Attempt ALL QUESTIONS IN SECTION A, ONE QUESTION FROM SECTION B and ONE QUESTION FROM SECTION C Linear graph paper will be provided.

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination ENERGY ENGINEERING PRINCIPLES ENG-5001Y Time allowed: 3 Hours Attempt ALL QUESTIONS IN SECTION A, ONE QUESTION FROM SECTION B and ONE QUESTION FROM SECTION C Linear graph paper will be provided. An Equation Information Sheet is provided at the end of the paper. You may also use an equation given in one question when answering other questions. Notes are NOT permitted in this examination. Do not turn over until you are told to do so by the Invigilator. ENG-5001Y Module Contact: Prof Lawrence Coates, MTH Copyright of the University of East Anglia Version: 1

2 2 SECTION A. 1. Figure Q1 shows a sketch of a vertical brake pedal subject to loads in the plane of the figure. Points B and C are fully fixed. BA is a spring component, CD and ADE are beams. Neglect the mass of the members. a) Draw a FBD of Part ADE and write down the force at Point A. b) Draw a FBD for the system (Parts BA, ADE, and CD) and calculate the bending moment at Point C. Figure Q1. Brake pedal 2. The aluminium beam of Figure Q2 is simply supported at points A and B. It is loaded by a distributed load of 340N/m on part BC. At point A the support can resist vertical loads in either direction. Determine the equations for the bending moment M(x) in the part BC. Then, plot the shear force diagram for the whole beam DABC. [10 marks] Figure Q2. Beam with distributed load ENG-5001Y Version 2

3 3 3. a) Plot a typical magnitude and phase of the steady-state response in the frequency domain for a damped single-degree-of-freedom system subject to harmonic excitation F0 cos( t). b) What are the Laplace Transform and the Fourier Transform? State the features and relationship between these two transforms. 4. Figure Q4 shows a plan view of a typical feature used in long straight pipelines that is designed to allow for safe temperature expansion without buckling. The diversion in a horizontal plane involves 4 identical right angled bends. a) For a particular pipe of 400mm diameter a flow of water of 0.8m 3 /s is entering at A under pressure of 300kPa. Focusing on bend A estimate the direction and magnitude of the force exerted on the bend as water flows round it. Hence deduce the forces on bends B, C and D and the overall force caused by the flow. You may neglect energy losses. [8 marks] b) Consider what happens when energy is lost as the flow passes through all four bends. How does this affect the overall force on the diversion? [2 marks] A D B C Figure Q4. TURN OVER ENG-5001Y Version 2

4 4 5. The vertical profile of horizontal mean wind speed is often modelled as a loglaw: U(z) = 1 d ln (z ) u 0.4 z 0 a) Sketch a typical profile and explain why, above a certain height, speed varies very little with z. Explain clearly the role of the parameters z0 and d. [3 marks] b) The vertical profile of mean wind speed at a site for a proposed wind turbine is assumed to behave as a log-law boundary layer with a roughness length of 0.03m and negligible displacement height. The mean wind speed at 10m height has been estimated as 7.5ms -1. Use this information to estimate the mean wind speed at 25m. [4 marks] c) If the proposed wind turbine tower has a diameter of 4m at a height of 25m what is the Reynolds number and the Strouhal number based on mean values (see information sheet)? Describe the significance of these two numbers in this context? What further data would be needed to be able to do an accurate analysis for checking the long-term safety of the turbine tower? [3 marks] ENG-5001Y Version 2

5 5 SECTION B. 6. a). Explain the basis and the significance of the Betz limit for the design of wind turbines. b). The theoretical power derived from a horizontal axis wind turbine is given by P = 1 2 C P ρ A swept U 3 Explain in what ways a power curve for a real turbine deviates from this theoretical curve, and why. c). Table Q6a shows a simplified distribution of daily mean wind speeds from three years of data. Construct a single representative annual probability distribution. Explain any assumptions that you make. [9 marks] d) Table Q6b shows a simplified power curve for a 50kW wind turbine. Using your results from part (c) and inserting appropriate values from table Q6b, estimate the Annual Energy Production, AEP. Show all units clearly and include an indicative calculation for your table. Express your answer in MWh/y. [6 marks] Table Q6a Wind Speed (m/s) < >15 Year 1 Year 2 Year 3 Number of Days Number of Days Number of Days Table Q6b Wind Speed (m/s) Turbine Output (kw) TURN OVER ENG-5001Y Version 2

6 6 7. The middle third rule may be derived for a rectangular slab carrying an eccentric load by considering the reactionary pressure distribution underneath it as analogous to the stress in a column. The following equations therefore avoid some of the mathematical complications. σ = F A + My = F (F e) y + I A I where the symbols have their usual meaning. a) Explain how the middle third rule for rectangular shapes can be useful in design. b) Construct a sequence of three sketch diagrams of a rectangular slab carrying a vertical eccentric load (equivalent to its weight and an additional point load) as shown in Figure Q7a, b and c such that the load is a) inside, b) at the limit of and c) outside the middle third. Show under each slab the linear pressure distribution provided by the ground. Explain why having the load outside the middle third doesn t necessarily lead to instability. [8 marks] a b Figure Q7 c c) A small concrete foundation slab 600mm by 600mm transfers a point load P to the ground. For one case the load is 150mm from the centreline. The bearing capacity of the ground (including safety factor) is limited to 120kPa. By considering vertical and moment equilibrium of the slab, find the largest value of P that the ground can sustain. [12 marks] ENG-5001Y Version 2

7 7 SECTION C. 8. In a wind turbine system, the main shaft (rotor shaft) and one blade (Figure 8a) can be simply modelled as a shaft and an eccentric force P due to the weight of the blade. Two forces P = 18kN and F = 15kN are applied to the shaft with a radius of R = 20mm as shown in Figure 8b. Following the steps from a to d, determine the maximum normal and shear stresses developed in the shaft using the Superposition Principle. a) Convert the force P at Point K into an equivalent force P and torque T at Point O. b) Calculate the second moment of area I and the polar moment of inertia J for the cross-section of the shaft. c) Determine the maximum bending moment Mmax and maximum shear force Vmax [8 marks] d) Determine the maximum normal stress σmax and shear stress ԏmax [7 marks] Figure Q8a. Rotor shaft and blades Figure Q8b. Simplified model TURN OVER ENG-5001Y Version 1

8 8 9. Consider a cantilever beam with the left end fixed in the wall in Figure Q9a. If the mass shown is suddenly dropped on the right end of the beam, the vibration of the cantilever beam can be approximated as a simple mass-spring system (shown in Figure Q9b) subject to the impulse function F δ(t) shown in Figure Q9c. Ignore the self-weight of the cantilever beam and assume that the mass will separate from the beam after the impact. a) What is the meaning of Duhamel s Integral? b) If this beam is considered as an undamped vibration system and the initial displacement and velocity are both zero, calculate the displacement response x(t) of the beam. [9 marks] c) Write down the governing equation of the above system subject to a unit step force where the damping of the system (c 0) is considered. [6 marks] d) Solve the equation using a Laplace Transform to determine the response in the s-domain. Assume that the initial conditions are x(0)=0 and x (0)=0. mass cantilever beam Figure Q9a. Cantilever beam F δ(t) F 0 Figure Q9b. Vibration model FigureQ9c. An impulse input of the form F δ(t) t d t END OF PAPER ENG-5001Y Version 1

9 9 Equation Information Sheet Generalised Bernoulli Velocity Potential Dispersion relation Morison s equation Breaking limits Strouhal number Continuity φ = gh 2ω φ t + p ρ + gz q2 = f(t) cosh k(h + z) sin(kx ωt) cosh kh ω 2 = gk tanh kh df = C M ρ(volume) U t + C D(Frontal Area) 1 2 ρu U H max λ = tanh kh, or St = f s D U H max h = ( Re ) Q = A 1 u 1 = A 2 u 3 = 0.78 Energy Linear momentum p 1 ρg + z 1 + u 1 2 2g = p 2 ρg + z 2 + u 2 2 2g + h f 12 F x = ρq(u 2 u 1) F y = ρq(v 2 v 1) Kinematic Viscosities for water is 1mm 2 /s or 1 x 10-6 m 2 /s. for air is 15 times for water Laplace Transform: TURN OVER ENG-5001Y Version 1

10 10 Centroids of common shapes of areas ENG-5001Y Version 1