Random Control: Averages per Loop, Frequency Resolution, Weighting, and Degrees of Freedom

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The objective of closed loop random vibration control testing is simple: recreate a broadband random vibration on a shaker table to find any durability related flaws in the product.

 

A typical vibration control system is shown in Figure 1. A Power Spectral Density function (PSD), determined to represent the real life vibration exposure, is used as the target vibration at the control accelerometer location.

 

vibco_system.pngFigure 1: Vibration control system monitors vibration levels on the control accelerometer and adjusts the output to the shaker to match a preset target.

The control accelerometer is positioned directly on the shaker table, where it is continuously measured and compared to a target reference control PSD spectrum. If there are any deviations from the target PSD, the drive output to the shaker is adjusted to reduce the error between the control and reference.

 

Seems like control should be easy, right? Not so fast! The control is being done on an averaged PSD of random data.  As a result, there is variation in the measured vibration, as shown in Figure 2.

 

PSD_with_Variation.pngFigure 2: During a random vibration test, the measured vibration (blue) has amplitude variation compared to the target (green). The alarms and abort limits (red and orange) are used to ensure the vibration levels are within acceptable levels of variation during the test.

In Figure 2, the average measured vibration (blue) does not perfectly match the target vibration (green) due to the variation in the average.

 

The amount of variation depends on how much averaging is done in the control loop. Given enough averaging time, the target and measured vibration at the control PSD can be identical.

 

However, using a long amount of averaging time means the control is less responsive to changes.  If the output to the shaker cannot be adjusted quickly enough, there might be difficulty matching the target vibration profile.

 

Key1_Random_Control.png

 

This article details the settings that affect the amplitude variation and control loop time in a vibration control test. 

 

The settings are described in the following sections:

  1. Inner and Outer Control Loop
  2. Inner Control Loop: Averages per Loop
  3. Frequency Resolution: Loop Time
  4. Outer Control Loop: Weighting
  5. Degrees of Freedom (DOFs)
  6. Tolerance Alarms and Aborts
  7. Reset Average at Level Setting

 

1. Inner and Outer Control Loop

 

The control logic used in a random control test of Simcenter Testlab is shown in Figure 3.  There are two loops: an inner and an outer control loop.

 

two_loops_random_control.pngFigure 3: The random control loop has an inner (light blue) and an outer loop (dark blue).

The control PSD is measured and compared against the target PSD (T).  The voltage drive (V) to the shaker is updated to minimize the difference between current control PSD and target according to Equation 1 below:

 

Equation1_Control_Loop.pngEquation 1: Random control loop equation where drive (D) is updated based on Inverse Transfer Function (Z) and target PSD (T).

Equation 1 consists of the following frequency based functions:

  • The Inverse Transfer Function (ITF or Z) based on drive spectrum (volts) divided by the measured control PSD (g's)
  • T is the target response PSD in g's of acceleration
  • D is the drive to the shaker in volts

 

For speed considerations, the inner control loop is executed continuously on the Simcenter SCADAS data acquisition hardware.  It constantly computes a linear average PSD on all the accelerometer channels, including the control accelerometer.

 

The outer control loop is performed on the host PC.  It does the following:

  1. Exponential Average – An exponential averaging is done on the control PSD, using the averaging results from inner loop.
  2. Abort Conditions – The averaged control PSD levels are checked to see if the test needs to be stopped based on exceeding the abort limits.
  3. Update Drive – The control signal from the SCADAS hardware to the shaker is updated. The update is based on an Inverse Transfer Function (ITF) calculated between the control accelerometer PSD (in g’s of vibration) and shaker drive signal (in volts).

 

Key2_Random_Control_Two_Loops.png

 

There are specific settings that govern the behavior of these control loops which are covered in the next sections.

 

2. Inner Control Loop: Averages Per Loop

 

In the inner control loop (Figure 4), an averaged Power Spectral Density (PSD) is calculated for each channel. The setting called Averages per Loop determines how many individual PSDs are acquired before an average is calculated.

 

inner_loop.pngFigure 4: The inner control loop (highlighted in light blue) of a random vibration control test.

The average PSD for each channel is calculated according to Equation 2:

 

Equation2_Inner_Loop_Average.pngEquation 2: Stable average of Power Spectral Densities (G) of the inner control loop are performed for each channel.

Where:

  • i is the average counter, ranges from 1 to M where M is the averages per loop
  • Gi-1 (with overbar) is the PSD from the previous average
  • Gi (with overbar) is the current average PSD
  • Gi is the current PSD measurement

The average being performed in Equation 1 is known as a stable average. A stable average yields the same result as a linear average:

  • In a linear average, the values are summed together and divided by the number of values after all measurements have been acquired. For example: (10+15+5)/3 = 10. 
  • A stable average continuously calculates the average after each measurement, so it does not need to be calculated at the end of the loop.

 

The averages per loop (M) are set under the “Advanced…” button of Random Control setup as shown in Figure 5.

 

averages_per_loop.pngFigure 5: Averages per loop parameter is set under the “Advanced…” button in Random Control setup.

By default, the averages per loop is 5 for the inner loop.  The averages per loop can be increased to reduce the amount of variation in the target PSD as shown in Figure 6. 

 

averages_per_loop_comparison.pngFigure 6: As the averages per loop is increased, with all other settings the same, the variation of the measured control Power Spectral Density (PSD) is reduced.

However, increasing the averages also increases the control loop time, making the test less responsive to changes and shifts in the test structure and shaker system that occur during the test.

 

Key3_Averages_per_Loop.png

 

This inner loop is executed in the memory of the SCADAS hardware as to make it as fast as possible.  The PSD calculated by the inner loop is not displayed in the software interface.

 

3. Loop Time: Frequency Resolution

 

The control loop time is equal to the averages per loop (M) multiplied by the acquisition time (T) for each average as shown in Equation 3. The acquisition time for each average is the inverse of the frequency resolution used in calculating the PSD.  It is the minimal amount of time needed, as there is overhead for other operations, etc.

 

Equation3_Control_Loop_Time.pngEquation 3: Control loop time is the averages per loop (M) multiplied by the acquisition time (T) for each average. The acquisition time (T) is inverse of the frequency resolution (delta f). There is also a small amount of fixed overhead time that needs to be added.

The frequency resolution is the spacing between data points in the frequency domain.  For example, the frequency resolution might be 2 Hz, meaning there is data points in the PSD spectrum at 0 Hz, 2 Hz, 4 Hz, 6 Hz, etc.

 

For more information on the inverse relationship between frequency resolution and acquisition time, see the Knowledge Base Article “Digital Signal Processing: Spectral Lines, Frequency Resolutions, etc.”.

 

The frequency resolution in Simcenter Testlab Random Control is set in upper right of the Setup worksheet as shown in Figure 7

 

frequency_resolution.pngFigure 7: Frequency Resolution is set in the upper right corner of the Random Setup worksheet.

The software will show all PSD measurements with the resolution entered in the control panel.  However, a finer frequency resolution might be employed in the inner control loop. To keep the control loop speed as short as possible, it is done within the firmware of the Simcenter SCADAS hardware, not in the software of the host PC.  The firmware has a fixed number of frequency resolutions that can be used.  The control loop time is based on the SCADAS front end hardware, regardless of the resolution specified in the software interface.

 

The actual hardware-based frequency resolution can be viewed by turning on the “Use FE resolution” option under the “Advanced…” button (Figure 8). The initials FE are short for Front End, which refers to the SCADAS hardware. 

 

Use_FE_Resolution.pngFigure 8: Click on the “Use FE Resolution” checkbox under the “Advanced…” button to see the hardware-based frequency resolution used during the inner loop PSD measurements.

After turning on the “Use FE resolution” checkbox, the resolutions shown in the software interface change to the frequency resolution that is actually used for the measurements in the hardware.  

 

The control loop time is shown in the field “Min Control Loop Time” under the “Advanced…” button as shown in Figure 9.

 

Control_Loop_Time.pngFigure 9: Based on the Averages per Loop and Frequency resolution, the “Min control loop time” is shown in the Advanced Control Setup menu.

The minimum control loop time field is a calculated (based on averages per loop and frequency resolution) and is not directly settable in the user interface.

 

Key4_Control_Loop_Time.png

 

How quickly the control PSD measurement is updated is determined by the control loop. The speed at which the output drive is corrected is based in part on the weighting factor, which is covered in the next section.

 

4. Outer Control Loop: Weighting

 

The average Power Spectral Density (PSD) calculated in the inner control loop is transferred to the outer control loop after all the averages are taken (see Figure 10).

 

outer_loop.pngFigure 10: Outer Control Loop (highlighted in blue) monitors for abort situations, updates the drive, and sends out the signal drive.

In the outer control loop, the inner control loop PSD is exponentially averaged and monitored for potential abort situations.  This is the actual control PSD which is shown in the software interface.

 

The drive to the shaker is also evaluated and updated based on this PSD. This is done by calculating an Inverse Transfer Function (ITF) between the drive and control accelerometer.  The ITF is a Frequency Response Function (FRF) in units of volts (from the drive output) per g (acceleration at the target).  The drive is altered based on the measured ITF, which tries to reduce the error between the outer loop control measurement and the target spectrum.

 

The control PSD is averaged as shown in Equation 4

 

Equation4_Outer_Loop_Average.pngEquation 4: The weighting factor (W) in the outer control loop equation determines the relative importance of the latest acquisition versus the previous average via exponential averaging.

Where:

  • j is the outer control loop count, which increases until the time duration of the test is reached
  • Gj+1 (with overbar) is the current average control PSD
  • Gj (with overbar) is the previous outer loop average control PSD
  • Gj+1 is the inner loop average control PSD
  • W is the exponential weighting factor

 

The weighting factor is set in the “Advanced Control Setup” menu as shown in Figure 11.

 

Weighting_Factor_menu.pngFigure 11: The weighting factor (W) is set in the Exponential Weighting field of the Advanced Control setup menu. 

The weighting factor (W) determines how much the previous averages influence the current average:

  • Low Weighting Factor: The previous averages have little influence on the current average. The most recent measurement will have a high amount of influence on the average. The lowest weighting factor that can be set is 1.
  • High Weighting Factor: The previous averages have high influence on the current average. The most recent measurement will have a low amount of influence on the average.

For a random control test, this means that a lower weighting factor shows more variation in the control PSD as shown in Figure 12.

weighting_factor_comparison.pngFigure 12: The control PSD has higher variation with lower weighting factors, all other test settings the same.

The weighting factor (W) influences how quickly the outer loop average PSD varies with a sudden change in system behavior.

 

For example, if the control accelerometer was disabled during the middle of a test, the average control PSD will not change as much when computed with a high weighting factor as shown in Figure 13.

weighting_factor_influence_on_PSD_bad_accel.pngFigure 13: Control accelerometer was disabled mid test. Control PSD (top, blue) computed with a low weighting factor deviates from the average more than the control PSD (bottom, brown) computed with a high weighting factor.

In both cases, the tests aborted in the same amount of time (there is a separate open threshold check also active on all the channels).  However, the control PSDs (blue versus brown curves in Figure 13) are quite different.

 

With a weighting factor of 90 (brown curve), the control PSD is close to the average. With a weighting factor of 5 (blue curve), the control PSD deviates more from the average.  The PSD with the lower weighting factor changed more rapidly in response to the accelerometer than the PSD computer with the higher weighting.

 

Key5_Weighting.png

 

High weighting helps reduce the amount of variation in the control PSD.  The total amount of expected variation based on the test settings is reflected in the Degrees of Freedom parameter, which is discussed in the next section.

 

5. Degrees of Freedom (DOFs)

 

The Degrees of Freedom (DOFs) is a function of the averages per loop and the weighting factor as shown in Equation 5:

 

Equation5_DOFs.pngEquation 5: Equation for Degrees of Freedom (DOF) as a function of averages per loop (M) and the weighting factor (W).

The Degrees of Freedom is used in random control test specifications as a check on the expected variation of the spectral data.  The higher the Degrees of Freedom, the lower the expected variation around the average control PSD.

 

A random control test uses a gaussian random when generating the drive signals. The table below (Figure 14) shows the relationship between the number of degrees of freedom and the confidence levels of a signal with a gaussian random distribution.

 

DOFs_versus_Confidence_Level.pngFigure 14: Table of Degrees of Freedom (left column) versus Confidence Level (top).

Using the table, it can be determined if test tolerances can be met in theory, before actual testing occurs. From statistics, a confidence level can be calculated on the gaussian signal distribution.  For 126 Degrees of Freedom, 99% of the averaged amplitude values in the spectrum will lie between +1.32 dB and -1.52 dB of the target. 

 

For more about dB values and how they work, see the Knowledge Base article “What is a Decibel?”.

 

After setting the weighting and averages per loop, the Degrees of Freedom are calculated and displayed in the Simcenter Testlab Random Setup worksheet (Figure 15).

 

DOF_Menu.pngFigure 15: In the Random Setup worksheet, the Degrees of Freedom (DOF) is displayed, which is based on the weighting (W) and averages per loop (M).

In random profile specifications, a minimal number of Degree of Freedoms is often specified.  For example, in MIL-STD 810, the Environmental Engineering Guidelines of the Department of Defense, a minimum of 120 Degrees of Freedom (Figure 16) are recommended for a random vibration control test.

 

MIL_STD.pngFigure 16: Minimum of 120 Degrees of Freedom recommended in MIL-STD 810 for Random Control testing.

This predicts that the 99% of the amplitude signal values will lie about +/- 1.5 dB from the average amplitude. 

 

Key6_DOFs.png

 

If the predicted variation is less than the alarm and abort limits, the test should be able to run without issues.  The actual variation is monitored during the test as described in the next section.

 

6. Tolerance Alarms and Aborts

 

Variation within the control PSD is monitored two different ways while the test is in progress (Figure 17): tolerance alarms and aborts.

 

alarms_and_aborts.pngFigure 17: The tolerance alarms (dotted orange) of +/- 3 dB around the vibration target are monitored during the test. If the tolerance alarms are exceeded, the test measurement will not proceed until the variation is reduced. If the abort limits (red) are reached (+/- 6 dB), the test will be stopped.

The monitoring works as follows:

  • Tolerance Alarms - Tolerance alarms will prevent the test measurement from proceeding to the next level in the test schedule until the variation is within the alarm tolerance. The default setting is for the alarm is +/- 3 dB around the target vibration. 
  • Abort – The test will be halted or stopped completely if the variation exceeds the abort limits. Default setting is +/- 6 dB around the target vibration.

 

In practice, a small number of frequency lines that exceed the alarm and abort limits can be tolerated.  In the Random setup worksheet, on the right side, the maximum alarm and abort lines that are allowed outside of the limits is specified (Figure 18).

 

 

This allows a few lines in the control spectrum to be outside the limits when proceeding to a new test level. The number of allowed lines is usually defined in the test specification.

 

Max_alarms_aborts.pngFigure 18: With these settings in the Random Setup worksheet, a maximum of ten frequency lines can be be outside of the alarm limits before the measurement proceeds. A maximum of five frequency lines can be outside of the abort limits before the test is halted.

When switching test levels, the average on the control PSD can be also be reset. This is discussed in the next section.

 

7. Reset Average at Level Switch

 

In the runup to the full level of a random vibration control test, there are several intermediate levels.  The average can be reset every time there is a change in level by changing “Reset exp. avg. on level switch” check box to ON in the Advanced Control setup dialog (Figure 19).

 

reset_average_menu.pngFigure 19: Under the “Advanced button…” of Random Setup worksheet, the “Reset exp. avg. on level switch” can be turned ON.

When changing levels, it is quite possible that the input/output relationship between the drive (voltage to shaker) and the output (control accelerometer response) changes.  If the test object and shaker system are linear, this relationship will remain fixed.  If the test object is non-linear, then the relationship can vary widely.

 

What exactly constitutes non-linear behavior between level changes?  Let’s say for a given frequency, that one volt of drive produces two g’s of vibration.  The relationship is 2 g per volt, or 2 g/V.  When the drive level is increased from one volt to two volts, if the structure and shaker system are linear, then four g’s of vibration should be produced.  Double the drive voltage, double the acceleration response.  If the system is non-linear, then perhaps five g’s of vibration are produced when the drive level is set to two volts.  The relationship is non-linear and harder to predict.

 

By starting the average “fresh” with a change in level, non-linear behavior can be detected and accounted for in the drive output in a timelier fashion.

 

Questions?  Email john.hiatt@siemens.com or contact Siemens PLM GTAC support.

 

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