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Root Mean Square (RMS) and Overall Level

Siemens Experimenter Siemens Experimenter
Siemens Experimenter

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There are several different ways that the term Root Mean Square (RMS) is used.

 

This article will go over the following:

  • RMS Amplitude Format: How the equivalent energy of a sine wave is expressed in a spectrum
  • RMS of a Spectrum: A representation of the overall energy in a spectrum – often called the overall level
  • Tracked RMS / Tracked Overall Level: Tracking the RMS of each average against a tracking parameter to see how the energy content changes
  • RMS in LMS Test.Lab

RMS Amplitude Format:

 

Amplitude formats describe how the amplitude of a sine wave is expressed in a spectrum.

The RMS amplitude format is valuable because it indicates the equivalent steady state energy value of an oscillating signal.

 

For example, a sine wave (like a pressure wave) oscillates above and below zero Pascal. The average value of the wave is zero. This is not a good representation of the effective energy of the wave. To get a better understanding of the equivalent steady state value, the RMS is used.

Figure 1: The RMS value is the equivalent steady state value of an oscillating signal. The red is the actual signal. The blue represents the perceived signal.Figure 1: The RMS value is the equivalent steady state value of an oscillating signal. The red is the actual signal. The blue represents the perceived signal.

The RMS amplitude format is calculated by squaring the peak amplitude (A) of the sine wave, diving it by two, and then taking the square root of that quantity. For a single sine wave, the RMS amplitude can be represented as 0.707*A.

Equation 1: Calculating RMS value of a single sine wave.Equation 1: Calculating RMS value of a single sine wave.

Look at the spectrum below. The curves are the exact same data but expressed with different amplitude formats. The black curve is expressed in peak format. The pink curve is expressed in RMS format. Notice that all amplitude levels of the pink curve are below the black curve (Figure 2).

 

In the zoomed in section, each spectral line is represented by a cross. When converting to RMS amplitude format, each spectral line is multiplied by ~0.707. For example, the spectral line at 694Hz has a peak amplitude of 1.000g. This corresponds to a RMS amplitude of 0.707g.

 Figure 2: The same spectral data can be displayed with different amplitude formats. In the zoomed area, each spectral line is represented by a cross. Notice that the peak spectral lines are multiplied by a fixed factor of ~0.707 to get the RMS values.Figure 2: The same spectral data can be displayed with different amplitude formats. In the zoomed area, each spectral line is represented by a cross. Notice that the peak spectral lines are multiplied by a fixed factor of ~0.707 to get the RMS values.

The RMS amplitude format is used to represent the equivalent steady state value of a sine wave at each spectral line.

 

RMS of a Spectrum:

 

Often, an RMS value of a spectrum is desired to be calculated. The RMS of a spectrum is a single number that represents the overall level of energy across a frequency range.

 

In the graphic below, the RMS of the spectrum is 54.08g.

Figure 3: The RMS vibration level of the spectrum is 54.08g.Figure 3: The RMS vibration level of the spectrum is 54.08g.

To calculate the RMS of a spectrum, the root sum square of all the spectral lines within the frequency range of interest must be calculated.

Equation 2: Equation for calculating the RMS of a spectrum.Equation 2: Equation for calculating the RMS of a spectrum.

The “A” term in the above equation represents the amplitude of a spectral line. It is important that this amplitude (A) value is in RMS format and has window correction factors taken into consideration (more about this below).

 

A_0 is the first spectral line under consideration, A_k is the last spectral line under consideration.

 

 Figure 4: A frequency spectrum in which the terms from Equation 2 are displayed. The frequency range of interest is f_1 to f_2.Figure 4: A frequency spectrum in which the terms from Equation 2 are displayed. The frequency range of interest is f_1 to f_2.

There are some important considerations that must be taken to ensure the calculated RMS value is correct:

 

1. The individual spectral line values (A) must be in Linear format.  If the units were squared or power units, they must be squared rooted before performing the calculation.  For example, an autopower of sound pressure with units of Pa2 must be square rooted to units of Pa.

 

2. Each individual spectral line must be in RMS format for the calculation. No matter what format is used for displaying the spectrum, RMS amplitude format is used when calculating the RMS of the spectrum. The RMS calculation produces the same result no matter how the spectrum is displayed.

 

Figure 5: The spectrum can be displayed in any format, but the software will change the format to RMS in the background for the calculation. The RMS calculation is the same no matter what amplitude format is used for visualization.Figure 5: The spectrum can be displayed in any format, but the software will change the format to RMS in the background for the calculation. The RMS calculation is the same no matter what amplitude format is used for visualization.

3. Window correction factors may need to be applied to the spectrum. If a window was used during acquisition, energy correction must be applied to the spectrum. The usage of windows to compensate for leakage can cause spectral lines to appear lower in amplitude / energy content than they really are. To get appropriate values for RMS spectral energy, energy correction must be applied.

 

Figure 6: Corrections must be applied so the amplitude of the windowed sine wave more closely matches the amplitude of the original sine wave. In the case of energy correction, spectral lines with a Hanning window are multiplied by 1.633. Spectral lines with a Flattop window are multiplied by 2.225.Figure 6: Corrections must be applied so the amplitude of the windowed sine wave more closely matches the amplitude of the original sine wave. In the case of energy correction, spectral lines with a Hanning window are multiplied by 1.633. Spectral lines with a Flattop window are multiplied by 2.225.

 

The RMS calculation has a pre-requisite of linear, RMS, energy corrected values.  

 

Figure 7: The RMS value calculated in LMS Test.Lab can be displayed in the curve legend.Figure 7: The RMS value calculated in LMS Test.Lab can be displayed in the curve legend.

The RMS value calculated in LMS Test.Lab will automatically convert data to energy, RMS, and linear values "behind the scenes", regardless of how it is displayed in the software interface.  This ensures consistent values for RMS calculations.

 

Users who want to manually calculate RMS must take these elements into consideration.

 

Tracked Overall Level:

 

The RMS value of a spectrum is often called the overall level. The overall level can be tracked versus speed or time to see how the amount of energy in the signal changes.

 

Figure 8: The overall level of a sound recording versus RPM.Figure 8: The overall level of a sound recording versus RPM.

When recording an event that is changing, it is often preferred to track the event on time or RPM. This means that for every increment of time or RPM a spectrum is calculated.

 

In the below example, a spectrum is calculated every 25 rpm between 3500 RPM and 4000 RPM. The overall level (RMS) for each of these spectrums is calculated.

Figure 9: A spectrum is calculated every 25 RPM. The RMS for each spectrum is also calculated.Figure 9: A spectrum is calculated every 25 RPM. The RMS for each spectrum is also calculated.

Then, these RMS values are plotted vs. RPM. This way, it is possible to see which RPM values have especially high (or higher than expected) energy content. 

Figure 10: The RMS and RPM values are plotted against one another. These values are the same as Table 1 in Figure 9. There is a data point every 25 RPM (represented by the crosses).Figure 10: The RMS and RPM values are plotted against one another. These values are the same as Table 1 in Figure 9. There is a data point every 25 RPM (represented by the crosses).

RMS in LMS Test.Lab:

 

This section will cover:

  • Changing Amplitude Format in LMS Test.Lab
  • Calculating the RMS of a frequency spectrum
  • Calculating the RMS over a specific frequency range
  • Calculating Tracked Overall Level in Test.Lab

Changing Amplitude Format in LMS Test.Lab:

 

To change the amplitude format in LMS Test.Lab, right lick on the y-axis of the plot, select “Processing” and then choose an amplitude format under “Spectrum & Section Scaling”.

 

Figure 11: It is possible to change the amplitude format in LMS Test.Lab.Figure 11: It is possible to change the amplitude format in LMS Test.Lab.

Calculating the RMS of a frequency spectrum:

 

To calculate the RMS of a spectrum in LMS Test.Lab, right click on the legend and choose “Options”.

Figure 12: Open the “Options…” window.Figure 12: Open the “Options…” window.

Next, calculate the RMS of the spectrum by selecting RMS in the “Calculated Content” tab, and the clicking the “Add to selection” arrow. Additionally, the Unit Label can be turned on in the lower right.

 

Figure 13: Add RMS to the Calculated Content.Figure 13: Add RMS to the Calculated Content.

The RMS value of the entire spectrum will now be displayed in the legend. Amplitude format and energy corrections are applied in the background as needed.

 

Figure 14: The RMS will be displayed in the plot legend.Figure 14: The RMS will be displayed in the plot legend.

Calculating the RMS over a specific frequency range:

 

Alternatively, the RMS value over a smaller frequency range can also be calculated.

 

Add a double cursor to the plot and position the cursor over a range of values.

 

Figure 15: Access the cursor menu by right clicking on the whitespace of the plot.Figure 15: Access the cursor menu by right clicking on the whitespace of the plot.

Then, right click on the cursor, and select Calculations -> RMS.

Figure 16: Right click on the cursor to access the calculations menu.Figure 16: Right click on the cursor to access the calculations menu.

The RMS value of the content between the cursors will be calculated and displayed in the cursor legend.

Figure 17: The RMS value is displayed in the cursor legend.Figure 17: The RMS value is displayed in the cursor legend.

Calculating Tracked Overall Level in Test.Lab:

 

To calculate a tracked overall level in LMS Test.Lab, ensure the measurement mode is set to “Tracked”.

 

Figure 18: Set the “Measurement mode” to “Tracked”.Figure 18: Set the “Measurement mode” to “Tracked”.

In the Section Settings dialog, make sure that “Overall level” is checked on under the “Overall Level” tab.

 

Figure 19: Check on the “Overall level” box.Figure 19: Check on the “Overall level” box.

The data will be saved in the Overall Level folder within the Sections folder.

 

Figure 20: The overall level will be saved into the sections folder and then the overall level folder.Figure 20: The overall level will be saved into the sections folder and then the overall level folder.

Conclusion:

It is important to understand the difference between RMS amplitude format, the RMS of a spectrum, and the tracked RMS.

 

RMS amplitude format represents the equivalent steady state value of a sine wave in a spectrum.

 

The RMS value of an entire spectrum represents the overall energy level. This value can be tracked to better understand how it changes.

 

Questions? Contact us.

 

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