LAWS OF GYROSCOPES / CARDANIC GYROSCOPE

Size: px

Start display at page:

Download "LAWS OF GYROSCOPES / CARDANIC GYROSCOPE"

Carmel Chandler
6 years ago
Views:

1 LAWS OF GYROSCOPES / CARDANC GYROSCOPE PRNCPLE f the axis of rotation of the force-free gyroscope is displaced slightly, a nutation is produced. The relationship between precession frequency or nutation frequency and gyro-frequency is examined for different moments of inertia. Additional weights are applied to a gyroscope mounted on gimbals, so causing a precession. EQUPMENT Gyro, Magnus type 1 Stopwatch, digital, 1/100 sec 1 Digital Stroboscope 1 TASKS 1) To determine the precession frequency as a function of the torque and the angular velocity of the gyroscope. ) To determine the nutational frequency as a function of the angular velocity and the moment of inertia. SET-UP AND PROCEDURE The gyroscope is set-up as shown in Fig. 1. There must be no additional masses on the axes or in the disc rotor). The disc should be at a state of neutral equilibrium in all spatial directions and therefore remain stationary in every position without swinging to and fro. Small corrections can be made by moving the two slotted compensating weights. To determine the precession frequency the additional masses are placed on the gyroscope axis mounting and gently screwed on. Using two different additional weights, the three combinations m 1,0), m, m 1 ) and m, 0) can be obtained. Page 1 of 7

2 Fig. 1: Gyroscope in cardanic mounting. The gyroscope is carefully set in motion with the starting handle. The angular velocity is measured with the stroboscope, and the precession frequency is determined with the stop watch. t is best to measure double the frequency with the stroboscope and half the precession period with the stop watch. The additional masses are removed to measure the nutation. The gyroscope is set in motion as before. A sharp tap with the hand on the gyro mounting movement about the inner gyroscopic axis produces a nutation. The nutation frequency can be measured with the help of a light barrier counter. Attach a light rod e.g. a short light plastic rod) to the gyroscope in such a way that as the gyroscope nutates, the rod blocks the counter periodically and is registered. With a stop watch, the number of times it blocks the counter within a time interval can be measured and hence the nutation frequency can be determined. The nutation frequencies are determined for different gyroscope velocities and for different additional weights symmetrically equal). Page of 7

3 THEORY AND EVALUATON Fig. : Moving frame of reference for the gyroscope. The equation of motion for a rigid body with an angular momentum L to which a torque T is applied, in the inertial system XYZ, is: dl T. The angular momentum of a gyroscope can be divided up into one part in the direction of the figure axis L, which stems only from the rotor, and a remainder L ns : L L s L ns. f the equation of motion is transformed to a moved reference system xy which rotates at an angular velocity Ω, with the origin at the centre of gravity, and whose axes are defined by the inner gimbal frame and the figure axis see Fig. ) we obtain: L s L ns Ω ) T L s L ns Page 3 of 7

4 The derivative against time must be formed in the rotating dashed symbol) system. f the angular momentum of the rotor in the direction of the figure axis is constant: L s ˆ ω s s ω, with the moment of inertia s of the rotor along the figure axis, and the angular velocity ω, and also if L s >> L ns, then we obtain the fundamental equation for gyroscope theory: L ns Ω L s T 1) The uncoupled equations for the components of 1), essential for precession and nutation, are obtained from 1): ) ) s ω 1 Ω x 1 Ω y T x ) s ω s ω y s ω T x T y disappears for weights placed on the rotor axis.) ntroducing the Euler angles θ and φ we obtain: Ω x θ Ω y sin θ φ Tx sin θ where m is the mass at a distance r from the origin which produces the torque due to gravitational acceleration. Page 4 of 7

5 f the precession frequency is small compared with the rotor frequency then: ω, y ) s ω cos θ << 1. This term can be neglected and we obtain: ) s ω 1 θ 0 1 ) s ω φ For the initial conditions where: θ t 0 0 φ t 0 φ 0 we have: θ y x φ 0 sin θ 0 sin t φ φ 0 cos t For the initial condition where: φ 0 the precession frequency is: ω pr and θ constant. φ 1 ) ω s Page 5 of 7

6 1. m kg. m m kg 3. m kg Fig. 3: Precession frequency as a function of the gyro frequency for different additional masses. From the regression line of the data of Fig. 3, using Y A B X we can obtain the exponents B 1, B and B 3. see )) f the figure axis of the rotating gyroscope is pushed with the hand, and therefore θ 0, the figure axis performs a periodic movement with the nutation frequency ω nu s ω x 3) y Page 6 of 7

7 From the regression line of the data of Fig. 4, using the linear Y A B X the gradients B 1, B and B 3 can be obtained. see 3)) 1. No masses applied. masses m kg) applied 3. masses m kg) applied Fig. 4: Nutation frequency f nu as a function of the gyro frequency f for different additional masses x, y ). SC Ng revised July 008 Page 7 of 7

Laws of gyroscopes / cardanic gyroscope

Laws of gyroscopes / cardanic gyroscope Principle If the axis of rotation of the force-free gyroscope is displaced slightly, a nutation is produced. The relationship between precession frequency or nutation frequency and gyrofrequency is examined