Non-linear stochastic flutter of aeroelastic structural systems

The main objective of t h i s investigation is to examine the linear and non-linear modal interactions of a two-degree-of-freedom aero-elastic structure subjected to a wide band random excitation. The linear analysis involves linear dynamic coupling and parametric random coupling. In terms of normal coordinates the response mean squares are obtained as functions of the system frequency ratio. The analysis shows that for modest values of mass ratio the first mode is suppressed when the natural frequencies of the two beams are identical. Furthermore, the system mean square responses are governed mainly by the external forced excitation, while the influence of the random parametric component is almost negligible.

The non-linear modal analysis involves quadratic non-linearity referred to as autoparametric coupling. This type of coupling gives rise to a new type of instability when the relationship between normal mode frequencies is linear. In the neighborhood of the internal resonance condition w2/w1=0.5 (where w1 and w2 are the normal mode frequencies of the system), a general differential equation of the response moments is derived and found to constitute an infinite hierarchy set. Two different closure schemes, based on a cumulant-neglect concept, are used to truncate the moment differential equations. The first is the Gaussian closure, which leads to fourteen coupled differential equations, while the second, known as the non- Gaussian closure, gives 69 coupled differential equations. These two sets of equations are solved numerically for the response moments. The Gaussian closure solution results in a quasi-stationary response, while the non-Gaussian closure solution gives a strict stationary response. The two solutions exhibit an exchange of energy between the two modes in such a manner that one mode acts as a vibration absorber of the second mode in the neighborhood of internal resonance condition w2/w1=0.5±0(e)f where e is a small parameter.

The influence of ranc3cam variation of the system parameters such as damping and stiffness is investigated. It is found that the damping variation has less effect on the random response of the structure than the stiffness variation. Numerical solutions for different initial conditions are obtained to find out if the system possesses more than one limit cycle. It is found that the initial conditions affect only the transition response, while the steady state response does not change by changing the initial conditions.