Modeling of a Gyroscope

Background and Motivation

• Device to measure or maintain orientation• Modeling a mechanical type of gyroscope• Based on the principles of angular momentum

Geometry and ConnectionsAll components are assumed rigid

Hinge Joints are used to connect different parts

Frame

Outer Gimbal

Inner Gimbal

Spinning Disc

Material

• Structural Steel (ρ = 7850 kg/m^3): Spinning Disc• Aluminum (ρ = 2700 kg/m^3): All other components

NOTE: Structural Steel is chosen for the Spinning Disc to increase its angular momentum at comparatively low rotational speed.

Operating Conditions

• Motion of outer Frame is prescribed in such a way that, at the center of mass:– No translation– Rotation: 2*sin(π*t) rad about (1,1,0) axis

• The Spinning Disc is rotating with respect to the Inner Gimbal:– Angular velocity: 100 rad/sec

Model Setup

• All components are assumed rigid, which makes it possible to model only one Hinge Joint between any two components instead of two.

• The joint axes passes though the origin (0,0,0) for all the Hinge Joints such that the center of the joints can be kept at the origin for all joints.

• This model is solved for 1 sec and the results are stored at the interval of 1e-2 sec.

• However, the maximum allowed time step is 5e-4 sec. This is needed as the Spinning Disc is rotating at very high speeds and completes almost 16 revolutions (100 rads) in 1 sec.

Results

Animation of gyroscope when disc is not spinning

Results

Animation of gyroscope when disc is spinning at 100 rad/s

Conclusions

• The results show that the Spinning Disc, when it is not spinning, experiences unbalanced forces and moments due to the motion prescribed to the outer Frame.– This is the reason why the Spinning Disc changes its orientation and does not remain

vertical.

• In the case when the Spinning Disc is rotating at high speeds, there is a large amount of angular momentum associated with this.– This angular momentum resists the unbalanced forces and moments and helps the disc

to maintain its orientation close to the vertical.