The rotor of a dial indicator is connected to a torsional...

The rotor of a dial indicator is connected to a torsional spring and a torsional viscous damper to form a single-degree-of-freedom torsional system. The scale is graduated in equal divisions, and the equilibrium position of the rotor corresponds to zero on the scale. When a torque of $2 \times 10^{-3} \mathrm{~N}-\mathrm{m}$ is applied, the angular displacement of the rotor is found to be $50^{\circ}$ with the pointer showing 80 divisions on the scale. When the rotor is released from this position, the pointer swings first to -20 divisions in one second and then to 5 divisions in another second. Find (a) the mass moment of inertia of the rotor, (b) the undamped natural time period of the rotor, (c) the torsional damping constant, and (d) the torsional spring stiffness.

Key Concepts

Single-Degree-of-Freedom Torsional System

A single-degree-of-freedom torsional system is a mechanical model that rotates about an axis and can be described using one independent coordinate, usually the angular displacement. This framework is fundamental for analyzing rotational dynamics where the system’s inertia, restoring torque (provided by the spring), and damping (often viscous) play dominant roles in its response to applied torques.

Mass Moment of Inertia

The mass moment of inertia is a measure of an object's resistance to changes in its rotational motion. It quantifies how the mass of a body is distributed with respect to the rotational axis and is a crucial parameter in determining the dynamic response of a torsional system when subjected to torques.

Torsional Spring Stiffness

Torsional spring stiffness describes the relationship between the angular displacement and the restoring torque in a torsional system. It is defined as the torque per unit angular displacement and is pivotal in establishing the natural frequency of oscillation in undamped systems.

Torsional Viscous Damping

Torsional viscous damping represents the resistance to rotational motion that is proportional to the angular velocity. This damping mechanism dissipates energy from the system, reducing the amplitude of oscillations over time and influencing the transient response following disturbances.

Undamped Natural Frequency (Time Period)

The undamped natural frequency is the rate at which a system oscillates when it is not subject to any damping forces, while the undamped natural time period is the corresponding time taken for one complete oscillation. These parameters are inherent to the system’s mass moment of inertia and stiffness, providing key insights into its dynamic behavior.

Transcript

Please give Ace some feedback