A component’s moment of inertia is its resistance to rotational acceleration.
Of course, that value partly depends on mass. But one more factor affects how much force is needed to get a component spinning: geometry. Also known as a radius of gyration, k expresses this part of inertia (normally determined from known component dimensions) in units of length. From there, calculate rotational inertia I = mass x k2.
Q: When is it necessary to study the inertial characteristics of cylinders?
A: They’re relevant whenever an apparatus must accelerate and response is crucial. They predict the performance of shafts and other spinning components — motors, clutches, and couplings. Inertial characteristics also affect speed and load regulation capabilities. One caveat: because the equations are good for modeling only one aspect of drive shafts, other mechanical drive component characteristics (such as backlash and stiffness) are also required.
Q: How are the results useful to designing open and closed control loops?
A: They help determine the potential response capabilities or limitations of open and closed control loops — such as torque, speed, tension, position, and level. More specifically, it’s critical to know rotational inertia for servo positioning, robotics, and cut-tolength applications that often require near-step, instant accelerations. (In these applications, it’s not uncommon that these drives be limited mechanically or electrically.)
Acceleration and deceleration criteria can vary significantly, even within a process. On paper-making machines, for example, forming, dewatering, pressing, and drying limit in-process rates of acceleration to 1.5 to 2.5 ft/min./sec. However, on the same system a paper winder with multiple set cycle times (and needing to keep ahead linear paper production) might regularly accelerate at 150 ft/min./sec. In these cases, inertial results help qualify the sizing of each mechanical and electrical drive component.
Q: What if a designer can tweak component dimensions and weight?
A: If a designer has the freedom (or the need) to adjust, for example, the shaft material or motor weight during the design process, then keeping density a variable makes sense. A more general form of the torque equation can be used. Note how the common terms
of mass and radius of gyration combine in the equation below.