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

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

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

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

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

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

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

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