AE 714 Aeroelastic Effects in Structures Term Project (Revised Version 20/05/2009) Flutter Analysis of a Tapered Wing Using Assumed Modes Method

Transcription

1 AE 714 Aeroelastic Effects in Structures Term Project (Revised Version 20/05/2009) Flutter Analysis of a Tapered Wing Using Assumed Modes Method Project Description In this project, you will perform classical flutter analysis for a straight tapered wing of a utility type aircraft using the assumed modes method. You may assume that the wing is subject to subsonic incompressible flow at the sea level. The computational steps involved in this project can be handled by scripts written in MathCAD or Matlab. You are not required to develop computer programs in FORTRAN or C. However, you may if you like. Use of Excel may seem promising at first, but since you cannot define complex numbers and do not have a built in eigenvalue solver, it is not recommended for use in this project. Figure 1 shows the typical wing dimensions. By using the structural data and assumed modes provided to you, you will first create a dynamic model of the wing. This step includes calculation of the mass and stiffness matrices using the given mode shapes and obtaining the natural frequencies. wing root locus of section ACs locus of section CGs wing tip Figure 1. The wing geometry, 10, 0.6., 1.5., 0.9, / The second step in this project involves the calculation of the unsteady aerodynamic forces acting on this wing. At this step, you will use the 2D strip theory and apply the Theoderson s solution at each wing section to evaluate the total aerodynamic lift and pitching moment using the definition of the virtual work. Once the equations are formulated for the aeroelastic system, you will proceed with the third step, which is the flutter calculation. Here, you will employ the k-method to determine the flutter speed and the frequency. In this project, you will form groups of 3 and work on your assigned tasks to perform all the required calculations and present them in two different reports. Reports must be computer typed, and contain relevant figures, references and equations. You should explain your steps in an organized fashion and show all the intermediate steps you performed during your analysis. Make sure that you do not simply rewrite the outlining steps described in this document. Be complete, organized and professional. You should communicate with your classmates as soon as possible to form the project teams and inform me regarding the team members by the deadline listed at the end of this project description. You may ask questions about this project and even verify your calculations with me throughout the semester. AE714 Term Project, Spring 2009, Dr. Seber. Page 1/5

2 Structural Data of the Tapered Wing For the wing to be analyzed, sectional properties at each wing strip can be calculated using the data provided in Table 1. You may assume that these values are provided to you by the structural design team, who would like to verify their preliminary design against aeroelastic instabilities using your help. Bending Rigidity, N.m 2 Torsional Rigidity, N.m 2 Mass/Length, 10.3 kg/m Section nondimensional radius of gyration (wrt SC), 0.55 Mass center location *, m Shear center location *, m Aerodynamic center *, / m LE AC SC CG Wing Torque Box x TE Secondary Wing Structure (Only contributes to section inertia) (* indicates that the locations are given with respect to the leading edge, LE) Table 1. Sectional properties of the reference wing section of unit chord length (c = 2b = 1 m) Application of the Assumed Modes Method to the Aeroelastic System Since this wing is slender (AR = 10), you may assume that it is possible to model it using the classical beam theory. Following the convention we have used in the class for the Exact treatment of Bending- Torsion Flutter, the total potential and kinetic energies of the wing are given by the following equations. In this project, you can ignore the damping in the structure. U EIywy, t dy GJyθy, t dy (1) T myw y, t dy S yw y, tθ y, tdy where and. Iyθ y, t dy (2) Note that since you have a tapered wing, sectional properties in these energy expressions can be assumed to be proportional to powers of the chord length. These dependencies are listed below. EIy EI cy GJy GJ cy my m cy (3a) (3b) (3c) However, you should assume that CG, SC and AC are always located at the same percentage of the chord length. In equations 1 and 2,, and, correspond to wing bending deflection and torsional rotation, which are both associated to the elastic axis. In this approach, it is possible to represent these AE714 Term Project, Spring 2009, Dr. Seber. Page 2/5

3 deformations using assumed shape functions and and generalized coordinates and as shown in the expressions below. wy, t y. w t θy, t ψ y. θ t (4a) (4b) where 1 3 1, 1 and 1 1, 1. You may check that these approximate shape functions are quite close to the exact uncoupled normal modes of a uniform beam, which should be adequate for our approximate calculations. Remember: Introduction of more terms into our displacement approximations (m=2, 3 and n=2,3 ) will increase the accuracy of our solution, as well as the time you would spend on this project. So let s eliminate this option for the time being. By applying the Rayleigh-Ritz method to discretize the kinetic and strain energies, the following expressions are obtained for the elements of the mass and stiffness matrices. M, m dy M, M, mbx ψ dy M, mb r ψ dy 5a-c K, EI, dy K, K, 0 K, GJψ, dy 6a-c Since the integrals given in equations 5 and 6 are hard to obtain analytically, you may use numerical integration techniques such as the trapezoidal rule. When performing the numerical integration you may use a section width of y = 0.60 m, i.e. 10 wing sections, with the properties evaluated at the middle. In order to model the unsteady aerodynamic loads, the Theoderson s solution will be used by assuming that the wing undergoes harmonic oscillations, i.e. wy, t w ye and θ y, t θ ψ ye. You should note that this step can be tricky, so be extra careful and dedicate enough time to it. The following equations can be used to represent the lift and pitching moments per unit span. L 2πρUbCk Uθ w b a θ πρb Uθ w baθ (7) M 2πρUb a Ck Uθ w b a θ πρb Ub a θ baw b a a θ (8) Then you may expand equations 7 and 8, and reorganize them to get equations 9 and 10. Hint: Compare your expressions to those given in the book by Hodges and Alvin, p152. But note that and. πρb ω k w kbψ θ e (9) M πρb ω k w kbψ θ e (10) By using the definition of the virtual work, the expressions above can be numerically integrated to evaluate the generalized aerodynamic force vector of the system as shown in the equations below. Note that since the chord length changes along the span, so will and of each wing strip, in addition to, and, i.e. all of these are functions of. (11) ω M πρb dy πρb ψ dy πρb ψ dy πρb ψ ψ dy w e θ ω A A A w e A θ (12) AE714 Term Project, Spring 2009, Dr. Seber. Page 3/5

4 By combining the expressions obtained, the equations of motion of the aeroelastic system can be developed. These equations can then be modified for the application of the k-method by introducing the artificial damping coefficient and the eigenvalue parameter 1 as shown in the expression below. 0 (13) where and are mass and stiffness matrices defined in equations 5 and 6, A and w θ. A A A The main steps involved in the application of the k-method are summarized below. 1. Assume incompressible flow ( 0) at standard sea level conditions. 2. Select the trial reduced frequency value for the reference wing strip with 2 1, i.e.. Hint: Small lead to high flight speeds, whereas large values lead to low flight speeds. The lowest limit on can be taken as 0.001, since the k-method does not work for Perform the numerical integration required to evaluate the matrix. You should note that for each wing strip the reduced frequency varies linearly and has to be calculated using cy. 4. After evaluating determine the value of the eigenvalue parameter. Hint: You may let your code evaluate the eigenvalues of the matrix to determine and. 5. Determine 1,, 0.5 and likewise,, and. Note: 0.5 factors in and expressions are due to the fact that the trial reduced frequency is assigned to a section with a unit chord length. 6. Add data points to your vs, and vs plots for the two aeroelastic modes in your system. 7. Repeat steps 2-6 until you determine the reduced frequency for which is equal to zero at an aeroelastic mode. This condition gives you the flutter speed and frequency. Remember: If is negative the system is stable, otherwise unstable. Deadlines and Reports to be Presented You must form your teams by March 13, 2009 and inform me with an . Preliminary Report: Structural Modeling of the Tapered Wing Due Date: April 17, 2009 In this report, you must perform all the steps required to create the structural model and apply the Rayleigh-Ritz method to evaluate the natural frequencies and mode shapes of the tapered wing. You may also present some of your initial work on formulating the unsteady aerodynamic loads. This step is recommended for the preliminary report, however it is not mandatory. Final Report: Aeroelastic Modeling and Flutter Calculations of the Tapered Wing Due Date: May 29, 2009 In this report, you must present your flutter calculations. This requires the formulation of the unsteady aerodynamic loads and application of the k-method. You may briefly summarize what you have done in the preliminary report for completeness. AE714 Term Project, Spring 2009, Dr. Seber. Page 4/5

5 Report Format Reports must be computer typed. You may use choose your own format for your reports, which could be based on your previous project reports at school or work. Make sure that you number each section, put equation numbers and introduce proper captions for figures and tables. Both reports must contain: 1. A title page with course name, project title, names of team members etc. 2. An introduction section with a brief summary of your work and important results. Also mention the responsibilities of each team member in this project. 3. A theory section where you describe in detail your technical approach. Provide the important intermediate steps of the formulation, equations, tables, plots etc. 4. A results section showing your flutter results. 5. A conclusion section. You may summarize your findings and discuss your results here. Is the calculated flutter speed permits the safe operation of the aircraft? If not do you have any recommendations about how one may improve the wing structure? You may also do some research and talk about how you may improve these calculations, i.e. compressibility effects, 3D flow correction etc. 6. A references section. 7. An appendix with you computer codes, some sample calculations, any hand-written material that you would like to present. 8. An electronic copy of your report and computer scripts that you have used. Useful References 1. AE714 class notes. 2. Chapter 4, Introduction to Structural Dynamics and Aeroelasticity, D. H. Hodges and G. A. Pierce 3. Chapters 4 and 9, Aeroelasticity, R. L. Bisplinghoff, H. Ashley, and R. L. Halfman. 4. Chapters 3 and 11, Introduction to Aircraft Aeroelasticity and Loads, J. R. Wright and J. E. Cooper. Prepared by Assist. Prof. Dr. Güçlü Seber, 04/03/2009. AE714 Term Project, Spring 2009, Dr. Seber. Page 5/5