1. Basic descriptions of physical data
Random Data Classification
Random data cannot be described by an explicit mathematical expression
8/1/2019
ME 508 Experimental Methods in mechanical Engineering

2. Basic descriptions of physical data
Random Data Classification
A single time-history representing a random phenomena is called a sample function (or a sample record if observed over a finite period of time).
The collection of all possible sample functions that the random phenomena can produce is called a random process or a stochastic process.
Random processes can be categorized into two main types
Stationary Random Process
Ergodic
Non-ergodic
Non-stationary Random Process
8/1/2019
ME 508 Experimental Methods in mechanical Engineering

3. Basic descriptions of physical data
Random Data Classification
Random Data
Stationary
Non-stationary
Ergodic
Non-ergodic
Special Classifications
8/1/2019
ME 508 Experimental Methods in mechanical Engineering

4. Basic descriptions of physical data
𝑡 1
𝑡 1 +𝜏
Stationary Random Processes
Consider a random physical phenomena, we can characterize the phenomena at any instant of time by averaging the instantaneous value over the collection of sample functions which describe the random process.
For the collection of sample functions 𝒙 𝒊 (𝒕), the mean value of the random process at a time instant 𝒕 𝟏 can be evaluated by adding the values 𝒙 𝟏 𝒕 𝟏 , 𝒙 𝟐 𝒕 𝟏 , …, 𝒙 𝑵 𝒕 𝟏 and divide by N.
We can also evaluate the correlation between the values of the random process at two different times (autocorrelation) by summing the product of the values of the sample function at two different times 𝒕 𝟏 , 𝒕 𝟏 +𝝉.
Mean:
𝝁 𝒙 𝒕 𝟏 = lim 𝑵→∞ 𝟏 𝑵 𝒌=𝟏 𝑵 𝒙 𝒌 𝒕 𝟏
Autocorrelation:
𝑹 𝒙 𝒕 𝟏 , 𝒕 𝟏 +𝝉 = lim 𝑵→∞ 𝟏 𝑵 𝒌=𝟏 𝑵 𝒙 𝒌 𝒕 𝟏 𝒙 𝒌 𝒕 𝟏 +𝝉
If 𝜇 𝑥 , 𝑅 𝑥 changes if 𝑡 1 is changed hence the process is non-stationary.
If 𝜇 𝑥 doesn’t change if 𝑡 1 is changed and 𝑅 𝑥 depends on 𝜏 only then it is weakly stationary.
If all moments (all possible correlations) don’t change if 𝑡 1 is changed then it is strongly stationary.
8/1/2019
ME 508 Experimental Methods in mechanical Engineering

5. Basic descriptions of physical data
𝑡 1
𝑡 1 +𝜏
Ergodic Random Processes
A stationary process is said to be ergodic if the time average of a sample function is equal to the ensemble average.
The time average is given by,
𝜇 𝑥 𝑘 = lim 𝑇→∞ 1 𝑇 0 𝑇 𝑥 𝑘 𝑡 𝑑𝑡
𝑅 𝑥 𝜏,𝑘 = lim 𝑇→∞ 1 𝑇 0 𝑇 𝑥 𝑘 𝑡 𝑥 𝑘 𝑡+𝜏 𝑑𝑡
For ergodic process the time averaged mean values are equal to the corresponding ensemble averaged values.
𝜇 𝑥 𝑘 = 𝜇 𝑥
𝑅 𝑥 𝜏,𝑘 = 𝑅 𝑥 𝜏
8/1/2019
ME 508 Experimental Methods in mechanical Engineering

6. Basic descriptions of physical data
Stationary Sample Records
Consider a sample record 𝑥 𝑘 (𝑡) obtained from the 𝑘𝑡ℎ sample function of a random process {𝑥(𝑡)}. We can evaluate the mean and autocorrelation function over a period T starting at time of 𝑡 1 as follows,
𝜇 𝑥 𝑡 1 ,𝑘 = 1 𝑇 𝑡 1 𝑡 1 +𝑇 𝑥 𝑘 𝑡 𝑑𝑡
𝑅 𝑥 𝑡 1 , 𝑡 1 +𝜏,𝑘 = 1 𝑇 𝑡 1 𝑡 1 +𝑇 𝑥 𝑘 𝑡 𝑥 𝑘 𝑡+𝜏 𝑑𝑡
If these average values vary with 𝑡 1 then the sample record is called non-stationary. Else if the time-averaged values don’t change significantly with 𝑡 1 Then the sample record is called stationary. A sample record obtained from an ergodic random process will be stationary.
8/1/2019
ME 508 Experimental Methods in mechanical Engineering

7. Basic descriptions of physical data
Basic Descriptive Properties of Random Data
Mean Square Value
Gives information on the intensity of the data
Probability Density Function
Furnishes data on the data amplitude
Autocorrelation Functions
Furnishes information about the data in time domain
Power Spectral Density Functions
Furnishes information about the data in frequency domain
This gives elementary information about the intensity of the data. It is defined as the time-averaged value of the data squared, consider a sample time-history record x(t), then the Mean Square Value, Ψ 𝑥 2 is
Ψ 𝑥 2 = lim 𝑇→∞ 1 𝑇 0 𝑇 𝑥 2 𝑡 𝑑𝑡
8/1/2019
ME 508 Experimental Methods in mechanical Engineering

8. Basic descriptions of physical data
Mean Square Value
The positive square root of the mean square is called the root mean square (rms) value.
Ψ 𝑥 2 = lim 𝑇→∞ 1 𝑇 0 𝑇 𝑥 2 𝑡 𝑑𝑡
The mean value is defined as
𝜇 𝑥 = lim 𝑇→∞ 1 𝑇 0 𝑇 𝑥 𝑡 𝑑𝑡
Variance:
𝜎 𝑥 2 = lim 𝑇→∞ 1 𝑇 0 𝑇 𝑥 𝑡 − 𝜇 𝑥 2 𝑑𝑡
Such that,
𝜎 𝑥 2 = Ψ 𝑥 2 − 𝜇 𝑥 2
Where 𝜎 𝑥 is the standard deviation.
8/1/2019
ME 508 Experimental Methods in mechanical Engineering