Figure $3$ shows a simplified model to simulate a recording head flying over a rough disk surface in computer hard disk drives. The head has mass $m$ and is supported by a suspension with stiffness $k_1.$ Moreover, the moving disk surface will generate an air bearing lifting the head slightly above the disk surface $($e$.$g$.,$ in the order of $20$ nm $).$ The air bearing is simplied as a linear spring with stiffness $k_2$ and damping coefficient $c.$ Let $x(t)$ be the roughness of the disk surface and serve as the input excitation to the head$/$suspension system. Moreover, $y(t)$ is the relative displacement of the head to the disk. In real hard disk drive applications, we want to keep $y(t)$ almost constant, so that the head can follow the disk surface to perform read$/$write operations.

$($a$)$ Show that the equation of motion is

$m{y}^{}+c{y}^{}+(k_1+k_2)y=-m{x}^{}-k_{1}x$

$($b$)$ Derive the frequency response function. Plot the magnitude and phase of the frequency response function. In plotting the frequency response function, let's define

${}_1=\sqrt[\phantom{2}]{\frac{k_1}{m}},{}_2=\sqrt[\phantom{2}]{\frac{k_1+k_2}{m}}$

Describe the motion of the head in the following three frequency ranges: and

$($c$)$ If we want to design the disk drive so that the head can follow the disk surface for a wide frequency range, how should we choose $k_1,k_2,$ and $m?$

$($d$)$ Now consider the case when the head flies over an isolated bump on the disk. The excitation from the bump to the head appears in the form of

$x(t)=\{\begin{array}{cc}hsin\frac{t}{T},& 0T\end{array}$

$2$

where $h$ is the height of the bump and $T$ is the time needed to fly over the bump. Find $x(),$ which is the Fourier transform of $x(t)$ in $(10).$ Plot the amplitude of $x()$ with respect to frequency $.$ Identify the frequencies where $|x()|$ is zero.

$($e$)$ Given the design in part $($c$),$ what is the minimum $T$ the disk can have without significantly exciting the head into large vibrati

Figure $3$: Suspension in computer hard disk drives