Engineering Tripos Part IB SECOND YEAR Part IB Paper 8: - ELECTIVE (2) MECHANICAL ENGINEERING FOR RENEWABLE ENERGY SYSTEMS Examples Paper 2 Wind Turbines, Materials, and Dynamics All questions are of Tripos standard Questions 1 and 2 relate to the Iskra/Vestas Guest lectures. 1. (a) Discuss the main differences between micro wind (<2m diameter), small wind and large wind (>10m diameter) with reference to their applications and energy yield. As a point of reference assume that a typical 5m turbine is rated at 5kW. (b) Discuss over-speed protection and methods of shedding surplus power making sure to include furling, active pitch regulation and mechanical brakes. (c) Discuss the benefits of designing a direct-drive wind turbine, i.e. without a gearbox. Why aren t all wind turbines direct drive? 2. (a) Identify in point form the engineering challenges associated with the following components of a large wind turbine: (i) foundations, (ii) tower, (iii) blades, (iv) hub, (v) nacelle (comprising gearbox, bearings, drive shaft, brake, generator, hydraulics, control system). (b) Discuss the challenges for designing wind-turbines to cope with harsh conditions (typhoon, lightning, extreme cold, extreme heat). (c) Give examples where simulation tools can be useful in the design of wind turbines.
Materials 3. (a) Discuss the importance of materials choice in wind turbine blade and tower design. (b) Consider a series of self-similar blades in which the plan form and cross-sectional shape scale with length L. The blades are subject to a uniform pressure loading along the length of the beam (i.e. a storm loading condition). Use simple beam theory to show that the peak root bending stress associated with this aerodynamic load does not depend on L, while the stress associated with self-weight scales as L. The selfweight stress you should consider is due to edge-wise bending when the blade is horizontal. Confirm your results using dimensional analysis. 4. Consider the design of a spar of length L as illustrated below with a linear variation of spar depth d with distance x from the tip, and a linearly tapering change in spar cross sectional area A, i.e. d d0 = x / L and A A0 = x / L, where the subscript 0 refers to conditions at the root. A total aerodynamic load W is uniformly distributed as a pressure acting on the triangular plan form. Derive the following expression for the mass required to give a tip deflection δ: 4 ρwl m δ = 2 12Ed 0 δ The spar material has Young's modulus E and fatigue strength σ f. Assume that spar skin thickness is much less than its depth in calculating the second moment of area. The corresponding mass to meet a strength constraint is given by ρ WL mσ = 6d σ 0 2 f Explain how these results can be used to identify whether stiffness or strength is the critical constraint for this type of design, and identify at what length L the crossover point lies, taking values of E = 45 GPa, σ f = 150 MPa, δ = 5m and d 0 = 0.05 L. Cross section 2d 0 2d x 2d L A/2
5. (a) Explain carefully, with a sketch, how rainflow counting is used to identify cycles of loading in a random signal. (b) Identify all the half-cycles present on the following diagram of stress as a function of time. Stress Time
6. A blade made of GFRP has fatigue properties which can be fitted by the expression N = S S where N is the number of cycles to failure under a given applied cyclic stress range S, with M = 9, S 0 = 2σ ts = 400 MPa. (a) The wind loading is estimated to give stresses in the critical area of the blade following the table distribution given below. Deduce the expected lifetime of the blade. 0 M Mean Stress S m (MPa) Alternating stress range S (MPa) 5-15 15-25 25-35 15-25 300 300 200 25-35 200 300 200 35-45 200 200 100 Number of cycles (in thousands) in a one month block (b) An alternative model fits the stress data by a Rayleigh probability density function φ, φ ( S) = 2 2 S exp S S with 10 6 loads per month and a mean stress range S = 30 MPa. Calculate the expected lifetime for this model of the loading. z 1 t Hint: ( z) t e dt Γ = is the Gamma function, equal to (z-1)! for positive integers. 0
Dynamics 7. A 40m diameter three-bladed wind turbine generates 166kW at 8m/s wind speed under which conditions the rotor turns at 30rpm. The rotor has a hub diameter of 4m. For simplicity the three blades may be considered to be divided up into three uniform 6m sections of mass 1200kg, 900kg and 300kg. Aerodynamic forces give rise to bending moments at the blade root of 104kNm flap-wise and 14kNm edge-wise. (a) Estimate the polar moment of inertia for the rotor. (b) Compute the gyroscopic couple acting on the rotor when the yaw rate is 10 deg/s. (c) What yaw rate will give rise to blade-root bending moments that match those due to aerodynamics? (d) Show on a sketch the position of a blade when its root is subject to maximum bending moment due to the sum of aero and gyro effects. Show directions of rotation and wind speed clearly on your diagram. 8. A 40m diameter three-bladed wind turbine generates 166kW at 8m/s wind speed under which conditions the rotor turns at 30rpm. The rotor has a hub diameter of 4m. For simplicity the three blades may be considered each to be uniform along their entire 18m length and each of mass 2400kg. Aerodynamic forces give rise to bending moments at the blade root of 104kNm flap-wise and 14kNm edge-wise. Assume that the turbine is operating at the Betz limit and that the generator is operating 100% efficiently. A gearbox is used to drive the generator at 240rpm. The gearbox comprises a single pair of herringbone gears on two parallel shafts at 900mm centre spacing. (a) During steady operation at 8m/s wind speed estimate (i) the thrust on the turbine, (ii) the forces acting on each of the two shafts in the gearbox. (b) The rotor is running steadily when an emergency stop is required. The rotor is brought to rest by reducing the speed steadily over a period of 35 seconds. (i) Explain why it is undesirable to stop the rotor instantaneously. (ii) How does the blade-root bending moment compare with that due to aerodynamics? (iii) You must decide whether to brake the rotor using a disc brake on the 30rpm shaft or to use the generator as an electric brake. How does your decision affect the forces acting on the shafts in the gearbox? (iv) How does the deflection of the tower due to braking loads compare with the deflection due to the steady operation thrust at 8m/s wind speed? Make simple assumptions for the geometry of the tower. 9. The rotor of a wind turbine is turning steadily at 30rpm. A gearbox is used to drive a three-phase fourpole generator at approximately 240rpm. The gearbox comprises a single pair of helical gears on two parallel shafts. The gear on the 30rpm shaft has 137 teeth and the gear on the 240rpm shaft has 17 teeth. (a) (i) What is the advantage of choosing such odd numbers of teeth on the gears? (ii) What are the advantages/disadvantages of prescribing the use of helical gears over spur gears? (b) What are the frequencies of vibration that can be expected by the drive train of the generator? (c) By what mechanisms does drive-train vibration manifest itself as airborne noise? How can this best be controlled?
Answers 4. 150 m 5. (b) 2-3, 3-3a, 4-5. 5-5a, 6-7, 7-7a, 1-8, 8-13, 9-10, 10-12b, 11-12, 12-12a, 13-14 6. (a) 58.8 years (b) 46.2 years 7. (a) 698000 kgm 2 (b) 383 knm (c) 4.1 deg/s 8. (a) (i) 31kN, (ii) 66kN (b) (ii) about double (iv) about 10% 9. (b) 0.5Hz and 4Hz shaft rotation speeds, 68.5Hz teeth mesh, 24Hz generator MPFS/HEMH/DDS May 2009