AAE 556 Aeroelasticity Lecture 17 Typical section vibration Purdue Aeroelasticity

Purdue Aeroelasticity Understanding the origins of flutter 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

A peek ahead at the final result coupled equations of motion dynamically coupled but elastically uncoupled mg = weight xq xq is called static unbalance and is the source of dynamic coupling Purdue Aeroelasticity

Purdue Aeroelasticity Lagrange and analytical dynamics an alternative to FBD’s and Isaac Newton z(t) is the downward displacement of a small portion of the airfoil at a position x located downstream of the shear center kinetic energy strain energy Purdue Aeroelasticity

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

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

Differential equation a trial solution Goal – frequencies and mode shapes Substitute this into differential equations Purdue Aeroelasticity

There is a characteristic equation here Purdue Aeroelasticity

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

Nondimensionalize by dividing by m and Iq Define uncoupled frequency parameters Purdue Aeroelasticity

Solution for natural frequencies Purdue Aeroelasticity

Solutions for exponent s These are complex numbers Purdue Aeroelasticity

solutions for s are complex numbers and Purdue Aeroelasticity

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

Natural frequencies change when the wing c.g. or EA positions change c.g. offset in semi-chords Purdue Aeroelasticity

Purdue Aeroelasticity Summary? Purdue Aeroelasticity