What is the formula for motor selection?

In this lesson, we will continue the discussion on the topic of motion control. All of us at RealPars, hope you have been following our previous lessons on motion control and have begun to benefit from these lessons. Now you will apply your base knowledge of Servo and Stepper motors in this lesson towards how to determine the correct servo motor size for your application.

In this lesson, you will learn the information required to help you learn how to size a motor based on payload, speed, acceleration and other requirements necessary for the desired application.

This section will help you with the procedures and the understanding of the units of measurement to select the optimum motor for a particular servo or stepper application. Read on to learn how to determine motor sizes for your projects.

The servo motor selection process requires a certain number of calculations for you to understand, as well as the units of measurement used throughout the equations for you to become familiar with.

Some learners will enjoy performing the motor selection calculations manually, (we will briefly illustrate and describe some of the calculations later on in this lesson), while others prefer, utilizing one of the many online programs available to help you with all the necessary specifications of the servo motor required for your project and expediate you through the servo motor selection process.

The targeted end result of this lesson is to provide you with the knowledge on how to obtain the Maximum Speed, Load Torque, and Moment of Inertia specifications to provide the manufacturer of servo and stepper motors with the help of using a software motor sizing tool.

Let’s begin with a review of a few key terms and the units of measurement used to determine your servo motor size.

1. Moment of Inertia

The manufacturer of servo motors will need to know the force required to move the load. The force to overcome the loads resistance is called Moments of Inertia.

Moments of Inertia quantifies the resistance of a physical object to angular acceleration. Moments of inertia is to rotational motion as mass is to linear motion.

For example, when traveling in a train or any other vehicle, have you noticed how you continue to move forward when it stops. Well, you just experienced Inertia!

In general, an object’s moment of inertia depends on its shape and the distribution of mass within that shape, as in, the greater the concentration of material away from the object’s geometric center, the larger the moment of inertia this object will have.

Moment of inertia varies depending upon the axis of rotation specified.

Another example of inertia, imagine an ice puck resting on a frozen pond. It takes a certain amount of force to set the puck in motion.

The greater the mass, the more force will be needed to move the puck. The same is true if the puck were sliding along the ice. It would continue to slide until a force is applied to stop the puck.

The more massive the puck is, the more force will be needed to stop the motion of the puck.

Both moment of inertia and inertia are measures; They measure resistance to a change in state. “Inertial mass” is defined as force required for “acceleration” and “moment of inertia” is defined as the torque required for “angular acceleration”.

The larger inertia of the object, the greater force you need to change the velocity in a given time.

The SI unit of measure for moment of inertia is “one kilogram-meter squared”. In equations, it is usually represented by the variable “I”.

1-1. Mass

Next, let us review a few examples of how to calculate the moments of inertia of various geometric shapes.

One of the variables of determining inertia is mass. Mass is defined as the amount of matter an object has. One of the qualities of mass is that it has inertia. To determine mass one way, you’ll need to know the density and volume of the load you are moving.

1-1-1. Density

Density measures how tightly the matter in an object is packed together. Each material has its own density, which you can look up online or in a textbook. The scientific unit of density is “kilograms per cubic meter”.

1-1-2. Volume

The Volume is the amount of space the object occupies. The volume of solids is measured in “cubic meters”.

Finally, Mass is equal to “Density Times Volume”. The variable “m” here is the symbol defined as the mass. Mass is a measure of how much inertia an object shows.

Great, now you are aware of some of the definitions of data required to move forward on How to Determine the motor size of your project.

The motor sizing software used in this lesson will help us with the calculations to simplify sizing the correct servo or stepper motor.

This manufacturer’s software tool also provides a utility to help you with the calculations such as moments of inertia or determining the mass.

Now let’s begin with the steps towards sizing your servo motor.

Many software tools offer many solutions for a screw, belt, rack and pinion, roll feed, index table, and arm movement control. With our sample application, we select the requirement for controlling the rotation of an “arm mechanism”.

2. Load Torque

Next, you’ll need to determine how much torque the motion application needs. Remember the torque is how much “muscle” it takes to rotate a mechanism, and comes from three different sources:

1. Accelerating the mechanism’s inertia,

2. Friction,

3. External forces such as pressing against an object or gravity.

This is the most difficult part to calculate accurately. Calculate the inertia of each component of the system and add the values.

The formulas for calculating rotational inertia of various shapes are readily available on the Internet.

In our sample program, we will determine the moment of inertia by completing the questions for rotating the arm in the vertical plane and defining the arm’s dimensions; “A” as 200 millimeters long and “B” and “C” as 50 millimeters in width and height.

We will not use gear box for speed reduction and torque control. The “density” of the material will be based on “aluminum” in which is determined by the software.

You may also with this sizing software change the calculation mode to “mass” and enter the mass in kilograms. We have selected the entry of 1.4 kg for this example. There are many online programs to assist in calculating the mass of an object.

Internally and behind the scenes, the software will multiply the acceleration by the load inertia to calculate the load’s acceleration torque.

The software will also calculate friction forces for sliding loads, gravitational forces for vertical loads, and any external forces.

Then each force is multiplied by the radius it is acting on (known as the “moment”) to calculate the torque.

The software then calculates the peak torque by adding up all the torque values in the worst-case scenario. This is typically when the fastest acceleration is occurring or when there is the most mass on the machine.

Adding up the torque values from external forces, gravity and friction to calculate the continuous torque requirement is tedious without the help of a software tool.

3. Speed

Moving along we will now define what the application’s motion profile looks like. We will use a servo type motor and use the travel amount calculation method. We are using what is called a trapezoidal profile and will only be required to provide the acceleration and deceleration speed of 1.5 seconds, the travel time of 3 seconds and the amount of movement in degrees.

We will rotate the arm three times (360 degrees times 3 is equal to 1080 degrees). After which the moment of inertia of 139.183 x 10-4 kg·m2 (kilogram meter squared) is calculated automatically.

For equipment that performs a repetitive operation, plot out the required motor speeds throughout the cycle. Be sure to allow for acceleration and deceleration time.

The next screen displays the calculated motor operation conditions that suggests the Maximum speed of 120 r/min, Load Torque of 1.098 N⋅m and finally the moment of inertia of 139.183 x 10-4 kg·m2.

Often the manufacturer provides additional support in selecting a motor with different classification questions.

Adding an electromagnetic brake for example will help hold the load in place when power is removed or at rest. And it’s always a good idea to add a safety factor to cover any additional changes for the future and load variations not determined.

At this point, the key criteria for choosing a servo motor have been defined and it is time to browse the manufacturer’s product selection guide to find the motor that matches these requirements.

Find a motor and drive that matches the supply voltage, has a rated speed, continuous torque, and peak torque rating that exceeds the values calculated.

If there is a motor that is a close match, you are finished. If not, using the supplied software gearing can be applied to match the motor and load more closely.

The program will allow you to print out the results of the move profile and all the calculations performed by the software.

Servomotors can produce their full rated torque from zero rpm up to many thousands of rpm. Few machines can take advantage of these speeds without gear reduction.

Gear reduction matches the servo to the load in three ways; reducing the speed, increasing the torque, and lowering the inertia ratio.

Speed is reduced proportional to the gear ratio, torque is increased proportional to the gear ratio, and most importantly, the inertia ratio is lowered by the square of the gear ratio.

Gearbox manufacturers list the inertia of the servo-grade gearboxes, making it easy to include the gearbox inertia into the torque and inertia calculations.

Most of the motors available likely are capable of much higher speeds than required. Divide the motor speed by the required speed and round down to get a starting gear ratio.

Then divide the required torque by the gear ratio to find the newly required torque. This will help you narrow the choices down to a few select motors.

Once you have the servo motor selected, choose a servo drive rated for the correct input voltage and with sufficient output current to drive the servo motor.

Servo drives can be controlled via several different interface types. Interfaces types you can choose from include pulse-and-direction digital control, analog control, and other servo networks.

A servo drive will provide your solution with high-speed control and feedback, a reduction in wiring, and overall superior diagnostics capabilities compared to other types of interfaces.

Finally, choose any options such as keyed motor shafts, shaft seals, holding brakes for vertical loads, or external braking resistors.

Selecting the best servo system for an application is a skill that improves with practice. When in doubt, it’s a good idea to verify your results with the manufacturer or distributor.

This concludes the blog post, How to determine the Motor Size for your application. I hope you have learned what’s required to move forward in creating your own motion control project.

This blog post is one of a series of blogs on motor motion control, so please check back with us soon for more motion control topics.

Thanks for reading and sharing our blog posts with your friends and colleagues.

Happy learning,

The RealPars Team

‍

For a detailed explanation of each formula, Click on the links below to go right to it.

Mechanical Formulas

Deg C = (Deg F - 32) x 5/9
Deg F = (Deg C x 9/5) + 32

Temperature Conversion Formula

In mechanical systems, all rotating parts do not usually operate at the same speed. Thus, we need to determine the "equivalent inertia" of each moving part at a particular speed of the prime mover.

The total equivalent WK2 for a system is the sum of the WK2 of each part, referenced to prime mover speed.

The equation says:

This equation becomes a common denominator on which other calculations can be based. For variable-speed devices, inertia should be calculated first at low speed.

Let's look at a simple system which has a prime mover (PM), a reducer and a load.

The formula states that the system WK2 equivalent is equal to the sum of WK2parts at the prime mover's RPM, or in this case:

Note: reducer RPM = Load RPM

The WK2 equivalent is equal to the WK2 of the prime mover, plus the WK2 of the load. This is equal to the WK2 of the prime mover, plus the WK2 of the reducer times (1/3)2, plus the WK2 of the load times (1/3)2.

This relationship of the reducer to the driven load is expressed by the formula given earlier:

In other words, when a part is rotating at a speed (N) different from the prime mover, the WK2EQ is equal to the WK2 of the part's speed ratio squared.

In the example, the result can be obtained as follows:

The WK2 equivalent is equal to:

Finally:

WK2EQ = 3200 lb.ft.2

The total WK2 equivalent is that WK2 seen by the prime mover at its speed.

Electrical Formulas     (For more formulas see "Formulas")

I = Amperes; E = Volts; Eff = Efficiency; pf = Power Factor; kVA = Kilovolt-amperes; kW = Kilowatts

What is IL with 245 volts (ELINE) applied to this motor?

IL @ 245 V. = 100 x 254V/230V

IL @ 245V. = 107 Amperes

Horsepower is work done per unit of time. One HP equals 33,000 ft-lb of work per minute. When work is done by a source of torque (T) to produce (M) rotations about an axis, the work done is:

When rotation is at the rate N rpm, the HP delivered is:

For vertical or hoisting motion:

Where:

For fans and blowers:

Or

Or

For purpose of estimating, the eff. of a fan or blower may be assumed to be 0.65.

For pumps:

Or

For estimating, pump efficiency may be assumed at 0.70.

The equivalent inertia of an adjustable speed drive indicates the energy required to keep the system running. However, starting or accelerating the system requires extra energy.

The torque required to accelerate a body is equal to the WK2 of the body, times the change in RPM, divided by 308 times the interval (in seconds) in which this acceleration takes place:

Where:

Or

The constant (308) is derived by transferring linear motion to angular motion, and considering acceleration due to gravity. If, for example, we have simply a prime mover and a load with no speed adjustment:

Example 1

The WK2EQ is determined as before:

If we want to accelerate this load to 1800 RPM in 1 minute, enough information is available to find the amount of torque necessary to accelerate the load.

The formula states:

In other words, 97.4 lb.ft. of torque must be applied to get this load turning at 1800 RPM, in 60 seconds.

Note that TAcc is an average value of accelerating torque during the speed change under consideration. If a more accurate calculation is desired, the following example may be helpful.

Example 2

The time that it takes to accelerate an induction motor from one speed to another may be found from the following equation:

Where:

The Application of the above formula will now be considered by means of an example. Figure A shows the speed-torque curves of a squirrel-cage induction motor and a blower which it drives. At any speed of the blower, the difference between the torque which the motor can deliver at its shaft and the torque required by the blower is the torque available for acceleration. Reference to Figure A shows that the accelerating torque may vary greatly with speed. When the speed-torque curves for the motor and blower intersect there is no torque available for acceleration. The motor then drives the blower at constant speed and just delivers the torque required by the load.

In order to find the total time required to accelerate the motor and blower, the area between the motor speed-torque curve and the blower speed-torque curve is divided into strips, the ends of which approximate straight lines. Each strip corresponds to a speed increment which takes place within a definite time interval. The solid horizontal lines in Figure A represent the boundaries of strips; the lengths of the broken lines the average accelerating torques for the selected speed intervals. In order to calculate the total acceleration time for the motor and the direct-coupled blower it is necessary to find the time required to accelerate the motor from the beginning of one speed interval to the beginning of the next interval and add up the incremental times for all intervals to arrive at the total acceleration time. If the WR2 of the motor whose speed-torque curve is given in Figure A is 3.26 ft.lb.2 and the WR2 of the blower referred to the motor shaft is 15 ft.lb.2, the total WR2 is:

And the total time of acceleration is:

Or

Figure A
Curves used to determine time required to accelerate induction motor and blower

Sales Orders are often entered with a note under special features such as:

-----"Suitable for 10 starts per hour"
Or
----"Suitable for 3 reverses per minute"
Or
-----"Motor to be capable of accelerating 350 lb.ft.2"
Or
-----"Suitable for 5 starts and stops per hour"

Orders with notes such as these can not be processed for two reasons.

1. The appropriate product group must first be consulted to see if a design is available that will perform the required duty cycle and, if not, to determine if the type of design required falls within our present product line.
2. None of the above notes contain enough information to make the necessary duty cycle calculation. In order for a duty cycle to be checked out, the duty cycle information must include the following:
   1. Inertia reflected to the motor shaft.
   2. Torque load on the motor during all portions of the duty cycle including starts, running time, stops or reversals.
   3. Accurate timing of each portion of the cycle.
   4. Information on how each step of the cycle is accomplished. For example, a stop can be by coasting, mechanical braking, DC dynamic braking or plugging. A reversal can be accomplished by plugging, or the motor may be stopped by some means then re-started in the opposite direction.
   5. When the motor is multi-speed, the cycle for each speed must be completely defined, including the method of changing from one speed to another.
   6. Any special mechanical problems, features or limitations.

Obtaining this information and checking with the product group before the order is entered can save much time, expense and correspondence.

Duty cycle refers to the detailed description of a work cycle that repeats in a specific time period. This cycle may include frequent starts, plugging stops, reversals or stalls. These characteristics are usually involved in batch-type processes and may include tumbling barrels, certain cranes, shovels and draglines, dampers, gate- or plow-positioning drives, drawbridges, freight and personnel elevators, press-type extractors, some feeders, presses of certain types, hoists, indexers, boring machines, cinder block machines, key-seating, kneading, car-pulling, shakers (foundry or car), swaging and washing machines, and certain freight and passenger vehicles. The list is not all-inclusive. The drives for these loads must be capable of absorbing the heat generated during the duty cycles. Adequate thermal capacity would be required in slip couplings, clutches or motors to accelerate or plug-stop these drives or to withstand stalls. It is the product of the slip speed and the torque absorbed by the load per unit of time which generates heat in these drive components. All the events which occur during the duty cycle generate heat which the drive components must dissipate.

Because of the complexity of the Duty Cycle Calculations and the extensive engineering data per specific motor design and rating required for the calculations, it is necessary for customer to refer to an electrical engineer for motor sizing with a duty cycle application.