Computational and Experimental Modal Analysis of a Clamped-Free Bar

I. Introduction

In this lab exercise you will analyze the modes of vibration of a clamped-free bar. Firstly, you will calculate the frequencies and modeshapes of bending and torsional modes for the bar analytically. Secondly, you will predict the vibratory behavior of the clamped bar computationally using finite element analysis. Thirdly, using the experimental technique of modal analysis with an impact hammer, you will measure the bar's the vibratory behavior. This lab is by no means a complete coverage of all the components of modal analysis, or of finite element analysis. It is merely intended to introduce you to the possibilities that such tools offer.

II. Theoretical Frequencies for a Clamped-Free Bar

A. Bending modes

The resulting allowed frequencies for bending motion are

$$ f_n = {{\pi c \kappa} \over {8L^2}}(1.194^2,\ 2.988^2,\ 5^2,\ 7^2,\ ...(2n-1)^2) \EQN bend1 $$

where for a beam with a rectangular cross section $\kappa \!=\! h/\sqrt{12}$ with $h$ being the beam thickness and $L$ being the beam length. The quantity $c \!=\! \sqrt{E \over \rho}$ is the speed of longitudinal waves in the beam. $E$ and $\rho$ are the Young's modulus and density of the beam. Notice that the higher frequencies are not harmonics of the fundamental (that is, higher frequencies are not integer multiples of the fundamental $n\!=\!1$).

B. Torsional modes

III. Experimental Modal Analysis Tools

A. Accelerometers

B. Impact hammers

The frequency range of excitation provided by a hammer is determined by the stiffness of the hammer-structure contact surfaces and the mass of the hammer head. There is a hammer-structure system resonance at a frequency given by $$ f_c = \sqrt{\hbox{contact stiffness} \over \hbox{impactor mass}} $$ above which it is difficult to deliver energy into the test structure. When the hammer hits the test structure, the resulting force pulse has a general sine-squared shape, as shown in \Fig{hammerhit}(a). The frequency spectrum of such an impulse is shown in \Fig{hammerhit}(b); the spectrum is essentially flat up to a certain frequency ($f_c$) after which it falls off significantly. There is a direct relationship between the duration of the impact and the cutoff frequency $f_c$; in order to raise the cutoff frequency one must shorten the impact duration. The impact duration is related to the stiffness (not hardness) of the contacting surfaces and to the mass of the hammer head. Stiffer surfaces and lighter hammers yield higher frequency ranges. Compliant surfaces and heavy hammers yield lower frequency ranges. Generally, you should use the softest tip possible; using too stiff a tip will result in energy being input into vibrations outside the frequency range of interest at the expense of vibrations within the desired frequency range.

The greatest difficulty in using an impact hammer to excite a structure is ensuring that each impact is essentially the same as all previous hits, not only in magnitude, but also in position and orientation to the normal to the surface. In addition, it is very important to avoid "double hits" which result when the hammer bounces against the surface. Double hits cause significant signal processing problems and contaminate measured data.

C. Frequency Analyzer

D. Software Package

IV. Finite Element Analysis

Finite element analysis is a computational solution of the vibration of a complex structure, performed by breaking the structure up into finite elements which are connected according to a defined mesh. There are books and books written about the finite element method, and even a couple of courses covering finite element techniques here at GMI. This lab is not an introduction to FEA. You will, however, use a finite element program to verify the modal analysis measurements you make. While there are many powerful finite element programs available in industry, and at GMI (IDEAS, ANSYS, NASTRAN, SYSNOISE) most of these programs have a steep learning curve. One simple program which is available on most of the PC's on the GMI campus is {\it weCan for Windows}; this lab will step you through this program in order to calculate the mode shapes and frequencies for a clamped-free bar.

Finite element analysis is also a very important tool in vibration analysis, and it is often used hand in hand with modal analysis. In fact, when studying a complex structure, it is generally a good idea to first perform a finite element analysis of a rough approximation of the structure. This not only indicates what types of motion and frequencies one might look for, but more importantly, it helps identify good and bad locations for accelerometer placement for modal analysis. If an accelerometer is placed at a location on the structure that does not vibrate at a particular frequency, then a modal analysis may not yield any useful data. Finite element analysis is a good preliminary computational tool to precede an experimental modal analysis.

V. Equipment

• rectangular beam (steel, $26\,\hbox{cm}\, \times 7\,\hbox{cm}\, \times 0.5\,\hbox{cm}$)
• C-clamps
• Accelerometer (PCB \#XXXXX)
• Impact Hammer (PCB \#XXXXXX)
• Amplifiers for accelerometer and hammer (PCB \#XXXXX)
• Frequency analyzer with two input channels, and 3.5-in floppy disk drive (HP 35760A)
• Modal analysis software (STAR MODAL) and 486X computer running Windows 3.1

VI. Procedure and Analysis

A. Theoretical frequencies

• Measure the dimensions of the bar. Take the bar to be steel with $E\!=\!19.5\times10^{10},\ \rho\!=\!7700,\ \sigma\!=\! 0.28,\ G\!=\!8.3\times10^{10}$, and approximate the mass by $M\!=\!\rho (L\times w\times h)$.
• Calculate the frequencies for the first 5 bending modes using \Eqs{bend1} and for the first 3 torsional modes using \Eq{torsional}.

B. Modal analysis

1. Analyzer and equipment setup

• Set up the HP 35670 Analyzer for 2 channel frequency response analysis between the force provided by the impact hammer and the acceleration output of the accelerometer. You can recall a previously saved setup from the floppy disk provided --- {\sl [Save/Recall]; [F5]`Recall State'; Filename = `MODAL.STA'; [F1]`Enter'}. The procedure for setting up the analyzer in this state is listed in the Appendix of this lab writeup.
• Once the analyzer is set up it will wait for you to press [Start] and then it will wait for a trigger signal from the hammer (Channel 1). If the hammer signal overloads the data is rejected. If the signal does not overload, then the analyzer will wait for you to manually accept each hit. The analyzer will display time data for both the hammer and the accelerometer so you can see if the hit is clean or not. The hammer data is windowed with a force window and the accelerometer data is windowed with an exponential window so as to force the response to zero within the time record. The window envelopes and their effects are not displayed on the analyzer though. After you have accepted a hammer hit, the analyzer will add the data to an average (total of 10 hits per measurement). Then the analyzer will display the coherence between the force and accelerometer signals (top display - trace A) and the transfer function (bottom display - trace B) which is essentially a ratio of the acceleration output to the force input.
• If the (larger) accelerometer is not already attached to the bar, then attach it, using a {\bf thin} layer of wax, to the underside of the beam at grid point \#1 as shown in \Fig{setup}.
• Make sure all cables are connected (hammer to amp to channel 1, accelerometer to amp to channel 2). Turn on the constant current power supplies for the hammer and accelerometer.

2. Data acquisition using impact hammer

• Start a measurement --- {\sl Measurement [Avg]; [F9]`Average Review'}. Press {\sl Measurement [Start]} to begin recording the first of 10 hits.
• Begin at node \#1 (at the corner of the bar). \underbar{\bf Gently} hit the hammer to the beam, keeping the hammer tip perpendicular to the beam. It may take several practice hits to get the hang of just how hard or how soft you need to hit the bar. The force transducer and accelerometer are very sensitive, so please to not hit the hammer as if you were driving nails! A relatively soft hit will yield very good data.
• After every hammer hit you will check to make sure the hit was clean (ie., no double hits) and you must manually accept [F8] or reject [F6] each hit. Be careful not to accept double hits, because they will screw up your data. You will have to wait a couple of seconds between hits for the analyzer to show you the time data, and then to average and display the frequency data after you have accepted a hit.
• Once you have accepted 10 good hits, the analyzer will display the message "Average Complete" above the top trace. Now you can save the data to a disk. You want to save the frequency response information, so make sure trace B is the active trace --- {\sl Display [Active Trace]; [F2]`B'}. To save the data to disk, press {\sl [Save/Recall]; [F1]`Save Data'; [F2]`Format'$\rightarrow$ `SDF'; [F1]`Save Trace'; [F9]`Into File'}
• Two things are very important when saving the data to disk. First, you need to save the data in SDF format in order for the modal analysis software to be able to read it. Secondly, the format of the filename you chose is critical in order for the modal analysis software to correctly interpret your entire set of measurements. The file name must have 8 characters. The first four characters represent the force coordinate. The last four characters represent the response coordinate. For example the file name for the measurement for hammer impact at node 1 with the accelerometer underneath node 1 would be "001ZM01Z.DAT" since the hammer location is node 1 in the z-direction and the accelerometer is at node 1 in the minus (M) z-direction. The file name for hammer impact at node \#2 would be "002ZM01Z.DAT", and so on. If you do not use this filename convention, the postprocessing will not work.
• Once you have saved the data move to the next node and start the process in step {$\square$} over again for nodes 2 through 39. (Once you get the hang of the hammer it will go pretty quickly.)
• When you are finished you should have a disk with 39 files on it with filenames ranging from "001ZM01Z.DAT" to "039ZM01Z.DAT".
• Turn OFF the power supplies for the hammer and accelerometer so that the batteries do not go dead.

Postprocessing of data using Star Modal software

• If Windows is not running, start it. Expand the {\bf SMS Apps} program group by double clicking on it. Double click on the "SMS STAR System" icon. The printer needs to be on in order for the program to run (a security key is connected to the printer port).
• A window will appear prompting you for a project name. Since this is the first time working on this project, click [Cancel].
• A flow chart should appear on the screen. If a flow chart does not appear, go to the pull down menu and select {\bf Gateway}$\rightarrow${\bf Turn Gateway On}. This should bring up the flow chart.
• Click on the "New Project" box. You will be prompted to enter a project name. Chose a name (up to 8 characters followed by a ".prj" extension) that matches the last name of one of your group members. Make sure the "Create New Directory" box is checked. Once the filename has been entered, click on [OK].
• A "Project Slate" window should now appear. The information in this window needs to be checked for accuracy. In the Description box you can replace "Project Information" with any information about your experiment. Since we are using a fixed response technique, make sure that the option {\bf Fixed Response} is checked. The box "Driving Point DOF" should read "1Z" for this experiment (location of accelerometer). Check to make sure that the "Measurement Units" is selected for "SI" units with Response in m/s$^2$ and Excitation in N. In the "Analyzer" box, Model should be set to HP35670, Channels to `2' and Address to `11'. When you have verified these settings, click [OK].
• Now you need to define the geometry of the test structure. Click on the {\bf Define Geometry} box. Two windows should appear: "Coordinates" and "Display Sequence". You can enter the geometry point by point and then connect the points one by one. Or, since a bar has simple geometry, you can let the computer generate the mesh for you. Leave the two windows open, but click on the mesh button on the button row at the top of the screen. The mesh button is just under the `T' of the pull down menu option "Tables".
• Fill in this window so that it looks like the figure below. \smallskip
  

  Press OK. The two tables should now have numbers in them; the table entries should make sense if you think how the grid is laid out. (The bar is 2" by 19.5" -- grid points along the length are 1.5" apart, while grid points along the width are 1" apart). You can save the coordinates information as "filename.crd" and the display sequence information as "filename.dsq". Close these windows by clicking on the [--] in the upper lefthand corner of each window.}

• You can check to see what the mesh looks like by clicking on the {\bf Show Structure} box. You should see a mesh that looks like the bar. Exit this window by clicking on the [--].
• Now would be a good time to save the work you have done so far by clicking on the {\bf Save Project} box.
• You are now ready to import the measured date from your floppy disk. (the floppy drive on the computer in the lab is drive b:) Minimize the STAR System by clicking on the down arrow in the upper right corner of the STAR System window. Open up the {\bf Disk Translator} program from the SMS Apps program group.
• You will be presented with a window labelled "STAR Data Disk Translator". In the pull down menu of this window select "Analyzer" and click on "HP Standard Data Format (SDF)". This will allow the software to recognize the data as coming from an HP analyzer.
• In the box labeled "DOF Labeling Method" select the command "Filename". This will instruct the software to use the filename convention to interpret and set the directions in which the measurements were taken.
• Find the box labeled "Source Directory". It should currently say "c:\\star" Click somewhere in this box and another smaller window will appear labeled "Source Directory". In this smaller window highlight the b: drive. You should see a list of the files on your disk. Click [OK]
• Find the box labeled "Destination Directory" and click somewhere in this box. Another small window should appear; in this smaller window replace the word "untitled" underneath "Project Name" with a name for your project data. Chose a different name than the actual project, make sure you select the directory you created when you opened STAR System, and make sure the "Create Subdirectory" box is checked. Then click [OK].
• Now click on the [Add All $\rightarrow$] button and the software will automatically select all the data files on your disk with an HP35670 data format and will move them to the "Files to Translate" box. Examine these filenames to make sure these are the files you stored on the analyzer earlier. Then click on the [Translate] button to begin translating the data files into a format the software can understand. This will take a few minutes.
• When the file translation process has completed, you will be prompted with a window "Data Translation Complete". Click [OK]. Then click on the [--] button in the upper left corner of the "START Data Disk Translator" window to close the translation program.
• Now it is time to identify the modes of interest from the data you have collected. Click on the box {\bf Identify Modes}. You will be presented with two windows: "Measurement" and "Curve Fit Panel". In the "Measurement" window, select the menu choice "File$\rightarrow$Open" and choose the subdirectory you just created in the data translation process. You should see a list of filenames (beginning with 001Z001Z.FRF). Click on the first filename and then click [OK]. You will see a warning message --- ignore this message by pressing [Yes]. From the pull down menu select "Trace$\rightarrow$Full Display" and then select "Axes$\rightarrow$Log Magnitude". The window should change to "Freq Response 1Z/1Z" with a frequency response function in it.
• The peaks in the frequency response function represent modes of the structure. The type of vibration (bending or torsional) will be identified shortly. You should count how many major peaks are present, and where they fall in frequency. Since not every mode is measured at every location, you should step through your data to make sure you account for all the modes. You can do this through the menu choice "File$\rightarrow$Next Measurement".
• Once you have identified the major peaks, it is time to curve fit the data. Move the cursor to the left side of a peak. Hold down the mouse button and move the cursor to the right side of a peak. When you lift up the mouse button you should see two vertical lines which surround the peak. These lines identify the frequency band over which you will do a curve fit of the data for this mode. Starting at the lowest frequency, put a band about the a single peak (or if several peaks are very clearly defined, you can group several peaks together).
• Once you have selected a frequency band, go to the "Curve Fit Panel" window. Set the "Band" to "1" -- this indicates the first band. Set the "Low Mode" to "1" -- this is the lower mode number within the above band. Set the "No. of Modes" equal to the number of modes within the band. Highlight "Polynomial" in the list at the left side of the window. This indicates the type of fit to the data. Make sure that "Freq \& Damp" and "Residues" are selected. Press [Set].
• Go back up to the "Frequency Response" window and select a new frequency band. Go down to the "Curve Fit Panel" window and set the "Band" to "2"; the "Low Mode" number should change automatically. Set "No. of Modes" equal to the number of modes within the new band. Press [Set]
• Continue until all major peaks have been included within a band.
• After the last peak has been [Set], then select "Autofit" and press [Fit]. A new window will pop up "Auto Fit Preferences". Press [OK] in this window. The computer will now automatically fit all of your data files to the frequency bands you have selected. This may take a few minutes. You may get an error message, just ignore it by pressing [OK]. Exit the fit windows by pressing [--] in the upper left corner.
• If you chose your frequency bands carefully, you should now have some animated shapes to look at. Click on the {\bf Show Structure} box. A window should appear with an animation of the lowest frequency mode of the bar. The mode number and frequency of the mode is displayed at the top of the window. You can check the other modes by adjusting the controls to the right of "Mode" in the "View Control Panel" window. You should see some modes which are definitely bending and some which appear to be more twisting (torsional). If you get lots of modes with garbage, then you should try to get a better fit -- go back and try to define your frequency bands more carefully.
• Save your project.
• You can get printouts of mode shapes by selecting the menu "Edit$\rightarrow$Still/Copy/Print". You can print the mode shapes to a file or to the printer.
• You should be able to find 3 bending modes and 2 torsional modes. Record the frequency and type for each mode, and if you can get a printout (or at least a sketch) of each mode.
• To exit STAR Systems, click on the [--] in the upper left corner.

C. Finite element analysis using weCan for Windows

• Follow the procedure for generating frequencies and mode shapes for a clamped-free bar as listed on the accompanying handout. This can be done on any of the PC's in the PC labs on the third floor of the Academic Building. You may also perform the finite element analysis on the computer in the lab. However, if you do so please save any files to floppy disk and not to the hard drive. Any student files saved to the hard drive may be removed. This finite element analysis should take less than half an hour to run through.
• Obtain plots of the first 12 mode shapes of the clamped-free bar. Identify the frequency and the type of each mode (bending, torsional, in-plane bending, etc.).

D. Analysis