Effect of Spinning Rotors on the Euler Equations

In evaluating the angular momentum h (see Sec. 4.3) it was tacitly assumed that the airplane is a single rigid body. This is implied in the equation for the velocity of an element (4.3,2). Let us now imagine that some portions of the airplane mass are spin­ning relative to the body axes; for example, rotors of jet engines, or propellers. Each such rotor has an angular momentum relative to the body axes. This can be computed from (4.3,4) by interpreting the moments and products of inertia therein as those of the rotor with respect to axes parallel to Cxyz and origin at the rotor mass center. The angular velocities in (4.3,4) are interpreted as those of the rotor relative to the air­plane body axes. Let the resultant relative angular momentum of all rotors be h’, with components (h’x, h[„ /г’) in FB, which are assumed to be constant. It can be shown that the total angular momentum of an airplane with spinning rotors is obtained simply by adding h’ to the h previously obtained in Sec. 4.3 (see Exercise 4.5). The equation that corresponds to (4.3,4) is then6

hB = I BwB + hj, (4.6,1)

Because of the additional terms in the angular momentum, certain extra terms appear in the right-hand side of the moment equations, (4.5,9). Those additional terms, known as the gyroscopic couples, are

In the L equation: qh’z — rh’y

In the M equation: rh’x — ph’z (4.6,2)

In the N equation: ph’y — qh’x

As an example, suppose the rotor axes are parallel to Cx, with angular momentum h’ = i/fl. Then the gyroscopic terms in the three equations are, respectively, 0, /fir, and —IClq.