1. Purdue Aeroelasticity
Lectures 25 & 26 Modeling and estimation energy methods – Rayleigh-Ritz flutter models
Lagrange equations
Generalized forces
Rayleigh-Ritz approximations
Virtual work
Purdue Aeroelasticity

2. Purdue Aeroelasticity
Objectives
Create models with few degrees of freedom but with reasonable accuracy
Set up aeroelastic problem with beam-like wing
Purdue Aeroelasticity

3. Energy in Structural Systems
Structural systems store and transfer energy
Energy methods are an alternative to Newton’s Laws for developing equations of motion – no FBD’s are required
Structural systems store energy as kinetic energy and strain energy
Energy acquired equals work done on the system
Purdue Aeroelasticity

4. Example – a spring, work and bookkeepingF d
External work equals internal force work.
Internal force (spring) work is negative. External work is stored as internal strain energy, U.
Purdue Aeroelasticity

5. Energy conservation Kinetic energy with respect to inertial referencex
Purdue Aeroelasticity

6. Time rate of change of energyx
Purdue Aeroelasticity

7. Purdue Aeroelasticity
Lagrangian mechanics for conservative systems with a finite number of degrees of freedom
Define the Lagrangian as L=T-U
Coordinates x1, x2, …xn
These are called generalized coordinates
Purdue Aeroelasticity

8. Continuous systems and the Rayleigh-Ritz method - a beam example
Compute kinetic energy and strain energy
The internal forces are accounted for in the strain energy portion of L=T-U
w(y,t)
Purdue Aeroelasticity

9. Purdue Aeroelasticity
Converting a continuous system to a system with a finite number of degrees of freedom using assumed deflection shapes
Approximate response as a series of “admissible functions.”
Purdue Aeroelasticity

10. Functional requirements
Comparison functions satisfy both the geometric and force b.c.’s
Admissible functions
Comparison functions
Eigenfunctions
Admissible functions satisfy the geometric b.c.’s
Eigenfunctions satisfy the geometric and force/moments b.c.’s and the governing differential equations for equilibrium
Purdue Aeroelasticity

11. Computing the strain energy in matrix form
Purdue Aeroelasticity

12. Kinetic energy and strain energy
Purdue Aeroelasticity

13. Lagrange’s equations identify mass and stiffness matrix elements
Purdue Aeroelasticity

14. An example - assumed deflection functions
Shape functions must at least satisfy geometric boundary conditions y
Purdue Aeroelasticity

15. Results – Lagrange’s equation
27% error
Purdue Aeroelasticity

16. A more detailed deflection function
This shape function satisfies both geometric and force/moment boundary conditions – it is the shape created by placing a uniform distributed load on a beam of length L. Note that it doesn’t matter where you reference your coordinates. y
Purdue Aeroelasticity

17. Purdue Aeroelasticity
Other functions y
Bending deflection due to uniform distributed load with coordinate system shown. w
Torsional deflection due to external, uniform distributed torque with coordinate system shown. y q
Purdue Aeroelasticity

18. First natural frequency and first three mode shapes
Purdue Aeroelasticity

19. Adding forces to the equation Real work vs. Virtual work
When forces are in equilibrium the virtual work is
The displacement dr is a virtual displacement
Virtual displacements are hypothetical and treated as infinitesimal, like real derivatives
Work is force times displacement in the direction of the force
Purdue Aeroelasticity

20. Virtual work and static equilibrium
If a particle is in static equilibrium then the sum of the forces is zero
If the particle could be artificially displaced in any direction then the work done by all the forces would be zero (surprise, surprise)
The artificial displacement is called a virtual displacement
This is called the Principle of Virtual Work and by itself is not very useful
Purdue Aeroelasticity

21. Generalized, external forces do virtual work
The internal forces are accounted for in the strain energy portion of L=T-U
Purdue Aeroelasticity

22. Purdue Aeroelasticity
Generalized forces are computed using our assumed functions – we need the loads and a definition of the displacements where the loads are applied
Why not dgi? dp(y)? Isn’t it possible that the load depends on the displacements?
Purdue Aeroelasticity

23. Defining the generalized forces
In aeroelasticity the loads depend on the displacements, but we “freeze” the loads during displacement variation.
Purdue Aeroelasticity

24. An example swept wing lift and development of generalized forces
Purdue Aeroelasticity

25. Lift expressions – swept wings
Purdue Aeroelasticity

26. Lift expressions – swept wings
Purdue Aeroelasticity

27. Purdue Aeroelasticity
Formation of equations of motion using the Lagrangian-uniform property wing
Purdue Aeroelasticity

28. Kinetic energy m(x,y) is the mass density per unit area
Save the integration until later, after differentiation
Purdue Aeroelasticity

29. Purdue Aeroelasticity
Lagrange’s Equations
Purdue Aeroelasticity

30. Purdue Aeroelasticity
The First Lagrange Equation differentiating the kinetic energy expression
Purdue Aeroelasticity

31. The First Lagrange Equation (The torsion equation)x
Purdue Aeroelasticity

32. The First Lagrange Equation
Purdue Aeroelasticity

33. The 2nd Lagrange Equation (bending)x
Purdue Aeroelasticity

34. The 2nd Lagrange Equation for bendingx
Purdue Aeroelasticity

35. Purdue Aeroelasticity
Final equations
Purdue Aeroelasticity

36. Purdue Aeroelasticity
Divergence
Purdue Aeroelasticity