Gyroscope Dynamics

X’,Y’,Z’ = body axis centered at center of gyrowheel

Z = global axis passing through center of gyrowheel

r = rotation around Z’ axis = gyrowheel rotation

p = rotation around X’ axis = nutation

q = rotation around Z axis = precession

Gyroscope Equations of Motion:

where: L = torque around X' axis

            M = torque around Y' axis

            N = torque around Z' axis

Assumptions for Analysis:

1) Igyrowheel >> Iframe

2) Neglect Friction

3) Constant Gyrowheel Rotation (r = r0 = constant)

Applying Assumption 1, the gyroscope equations simplify to:

IxxP - (Iyy-Izz)qr0 = L         (1)

IyyQ - (Izz-Ixx)rp = M           (2)

IzzR - (Ixx-Iyy)pq = N           (3)

where P = acceleration of nutation = time derivative of p

           Q = acceleration of precession = time derivative of q

           R = acceleration of gyrowheel rotation = time derivative of r

Case 1: Fixed Nutation (p = 0)

For fixed nutation, equation (1) simplifies to:

-(Iyy-Izz)qr0 = L0 

→ q = -L0/(Iyy-Izz)r0

        where L0 = constant applied torque around X' axis

For a symmetric gyrowheel:

Iyy = mr2

Izz=0.5*mr2

→ (Iyy-Izz) = -0.5*mr2

    where: m = mass of gyrowheel

                  r = radius of gyrowheel

Therefore:

q = L/(0.5*mr2r0)   (4)

From equation (4) we can see that with nutation fixed:

• • The rate of precession will decrease for greater gyrowheel rotational speed

  • The rate of precession increases for greater torque applied

  • The rate of precession will decrease for greater gyrowheel mass and radius

Case 2: Precession fixed (q = 0)

With precession fixed, equation (1) simplifies to:

IxxP = L

→ P = L/Ixx

And more specifically:

→ P(t) = L(th)/Ixx

    where th = the angle of nutation

The applied torque is assumed to be a function of the angle of nutation because the direction of the force causing the nutation is not assumed to change as the angle of nutation increases. From the characteristics of this relationship one can see the magnitude of the torque is dependent upon the cosine of the nutation angle meaning equation (1) further simplifies to:

P(t) = Locos(th)/Ixx (5)

    where Lo = constant torque applied about the X' axis

                  Ixx=mr2

This leaves us with a 2nd order nonlinear ODE which is very difficult to solve numerically. However, equation (5) still tells us that for fixed precession:

• the rate of nutation will be sinusoidal

• the rate of nutation will increase for larger torque applied

• the rate of nutation will decrease for gyrowheels with larger inertia values