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

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

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

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

Energy conservation Kinetic energy with respect to inertial reference x Purdue Aeroelasticity

Time rate of change of energy x Purdue Aeroelasticity

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

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

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

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

Computing the strain energy in matrix form Purdue Aeroelasticity

Kinetic energy and strain energy Purdue Aeroelasticity

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

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

Results – Lagrange’s equation 27% error Purdue Aeroelasticity

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

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

First natural frequency and first three mode shapes Purdue Aeroelasticity

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

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

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

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

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

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

Lift expressions – swept wings Purdue Aeroelasticity

Lift expressions – swept wings Purdue Aeroelasticity

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

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

Purdue Aeroelasticity Lagrange’s Equations Purdue Aeroelasticity

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

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

The First Lagrange Equation Purdue Aeroelasticity

The 2nd Lagrange Equation (bending) x Purdue Aeroelasticity

The 2nd Lagrange Equation for bending x Purdue Aeroelasticity

Purdue Aeroelasticity Final equations Purdue Aeroelasticity

Purdue Aeroelasticity Divergence Purdue Aeroelasticity