The sketch is a model for airplane wing vibration. The center of mass is at G. U

Question

The sketch is a model for airplane wing vibration. The center of mass is at G. Use coordinates x, theta with positive directions as indicated. The wing has mass m and moment of inertia J. a) Derive the equation(s) of motion two ways. Your answer should have as many equations of motion as the number of degrees of freedom. i. Free-body diagrams. Newton's 2^nd law, and M =J theta. ii. Lagrange's equations b) Linearize your equations of small motions, that is. assume any products of x, theta, x, theta are negligible (use above Taylor series for sine and cosine).

Explanation / Answer

consider a non uniform wing ( chord, beam stiffness, and mass varying a long the span) free to move and bend vertically and subjected to a sinusoidal vertical gust at a spanwise element of width a centered at y = y8 . the balance of forces then requires that

Fs - Ft - Fm + Fg d( y , y*)

where Fs is the force due to beam stiffenss , F i i sthe inertial frce, and Fm , and fg are aerodynamic forces due to translatory wing motion and due to vertical gst.the function d9y , y ) selects the portion of the wing which is subjeced to the gust and is xero every where except between y* +- delta/2 where it has the value of unity.

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