AAE 556 Aeroelasticity Lecture 21 Modal coordinates and flutter Purdue Aeroelasticity

Modal analysis objectives Accurate flutter analysis with few degrees of freedom Identification of interaction in terms of physically meaningful motion – vibration mode shapes obtained from testing or analysis Key words Modal orthogonality generalized mass & stiffness Purdue Aeroelasticity

Reducing the number of system degrees of freedom Vibration equations with quasi-steady loads This can be a really big eigenvalue problem with “n” degrees of freedom Let’s reduce the problem to “m” degrees of freedom and retain accuracy Purdue Aeroelasticity

Purdue Aeroelasticity Begin with eigenvalue/eigenvector analysis for free vibration frequencies and mode shapes Solve for eigenvectors (mode shapes) construct a modal matrix from the eigenvectors Modal matrix Eigenvectors (mode shapes) Purdue Aeroelasticity

Purdue Aeroelasticity Define response in terms of modal amplitude coordinates, h(t), and eigenvectors System response Modal matrix This series can be truncated to reduce DOF number from n to m (m<n) We go from n actual displacements (the x’s) to a m displacements (the h’s) Purdue Aeroelasticity

Example – String with tension S and three equal masses, m S=tension Purdue Aeroelasticity

Purdue Aeroelasticity Any general deflection can be constructed as a summation of normal modes Purdue Aeroelasticity

Use new (modal) coordinates to define motion Purdue Aeroelasticity

Redefine motion in terms of modal coordinates Pre- and post-multiply by the modal matrix Why? Purdue Aeroelasticity

Purdue Aeroelasticity Compute matrix products involving the mode shapes and the mass matrix The generalized masses and orthogonality Results Purdue Aeroelasticity

Generalized masses (in general) we diagonalize the mass matrix Generalized mass matrix definition Diagonal matrix Purdue Aeroelasticity

Stiffness matrix multiplication Purdue Aeroelasticity

Stiffness matrix multiplication showing orthogonality Purdue Aeroelasticity

Generalized stiffness matrix Diagonal matrix Purdue Aeroelasticity

Purdue Aeroelasticity Final result - a new set of “modal” coordinates gives a set of totally decoupled equations Purdue Aeroelasticity

System with modal coordinates The matrix order can be reduced from n degrees of freedom to m degrees of freedom so that the matrix is smaller but still has a great deal of information content Purdue Aeroelasticity

The eigenvalue problem is decoupled into a set of n (or m) equations Purdue Aeroelasticity

The original problem, but this time in modal form Purdue Aeroelasticity

General aeroelastic system with aero displacement dependent loads The matrix order can be reduced from n degrees of freedom to m degrees of freedom so that the matrix is smaller but still has a great deal of information content Purdue Aeroelasticity

Purdue Aeroelasticity A problem for homework Purdue Aeroelasticity

Purdue Aeroelasticity Do the following Let airspeed be zero Use MATLAB to compute the natural frequencies and the mode shapes fi Construct the 2x2 modal displacement matrix Fij Perform the matrix multiplication to find the modal mass and modal displacement matrices Identify the modal masses and modal frequencies Solve for the aerodynamic modal matrix. Purdue Aeroelasticity