AAE 556 Aeroelasticity Lectures 22, 23 Typical dynamic instability problems and test review Purdue Aeroelasticity

How to recognize a flutter problem in the making Given: a 2 DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements Q=0 Q is a real number If p12 and p21 have the same sign (both positive or both negative) can flutter occur? Q not zero Purdue Aeroelasticity

The modified determinant Purdue Aeroelasticity

If flutter occurs two frequencies must merge FLUTTER – Increasing Q must cause the term under the radical sign to become zero. For frequency merging flutter to occur, p12 and p21 must have opposite signs. Purdue Aeroelasticity

Purdue Aeroelasticity If one of the frequencies can be driven to zero then we have divergence Divergence requires that the cross-coupling terms have the same sign Purdue Aeroelasticity

Aero/structural interaction model TYPICAL SECTION What did we learn? Purdue Aeroelasticity

Divergence-examination vs. perturbation Purdue Aeroelasticity

Perturbations & Euler’s Test ...result - stable - returns -no static equilibrium in perturbed state ...result - unstable -no static equilibrium - motion away from equilibrium state ...result - neutrally stable - system stays - new static equilibrium point Purdue Aeroelasticity

Stability equation is original equilibrium equation with R.H.S.=0. The stability equation is an equilibrium equation that represents an equilibrium state with no "external loads" – Only loads that are deformation dependent are included The neutrally stable state is called self-equilibrating Purdue Aeroelasticity

Multi-degree of freedom systems From linear algebra, we know that there is a solution to the homogeneous equation only if the determinant of the aeroelastic stiffness matrix is zero Purdue Aeroelasticity

Purdue Aeroelasticity MDOF stability Mode shapes? Eigenvectors and eigenvalues. System is stable if the aeroelastic stiffness matrix determinant is positive. Then the system can absorb energy in a static deformation mode. If the stability determinant is negative then the static system, when perturbed, cannot absorb all of the energy due to work done by aeroelastic forces and must become dynamic. Purdue Aeroelasticity

Three different definitions of roll effectiveness Generation of lift – unusual but the only game in town for the typical section Generation of rolling moment – contrived for the typical section – reduces to lift generation Multi-dof systems – this is the way to do it Generation of steady-state rolling rate or velocity-this is the information we really want for airplane performance Reversal speed is the same no materr which way you do it. Purdue Aeroelasticity

Control effectiveness reversal is not an instability - large input produces small output opposite to divergence phenomenon Purdue Aeroelasticity

Steady-state rolling motion Purdue Aeroelasticity

Purdue Aeroelasticity Swept wings Purdue Aeroelasticity

Purdue Aeroelasticity Divergence Purdue Aeroelasticity

Purdue Aeroelasticity Lift effectiveness Purdue Aeroelasticity

Purdue Aeroelasticity Flexural axis Flexural axis - locus of points where a concentrated force creates no stream-wise twist (or chordwise aeroelastic angle of attack) The closer we align the airloads with the flexural axis, the smaller will be aeroelastic effects. Purdue Aeroelasticity

How to recognize a flutter problem in the making Given: a 2 DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements Q=0 Q is a real number If p12 and p21 have the same sign (both positive or both negative) can flutter occur? Q not zero Purdue Aeroelasticity

If flutter occurs two frequencies must merge FLUTTER – Increasing Q causes the term under the radical sign to be zero. For frequency merging flutter to occur, p12 and p21 must have opposite signs. Purdue Aeroelasticity

If one of the frequencies is driven to zero then we have divergence Divergence requires that the cross-coupling terms are of the same sign Purdue Aeroelasticity

Purdue Aeroelasticity Fuel line flutter A hollow, uniform-thickness, flexible tube has a mass per unit length of m slugs/ft. and carries liquid fuel with density r to a rocket engine. The fuel flow rate is U ft/sec. through a pipe cross-section of A. The tube is straight and has supports a distance L apart, the tube bending displacement is approximated to be Unknown amplitudes of vibrational motion The free vibration frequencies when the fluid is not flowing are: Purdue Aeroelasticity

Purdue Aeroelasticity Fluid flow creates system coupling, but through the velocity, not the displacement Find the divergence speed Estimate the flow speed that flutter occurs, if it occurs Purdue Aeroelasticity

Purdue Aeroelasticity Divergence is found by computing the determinant of the aeroelastic stiffness matrix Purdue Aeroelasticity

Purdue Aeroelasticity Assume that coupling leads to flutter and find an estimate of the merging point Harmonic motion? Purdue Aeroelasticity

The frequencies are approximated Purdue Aeroelasticity