A STUDY OF METHODS FOR THE
EXPERIMENTAL ANALYSIS OF VIBRATING STRUCTURES
by
Ashwin Balasubramanya
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment of
the requirements for the degree of
Master of Science
Mechanical Engineering
2005
r
A
\
This thesis for the Master of Science
degree by
Ashwin Balasubramanya
has been approved
by
John Trapp
Date
Mohsen Tadi
Balasubramanya, Ashwin (M.S., Mechanical Engineering)
A Study of Methods for the Experimental Analysis of Vibrating Structures.
Thesis directed by Professor Samuel W. J. Welch
ABSTRACT
Currently experimental analysis is used to characterize the dynamic
behavior of a vibrating structure in terms of natural frequencies, mode
shapes and damping. The thesis studies the experimental modal analysis
and experimental finite element analysis to develop a scheme, which can be
used to continue the analysis of complicated structures.
The mathematical model to predict the transient behavior of a vibrating
beam subjected to an impact load is derived and the solution is used to
compare the data extracted from the experimental modal analysis of the
structure. It is shown that the comparison between the measured and
predicted responses can be used to identify errors in the modeshape
parameters extracted from the experimental modal analysis.
The comparisons are discussed and the validated experimental method is
extended for a complicated structure such as a BAJA car frame. The
developing concept of experimental finite element analysis is studied at this
stage. Understanding the tools of the analysis and using them to extract the
parameters from the complex structure summarizes the study.
The thesis attempts to explain some concepts about how structures vibrate
and the use of some of the tools to solve structural dynamic problems. It can
be viewed as a tutorial for the use of the National Instruments data
acquisition package and experimental analysis techniques in the analysis of
experimental dynamic data for the purpose of the identification of the
physical characteristics of the measured system.
This abstract accurately represents the content of the candidates
thesis. I recommend its publication.
Signed
IV
DEDICATION
I dedicate this thesis to my parents. If not for them I would not have been
where I am today. My deepest gratitude goes to them for believing that I
could actually come this far and get this degree. I truly thank them for all
their support and blessings.
ACKNOWLEDGEMENT
My sincere and profound thanks to my advisor, Dr. Welch, for introducing
me to Vibrations and providing me with an opportunity to study the area
under his tutelage. I cherish his guidance and patience through out the
period of my existence at the University of Colorado at Denver as a
graduate student. Also, I wish to thank the Mechanical engineering
department for providing me with the space and equipments that was
required for this thesis.
TABLE OF CONTENTS
Figures...........................................................viii
Tables...............................................................x
Chapter
1. Introduction......................................................1
2. Method and Equipment Used.........................................4
2.1 Setup and Measurement............................................5
3. Validation.......................................................12
3.1 Theory..........................................................13
3.2 Analytic Solution...............................................19
3.3 Experimental Analysis...........................................24
3.4 Experimental Modal Analysis.....................................29
4. Experimental Finite Element Analysis of a BAJA Car Frame........47
4.1 Finite Element Model of BAJA Car Frame..........................49
4.2 Modal Test of BAJA Car Frame....................................54
5. Summary and Scope for Further Research...........................67
Appendix
A: Algorithm to Solve the Analytical Model for a Vibrating Beam....69
B: Characteristic Equation for a FreeFree Beam.....................72
C: Algorithm to Generate the Waterfall Plot of Vibrating Sections...73
References..........................................................74
vii
FIGURES
Figure 2.1: The LabVIEW 6i algorithm developed for the data acquisition
from the vibrating test structure for the thesis.................9
Figure 3.1: A freefree aluminum beam being excited at a distance a
and displacement measured at a distance x from the free end...19
Figure 3.2: Freefree beam suspended with elastic chords to
approximate the boundary condition..............................24
Figure 3.3: Front panel of the LabVIEW 6i algorithm with the
displacement and the transfer function plotted using the signal
from one accelerometer..........................................27
Figure 3.4 (a) Comparison between the experimental data in solid
lines with the regenerated data using equation 3.19.............38
Figure 3.4 (b) Comparison between the Analytical Solution and the
regenerated data using equation 3.19............................38
Figure 3.5(a): Comparison between the experimental data in solid
lines with the regenerated data using equation 3.19.............39
Figure 3.5(b): Comparison between the Analytical Solution and the
regenerated data using equation 3.19............................39
Figure 3.6(a): Comparison between the experimental data in solid
lines with the regenerated data using equation 3.19.............40
Figure 3.6(b): Comparison between the Analytical Solution and the
regenerated data using equation 3.19............................40
Figure 3.7(a): Comparison between the experimental data in solid
lines with the regenerated data using equation 3.19.............41
Figure 3.7(b): Comparison between the Analytical Solution and the
regenerated data using equation 3.19............................41
viii
Figure 3.8: x = 22 inches, a = 10 inches Comparison between the
analytical solution in blue and the experimental data in red.....43
Figure 3.9: x = 18 inches, a = 12.125 inches Comparison between
the analytical solution in blue and the experimental data in red.44
Figure 3.10: x = 7.1875 inches, a = 12.125 inches Comparison
between the analytical solution in blue and the experimental
data in red..........................................................45
Figure 3.11: x = 20.2813 inches, a = 25 inches Comparison
between the analytical solution in blue and the experimental
data in red.....................................................46
Figure 4.1 :(a) BAJA car frame suspended for modal testing.
(b) Bungee chords used to suspend the frame using the front
end of the frame, (c) The Nl PXI device with the signal conditioners.
(d) The displacement of accelerometers over the section of
the frame that is being considered..............................56
Figure 4.2: (a) MATLAB waterfall plots for the isolated frequencies
of 54 and 58 Flz. (b) and (c) The corresponding finite element mode
shape with the measurement area specified by the circle.....62
Figure 4.3: (a) MATLAB waterfall plots for the isolated frequencies
of 77 and 80 Hz. (b) and (c) The corresponding finite element mode
shape with the measurement area specified by the circle.....63
Figure 4.4: (a) MATLAB waterfall plots for the isolated frequencies
of 118 Hz. (b) The corresponding finite element mode shape
with the measurement area specified by the circle...............65
Figure 4.5: (a) MATLAB waterfall plots for the isolated frequencies
of 127 Hz. (b) The corresponding finite element mode shape
with the measurement area specified by the circle...............66
IX
TABLES
Table 3.1: Boundary conditions for the free beam. It includes the
shear force and bending moment....................................14
Table 3.2: Comparison of the natural frequencies of the beam...........28
Table 3.3: Tabulating the amplitude spectrum measured at 3 points
on the beam. The response to an impact load due to a hammer
strike at each of the 3 points is recorded.............................32
Table 3.4:Tabulating the phase spectrum measured at three points
on the beam. The response to an impact load due to a hammer
strike at each of the three points is recorded.........................33
Table 4.1: Converged results listing the natural frequencies of the
finite element model..............................................53
Table 4.2: Comparison between the natural frequencies converged
by the finite element model and extracted from the modal test.....58
x
1. Introduction
Vibration problems can be simple or complicated. Vibration theory rarely
applies directly to a real structure with the exception of those structures
whose dynamics can be described accurately by a finite number of partial
differential equations. This is because theory is usually developed for an
idealized version of a real problem with various assumptions. The process
of idealizing a real structure before analysis can proceed is mathematical
modeling for the structure. [3]
Rapid development in the measurement and analysis techniques since the
early 1960s have provided tools and methods that are considerably in
advance of their practical application. The experimental study of structural
vibrations has always provided a major contribution to the efforts of
engineers to understand and to control the many vibration phenomena
encountered in practice. Experimental observations have been made for
two major objectives: [2]
Determining the nature and extent of vibration response levels.
Verifying theoretical models and prediction.
There are a number of structures ranging from turbine blades to
suspension bridges for which structural integrity is of paramount concern
1
and for which a thorough and precise knowledge of the dynamic
characteristics is essential. There is an even wider set of components or
assemblies for which vibrations is directly related to performance either by
virtue of causing temporary malfunctions during excessive motion or by
creating disturbance or discomfort including that of noise.
Modal testing is a tool which is used to encompass the processes involved
in testing components or structure with the objective of obtaining a
mathematical description of their dynamic or vibration behavior. It includes
both data acquisition and subsequent analysis of the data
In this thesis attempt has been made to understand the intricacies of
modal testing by comparing the test results with the analytical results
obtained through vibration models. Initially the data acquisition package is
set and an algorithm has been developed using LabVIEW 6i to acquire the
data through the data acquisition device. Also the algorithm performs the
necessary mathematical operations to convert the raw data acquired into
significant data.
The analytical model for flexural vibration is studied and the particular
solution for a freefree beam vibrating under an impulse load is derived
using Duhamels integral. In the III chapter the solution is explained with
the assumptions and equations obtained from various references specified
and the result is compared with the modal parameters extracted through
2
the data acquisition package used. The test preparations and
considerations are also discussed in the chapter. During the course of the
discussions experimental modal analysis is studied in detail. The results of
the comparison form a basis to extend the modal analysis procedure to a
complex structure such as a car frame. In chapter IV a numerical model of
the car frame is developed using a finite element package and the natural
frequencies and the mode shapes are compared with the experimental
modal model. In this chapter the concept of experimental finite element
analysis is studied and understood to modify the concepts for our
advantage.
The thesis aims at combining these comparisons to verify the
experimental procedure used. The attempt is to build a versatile platform
that could be altered easily and used to solve complex problems in the
field of structural dynamics.
3
2. Method and Equipment Used
There are two types of vibration measurement:
Those in which just one parameter is measured (usually a response
level)
Those in which both input and response output are measured (used
in this thesis)
In the fundamental relationship response =modal properties X input it is
seen that only when two of the three terms are measured, can we define
completely what is going on in the vibration of the test object. Accordingly,
in the test, data from the exciting medium and the test structure are both
recorded for analysis.
The experimental setup can be broken into three major divisions.
Excitation mechanism
Transducers and data acquisition tools that measures and acquires
the data from the hammer (input force) and the test structure.
The algorithm and analyzer to process the signal and to derive the
Frequency Response Function (FRF) from the measured
parameters
4
The excitation method used is Single point excitation in which a
straightforward approach is adapted and the structure is excited at a
single point. The point of excitation maybe varied around the structure
along the course of the experiment. This type of measurement is often
referred to as Mobility measurement: [2]
2.1 Setup and Measurement
A relatively simple means of exciting the structure into vibration is through
a hammer. The use of a hammer is advantageous because of the different
frequency ranges that are excited by its impact. The frequency range that
is effectively excited by the hammer is controlled by the stiffness of the
contacting surface and the mass of the hammerhead. There is a system
. , . , iL_ . I Contact Stiffness
resonance at a frequency given by the relation , above
\ Hammer Mass
which it is difficult to deliver energy into the test structure. A soft tip is used
in order to inject all the input energy into the frequency range of interest.
Using a stiffer tip than necessary will result in energy being input to
vibrations outside the range of interest at the expense of those inside that
range.
To measure the response, accelerometers are placed on the structure.
Care is taken to avoid placing them on one or more of the structures
5
nodes, which would cause difficulties in making an effective measurement
of the modes. Most modal tests require a point mobility measurement as
one of the measured frequency response functions. In order to measure
true point mobility both the force and acceleration transducers should be
at the same point on the structure. This can be achieved by exciting the
structure in line with the acceleration transducer but on the opposite side
of the structure. This is possible only for cases like the beam in this thesis
when the structure is locally thin. [2], In large structures the structures are
excited as close as possible to the transducers. The accelerometers,
which are a type of piezoelectric transducers, are used for mobility
measurements. Due to the fact that the piezoelectric crystals in the
accelerometers produce very small electrical charge that cannot be
measured by the analyzer a signal conditioners are used to boost the
charge. The signal conditioners also eliminate noise at the initial stage of
data acquisition. The accelerometers signal conditioners and the impact
hammer used in manufactured by PCB piezotronics and are company
calibrated for accuracy.
The accelerometers and the hammer are connected to the National
InstrumentsData Acquisition (NIPXI) device through a signal conditioner
to amplify the signals generated. NIPXI (PCI extensions for
6
Instrumentation) is a PCbased platform for measurement and automation
systems. These systems serve applications such as manufacturing tests,
military and aerospace, machine monitoring, automotive, and industrial
tests. The system used for the thesis consists of three components a
signal acquisition module, a measurement module and a highspeed
computer (analyzer) to perform the necessary analysis.
SIGNAL ACQUISITION MODULE: The National Instruments PXI4472
is an 8channel dynamic signal acquisition module for making high
accuracy frequencydomain measurements. The eight input channels
of the Nl PXI4472 simultaneously read signals and can obtain a
variety of accurate time and frequency measurements.
MEASUREMENT MODULE: The National Instruments PXI4220 is
capable of measuring highspeed strain and voltage inputs up to 100
V, making it ideal for highspeed structural tests. This module
incorporates builtin signal conditioning so it can measure sensors and
highvoltage signals directly.
COMPUTER: The National Instruments PXI8176 embedded controller
is a highperformance PC platform for modular instrumentation and
data acquisition applications.
7
NTs LabVlEW\s a graphical software system used to develop high
performance scientific and engineering applications. It can acquire data
and control devices through the devices explained as well as plugin I/O
boards. LabVIEW programs, called Virtual Instruments (Vis), are created
using icons instead of conventional, textbased code. A VI consists of a
front panel and a block diagram. The front panel (with knobs, switches,
graphs, and other virtual control tools) is the user interface. The block
diagram, which is the executable code, consists of icons that operate on
data connected by wires that pass data between them. LabVIEW6i\s
used to analyze the data acquired from the hardware. The software has a
vast library of built in mathematical functions that can be used to perform
the necessary signal processing operations.
Figure 2.1 is a screen shot of one such algorithm that is used for the
thesis. The purpose of this algorithm is to perform all the operations
discussed. There are three subroutines that are used to perform the
acquisition. These subroutines or Vis as they are called are built in
functions programmed to perform the intended operations. Combination of
these subroutines would provide us with the necessary algorithm. The
very basic thought given to the algorithm is towards the fact that the
acquisition should be uniform. The exciting force should be applied at a
8
known time so that the acquisition can be uniform. For this reason the
algorithm waits for the hammer to trigger the acquisition. The details of the
algorithm are discussed in the following sections.
fe: trigger_with_.hammer.vi Diagram *
Fie Edit Operate look frowse Window help
13pt Application Font
[Trigger Channell
GH
[Trigger Level I
cofinc
r i&al
41 Confal 41 Start 1
LJ
2^
Figure 2.1: The LabVIEW 6i algorithm developed for the data acquisition
from the vibrating test structure for the thesis.
As seen in the figure the Al Config subroutine is used to configure all the
input parameters and create a task ID that would be supplied to the Al
Start subroutine. The input to configure the acquisition is the device #
which is the address of the Nl PXI 4472 that is assigned initially during
the setup. The subroutine is programmed to acquire signals through all
9
the 8 channels of the acquisition device. The time for which the signal is
acquired and also the range of frequencies that has to be studied are
controlled using the sample rate and the number of samples to be
acquired. The task ID consisting of these data as input to the Al start
subroutine which would perform the mentioned acquisition after the
specified trigger. The set up is set to trigger using the signal through
channel 7. The point to be considered here is that that Channel 7 (8th
channel) is set to read voltage with an intention to couple the impact
hammer with the channel. All other channels are programmed to read the
accelerometers. The advantage of specifying these details is that the
package performs the necessary conversions and provides the data in gs
(1 g = 9.81 m/sA2) for the first 7 channels. For the 8th channel the data
acquired is in Voltage.
Various parameters of the signal acquired through Channel 7 can be
used to trigger the acquisition. A set voltage on the rising or falling slope of
the input signal can be used to trigger the acquisition. Also the algorithm is
programmed to wait for a specified period of time for the trigger to occur.
These details are added to the task ID and the ID is input to Al Start
subroutine. The algorithm is now ready to acquire the signal from
Channels 06 when triggered by Channel 7. The acquired signal
10
consists of an array of waveforms and is indexed using Index waveform
array.vi. Once the signals have been indexed and separated the
necessary mathematical operations are carried out using the transfer
function. It can also be seen that the isolated signals are converted to the
respective dimensions before directing them to the transfer function. There
are two inputs to the transfer function one is the stimulus signal, which is
force and the second is the response signal, which is acceleration. The
transfer function VI uses discrete Fourier transform to convert the two
signals into their frequency response functions. Once the conversion is
made the output provides the ration of the response to the stimulus signal.
The following chapters of the thesis aim at analyzing this response
function and extracting the modal parameters from them. It can be stated
here that the very basic step of any modal analysis procedure would be to
obtain the frequency response function from the vibrating test structure.
11
3. Validation
The objective of this chapter is to verify the correct use of the NTs Data
Acquisition device and the algorithms developed in LabVIEW 6ito acquire
the vibration signals through the setup. The idea is to compare the
experimentally extracted parameters with the analytical model of a
vibrating beam with an intention to verify the DAQ package and the modal
analysis technique adapted. To start with, the partial differential equations
that model the flexural displacement in beams are reviewed. The
analytical model of a freefree beam experiencing flexural vibration caused
by an impulse load is developed and solved to obtain the Frequency
response. The response obtained is then used to calculate the modal
parameters and compared with the experimental results. This chapter of
the thesis refers to Mechanical and Structural Vibrations Theory and
Applications by Jerry H. Ginsberg [3] to solve the analytical model of the
freefree beam experiencing flexural vibration caused by an impulse force.
Initially, the partial differential equations for both flexural and longitudinal
vibrations are discussed. However, the particular solution for a freefree
case of the beam excited by an impulse load is derived only for flexural
vibration. This is because in the later stages of the validation process
where experimental modal analysis is studied, flexural vibration has more
12
flexibility for the analysis. To apply the concepts of modal analysis it
requires that there would be a larger cross sectional area, which would
provide more options to position the accelerometers. In case of
longitudinal vibration the area of cross section is very low compared to the
length of the beam and hence it is ruled out for modal analysis. However,
a single accelerometer is sufficient to identify the natural frequencies of
the beam and hence the beam is analyzed only for the natural frequencies
under the longitudinal case.
3.1 Theory
The study of the straight beam is based on BernoulliEuler model for
deformation, which is referred to as the Classical Beam theory. The
differential equation involving the relation between force and
displacement, which has been derived is given by [3]
dx
FA 5u
EA
dx
 pAii = fx ~ for longitudinal vibration
3.1(a)
dx2
El
32w>'
Hx2
+ pAw = fz ~ for flexural vibration
3.1(b)
The abovementioned equations form the basis for analysis of extensional
and flexural motions. The solution of the field equation requires
specification of the boundary conditions. These are drawn from the
13
geometric conditions and from natural conditions that the internal force
resultants must satisfy. The boundary conditions that apply in the case of
a freefree beam are listed in table 3.1. [3]. In the boundary conditions
listed 'w'and V are the displacements for flexural vibration and
longitudinal vibration respectively. The beam considered in the thesis is
analyzed under freefree boundary conditions and hence the table lists
only the relevant boundary conditions.
Boundary Conditions @ @ X = 0 @ X = L
Flexural vibrations y:0
dx2 dx3
Extensional vibrations ^ = 0
dx
Table 3.1: Boundary conditions for the free beam. It includes the shear
force and bending moment.
The boundaries of a beam experiencing flexural motion have no bending
moment and no shear force which is which is visible in the table. In case
of longitudinal vibrations the boundaries experience a constant
displacement and hence= 0. These boundary conditions are also used
dx
14
while finding the eigen solution for the vibrating beam that would precede
the derivation of the modal response equation.
The solution for the modal response equation is initiated by using the
normal modes as the basis functions [3]
To begin with, the eigensolutions of the field equations are solved for. The
eigensolutions are the modal properties that form the foundation for
evaluating response. To obtain the vibration modes y/ the field equation is
solved in the absence of the external loads. Thus the solution of the
homogeneous partial differential equations 3.1(b) are solved with the
following substitutions [3]
wl
M
3.2
~ where the summation ^ extends over all modes f.
u 
~ longitudinal vibration 3.3(a)
~ flexural vibration
 3.3(b)
15
The general solution for the above equations are given by [3]
V,ong
= C, sin r + C2 cos f
a a
l L) l L)
3.4(a)
f
aco
v
pAL
EA
2 "\
1/2
~ Longitudinal vibration
y'flex=Cl sin
( x^ { A ( A r
a + C, cos a + C, sinh a + C, cosh a
l L) l l) j l L) l L)
3.4(b)
a =
' pAL4a)2^
1/4
~ Flexural vibration
Now the modal coordinates tj are to be derived and the solution may be
obtained from the differential equation of the modal coordinates. The field
equation for flexural vibration equation 3.1(b) is manipulated by multiplying
the equation with one of the normal modes y/k, and integrated over the
interval 0
technique. As mentioned earlier the beam is not solved for longitudinal
vibration and the equations that are used to calculate the natural
frequencies are obtained through the reference [3].
For flexural vibrations the portion of the manipulated equation containing
the derivatives has to be integrated by parts twice to obtain [3]
16
d
El
\
d2w^
dx2
dyk
dx
El
\dx J
t
d2Vk
dx2
2, A
Â£7
dlw
dx2
dx+ pAy/kwdx= fz^kdx
3.5
The boundary conditions enable us to eliminate the derivatives at the
boundaries. As a result the first two terms of the equation are dropped and
the resulting equation is [3]
t
d2Vkr
dx2
El
d2w^
dx2
dx+^pA y/k wdx = f fzWkdx
3.6
The next step is to substitute the modal series equation 3.2 into this
relation. Because 77.is independent ofx, we get [3]
I
f pa
3.7
The mass and stiffness orthogonality conditions for flexural vibrations are
considered here which form the coefficients of 77, and77, The conditions
are [3]
3.8(a)
 pA i//ki//jdx = Sjk ~ for mass orthogonality
d y/k d y j 2
rT~dx = (o2Sjk
dx2 dx2 J 1
~ for stiffness orthogonality
3.8(b)
17
The terms of the summation in equation 3.7 in which j k, the mass and
orthogonality conditions equal to zero and for the terms in which j = k, the
conditions equal 1 and co) respectively.
Applying the summation to the conditions and reducing it leads to the
differential equation [3]
iij + cOj2T}j = Qj 3.9
where Q} = jfxy/jdx is the generalized force. If the damping is significant
equation 3.9 would be [3]
fjj + 2 Â§. cojjjj + o)j2tJj = Qj 3.10
The abovederived differential equation is solved for the particular case of
the freefree beam in the following topics. With the general solution of the
beam being reviewed, attention is now given to the particular case of a
freefree beam under impulse load, which is used in the thesis.
18
3.2 Analytic Solution
Impulse hammer
V
 X V
Accelerometer reading
Figure 3.1: A freefree aluminum beam being excited at a distance a and
displacement measured at a distance x from the free end.
Theoretically any structure possesses rigidbody modes and each of these
modes will have a natural frequency of 0 Hz. These modes, which are used
to determine the mass and inertia properties of the structure, are exhibited
by the structure only when they are freely suspended in space. Also it is
difficult to generate the condition where the body is grounded, as no
structure is rigid enough to provide the necessary grounding.
The boundary conditions for a freefree beam experiencing flexural vibration
from table 3.1 are applied to the general homogeneous solution for mode
shape given by y/j in equation 3.4(b). The function is derived to find the
d2u/ 9 V,
bending moment r^and the shear stress^. The derived functions
dx2 Bx
are evaluated at the boundaries to solve for the constants. Consequently
19
the relation between the arbitrary constants of the general solution is given
by C,= C^ and C2= C4. These conditions are used while continuing with the
solution and the results of these substitutions are
sinh('y) sin(ar;.) cosh(ar;) cos(cr;) "c," 'O'
cosh(or;) cos(r;) sinh(cr;) + sin(or;) c2_ 0
For a nontrivial solution, the square matrix in the above equation must not
be invertible, thus the determinant of the matrix is equated to zero. The
conditions lead to the characteristic equation
cos(ay )cosh(a;.) = 1 3.11
ocj is solved for using the algorithm in appendix B The algorithm uses
Newtons method to solve for the values of a.. The equation will always be
transcendental, having an infinite number of roots,a,(a2(a3(.... [3]. In the
case of the freefree beam there are two rigid body modes present which
refers to a, ,r = 0,ai = 0. These rigid body modes are briefly discussed later
with the derivation of Modal equation. The roots obtained by the algorithm
are used to calculate the natural frequencies of the vibrating beam using the
relation between aj and o)j given in equation 3.2.
Applying the relation derived between the modal coefficients the particular
20
solution for the mode shape of a freefree beam during flexural vibration is
henceforth given by
f r a f A f x\\
sin a + sinh a, + R. cos or, + cosh a,
K 1 L) k J l). J l 1 L) l 1 L) J
~ where R.
[sin(ay)sinh(ay)J
[cos(cry) cosh(or;)]
After deriving the mode shape, the equation for the modal coordinates 77,
has to be derived and the solution could be obtained from the differential
equation of the modal coordinates.
In what follows, the uncoupled modal equations are used to obtain an
analytic solution for flexural vibration of a freefree beam.
In the generalized force term Qj = jfxy/jdx of equation 3.10 we have
fx = F0S(t)S(xa) for an impulse load. Here damping is added to the
equation. The addition of the damping ratio is justified because in the
physical world every structure has some amount of damping. Considering
an arbitrary value for the damping ratio Â£, the solution for the modal
coordinates is obtained using Duhamels integral and is given by the
equation 3.14
Tjj = ^1 e~4c0jl sin(o)dt)  3.13
co,
21
Thus substituting the derived equations in equation 3.2 we arrive at
3.14
The Fast Fourier transform of the above function would result in the transfer
function
The abovederived equation is used to plot the frequency response of the
beam and will be compared with the experimentally obtained parameters.
The beam considered vibrates harmonically and hence structural damping
is considered here. The referred text explains the modification of the
transfer function with which the structural damping can be accounted.
Structural damping is associated with a damping force that is proportional to
the magnitude of the elastic force. The damping loss factor y is the factor of
proportionality between the elastic and damping forces. The derived transfer
function for a structurally damped system is [3]
[cfjui *(***) ~ y y'jWy'M)
F0 ^cof+d^+ico)2
3.15
where cod = a)] ^1 %2
[G(
3.16
22
Thus the structural damping model replaces l^cojco in the denominator
with (61jy.
This may be interpreted as stating that the modal damping ratios are: [3]
1
^=7r 317
2 co
In other words, rather than being independent of the excitation frequency,
the effective damping ratio for each mode decreases inversely to the ratio of
the excitation frequency to the modes natural frequency. The damping loss
factor / for Aluminum is 0.01. It was observed that with a constant damping
ratio the response of the beam varied from the analytical model.
Considering the fact that the beam was suspended using an elastic chord to
approximate the boundary conditions it can be argued that the damping
ratio is not constant throughout the structure. The beam might have a varied
damping factor from the free end through the suspended end. Structural
damping provided results, which converged with the experimental data, and
hence it is considered as better representation for this case.
With the analytical solution for the particular case derived, focus is now
shifted to the experimental analysis. Experimental analysis consists of two
parts the data acquisition and the experimental modal analysis. Following
23
the procedures of the experimental analysis the comparison plots are
produced using MATLAB which illustrates the argument of the present topic.
3.3 Experimental Analysis
The test structure used is an Aluminum beam, which is suspended with an
elastic chord inserted through a 0.0787 hole drilled at the end of the beam
as shown in figure 3.2. The chord used to suspend the beam is a soft spring
wire (fish wire). The suspension wire is set normal to the primary direction of
vibration with an intention to avoid the spring stiffness that would be added
to the structure if suspended along the direction of vibration. Locations are
marked along the axis plane of the beam for accelerometer placement for
transfer mobility.
Figure 3.2: Freefree beam suspended with elastic chords to approximate
the boundary condition.
24
The algorithm in figure 2.1 is a screen shot of the LabVIEW 6i algorithm that
is generated to acquire the signal from the vibrating beam. A Vinyl tipped
impulse hammer connected to Channel 7 of the PXI device is used as the
analog trigger. When the algorithm is executed, the PXI device waits for a
period of ten seconds for the trigger to occur. An accelerometer is attached
to the hammer, which generates an impulse signal when the hammer
strikes the beam. A rising edge of the impulse signal generated acts as the
trigger and initializes the data acquisition. A trigger level of 0.1 V is
introduced to avoid acquiring noise from the surroundings that would be
picked by the hammer. Once the algorithm is activated the NIPXI device
acquires two thousand samples of the vibrating signal from the beam at a
rate of two thousand samples per second. The number of samples used is
chosen over a series of tests. It is observed that the impact hammer used
excites the lower modes and the higher frequencies are not distinctly
excited. With a rate of two thousand samples a second, the first four modes
of the beam are recorded and the comparison is performed for the four
modes. The acquired signals i.e., the impulse and the response signal are
directed to the transfer function and the Frequency Response of the signal
is generated. Figure 3.3 is the front panel of the LabVIEW algorithm after
the acquisition.
25
An important computation incorporated in the algorithm is the conversion of
the obtained experimental data to relevant data. The Channels 0 6 on the
Nl PXI device are configured to measure through an accelerometer. The
specifications for these channels require the input of the sensitivity and the
range of the PCB303A03 accelerometers that would be coupled with these
channels. Channel 7 on the Nl PXI device is configured to measure
voltage. It is coupled with the PCB 086B01 impulse hammer.
When the device acquires data through the Channels 06 the
Itl
measurement displayed is in gs where lg =9.81. Consequently the
sl
output has to be multiplied by 9.81 to obtain the acceleration at the point of
interest. The sensitivity of the impulse hammer coupled with Channel 7 is
mV
9.4 and since the measurement is in Volts the conversion factor to be
N
introduced in is which would be multiplied with the output. These
details are considered and incorporated in the algorithm. Thus the outputs
from all eight channels are known and the algorithm developed converts the
data acquired to relevant quantities before it is input to the transfer function.
The output of the transfer function is a plot of the Transfer function
w(x,t)
vs. Frequency. The output plot is stored in an excel file which is read by
26
MATLABXo compare the analytical solution and the experimentally obtained
plot on a single window.
Yet another aspect to be considered is ensuring that each impact is
essentially the same as the previous one, not so much in magnitude as in
position and orientation relative to the normal to the surface. Multiple
impacts and Hammer bounce are avoided as these create difficulties while
processing the signal.
Ito*
_______________________________________________i ir
Figure 3.3: Front panel of the LabVIEW 6i algorithm with the displacement
and the transfer function plotted using the signal from one accelerometer.
27
The waveform graph on the ACQUIRED SIGNAL window is the sampled
signal from the beam. The above picture is a screen shot of the signal and
the transfer function acquired from one accelerometer out of the eight
possible channels. The graph on the TRANSFER FUNCTION window is
the frequency response of the beam. The transfer function is generated
after the required conversions are made and the values can be directly
compared with the analytical model. The first five natural frequencies of the
beam obtained from the above experiment are listed in the table below and
are compared with the natural frequencies calculated analytically.
FLEXURAL VIBRATION
Frequency in Flz
h = 0.64 inch, w = 1 inch h = 1 inch, w = 0.64 inch
Analytical Experimental Analytical Experimental
0.00 0.00 0.00 0.00
82.76 86.00 131.36 137.00
228.13 237.00 362.11 374.00
447.23 462.00 709.88 731.00
739.23 762.00 1173.50 1200.00
LONGITUDINAL VIBRATION
Analytical Experimental
0.00 0.00
2456.10 2584.00
4912.20 5167.00
7368.30 7750.00
9824.40 10332.00
Table 3.2: Comparison of the natural frequencies of the beam.
28
It is seen that the experimental values and the analytical values are close
for lower frequencies and the variance increases as the frequency
increases. A part of the variances can be accounted to the boundary
conditions of the beam. As mentioned earlier it is difficult to generate a
grounded end condition of the beam in the lab. Though the freefree
condition was almost achieved it is seen that as the response is recorded
closer to the suspended end the variance increases.
3.4 Experimental Modal Analysis
The modal test is considered complete with the analysis of the data
acquired and its comparison with the analytical model of the beam. After the
completion of the first phase in the modal test, i.e., completion of the
measurement of the raw data attention is diverted to the analysis, which
must be undertaken in order to achieve the objectives. A major part of this
analysis consists of curve fitting the theoretical expression of an individual
FRF to the actual measured data obtained by one of the methods. This
phase of the modal test procedure is often referred to as experimental
modal analysis or modal analysis.
An initial check of the data obtained would save time by indicating the bad
data even before the analysis is started. To continue with the discussions, it
can be mentioned here that the plots would highly depend on the type of
29
measurement that is used namely point mobility (drive point) measurement
or transfer mobility measurement. With point mobility the drive point of the
beam and its response measurement are at the same location where as
with transfer mobility they would be different. The plots generated differ with
the presence of antiresonances in the case of point mobility. In this case all
resonance peaks are separated by antiresonances and the peaks in the
imaginary part of the frequency response function must all point in one
direction. The phase loses tt degrees of phase as the plot passes over a
resonance and gains tt degrees as the plot passes over antiresonance [1].
Apart from this the plots can be checked if they are mass dominated or
stiffness dominated. Plots that do not behave in any of the above mentioned
patterns can be considered as not good data and should be rerecorded.
With this knowledge the data can be validated at a preliminary stage and
the process of the process of modal analysis can be continued to calculate
the required parameters.
The figures that follow reinforce the above discussions for a freefree beam.
Three points are chosen on the beam and the responses are recorded for
hammer impacts at each of these points. Figure 3.3 illustrates the amplitude
response of each of the cases. It can be seen that in only the elements
along the principle diagonal of the table there is antiresonance between the
two peaks. These are elements formed by exciting the point of
30
measurement that is called point mobility. One other characteristic of the
table is the table should be symmetric along the principle diagonal. This is
because in transfer mobility, as it is called, the data acquired from Point 2
when Point 1 is excited should be similar to data acquired from Point 1
when Point 2 is excited.
The peaks that appear on the plots as resonance can be verified using the
phase plots too as discussed above. The table 3.4 lists the phase plots for
the exact same cases listed in table 3.3.
31
Table 3.3: Tabulating the amplitude spectrum measured at 3 points on the
beam. The response to an impact load due to a hammer strike at each of
the 3 points is recorded.
32
Table 3.4:Tabulating the phase spectrum measured at three points on the
beam. The response to an impact load due to a hammer strike at each of
the three points is recorded.
It can be noticed that there is a p degree phase loss at the point of
resonances. This helps us in identifying the resonant frequencies when
there is a lot of noise prevailing in the test environment. The table 3.4 like
the previous table has to be symmetric along the principle diagonal. The
characteristic of the table, which is missing, is that along the diagonal
elements there should have been a gain of 180 degrees at frequencies of
antiresonance. The absence can be attributed to the fact that the
33
antiresonances present in the table 3.3 are tending features of the plots and
are not complete due to reasons of approximated boundary conditions and
physical material properties.
Modal analysis, which is the analysis of the acquired data, is a path that
leads to the systems modal properties. Experimental modal analysis is
essentially a curve fitting procedure. The FRF data that is acquired from the
vibrating system is analyzed and is compared with a mathematical model to
calculate one of the three models namely Spatial model, Modal model or
Response model[2]. A Spatial model is of mass, stiffness and damping
properties, a Modal model calculates the natural frequencies and the mode
shapes and a Response model consists of the frequency response
function. In this thesis a modal model has been developed using the FRF
data from the vibrating freefree beam. The FRF data is analyzed to extract
the natural frequencies and also the mode shapes by explaining the method
used to extract the modal constants from the frequency response function.
When a structure is lightly damped, it becomes difficult to obtain accurate
FRF data near the resonances. The Nyquist plot of an FRF is not very
useful since all FRF data points are amassed along the real axis. The other
methods of curve fitting are not useful anymore. The alternative method is to
use FRF data away from the resonances. To start with the FRF data from
the LabVIEW algorithm is recorded over the frequency range of interest.
34
The resonances can be easily spotted for such lightly damped structures
and the corresponding natural frequencies are recorded. Now data points
are marked on the FRF and the corresponding frequencies are noted. The
data points considered are between resonances and are away from the
peaks.
The theory behind this as explained by D. J. Ewins is quite simple. Using
the structural damping model, we know the FRF can be written as: [4]
a
jk
() = Z
ljk
,=i o)2rco2 + j7jrco;
3.18
As the structure is lightly damped, the equation can be approximated into:
[4]
a^=t^
3.19
This approximation leads to a convenient way to estimate modal constants
of the FRF. At a measured frequency Q,, the FRF can be written as: [4]
i) =
1
1
1
co2 Qf o)\ Qj* (o; Q,
2 Ajk
n Ajk
3.20
35
By using the FRF data at different frequencies, we can form the following
matrix equation:[4]
1 1 f A ]
co2 Q2
1 1 > 2 jk
co2 Q22 co2
Representing the square rectangular matrix by/?(Q), the modal constants
can be estimated as:[4]
M=Wa)}"'
By using the FRF data at different frequencies'Q', the solution for the
unknown modal constants rAjk in terms of the measured FRF data points
and the previously identified natural frequencies may be obtained. The
points chosen for the individual FRF measurements play a prominent role in
the accuracy of the method. The points chosen are distributed throughout
the frequency range and include as many antiresonances as are available.
At antiresonance the theoretical model will exhibit a zero response: hence it
is possible to supply a null value for the appropriate data in equation. This
means that only one FRF data point may be required from the
measurements, all the other points being set to be identically zero even
36
though from the measurements on a real structure their values would be
extremely small, but finite.
In the following figures a comparison has been made between the three
different sets of data namely the Experimental data which is obtained from
the LabVIEW algorithm, the Regenerated data which is the recreated data
using equation number 3.18 and the Analytical solution which is the result
of the algorithm solving equation 3.16. In each of the comparisons there are
two plots. Plot (a) of the comparison compares the experimental data and
the regenerated data and plot (b) compares the regenerated data and the
analytical data. The comparisons are plotted using the MATLAB algorithm
listed in the appendix. Before analyzing the plots it can be assumed that
there would be some amount of variances due to reasons discussed
through this chapter. The Yaxis of these plots is in the semilog format
because in linear plots, which are listed in figure 3.4 to figure 3.7, the
presence of antiresonances and the drops are not visible. The figures with
the linear plots are used to compare the amplitude variance between the
analytical model and the experimental data.
37
gg?
jCtf QAi^ A ^ / & 0 6
Experimental) Vs Regenerated()
I
,\
' \
V
\!
l
V /
\ /
\ /
1 /
U
A 17 Inches
X 1.5 Inches
A
/ \
/
300 400 S00 BOO 700 800 900 1000
FREQUENCY
Figure 3.4 (a) Comparison between the experimental data in solid lines with
the regenerated data using equation 3.19
JD d* B A A~> s\ & O
ANALYTICAL SOLUTION
REGENERATED PLOT
Figure 3.4 (b) Comparison between the Analytical Solution and the
regenerated data using equation 3.19
38
c? B 4 k A / & Â£>r>
Figure 3.5(a): Comparison between the experimental data in solid lines with
the regenerated data using equation 3.19
Figure 3.5(b): Comparison between the Analytical Solution and the
regenerated data using equation 3.19
39
]}Q fi* B A
Figure 3.6(a): Comparison between the experimental data in solid lines with
the regenerated data using equation 3.19
Figure 3.6(b): Comparison between the Analytical Solution and the
regenerated data using equation 3.19
40
Figure 3.7(a): Comparison between the experimental data in solid lines with
the regenerated data using equation 3.19
Figure 3.7(b): Comparison between the Analytical Solution and the
regenerated data using equation 3.19
41
Figure 3.4 is the data acquired to identify the natural frequencies of the
vibrating beam. It is seen that the analytical solution and the regenerated
data tend to be similar even though there are slight variations in the
experimentally obtained data. In the figures 3.5 through 3.7 the comparison
is done for the first two modes that were obtained to build table 3.3 and 3.4.
It is seen that the three models are almost similar except for the initial
values and a slight variance through the plots. The variances can be
attributed to the analytical solution and also the triggering signal. In the
analytical solution the rigid body modes are not considered and hence the
initial value of the solution apparently tends to zero. The presence of rigid
body modes would produce a displacement at zero frequency, which is
visible in the experimental data. Also the slight variance through out the
data may be attributed to the excitation force. In theory the impulse is
applied at time is zero, which is the ideal case. In the experimental model
the ideal case of time is zero is not possible because with the algorithm a
fixed level of voltage would trigger the acquisition. When the triggering
voltage occurs, a part of the peak is lost and the rising slope continues after
the trigger. Also since the accelerometer is fixed to the hammer, it records
the minor motions induced in the hammer due to the impact. These minor
motions are input to the transfer function as stimulus signal while actually
they are not applied to the beam. Considering these factors it can be said
42
that the plots obtained are similar and satisfy the validation process.
Further, the amplitude spectrum of the analytical solution and the
experimental data compared in figures 3.8 through 3.12 provide the
justification to believe that the algorithm and the setup used for the thesis is
valid and can be extended for further work in the area.
Ffe Edt Aaf toMft Tecfa Wndw
Jo lias h / /
Analytical(blue) Vs Experimental(red)
Figure 3.8: x = 22 inches, a = 10 inches Comparison between the
analytical solution in blue and the experimental data in red.
43
fib Edt JTMrt Tools Whdm Hefc
DÂ£Qd \ k / &&{
Analytical(blue) Vs Experimental(red)
X18 Inches
A= 12.125 Inches
FREQUENCY
Figure 3.9: x = 18 inches, a = 12.125 inches Comparison between the
analytical solution in blue and the experimental data in red.
44
Figure 3.10: x = 7.1875 inches, a = 12.125 inches Comparison between
the analytical solution in blue and the experimental data in red.
45
mmmBAEMZSSr?
Cdt incert Took Window He*>
.Q^BS u^ / & &
Analytical(blue) Vs Experimental(red)
Figure 3.11: x = 20.2813 inches, a = 25 inches Comparison between the
analytical solution in blue and the experimental data in red.
46
4. Experimental Finite Element Analysis of a
BAJA Car Frame
In the previous chapter the methods and the algorithms used for the modal
test and analysis have been validated using the simple freefree beam case.
With the gained knowledge the modal techniques will now be extended to a
complex structure. A BAJA car frame was used for the analysis. In the
process of applying the analysis to the car frame experimental Finite
element analysis was studied. Until very recently, Finite Element Analysis
(FEA) and Experimental Modal Analysis (EMA) have been very separate
engineering activities aimed at solving a common problem. Now the two are
converging and powerful new tools for solving noise and vibration problems
are emerging as a result the recent term Experimental FEA has emerged.
[5]
Experimental FEA as defined by the referred work [5] is an integrated
collection of modeling and analysis tools designed to be used by the
experimentalist. It includes the facility to quickly build simple smallscale
finite element models and solve them for natural frequencies and mode
shapes. The intent was not to replace the traditional tools, which are far
more detail focused. Rather, Experimental FEA is intended to augment the
experimental understanding of structures by aiding us to perform better
47
modal tests, faster. Thus experimental FEA can be stated essential as it
provides answers to questions that arise before modal testing or in other
words is essential for test planning.
Some of the questions that could be answered using experimental FEA are:
[5]
What bandwidth must be used in the test of a new structure?
How many modes must be identified within this bandwidth?
Can all of these modes be identified from a single reference test, or
must multipleinput multipleoutput (MIMO) measurements be used?
Where should the shakers (or the reference accelerometers for an
impact test) be placed to achieve the best measurement results?
How many responses or impact points must be measured in order to
draw clear and unambiguous animations of every mode?
What is the minimum number of frequency response functions
(FRFs) that must be measured to validate a largescale FEM model
of a new structure?
Where these responses should be measured?
In simple terms Experimental finite element analysis is a procedure in which
a basic finite element model of the proposed structure is studied to identify
parameters, which would help in the performance of a successful modal test
48
[5]. In this thesis I have used ALGOR to build a finite element model of the
BAJA car frame. The finite element package used is a traditional package
and the concepts of experimental finite element analysis applied are
explained as the chapter proceeds.
4.1 Finite Element Model of BAJA Car Frame
To begin with the dimensions of the BAJA car frame were measured and
the coordinates were calculated for the finite element model. The car frame
is built using alloy steel and has a tubular cross section with 0.125 wall
thickness. Various lengths of tubes are cut and welded together at the
required angles. FEMPRO is used to generate the model using the
dimensions measured along the centerline of these tubes. The model is
analyzed using ALGOR the finite element package used in the thesis. The
model is constructed using straight lines that are defined as beam elements
with a tubular cross section for the analysis. The model was analyzed for a
free free case and the natural frequencies were recorded. Only a response
model of the structure is built in this chapter. The data collected consists of
the natural frequencies of the structure. Also the screen shots of the few
modes that have distinct amplitude response are recorded for comparison
with the waterfall plots later in this chapter.
49
The preliminary work proceeding to the experimental modal testing of the
structure was to use the finite element model to identify the degree of
freedom on the structure that has to be excited and also the degree of
freedom that has to be located for response measurement. This is where
the shape product vector, a concept of the experimental Finite element
analysis was studied. With the shape product vector we can identify the
optimum degree of freedom that can be excited or observed for all the
modes of the structure. At each degree of freedom, the shape product is the
product of modal coefficients for all modes at the particular degree of
freedom. If the degree of freedom is a node for any one of the modes then
the shape product is zero. By plotting the function we can identify the
degree of freedom where all of the modes are active. These locations would
be the best choices for shakers/impact hammer or reference accelerometer
installation. The shape product vector is generated by exclusive
experimental finite element analysis packages like Vibrant technologies
MEscope VES. In this thesis the concept of the vector is only studied and
not applied to analyze the car frame. However, the understanding of the
concept provides a direction to look at during the test planning.
As mentioned earlier ALGOR is not a part of experimental finite element
analysis package and hence does not consists of tools to find the shape
product and other related parameters that would be considered helpful in
50
the procedure. Experimental finite element analysis packages consist of
tools that only concentrate on the Modal applications. Their objective
focuses to provide a quick analysis of the structure in order to design the
modal test scientifically. It avoids the trial and error method that is used to
identify the degree of freedom that excites all the modes of vibration and
hence saves time and produces better testing results.
With the understanding of the concept and with the available tools, the
saved screen shots of the ALGOR model were used to analyze the results.
The point to remember here is if there is a node at an identified degree of
freedom than the corresponding value of the shape product vector is zero.
The screen shots of the first few modes recorded were studied to eliminate
the degrees of freedom with nodes and points with minimal displacement.
The elements with maximum displacements are noted and the degrees of
freedoms on these elements, which have maximum displacement or high
relative displacements, are identified. These degrees of freedom would be
the elements of the shape product vector with maximum values. This is
because the shape product vector would have a nonzero value at points
where the displacements are not nearing zero. Points are thus located and
listed on the structure that can be used to identify the modes and also
record the natural frequencies. Though these points are listed and will be
used for the modal test, it is not the proper way to shortlist the points. As
51
mentioned earlier here only the concept of the shape product was used and
not the shape product itself. However, the process saves time as it
eliminates a few trials that would have been carried out to identify such
points. At the end of the discussion a significant amount of time was spent
during the test phase to identify the points.
Table 4.1 lists the natural frequencies of the car frame calculated by the
finite element model. The results are converged for accuracy and have been
tabulated. It can be noticed that the frequencies are denser as we go higher
along the scale and also since the structure was tested under freefree
condition, there should be six rigid body modes. This is apparent from the
table.
52
Base Model Divide by 4 Divide by 16 Divide by 64 Divide by 128
1st Nat Freq 1.59E11 1.59E11 1.59E11 1.59E11 1.59E11
2nd Nat Freq 1.59E11 1.59E11 1.59E11 1.59E11 1.59E11
3rd Nat Freq 1.59E11 1.59E11 9.63E05 1.59E11 6.90E04
4th Nat Freq 1.59E11 1.32E11 1.09E04 1.59E11 1.06E04
5th Nat Freq 1.48E05 4.07E05 1.47E04 1.65E04 1.31E03
6th Nat Freq 2.51 E05 1.15E04 1.54E04 4.42E04 2.37E03
7th Nat Freq 45.042 50.894 51.1615 51.1779 51.1787
8th Nat Freq 52.1227 54.5692 54.5605 54.5599 54.5599
9th Nat Freq 70.8906 76.3834 76.5425 76.5523 76.5527
10th Nat Freq 77.3258 83.1797 83.2559 83.2609 83.2611
11th Nat Freq 102.548 103.743 103.667 103.662 103.662
12th Nat Freq 106.765 109.915 109.784 109.771 109.771
13th Nat Freq 112.678 111.011 110.97 110.971 110.971
14th Nat Freq 117.344 117.85 117.89 117.893 117.893
15th Nat Freq 122.855 123.539 123.338 123.326 123.325
16th Nat Freq 124.257 126.112 126.217 126.223 126.223
17th Nat Freq 129.223 133.141 133.268 133.275 133.275
18th Nat Freq 147.415 148.418 148.205 148.185 148.184
19th Nat Freq 151.715 151.682 151.831 151.837 151.837
20th Nat Freq 153.544 154.747 154.767 154.76 154.76
21st Nat Freq 157.531 157.978 157.85 157.837 157.836
22nd Nat Freq 167.316 160.793 160.641 160.627 160.626
23rd Nat Freq 170.945 165.09 165.555 165.56 165.56
24th Nat Freq 178.857 165.717 166.004 166.011 166.011
Table 4.1: Converged results listing the natural frequencies of the finite
element model.
53
4.2 Modal Test of BAJA Car Frame
After the listing of the excitation and the response measurement points the
next step was to conduct the modal test on the car frame. The analysis of
the car frame was with free boundary conditions and hence the boundaries
had to be approximated for the test. Using the finite element model plots
from ALGOR the stiff end of the structures is identified. The area of the
structure, which has minimal displacements in all the modes under
consideration, is identified and used to suspend the structure. The
displacement being minimal at these sections of the frame the stiffness
generated by the suspension chords will affect the results of the test by a
relatively smaller factor. Bungee chords are used to suspend the car frame.
The front end of the frame is identified for the reason mentioned and the
bungee chords are inserted through this end. The frame is excited at the
listed points using an impact hammer with aluminum tip. A few points on the
list of degrees of freedom identified using the finite element model that are
close to the suspended end are eliminated initially with an intention to avoid
the effect of the suspension chords stiffness on the response. The band of
frequencies that would be excited by the hammer would be at the lower end
of the frequency spectrum, considering the hardness and mass of the
hammerhead compared to the whole frame. This being said only
frequencies less than 175 hertz is recorded using the data acquisition
54
system. The sampling rate chosen is also supported by the conclusion of
our previous chapter, which specifies that the accuracy of the algorithm and
the tests is higher for the lower frequencies of the spectrum. In order that all
the frequencies within the range of interest are recorded the vibration signal
is sampled at 350 samples a second.
The picture in figure 4.1(a) and (b) illustrates the suspension of the car
frame. As mentioned earlier the bungee chords are used to suspend the car
frame through the front end considering the stiffness of the end. The
accelerometers are placed at chosen sections of the frame to record the
displacement. These sections are chosen based on the finite element
model. The elements of the car frame which exhibit high and distinct
response is identified using the model. The idea was to produce a waterfall
plot of the displacement along the section of the car frame and hence the
accelerometers available are placed as shown in the figure 4.1(d). The
distance between the accelerometers is six inches. The process is repeated
so that the response is recorded for the whole section chosen. The sections
chosen are measured along two planes. These planes and the sections
themselves can be specified using the pictures of the finite element model
provided in the figures 4.2(c) to 4.5(c). In figure 4.2 and 4.3 the
accelerometers are placed perpendicular to the plane of the specified circle
and in figure 4.4 and 4.5 they are placed in the plane of the specified circle.
55
(b) (c) (d)
Figure 4.1 :(a) BAJA car frame suspended for modal testing, (b) Bungee
chords used to suspend the frame using the front end of the frame.
(c) The Nl PXI device with the signal conditioners, (d) The displacement of
accelerometers over the section of the frame that is being considered.
56
Theoretically one reference point to place the accelerometer is sufficient to
capture all the natural frequencies of the beam unless there are repeated
modes in the structure. The choice of this reference point traces back to
the previous discussion about shape function. We can assume that the
response would be similar to a lightly damped structure such as the beam
used for validation unless the reference point chosen is away from the
joints. In most of the structures these joints provide the damping and the
damping ratio is higher close to the joints. One accelerometer is sufficient
to provide details to calculate the frequency response function of the
structure, but in this thesis more points are used with an intention to
analyze the mode shapes. The measurement is done with sets of 3
accelerometers at a time due to hardware limitation. For the second set of
data recorded the first of the three accelerometers are placed on the third
accelerometer position of the previous set. The acquisition is continued
until the same values are recorded at the common position while using the
accelerometer from the second set. The natural frequencies recorded by
the modal test and also the FEA model is listed in table 4.2. As seen in the
table the frequencies obtained by the two methods do not coincide due to
reasons of approximation. Also other factors causes the discrepancies 
material properties, presence of welded joints and others discussed in the
previous chapter.
57
Converged FEA Results Experimental Results
1st Nat Freq 1.59E11 0
2nd Nat Freq 1.59E11 0
3rd Nat Freq 6.90E04 0
4th Nat Freq 1.06E04 0
5th Nat Freq 1.31E03 0
6th Nat Freq 2.37E03 0
7th Nat Freq 51.1787 54
8th Nat Freq 54.5599 58
9th Nat Freq 76.5527 77
10th Nat Freq 83.2611 80
I 11th Nat Freq 103.662 97
12th Nat Freq 109.771 105 I
13th Nat Freq 110.971 109
14th Nat Freq 117.893 118
15th Nat Freq 123.325 125
16th Nat Freq 126.223 127
17th Nat Freq 133.275 133
18th Nat Freq 148.184 141
19th Nat Freq 151.837 147
20th Nat Freq 154.76 157
21st Nat Freq 157.836 161
22nd Nat Freq 160.626 164
23rd Nat Freq 165.56 166
24th Nat Freq 166.011 169
Table 4.2: Comparison between the natural frequencies converged by the
finite element model and extracted from the modal test.
58
Due to the variations seen an attempt is made to relate the natural
frequency with the corresponding mode shape by using the waterfall plot
of the response functions over a series of points in line on a chosen
section of the structure. The FEA screen shots are used to locate sections
that have distinct shapes. Accelerometers are placed at 6inch intervals
over selected sections where the response is distinct. The response at
each of these points is recorded. The data thus collected over the length
of the section are mesh plotted as a waterfall plot together as shown in the
figures 4.2(a) to figure 4.5(a). The natural frequencies that are closely
placed have been isolated and have been grouped with the corresponding
screen shots of the FEA model. These comparisons are made for the
modes whose frequencies are excited with the impact hammer used. The
comparisons provided me with a tool to relate the natural frequency
obtained by the two methods.
In the following figures a comparison between the response measured
and the finite element model is discussed. The input to all the channels of
the data acquisition device are recorded and processed to obtain the
frequency response function. These functions are read by the MATLAB
algorithm and are meshed together as shown in the following figures.
In the plots group (b) of each figure is the top profile of the particular mode
under consideration and group (c) is the side profile. In figure 4.2(c) the
59
circle denotes the section that is being considered for the comparison and
as mentioned earlier the displacement perpendicular to the plane of the
circle is measured. It can be seen from the top profile of the structure the
displacement is smaller in the part of the section closer to the frontend
and increases as the distance from the frontend increases. This is the
case for both the 7th and 8th mode. The MATLAB plot in figure 4.3(a)
illustrates a similar characteristic. The 1st accelerometer was placed close
to the frontend and the remaining positions moved along the length of the
section with a 6inch gap between them. The profile of the waterfall plot
relates to the profile seen in the finite element model. This leads to the
conclusion that the frequencies obtained from the experimental model
listed in the table for the first two modes are for the same modes that they
are listed against though there is a small deviation.
In figure 4.3 a similar analysis is done for the 9th and 10th modes. It can be
seen in the finite element model that the section under consideration
displays a sinusoidal type displacement and this can be seen in the
waterfall plot. The waterfall plot provides the absolute values of the signal
acquired through the setup and hence we see a depression around the
intermediate accelerometer positions that were placed closed to the area
of the section that is a node for the particular mode.
60
The presence of a node in the acquired signal is because of the intent of
the current work. My previous discussion about using a single
accelerometer was used to experimentally identify the natural frequencies
of the test structure and compare them with the natural frequency
obtained from the numerical model of the structure. My current work is to
relate the two lists to one another, as there are deviations.
61
Figure 4.2: (a) MATLAB waterfall plots for the isolated frequencies of 54
and 58 Hz. (b) and (c) The corresponding finite element mode shape with
the measurement area specified by the circle.
62
(a)
!
Figure 4.3: (a) MATLAB waterfall plots for the isolated frequencies of 77
and 80 Hz. (Id) and (c) The corresponding finite element mode shape with
the measurement area specified by the circle.
63
In figures 4.4 and 4.5 the rear section of the car frame is considered, which
the circles on the side profile of the figures illustrate. 6 points are identified
along the length of the section. The displacements along the plane of the
circle are measured. It can be seen in figure 4.4(c) that the displacement is
les at the first position of the accelerometer and after an initial dip it
increases as the distance from the top element increases. The waterfall plot
in 4.4(a) illustrates the mode shape acquired through the experimental
setup and it has a similar characteristic.
With figure 4.5 (c) the displacement is minimal and has a smooth increase
as the distance increases and the waterfall plot illustrates the experimental
values of the mode shape. These two comparisons over a different section
of the car frame illustrates that the natural frequencies listed in table 4.2
using the two different models belong to the same 14th and 16th modes
respectively. The modes chosen for comparison have a distinct
displacement profile in the finite element model. The frequencies compared
are at the lower end of the table as the sampling rate is 350 samples a
second. With this sampling rate the frequencies close to 175 Hz is not
considered accurate which can be deduced by Nyquists definition for the
sampling rate. With the comparisons and the data acquired the modal test is
considered to be a success and can be used for further tests in the field.
64
i
....
i
(a)
Figure 4.4: (a) MATLAB waterfall plots for the isolated frequencies of 118
Hz. (b) The corresponding finite element mode shape with the
measurement area specified by the circle.
65
(b) (c)
Figure 4.5: (a) MATLAB waterfall plots for the isolated frequencies of 127
Hz. (b) The corresponding finite element mode shape with the
measurement area specified by the circle.
66
5. Summary and Scope for Further Research
The two test structures used freefree beam and the BAJA car frame
were excited at various locations and the responses were compared with
the corresponding predicted response each time. Extensive analyses of
the comparison were conducted to device and validate a basic test
method. The test method developed can be used to extract the required
modal parameters and thus the physical properties of any complicated test
structure in the future. The knowledge and understanding the various
concepts and tools of EMA and experimental FEA can be used to
scientifically design a successful modal test. This in turn avoids the time
loss, which is usually incurred because of trial and error methods.
Though the tests were compared and validated using many plots and
extensive analysis, additional confidence can be gained by comparing the
derived mathematical model from the extracted modal parameters. Tests
can be conducted to characterize the dynamic behavior of a structure in
terms of natural frequencies, mode shapes and damping. Using the
parameters extracted from this analysis a mathematical model can be
generated to predict the transient behavior of the structure subjected to an
impact. It can be shown that a comparison between the measured and
67
predicted responses can be used to identify errors in the modeshape
parameters extracted from the experimental modal analysis.
68
APPENDIX
Appendix A: Algorithm to Solve the Analytical Model for a Vibrating Beam
load data.xls;
lab = abs(data);
nfreq = [85 236]; % Natural Frequencies located using LabVIEW
omega = [60 142]; % 'N' Points to obtain 'N' Modal constants
Mod_Consts = [lab(omega(1)) lab(omega(2))]';
% Modal constants picked graphically at omega using the LabVIEW plot
for k = 1 :length(omega)
for n = 1 :length(omega)
Non_lnv_Matrix(n,k)= 1/(4*piA2*(nfreq(k)A2omega(n)A2));
end
end
lnv_Matrix = inv(Non_lnv_Matrix);
Modal_Constants = lnv_Matrix*Mod_Consts;
for k = 1:length(nfreq)
for w = 1:350
freq(w) = w;
dof(w) = k;
matrix(w,k) = 1/(4*piA2*(nfreq(k)A2wA2));
end
end
FRF_data = abs(matrix*Modal_Constants);
FRF_data_acc = FRF_data;
% Conversion to the acceleration plots from the acquired signal
for n = 1:350
FRF_data_disp = FRF_data_acc/(2*pi*pi*freq(n)*freq(n));
end
%dimensions of the freefree beam converted to SI Units
L = 38.4375*0.0254;
h = 0.63*0.0254;
b= 1.0*0.0254;
rho = 2710;
E = 62.1 e9;
A = h*b;
Iner = b*hA3/12;
69
nroot = 20;
kk = 0;
mass = rho*L*h*b;
%get roots
for k = 1 :nroot
if k > 1
w(k) = w(k1) + 1.0;
else
w(k) = 0.0;
end
testom = fzero('beam_Freq',w(k));
fuk = beam_freq (testom);
if abs(fuk) < 1,0e6
kk = kk + 1;
alpha(kk) = testom;
if kk > 1
for nn = 1:kk1
if abs((alpha(kk)alpha(nn))/alpha(kk)) < 1.0e12
kk = kk1;
end
end
end
if alpha(kk) < 0
kk = kk1;
end
end
end
nroot = kk;
constl = sqrt(E*lner/(rho*A*LA4));
for n = 1:nroot
wn(n) = (alpha(n)A2)*const1;
freqn(n) = wn(n)/(2*pi);
end
Ien1 = 18*0.0254;
Ien2 = 1.5*0.0254;
for p = 2:nroot
const = (((sin(alpha(p)))(sinh(alpha(p))))/((cos(alpha(p)))
(cosh(alpha(p)))));
alp = alpha(p);
70
m_shape1(p) = sin(alpha(p)*len1/L) + sinh(alpha(p)*len1/L) + const *
(cos(alpha(p)*len1/L) + cosh(alpha(p)*len1/L));
m_shape2(p) = sin(alpha(p)*len2/L) + sinh(alpha(p)*len2/L) + const *
(cos(alpha(p)*len2/L) + cosh(alpha(p)*len2/L));
end
freqn = round(freqn);
lam = 0.0015; %coefficient of damping
for n = 1:350
freq(n)= n1;
disp(n)=0;
acc(n) = 0;
for j = 1:nroot
zeta(j) = 0.4*lam*(freqn(j)/freq(n));
freqd(j) = freqn(j)*sqrt(1zeta(j)A2);
disp(n) = disp(n)+ (m_shape1(j)*m_shape2(j)/((2*pi*freqd(j))A2
(2*pi*freq(n))A2));
acc(n) = acc(n)+
(m_shape1(j)*m_shape2(j)/((2*pi*freqd(j))A2+((zeta(j)*2*pi*freqn(j))+(2*i*pi
*freq(n)))A2))*4*pi*pi*freq(n)*freq(n);
end
end
figure
semilogy(freq,abs((acc))I'',freq,FRF_data_acc,'')
xlabel ('FREQUENCY/FontsizeM 8)
ylabel (XO/F'/FontsizeMS)
title ('Analytical() Vs Regenerated()','Fontsize',18)
figure
semilogy(freq,FRF_data_acc,'',freq,lab,'')
xlabel ('FREQUENCY'.'FontsizeM 8)
ylabel (XO/F'.'Fontsize'.IS)
title ('Experimental() Vs Regenerated()','Fontsize,,18)
71
Appendix B: Characteristic Equation for a FreeFree Beam.
function f = beam_freq(alpha)
f = cos(alpha)*cosh(alpha) 1.0;
72
Appendix C: Algorithm to Generate the Waterfall
Plot of Vibrating Sections
for n = 1 :length(name)2
filename= name(n+2).name;
file = load(filename);
len(n) = length(file);
per = len(n)/nsamp;
afile = zeros(nsamp,1);
for nn = 1:per
for mm = 1 :nsamp
p = (nn1)*nsamp+(mm);
afile(mm)=afile(mm)+file(p);
end
end
afile = afile/per;
filea(:,n)=afile;
end
w = [1 6 1 175 0 5];
figure
mesh(filea(:,:))
axis (vv)
73
REFERENCES
[1] Avitabile, Peter., Experimental Modal Analysis A Simple Non
Mathematical Presentation, Sound and Vibrations Control Magazine,
pp. 1 4, February 2001;
http://www.sandv.com/downloads/0101 avit.pdf.
[2] Ewins, D. J., Modal Testing: Theory and Practice, Research Studies
Press Ltd., England, June 1988 (Reprint).
[3] Ginsberg, Jerry H., Mechanical and Structural Vibrations Theory and
Applications, John Wiley & Sons, Inc., NY, 2001.
[4] He, Jimin and Fu, ZhiFang., Modal Analysis, ButterworthHeinemann,
Woburn, MA, 2001.
[5] Lang, George Fox., Experimental FEA...Much More Than Pretty
Pictures. Sound and Vibrations Control Magazine, pp. 12 17,
January 2005; http://www.sandv.com/downloads/0501lang.pdf.
74
