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Motor Fundamentals

Motor Fundamentals


Until now, we have presented the electrical characteristics of motors to acquaint you with the fundamentals of motor action and the effects of direct and alternating current on motor design and operation. Electrical characteristics affect a designer’s decisions on which motor to choose for any given application.

Equally important in understanding motor operation are the mechanics and performance characteristics of electric motors. Mechanics encompasses the rules which govern the motion of objects, in particular:

a). the force which must be applied to start an object moving or to stop it, and
b). the opposing forces which must be overcome before movement can begin or end.

Other factors such as speed, acceleration and amount of displacement all play a part in determining which motor is best suited to perform a task.

Translational Motion
The movement of a uniform object in a straight line is referred to as translational motion. The three parameters of translational motion are displacement, velocity and acceleration.

The change in position of an object is known as displacement. It is a vector quantity with both magnitude and direction and is shown mathematically as:
where Ox is the total displacement, xf is the object’s final position and xi is the object’s initial position.

The rate at which an object’s position changes with time is its velocity. There are two types of velocity: average and instantaneous. Average velocity is the net displacement divided by the elapsed time:
where d is the net displacement and t is the elapsed time to make the displacement, tf is the final time and ti is the initial time.

At any instant in time the velocity of an object may exceed the average velocity, so it is sometimes necessary to know the instantaneous velocity:

Frequently, the terms speed and velocity are used interchangeably. Velocity can be positive or negative. Speed is equal to the absolute value of the instantaneous velocity and is always expressed as a positive number:

As an object begins to move, its velocity changes with respect to time. This is called acceleration. Like velocity, acceleration is expressed in average and instantaneous quantities. Average acceleration equals:
where Dv is the difference between the object’s final and initial velocities, and Dt is the elapsed time.

The instantaneous acceleration is defined by the following formula:
Rotational Motion
Motors can be used to move objects in a straight line, which is why a brief overview of translational motion was given. But motor design and application focuses heavily on rotational motion around an axis. The same principles of displacement, velocity and acceleration also govern rotational motion. In many motion control applications, it often becomes necessary to transform linear motion into rotational motion or vice versa.

Angular Displacement:
For rotational motion, displacement is expressed in radians, degrees or revolutions because the displacement occurs in reference
to a rotational axis (one radian = 57.3°, one revolution = 360° = 2π radians.) Angular displacement is expressed as:
where θl is the object’s initial angular position relative to the axis and θ2 is the final angular position.

Angular Velocity:
Angular velocity is expressed in radians / second, revolutions / second, or revolutions / minute (RPM). It is the rate at which an
object’s angular displacement changes with respect to time. Like translational velocity, it can be expressed as an average or instantaneous quantity.

The formula for average angular velocity is:
where Δθ is the net angular displacement between the initial position and final position and Δt is the elapsed time.

Instantaneous angular velocity is expressed as follows:
where v = circumferential linear velocity.

Angular Acceleration:
When an object’s angular velocity changes with respect to time, it is undergoing angular acceleration. Average angular acceleration
is expressed as:
An object’s instantaneous angular acceleration an be calculated as:
Statics and Dynamics
The previous discussion focused on the motion of an object either in a straight line or about an axis. But other factors must be considered when discussing motion. The size and weight of an object determine the amount of force needed to move it or stop
it. Other factors such as friction also play a role in determining the amount of force needed to move an object. We will now
center our attention on these other factors.

Mass is the property of an object that determines its resistance to motion. It is a factor of the object’s weight (W) and its acceleration due to gravity (g). Mass is the quantitative measure of inertia. It is the mass of an object that requires a force to move it. It is usually expressed in kilograms or pounds (mass)*.

* The pound-mass is a body of mass (0.454 kg). The pound-force is the force that gives a standard pound-mass an acceleration equal to the standard acceleration of gravity (32.174 ft/sec.).

In a linear system:

The fundamental measure of an object’s motion is momentum. In a linear system, it is the product of the object’s mass and linear velocity and is expressed in newton-seconds or pound seconds:

The push or pull on an object that causes it to move or accelerate is called force. It is directly proportional to the object’s mass and acceleration:
where M is the object’s mass and a is the acceleration.

Rotational Inertia:
In linear motion, the inertia of an object is represented by the object’s mass:
It is the mass which tells us how large a force will be required to produce constant acceleration. The rotational analog of this formula is:
This formula tells us how much torque (T) is required to produce angular acceleration (a). The moment of inertia (I) can be defined as the mass of the object times the square of the distance (r) from the rotational axis (see figure below):
The moment of inertia of a hoop containing many small masses on its circumference.
The moment of inertia can be calculated for any object this way but calculus is usually needed for the summation. The figure below
shows the values of I for several familiar shapes used in mechanical systems.

The figure below shows that the moment of inertia is always the product of the object’s mass and the square of a length.
Moments of inertia for familiar objects.
For a hoop, I = Mr2. This leads to a general formula:
where k is the radius of rotation at which the entire mass of the object should be concentrated if the moment of inertia is to
remain unchanged. A more standard term for this length is the radius of gyration.
Motor Load and Torque Characteristics
The principles we have just discussed can be applied specifically to motor applications. A motor cannot be selected until
the load to be driven and the torque characteristics are determined.

Motor Load:
The term “motor load” can refer to horsepower (hp) required by the driven object or machine. Motor load in hp can be expressed:
where r (in feet) is the radius at which the force (F, in pounds) is applied and N is revolutions per minute.
Where torque (T) is expressed in lb-ft., or if T is expressed in oz-in., then:
Motor load is best described as the torque required by the load. The torque requirement may be dependent upon speed
as well. Various conditions place specific demands on torque requirements and they are discussed next.

Breakaway Torque:
This is the torque required to start the shaft turning and is usually the torque required to overcome static friction.

Accelerating Torque:
This torque may be expressed in percent of running torque. It is the amount of torque needed to accelerate the load from standstill
to full speed, and to overcome friction, windage, product loading and inertia.

Peak Torque:
Peak torque is the maximum instantaneous torque that the load may require. High peaks for brief periods are acceptable, but if an application requires sustained torque higher than a motor’s peak rating, a different motor should be considered.

Constant Torque:
A load with a horsepower requirement that varies linearly with changes in speed is said to have constant torque requirements.
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