffness has an impact on natural frequencies of the overall radical translational modes, while the axial and twist coupling stiffness only affect the natural frequencies of the overall axial translational–rotational modes.

Three-Dimensional Modeling and Structured Vibration Modes of Two-Stage Helical Planetary Gears Used in Cranes

The dynamic investigation of helical planetary gears plays an important role in structure design as the vibration and noise are perceived negatively to the transmission quality. With consideration of the axial deformations of members, the gyroscopic effects, the time-variant meshing stiffness, and the coupling amongst stages, a three-dimensional dynamic model of the two-stage helical planetary gears is established by using of the lumped-parameter method in this paper. The model is applicable to variant number of planets in two stages, different planet phasing, and spacing configurations. Numerical simulation is conducted to detect the structured vibration modes of the equally spaced systems. Furthermore, the unique properties of these vibration modes are mathematically proved. Results show that the vibration modes of the two-stage helical planetary gears can be categorized as five classes: the rigid body mode, the axial translational-rotational mode, the radical translational mode, and the 1st-stage and the 2nd-stage planet mode.

Dynamic modeling and vibration characteristics of a two-stage closed-form planetary gear train

Planetary gears are widely used in modern machines as one of the most effective forms of power transmission. However, the vibration resulting from gear meshing may shorten the machine’s service life. In this paper, a translational–rotational dynamic model of a two-stage closed-form planetary gear set under consideration of the rotational and translational displacements is first presented to investigate the dynamic response and to avoid resonance. The dynamic equations are formulated into matrix form for the calculation of the natural frequencies and mode shapes. The corresponding 3D model is also proposed to accurately confirm the physical parameters. The effect of planet number in each stage and coupling stiffness on natural modes is then analyzed. Finally, the lumped-parameter model is compared with the finite element simulation. The results show that mode types can be classified into five groups: a rigid body mode, rotational modes, translational modes, the first-stage and the second-stage planet modes. Natural frequencies of rotational and translational modes vary monotonically with planet number. Coupling-twist stiffness has a significant impact on translational modes and the coupling-translational stiffness only affects rotational modes.

Modal Analysis of Double-Helical Planetary Gears With Numerical and Analytical Approach

The vibration modal properties of double-helical planetary gear (DHPG) system with three-dimensional motion are investigated with a combined use of numerical and analytical approach in this paper. The lumped-parameter model of the DHPG considering stiffness coupled between the left gear and the right gear is developed. Load-sharing with journal bearings is accepted in this planetary gear system, so that four stiffness coefficients can be applied to describe the dynamic behavior between the planet gear and the carrier. The model has three planar degrees of freedom for the carrier and an added axial degree of freedom for all gears, considering the effect of axial dynamic forces. The vibration equations are obtained according to Lagrange equation. A modal type distribution map is plotted initially to simplify modal classification. With the application of this modal type distribution map, all vibration modes are categorized distinctly into three essentially different types of modes including planet mode (PM), rotational-axial mode (RAM), and planer-translational mode (PTM). Unique characteristics of these vibration modes, such as, eigenvalue number, multiplicity of natural frequencies and deflection relations, are deduced and proved analytically. For each type of vibration modes, the reduced-order eigenvalue problems are derived.

An analyzing method of coupled modes in multi-stage planetary gear system

This paper analyzed the natural frequency and coupled mode characteristics in multi-stage planetary gear system. A dynamic model of multi-stage planetary gears with general description is established based on the lumped mass method. Solving the associated eigenvalue problem, the vibration modes of the system with identical, equally spaced planets are classified into rotational, translational and planet modes with unique properties. A criterion to distinguish the dominant vibration stage for the two coupled mode types, rotational mode and translational mode, are provided by comparing the eigensensitivity to the component parameters of different planetary stages. Taken the dominant vibration stage as an independent single-stage planetary gear system, the natural frequencies and vibration modes are observed to be similar to the multi-stage. It is shown that the coupling in the multi-stage planetary gear system grows as the frequency reduces. In rotational modes and translational modes, the highest eigensensitivity to mesh stiffness and planet bearing stiffness of each stage are associated with different frequencies in the higher-frequency range, while the highest eigensensitivity to bearing stiffness of carriers, rings, and suns occurred in the lower-frequency range.

1210 ハイブリッド自動車の動力分配を想定した遊星歯車の3軸トルク計測に基づく考察(歯車(1))

Planetary gear set is widely used in hybrid vehicle as a power distribution system or in electric vehicle as a high reduction system, but its noise and vibration problem has emerged. There are many reports dealing with two-axes driving or displacement of planet gear under its condition, but we can see few reports dealing with about three-axes driving. In the present report, three-axis driving is tested with three-axes torque measuring and dynamic characteristic is investigated focusing on planet gear. As a result, it's confirmed that transition forces around planet gear are balanced and torque is also balanced around instantaneous rotation center. Meshing frequency in this condition are harmonics of two meshing frequencies, which are meshing frequency of ring gear and meshing frequency of planet gear. Input power of ring gear is distributed to carrier and sun gear through planet mode, star mode and solar mode.

Model analysis and verification of 2K-H planetary gear system

The planetary gear system is the important part of the cutting unit in a coal shearer, and its performance is directly related to the output efficiency in coal mines. Analyzing the natural frequencies and vibration modes of a planetary gear system in a coal shearer is helpful for designing and monitoring the planetary gear system. So a lateral-torsional coupled free vibration model is proposed with the shaft stiffness of the sun gear based on the analysis of a purely torsional model. The solutions are determined by using the modal superposition method. Taking the planetary gear system in the cutting unit of a YBCS2-400 type coal shearer as an example and studying the natural characteristics, the vibration modes of the undamped system could be classified into three types – rotational mode, translational mode and planet mode. The associated reduced-order eigenvalue for each type of vibration mode is derived and the analytical expression of natural frequency is presented for the planet mode. The established model is validated by comparing it with the results of a finite element model, and the effect on the vibration frequencies of the transmission and width ratio has been obtained by means of the finite element model.

Dynamic Analysis of Helical Planetary Gear Train

A dynamic model for helical planetary gear train (HPGT) is proposed. Based on the model, the free vibration characteristics, steady-state dynamic responses and effects of design parameters on system dynamics are investigated through numerical simulations. The free vibration of the HGPT is classified into 3 categories. The classified vibration modes are demonstrated as axial translational and torsional mode (AT mode), radial translational and rotational mode (RR mode) and planet mode (P mode) followed by the characteristics of each category. The simulation results agree well with those of previous discrete model when neglecting the component flexibilities, which validates the correctness of the present dynamic model. The steady-state dynamic responses indicate that the dynamic meshing forces fluctuate about the average static values and the time-varying meshing stiffness is one of the major excitations of the system. The parametric sensitivity analysis shows that the impact of the central component bearing stiffness on the dynamic characteristic of the HPGT system is significant.

ANALYTICAL MODELING OF PLANETARY GEAR AND SENSITIVITY OF NATURAL FREQUENCIES

This work develops an analytical model of planetary gears and uses it to investigate their natural frequencies and vibration modes. The model admits three planar degrees of freedom for each of the sun, ring, carrier and planets. Vibration modes are classified into rotational, translational and planet modes. The natural frequency sensitivities to system parameters are investigated for tuned (cyclically symmetric) planetary gears. Parameters under consideration include support and mesh stiffnesses, component masses, and moments of inertia. Using the well-defined vibration mode properties of tuned planetary gears, the eigen sensitivities are calculated and expressed in simple exact formulae. These formulae connect natural frequency sensitivity with the modal strain or kinetic energy and provide efficient means to determine the sensitivity to all stiffness and inertia parameters by inspection of the modal energy distribution.

VIBRATION ANALYSIS OF PLANETARY GEAR SYSTEM

In this work a dynamic model of a planetary gear transmission is developed to study the sensitivity of the natural frequencies and vibration modes to system parameters in perturbed situation. Parameters under consideration include component masses ,moment of inertia , mesh and support stiff nesses .The model admits three planar degree of freedom for planets ,sun, ring, and carrier. Vibration modes are classified into translational, rotational and planet modes .Well-defined modal properties of tuned (cyclically symmetric) planetary gears for linear ,time-invariant case are used to calculate eigensensitivities and express them in simple formulae .These formulae provide efficient mean to determine the sensitivity to stiffness ,mass and inertia parameters in perturbed situation.

Natural Frequency Clusters in Planetary Gear Vibration

This paper investigates how the natural frequencies of planetary gears tend to gather into clusters (or groups). This behavior is observed experimentally and analyzed in further detail by numerical analysis. There are three natural frequency clusters at relatively high frequencies. The modes at these natural frequencies are marked by planet gear motion and contain strain energy in the tooth meshes and planet bearings. Each cluster contains one rotational, one translational, and one planet mode type discussed in previous research. The clustering phenomenon is robust, continuing through parameter variations of several orders of magnitude. The natural frequency clusters move together as a group when planet parameters change. They never intersect, but when the natural frequencies clusters approach each other, they exchange modal properties and veer away. When central member parameters are varied, the clusters remain nearly constant except for regions in which natural frequencies simultaneously shift to different cluster groups. There are two conditions that disrupt the clustering effect or diminish its prominence. One is when the planet parameters are similar to those of the other components, and the other is when there are large differences in mass, moment of inertia, bearing stiffness, or mesh stiffness among the planet gears. The clusters remain grouped together with arbitrary planet spacing.

Influence of Bearing Stiffness of Ring on Natural Characteristics of Spur Planetary Gears

Abstract A dynamic model is constructed for 2K-H spur planetary gear set in the carrier-attached coordinate system. By providing the modeling details and comparisons with previous planetary gear models, this paper clarifies discrepancies in the carrier-attached coordinate of the planets and corrects errors in previously published models. The modal properties of spur planetary gears with equally spaced planets are categorized and mathematically proved. A dynamic model, which corrects omissions of previous work and ensures analysis reliable, is formulated to obtain the equations of vibration of the planetary gears. The influence of the bearing stiffness of central components on the vibration mode characteristics of the planetary gear is examined. The results show that spur planetary gears have exactly three types of vibration modes. The investigation into component stiffness reveals that the stiffness of the ring only influences rotational and translational modes, while the circumferential stiffness and the radial stiffness have no effect on the planet modes.

Analysis of Modal Properties of Spur Planetary Gears

A translational-rotational-coupling dynamic model has been built in the carrier-attached coordinate system.Differential equations of the system have been derived, and the natural frequencies and vibration modes of the planetary gear set have been obtained through solution of the associated eigenvalue problem. Based on the properties of the transmission system, the vibration modes of 2K-H spur planetary gear set can be classified into three categories, i.e., translational mode along radial direction, rotational mode, and planet mode.

Free Torsional Vibration Characteristics of a Closed-Form Planetary Gear Set

A purely torsional dynamic model of closed-form planetary gear set is developed to investigate its natural frequency and free vibration modes. The closed-form planetary gear set considered consists of two-stage planetary gear connected by high-speed carrier and ring gear. Three identical planet gears are equally spaced in each stage. Based on the precondition, the natural modes are classified into three types of vibration modes: overall modes，high-speed planet mode and low-speed planet mode

Planetary gear modal vibration experiments and correlation against lumped-parameter and finite element models

Experimental modal analysis techniques are applied to characterize the planar dynamic behavior of two spur planetary gears. Rotational and translational vibrations of the sun gear, carrier, and planet gears are measured. Experimentally obtained natural frequencies, mode shapes, and dynamic response are compared to the results from lumped-parameter and finite element models. Two qualitatively different classes of mode shapes in distinct frequency ranges are observed in the experiments and confirmed by the lumped-parameter model, which considers the accessory shafts and fixtures in the system to capture all of the natural frequencies and modes. The finite element model estimates the high-frequency modes that have significant tooth mesh deflection without considering the shafts and fixtures. The lumped-parameter and finite element models accurately predict the natural frequencies and modal properties established by experimentation. Rotational, translational, and planet mode types presented in published mathematical studies are confirmed experimentally. The number and types of modes in the low-frequency and high-frequency bands depend on the degrees of freedom in the central members and planet gears, respectively. The accuracy of natural frequency prediction is improved when the planet bearings have differing stiffnesses in the tangential and radial directions, consistent with the bearing load direction.

The Three-Dimensional Dynamics for Toroidal Drive

In this paper, a model to simulate the dynamic behavior of the toroidal drive is developed. The three-dimensional dynamic model includes all six rigid body motions of the stator, worm, rotor and the planets. Using the model, the natural frequencies and vibration modes of the drive system are investigated. The vibration modes are classified into single modes and coupled modes. The single modes include planet mode, worm mode and stator mode. The vibration and frequency characteristics of different modes are obtained. The relation between modes and half cone angle of the planet tooth is discussed. The relation between vibrations and bearing stiffness is also discussed. When the bearing stiffness is about 10 times of the mesh stiffness, some vibration displacements of the drive system are quite small and can be neglected. Meanwhile, the dynamic equations for the drive system can be simplified.

Modal analyses of herringbone planetary gear train with journal bearings

A generalized dynamic model for herringbone planetary gear train (HPGT) is developed to investigate its modal properties. The model includes the axial vibration of two helical ring gears in addition to three planar degrees of freedom for the carrier and all gears. Four stiffness coefficients are applied to describe the asymmetry and the interaction of the oil film stiffness of journal bearings for supporting the planets. Vibration modes are classified into rotational and axial mode, translational mode, planet mode, rotational and axial ring mode and translational ring mode. For each type of mode, the reduced-order eigenvalue problems are derived according to the modal properties. The formulas for calculating the modal strain and kinetic energy distributions are also given for each mode. The proposed dynamic model and analysis methods can be applied to HPGT with any number of planets. Only when the asymmetric interaction exists in journal bearings, will the dramatic change of mode shape for translational mode occur. The new relations between deflections of planets in translational mode are also derived in this research.