The moment of inertia of the load is a key measure for sizing and tuning servomotors. Traditional sizing methods use mathematical equations to accurately estimate the inertia of the load based on dimensions, materials, and densities. This method is especially valid on machines that are still in the design phase with data readily available.

Authored by:

Matt Pelletier
Product Training Engineer
Yaskawa Electric America Inc.
Waukegan, Ill.

Edited by Robert Repas
robert.repas@penton.com

Key points
• Retrofit motors are often oversized to assure proper operation with assumptions of questionable accuracy.
• The load inertia parameter in many modern drives must still be set for optimal tuning.
• Any servomotor can supply the key values needed to calculate inertia load on an existing machine.

Resources
Yaskawa Electric America Inc., yaskawa.com
Sigma II parameter calculator in Excel, tinyurl.com/8geva9
eLearning module: Inertia ratio by graphical analysis, tinyurl.com/7567al

But what if the value wasn’t known for a given machine? The design data may be long lost or operating conditions may have changed invalidating the original data. Design revisions may leave original sizing assumptions of questionable accuracy once the motor is installed.

Retrofits can prove annoyingly tedious in this regard. The engineer must either research component specifications or disassemble the machine and directly measure the components. Often the motor is simply oversized to avoid this, adding to the cost of the retrofit and possible cancellation because of the higher expense.

While oversizing the motor does typically make tuning a whole lot easier, it still doesn’t tell you the inertia ratio. In many advanced servomotor systems, such as Yaskawa’s Sigma-5 series, load inertia exists as a parameter for the servo’s mechanicalsystem model and must still be set for optimal tuning.

But there is a way to avoid all of the hassle and yet obtain an accurate inertia load value. It’s possible to use any servomotor to get the key values needed to calculate inertia load on an existing machine.

This method of determining load inertia is referred to as inertia by graphical analysis. It does not depend on any manufacturer or special software and can use either the existing servomotor or a new retrofit motor. With a little time and a few fundamental calculations, anyone can capture that elusive load inertia and proceed confidently with sizing and tuning.

The idea behind inertia by graphical analysis is a simple one: work the equations of motor sizing backwards. When sizing a motor, inertia is used to calculate the torque and speed profile that helps select the appropriate motor. With inertia by graphical analysis, the torque and speed profile of the installed motor is used to find the inertia of the load.

The key equation is the rotary version of Newton’s second law:

TA = (JM + JL) × α

where TA = the torque it takes to accelerate, JM = the inertia of the motor, JL = the inertia of the load, and α = the motor’s measured acceleration. For inertia by graphical analysis, solve the equation for JL:

JL = (TA / α) - JM

In this equation, JM can be found in manufacturer specifications. The unknowns TA and α are determined from a graphic plot of motor acceleration with an applied constant torque. The value of TA is measured directly on the graph by subtracting the friction torque (TF) from the maximum or peak torque (TP.)

TA = TP - TF

Acceleration is calculated by dividing the change in speed (Δω) by the change in time (Δt.)

α = Δω / Δt

Let’s walk through an example. First, the motor specifications are obtained from the manufacturer documentation. Fortunately, this is easy today through use of the Internet. For this example, the motor specifications are:

Motor inertia (JM) = 2.59 ×10-5 kg-m2
Torque constant (TC) = 0.435 Nm/Arms (100%)
Rated torque (TR) = 0.637 Nm
Horizontal ball-screw load, no external forces.

Next we plot the motor-torque diagram. Yaskawa’s SigmaWinPlus software serves in this example to capture the graph data, but any graphing software or external digital oscilloscope may be used.

The first graph shows torque was not held constant during acceleration, so it is not suitable as a measurement for TP. However, it does show that the steady-state torque at the motor’s maximum speed is 0.134 Nm, or 21% of the rated torque value. This is the torque needed to overcome friction, TF.

A second test run gave values suitable for calculating acceleration. Amplifier current was kept just below the point of highest torque to ensure the motor accelerates under a constant torque.

For this example, the maximum current setting limited torque to 0.255 Nm (40%), which becomes the maximum or peak torque (TP). Acceleration torque (TA) is then calculated as:

TA = 0.255 Nm - 0.134 Nm
TA = 0.121 Nm

With the graph now showing measurable values of speed and time, it’s possible to calculate the acceleration rate. The change of speed was measured at 200 rpm (from 600 to 800 rpm) within a time change of 0.0127 sec (12.7 msec.) The acceleration rate is then calculated as:

α = (200 rpm × 2π / 60) / 0.0127 sec
α = 1,646 rad/sec2

This is all the information necessary to find the load inertia and to apply Newton’s law.

JL = (0.121 Nm / 1,646 rad/sec2) - 2.59 × 10-5
JL = 4.76 × 10-5

And the inertia ratio (JRatio) is then calculated as:

JRatio = JL / JM
JRatio = 4.76 × 10-5 / 2.59 ×10-5
JRatio = 1.84

As technology moves forward, advanced algorithms will let servoamplifiers measure and calculate the As technology moves forward, advanced algorithms will let servoamplifiers measure and calculate the inertia ratio automatically. But finding the inertia by graphical analysis will still be useful for years to come.