## The Torsional Vibrating Mass Gyro

A schematic of a single, torsional vibrating mass angular rate sensor is shown in Figure 2.7-2. This structure is analogous to one haltere. This particular example (1) introduces and illustrates Lagrange's method of analyzing complex, dynamic bio-mechanical systems (Cannon, 1967); (2) allows extension of the analysis of a single, torsional, vibrating-mass, angular-rate sensor to a paired structure in which the masses vibrate 180° out of phase (a crude model for the paired halteres of a complete y).

FIGURE 2.7-2 Schematic (top view, side view) of a single torsional vibrating rate sensor. A mass M on the end of a rod of length R is caused to vibrate sinusoidally through an arc of ± . m. See analysis in text.

A haltere is attached and pivoted at its small end to the insect's body below and behind each wing. It is basically a mass at the end of a stiff rod that vibrates in one plane at the wing-beat frequency when the insect ies. To derive a mathematical description for the torques at the base of the vibrating mass, this section uses the well-known LaGrange method (Cannon, 1967), which is based on Newtonian mechanics. For a complex mechanical system one can write N equations of the form:

dt :• qk: • qk k where qk is the kth independent coordinate. The qk can be rotational or translational. Qk is the kth moment (torque) if rotational dynamics are considered, or the kth force if the dynamics involve linear translation. T is the total kinetic energy of the system, and U is the total potential energy of the system. L = T - U. L is called the system's "LaGrangian function."

(yaw left)

Haltere

FIGURE 2.7-3 Schematic of ying insect sho wing the left haltere vibrating in the xz plane.

(yaw left)

Haltere

FIGURE 2.7-3 Schematic of ying insect sho wing the left haltere vibrating in the xz plane.

Consider Figure 2.7.3 in which the linear ight velocity vector is along the y axis, and for simplicity, let the halteres vibrate in the XZ plane. In this example, the independent coordinates are ., the angle of the vibrating mass in the XZ plane with the z axis, and the input rotation around the Z axis (pitch up is shown). The system potential energy is U = (1/2)K..2 J. K. is the spring constant of the torsion spring that restores the resting mass to align with the Z axis. A muscular "torque motor" (not shown) keeps the mass vibrating sinusoidally in the XZ plane so that . (t) = . m sin(. m t).

There are two components to the system kinetic energy:

Here Jo is the moment of inertia of the haltere (mass on the end of a rod of length R): It is well-known that Jo = MR2. r is the projection of R on the XY plane: r = R sin(.). (r|) is the tangential velocity of the mass M in the XY plane due to the input rotation rate. Thus, the LaGrangian of the system is: