Frequency response When a linear system is subjected to a sinusoidal input, its steady state response is also a sustained sinusoidal wave, with the same frequency The frequency response of a system is defined by the relationship between a sinusoidal input and the corresponding sinusoidal output for the system over a range of input frequencies

Aeroelastic Analysis of Wing Flexure & Flutter We analyze the oscillatory tendencies of a wing to bend and “flap” by assuming a sinusoidal oscillation of the aerodynamic force that is flexing the wing. This is the input excitation. We measure the deflection of the wing tip as the output and we analyze the “Gain” (Amplitude) and Phase Angle of the output waveform with respect to the input waveform.

Frequency Response Analysis The plots on the previous page show different gains and phases, but all at the same frequency. Any complex response is composed of many different frequencies of oscillations over the same period of time. Here we can see the additive effect of having many frequencies mixing at the same time.

Frequency Response (Bode) Plots The most effective way to visualize how any given airplane wing design responds to a wide range of input frequencies is by plotting how the Amplitude (Gain) and Phase of the output varies over a range of low to high frequencies. The Gain vs. Frequency plot uses a log-base-10 scale for amplitude response, and the Phase vs. Frequency plot uses a linear scale for the output phase angle difference from the input. w1 w2 20*log(G) (dB) 0.1 Hz 1.0 Hz 10 Hz 180 90 Phase Lag (Deg) -90 -180 0.1 Hz 1.0 Hz 10 Hz

Natural Frequency, Resonance & Damping Natural Frequency – The frequency at which an object will naturally oscillate after the object has been disturbed by an external force. When you let a pendulum go it will oscillate back and forth at its natural frequency. For an aircraft wing imagine if you displaced the wing tip by 1 or 2 feet, and then released it. The wing would flex up and down in a decreasing sinusoidal motion. The frequency of this motion is the wing’s oscillatory natural frequency. Resonance – If the oscillation of external aerodynamic forces (inputs) on a wing occur at the wing’s natural frequency, a dangerous “standing wave” oscillation (resonance) occurs. The wingtip’s displacement oscillation (output) can easily become larger in magnitude than the input force would normally cause at some other oscillation frequency. A divergent resonance can lead to catastrophic failure (fracture) of the wing. NOT GOOD! Damping – Any tendency in a physical system to decrease the amplitude of an oscillatory mode. Damping can be natural or artificial. Natural damping factors of a wing are its weight and stiffness. Artificial damping can be added to a system by means of using feedback sensors to limit inputs (e.g. aileron commands).

non dimensional frequency (w/wn ) Bode Diagram For Typical Second Order System 1 10 Thick line = asymptotic approximation 1 10 non dimensional frequency (w/wn )

System Stability & Gain Margin Gain Margin – The amount of Gain between system response and 0 dB point when the system’s phase lag = 180 Deg. (negative = margin above 0 dB)

System Stability & Phase Margin Phase Margin – The phase shift required for a system operating at unity Gain (= 0 dB) to reach -180 phase lag. (negative = phase lag less than -180 at 0 dB Point).

Controls Solutions: Notch Filter & Low Pass Filter