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tokens: Again we note that has a small constant component, , in the direction of the body frame and a big one of modulus describing a circle in the plane of the body frame. The angular velocity in the body frame is given by It follows from Eqs. ( 3 )and ( 5 )that . Since the main axis, , of the ellipsoid is much greater than the other axes, and , we find that . Moreover, for low values of , , and therefore is also small. Finally let us note that when , then and both, angular momentum and the angular velocity are aligned on the axis. Since the angular momentum, , is preserved in an inertial reference frame, we take the inertial system so that the axis is aligned with . The major axis of the bar has to keep an angle with respect to , and then also measures the angle between the main axis of the bar and the plane of the inertial reference system. For this reason, from now on, it will be referred to as the tilt angle of the motion of the bar (see bottom panel of Fig. 2 ). In the inertial frame the major axis of the bar describes a cone about the axis, while and are slightly misaligned since the bar is also rotating about its major axis. When , the major axis of the bar rotates inside the plane, there is no rotation of the bar about its major axis, and and are again aligned. We define now a new non-inertial reference frame henceforth called the precessing reference system. In this reference frame, the axis is aligned with the major axis of the bar, but the bar is not fixed as in the body frame but rotating around the axis. In the body reference system, the angular momentum and angular velocity vectors rotate around the main axis of the bar ( axis in the body frame) with angular speed . The precessing reference frame rotates with respect to the body frame about the axis they share. This is, these frames are related by means of a time-dependent rotation of axis and angular velocity , in such a way that and are constant in the precessing reference system, with values of We are interested in the equations of motion of our galactic model in the precessing reference frame. To compute them, we need the angular velocity of the precessing frame with respect to the inertial one. This angular velocity, , is the sum of the angular velocity of the body, , and the angular velocity of the body axes in the precessing frame. Therefore, and taking the value of into account, we obtain where (see Chapter 2 of Sánchez-Martín 2015 for more details about the rotations). Finally, as a complement, we briefly discuss the motion of the bar in the inertial frame. According to Poinsot’s theorem (see Arnold 1989 ), the bar, which is described by an ellipsoid, rolls without slipping on a fixed plane normal to the angular momentum . If the ellipsoid has axial symmetry, as in our case, this motion is the superposition of a rotation of the ellipsoid along its symmetry axis with constant angular velocity, , and a precession with constant pattern speed around the axis of the angular momentum.