Timoshenko beam elements show a increase in the flutter velocity and decrease in flutter frequency. The flutter modes correspond to the 3rd Bending and 1st Torsion mode frequencies. From table 5.8 , it can be seen that the 3rd bending mode frequencies significantly come down for Timoshenko beam elements. Thus there is a increased gap between the bending and torsion mode frequencies leading to increased flutter velocities and decreased flutter frequencies. The RIE predicts higher flutter velocities and lower flutter frequencies over IIE. This is because of the delayed convergence characteristics of the RIE. 6 SENSITIVITY ANALYSIS The following section is dedicated for the development of an analytical expression for the derivative of flutter velocity with respect to the design parameters of the aircraft wing, which is modeled as a uniform cantilever beam. The design variables xd considered in this analysis are Lengthof the beam(l),Bending Rigidity (EI), Torsional Rigidity (GJ), Mass per unit length () and Inertia per unit length (Im). 6.1 Analytical expression for the derivative of flutter velocity with respect to the design variables: Using the state space method, the eigenvalue eigenvector problem has been defined as, (6.1) For a beam whose generalized matrices are truncated to m modes (6.2) where [S] is a matrix composed of aerodynamic stiffness and aerodynamic damping matrices and is the vector of the natural coordinates. These matrices depend upon the free stream air
Timoshenko beam elements show a increase in the flutter velocity and decrease in flutter frequency. The flutter modes correspond to the 3rd Bending and 1st Torsion mode frequencies. From table 5.8 , it can be seen that the 3rd bending mode frequencies significantly come down for Timoshenko beam elements. Thus there is a increased gap between the bending and torsion mode frequencies leading to increased flutter velocities and decreased flutter frequencies. The RIE predicts higher flutter velocities and lower flutter frequencies over IIE. This is because of the delayed convergence characteristics of the RIE. 6 SENSITIVITY ANALYSIS The following section is dedicated for the development of an analytical expression for the derivative of flutter velocity with respect to the design parameters of the aircraft wing, which is modeled as a uniform cantilever beam. The design variables xd considered in this analysis are Lengthof the beam(l),Bending Rigidity (EI), Torsional Rigidity (GJ), Mass per unit length () and Inertia per unit length (Im). 6.1 Analytical expression for the derivative of flutter velocity with respect to the design variables: Using the state space method, the eigenvalue eigenvector problem has been defined as, (6.1) For a beam whose generalized matrices are truncated to m modes (6.2) where [S] is a matrix composed of aerodynamic stiffness and aerodynamic damping matrices and is the vector of the natural coordinates. These matrices depend upon the free stream air
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