In this eLearning module we'll look more in depth at several sizing factors...

...including torque, the motion profile, the load inertia and how to reduce it...

...the effects of gearing, regeneration and machine efficiency.

When appropriate we'll look at examples of equations that are used...

...as well as practical suggestions for motor sizing and machine design.

We'll start by looking at the components of the torque profile.

For a horizontally oriented actuator with no external forces...

...there are just two components to torque that are considered when sizing the motor...

...friction torque and acceleration torque.

The friction torque is modeled as one constant level of torque that is always present.

This is the baseline torque.

The acceleration torque is also a constant level but is only present during acceleration and deceleration.

Acceleration torque is added to the baseline friction torque while the motor accelerates.

Acceleration torque is subtracted from the baseline friction torque while the motor decelerates...

...because friction torque helps decelerate the motor.

If the acceleration rate and deceleration rate are the same as shown here...

...then the amount of torque that is added or subtracted is the same.

The sum of these torque components produces the torque profile.

A force due to friction is present on the load bearing surface as a load moves linearly.

This friction force is proportional to the coefficient of friction 'µ'...

...which is often given by the manufacturer of the linear bearing.

The coefficient of friction is a value between zero and one.

Translated through the actuator...

...this force is seen as a frictional torque and is quantified by the following equation.

This equation shows that a heavier load will produce more friction torque...

...as well a higher coefficient friction.

Increasing the pitch or revs/in of the actuator will reduce the friction torque.

The efficiency of the actuator also affects the value of friction torque.

We'll discuss efficiency later.

The equation for acceleration torque comes from Newton's second law F = ma.

Translated to the rotational world of servo motors...

...force becomes torque, mass becomes inertia...

...and the units of acceleration change as represented by the symbol 'α'.

The efficiency of the mechanism must also be considered.

Besides efficiency the acceleration torque depends on two other factors...

...the total system inertia and the acceleration rate.

By keeping these factors low, the acceleration torque will also be kept low.

In turn keeping the size and cost of the motor low.

One way to keep the acceleration torque low is to keep the acceleration rate low.

Ultimately, the acceleration rate is determined by the motion program in the upper level controller.

The other way to reduce the acceleration torque is to reduce the system inertia.

The system inertia can be divided into two components...

...the motor and the load.

The torque required can be reduced by selecting a smaller motor with lower inertia.

However, by doing so the inertia ratio will increase.

So it is preferable to take measures to reduce the load inertia.

In many cases the motion profile for the application is fixed.

But if there is any flexibility in designing the motion profile at the time of motor sizing.

Use that opportunity to design a profile that minimizes the acceleration torque...

...thereby minimizing the motor size.

As shown in the previous slide...

...minimizing the acceleration torque is most commonly accomplished by decreasing the acceleration rate...

...while increasing the acceleration time and top speed of the move.

When the applications maximum speed, distance and move time are known...

...the slowest possible acceleration for a symmetric trapezoidal profile can be found with this equation.

It is important to use the worst-case profile for sizing.

This means to size to the move profile...

...with the highest speed and acceleration that you ever expect the machine to experience.

Additionally, if the motor is at rest during part of the Machine cycle....

...this dwell time must be included in the motion profile in order for the RMS torque to be calculated correctly.

Longer dwell times mean a lower RMS torque and can have a significant impact on motor sizing.

The torque is calculated for each part of the profile as well as the maximum and RMS torque.

A final consideration is the motion profile type.

For the purposes of sizing a trapezoidal speed profile is always used...

...even though it may be possible to program s-curve or parabolic acceleration profiles in the controller.

The load inertia is perhaps the most important consideration when sizing a servo.

Reducing the load inertia reduces both the required torque and inertia ratio providing a double advantage.

Remember that all objects moved by the motor contribute to the load inertia.

Among these parts are those that move linearly and those that rotate.

Rotary and linear inertia components each have specific details to consider.

The linear component of the load inertia for a ball screw application is given by the following equation.

Other mechanical systems with linear inertia follow a similar equation.

The linear inertia is proportional to the total weight that is being moved...

...which includes the weight of the payload and the weight of the table itself.

Reducing the weight of the linear components...

...will proportionally reduce the linear inertia contribution to the load inertia.

Constants for the force of gravity and PI(π) are used in the equation to convert units.

But the most useful observation is that the pitch of the screw reduces the linear inertia component...

...by the square of the pitch.

For example, a ball screw with twice as many rev/in reduces the load inertia by 2 squared or 4 times lower.

But more rev/in also means that the motor will have to accelerate itself...

...and any rotary components faster and to a higher speed...

...in order to maintain the speed profile of the load.

Even so, many applications can benefit from a sizing perspective by using a ball screw with a different pitch.

One common misconception is that friction and machine efficiency increase the inertia.

It is evident by this equation that friction and efficiency do not have any effect on the inertia.

remember that inertia is the rotor equivalent to mass...

...and no amount of friction can change an object's mass or inertia.

Anything that is rotated by the motor whether directly or indirectly contributes to the rotary inertia of the load.

In a ball screw system...

...these components include the screw itself as well as the coupling.

The inertia for each individual rotary component is given by an equation similar to the following equation...

...for the inertia of a solid cylinder.

The equation shows that the inertia is proportional to the length and density.

Again the factors of PI(π) and the force of gravity are included...

...but the dominating factor that determines the inertia of a rotating object is that object's radius.

The inertia is proportional to the radius to the fourth power.

To put this into perspective consider an object with a given radius and inertia.

If the radius is reduced to just 85% of the original the inertia will reduce to 50%.

Notice that once again...

...machine efficiency and coefficient of friction do not play a part in determining the rotary inertia.

Oftentimes the inertia ratio is the limiting sizing factor.

A much smaller motor may have plenty of torque and speed but the inertia ratio is too high.

An engineer wants to offer a competitive cost while still ensuring that the motor is not too small.

So the question often becomes, what is the maximum acceptable inertia ratio for this application?

Such a wide variety of servo applications exist that there is only one answer that's always true.

It depends on the application.

But we can attempt to explain some general trends.

As mentioned in the e-Learning module servo sizing part 1...

...performance tends to increase as inertia ratio decreases.

High performance implies multiple high-speed moves per second...

...requiring fast settling times around 10 milliseconds or less.

But inertia ratio is only one factor contributing to high performance.

Perhaps the most important performance factor is the mechanical construction.

If a machine is not rigidly and tightly built...

...it will not be able to achieve high performance no matter what the inertia ratio is.

But in all machines a higher inertia ratio...

...lowers the natural resonant frequencies of the system as modeled by the following equation.

In machines that are not rigidly designed and constructed...

...the value of 'Ct' in the equation is very low.

This lowers the resonant frequency such that it begins to interfere with the machines operation.

Resonance can often be mitigated by adjusting the tuning gains...

...applying notch filters or any other combination of tuning techniques...

...but sometimes the only solution is to increase the motor inertia...

...so that the machines natural resonant frequency rises.

Due to the many unknowns about a machine that may still be in the design phase itself.

Sizing for a lower inertia ratio is used as a safety factor...

...to compensate for possible compliance in the mechanical construction...

...to ensure high performance and to assist in tuning.

So when limited information is available follow these simple guidelines.

Inertia ratios around 5:1 typically allow fine performance in most applications.

While 2:1 or lower is suitable for the highest performance machines.

10:1 and higher can still produce fine results.

With the higher inertia ratio it is likely that more tuning will be required...

...and even after tuning, the performance will still not be as high as it would have been if a larger motor was used.

If the inertia ratio is higher than desired, the easiest solution is to select a larger motor.

It may also be possible to decrease the load inertia either linear or rotary.

The most effective methods are to increase the pitch of the rotary to linear conversion mechanism...

...and to reduce the diameter of any rotary components.

Alot can also be done to lower the inertia ratio by adding a gearbox to the motor.

A gearbox can make all the difference for applications that require high torque and low speed.

Making it possible to use a much smaller motor...

...while at the same time achieving an acceptable inertia ratio.

Torque produced by the output of a gearbox is magnified by the gear ratio.

A 2:1 gearbox allows a motor to produce twice as much torque.

A five-to-one gearbox produces five times the torque, etc.

Realistically, these torque values are slightly lower due to the gearbox efficiency.

But this increase in torque is made possible...

...because the speed out of the gearbox is reduced by the gear ratio.

The motor will be required to turn faster and accelerate faster when a gearbox is connected...

...in order to achieve the same output speed and acceleration to the machine.

The load inertia is reduced even more by the square of the gear ratio.

So a 2:1 gearbox reduces the load inertia by four times...

...and a 5:1 gearbox reduces the load inertia by twenty five times.

However, a gearbox does have an inertia itself plus it adds another mechanical element to the system...

...decreasing the mechanical rigidity, adding backlash, requiring maintenance and can be quite expensive.

In many cases the additional cost of a high-quality gearbox exceeds the additional cost of a larger motor.

So consider adding a gearbox to the system when the motor is excessively over-sized...

...due to torque or inertia ratio constraints and when the motor operates at a low speed.

Regeneration means that the motor is generating energy rather than using energy.

This happens during deceleration because the momentum of the load...

...forces the motor to move in the direction opposite to that in which torque is being applied.

Basically the kinetic energy stored in the machine is converted to electrical energy by the motor.

Some of this energy can be stored in the DC bus capacitor...

...but the excess energy must be burned off through a resistor and is lost as heat.

Several factors increase the regeneration energy.

The kinetic energy stored in the system increases with higher speed and inertia.

So to reduce regeneration, try to reduce the load inertia...

...and try to design a system that uses a motion profile with a lower speed.

High friction helps the system decelerate without motor torque.

So applications with low friction can also produce greater regeneration energy.

A fast deceleration rate means that alot of energy is dumped back into the amplifier quickly...

...which also increases regeneration.

It is not uncommon for vertically oriented applications to require regeneration resistors.

In vertical applications the whole torque profile is offset by a constant torque required to overcome gravity.

Downward deceleration's require even more torque...

...increasing the regeneration energy even further.

A counterbalance may be used to offset the torque due to gravity decreasing the regeneration energy.

If these factors are reduced the regeneration energy will also be reduced...

...and it may be possible to avoid the use of an external regeneration resistor.

But if regeneration energy is present this does not mean that the motor is under-sized.

Most amplifiers include a built-in regeneration resistor that is sufficient for the majority of applications.

Efficiency is the measure of the effectiveness of a mechanism to transmit torque from its input to its output.

The ideal machine is 100% efficient but due to frictional losses no machine is ideal.

In the example of a ball screw...

...internal frictional losses in the bearings and in the nut lead to reduced efficiency in the mechanism.

So, when the motor produces a certain amount of torque, a percentage of it is lost inside the machine.

The percentage that is transmitted through the machine is the Machine efficiency.

The efficiency increases the input torque required to overcome friction...

...as well as the torque required to accelerate.

The efficiency is normally specified by the manufacturer.

If it is not given, typical mechanism efficiencies are listed in the following table.

The efficiency is an important specification...

...as it can have a very significant effect on the torque required by the motor...

...and therefore on the size of the motor.

This concludes the Servo Sizing part 2 eLearning module.