The motion of a lifting surface about the steady flight path due to atmospheric turbulence can be represented by the equation

\[\ddot{x}(t)+2 \zeta \omega_{n} \dot{x}(t)+\omega_{n}^{2} x(t)=\frac{1}{m} F(t)\]

where \(\omega_{n}\) is the natural frequency, \(m\) is the mass, and \(\zeta\) is the damping coefficient of the system. The forcing function \(F(t)\) denotes the random lift due to the air turbulence and its spectral density is given by [14.17]

\[S_{F}(\omega)=\frac{S_{T}(\omega)}{\left(1+\frac{\pi \omega c}{v}\right)}\]

where \(c\) is the chord length, \(v\) is the forward velocity of the lifting surface, and \(S_{T}(\omega)\) is the spectral density of the upward velocity of air due to turbulence, given by

\[S_{T}(\omega)=A^{2} \frac{1+\left(\frac{L \omega}{v}\right)^{2}}{\left\{1+\left(\frac{L \omega}{v}\right)^{2}\right\}^{2}}\]

where \(A\) is a constant and \(L\) is the scale of turbulence (constant). Find the mean square value of the response \(x(t)\) of the lifting surface.