Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

129AppendicesAppendix AA.1 The Convolution Integral TheoremAlso called the Superposition Theorem or Duhamel’s integral theorem [57].LetX 1() s to be the Laplace transform of ( t)Then the functionxt() t x ( τ ) x ( t τ ) dτ= ∫3 1 2−τ = 0x 1, and X 2( s)be the transform of ( t)x 3() t whose transform is the product ( s) X ( s) X (s= isX3 1 2)x 2.The theorem of this section helps with the calculation of transient responses of linearsystems to complicated forcing functions. The general response appears as thesuperposition of responses to a sequence of steps or impulses that represent the actualforcing function.