Elsevier

European Journal of Mechanics - A/Solids

Volume 47, September–October 2014, Pages 45-57
European Journal of Mechanics - A/Solids

Nonlinear dynamics and stability of wind turbine planetary gear sets under gravity effects

https://doi.org/10.1016/j.euromechsol.2014.02.013Get rights and content

Highlights

  • We investigate planetary gear dynamics considering gravity effects.

  • We develop an analysis tool that could minimize planetary gear vibration.

  • Gravity is a fundamental vibration source in wind turbine planetary gears.

  • Gravity causes tooth wedging and nonlinear bearing-raceway contacts.

  • Optimizing carrier-bearing clearance could prevent gear tooth wedging.

Abstract

This paper investigates the dynamics of wind turbine planetary gear sets under the effect of gravity using a modified harmonic balance method that includes simultaneous excitations. This modified method along with arc-length continuation and Floquet theory is applied to a lumped-parameter planetary gear model including gravity, fluctuating mesh stiffness, bearing clearance, and nonlinear tooth contact to obtain the dynamic response of the system. The calculated dynamic responses compare well with time domain-integrated mathematical models and experimental results. Gravity is a fundamental vibration source in wind turbine planetary gear sets and plays an important role in the system dynamic response compared to excitations from tooth meshing alone. Gravity causes nonlinear effects induced by tooth wedging and bearing-raceway contacts. Tooth wedging, also known as a tight mesh, occurs when a gear tooth comes into contact on the drive-side and back-side simultaneously and it is a source of planet-bearing failures. Clearance in carrier bearings decreases bearing stiffness and significantly reduces the lowest resonant frequencies of the translational modes. Gear tooth wedging can be prevented if the carrier-bearing clearance is less than the tooth backlash.

Introduction

The National Renewable Energy Laboratory (NREL) Gearbox Reliability Collaborative (GRC) was established by the U.S. Department of Energy in 2006. Its key goal is to understand the root causes of premature gearbox failures (Musial et al., 2007) through a combined approach of dynamometer testing, field testing, and modeling (Link et al., 2011), resulting in improved wind turbine gearbox reliability and a reduction in the cost of energy. As a part of the GRC program, this paper investigates gravity-induced dynamic behaviors of planetary gear sets in wind turbine drivetrains that could reduce gearbox life. Planetary gear sets have been used in wind turbines for decades because of their compact design and high efficiency. Despite these advantages, planetary gear sets generate considerable noise and vibration. Vibration causing high dynamic loads may result in gear tooth and bearing failures (Musial et al., 2007). Fatigue failures are a concern in long life-cycle applications. Analyzing the dynamics of wind turbine planetary gear drivetrains is important in improving gearbox life and reducing noise and vibration.

The majority of wind turbines use a horizontal-axis configuration; thus, gravity becomes a periodic excitation source in the rotating carrier frame. Prior study of gravity by the authors was performed with a static analysis and focused on the effect of gravity upon bearing force and tooth wedging in a planetary spur gear set (Guo and Parker, 2010). It was found that tooth wedging, an abnormal contact situation where the tooth is in contact with both the drive-side and back-side flanks simultaneously, was caused by gravity. Tooth wedging increases planet-bearing forces and disturbs load sharing among the planets, which could lead to premature bearing failure.

Significant in-plane translational gear component motions in planetary systems lead to tooth wedging. It is the combined effect of gravity and bearing clearance nonlinearity. Bearing clearance results in greater translational vibration, while gravity is the dominant excitation source causing the large motions that lead to tooth wedging. For heavy planetary gear sets, tooth wedging is likely to occur. Tooth wedging in planetary gear sets leads to unequal load sharing and excessive planet-bearing loads by disturbing the symmetry of the planet gears. This may cause bearing failure and tooth damage (Guo and Parker, 2010).

Research on tooth separation is well-established for automotive and helicopter applications. Tooth separation was observed in spur gear pair experiments (Blankenship and Kahraman, 1996). Botman (1976) experimentally observed tooth separation in planetary gear sets. Using finite element and lumped-parameter models, Ambarisha and Parker (2007) predicted tooth separation and other nonlinear phenomena in a planetary gear set in a helicopter gearbox. Velex and Flamand (1996) investigated tooth separation at critical speeds. Bahk and Parker (2011) derived closed-form solutions for the dynamic response of planetary gear sets with tooth separation based on a purely torsional model.

Nonlinear dynamics induced by bearing clearance has been studied for relatively small geared systems. Kahraman and Singh (1991) observed chaos in the dynamic response of a geared rotor-bearing system with bearing clearance and backlash. Gurkan and Ozguven (2007) studied the effects of backlash and bearing clearance in a geared flexible rotor and the interactions between these two nonlinearities. Guo and Parker (2012a) investigated the nonlinear effects and instability caused by bearing clearance in helicopter planetary gear sets. Dynamic effects of bearing clearance in wind turbine planetary gear sets have not been studied in the past because the wind turbine operating speed was believed to be well below the frequency range of drivetrain dynamics. However, bearing clearance reduces some gearbox resonances significantly.

The finite element program developed by Vijayakar (1991) uses a combined surface integral and finite element approach to capture tooth deformation and contact loads in geared systems. This finite element model includes bearing clearance, tooth separation, tooth wedging, fluctuating mesh stiffness, and gravity. Numerical integration is widely adopted to compute dynamic responses of mechanical systems in the time domain. Ambarisha and Parker (2007) used numerical integration to study nonlinear dynamics and the impacts of mesh phasing on vibration reduction of planetary gear sets. Velex and Flamand used numerical integration results of a planetary gear set with time-varying mesh stiffness as a benchmark to evaluate results from a Ritz method.

The harmonic balance method (Thomsen, 2003) calculates nonlinear, frequency domain, steady-state response of mechanical systems. Zhu and Parker (2005) used this method to study clutch engagement loss in a belt-pulley system. Al-shyyab and Kahraman (2005a,b) investigated primary resonances, subharmonic resonances, and chaos in a multimesh gear train caused by fluctuating gear mesh stiffness. Bahk and Parker (2011) employed harmonic balance to analyze planetary gear dynamics based on a purely rotational model. Use of the harmonic balance method reduces computational time for lightly damped or physically unstable systems by avoiding the long transient decay time before a steady state is reached. Compared to numerical integration and finite element analysis, which are widely adopted approaches to compute dynamic responses, the computation time of the harmonic balance method is one–two orders of magnitude lower. Harmonic balance often employs arc-length continuation (Nayfeh and Balachandran, 1995) and Floquet theory (Raghothama and Narayanan, 1999; Seydel, 1994) to calculate nonlinear resonances in the dynamic response, including unstable solutions that numerical integration and finite element analysis are unable to obtain. The established harmonic balance formulation is only suitable for systems with one fundamental excitation frequency and its higher harmonics. However, wind turbine drivetrains have simultaneous internal and external excitations, including fluctuating mesh stiffness, gravity, bending-moment-induced excitations in the rotating carrier frame, wind shear, tower shadow, and other aero-induced excitations.

The major objectives of this study are to: 1) develop a modified harmonic balance method to obtain the dynamic response of wind turbine planetary gear sets considering gravity, fluctuating mesh stiffness, bearing clearance, and nonlinear tooth contact; 2) validate the proposed method by comparing the calculated results against experimental data and a numerical integration approach; and 3) investigate the gravity-induced dynamic behaviors using the developed approach, which includes tooth wedging, tooth contact loss, and bearing-raceway contacts.

Section snippets

Gearbox description

This study investigates both a 750-kW wind turbine planetary gear (PG-A) used by the GRC (Link et al., 2011) and a 550-kW wind turbine planetary gear (PG-B) (Guo and Parker, 2010; Larsen et al., 2003; Rasmussen et al., 2004). These drivetrains have a main bearing that supports the main shaft and rotor weight, and two trunnion mounts that support the gearbox. These two gearboxes have similar configurations and are representative of the majority of three-point-mounted wind turbine drivetrains.

Lumped-parameter model for planetary gear sets

A previously developed and validated lumped-parameter model was adopted for this paper (Guo and Parker, 2010, 2012a). As depicted in Fig. 2(a), the carrier, ring, sun, and planets are rigid bodies, each having two translational and one rotational degree of freedom. The carrier rotating frame is used as the general coordinates for all of the components of planetary gear sets. This two-dimensional model has 3(N + 3) degrees of freedom, where N is the number of planets. The model includes gravity,

Extended harmonic balance method

The extended harmonic balance method is used to obtain the dynamic responses of the model in Eq. (1). The formulation includes two excitation sources with excitation frequencies Ω1 and Ω2. The coupling effects between these two excitations are considered by including their side bands. Other excitation sources can be considered in a similar way. The response z is expanded into a Fourier series and assumed to include the R1, R2, R3R4R5, and R6R7R8 harmonics of excitation frequencies Ω1 and Ω2 and

Model validations

Experiments on PG-A provide a benchmark for the lumped-parameter model using the extended harmonic balance method; however, the relevant available experimental data for PG-A is only at a single speed. Thus, results from the extended harmonic balance method are also compared to the time-integrated results of the lumped-parameter model of PG-B within the speed range of 0–200 Hz.

Dynamic response of planetary gear sets with gravity

Dynamic analyses of PG-A and PG-B were performed using the extended harmonic balance approach. The response included 100 harmonics of the carrier frequency, three harmonics of the mesh frequency, and eight upper and lower side bands of the mesh frequency harmonics. These parameters were selected based on the energy distribution in the frequency spectrum of the experimental data in Fig. 5.

Vibration modes of two-dimensional planetary gear sets include rotational modes with distinct natural

Conclusions

A harmonic balance method was developed to calculate the dynamic response of wind turbine planetary gear sets. Results obtained compare well with the GRC experiments of PG-A and the numerically integrated results of the lumped-parameter model of PG-B. This approach avoids protracted time integration simulations by calculating the dynamic response in the frequency domain. The time savings of this method make it suitable for parametric studies, which could be used to tune the planetary gear

Acknowledgment

This work was supported by the U.S. Department of Energy under Contract No. DE-AC36-08GO28308 with the National Renewable Energy Laboratory.

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