(Solved): The motion of a lifting surface about the steady flight path due to atmospheric turbulence can ...

The motion of a lifting surface about the steady flight path due to atmospheric turbulence can be represented by the equation $\ddot{x}(t)+2?{?}_n\stackrel{?}{x}(t)+{?}_n^{2}x(t)=\frac{1}{m}F(t)$ where ${?}_n$ is the natural frequency, $m$ is the mass, and $?$ 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] $S_F(?)=\frac{S_T(?)}{\left(1+\frac{??c}{v}\right)}$ where $c$ is the chord length, $v$ is the forward velocity of the lifting surface, and $S_T(?)$ is the spectral density of the upward velocity of air due to turbulence, given by $S_T(?)=A^2\frac{1+{\left(\frac{L?}{v}\right)}^2}{{\left\{1+{\left(\frac{L?}{v}\right)}^2\right\}}^2}$ 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.