# wagner

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7/4/2016 wagner http://soliton.ae.gatech.edu/labs/windtunl/classes/unstaero/wagner/wagner.html 1/2 Arbitrary Surface Motion:The Wagner Function Assume that the response to an arbitrary airfoil motion in the linear regime can be built up as a superposition of responses to a series of step changes in normal wash. Response of a linear system: Indicial Admittance Apply a unit step force 1(t) to the mass m inthe +ve x direction:The system was initially in rest at unloaded equilibrium: ; .The motion is described by General solution: where Applying the initial conditions, we get: ; .Thus the response of the linear system to a unit step function is: . This is called indicial admittance . The particular functional form of A depends on thelinear system being considered. The response to an arbitrary force f(t) can be built up from this. Response at time to a step load D f applied at time t+ D t is:Summing up,As , This is the Duhamel integral, in terms of the indicial admittance and the derivatives of the forcing function. Theforcing function may not be available in analytic form, so we need an alternate form:Integrate by parts, using: where , so that , so that  7/4/2016 wagner http://soliton.ae.gatech.edu/labs/windtunl/classes/unstaero/wagner/wagner.html 2/2  Note that:Substituting,Thus if the indicial admittance function is known, the system response can be determined from this. Indicial Admittance for an airfoil in pitch+plunge: Wagner Function Unsteady lift associated with the Theodorsen function acs at the quarter chord, and is due to the normal wash at3/4 chord. Take the normalwash as a step function: . Take the Fourier transform of w: . Circulatory lift coefficient is: (taking the inverse Fourier transform). Define as the number of semichords traveled in time t. Then . This can be writen as where f(s) is the Wagner function.Also, .  Note: The Wagner function is the indicial admittance for the normalized circulatory lift associated with a stepchange in normalwash at the 3/4-chord.  This is approximated by the following expressions: . These expressions are compared in the linked table and figure.
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