The system in Figure $1$ is a simple model for a rate gyro, which uses gyroscopic effects to

measure yaw rates. The rotor, which spins at a constant rate $\backslash$phi $\_($o$)^()$ about the body$-$fixed $3$ axis,

reacts to $\backslash$psi $\_($o$)^(),$ the yaw rate of the frame, by a nutation deflection, $\backslash$theta $.$ For this problem, you may

assume the following: the spring k limits $\backslash$theta to small angles, $\backslash$psi $\_($o$)^()$ is constant and purely vertical,

and the damper provides a torque of b$\backslash$theta $\_($o$)^()$ that resists rotation about the $1$ axis. In this problem,

you will make use of the axis of symmetry $($of the suspended body$)$ and define two different

vec$(\backslash$omega $)\text{'}$s$,$ one for the body and one for the $1-2-3$ axes $($which will be different$).$

A$\backslash$theta it is given by$,$

I$\_(1)\backslash$theta $^()+$b$\backslash$theta $^()+[($I$\_(1)-$I$\_(3))\backslash$psi $\_($o$)^()^(2)+$kL$^(2)]\backslash$theta $-$I$\_(3)\backslash$psi $\_($o$)^()\backslash$phi $\_($o$)^()=0$

Note: this is a common equation that you would study in vibration class.

B$\backslash$theta and

the yaw rate $\backslash$psi $\_($o$)^()\backslash$theta is constantkL$^(2)($I$\_(1)-$I$\_(3))\backslash$psi $\_($o$)^()^(2),$ that is$,$ the spring is very stiff, that there is a linear

relationship between $\backslash$theta and $\backslash$psi $\_($o$)^().$

Figure $1.$