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Thus the M$ moment lies in the XY plane and contains a dc term proportional to $, and also double-frequency terms in $ and $. The desired angular velocity term is larger by an amount . m, which can be 900 r/s in mosquitos.

If it is assumed that the sensory neurons associated with the base of each haltere respond to the twisting torque, M$, induced by $ and $, then the sensory output will contain a pulse frequency code proportional to pitch angular acceleration and, more interestingly, pitch angular velocity at twice the wing-beat frequency. The same haltere can be shown to respond to angular acceleration and velocity in the yaw direction.

Refer to Figure 2.7.3. To simplify the yaw calculations, assume that there is no restoring spring torque (K. = 0). The Lagrangian is thus the kinetic energy of the system:

where .a' is the tangential velocity of the mass M due to yaw angular velocity, a. The instantaneous effective radius of the mass (projected onto the z axis) is . = R cos(. ). The forced oscillatory displacement of the haltere can be written: . (t) = . m sin(. m t). Thus, the Lagrangian can be rewritten as

L = (l/2)Jo. 2 + (l/2)Ma 2R2 cos2 (. ) = (l/2)Jo. +(l/2)MR2a2 (1/2)[l + cos(2. )]

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