AAE 556 Aeroelasticity Lecture 16 Dynamics and Vibrations Purdue Aeroelasticity

Aeroelasticity – the challenge the shapes to come New systems, vehicles, shapes, environments & challenges integrating aerodynamics, structures, controls and actuation to create unbeatable systems

In the future Future aircraft will be Multi-purpose and robust Automated and robotic Semi-to-fully autonomous Able to change state Aircraft will be able to Cope with environmental change, both man-made and due to nature Self-repair Use nontraditional propulsion

…but the challenges loom large Conceptual design of highly integrated systems with distributed power and actuation Redefining aeroelastic stability concepts for structures that lock and unlock, move, stay fixed and then move again Calculating loads and transient response Developing test plans for multi-dimensional structural configurations Assigning risk

Low speed and high-speed flutter Gloster Grebe Handley Page O/400 X-15 Electra Purdue Aeroelasticity

Unusual flutter has become usual Purdue Aeroelasticity

Flutter is a dynamic instability it involves energy extraction Purdue Aeroelasticity

Understanding the origin - Typical section equations of motion - 2 DOF Plunge displacement h is positive downward & measured at the shear center xq measured at the shear center from static equilibrium position Purdue Aeroelasticity

Coupled Equations (EOM) are dynamically coupled but elastically uncoupled mg = weight xq xq is called static unbalance and is the source of dynamic coupling Purdue Aeroelasticity

Prove it! Lagrange steps up to the plate z(t) is the downward displacement of a small potion of the airfoil at a position x located aft of the shear center kinetic energy strain energy LaGrange's equations promise Purdue Aeroelasticity

Kinetic energy integral simplifies Sq is called the static unbalance m is the total mass Iq is called the airfoil mass moment of inertia – has 2 parts Purdue Aeroelasticity

Equations of motion for the unforced system (Qi = 0) EOM in matrix form, as promised Purdue Aeroelasticity

Diff Eqn. trial solution -separable Goal – frequencies and mode shapes substitute into coupled differential eqns. Purdue Aeroelasticity

Purdue Aeroelasticity The eventual result Goal – frequencies and mode shapes Purdue Aeroelasticity

divide by exponential time term But – we have a derivation first collect terms into a single 2x2 matrix divide by exponential time term Matrix equations for free vibration Purdue Aeroelasticity

The time dependence term is factored out Determinant of dynamic system matrix set determinant to zero (characteristic equation) Purdue Aeroelasticity

Define uncoupled frequency parameters nondimensionalize Define uncoupled frequency parameters Purdue Aeroelasticity

Solution for natural frequencies Purdue Aeroelasticity

Solution for exponent s Purdue Aeroelasticity

solutions for w are complex numbers and Purdue Aeroelasticity

Example configuration 2b=c and and New terms – the radius of gyration Purdue Aeroelasticity

Natural frequencies change value when the c.g. position changes c.g. offset in semi-chords Purdue Aeroelasticity