Short communicationAnalytical determination of back-side contact gear mesh stiffness
Introduction
Back-side contact in a gear mesh refers to contact on the surfaces of a gear that are not used to transmit power. Recent studies on gear dynamics [1], [2], [3], [4] show that it is possible for tooth wedging (or tight mesh), that is, simultaneous drive-side and back-side contact, to happen in applications such as wind turbine gearboxes. Tooth wedging in wind turbines results from the combined effect of gravity and bearing clearance nonlinearity, and it proved a likely source of gearbox bearing failures in a particular case. For better understanding of the impact of tooth wedging on gearbox failures, it is necessary to have a model that includes accurate description of the back-side contact mesh stiffness.
Besides tooth wedging, anti-backlash (or scissor) gears are another case for back-side contact to occur. To minimize the undesirable characteristics caused by backlash, anti-backlash gears eliminate the backlash by using a preloaded spring to force the fixed part of the driving gear to contact the drive-side of the driven gear teeth and, simultaneously, force the free part of the driving gear to contact the back side of the driven gear [5]. Accurate modeling of back-side contact mesh stiffness is necessary to analyze such systems.
Mesh stiffness variation and its impact on gear mechanics have been extensively investigated. Mesh stiffness variation is the source of static transmission error fluctuations. Munro and his team experimentally investigated gear tooth mesh stiffness throughout and beyond the path of contact [6]. Blankenship and Kahraman experimentally and analytically studied a single degree of freedom gear pair driven by time-varying mesh stiffness variation; they showed contact loss and back-side contact that are subject to a symmetric backlash condition [7]. The same system was investigated with analytical and finite element models [8]. Lin, Liu, and Parker analyzed mesh stiffness variation instabilities in two-stage gear systems [9], [10], as well as in simple planetary gear systems [11]. Their studies showed that parametric excitation from time-varying mesh stiffness causes instability and severe vibration under certain operating conditions. They applied a perturbation method to analytically determine the instability conditions. Velex and Flamand extended the research scope to planetary gear trains and studied their dynamic responses with varying mesh stiffness [12]. Wu and Parker [13] extended the study on parametric instability to planetary gears with elastic continuum ring gears. Sun and Hu [14] investigated mesh stiffness parametric excitation and clearance nonlinearity for simple planetary gears. Bahk and Parker [15] derived closed-form solutions for the dynamic response of planetary gears with time-varying mesh stiffness and tooth separation nonlinearity based on a purely torsional planetary gear model. They extended this to systems with tooth profile modifications [16]. Guo and Parker [1] modeled and analyzed a simple planetary gear with time-varying mesh stiffness, tooth wedging, and bearing clearance nonlinearity. Although back-side contact is included in their model, the average value of the periodic mesh stiffness on the drive-side is used to approximate the back-side mesh stiffness, which is a simplified description of the back-side mesh stiffness.
Despite the abundance of literature on mesh stiffness variation and gear dynamics, no studies have derived the back-side mesh stiffness in their analytical model. One possible reason is that the usual symmetry of the gear teeth ensures that the contact ratios, mesh periods, and average mesh stiffnesses over the mesh period are the same for drive- and back-side contact. This may lead to the mistaken conclusion that the back-side mesh stiffness is the same as the drive-side one. For example, Kahraman and Blankenship performed experiments on the nonlinear response of spur gear pairs with varying involute contact ratios [17], [18]. The back-side contact is assumed to be the same as the drive-side contact in their study. In the tight mesh case shown in Fig. 1, the back-side mesh stiffness, however, is not equivalent to the drive-side one, because the back-side contact is along the back-side line of action (the dashed line in Fig. 1) and the number of gear teeth in contact along the back-side line of action is not always equal to that along the drive-side line of action (the solid line in Fig. 1). Fig. 2 illustrates one such case (the simulation results are from Calyx [19], a multi-body finite element/contact mechanics program with precise gear tooth contact capability). There are two pairs of gear teeth in contact along the back-side line of action, while only one pair of teeth is in contact along the line of action. Therefore, the back-side mesh stiffness differs from the drive-side mesh stiffness at this moment.
Section snippets
Derivation of back-side mesh stiffness
The drive-side mesh stiffness refers to the stiffness of the nominally contacting teeth at a mesh in the direction of power transmission. It varies as the number of teeth in contact fluctuates with the gear rotation. The stiffness acts along the line of action. The period of its variation is known for the given rotation speed. Mesh stiffness variation functions are often approximated by Fourier series in analytical studies. They can be accurately calculated by finite element software. The
Conclusion
This study investigates the relationships between the drive- and back-side mesh stiffnesses for arbitrary gear pairs and anti-backlash gear pairs. The impact of backlash and center distance changes on the phase lag in the back-side mesh stiffness variation function is analytically determined. The resulting formulae are useful for the static and dynamic analysis of gear systems that involve back-side gear tooth contacts, including anti-backlash gears.
Nomenclature
- T
- Mesh period
- rdr
- Pitch radius of the driving gear
- rdn
- Pitch radius of the driven gear
- Zdr
- Tooth number of the driving gear
- Zdn
- Tooth number of the driven gear
- k(t)
- Time-varying drive-side mesh stiffness
- kb(t)
- Time-varying back-side mesh stiffness
- 2b
- Nominal backlash along the pitch circle
- p
- Circular pitch
References (22)
- et al.
Dynamic modeling and analysis of a spur planetary gear involving tooth wedging and bearing clearance nonlinearity
Eur. J. Mech. A Solids
(2010) - et al.
Steady state force response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity
J. Sound Vib.
(May. 1995) - et al.
Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons
J. Sound Vib.
(Oct. 2000) - et al.
Planetary gear parametric instability caused by mesh stiffness variation
J. Sound Vib.
(Jan. 2002) - et al.
Nonlinear dynamics of a planetary gear system with multiple clearances
Mech. Mach. Theory
(Dec. 2003) - et al.
Analytical investigation of tooth profile modification effects on planetary gear dynamics
Mech. Mach. Theory
(December 2013) - et al.
Nonlinear dynamics of planetary gears using analytical and finite element models
J. Sound Vib.
(2007) - et al.
Dynamics of a wind turbine planetary gear stage
- F. Rasmussen, K. Thomsen, T. Larsen, The gearbox problem revisited. Risoe fact sheet aed-rb-17 (en), Risoe National...
- et al.
Gearbox loads caused by double contact simulated with hawc2
Investigation of Transmission Error, Friction, and Wear in Anti-backlash Gear Transmissions: A Finite Element Approach
Cited by (26)
Analytical determination of back-side contact force for paralleled beveloid gear
2024, Mechanism and Machine TheoryThe global behavior evolution of non-orthogonal face gear-bearing transmission system
2022, Mechanism and Machine TheoryEvaluation model of mesh stiffness for spur gear with tooth tip chipping fault
2021, Mechanism and Machine TheoryCitation Excerpt :Xue et al. [32] and Luo et al. [33] analyzed the influence of the variation of gear center distance on the mesh stiffness. Under idling and light loading conditions, gear back-side tooth may come into contact, Yu et al. [34] and Guo et al. [35] studied the relation between the back-side mesh stiffness and drive-side mesh stiffness. Luo et al. [36] presented a new method for calculating mesh stiffness of spalling fault gears, which does not depend on specific geometry.
Experimental measurement of mesh stiffness by laser displacement sensor technique
2018, Measurement: Journal of the International Measurement ConfederationCitation Excerpt :Chen and Shao [4] included gear rim deformation in mesh stiffness evaluation. Guo and Parker [23] investigated analytically back-side contact gear mesh stiffness. Mohammed et al. [9] used a method to estimate the gear mesh stiffness for large crack sizes.
Dynamic modeling of gearbox faults: A review
2018, Mechanical Systems and Signal ProcessingInfluence of the addendum modification on spur gear back-side mesh stiffness and dynamics
2017, Journal of Sound and VibrationCitation Excerpt :This model ignores the phase shift between the drive-side and back-side mesh stiffness. To cope with this challenge, the time-varying asymmetric mesh stiffness model was proposed by Guo [1] and Chen [2]. They have derived analytical equations to calculate the back-side mesh stiffness based on the drive-side mesh stiffness respectively.