Glossary entry (derived from question below)
English term or phrase:
weighted mean root square
English answer:
weighted root mean square
Added to glossary by
Ken Cox
Dec 18, 2004 16:44
19 yrs ago
5 viewers *
English term
United States
Translation by an Engineer & MBA!
Native in: Indonesian 
Works in: English to Indonesian, Japanese to Indonesian, German to Indonesian, and 3 more.
weighted mean root square
English
Tech/Engineering
Physics
physics or mathematical equation
The whole text:
Vibration level: the weighted mean root square acceleration to which the hands are subjected is less than the 2.5 m/s2
Vibration level: the weighted mean root square acceleration to which the hands are subjected is less than the 2.5 m/s2
Responses
| 4 +3 | weighted root mean square |
Ken Cox
|
| 4 | COMMENT (not for grading) |
Tony M
|
| 4 | vide infra |
Richard Benham
|
Responses
+3
3 hrs
Selected
weighted root mean square
With all due respect for Richard's erudition (and that is intended without the least trace of irony), I strongly suspect that 'mean root square' is an 'alternative' name for 'root mean square'. The Google hits for "mean root square" are dominated by non-native-English sites and sites with contents originating from non-native-English sources (with a fairly high concentration of contents originating from Russia and other Slavic countries), and it's not hard to find sites where the terms are used as synonums.
As 'root mean square' (often abbreviated 'rms') is a well-established term and concept in many fields (electronics and statistics in particular), it is highly likely that that is what is meant here. In theory, there could be such a thing as 'mean root square', but as Richard points out, that would be the same as the mean absolute value, which is also an established term, and which one would thus expect to be used by prefererence.
A weighted rms value is a value that is computed using a weighting function, which is a general name for any function that 'shapes' the computation. There are many different types of weighting functions.
In this case, it may be that the acceration was measured using an electronic transducer (accelerometer) and an ac voltmeter with rms capability (quite commonly available nowadays) and a (selectable) weighting function (which in practice effectively amounts to a filter that causes the level of the measured signal to depend on the frequency of the signal, in a defined manner). In statistics, rms values are used because they measure only the magnitude of the measured quantity and attach more weight to large magnitudes than small magnitudes, while in electronics, rms measurements are used because they accurately represent the effective power of the measured signal.
From the Wikipedia entry for 'root mean square' (equations omitted):
In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values.
...
The RMS value of a function is often used in physics and electronics. For example, we may wish to calculate the power P dissipated by an electrical conductor of resistance R. It is easy to do the calculation when a constant current I flows through the conductor.
...
But what if the current is a varying function I(t)? This is where the rms value comes in. It may be shown that the rms value of I(t) can be substituted for the constant current I in the above equation to give the mean power dissipation, thus
...
In the common case of alternating current, when I(t) is a sinusoidal current, as is approximately true for mains power, the rms value is easy to calculate from equation (2) above. The result is:
...
The RMS value can be calculated using equation (2) for any waveform, for example an audio or radio signal. This allows us to calculate the mean power delivered into a specified load. For this reason, listed voltages for power outlets (eg. 110V or 240V) are almost always quoted in RMS values, and not peak-to-peak values.
http://en.wikipedia.org/wiki/Root_mean_square
As 'root mean square' (often abbreviated 'rms') is a well-established term and concept in many fields (electronics and statistics in particular), it is highly likely that that is what is meant here. In theory, there could be such a thing as 'mean root square', but as Richard points out, that would be the same as the mean absolute value, which is also an established term, and which one would thus expect to be used by prefererence.
A weighted rms value is a value that is computed using a weighting function, which is a general name for any function that 'shapes' the computation. There are many different types of weighting functions.
In this case, it may be that the acceration was measured using an electronic transducer (accelerometer) and an ac voltmeter with rms capability (quite commonly available nowadays) and a (selectable) weighting function (which in practice effectively amounts to a filter that causes the level of the measured signal to depend on the frequency of the signal, in a defined manner). In statistics, rms values are used because they measure only the magnitude of the measured quantity and attach more weight to large magnitudes than small magnitudes, while in electronics, rms measurements are used because they accurately represent the effective power of the measured signal.
From the Wikipedia entry for 'root mean square' (equations omitted):
In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values.
...
The RMS value of a function is often used in physics and electronics. For example, we may wish to calculate the power P dissipated by an electrical conductor of resistance R. It is easy to do the calculation when a constant current I flows through the conductor.
...
But what if the current is a varying function I(t)? This is where the rms value comes in. It may be shown that the rms value of I(t) can be substituted for the constant current I in the above equation to give the mean power dissipation, thus
...
In the common case of alternating current, when I(t) is a sinusoidal current, as is approximately true for mains power, the rms value is easy to calculate from equation (2) above. The result is:
...
The RMS value can be calculated using equation (2) for any waveform, for example an audio or radio signal. This allows us to calculate the mean power delivered into a specified load. For this reason, listed voltages for power outlets (eg. 110V or 240V) are almost always quoted in RMS values, and not peak-to-peak values.
http://en.wikipedia.org/wiki/Root_mean_square
Peer comment(s):
| neutral |
Richard Benham
: [...]//You haven't actually ever specified the mathematical relationships here. Do you mean a weighted mean of RMS values, the square root of a weighted mean of squared RMS values, the square root of a weighted mean of squared values, or what?.
14 mins
|
|
Apology accepted (but still, why invent a new term when a commonly used term already exists?).
|
|
| agree |
Alaa Zeineldine
: agree and I think the author just shuffled the words in error.
54 mins
|
| agree |
Tony M
: Well done, Kenneth --- you have summed up very elegantly what I had laboriously typed into my own much-awaited (!) answer, until the moment when it was whisked off into the aether with one brush of the 'Submit' button...
1 hr
|
| agree |
airmailrpl
: - Dusty was had by the hyperspace gremlins
12 hrs
|
4 KudoZ points awarded for this answer.
Comment: "Thank you"
5 hrs
COMMENT (not for grading)
I think there is little I can add to Kenneth's excellently-argued answer, so I'll just use this opportunity to explain what I mean for RB's benefit.
Yes, indeed, there IS another place you can do 'weighting' --- and it is very common in acoustic, audio and vibration work.
Imagine the situation where you first calculate rms values for a number of different freqeuncies; THEN you multiply each of these by a 'weighting factor' according to (for example) their relative importance, before adding them all together (or otherwise combining...) to arrive at a single overall figure that can be used as a (more or lesss..) meaningful global performance indicator.
This sort of weighting procedure is used, for example, in audio or acoustic noise measurements, to create 'realistic' noise figures that more closely reflect true 'nuisance values' --- so we very commonly get 'DIN A weighted' and 'B weighted'
So I honestly believe that, if we accept the quirky word order for 'mean root square' as = 'root mean square', there is no need to look any further to 'correct' the word order for it to fit in a vibration analysis context.
--------------------------------------------------
Note added at 14 hrs 44 mins (2004-12-19 07:28:32 GMT)
--------------------------------------------------
Well, as I said in my peer response, RB of course has far greater mathematical knowledge than I, so LISTEN TO HIM!
All I know is that this appied maths technique of weighting rms values IS very commonly used, rightly or wrongly, in my own specialist field of audio, electronics etc. My \'otherwise combining...\' was just a catch-all to cover myself from some clever-clogs coming along and telling me I\'d left something out.
In audio (acoustics, mechanics, vibration, etc.) what we often do is calculate a series of rms values --- say at different frequencies --- and then each value is multiplied by a \'weighting factor\' that reflects its relative importance in the final outcome. Then an arithmetic average (usually, I believe) is taken of these weighted values to give, as I said before, one overall \'average rms figure\' deemed to be more or less representative of the unit\'s specification.
So it seems to me that taking the rms of a number of varying functions, and then weighting these values and taking a simple average of these, would no way produce the same result as performing the calculation in a different order. But even if it did, the fact of the matter is that in our industry, maybe just for practical reasons, that is the way round it is done, to arrive at a figure that is well known and familiar to all of us by the name in Asker\'s context (well, more or less, anyway!)
So expert maths aside, Richard, I promise you this is a standard term that is frequently encountered in our industry, and I hope all this discussion has helped answer Asker\'s question, at least for their intended purpose.
Yes, indeed, there IS another place you can do 'weighting' --- and it is very common in acoustic, audio and vibration work.
Imagine the situation where you first calculate rms values for a number of different freqeuncies; THEN you multiply each of these by a 'weighting factor' according to (for example) their relative importance, before adding them all together (or otherwise combining...) to arrive at a single overall figure that can be used as a (more or lesss..) meaningful global performance indicator.
This sort of weighting procedure is used, for example, in audio or acoustic noise measurements, to create 'realistic' noise figures that more closely reflect true 'nuisance values' --- so we very commonly get 'DIN A weighted' and 'B weighted'
So I honestly believe that, if we accept the quirky word order for 'mean root square' as = 'root mean square', there is no need to look any further to 'correct' the word order for it to fit in a vibration analysis context.
--------------------------------------------------
Note added at 14 hrs 44 mins (2004-12-19 07:28:32 GMT)
--------------------------------------------------
Well, as I said in my peer response, RB of course has far greater mathematical knowledge than I, so LISTEN TO HIM!
All I know is that this appied maths technique of weighting rms values IS very commonly used, rightly or wrongly, in my own specialist field of audio, electronics etc. My \'otherwise combining...\' was just a catch-all to cover myself from some clever-clogs coming along and telling me I\'d left something out.
In audio (acoustics, mechanics, vibration, etc.) what we often do is calculate a series of rms values --- say at different frequencies --- and then each value is multiplied by a \'weighting factor\' that reflects its relative importance in the final outcome. Then an arithmetic average (usually, I believe) is taken of these weighted values to give, as I said before, one overall \'average rms figure\' deemed to be more or less representative of the unit\'s specification.
So it seems to me that taking the rms of a number of varying functions, and then weighting these values and taking a simple average of these, would no way produce the same result as performing the calculation in a different order. But even if it did, the fact of the matter is that in our industry, maybe just for practical reasons, that is the way round it is done, to arrive at a figure that is well known and familiar to all of us by the name in Asker\'s context (well, more or less, anyway!)
So expert maths aside, Richard, I promise you this is a standard term that is frequently encountered in our industry, and I hope all this discussion has helped answer Asker\'s question, at least for their intended purpose.
Peer comment(s):
| neutral |
Richard Benham
: [...].//It is the "otherwise combining" [...]// Ote: The difference in the two values in my contrived example was 6%--which is next to nothing in the present context, I was writing as a pure mathematician, not an applied m. or engineer.....
5 hrs
|
|
I have to bow to your superior mathematical knowledge, Richard (I stopped at A-level!) Please see my added comment... /// You're too quick for me! It's there now!
|
14 hrs
Richard Benham
France
France
Native in English
&
English
Works in field
France
When quality counts
Native in: English
Works in: German to English, French to English, English
Related KudoZ points
:
4
vide infra
While I am weighting (sorry) for various people to answer my questions, I shall make the following observations.
Kenneth tells that "A weighted rms value is a value that is computed using a weighting function". He does not favour us with his views as to exactly what the weighting function is applied to. Similarly, Dusty talks about "adding them [the weighted values] all together (or otherwise combining...)", without any amplification as to what this "otherwise combining" might be.
Let me make something clear. The most meaningful way of combining RMS values is to square them, apply the weighting function, and then take the square root of the result. Equivalently, you could apply a different weighting function (the coefficients being the square roots of the ones in the previous function). sqaure the weighted values, add them (or take the mean, or whatever) and then take the square root of the result.
Why is this more meaningful than taking the arithmetic mean of RMS values? Consider the following (trivial) example. Suppose you have n data points with a value of 2 and 2n data points with a value of 1. The mean-square value for these points is 2 times 1 squared plus 2 squared, all divided by 3, which comes to 2. So the RMS value is root 2, or around 1.41.
Now suppose you are just told that there is a sample of size n with an RMS value of 2 and a sample of size 2n with an RMS of 1. Can you calculate the RMS value for the combined population? Yes, you can. However, you DON'T do so by taking the (weighted)arithmetic mean of the two samples. If you did this, you would get 4/3 or approx. 1.33, which is slightly WRONG. If, however, you SQUARE the values, take the weighted arithmetic mean, and then take the SQUARE ROOT, you get the correct value of root 2 or 1.41. Note also that if the data had been grouped differently, so that one sample had n values of 1 and n of 2, while the other sample had n values of 1, applying the weighted arithmetic mean of the RMS values would give 1.39, which is different again from the other two values.
The moral of the story is that, if you want to combine RMS values in a way that is not subject to grouping artifacts, the way to do it is to square first, weight, do your linear operations (mean, adding up,...) and then take the square root.
The moral of this moral is that doing this is STRICTLY MATHEMATICALLY EQUIVALENT to applying a corresponding weighting function to the individual data points, which is how my previous (now hidden) answer explained it.
--------------------------------------------------
Note added at 19 hrs 14 mins (2004-12-19 11:58:38 GMT)
--------------------------------------------------
Hello. I am not sure whether the asker wanted a better term, or an explanation. I was trying to provide an explanation.
Now that the Dust has settled, I can sum things up as follows. Taking the weighted mean of the squares and taking the square root is the same as taking some RMS values and then squaring them, taking a weighted sum, and taking the square root (the weights will be different, of course). Both procedures are perfectly legitimate, both in theory and in practice.
Taking a bunch of RMS values and producing a weighted arithmetic mean is subject to grouping artifacts, and so therefore not entirely legitimate in theory. However, even in my example, which was contrived to illustrate the inaccuracy that could result, the error was only around 6%. If an acceleration of 2.5 m/s2 is considered safe, then 6% more than that, 2.65 m/s2, is hardly likely to break your wrist. So there is probably no harm in using the weighted mean of a bunch of RMS values as a rough-and-ready measure of the performance of, say, a pair of vibration-absorbing gloves.
As to the terminology, \"weighted RMS\" seems to be what it\'s called.
I am grateful to Dusty for pointing out the real-world engineering practice.
Kenneth tells that "A weighted rms value is a value that is computed using a weighting function". He does not favour us with his views as to exactly what the weighting function is applied to. Similarly, Dusty talks about "adding them [the weighted values] all together (or otherwise combining...)", without any amplification as to what this "otherwise combining" might be.
Let me make something clear. The most meaningful way of combining RMS values is to square them, apply the weighting function, and then take the square root of the result. Equivalently, you could apply a different weighting function (the coefficients being the square roots of the ones in the previous function). sqaure the weighted values, add them (or take the mean, or whatever) and then take the square root of the result.
Why is this more meaningful than taking the arithmetic mean of RMS values? Consider the following (trivial) example. Suppose you have n data points with a value of 2 and 2n data points with a value of 1. The mean-square value for these points is 2 times 1 squared plus 2 squared, all divided by 3, which comes to 2. So the RMS value is root 2, or around 1.41.
Now suppose you are just told that there is a sample of size n with an RMS value of 2 and a sample of size 2n with an RMS of 1. Can you calculate the RMS value for the combined population? Yes, you can. However, you DON'T do so by taking the (weighted)arithmetic mean of the two samples. If you did this, you would get 4/3 or approx. 1.33, which is slightly WRONG. If, however, you SQUARE the values, take the weighted arithmetic mean, and then take the SQUARE ROOT, you get the correct value of root 2 or 1.41. Note also that if the data had been grouped differently, so that one sample had n values of 1 and n of 2, while the other sample had n values of 1, applying the weighted arithmetic mean of the RMS values would give 1.39, which is different again from the other two values.
The moral of the story is that, if you want to combine RMS values in a way that is not subject to grouping artifacts, the way to do it is to square first, weight, do your linear operations (mean, adding up,...) and then take the square root.
The moral of this moral is that doing this is STRICTLY MATHEMATICALLY EQUIVALENT to applying a corresponding weighting function to the individual data points, which is how my previous (now hidden) answer explained it.
--------------------------------------------------
Note added at 19 hrs 14 mins (2004-12-19 11:58:38 GMT)
--------------------------------------------------
Hello. I am not sure whether the asker wanted a better term, or an explanation. I was trying to provide an explanation.
Now that the Dust has settled, I can sum things up as follows. Taking the weighted mean of the squares and taking the square root is the same as taking some RMS values and then squaring them, taking a weighted sum, and taking the square root (the weights will be different, of course). Both procedures are perfectly legitimate, both in theory and in practice.
Taking a bunch of RMS values and producing a weighted arithmetic mean is subject to grouping artifacts, and so therefore not entirely legitimate in theory. However, even in my example, which was contrived to illustrate the inaccuracy that could result, the error was only around 6%. If an acceleration of 2.5 m/s2 is considered safe, then 6% more than that, 2.65 m/s2, is hardly likely to break your wrist. So there is probably no harm in using the weighted mean of a bunch of RMS values as a rough-and-ready measure of the performance of, say, a pair of vibration-absorbing gloves.
As to the terminology, \"weighted RMS\" seems to be what it\'s called.
I am grateful to Dusty for pointing out the real-world engineering practice.
Discussion