ME 3901 Engineering Experimentation

Mechanical Engineering Department

Worcester Polytechnic Institute

Error Analysis

· Uncertainty Analysis for Density

· Hoop Stress Error Analysis

ME3901 is composed of several experiments

Need to learn:

How to measure

What tools/meters to use

The consequences of measurement errors and uncertainty

Data can occur as:

Single measured data – Some uncertainties (meter dependent) may not be discovered by repetition

Ex: Measure fluid T with a thermometer many times. These repeated measurements address the uncertainty in your ability to read the thermometer. But no information is available on the meter itself.

Multi-sampled data – Measure T with several meters/devices

Thermometers

Thermopiles

Thermocouples T/Cs

Thermistors

Three types of error that can cause uncertainty in experimental measurements.

1.) Gross Blunders: Construction, Application, Staffing flaws

2.) Fixed Errors: Systematic errors, Bias errors – repeated readings to be in error by approximately the same amount, but for some unknown reason(s). Frequently, the systematic error is quantified as the (Average of the measured readings - True Value). (Unfortunately, the True Value if frequently unknown).

3.) Random Errors: Fluctuations – electronic, personnel, random phenomena frequently statistically distributed; at times difficult to separate from fixed errors. Frequently, the random error is quantified as the (Individual measured reading - Average of all measured readings).

If a measurement has small systematic errors it is Accurate.

If a measurement has small random errors it is Precise.

Frequently a systematic error can be mitigated and/or estimated, i.e. calibrate to remove systematic error

How to report or ‘analyze’ errors?

Common Sense

(variable – who’s common sense!)

Largest error

Worse case error (Sum of all errors)

It is equally inaccurate to over-estimate as it is to under-estimate uncertainty.

A Least Squares Error Analysis

- Most robust

- Most widely accepted

Consider

P = 100 kPa

P = 100 kPa +/- 1 kPa

P = 100 kPa +/- 1 kPa (20 to 1)

(This latter statement requires total lab experience to put odds on uncertainty)

Most Results (R) are derived from multiple measurements, each of which has uncertainty

R = R(x₁, x₂, x₃, … x_N)

where x_i are Independent variables

Want the uncertainty of the result w_r

In terms of the uncertainty of the independent measurements, w_i

That is,

A Least Squares Uncertainty Estimate

Consider the problem – measure the density, r, of a fluid.

Are these measurements really independent?

Balance scale f(ruler)

But l, d, and h ARE dependent

What is the uncertainty in V?

Let the ruler be ‘shorter’ than actual

then the readings of l, d, and h will all be too large by approximately the same amount

i.e. a Systematic Error exists

( Much Larger Term!)

Independent Errors:

Sum ( the squares of the error contributions)

Dependent Errors:

Square (the sum of the errors) ß Much Larger

Dependent/Independent Error Conclusions

Dependent Errors cause greater total errors compared to Independent Errors.

Thus, if one had 3 measuring devices (different brand rulers) with similar uncertainties it would be significantly better to measure the l, d, and h with independent measuring devices!