Calculus of the response of a wing due to an external gust These equations define are our general equation and solution for a 1DOF mass-spring-damper system that is exposed to an external gust of amplitude 𝑓0 and frequency Ω. By defining the 𝛺 frecuency ratio 𝜔 ̃ = 𝜔 we can define our phase and amplitude as follows: 𝑛 As we can see we have 2 variables in these equations, the damping ratio (DR) and the frequency ratio. If we plot this respect to their frequency ratio and with multiple DR, we get the following: As we can see from these graphs, the more in phase we are with our natural frequency the more pronounced the response is to the applied force. This effect is only magnified if we don’t have the appropriate amount of damping (i.e., DR<1) to compensate for these instabilities. If we were to deal with a problematic response frequency we usually cant do much about our operating frequency or forces that our system is submitted to. Thus we must opt for increasing the natural frequency of our system. This is typically done by increasing the inertia, and thus the flexural rigidity of our system. In a wing this would be the equivalent of increasing the cross section by increasing the chord and or thickness of our profile.
