7
Wind turbine gearbox design with drivetrain dynamic analysis
Abstract
This chapter focuses on the fundamentals of designing a gearbox for a wind turbine and analysis of the associated drivetrain dynamics. First of all, some aspects of early stage design considerations and typical gear layout arrangements (ie, parallel shaft and planetary systems) are discussed. The preliminary design considerations lead to major decisions, such as on the configuration of the gearbox (modular or integral), number of stages, type of layout (parallel or planetary), etc. Following the discussion on early stage design considerations more detailed design principles at the development and production stages are presented. Some of the recommendations following ISO/IEC 61400-4 are also highlighted. Aspects on gear manufacture and inspection are also briefly discussed. Finally, analysis of drivetrain dynamics, variable loading on gearbox and fatigue design of gear teeth are presented. A simple two-mass model of the drivetrain shaft has been presented in the context of dynamic analysis and the importance of electromechanical coupling has been highlighted, including the generator dynamics.
Keywords
Concept design; Drivetrain dynamics; Electromechanical coupling; Fatigue design; Inspection and maintenance; Load spectrum; Parallel gear shaft; Planetary gear; Speed-up ratio; Variable loading7.1. Introduction
The gearbox (GB) is an integral part of the drivetrain in large-scale wind turbine generator systems (WTGS). Gearboxes are used in the majority of WTGS with synchronous or induction generators installed for electricity generation [1]. Small wind turbine rotors may turn at speeds of the order of hundreds of revolutions per minute and may not require a gearbox. In these cases, known as direct drive, permanent magnet generators or synchronous generators with a sufficiently high number of poles may be directly coupled to the rotor. However, large wind turbines turn more slowly and a gearbox is a practical necessity. Horizontal axis wind turbines (HAWT) with power ratings in excess of 500 kW have a dedicated ISO/IEC standard for gearbox design [2]. Fig. 7.1 shows a schematic layout of the main subsystems found in a typical HAWT. The gearbox is coupled to the rotor through the low-speed shaft and is connected to the generator through the high-speed shaft. It is the convention in GB design to distinguish between the rotor side (RS) and the generator side (GS). The basic function of the gearbox is to transmit mechanical shaft power from the rotor side, running at low speeds (approximately10–18 RPM for offshore wind turbines), to the generator side, running at higher speeds (1800 RPM for 60 Hz or 1500 RPM for 50 Hz assuming a four-pole synchronous generator) [1,3]. For safety reasons a mechanical brake may be installed on the high-speed shaft. However, for installation or maintenance, a rotor lock system is also installed on the low-speed shaft (main shaft).
Statistics have shown that operating and maintenance (O&M) costs for HAWT are significant. Márquez et al. [4] reported that for a 750-kW turbine over a 20-year operating life, the O&M costs are 25–30% of overall energy generation [5] or 75–90% of investment cost [6]. In addition, Márquez et al. [4] reported data from [7] that suggests that larger turbines fail more frequently. Van Bussel et al. [8] also stated that 25% of O&M costs are due to failures of main components. It is clear that all components and subsystems contribute the O&M costs; however, as highlighted by Sheng [9], the gearbox can be one of the most costly WTGS subsystems to maintain over the 20-year operating life. Data provided by [10] (reported in Igba et al. [11]) show that annual gearbox failure rates may be relatively low when compared to other subsystems, but the downtime caused by the gearbox failure is relatively high and is, therefore, a major cost driver.
Given this context, we can see the importance of gearbox design and the related operational issues. Since WT gearbox is a mature technology, the design process tends to be iterative, but from the viewpoint of categorising the design process, we can discuss a generic design life cycle as shown in Fig. 7.2 (reported in Igba et al. [11]). The life cycle consists of five stages: concept, development (detailed design), production, utilisation and support, and retirement. At the concept stage, the design engineer is required to understand the technical problems at hand. A problem statement should be outlined within a design requirements document or product design specification. Several concept variants may provide a working solution to the problem. It is typical that several concept variants are evaluated against technical or economic criteria. The most appropriate concept is proposed for the development (or detailed design) stage. The detailed design stage considers all of the gearbox components and subsystems. The detailed design includes the integration of the various systems and would typically culminate in a verification or validation test at component or subsystem level. Once the development stage is completed and verified, the design will proceed to the production stage. The production stage may be viewed as component manufacture and integration (assembly). The utilisation and support stage is typically the longest stage of the life cycle. The retirement stage involves decommissioning of the gearbox and recycle or salvage of components.
Figure 7.2 Generic design life cycle. Adapted from Igba J, Alemzadeh K, Durugbo C, Henningsen K. Performance assessment of wind turbine gearboxes using in-service data: current approaches and future trends. Renewable Sustainable Energy Rev 2015;50:144–59.
In this chapter we shall review the standard approach to the design of WTGS gearboxes and demonstrate a WTGS drivetrain dynamic analysis. As mentioned above, a designated standard exists for gearbox design in WTGS with power ratings in excess of 500 kW (ISO/IEC 61400-4). This standard was jointly prepared by IEC technical committee 88: ‘Wind turbines’ and ISO technical committee 60 ‘Gears’. This standard is part 4 of the series of standards for WTGS, IEC 61400, prepared by IEC. The standard applies to WTGS installed onshore or offshore. The gearbox standard (ISO/IEC 61400-4) applies to gears and gear elements on the main power path, auxiliary gearing (for example, power take off or yaw system drives) is excluded or covered elsewhere.
This chapter will review ISO/IEC 61400-4 in the context of the design life cycle stages presented in Fig. 7.2. It should be noted that ISO/IEC 61400-4 presents a flowchart that highlights the iterative nature of GB design at the detailed stage, however the generic design life cycle still applies in a broad sense. Specifically, at concept stage (Section 7.2), we shall present the basic working principles and concept variants of typical WTGS gearboxes; at development stage (Section 7.3), we shall review the standard requirements for gears, bearings, structural elements, and lubrication and sealing in WTGS gearboxes; and at production stage (Section 7.4), we shall review the issues around component manufacture and assembly of gearboxes. Section 7.5 describes a DTD modelling tool that simulates the gearbox performance under dynamic loading events. Such tools are essential in the prediction of WTGS performance at the development and detailed design stage.
Finally, some conclusions are made on the overall design process for the WTGS gearbox. Issues relating to reliability are discussed. The requirement to have fault detection and condition monitoring systems on the WTGS gearbox is briefly highlighted.
7.2. WTGS gearbox design – concept stage
Before setting out on a design exercise, the engineer must understand the problem at hand. To aid with this understanding, this section begins with an overview of the basic operation of the WTGS gearbox. The section will follow on to review some early stage design considerations and lead onto the concept variants that the designer may consider at the early stage of the design. To provide an overview, the diagram in Fig. 7.3 outlines the steps involved in the concept design of a gearbox.
7.2.1. Basic operation of the WTGS gearbox
In WTGS, the gearbox is speed-increasing, where the defining parameter is the gearbox speed ratio, u. The speed ratio relates the input speed on the rotor side, ΩIN, to the output speed on the generator side, ΩOUT.
[7.1]
The speed-up ratio may be cited as a ratio (X:1) or as a number (where ηg > 1 in the case of speed-increasing gearboxes). Taking the example where a rotor is rotating at 50 RPM and the generator is rotating at 1500 RPM, the speed-up ratio is 30:1 or ηg = 30. In WTGS, the gear box speed ratio may be in the vicinity of 100:1.
From a power perspective, the gearbox receives power on the rotor side, PROTOR, and delivers power on the generator side, PGB. The gearbox will experience power loss, PLOSS, as heat is generated due to friction between parts with relative motion (shaft seals, gears, bearings, etc.) and viscous losses (lubrication system). Conservation of energy will apply to give
[7.2]
However, the gearbox output power is usually related to the power from the rotor by the gearbox efficiency which is mainly affected by gear loss and oil churning loss, ηM.
[7.3]
Furthermore, the relationship between the gearbox power on the generator side and the wind power, PWIND, is given by
[7.4]
where CP is known as the power coefficient for the rotor. Theoretically, the power coefficient for a rotor cannot exceed the Betz limit (equal to 16/27, approximately 0.593). Typically, the power coefficients for HAWT are in the range 0.3–0.5. Fig. 7.4(a) shows some typical wind tunnel test data for an HAWT rotor. The plot on the left shows rotor power versus rotor speed for three different wind speeds. At each wind speed, the rotor power has a peak value at a specific rotor speed. As the wind speed increases, the overall power levels show an increasing trend. The same data as Fig. 7.4(a) are used to generate Fig. 7.4(b). Here, the classic plot of power coefficient, CP, versus tip-speed ratio, λ, is shown. Similar to rotor power, the power coefficient has a peak value at a specific tip-speed ratio.
Torque is an important consideration in gearbox design. The torque is required to determine the gear forces at the contact points where gears mesh with each other. In addition, the torque applied to the gear train must be reacted by the structural components in the gearbox: the housing, the mounting points, flange couplings, torque arms, etc. Since the shaft power is the product of torque and rotational speed we can expand Eq. [7.3] to give
[7.5]
where TROTOR and TGB are the torques on the rotor side (ie, gearbox input) and the generator side (ie, gearbox output), respectively.
By rearranging Eq. [7.5], we get the following relationship
[7.6]
Recognising that the speed relationship on the right-hand side of this equation is a kinematic constraint described by Eq. [7.1], we come to the important conclusion that the effect of power loss through the gearbox is manifested as a reduction in the ideal torque output (where the ideal torque is TRotor/ηg).
[7.7]
Following on from Fig. 7.4, Fig. 7.5 shows how the torque–speed relationships change through the gearbox from the rotor side to the generator side. The wind tunnel power data from Fig. 7.4 were used to get the rotor torque–speed curves on the left-hand side. As the wind speed increases, the torque levels increase. However, assuming a constant wind speed, as the shaft speed increases the torque decreases. To aid with the discussion, lines of constant power are superimposed onto the plot area. Each line of constant power represents the peak rotor power at the respective wind speed. These curves show that, even with constant power, the torque must decrease as the shaft speed increases. Eq. [7.7] is applied to the rotor torque–speed curves to give the gearbox torque–speed curves on the bottom right of the plot. It can be clearly seen that in the case of a speed-increasing gearbox, the specific speed values and the range of speed values are increased. On the other hand, the torque levels and the range of the torque values are decreased. Additionally, the torques at the gearbox GS are further decreased by the mechanical efficiency. If the gearbox had a 100% mechanical efficiency, the GB torque curves would be tangential to their respective lines of constant power, but this is not the case due to the power losses in the GB.
Following this analysis, a few important observations can be made in relation to the final design. Because the rotor is running at the lowest speed, it is also delivering the highest torque. Hence, the diameter of the low-speed shaft is large compared to the diameter of the high-speed shaft. The low-speed shaft requires a large, sturdy bearings set to handle the relatively large torques and forces exerted on the rotor. It should be noted that there is a distinct advantage to having the mechanical brake on the high-speed shaft. A torque applied by the mechanical brake on the high-speed shaft is amplified through the gearbox onto the rotor by the inverse effect to that just discussed here.
7.2.2. Early stage design considerations
At the outset, the gearbox designers must develop the gearbox conceptual design with inputs such as input torque, gear ratio, interface requirement, maximum envelope, maximum weight, and manufacturability, etc., from the wind turbine manufacturer and other key suppliers in the design process. Design for reliability is a particular goal, with a design life of 20 years as a target. ISO/IEC 61400-4 recommends that all relevant parties should engage at an early stage to complete a critical system analysis. The standard suggests a list of topics for consideration in the system analysis. A Failure Modes and Effects Analysis (FMEA) is suggested [12,13].
The gearbox design must take into account the complete system architecture and, in particular, the interface requirements with other subsystems in the WTGS. The gearbox will have mechanical interfaces with other parts of the WTGS, for example, mechanical couplings to the low-speed shaft, the high-speed shaft, and the nacelle mainframe. The gearbox will have a lubrication system and will interface with external lubrication components such as reservoirs, coolers, pumps and filters. Other interfaces may exist which relate to the flow and control of sensory information or signals to and from the gearbox. All interfaces must be identified at the early stage of the design.
In some instances the interfaces may overlap due to integration in the system architecture. Consider, for example, the low-speed shaft. A modular design concept (as shown in Fig. 7.1) may be adopted, whereby the low-speed shaft runs on two separate bearings mounted externally from the gearbox. An alternative modular configuration, known as a drivetrain with a three-point supporting structure (suspension), has one bearing on the low-speed shaft and another integral to the gearbox. In this case the bearing internal to the gearbox will assist with reacting the bending moments and forces on the main shaft. A fully integrated drivetrain is also possible, whereby the internal gearbox bearings on the rotor side will carry all of the low-speed shaft loads.
An important aspect in engineering design of any machine is the development of relevant documentation. ISO/IEC 61400-4 gives a list of recommended documentation titles for the entire design cycle. A product design specification should be outlined at the early stage in any design process. Among the recommendations, the ISO/IEC 61400-4 standard suggests that two initial documents be prepared and issued by the wind turbine manufacturer: (1) a general specification and (2) a load specification. The general specification would outline general details on technical performance along with any supplementary information. The load specification would provide information for detailed design in respect of design loads, fatigue loads, extreme loads, etc.
7.2.3. Concept variants for WTGS gearbox
Two basic gear layout arrangements have dominated WTGS gearbox applications: (1) gears on parallel shafts and (2) planetary gear systems (also known as an epicyclic gear system). A typical gearbox speed ratio in large WTGS may be around 30:1 to 100:1. This represents a significant change in rotational speed. A single gear stage is limited to give a speed ratio of approximately 6:1 [3]. Hence, most WTGS gearboxes require multiple stages to the desired overall speed ratios. For example, to achieve a speed ratio of 100:1, three stages could be employed.
Fig. 7.6 shows a schematic for a single-stage, parallel shaft gear arrangement. The input gear (known as the driving gear) has a number of teeth equal to NA, whereas the output gear (known as the driven gear) has a number of teeth equal to NB. The speed ratio for this particular setup is given as
[7.8]
It is clear that to obtain a speed-increasing ratio (r > 1) then the number of teeth on the input gear must be greater than the number of teeth on the output.
Fig. 7.7 shows an arrangement for a two-stage, parallel shaft arrangement that will obtain a higher overall speed increase. Power, PIN, is applied at the input, power, POUT, is delivered from the output gear. In this case, an intermediate shaft is introduced to carry shaft power from the input gear to the output gear. The overall speed ratio is given as the product of each gear stage.
A planetary gear set can give a higher power-to-weight ratio than a parallel shaft gear set while keeping the input and output shaft coaxial. The planetary gear system is named after its likeness to an orbital planetary system. A gear in the centre is called the sun gear, the gears rotating around the sun are called planet gears, and the gear on the outside is called a ring gear. The planet gears are usually equally spaced around the sun gear and are connected to each other by planet pins onto a planet carrier. The ring gear is an internal gear with all other gears being external gears. To increase the speed, the ring gear is held in position (ie, fixed), input power applied to the planet carrier, and output power taken from the sun gear. Fig. 7.8 shows how the gears are arranged. The planetary gear set benefits from power branching, that is, the transmission forces are shared among multiple planet gears. Power branching helps to reduce the loads on individual gear teeth. The planetary gear system is well-suited to carrying high torques.
The number of planet gears and the number of gear teeth are not arbitrary [14]. The pitch for the sun, planets, and ring gears must match. With an equally spaced planetary arrangement with n planet gears, the following relationship must hold:
[7.9]
where NR is the number of teeth on the ring gear, NS is the number of teeth on the sun gear, and NP is the number of teeth on the planet gears. In addition, the following relationship must hold between gear teeth
[7.10]
The gear ratio from a fixed ring, planetary arrangement is given by the relationship
[7.11]
Planetary gear stages may be used in series with parallel axis gear stages to get the overall desired ratio. As mentioned above, two or three stages may be required to get the overall ratio. Typically, due to the higher torque requirement on the rotor side, planetary gear sets are used in the first and/or second stages of the drivetrain.
For large offshore wind turbines, the use of a split torque planetary gearbox is currently gaining popularity. Split torque systems have the advantage of increasing power density. These systems can transmit a very high torque in a very small space, which has made such systems attractive for offshore wind turbine applications. These gear systems use the principle of division of the transmission force between several contact areas, which in turn increases the contact ratio.
7.3. WTGS gearbox design – development stage
After the conceptual design has been selected and the major decisions have been made (modular or integrated layout, number of stages, planetary or parallel axis, etc.) the detailed design stage must begin. The FMEA, if conducted, should have led the designer to consider the major analyses required to make the design safe. ISO/IEC 61400-4 makes recommendations on determining the drivetrain operating conditions and loads. Recommendations are made regarding the use of computer simulations of drivetrain loads. In particular, ISO/IEC 61400-4 outlines the practices for determining time series, fatigue, and extreme loading scenarios. Fig. 7.9 is adapted from ISO/IEC 61400-4 and gives a suggested design process workflow for WT gearbox.
Figure 7.9 Design process flowchart – development stage. Adapted from ISO/IEC 61400-4. Wind turbines – part 4: design requirements for wind turbines gearboxes. 2012.
The ISO/IEC standard makes particular recommendations in several design aspects, namely; gearbox cooling; gears; bearings, shafts, keys, housing joints, splines and fasteners; structural elements; and lubrication.
7.3.1. Gear design
Internally within the gearbox, the gears closest to the rotor side carry the highest torques. The torque gets progressively smaller as the rotational speed of the gear shafts increase through each gear stage. Hence, the gear stage closest to the rotor side is usually designed to take greater torque than the gear stage at the generator side. This aspect of operation impacts upon the design of the internal components of the gearbox. As mentioned above, planetary gear stages can carry greater torque due to power branching between planet gears and, therefore, are sometimes the preferred option at the rotor side. Otherwise, gear stages with identical ratios (assuming identical pitch, helix angle, and number of teeth) will have differing torque ratings due, primarily, to the width of the gear in the axial direction – narrow gears will carry less torque than wide gears. The increase in speed through the gear stages will impact upon the bearing selection.
Gears may be manufactured as spur gears or helical gears. Spur gears are the simpler of the two gear types: the gear teeth run parallel to the gear shaft in the axial direction. Helical gears have gear teeth that run along the gear shaft at a particular angle, called the helix angle. The introduction of the helix angle allows for a smoother engagement of the gears. This is because the contact ratio – the average number of teeth in contact at any time – is typically greater for helical gears than for spur gears. However, due to the reaction of forces on the helix gears, the shaft and bearings must be designed to counteract greater axial thrust forces. Another gear configuration that is used to counteract the axial thrust is called herringbone, whereby two counteracting helices are used on a single shaft.
The application of ISO/IEC 61400-4 requires that several other general gear-related standards be applied. Table 7.1 shows the typical modes of failures that can occur and the relevant standard that should be adhered to.
ISO/IEC 61400-4 makes some specific recommendations related to gear rating factors. For example, the dynamic factor, KV, for WTGS gearboxes should be calculated using method B from ISO 6336-1 and be subject to a minimum value of 1.05 unless proven otherwise by measurement. Dynamic factor can significantly affect gear rating.
The gear load-sharing factor or mesh load-sharing factor, Kγ, applies specifically to gear stages that incorporate power branching, ie, planetary gear stages. Inaccuracies in the gears can cause deviations in the load splitting and the mesh load factor takes account of these deviations based on n, the number of planets gears. Table 7.2 provides data on the recommended mesh load factors.
The gear mesh load distribution factor takes into account non-uniform load distribution over the gear face width on the surface stress (KHβ) and on the tooth root stress (KFβ). ISO 6336 gives procedures for calculating the face load factors and these procedures are generally applicable for WTGS gearbox design. However, ISO/IEC 61400-4 gives some particular recommendations for calculating the face load factor for surface stress, KHβ. Specifically, the gear mesh misalignment, fma, (given in millimetres) is calculated based on specific recommendations that differ from the procedures given in ISO 6336. The minimum recommended vale of KHβ is 1.15.
Table 7.1
Applicable gear standards for gear component design
| Failure mode | Relevant details (as cited in ISO/IEC 61400-4) |
| Gear pitting | ISO 6336 series Minimum safety factor, SH = 1.25 |
| Bending | ISO 6336 series Minimum safety factor, SF = 1.56 Life factors, ZNT and YNT, determined using 0.85 × 1010 cycles |
| Scuffing | ISO/TR 13989-1 DIN 3990-4 ANSI/AGMA 925-A02 + ISO/TR 13989-2 |
| Micropitting | ISO/TR15144-1 |
| Static strength | ISO 6336 series Evaluated at extreme torque using static life factors YNT and ZNT Minimum safety factor for root bending, SF > 1.4 Minimum safety factor for surface durability, SH > 1.0 |
| Subsurface-initiated fatigue | DNV classification note 41.2 (subclause 2.13) |
7.4. WTGS gearbox design – production stage
There are many issues that can arise at the production stage that can affect the performance and reliability of the WTGS gearbox in service. In a broad sense, the production stage may be considered as component manufacture and assembly. A quality plan should be in place to ensure that no adverse effects occur due to production.
Many processes are used in the manufacture of a single component: forming processes (casting and forging), subtractive processes (cutting and grinding), and surface treatment (polishing, heat treatment, etc.) to name a few. Finished components always have some deviation from the intended form. These deviations could be geometrical in nature due to the accuracy and precision limitations of the manufacturing processes. Variations also occur in the material's physical properties due to chemical or microstructural variations in the material which, again, are related to the processing of the material. Materials may also suffer from defects within the material: flaws, voids, inclusions, etc. The designers must develop a robust design that can tolerate these minor variations. Manufacturing engineers must ensure that the components are supplied within the allowable tolerances as outlined within the specifications. These aspects should form part of the quality plan. To allow for specified variations in geometry, designers should use geometric dimensioning and tolerancing standards for all components [15]. Minimum material requirements are usually set out in accordance with a standard, for example, gear components must comply with the requirements of ISO 6336-5, steel used in bearings must meet ISO 683 requirements.
Assembly processes can be permanent (for example, welding) or non-permanent (screwed fasteners, etc.). Most of the assembly processes used in gearbox manufacture are the non-permanent type. This allows for easier maintenance during the operating stage and easier disassembly at the retirement stage. Application of the correct torque to all bolted joints is essential to the assembly. In some instances, as with tapered roller bearings, precompression of the bearing is also important. This preload is achieved by applying torque to a tightening nut which compresses the bearing set. All of these torques must be applied appropriately. In the housing joints, it is recommended that split plane housing joints should have positive locating devices such as dowel pins to aid with assembly. In the case of planetary gear sets it is recommended that the ring gear joint should be capable of carrying the maximum operating load by friction only (with some safety margin). If the friction generated by bolt torque alone is insufficient, then the designer should consider the application of solid pins to carry the loads at the flange. In the case of pins, the contribution of friction should be ignored from the calculations. This feature of ISO/IEC 61400-4 represents specific, conservative design requirements that have been determined through the experience of iterative design cycles. In some cases accuracy may be lost during the assembly processes, a phenomenon known as tolerance stack up. Designers need to be aware of tolerance stack up during design of an assembly.
7.4.1. Gear manufacture and inspection
ISO/IEC 61400-4 has specific requirements related to gear manufacture. The method and processing of all gear elements should be specified as a matter of course. Grinding notches should be avoided in the gear cutting process; however, if gear notches do occur during the manufacture process, then FEA methods or the YSG factor from ISO 6336-3 should be used to determine the reduction in tooth bending stress. Rejection criteria are clearly outlined in ISO/IEC 61400-4.
Otherwise, gear accuracy should be specified in accordance with ISO 1328-1. The standard ISO 1328-1 sets out 11 grades of gear accuracy. The tolerance values increase with each increase in grade level. ISO/IEC 61400-4 gives maximum accuracy grades as level 6 for external gears and level 7 for internal gears (with some specific allowances to grade 8 for nitrided internal gears). Improved tolerances will ensure smoother performance but the cost factors associated with increased accuracy should be considered. It should be noted that the accuracy grades apply to assembled gears – if gears loose there accuracy during assembly, then grinding after assembly should be considered.
The surface finish required on gear components is specified as Ra = 0.8 μm for external gears and Ra = 1.6 μm for internal gears. Improving the surface finish reduces the risk of micropitting on the gear. It should be noted that shot-peening of gear flanks is not permitted as a final operation; this is because the surface finish may be compromised.
ISO/IEC 61400-4 makes specific recommendations for surface temper after grinding. In particular, a 100% sampling plan is recommended. This requirement is provided to ensure that all gears have the appropriate surface qualities, such is the importance of surface finish. Poor gear tooth surface finish can cause micropitting. Through experience, the following surface roughness, Ra, values are recommended: Ra < 0.7 μm for high-speed and intermediate pinion and gears; Ra < 0.6 μm for low-speed pinion and gear; and Ra < 0.5 μm for low-speed sun and planet. It is important that the gear surface roughness measurement compensates for the involute form of the gear tooth. A skid style stylus pick-up is often used for gears. The stylus and the skid are situated side-by-side for improved tracking of the involute form. The skid radius on a gear tooth pick-up is around 0.8 mm and is typically smaller than that found on a general-purpose pick-up.
Surface crack detection is also advised whereby magnetic particle, fluorescent magnetic particle penetrant or dye penetrant inspection methods may be used as agreed by the supplier and customer (ISO 6336-5 should be referred to in this instance).
7.5. Drivetrain dynamic analysis
ISO/IEC 61400-4 recommends, as a minimum, that drivetrain analysis be performed to verify the WTGS aero-elastic response. An example of a WTGS drivetrain dynamic (DTD) model, which includes the gearbox subsystem, is discussed next.
7.5.1. Variable loading
The torque level in a wind turbine gearbox is variable. This fluctuation will normally vary between zero and rated torque as the wind speed varies. Excursions above rated torque are possible on a fixed-speed pitch-regulated machine. This can be attributed to the slow pitch response. The torque time histories will be subject to dynamic magnification and can in fact excite the drivetrain leading to resonances. Transient events such as braking can induce infrequent large-magnitude torques but of short duration. This effect of producing large-amplitude torques can be mitigated if the brake is fitted to the low-speed shaft. The normal practice to calculate the load–duration curves is by combining the power curve with the distribution of instantaneous wind speeds. The computation of distribution of instantaneous wind speeds is carried out by superposing the turbulent variations around each mean speed on the Weibull distribution of hourly means.
7.5.2. Drivetrain dynamics
Rotational sampling by the blades is the source of torque fluctuations in wind turbines. Since this phenomenon is caused by the effect of rotating blades, the frequencies generated are at blade-passing frequency and multiples thereof. Hence, all wind turbines experience aerodynamic torque fluctuations at these frequencies. These fluctuating torques interact with the dynamics of the drivetrain, with the possibility of dynamic amplification and hence modifying the torque transmitted. If a fixed-speed wind turbine with an induction generator is considered, the resulting drivetrain torque variability can be assessed by dynamic analysis of a suitable drivetrain model. Such a model consists of the following elements connected in series:
• a body with rotational inertia and damping (representing the turbine rotor);
• a torsional spring (representing the stiffness of the gear box and the high-speed shaft coupling);
• a body with rotational inertia (representing the inertia of the mainshaft, rotor lock disk and generator rotor);
• a torsional damper (modelling the resistance produced by slip-on induction generator);
• the electrical grid.
These inertias, stiffness and damping must all be referred to the same shaft.
For carrying out a detailed stress analysis of the drivetrain an accurate multibody model of the system is required. A detailed model provides information on deflection, stresses generated and damage induced. However, if dynamics related to the modes of vibration and development of control algorithm is the focus of attention then simplified models may be used instead. Such a simplified dynamic model is described in Section 7.5.3 which is capable of capturing the natural frequencies and modes of vibration and also provides information on the states related to torsional oscillations.
7.5.3. Two-mass drivetrain shaft model
Drivetrain components are fundamentally responsible for transmitting the generated aerodynamic torque generated on the rotor from the rotor hub to the wind turbine generator [16]. The drive torque is filtered by the drivetrain and is converted into mechanical torque, which in turn drives the generator shaft. In this section, the drivetrain is modelled by a simplified two-mass mechanical system as shown in Fig. 7.10. In comparison with higher-order multimass drivetrain models, the two-mass system has been found to provide enough accuracy for the transient stability analysis of wind turbine generation systems [17]. Muyeen et al. [18] have carried out a comparative study among different types of drivetrain modelling. Their study concluded that higher-order models (six-mass and three-mass) of drivetrains can be reduced to a two-mass model without significant loss of accuracy with respect to response exhibited under a network disturbance.
Figure 7.10 Torsional vibration model of the WTGS drivetrain. After Basu B, Staino A, Basu M. Role of flexible alternating current transmission systems devices in mitigating grid fault-induced vibration of wind turbines. Wind Energy 2014;17: 1017–33. http://dx.doi.org/10.1002/we.1616. Copyright © 2013 John Wiley & Sons, Ltd.
In the simplified model considered the following elements are included to represent the drivetrain dynamics:
• a mass with rotational inertia 
(representing the turbine rotor, ie, the blades);
(representing the turbine rotor, ie, the blades);• a torsional spring and a torsional damper (representing the low-speed shaft);
• an ideal gear box with speed-up ratio 
;
;• a rigid high-speed shaft;
• a mass with rotational inertia 
(representing the generator rotor).
(representing the generator rotor).A linear torsional spring with stiffness coefficient
is used to represent the low-speed shaft and a torsional damper of damping coefficient 
represents the damping of the drivetrain.
is used to represent the low-speed shaft and a torsional damper of damping coefficient represents the damping of the drivetrain.The equation governing the dynamics of the blades (turbine rotor) is given by
[7.12]
In Eq. [7.12], 
and 
are the aerodynamic torque generated by the wind and the torque at the low-speed shaft, respectively. A small amount of damping, represented by 
, may be also added in order to model friction dissipation effects.
and are the aerodynamic torque generated by the wind and the torque at the low-speed shaft, respectively. A small amount of damping, represented by , may be also added in order to model friction dissipation effects.The dynamics of the shaft is represented by the following equation:
[7.13]
where 
is the rotor angular position; 
and 
are the low-speed shaft angular deviation and speed, respectively. The transmission speed-up ratio is given by (on the assumption of an ideal gearbox):
is the rotor angular position; and are the low-speed shaft angular deviation and speed, respectively. The transmission speed-up ratio is given by (on the assumption of an ideal gearbox):
[7.14]
In Eq. [7.14], 
and 
denote the generator shaft angular position and speed respectively, and 
is the drive torque provided as an input to the generator. Finally, the following differential equation represents the dynamics of the mechanical part of the generator
and denote the generator shaft angular position and speed respectively, and is the drive torque provided as an input to the generator. Finally, the following differential equation represents the dynamics of the mechanical part of the generator
[7.15]
where 
is the electromagnetic torque developed and 
is the associated damping on the generator side.
is the electromagnetic torque developed and is the associated damping on the generator side.For the purpose of studying the induced vibration, the relevant degree-of-freedom describing the drivetrain dynamics associated with the torsional mode of the flexible shaft system has been considered. The equation governing the drivetrain torsional oscillation is
[7.16]
where 
represents the angular difference between the two ends of the low-speed shaft.
represents the angular difference between the two ends of the low-speed shaft.In addition to aerodynamic fluctuations, occurrence of faults in the electrical subsystem of the wind turbine may also induce torsional drivetrain oscillations [19].
7.5.4. Coupled electromechanical interaction
The importance of coupled electromechanical effects has been highlighted in the available literature [20–24]. These studies stress the necessity of a detailed mechanical model, to understand the mechanical response of wind turbines to an electrical disturbance. In addition, a detailed electrical model is also essential as the fault generating the disturbance is an electrical phenomenon. The detailed mechanical and electrical models are then needed to be coupled together. In order to do so, it is required to identify the parameters which are be to communicated or exchanged between the two detailed models (electrical and mechanical).
The dynamics of the flexible rotor blades are governed by the aerodynamic load. The corresponding equations governing the variable-speed rotor dynamics are a function of the blade rotor speed and accelerations. The generated aerodynamic torque 
forms an input to the drivetrain. On the electrical side, the dynamics of the generator is governed by Eq. [7.15]. For this equation, the torque from the drivetrain (high-speed shaft) serves as the input. Subsequently, the generator speed and accelerations, and the electrical torque are the output. Hence, the mechanical drivetrain and gear system form the connecting link between the rotor blades and the electrical generator. This is illustrated in Fig. 7.11. The motion of the two-mass drivetrain model with the shaft is described by Eqs [7.12]–[7.14] and Eq. [7.16]. These equations lead to the following second-order linear differential equation which represents the torsional oscillation of the drivetrain with the aerodynamic torque, and the generator speed and accelerations as the input.
forms an input to the drivetrain. On the electrical side, the dynamics of the generator is governed by Eq. [7.15]. For this equation, the torque from the drivetrain (high-speed shaft) serves as the input. Subsequently, the generator speed and accelerations, and the electrical torque are the output. Hence, the mechanical drivetrain and gear system form the connecting link between the rotor blades and the electrical generator. This is illustrated in Fig. 7.11. The motion of the two-mass drivetrain model with the shaft is described by Eqs [7.12]–[7.14] and Eq. [7.16]. These equations lead to the following second-order linear differential equation which represents the torsional oscillation of the drivetrain with the aerodynamic torque, and the generator speed and accelerations as the input.
Figure 7.11 An electromechanically coupled system. After Basu B, Staino A, Basu M. Role of flexible alternating current transmission systems devices in mitigating grid fault-induced vibration of wind turbines. Wind Energy 2014;17: 1017–33. http://dx.doi.org/10.1002/we.1616. Copyright © 2013 John Wiley & Sons, Ltd.
[7.17]
The solution of the second-order linear differential equation provides the required quantities of interest, such as drivetrain torsional angular displacement and the angular velocity with prescribed initial conditions. However, for solving Eq. [7.17], appropriate initial conditions are required. Initial conditions that can be used for solving the equation without any loss of generality are zero values for drivetrain torsional states (ie, no initial twist) and rated generator speed with no acceleration corresponding to the rated rotor blade speed. Using Eqs [7.13] and [7.14], the high-speed shaft torque can then be computed. Also, blade rotor speed and accelerations can be computed using Eq. [7.12]. Thus, the outputs generated are the high-speed shaft torque, and the blade rotor speed and accelerations. The high-speed shaft torque is fed into the generator to update the generator dynamics, while the blade rotor speed and accelerations form inputs to the rotor blade dynamics equations to update the system matrices and to compute the variable-speed rotor blade dynamics. The scheme of the computation described with the outlined electromechanical coupling incorporates the effect of (1) flexible edgewise blade dynamics, (2) variable rotor blade speed, (3) two-mass drivetrain with shaft model and (4) generator dynamics.
7.5.5. Effect of variable loading on fatigue design of gear teeth
Fatigue design of gear teeth is guided by two factors. The contact stresses generated on the flanks and the bending stresses generated at the root, must both be within acceptable limits. The compression stress (for Hertzian contact) between a pair of spur gear teeth in contact at the pitch point (ie, at the point on the line joining the gear centres) is given by
[7.18]
where 
is the force between gear teeth at right angles to line joining the gear centres; 
is the gear face width; 
is the pinion pitch diameter of the pinion of the driving or input gear; 
is the gear ratio (greater than unity); 
is the pressure angle – that is, that angle at which the force acts between the gears – usually 20 degree to 25 degree, E is the elastic modulus; ν is the Poisson's ratio.
is the force between gear teeth at right angles to line joining the gear centres; is the gear face width; is the pinion pitch diameter of the pinion of the driving or input gear; is the gear ratio (greater than unity); is the pressure angle – that is, that angle at which the force acts between the gears – usually 20 degree to 25 degree, E is the elastic modulus; ν is the Poisson's ratio.The maximum bending stress at the root of gear teeth is given by
[7.19]
where h is the maximum height of single-tooth contact above the critical root section; t is the tooth thickness at the critical root section; KS is the root stress concentration factor.
For the case of the gears operating at rated torque it is sufficient to show that the resultant bending stress multiplied by a suitable safety factor is less than the endurance limit stress multiplied by a number of factors (such as life factor and a number of stress modification factors). A similar procedure also has to be followed for the contact stress.
For design against fatigue or calculation of fatigue damage the predicted turbine load spectrum should be used. This should also include dynamic effects. For the fatigue design calculation it is necessary to compute the design equivalent torque at the endurance limit. This computation is normally carried out with the aid of Miner's rule for fatigue damage and calculating the infinite life torque for which the design torque spectrum yields a damage index of unity in conjunction with the prescribed S–N (load vs. number of cycle) curve for the material concerned. In this scenario, the life factor can be set to unity as it has been indirectly accounted for in the infinite life torque calculation. Codes (such as BS 436) prescribe specimen torque–endurance curves which can be used for gear tooth design. The design infinite life torque can be calculated from the load-duration spectrum according to the formula:
[7.20]
where 
is the number of cycles of torque of magnitude Ti. Torques of magnitude less than T∞ are not considered in the computation. The number of cycles at the lower knee of the torque–endurance curves N∞ is 3 × 106 for tooth bending. However, this is generally higher for contact stress and varies according to the type of material used. The index m, for the torque–endurance curve-related contact stress, is half that of the contact stress–endurance curve as torque is proportional to the square of the contact stress. More details on the fatigue design of gear teeth, bearings and shafts are available in Burton et al. [16].
is the number of cycles of torque of magnitude Ti. Torques of magnitude less than T∞ are not considered in the computation. The number of cycles at the lower knee of the torque–endurance curves N∞ is 3 × 106 for tooth bending. However, this is generally higher for contact stress and varies according to the type of material used. The index m, for the torque–endurance curve-related contact stress, is half that of the contact stress–endurance curve as torque is proportional to the square of the contact stress. More details on the fatigue design of gear teeth, bearings and shafts are available in Burton et al. [16].7.6. Conclusions
The role of the gearbox in WTGS has been highlighted. A particular focus was given to the design of wind turbine gearboxes. A generic life cycle was presented, starting at the concept stage, going through the development stage, the production stage, the operation and maintenance stage and ending with the retirement stage. As it is the most relevant standard in the area, ISO/IEC 61400-4, Design requirements for wind turbine gearboxes, was reviewed in some detail. In particular, the elements outlined in ISO/IEC 61400-4 were categorised according to the generic life cycle. A focus was provided on the concept stage, the development stage (detailed design) and the production stage. A drivetrain dynamic analysis has been outlined. This type of analysis is a minimum requirement contained within ISO/IEC 61400-4.
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