Analytical coupling characterization of multi-stage planetary gear free vibration considering flexible structure

A dynamic model of double-mesh helical gear set was established including the torsional vibration, axial vibration and time varying mesh stiffness. The natural modes were systematically analyzed and classified into three types: rotational vibration mode, axial vibration mode and overall vibration mode. The influence of time varying mesh stiffness and the dynamic responses of the gear set were also investigated. The results showed that several natural frequencies are excited in the dynamic response and there is a frequency sensitive area which may cause large dynamic load.

A coupling dynamics analysis method for a multistage planetary gear system

Studies on the planetary gear have attracted considerable attention because of its advantages, such as compactness, large torque-to-weight ratio, vibrations, and high efficiency, which have resulted in its wide applications in industry, wind turbine, national defense, and aerospace fields. We have established a novel dynamic model of the planetary gear transmission by using Newton’s theory, in which some key factors such as time-variant meshing stiffness, phase relationships, and tooth contact characteristics are considered. The influences of gear axial tipping, operating conditions, and the meshing phase on the contact characteristics and the dynamic characteristics were researched systematically. It was found that the contact area of the tooth surface was moved due to the axial gear tipping, which obviously affected the meshing stiffness. With the increase in the inclination angle of the sun gear, the meshing stiffness decreases, which produces an evident influence on the high natural frequency in the planetary transmission system. In terms of the dynamic characteristics of the system, the component of rotating frequency appeared in the dynamic meshing force of the sun gear and the planetary gear. Moreover, the floating track of the center wheel varied significantly and exhibited an oval distribution as the inclination angle of the sun gear changed. When the inclination angle of the sun gear increased, the rotating frequency component increased significantly, but the other meshing frequency components remained unchanged; meanwhile, the deformation of the floating track also increased. If the inclination angle of the sun gear changes, the vibration state of the system and the collision impact could become more serious, and the lifetime of the planetary transmission system will reduce. Furthermore, when the load was increased, we found that the gear-tooth contact zone transformed from line contact to surface contact, the meshing stiffness increased, the effect of high natural frequency on the planetary transmission system became more evident, but its low-order natural frequency remained stable. With regard to the dynamic characteristics of the system, the components of the major frequency at the external gearing remained unchanged, but the rotation frequency of the sun gear and the meshing frequency amplitude increased linearly with the increase in load. In conclusion, the variation in the meshing stiffness of the planetary gear system had minor impact on the low-order natural frequency, but had a significant impact on the high natural frequency of the planetary transmission system due to the phase variation of the gear.

Modulation sideband analysis of a two-stage planetary gear system with an elastic continuum ring gear

The effect of interactive damping on the sensitivity of flexural and axial vibration modes of scanning near-field optical microscope (SNOM) with a tapered optical fiber probe has been analyzed. The interaction of the SNOM probe with a sample surface is modeled by a combination of a spring and a dashpot in the flexural direction and a similar combination in the axial direction. An approximate form for the sensitivities of both modes was derived by using the Rayleigh–Ritz method. The results show that the interactive damping will decrease the sensitivities of both flexural and axial vibration modes when the contact stiffness is low. The more the damping effect, the lower the sensitivities are. In addition, when the contact stiffness was low, the flexural sensitivity of the tapered probe slightly increased as the tapered angle decreased. However, the axial sensitivity apparently decreased as the tapered angle decreased. When the contact stiffness became higher, the sensitivities of both flexural and axial vibration modes increased as the tapered angle increased.

Two time varying approaches are executed in analyzing dynamics for an involute planetary gear system, which respectively use a conventional discrete model of the equivalent mass-damping-spring elements and a continuous geometry model by the finite element method. In the discrete approach, the tooth number, position, and phasing difference of the meshing tooth pairs are described by time varying and nonlinear meshing stiffnesses. Natural frequencies, deformations, meshing forces, fillet stresses, and dynamic factors can be calculated by using the Jacobi transformation and the Runge-Kutta integration. In the continuum approach, dynamics of the planetary gear system is analyzed using the software, LS-DYNA. The approach of the continuous geometry model can incorporate the time varying properties intrinsically. In this continuum study, not CAD models, high quality mesh elements of the planetary gear system are automatically generated directly using the derived tooth profile equations. After assigning initial and boundary conditions, dynamic responses for the planetary gear system are solved. Natural frequencies and fillet stresses of the both approaches are verified by each other comparison. Potentially, the continuum approach can extensively and sophistically analyze dynamics problems of the planetary gear systems.

Sensitivity analysis of scanning near-field optical microscope probe

PurposeIrregular windy loads are loaded for a wind turbine. This paper aims to determine the form of gear failure and the working life of the gear system by assessing the dynamic strength of gears and dynamic stress distribution.Design/methodology/approachThe helical planetary gear system of the wind turbine growth rate gearbox was investigated, and while a variety of clearance and friction gear meshing processes were considered in the planetary gear system, a finite element model was built based on the contact–impact dynamics theory, solved using the explicit algorithm. The impact stress of the sun gear of the planetary gear system was calculated under different loads. An integrated planetary gear meshing stiffness, and the error of system dynamic transmission error were investigated when the planetary gear meshes with the sun or ring gears.FindingsThe load has little effect on the sun gear of the impact stress which was known. The varying stiffness is different while the planetary gear meshes with the sun and ring gears. There were differences between the planetary gear system and the planetary gear, and with load, the planetary gear transmission error decreases.Originality/valueThis study will provide basis knowledge for the planetary gear system.

Dynamic simulation of planetary gear set with flexible spur ring gear

Drillstring and down-hole tool failure usually results from failing to control one or more of the vibration mechanisms. The solution starts with the ability to measure different modes of vibration, hence identifying different vibration mechanisms. Lateral, torsion and axial are vibration modes that take place when drill pipes run into problems downhole. Due to the three modes of vibration mechanisms such as bit bounce, stick-slip, lateral shocks, bit and bottom hole assembly (BHA) whirl, parametric and torsional resonance occur. Understanding the causes of the destructive loads is the main step towards developing approaches to prevent or reduce their effects, hence improving drilling performance. Vibration modes and mechanisms lead to failure of the drill pipes, BHA and drill bits. Drill pipes fatigue failure is very common due to capability of producing all vibration modes and mechanisms. Drill pipe and downhole tool assembly failure usually result from failing to have power over one or more of these vibration mechanisms. A novel in house experimental setup has been developed to mimic downhole axial, lateral and torsional vibration modes and mechanisms in drilling operations. In this paper, we focus on the design and construction of the testing facility. A number of tests were conducted to validate the capability and performance of the test setup. Drill pipe fatigue failure due to lateral cyclic stresses induced in the drill pipe has also been investigated and presented in this paper. The results show that operating on a rotation speed higher than 90% of the drillstring critical speed leads to yielding in the drillstring.

Dynamic stability of liquid-filled cylindrical shells under horizontal excitation, part I: Experiment

This study presents a novel and efficient modal analysis framework for thick cylindrical structures with complex geometry and material variability, subject to arbitrary boundary conditions. The methodology is applied to an electric motor (e-motor) stator assembly modelled as a thick cylindrical shell incorporating stator teeth, windings, and housing or cooling jacket effects. The model accommodates both continuous and piecewise variations in material properties and thickness. Based on First-Order Shear Deformation Shell Theory (FSDST), it accounts for shear deformation, rotary inertia, and trapezoidal stress distributions, enabling accurate prediction of axial, circumferential, torsional, and bending vibration modes. A segmentation approach is used along the axial direction, with artificial massless springs enforcing continuity and permitting general boundary conditions. Displacement fields are constructed using orthogonal Jacobi polynomials, and the eigenvalue problem is solved via the Rayleigh–Ritz method. Notably, the methodology allows accurate and efficient prediction of axisymmetric (breathing) modes in thick, non-uniform cylindrical shells – a capability rarely addressed in existing literature, despite its importance in Noise, Vibration, and Harshness (NVH) analysis. Validation against Finite Element Analysis (FEA) and Experimental Modal Analysis (EMA) on both generic cylindrical shells and a real Permanent Magnet Synchronous Machine (PMSM) stator shows excellent agreement, with natural frequency deviations typically below 5% and highly consistent mode shapes. The framework also achieves over 95% reduction in computational time compared to FEA, establishing it as a highly adaptable and practical tool for vibration analysis in electric motor design and NVH engineering applications.

This paper investigates the axial vibration of single-walled carbon nanotubes (SWCNTs) with fractional damping based on doublet mechanics. The Kelvin–Vigot model is used to incorporate damping effect for CNTs. By solving the equation of motion, the relation between natural frequency with scale parameter and fractional order is derived in the axial mode of vibration. It is shown that fractional order and scale parameter play significant roles in the axial vibration behavior of SWCNTs. Such effects decrease the natural frequency compared to the predictions of the classical continuum mechanics models and also ignores the damping effects. These effects on the natural frequency are more apparent in higher mode numbers and lower tube lengths and radii. Results for complex roots of characteristic equation obtained for a SWCNT without viscoelastic foundation, where imaginary parts represent damped frequencies, were compared with the results found from molecular mechanics simulations and a good agreement was achieved.

Helicopter transmission system contains different types of gear sets, such as planetary gear set, bevel gear set, parallel shaft gear set and so on. The vibration characteristics of each gear set type has been widely investigated. However, few studies focus on the coupled vibration modes related to two or more gear sets. The systematic dynamic model of the planetary gear set-bevel gear set coupling system (PBCS) is constructed in this study. On this basis, the modal analysis is carried out, and the influences of the coupling effect on the natural frequencies and modes of PBCS is obtained. The results have certain theoretical significance for the dynamic modeling, vibration analysis, fault diagnosis of helicopter transmission system.KeywordsPlanetary gear setBevel gearSystematic dynamic modelHelicopter transmission systemModal analysis

The mesh stiffness of gear teeth is one of the major sources of excitation in gear systems. Many analytical and finite element methods have been proposed in order to determine the mesh stiffness of gears especially parallel axis spur gears. Most of these methods are not trivial because they involve complicated analyses which incorporate parameters like gear tooth error, gear spalling sizes and shapes, nonlinear contact stiffness and sliding friction before mesh stiffness can be determined. In this work, a method is proposed to determine the sun-planet and ring-planet mesh stiffnesses of a planetary gear system. This approach involves fitting a relationship between the measured natural frequencies from an experimental modal test and natural frequencies predicted using an analytical model of a planetary gear. This method is relatively easier compared to the existing methods which involve complicated analyses. For this study, the average mesh stiffness estimated is 12.5 MN/m.