In Task 2 you demonstrate your knowledge of analytical analysis of continuous systems and your ability in employing your analytical skills in problem solving and decision making in the context of a real-life problem. Background Wind turbine blades behave as a beam as well as a torsional member. As a beam, blades can vibrate in two directions, namely, flapwise and edgewise and as a torsional member, blades can vibrate about their axis (see Figure 1). Flapwise direction Edgewise direction blade axis Figure 1-Top: flapwise and edgewise directions; Bottom: rotation about blade axis (normal to the plane) Large wind turbine blades are subjected to a variety of dynamic loads F(t) which have a wide spectrum of frequencies. However, those dynamic loads which have enough energy intensity to cause excitation have a frequency of fr(t) < 0.3 Hz. You are leading a team of design engineers and are in charge of the structural design of a 5MW wind turbine blade. Your team has sent you the initial design with the blade structural data as shown in Table 1 (also available in excel format 'A1T2_Table1' from the Assessment folder). The project is behind its schedule and you need to make a quick decision whether the dynamic behaviour of the initial design is satisfactory or not without conducting a thorough finite element analysis. Table 1- Distributed Blade Structural Properties span location (measured from blade root) r(m) Flapwise flexural rigidity unit length pA Elnap (Nm) Edgewise flexural rigidity Blade density per Torsional rigidity EA(N) Eledge (Nm) GJ(Nm) 1.5 678.935 1.81E+10 1.81E+10 5.56E+09 9.73E+09 1.7 678.935 1.81E+10 1.81E+10 5.56E+09 9.73E+09 2.7 773.363 1.94E+10 1.96E+10 5.43E+09 1.08E+10 3.7 740.55 1.75E+10 1.95E+10 4.99E+09 1.01E+10 4.7 740.042 1.53E+10 1.98E+10 4.67E+09 9.87E+09 5.7 592.496 1.08E+10 1.49E+10 3.47E+09 7.61E+09 6.7 450.275 7.23E+09 1.02E+10 2.32E+09 5.49E+09 7.7 424.054 6.31E+09 9.14E+09 1.91E+09 4.97E+09 8.7 400.638 5.53E+09 8.06E+09 1.57E+09 4.49E+09 9.7 382.062 4.98E+09 6.88E+09 1.16E+09 4.03E+09 10.7 399.655 4.94E+09 7.01E+09 1.00E+09 4.04E+09 11.7 426.321 4.69E+09 7.17E+09 8.56E+08 4.17E+09 12.7 416.82 3.95E+09 7.27E+09 6.72E+08 4.08E+09 13.7 406.186 3.39E+09 7.08E+09 5.47E+08 4.09E+09 14.7 381.42 2.93E+09 6.24E+09 4.49E+08 3.67E+09 15.7 352.822 2.57E+09 5.05E+09 3.36E+08 3.15E+09 16.7 349.477 2.39E+09 4.95E+09 3.11E+08 3.01E+09 17.7 346.538 2.27E+09 4.81E+09 2.92E+08 2.88E+09 19.7 339.333 2.05E+09 4.50E+09 2.61E+08 2.61E+09 21.7 330.004 1.83E+09 4.24E+09 2.29E+08 2.36E+09 23.7 321.99 1.59E+09 4.00E+09 2.01E+08 2.15E+09 25.7 313.82 1.36E+09 3.75E+09 1.74E+08 1.94E+09 27.7 294.734 1.10E+09 3.45E+09 1.44E+08 1.63E+09 29.7 287.12 8.76E+08 3.14E+09 1.20E+08 1.43E+09 31.7 263.343 6.81E+08 2.73E+09 8.12E+07 1.17E+09 33.7 253.207 5.35E+08 2.55E+09 6.91E+07 1.05E+09 35.7 241.666 4.09E+08 2.33E+09 5.75E+07 9.23E+08 37.7 220.638 3.15E+08 1.83E+09 4.59E+07 7.61E+08 39.7 200.293 2.39E+08 1.58E+09 3.60E+07 6.48E+08 41.7 179.404 1.76E+08 1.32E+09 2.74E+07 5.40E+08 43.7 165.094 1.26E+08 1.18E+09 2.09E+07 5.31E+08 45.7 154.411 1.07E+08 1.02E+09 1.85E+07 4.60E+08 47.7 138.935 9.09E+07 7.98E+08 1.63E+07 3.76E+08 49.7 129.555 7.63E+07 7.10E+08 1.45E+07 3.29E+08 51.7 107.264 6.11E+07 5.18E+08 9.07E+06 2.44E+08 53.7 98.776 4.95E+07 4.55E+08 8.06E+06 2.12E+08 55.7 90.248 3.94E+07 3.95E+08 7.08E+06 1.82E+08 56.7 83.001 3.47E+07 3.54E+08 6.09E+06 1.60E+08 57.7 72.906 3.04E+07 3.05E+08 5.75E+06 1.09E+08 58.7 68.772 2.65E+07 2.81E+08 5.33E+06 1.00E+08 59.2 66.264 2.38E+07 2.62E+08 4.94E+06 9.22E+07 59.7 59.34 1.96E+07 1.59E+08 4.24E+06 6.32E+07 60.2 55.914 1.60E+07 1.38E+08 3.66E+06 5.33E+07 60.7 52.484 1.28E+07 1.19E+08 3.13E+06 4.45E+07 61.2 49.114 1.01E+07 1.02E+08 2.64E+06 3.69E+07 61.7 45.818 7.55E+06 8.51E+07 2.17E+06 2.99E+07 62.2 41.669 4.60E+06 6.43E+07 1.58E+06 2.13E+07 62.7 11.453 2.50E+05 6.61E+06 2.50E+05 4.85E+06 63 10.319 1.70E+05 5.01E+06 1.90E+05 3.53E+06 4 Assignment 1 Task Apply necessary simplifying assumptions to the problem at hand and use the analytical methods that you have learned in this course to analyse the blade. Based on the results of the analysis you respond to the team. Which one of the following is your response? (a) the results of analytical analysis confirm that the dynamic behaviour of the initial design is satisfactory (explain why) (b) the results of analytical analysis neither confirm nor reject the suitability of the design, hence an accurate analysis (using FEA) is required to make a decision (explain why) (c) the results of analytical analysis confirm that the dynamic behaviour of the initial design is not satisfactory and the design needs to be changed (explain why and make a list of suggestions for changing the design) Report your analysis with all details, such as, deriving/identifying boundary conditions, applying boundary conditions and solving characteristic equation/frequency equation. You also need to report the assumptions that you need to make to be able to use the analytical methods that you have learned in this course for the dynamic analysis of the blade, and why you need these assumptions a comprehensive discussion on how realistic your assumptions are a comprehensive discussion on how your simplifying assumptions affect the result of analysis, hence your decision Notes: 1. Table 1 is all the data that you have. 2. You can use MATLAB to solve nonlinear equations.