Rotational Dynamics and Multi-Body Systems

Angular Motion and Rotational Kinematics

Rotation is simply linear motion with an extra dimension: angle around a circle. The same physics is at work, but viewed from different vantage points in the coordinate system. Instead of position (x, y), we have angle θ, instead of velocity (v_x, v_y), we have angular velocity ω (radians per second), and instead of acceleration (a_x, a_y), we have angular acceleration α.

The same set of equations apply for this case, except that we now use angular quantities instead of linear ones. The equation for velocity for constant angular acceleration is ω = ω₀ + αt. The equation for the traveled angle is θ = θ₀ + ω₀t + ½αt².

Every spinning object (motors, gears, turbines, wheels, ...) uses these equations. After learning linear kinematics, rotational kinematics is a breeze.

Torque and Rotational Effect of Forces

Force can accelerate objects in a straight line, but torque can make them spin. Grab a door handle and it rotates. Grab it near the hinge and it doesn't move. Same amount of force, vastly different torque. Torque is proportional to the force applied and where on an object you apply that force.

R is the distance from the axis of rotation to where the force is applied. F is the force applied to the object. The cross product of two vectors gives a value that only corresponds to the component of the two vectors that is perpendicular to both of them. In the case of force and radius, only the force applied in the tangential direction (perpendicular to radius) will result in torque. A force pushed radially inward toward the axis of rotation will have zero effect on torque. The maximum torque would be achieved by applying the force in a completely tangential direction.

Torque is an unfamiliar quantity to many, showing up only when it is too late. Torque affects bolted joints, motor shafts, steering systems and generally anything that is attached to something else. Design specifications often don't just give you the allowed forces but also the allowed torque. The common notation for torque is to put "Nm" after the torque, e.g. "100 N·m". Note that just because something can handle 100N of force doesn't mean it can handle 100 N·m of torque.

Newton's Second Law for Rotational Motion and Moment of Inertia

Motion can be linear or rotational. For linear motion: F = ma. For rotational motion: Στ = Iα. Net torque is equal to moment of inertia times angular acceleration. The same structure using rotational variables is used for this equation.

In this video we'll look at the second of the physical parameters we can use to describe objects. The first was mass, and mass resists linear (straight line) acceleration. This second physical parameter resists angular (rotational) acceleration. That means it resists change in rotation rate. And, unlike mass, which is just a number (2 kg, for example), this physical parameter is different for different distributions of the same mass. So the moment of inertia of a rigid rod is different from the moment of inertia of a disk of the same mass.

Two objects of equal mass are constructed as a compact sphere and a hollow ring. The ring has more mass far from the axis of rotation than does the sphere. Explaining why the ring would move more slowly when the same torque is applied to each object. This characteristic, which is true for any object, is why flywheels are constructed with mass at the rim in order to achieve high moments of inertia in order to store rotational energy and to decrease changes in rotational speed.

The moment of inertia affects the vibration frequency and time constant of a system. When a different configuration is made, it will behave differently. People often associate inertia with weight but mass also plays a very important role in where it is located in the system.

Angular Momentum and Its Conservation

Momentum is the product of an object's mass and its velocity, or p = mv. Similarly, for any object that is spinning or rotating (i.e., revolving about a fixed point), angular momentum (L) is the product of an object's moment of inertia (I) and angular velocity (ω), L = Iω. If torque is zero, an object's angular momentum will remain constant, a conservation law similar to that of linear momentum.

This principle can be observed in many things such as complex machinery, gyroscopes, how a spacecraft maintains attitude, and the dynamics of a vehicle. Increasing the mass distribution of a spinning object will cause it to change speed automatically in order to maintain balance, and this stability is a result of angular momentum conservation, requiring external torque to alter its rotational speed.

Rigid-Body Motion and Coupled Systems

Most physical objects we interact with in the real world will undergo physical deformation while being subjected to load. In contrast to this reality, the rigid body assumption treats physical objects as if they were rigid. This is a reasonable approximation in many scenarios, however, particularly when deformation is negligible compared to the object's motions. For example, when rotating a real-world object, both its rotation and translation can be studied with the rigid body assumptions without ever having to consider the internal stresses that would occur in reality.

All fun starts from one equation, the single particle (or object at rest), where F = ma rules the day. Things get even more fun when we move to a rigid body, where we suddenly need a second equation, ΣF = ma (for translation of the center of mass) and Στ = Iα (for rotation about the center of mass). It's a simple rule: some forces cause translation and others cause rotation. Some forces even cause both.

The systems we encounter in life are typically multiple bodies that interact, and that is the key characteristic that makes their dynamics so complicated. Examples include: mechanical systems composed of chains of rigid bodies such as linkages, gear systems, vehicles, and various types of robots, all of which require us to compute the dynamic behavior of interacting rigid bodies. (This is the crux of mechanism design and multibody simulation.) The rigid-body assumption makes this problem much easier, as it significantly reduces the number of variables with which we must work, but there is still no easy formula for describing what these systems do. Motion of one body causes the other bodies to move as well. This coupled motion must be included in a model of the system behavior, resulting in a set of simultaneous equations, as opposed to an easy formula. Thus, dynamics can be a difficult field, but the difficulties are usually characteristic of and relevant to real systems.