Simcenter Testing Solutions Simcenter Testlab: Calculating Angular Difference

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Simcenter Testlab can be used to measure angular position (degrees) and rotational speed (revolutions per minute or rpm) from multiple and different types of encoder devices.  Math can be performed between measurements made from different devices to determine the relative rotational speed and angular positions at locations along a drive system.

An example drive system with an electric motor is shown in this article. In this system, as the speed of the motor changes, an unwanted vibration occurs.  An illustration of the system is shown in Figure 1
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Figure 1: Drive system with electric motor, transmission, and final drive. Electric motor has a resolver, while the final drive has an encoder.
To diagnose the issue, the windup (i.e., the difference in angular displacement in degrees) between the electric motor to the final drive encoder is calculated. The steps for performing these calculations using Simcenter Testlab are also explained in depth.

1.    Overview
2.    Resolver
3.    Rotational Speed versus Rotational Angle
4.    Encoder: Moments versus RPM
5.    Angular Difference Results

1. Overview

Equation 1 is used to calculate the the relative angular displacement between the electric motor shaft rotation and the final drive rotation:
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Equation 1: Equation to determine the relative angular displacement between the electric motor and final drive.  Can be calculated in degrees, radians, etc.
Because there is a geared transmission unit between the electric motor and final drive, the gear ratio must be used to adjust the electric motor angular position relative to the final drive in order to calculate the difference.

The formulas used in the Simcenter Testlab Time Signal Calculator to calculate the angular difference are shown in Figure 2.
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Figure 2: Formulas to calculate the angular difference.
The equations CH18 and CH 20 are not need but are performed to verify the correctness of the results.  This is explained in the upcoming section "Rotational Speed versus Rotational Angle".

See the knowledge article "Time Signal Calculator Tips!" for information on using formulas for calculations.

The formulas are further explained in the next sections.

2. Resolver

This section details the formulas (Figure 3) used to calculate the angle position and rotational speed from the resolver on the electric motor.
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Figure 3: Highlighted formulas calculate the angular position and rotational speed from the resolver on the electric motor.

Electric motors are often equipped with resolvers (Figure 4) .  Resolvers are electromagnetic transducers used to measure the angle of rotation of the electric motor main shaft.
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Figure 4: Electric motor resolver (black with copper windings).
There are several different types of resolvers, but most have three key signals:
  • EXCITATION: Excitation voltage with carrier frequency.
  • SINE: Sine wave proportional to one rotation of the shaft. The excitation frequency is modulated by this sine wave.
  • COSINE: Cosine wave proportional to one rotation of the shaft. The excitation frequency is modulated by this cosine wave.
Examples of these three signals measured from a resolver are shown in Figure 5.
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Figure 5: Excitation (red), sine (green), and cosine (blue) measured signals from a resolver. When viewed in detail (bottom) the signals are actually composed of amplitude modulated sinusoids, while the overall (top) signals appear as a sine, cosine, etc.
The signals are voltages composed of sine waves.  The sine and cosine signal are amplitude modulated while the excitation signal is not. 

The Simcenter Testlab Time Signal Calculator has specific functions for calculating rpm or angle from these resolver signals as shown in Figure 6.
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Figure 6: Simcenter Testlab Time Signal Calculator “RESOLVER_TO_RPM” input menu.
The inputs to the method are as follows:
  • function1: Cosine signal.
  • function2: Sine signal.
  • function3: Excitation signal.  If not measured, enter zero.  The carrier frequency will need to be manually input.
  • carrier_frequency: Manually entered carrier frequency of the excitation. Only used in function3 is set to zero.
  • resolver_type: Enter 1 for variable reluctance and 2 for electromagnetic resolver.
  • number_of_poles: Number of poles on the resolver. Used to scale the rpm output.
The rate of the modulation of the sine and cosine is proportional to the speed of the motor. Figure 7 shows the results of the calculations.
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Figure 7: The resolver sine (top, green) and cosine (blue) signals are used to calculate the rotational speed (rpm) and/or angular position (degrees) of the electric motor.

The faster the rate of the amplitude modulation of the sine and cosine signals, the higher the speed of the electric motor.

3. Rotational Speed versus Rotational Angle

In the example calculations shown in this article, both rpm and angle are calculated from encoder or resolver data.  The angle data is used in the calculation of the degree difference, while rpm data is used as verification that the parameters used in the formula are correct.

To support this, there are similar functions in the Simcenter Testlab Time Signal Calculator that can output either angle or rpm (Figure 8) from the same encoder data.
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Figure 8: Similar functions (highlighted) from the Time Signal Calculator that output either rpm or angle data from the same input data.

It is not uncommon to know at least the approximate rpm of the spinning systems. This rpm knowledge can be used to verify that all entries used in the formula were correct (Figure 9).  
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Figure 9: Rotational speed (top, blue) of the electric motor over 25 rpm sweeps of the motor.  By verifying the rpm is between 70 and 850 as expected (top, blue), there is greater confidence in the calculated angular position (bottom, green) calculated using the same settings.
By verifying the rpm is as expected, then there is greater confidence that the entries for number of pulses, number of poles, type of resolver, etc are correct.

4. Encoder: Moments versus RPM

The angular position of the final drive is calculated from the highlighted formulas in Figure 10:
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Figure 10: Highlighted formulas calculate final drive angular position and rotational speed from encoder.

The final drive angular position was measured using an encoder.  Encoder is a generic name for a device that outputs a series of voltage pulses proportional to speed of a piece of rotating machinery. The time between pulses is used to calculate the rotational speed.  As the rotational speed increases, the time between pulses becomes smaller.

Examples encoder devices include: lasers, optical probes, magnetic pickups, and incremental encoders are shown in Figure 11.
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Figure 11: Types of encoders include magnetic pickups, lasers, and incremental encoders.

When measuring an encoder (produces multiple pulses per revolution) with Simcenter Testlab and Simcenter SCADAS hardware, two pieces of information are recorded: the time stamps of each pulse crossing (the “raw” trace), and the rpm that is calculated from these crossings. 

An example of the resulting measurement is shown in Figure 12.
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Figure 12: Encoder data acquired by a Simcenter SCADAS has both a rpm channel and a “raw” channel consisting of the time stamps from pulse crossings.

The two recordings from the encoder measurement (calculated rpm and raw data) are shown in Figure 13:
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Figure 13: In Simcenter Testlab, an encoder measurement includes both the rpm data (left) and the crossing time stamps (right).
The raw speed signal consists of the time stamps (or moments in time) that each encoder pulse occurred.  It is the most compact and precise form for storing the rpm or angle data.

In the Simcenter Testlab Time Signal Calculator, both rpm and angle data can be calculated from these time stamps or moments. Functions that have the word “MOMENTS” utilize the raw time stamps for their calculations (Figure 14).
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Figure 14: Time signal calculator functions with the word MOMENTS (highlighted) utilize the raw time stamps in their calculations.

When available, it is usually best to use the MOMENT based functions for calculating rpm or angle from an encoder.  An example with a 1000 pulse per revolution encoder is shown in Figure 15.
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Figure 15: Calculation of the final drive (1000 pulses per revolution) angular position from encoder recording.
Figure 16 shows the results of the encoder calculations for the final drive:
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Figure 16: Rotational speed (top, blue) and angular displacement (bottom, magenta) from encoder on final drive.

More information about using encoders in Simcenter Testlab can be found in the following articles:
With both the electric motor resolver and final drive encoder angular positions calculated, the angular difference is calculated in the next section.

5. Angular Difference Results

Formulas for subtracting the angular position of the electric motor and final drive are shown in Figure 17:
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Figure 17: Highlighted formulas calculate the difference between angular position of electric motor and final drive (adjusted by gear ratio).

The electric motor (which drives the rotating system) and final drive have a gear ratio of approximately nine (9x) as shown in Figure 18
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Figure 18: Top – Rpm of electric motor (black) and final drive (blue) shows a 9x ratio.  Bottom – After adjustment by the ratio, the angular position of the electric motor (green) and final drive encoder (magenta) overlay closely.
Knowing the exact ratio (in this case 9.04545) will yield the most accurate results. After adjusting by the ratio, the two angular position traces are subtracted as shown in Figure 19.
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Figure 19: Angular difference (red) between electric motor angular position (green) and final drive angular position (magenta).  Vibration measured near the drive unit (bottom, orange) has peak levels of vibration that corresponds to deviations in the angular difference.
After subtracting, the angle difference shows brief “glitch” or “judder”.  This corresponds to areas of higher vibration in a nearby accelerometer measurement. The exact cause of these “glitches” to be investigated and understood.

Hope this article is helpful!  Questions?  Email

Related Rotating Machinery Dynamics links:

KB Article ID# KB000048695_EN_US



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