ON THE VISCOELASTIC BEAM DAMPER FOR A FREELY PRECESSING GYROSCOPE

The effect of viscoelastic beam dampers on the removal of the wobble motion of a freely precessing gyroscope is analyzed. Hamilton's Principle for a deformable body having large rigid body rotational motion is derived. The equations of motion are non-linear in nature, so the elastic-viscoelastic correspondence principle is inapplicable. The stress variable method is introduced to overcome the difficulty of integrating the strain energy integral appearing in the variational equations, if the constitutive equations contain the time derivatioves of stress. The numerical solutions show that, the wobble motion of the gyroscope will be eliminated most rapidly if the natural frequency of the beam damper is tuned to equal the nutation frequency of the gyroscope. The decay time constant is a concave function of damping ratio, so large damping also has an adverse effect. Therefore, the peak of the damping factor of frequency-dependent materials should not be located at the nutation frequency. For the simple case in which the single-beam damper is made of a Kelvin type material, an approximate analytical solution is obtained by using the method of multiple scales. The equations, which relate the minimum time constant and the critical damping ratio to the material constants of the damper and the parameters of the system, are obtained in the explicit form. Besides being consistent with the numerical solutions about the above mentioned results, the perturbation solution shows the other important results that for the tuning case, the minimum time constant is inversely proportional to the length of the beam, the nutation frequency of the gyroscope, and the square root of the mass of the tip body. © 1992 Academic Press Limited.