Case Study: Experimental Data Analysis
Estimation of Modal Parameters from available experimental data
A three degree of freedom vibration experimental model, as shown in the Figure 1(a) here, is subjected to frequency sweep excitation. The encoders recorded the corresponding displacements of the individual masses. The associated mathematical model of the experiment is shown in Figure 1(b).
Figure 1: (a) Experimental setup for forced vibration analysis, (b) Mathematical model for the experiment for the experiment which means these dampers and springs were not connected, as seen in Figure 1(a).
The experimental data is saved in a text file "3DoFtestdata.txt" and is available to you for the experimental data analysis.
This file hasfive columns of data. The first column has data corresponding to time (in sec), the second column contains excitation force data (in N) and the third to fourth columns have encoder data (sensor) which contains the displacement (in m) of three masses respectively.
In this case study you need to analyze the experimental data and extract modal parameters (natural frequencies, damping ratio and mode shapes) by presenting the results as outlined here.
Hints:
• Use the "load(‘filnename.extension')" to load the data file into the MATLAB workspace. The data will be stored in a matrix format and it is important to know identify what each column represents. Note that you may want to assign the loaded data to a variable such as "A = load(‘filename.extension')".
• Once you've found your data, open the file. The data in the text file is stored as follows (time, force, mass 1 position, mass 2 positionand mass 3 position).
• In order to extract a column of data from data matrix (your file), you can use the following command: x5=A(:,5) extracts the 5th column from a matrix A and save as a variable x5.
(i) Plot the time vs. force and time vs. mass displacements and summarize your observations of the excitation force and the response of individual masses.
(You can use subplot command)
(ii) Using the FFT, create Frequency Response transfer functions between the excitation force and response (h11(ω), h21(ω), h31(ω)). Plot the absolute values of these three frequency response functions against the frequencies(0 - 7 hz.). Also, summarize your observations on the transfer functions.
Note: refers to transfer function between the excitation at 1st DoF and response at 2nd DoF.
Hint: For computing the frequency response function refer toModule 8 video and slides on Fourier Transform and FFT.
(iii) Based on the transfer function h11(ω) in part (ii) approximate the natural frequencies and corresponding damping ratios by "peak picking method". Show your calculations.
Hint: For FFT and damping ratio calculation from the frequency response function refer toModule 8 video and slides on Fourier Transform and FFT.
(iv) By plotting the imaginary values of Frequency Response transfer functions (h11(ω), h21(ω), h31(ω)) estimate the three mode shapes.
Note: Mode shape discussions and estimation refer to Module 12 videos (Experimental Modal Testing and Modal data extraction from the mode shape).
Case Study: Simulation and Design for Vibration Based Energy Harvesting
An electro-mechanical vibration based energy harvester device is shown in the Figure here, which converts the relative motion x(t) = u(t) - y(t) between the structure movement y(t) and the motion of harvester mass u(t) into the useful electrical energy.
Figure: Schematic of energy harvester for mathematical modeling and design
(b) For the given parameters:
m = 8.4 x10-3 kg, c = 0.1541 N.s/m, k = 2.5x104 N/m,
C0 = 18.9 x10-9 F, R = 30000 Ω, α = 1.52 x10-3 N/Volt,
Plot the absolute values transfer functions (i) X(s)/s2Y (s) and (ii) |V(s)/s2Y(s)| against the frequency, f (in hertz) with s = jω
where j = √-1 and ω is the frequency in rad/sec and f = ω/2Π is frequency in Hz.
Choose the frequency range for simulation:
f = (0 0.01 499.99 500) hz.
(c) The electro-mechanical force factor α depends type of piezoelectric materials. Suppose four different materials with:α = 1.52x10-3 (0.5 1 2 4) N/Volt , are being evaluated while keeping all the others parameters in (b) to be same. Then plot the absolute value of the transfer functions |V (s)/s2Y (s)| against the frequency, f (in hertz) for these three materials.
Choose the frequency range for simulation: f = (0 0.1 499.9 500) Hz.
(d) Summarize your observations of plots in (b) and (c) in relation to the design of harvester.
Time domain analysis
(e) By choosing the three state variables: x1 = x, x2 = x, x3 = V obtain the three state equations corresponding to equations (1) and (2). Show how you obtained these state equations and write them in the matrix form shown below.
(f) Assuming the harvester system is subjected to the zero initial conditions x(0) = x(0) = V (0) = 0 m and subjected to the structural acceleration environment y = As sin (ωst ) where, As is the acceleration magnitude of the structure and ωs is the excitation frequency at which the structure is vibrating.
By using the state equations (3) in part (e) and values from part (c), compute the voltage V (t) using Matlab ode45 function when the amplitude of the acceleration is As = 9.81 m/sec2 and excitation frequency is 275 Hz and plot the voltage response (millivolt) for t = (0 0.001 0.002 0.999 1) sec.
(i) What is the maximum value of voltage output:
(ii) What is the rms (root mean square) value of voltage output:
(iii) What is the maximum harvester power output ( P = V2max/R ) in mW (mili-watt):
(g) By repeating (f) obtain and plot the maximum harvested power output for different value of resistance in the piezoelectric circuit
R = (10x103 20x103 30x103 106 )Ω .
Hint: Use maximum harvester power output ( P = V2max use "semilogx" command instead of "plot" in Matlab.
(i) What value of resistance produces the maximum power output (in mili-Watt)
(ii) Qualitatively explain what are the parameters can influences the power output of energy harvester for a given operating condition (vibration of structure at the which harvester is mounted)
Attachment:- data set.rar