Motion of a torsion pendulum immersed in a linear viscous liquid. Influence of wave phenomena

• Weak damping of a linear oscillator does not exist.
• The root locus of a linear oscillator is not a circular arc.
• A Newtonian fluid can be completed with relaxation.

A cylindrical pendulum, which is suspended by an elastic rod or wire and immersed in a viscid liquid in a cylindrical container, can undergo rotating oscillations. The propagation velocity of vorticity perturbations in the gap between the cylinder and the container is assumed to be finite. This means that the acceleration of the liquid is characterized by not only the viscosity but also by a relaxation time constant. The propagation velocity of elastic torsional waves in the rod is assumed to be finite. The equations that describe the motion of such a complicated compound system are linear and have been solved in closed form. The solution shows that there is a considerable deviation between the exact solution and the simple quasi-steady solution. The most remarkable conclusion is that the classical quasi-steady solution for very weak damping is incompatible with the general solution. Propagation of elastic waves in the suspension rod and propagation of vorticity waves in the liquid have a great influence on the rotational motion of the pendulum. The purpose of this study is to formulate the criteria that make the classical quasi-steady analysis valid. The derived solution permits also measurement of viscosity and a conceivable relaxation coefficient of the liquid as well.