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DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS
MICHAEL A. KOPLOW
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
Michael A. Koplow
I would like to express my sincere gratitude for everyone who has helped make
this thesis possible. My greatest appreciation to all of my committee members
for their insight and comments. Special thanks go to Dr. Mann and Dr. Sankar
for their advice and confidence. I would also like to thank Raul Zapata, Abhijit
Bhattacharyya, Ryan Carter, and the MTRC for their time, effort, and energy
during this work.
For my dad, thank you for your love and guidance. You have made this all
possible. I hope my thoughts and inspirations come as free flowing for me as they
did for him; for he is the spirit that guides me. Finally, I would like to thank my
girlfriend, Briana, my sister, Sarah, my brother, David, and my mother for all their
support during this project. Through their continued love and support, this project
was a success.
It is common sense to take a method and try it; if it fails, admit it frankly and
try another. But above all, try something. Franklin D. Roosevelt
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................... ...... iii
LIST OF TABLES ...................... ......... vi
LIST OF FIGURES ................... ......... vii
ABSTRACT ...................... ............. ix
1 INTRODUCTION .................... ....... 1
1.1 Introduction to the Problem .......... ....... .... 1
1.2 Machining and the Material Removal Process ............ 2
1.3 Application to Industry ................... ... 2
1.4 Literature Survey ........................... 3
2 EXPERIMENTAL MODAL TESTING ................... 7
2.1 Dynamic Response of Linear Systems ................ 7
2.2 Impact Testing Overview ........... ............ 8
2.3 Contact Sensor Mass Loading Effects ... . .... 13
3 DYNAMIC RESPONSE PREDICTION OF CONTINUOUS BEAMS 15
3.1 Derivation of the Equation of Motion ............ 15
3.2 Dynamic Response Prediction of Uniform Beams . .... 17
3.3 Experimental Response of Uniform Beams . . ..... 20
4 DYNAMIC RESPONSE PREDICTION OF DISCONTINUOUS BEAMS 23
4.1 Receptance Derivation for Discontinuous Beams with Aligned Neu-
tral Axes .... .......... ............ 23
4.1.1 Discontinuous stepped beam solution for force excitation
at location C . . . . .... .. 23
4.1.2 Discontinuous stepped beam solution for force excitation
at location A ................ .... .. 28
4.1.3 Extension of the analytical solution for applied couples .29
4.1.4 Comparison of the analytical solution to receptance coupling 31
4.1.5 Experimental verification of the stepped beam solution .38
4.2 Receptance Derivation for Discontinuous Beams with Misaligned
Neutral Axes ..... . ......... 40
4.2.1 Discontinuous misaligned beam solution for force excita-
tion at location C ........... ... ..... 41
4.2.2 Experimental study of the misaligned neutral axis solution 45
5 STABILITY OF LAYER REMOVAL PROCESS . . 49
5.1 Limiting Chip Width for Machining Process . . ... 49
5.2 Mode Shape Analysis as a Function of the Notch Height . 52
6 CONCLUSIONS AND FUTURE WORK ................. .. 57
REFERENCES ............. .. ............ ... 60
BIOGRAPHICAL SKETCH .............. . .. 63
LIST OF TABLES
3-1 Euler-Bernoulli beam notation. ................ ..... 16
3-2 Boundary conditions for classical beam ends. ............ ..18
3-3 C'!i teristic equations for the free vibration of uniform Euler-Bernoulli
beam s. . . . . . .. . . 19
3-4 Beam primary receptances. .................. .... 19
4-1 Notation for force excitation at position A .............. ..30
4-2 Discontinuous notched beam continuity conditions. .......... ..42
4-3 Axial vibration boundary conditions for classical beam ends. . 43
4-4 Notation for FRF with force excitation at position C including a mis-
aligned neutral axis. .................. .... 45
LIST OF FIGURES
1-1 Alcoa testing procedures .............. ... ... .. .. 3
2-1 Signal processing overview ............... .. 9
2-2 Comparison of different modal hammers for: (a) a force measurement
in the time domain and (b) a force amplitude measurement in the
frequency domain. .................. .... 11
3-1 Schematic of a fixed-free forced beam. ................ .. 15
3-2 Free body diagram of a beam element. ............... 16
3-3 Schematic of a uniform beam subjected to a force of amplitude F
and frequency u, applied at x L. ................. 20
3-4 Experimental setup for FRF testing on a uniform beam. . 21
3-5 Comparison of experimental (solid) and analytical (dashed) FRFs for
the uniform beam. .................. .... 22
4-1 Schematic of the stepped beam with aligned neutral axis and free bound-
ary conditions at locations A and C. ............... 24
4-2 Schematic of a stepped beam subjected to: (a) a force of amplitude
F and frequency w, applied at location C, (b) a force of amplitude
F and frequency w, applied at location A,(c) a couple of amplitude
M and frequency w, applied at location C, and (d) a couple of am-
plitude M and frequency u, applied at location A. . ... 24
4-3 Receptance coupling components (a) and assembly (b) models for ex-
citation at C. . . . . . .. . 32
4-4 Receptance coupling components (a) and assembly (b) models for ex-
citation at 3. . . . . . .. . 35
4-5 Beam dimensions for comparison of the stepped beam analytical so-
lution to receptance coupling and experiment. Dimensions are in
(m m ). . . . . . .. . . 36
4-6 FRF comparison between analytical (solid) and receptance coupling
(dashed) methods when forced at position C. . . 36
4-7 FRF comparison between analytical (solid) and receptance coupling
(dashed) methods when forced at position A. . . 37
4-8 Experimental setup for FRF testing on a stepped beam ....... ..38
4-9 Comparison of experimental (solid) and analytical (dashed) FRF when
forced at position C. .................... ...... 39
4-10 Comparison of experimental (solid) and analytical (dashed) FRF when
forced at position A. .................... ..... 40
4-11 Schematic of a discontinuous notch beam with a misaligned neutral
axis and free boundary conditions at locations A and C. ..... ..41
4-12 Free body diagram of (a) forces and (b) displacements for a discon-
tinuous notched beam with a misaligned neutral axis. . ... 41
4-13 Dimensions for analytical study of beam with jump discontinuity. Di-
mensions are in (mm). ............... .... .... 46
4-14 Comparison of the analytical FRF with a misaligned neutral axis (solid)
to the analytical FRF with an aligned neutral axis (dashed) when
forced at position C. .................... ...... 47
4-15 Dimensions for experimental study of beam with a jump discontinu-
ity forced at the end position. Dimensions are in (mm). . 47
4-16 Comparison of experimental (solid) and analytical (dashed) FRF for
3 sectioned notch beam with forcing at the end location. . 48
5-1 Schematic of the clamped-free notched beam during machining. Di-
mensions are given in (mm). .............. ...... 51
5-2 Analytical FRF for the notched beam with fixed-free boundary con-
ditions. ............... ............. .. 52
5-3 Experimental mode shapes as a function of the notch height for Al-
coa testing conditions. ................ .. .... 53
5-4 Analytical mode shapes as a function of the notch height assuming
fixed-free boundary conditions. ................ .... 54
5-5 Analytical mode shapes as a function of the notch height assuming
compliant-free boundary conditions. ............. .. .. 55
5-6 Comparison of limiting chip thickness, bum, as a function of the notch
depth for: (a) experiment, (b) a fixed-free model, and (c) a compliant-
free model. ............... ......... .. .. 56
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS
Michael A. Koplow
C'!I wi': Brian P. Mann
Major Department: Mechanical and Aerospace Engineering
The dynamic response of discontinuous structures is often of vital importance
in the design of many engineering applications. In many cases, it is preferable to
have an analytical model of the system which can reduce the amount of design,
' in.- and manufacturing of products. This work grew out of the need to examine
the dynamic response of a discontinuous beam used in an industrial application.
As milling operations were being performed on the beam, the natural frequencies
of the beam would shift, leading to unstable vibrations of the cutting process. The
goal of this research was to ai 1v.. the dynamic response and characterize the
stability of the discontinuous beam.
The present work considers beams with two types of discontinuities. The first
is that of a stepped beam with an aligned neutral axis. The second is that of the
notched beam which contains a jump discontinuity and a misalignment of the indi-
vidual beam segments neutral axes. The discontinuous beam is modeled as separate
uniform Euler-Bernoulli beams with continuity conditions at the discontinuity. The
analytical results are compared to receptance coupling substructure analysis and
experiment. Results show that the stepped beam model produces very accurate
results compared to other analytical techniques and experiment. Results for the
notched beam show errors due to neglecting shear and rotary inertia components of
the beam segments.
A stability analysis is performed considering the workpiece to be the most
flexible portion of the cutting operation. Additionally, a study of the notch height
is performed to analyze the change in dynamic response as a function of the
material removal process.
1.1 Introduction to the Problem
Structural dynamics is widely used in research and in industry to make
accurate predictions of the response of many different structures. While the
modeling and dynamic response predictions for continuous structures has been well
developed, there are relatively few techniques available for modeling discontinuous
structures. Difficulties often arise in the modeling of structures with complex
geometry; i.e. structures containing joints, connections, or notches. In many cases,
these structures are either modeled with finite element packages or tested using
experimental work pieces. Design using these methods are often time consuming
and costly and thus it is often beneficial to have analytical solutions for structural
Beams provide a fundamental model for the structural elements of many
engineering applications. For instance, helicopter rotor blades, spacecraft antennae,
and robot arms are all examples of structures that may be modeled with beam-
like elements [1; 2]. The work presented in this thesis grew out of the need to
examine an industrial machining process where the dynamic response of a beam-
like structure was the primary limiting factor. This material removal process
additionally presented two unique challenges: (1) a change in the beam's dynamic
response and machining stability limit as each 1-,-r of material was incrementally
removed; and (2) a discontinuity in the beam structure which prevented direct
application of conventional beam theory. The goal of this work is to present
analytical solutions for the dynamic of a discontinuous beam that were developed
to better understand the aforementioned industrial process.
1.2 Machining and the Material Removal Process
Machining is the most important manufacturing process in terms of time
and money spent. Machining involves the process of removing material from a
workpiece in the form of chips. Researchers have expended many efforts to identify
the limits of stability and safe cutting conditions, depths of cut and spindle speeds,
for milling operations. The goal is to prevent chatter, or undesired large vibrations.
C'! ,I I i r, related to the dynamics of the structure during i, 1 i ini.- will adversely
affect the quality of the produced surface, and may lead to increased tool wear and
tool failure. The mechanism for chatter is commonly identified as the regeneration
effect [3-5]. In most models, chatter occurs due to the interactions between the tool
and the wavy surface left on the workpiece from previous revolutions.
Stability analyses of machining in literature show frequency diagrams labeling
stable and unstable depths of cut as a function of various spindle speeds. The
stability limits are obtained assuming that chatter occurs due to dynamics of the
spindle holder and tool resonance frequencies. The tool is usually considered the
most flexible part of the dynamic system. However, during milling operations on
beams, the natural frequencies of the workpiece shift, causing the beam to become
the most flexible part of the system. This shift may also result in chatter. This
type of chatter can occur even after several successful passes of stable material
removal have been performed.
1.3 Application to Industry
Alcoa, a in i ii aerospace aluminum manufacturer, has implemented a 1i,-,r
removal process to experimentally extract the residual stress of various aluminum
alloys as shown in Fig 1-1. The test procedure requires the removal of a 1-,-r of
material, using a machining process called milling, and the measurement of the
workpiece static displacement. Static displacement measurements are used to
estimate the remaining residual stress in the material. The milling process and
static measurement cycle is repeated several times to determine the residual stress
at each l-v-r of the workpiece material.
Figure 1-1: Alcoa testing procedures.
The concern is that the large amplitude vibrations can occur during the
machining process; these vibrations are likely to cause (1) a shift in the transducer
zero location or create an offset from the transducer measurement; (2) the nominal
depth of cut will be different than the actual depth of cut due to relative movement
between the tool and the workpiece; and (3) transverse tool vibrations will either
remove more or less material than anticipated (i.e. a larger/smaller slot than the
one used for residual stress calculations). The uncertainty created from these
factors will inevitably diminish the ability of Alcoa to correlate the initial residual
stress distributions to changes in material processing. Therefore, the primary
concern of this work is to reduce the large amplitude vibrations by analyzing the
machining stability limits of the workpiece at various stages of the material removal
1.4 Literature Survey
A literature survey has shown that the dynamic response for the transverse
vibration of continuous Euler-Bernoulli beams has been well studied using both
modal superposition techniques  and receptance techniques . Modal super-
position requires first solving the eigenvalue problem for the free vibration of the
structure without damping. As the name -ii-.-- -i- the forced vibration solution is
obtained by assuming orthogonality of the modes and then summing up the indi-
vidual responses of each mode. Receptance techniques solve for the forced vibration
solution by assuming a solution for the mode shape functions and by applying
forces directly into the boundary conditions. Receptance techniques do not require
the two step process of modal superposition, but are limited somewhat by forcing
the applied forces into boundary conditions.
The static behavior of Euler-Bernoulli beams with jump discontinuities has
been studied using generalized solutions [8-10]. In these methods, the disconti-
nuities are modeled as delta functions at the point of discontinuity. The authors
investigate the means by which the discontinuities can be applied to the governing
differential beam equations. While these authors do a superb job at modeling the
discontinuities in the static sense, they do not apply their formulations for dynamic
Several authors have studied the free vibration of stepped beams with aligned
neutral axes. The discontinuous structures have previously been treated to find
natural frequencies and mode shapes expressed as determinants equated to
zero [11-14]. These works analyze the boundary conditions and continuity con-
ditions to solve for the system frequency equations. Jang and Bert [11; 15] obtained
the first exact results of the frequency equation for stepped beams with classical
boundary conditions. Maurizi and Belles  extended the work of Jang and Bert
to include the effects of elastically restrained boundary conditions. De Rosa and
coworkers [13; 16; 17] analyzed the free vibration of stepped beams with elas-
tic supports including an intermediate support, and the effects of concentrated
masses. N i.,1. waran [14; 18; 19] considered the effects of multiple beam spans,
a non-symmetrical rigid body at the discontinuity, and applied static axial forces.
Tsukazan  studied the use of a dynamical bases for computing the beam modes.
While all of these previous works have treated the free vibration case, very little
work has been presented on the forced vibration case.
Alternative coupling techniques, such as receptance coupling substructure
synthesis [21-23], can also be used to examine the dynamic behavior of discontin-
uous beams. Substructuring methods allow the prediction of assembly frequency
response functions (FRFs) using FRFs from individual components obtained ei-
ther analytically or experimentally. The solution forms a two by two matrix of
the primary receptances of the individual beam components for each frequency.
The technique requires an inversion of the two by two matrices per frequency. For
high-resolution FRFs, the solution becomes computationally expensive.
In this research, an analytical solution for the dynamic response of discontinu-
ous beams is considered. Two different discontinuities are considered: (1) a stepped
beam with both aligned neutral axes and (2) and notched beam consistent with
the material removal process detailed earlier. The analytical results are verified
by receptance coupling methods and via experiment. One limitation of this work
is that a partial differential equation is needed to obtain an assumed mode shape
solution. Also, it is required to have information about the continuity conditions
between individual components. The presented work can easily be extended to
beams with n-beam sections and different classical boundary conditions.
The organization of the thesis is as follows. C'!i lpter 2 details a background
of information concerning experimental modal testing. The chapter outlines the
experimental procedure, data analysis techniques to obtain frequency response
measurements, and techniques to eliminate mass loading effects of experimental
data due to contact sensors. ('!i lpter 3 gives a derivation of the uniform Euler-
Bernoulli beam as well as the procedure to obtain frequency response functions for
classical boundary conditions. The .111 Iiltical results are verified by experiment.
('!i lpter 4 extends the analysis of the uniform beam to include discontinuities.
The cases of stepped and notched beams are considered. Results are verified using
receptance coupling techniques and experiment. C'! Ilpter 5 analyzes the stability
of the milling process for discontinuous beams. Additionally, the chapter examines
the dynamic behavior of the beams as a function of the notch height using both
experimental and analytical data. Finally, ('! Ilpter 6 summarizes the conclusions
and provides recommendations for future work.
EXPERIMENTAL MODAL TESTING
This chapter details the basic operations for acquiring experimental frequency
response measurements. The goal of this chapter is to provide an overview of
experimental modal analysis techniques. Specifically, this chapter provides infor-
mation about frequency response measurements, methods for obtaining time series
measurements, and the data analysis techniques required to convert time domain
measurements to frequency domain measurements. The discussion is followed by a
brief description of mass loading effects due to contact sensors.
2.1 Dynamic Response of Linear Systems
The impulse response function, h(r), can fully describe the dynamic response
of a linear system. The impulse response is the output of the system due to a
corresponding unit impulse applied at any time r. The output y(t), for any input
x(t), is given by the convolution integral 
y(t) h(r)x(t 7)dr. (2.1)
The convolution integral of the input and impulse response is usually very difficult
to solve in the time domain. Converting the time domain signal into the frequency
domain allows for easy computation of Eq. (2.1). As will be shown, the convolution
integral in the time domain becomes simple algebra in the frequency domain.
The Fourier transform is used to convert the time domain impulse response
function h(r) into the frequency domain frequency response function (FRF). For a
physically realizable system, the frequency response function, given by Bendat and
Piersol , is
H(f) j= h(r)e-j2fLdr, (2.2)
where j is the imaginary term. By definition, the frequency response function is
defined as the Fourier transform of the impulse response function. The Fourier
transform is typically applied in many computational packages using the Fast
Fourier Transform (FFT) which restricts the limits of the integral to a finite time
interval. Taking the Fourier transform of Eq. (2.1) yields
Y(f) H(f)X(f), (2.3)
where the capital letters denote Fourier transforms and f denotes frequency
dependence. The common notation is to use lower case letters to represent time
domain signals, h(r), while capital letters are used to represent frequency domain
signals, H(f). Equation (2.3) shows the relationship between the frequency
response function H(f) and the input and output.
In practice, the term FRF is often used interchangeably with the term transfer
function. However, this is a misnomer as there is a subtle difference. Transfer
functions, as typically applied in control theory, are Laplace transforms of the
impulse response function. The difference is found in the integration of Eq. (2.2).
Rather than integrating only the imaginary variable jf, Laplace transforms
integrate a complex variable s = a + jf. Therefore Laplace transforms account
for transient and steady state responses whereas the Fourier transform assumes
an invariant signal. Laplace transforms can be thought of as more general because
they plot the poles and zeros on the complex plane. Fourier transforms ignore the
real portion and are only concerned with the jf axis of the complex plane.
2.2 Impact Testing Overview
The following section describes a method for obtaining frequency response
functions from a physical system. The discussion is limited to impulse inputs
from impact hammers and output responses measured by accelerometers. In this
method, both the input and output responses are measured. From Eq. (2.3),
the frequency response function is simply the response output divided by the
force input in the frequency domain. Extensions for other types of excitation and
measurement can be found in literature [25; 26].
Figure 2-1 shows an illustration of the process to obtain the frequency re-
sponse function from time series data. The steps are broken down into a sequence
for the input (hammer/force measurement) and the output (accelerometer mea-
surement). An overview of the modal testing process is listed below. A detailed
explanation of each step follows.
Time domain measurements are obtained using a data acquisition system and
modal testing equipment.
Windows are applied to clean the data and avoid leakage.
The fast Fourier transform is used to convert the input and output data into
the frequency domain.
The FRF is obtained as the output over the input. The results are averaged
over multiple impacts to ensure good coherence.
TIME DOMAIN ANTI-ALIASING WINDOWING APPLY FFT AVERAGING REAL AND
DATA FILTER IMAG FRF
I-^L -/ \p
I- I_ 'r I I
Figure 2-1: Signal processing overview.
The first step is to acquire time series measurements using a data acquisition
system, a modal hammer, and a transducer. For the purposes of this discussion,
it is assumed to have only one input and one output, but multi- input and output
systems are possible. During the analog to digital conversion process, an anti-
aliasing filter is applied to remove any high frequency signals that may exist in the
data. The Nyquist frequency states that the highest possible observable frequency
is equal to half of the sampling frequency. Therefore, the sampling frequency must
be greater than twice the maximum frequency of interest present in the signal. If
the data are sampled at too low of a rate, the signal will be aliased and the correct
frequency content will not be observable. For reconstructing the true signal, a
general rule of thumb is to sample at 5-10 times the highest frequency of interest.
If the signal has aliased, it is not possible to reconstruct the true signal and the
signal is unusable.
The impact hammer is a popular excitation system because it is easy to
implement. The energy applied to the structure is directly related to the hammer
mass. Hammers range in size from a few ounces to several pounds with varying
contact tip materials. The frequency content depends on the hammer mass and
the contact stiffness. The choice of modal hammer depends on the application
and desired frequency range. Figure 2-2 shows a comparison of the effects of
various modal hammers. Softer tips will excite a larger amplitude of motion, but
will contain a smaller frequency bandwidth. Stiffer tips typically excite a larger
frequency range, but will contain less amplitude. Additionally, larger hammers will
provide more energy and will thus excite for longer time. Smaller hammers will
show a more refined impulse that dissipates more rapidly.
Using a modal hammer, the input should ideally show a single impulse. How-
ever, double hits (or multiple impacts) can occur and are sometimes unavoidable.
The double hit problem can often be minimized by selecting the smallest possible
hammer. The output should ideally show a damped response with transients that
decay to zero. Depending on the sampling rate, boundary conditions, and number
Figure 2-2: Comparison of different modal hammers for: (a) a force measurement in
the time domain and (b) a force amplitude measurement in the frequency domain.
of samples taken, this may or may not be the case. To prevent leakage in the data,
it is alv--x preferable to allow the system to naturally damp out.
Next, windows are applied to remove static and random noise from the signal.
For the input signal, as stated above, the force should be a perfect impulse at one
instant and equal to zero everywhere else. Impulses are finite in duration and the
FFT of the impulse provides the input frequency spectrum. Therefore any value
present, excluding the impulse, can be regarded as noise and will be eliminated. It
is important to retain any double hits because they are integral to the system and
should not be erased from the signal. To remove input noise, multiply the signal by
a square wave filter (force window) that is equal to one during the impulse and zero
everywhere else. Because only noise components are removed, the force window will
not add any artificial effects.
For the output signal, the signal should begin at zero (a number of precursor
scans taken before the impact) and end at zero after the impact has damped out.
The Fourier transform requires that a signal be periodic in order to obtain the
FFT. While accelerometer tests are not periodic, forcing the signal to zero at
the ends will result in an accurate transformation. If the signal has not naturally
attenuated to zero, the Fourier transform will distort the frequency domain signal.
The distortion occurs in the form of leakage which is a smearing of the frequency
content over a wide range . To prevent leakage, the sample should be allowed
enough time for the signal to naturally attenuate to zero. In several instances it
becomes impractical to allow for naturally system decay due to file size limitations
or time constraints. In these cases, an exponential window can be applied to
force the signal to zero. Although it is sometimes necessary to use an exponential
window to prevent FFT distortion, the exponential window should be used with
caution as it will add artificial damping into the system.
Once the signals have been processed to reduce noise and leakage, the next
step is to convert the time domain signal into the frequency domain using the FFT
transform algorithm. The FFT provides the complex (real and imaginary) valued
linear Fourier spectrum of the input and output signals. The ratio of the output
over input spectrums at this point is called the accelerance function given by J.A
The accelerance function has resulted because the output was measured using
an accelerometer. Converting the accelerance function to a receptance function
requires the following. Consider a sinusoidal input given by
F(t)= Rsint, (2.4)
where R is the amplitude of excitation and a is the frequency term in the units
of rad/s. From linear system theory, a sinusoidal input will result in a sinusoidal
output of the same frequency. Therefore, the output will have a response of the
same form as Eq. (2.4). Differentiating the response twice yields -W2F. Converting
the system from accelerance to receptance requires dividing by -C2. The measured
FRF is the ratio of output over input divided by -c2,
The results are averaged over multiple impacts to improve accuracy and reduce
random noise. The system response can then be plotted in terms of the real and
imaginary responses or similarly magnitude and phase plots. The final step is
to determine the accuracy of the experiments. The coherence function  is
one measure that can be used to gage the effectiveness of the impact tests. The
coherence function is a real valued quantity which provides a measure of the linear
dependence of two impact tests as a function of frequency. For a linear system, the
coherence function, 72, is given by Bendat and Piersol  as
2 IGy(f)12 (2.6)
where Gxy is the cross power spectrum between the input and output signals,
Gxx is the input power spectrum, and Giii is the output power spectrum. The
value of the coherence function will be between 0 and 1, whereby a value of 0
corresponds no relationship between two signals and a value of 1 corresponds to a
perfectly linear relationship. There are several causes of poor coherence functions.
The coherence will drop as a result of nonlinearities in the system and due to poor
signal to noise ratios for anti-resonance frequencies. Furthermore, the coherence
will be poor if the user does not impact the same location with the same force. It
is important to note that the coherence must be calculated using data containing
multiple averages. A test with only one impact will misleadingly show perfectly
linear data because the coherence will measure the single test onto itself.
2.3 Contact Sensor Mass Loading Effects
Discrepancies between measured and theoretical FRFs are partly due to mass
loading effects due to the added inertia of the accelerometer. Several authors [25;
27; 28] have shown that measured and predicted FRFs may be compensated to
include the additional dynamics of the sensor. The correction for a driving point
FRF, or direct impact FRF, from Ashory , is given by
At 1 (2.7)
1 1 MA7
where A11 represents a direct impact accelerance FRF, the super script m rep-
resents the measured accelerance, the super script t represents the theoretical
accelerance function without the additional inertia, and M is the extra mass of the
accelerometer in (kg). This form of correction is known as mass cancelation. As
shown in Eq. (2.7), the mass loading effect is frequency dependent. Equation (2.7)
is used to remove the effects of the additional inertia added by the attached sen-
sor. In this formulation, the experimental results are altered to resemble the true
vibrations of the beam if the sensor were not attached.
Due to limited bandwidth and noise in the measured signal, applying Eq. (2.7)
will distort some modes in the experimental data. Although the signal will shift
to the correct frequency response, the damping ratio is incorrectly shifted and will
distort some modes in the FRF. To avoid these problems, it becomes easier to
shift the theoretical response to the experimental response for comparison. In this
formulation, the theoretical response is shifted to resemble the vibrations of the
beam as if the additional dynamics were included. The theoretical result can be
compensated to include the accelerometer mass by 
A 11 (2.8)
An 1 + MA\, '4
where the terms are the same as defined above. Because theoretical models contain
no noise and can be set to include a very fine resolution, the modes will shift to the
correct locations without distortion. For transfer FRFs, those that are measured in
a different location than sensed, the correction is given by
A77j 1 At (2.9)
12 1 + MAt'
where A12 represents a transfer accelerance FRF measured at one location and
forced at another. As discussed previously, the accelerance result can be trans-
formed into receptance by dividing by -L2.
DYNAMIC RESPONSE PREDICTION OF CONTINUOUS BEAMS
This chapter develops the equations of motion for continuous, uniform beams
and reviews the method to acquire receptance functions for the case of excitation
at the boundary conditions. For this analysis, the system will be modeled using the
Euler Bernoulli beam theory, neglecting shear and rotary interia. The beams are
modeled using receptance techniques whereby external forces are applied directly
to the boundary conditions. The discontinuous beam formulation will be shown in
later chapters to be an expansion of the uniform beam.
3.1 Derivation of the Equation of Motion
This section derives the equation of motion for an Euler-Bernoulli beam.
Figure 3-1 shows a typical fixed-free beam with an applied transverse force.
Figure 3-2 shows the free body diagram for a differential beam element with a
constant cross sectional area, where the beam notation is defined in Table 3-1.
Figure 3-1: Schematic of a fixed-free forced beam.
The shear force and bending moments illustrated in Fig 3-2 show the positive sign
convention for the beam element. Positive shear and bending moments are assumed
to produce upward displacements and rotations.
Q \ dx
Figure 3-2: Free body diagram of a beam element.
Table 3-1: Euler-Bernoulli beam notation.
f(x,t) = Applied transverse force as a function of space (x) and time (t)
Q = Shear force acting on the cross section
M = Internal bending moment
p = Mass density (kg/m3)
A = Cross sectional area (m2)
E = Young's Modulus (Pa)
I = Area moment of inertia about the neutral axis (m4)
For the Euler-Bernoulli beam analysis, there are 2 underlying assumptions.
The first assumption is that the beam is long and slender. The length of the
beam is assumed to be much greater than the height, such that the shear force is
dominated by the bending stresses. As a result, the shear stresses and rotatory
inertia terms are considered negligible. Therefore Q = OM/8t. The second
assumption is a small slope of deflection curvature of the beam. This is the
assumption of small angles. For small angles, it can be shown that 0 = 9v/ax. In
practice, the Euler-Bernoulli approximation is valid when the beam length is at
a minimum of 5 to 10 times its height. In these cases, the Euler-Bernoulli beam
shows very accurate results for the lowest modes. Errors will begin to acrue for
higher modes. However, the a i1,J ,i -i; presented in this work is most interested in
the fundamental mode vibration because the small amplitude of the higher modes
is less important.
Summing the forces on the differential element and using Newton's second law,
it follows that
Fy = ma, => Q dx + f(x,t)dx = pAdx (3.1)
O1 x at2
Application of the first assumption of slender beams to Eq. (3.1) results in
82M /1 82(x. (.)
x + f(x, t) p A t2 (3.2)
From bending theory, recall M = EI Applying the second assumption for
small deflections, it follows that
-02 02vU 02v(, t)
2 (EI82) + fa) (x,) pA N (3.3)
Assuming only uniform beams, Eq. (3.3) may be rewritten as
pA + El f( xt) (3.4)
A t2 + X2
Equation (3.4) represents the equation of motion for transverse vibration of
a uniform Euler Bernoulli beam. The result is a fourth order partial differential
equation dependant on space and time.
3.2 Dynamic Response Prediction of Uniform Beams
The solution to Eq. (3.4), when subjected to an input of frequency w (rad/s),
can be separated into a solution in space and time
v X(x) sin ut. (3.5)
Substitution of Eq. (3.5) into Eq. (3.4) yields dependence upon the spatial
4( 34X(x) = 0, (3.6)
34 w2pA (3
EI(1 + i) '
is the solution to the eigenvalue problem and r] is a non-dimensional structural
damping factor. The general mode shape solution to X(x) is
X(x) = asin 3x + bcos x + c sinh 3x + dcosh 3x, (3.8)
where a, b, c, and d are constants determined by suitable boundary conditions. The
free vibration solution is written as a 4 by 4 determinant obtained by applying 4
boundary conditions to Eq. (3.8). The boundary conditions for classical conditions
are listed in Table 3-2. The frequency equation solution becomes a transcendental
function where 3 is the unknown quantity. Values for 3 are determined by roots of
the transcendental equation. Because the equation is transcendental in nature, the
roots are not easily obtained. The values may also be found by the zero crossings
in a plot of the equation as a function of 3. The natural frequencies are then
determined solving for w in Eq.(3.7). The characteristic equations for the free
vibration problem for fixed-free and free-free beams are given by Balachandran and
Magrab  and are listed in Table 3-3
Table 3-2: Boundary conditions for classical beam ends.
v = 0 for a fixed end,
El D = E17 0 for a free end,
v =EI s = 0 for a pinned end, and
H1 El 0 for a sliding end.
The forced vibration solution  is obtained by equating applied forces into
the boundary conditions. Applied forces are equated to the shear force while
Table 3-3: C'!i o .:teristic equations for the free vibration of uniform Euler-
Fixed-Free Beam: cos 3L cosh 3L + 1 = 0
Free-Free Beam: cos f3L cosh fL 1 = 0
applied couples are equated to the bending moment. The signs on the forces are
determined by the positive sign convention as shown in Fig 3-2. The FRF for a
uniform beam is obtained by solving a set of four equations with four variables.
Note that for any position of a uniform beam, it is possible to model the response
as a 2 by 2 matrix of its primary receptances. These receptances form the transfer
functions of the beam, as listed in Table 3-4.
Table 3-4: Beam primary receptances.
1 Translation due to an applied force v/F
2 Translation due to an applied moment v/ \
3 Bending due to an applied force 0/F
4 Bending due to an applied moment 0/ \!
Consider the free-free uniform beam shown in Fig 3-3. The desired FRF is
the direct receptance at the location x = L due to an applied force. The boundary
conditions for this case are
At x = 0 El = 0, (3.9a)
El- 0, (3.9b)
At x = L EI =L 0, (3.9c)
E = -F sin t. (3.9d)
Applying the boundary conditions given in Eq. (3.9) to Eq. (3.8) results in the
following FRF solution 
v sin fL cosh fL cos /L sinh f3L
F EI(1 + i)/33(cos 3L cosh fL 1) '
where the denominator, cos 3L cosh 3L = 1, forms the frequency equation whose
roots determine the natural frequencies of the system.
Figure 3-3: Schematic of a uniform beam subjected to a force of amplitude F and
frequency a/, applied at x=L.
3.3 Experimental Response of Uniform Beams
This section provides experimental verification for the uniform beam recep-
tance functions. The experiment consists of the uniform beam of aluminum 7051
which is 393.7 [mm] long with a cross section that is 25.4 [mm] wide and 19 [mm]
tall. Free-free boundary conditions were obtained by hanging the beam with a taut
nylon -1 ii.- rigidly attached to the end of the beam via a thin piece of plexi-glass
as shown in 3-4. The free boundary conditions were applied because they provide
very accurate and repeatable results. Fixed boundary conditions are very difficult
to experimentally obtain because there is .l1.- i,- some measure of compliance in
the connection. Experiments were conducted by forcing the beam with a modal
hammer and obtaining the response with a low mass accelerometer mounted onto
the beam. Mass loading effects due to the contact sensor were corrected using
Eq. (2.8). The accelerometer mass was measured to be m = 0.8 grams. By com-
parison, the total mass of the beam is 0.54 [kg]. The material has a density of
p = 2830 [kg/m3] and a Young's modulus of E = 71 [GPa]. Structural damping
was obtained as r = 0.0003 via a best fit approximation to the experimental data.
impact I I so
hammer (U -
Figure 3-4: Experimental setup for FRF testing on a uniform beam.
Figure 3-5 shows the results for the experiment for the first 2 modes for direct
FRFs at location x = L. The data show modes at 630 Hz and 1706 Hz for the
experimental test. As the data show, the experimental results are in excellent
agreement with the analytical predictions. Results show that analytical predictions
have higher natural frequencies than the experimental measurements. Because
damping is fit to the entire structure, it does not perfectly match for each mode.
In this case, the length to height ratio was 21:1 and the results show that the
Euler-Bernoulli beam approximation is capable of modeling the response of the first
2 modes. It is expected that higher modes would show greater errors.
x 103 Total Response
500 1000 1500
x 103 Mode 1
600 620 640 660
600 620 640 660
and analytical (dashed) FRFs for
Figure 3-5: Comparison of experimental (solid)
the uniform beam.
RESPONSE PREDICTION OF DISCONTINUOUS BEAMS
This chapter develops the receptance functions for the dynamic response of
discontinuous beams. For brevity, the derivation will applied to the case of free
boundary conditions at the end locations with one change in cross section. The
discontinuity is treated by assuming two separate uniform Euler-Bernoulli beams
coupled with continuity conditions at the joint between beams. The problem is
solved as a boundary value problem with 8 unknown constants. This method
can be easily expanded for beams containing different boundary conditions or
additional uniform sections. The case of a notched beam with an unaligned neutral
axis is treated with a coupling of the transverse bending and axial vibrations.
4.1 Receptance Derivation for Discontinuous Beams with Aligned Neutral Axes
This section develops the receptance functions for a discontinuous beam
with an aligned neutral axis and the case of free boundary conditions at the end
locations with one step change in cross section as shown in Fig 4-1. The solution
can be viewed as expansion of the uniform beam receptance derivation whereby
the individual sections are modeled as separate beams with continuity conditions
applied at the joints. The following section will solve for the cases of force and
couple excitation as shown in Fig 4-2. The results are compared to an alternative
solution using receptance coupling and to experiment.
4.1.1 Discontinuous stepped beam solution for force excitation at location C
This section develops the frequency response function for force excitation at
position C as shown in Fig 4-2(a). The solution for the first beam section (A-B) is
< L1 -- L >
A B C
Figure 4-1: Schematic of the stepped beam with aligned neutral axis and free
boundary conditions at locations A and C.
F sin(o t)
A B C
Figure 4-2: Schematic of a stepped beam subjected to: (a) a force of amplitude F
and frequency w, applied at location C, (b) a force of amplitude F and frequency
w, applied at location A,(c) a couple of amplitude M and frequency w, applied at
location C, and (d) a couple of amplitude M and frequency w, applied at location
Xi(xl) = c1 sin ixl + c2 cos/j3ll C 3 sinh/ Pix + c4 cosh/ Pix ,
where the subscript 1 refers to the (A-B) beam section. The (A-B) beam sectional
properties are given by El, II, pi, A1, and 31. As with the uniform beam, /1 is
Ell(l+i ). Applying the free boundary condition at location A
written as p04
Substituting Eq. (4.2) into Eq. (4.1) yields cl c3 and c2 = c4. The resulting
Xl(xi) = ci (sin oixl sinh 3lxl) + c2 (cos 0 cosh Plxl)
The solution for the second beam section (B-C) is given by
X2 (2) = C5 sin 32x2 + c6 cos 2x2 + c7 sinh 32x2 + cs cosh 32x2,
where the subscript 2 refers to the (B-C) beam section. The (B-C) beam sectional
properties are given by E2, 2, p2, A2, and 32, which is given by 23- 2= 22(1+*
is understood that 31 and 32 are functions of frequency and the explicit notation
has been left out.
The continuity conditions at location B for the given case of a colinear neutral
axis state that the deflection, slope, bending moment, and shear force are equal
for the opposite sides of the joint. The analytical expressions for the continuity
dE2 2 X
Applying the continuity equations yields
FicI + F3C2 0 1 0
F3C1 F2C2 21121 0 /321121
FicI + F4C2 0 3021121 0
-F4c1 + FC2 321 0 3 P21
where the undefined terms in the above matrix are
+ sinh 3iL ,
- sinh/3iL ,
+ cosh3iL1 ,
- cosh/3iL ,
121 = and 21 -0 (4.8)
Constants c5, C6, C7, and cs are eliminated by solving Eq. (4.6). The solution for
X2(x) may now be expressed in terms of the remaining unknown constants cl and
X2 (x2) =C1 (T1 sin 32x2 + T2 cos /2x2 + T3 sinh /322 + T4 cosh /322)
+ C2 (VI sin /322 + V2 cos 32X2 + V3 sinh /322 + V4 cosh 32X2)
where the undefined terms are
T =2F + 3 (4.10a)
T2 -= 2+ (4.10b)
T3- F + (4.10c)
T4 --1 + (4.10d)
2I21, 1 2
Vl 22 1 =2 (4.11a)
V2 1 + T3 (4.11b)
212P221 2 '
V3 2I F1 2 (4.11c)
V4 -- 2+ (4.11d)
12 02 1 2 2
Constants cl and c2 are determined by the boundary conditions at location C.
The boundary conditions at location C require
= 0, (4. 12a)
E212 *( = F sin uLt. (4.12b)
The boundary conditions state that the bending moment is equal to zero while
the shear force is equal to the applied impulse load. Applying the conditions of
Eq. (4.12) to Eq. (4.9) yields
cIZ1 + c2Z2 0, (4.13a)
c1Z3 + c2Z4 = 1 (4.13b)
where the relationships for Z1 to Z4 are
Zi -Ti, -T2/3 T3/3 T4 sin 2L2
Z2 -Vi2 -V2 V3 sinh P2L2
Z3 -T13 T2/33 T33 T4 3 cos 32L2
Z4 -2 V V V303 V403 COSh 2 L2
Z4 -V3 V2/ V/3 V/3 cosh032L2
Solving Eq. (4.13) yields the frequency response solution
[Z2 (T1 sin 0x2 + T2 cos 02X + T3 sinh 02X2
F (1 + iq/)E212(ZiZ4 Z2Z3)
+T4 cosh /2X2) Z1 (Vi sin /2x2 + V2 cos 2x2 + V3 sinh 02x2 + V4 cosh 22)] ,
where the compound beam is forced at position C, x2 represents the spatial
output location, and r represents the structural damping factor. The denominator
Z1Z4 Z2Z3 = 0 forms the so called frequency equation whose roots are the natural
frequencies of the system.
4.1.2 Discontinuous stepped beam solution for force excitation at location A
This section develops the frequency response for force excitation at position A
as shown in Fig 4-2(b). The continuity conditions are the same as discussed above,
however the boundary conditions at location A require
a2V 0) (4.16a)
E V (0) Fsin t (4.16b)
The sign change on the forcing term is due to the free body sign convention as
shown in Fig 3-2. The boundary conditions at location C now require
2t, (L2) 0' ,(L2)
(2 = 0. (4.17)
Using the same procedure as outlined before, the response of the compound
beam to force excitation is obtained. However, for loading at position A, the order
of the procedure is reversed. In this case, the boundary conditions at location C
are applied first, then the continuity conditions at location B, and then finally
the boundary conditions at location A. Using the method as outlined before, the
S( i)E (ZZ ZZ7) [Zs (V5 sin 3iAx + V6 cos 31 + V7 sinh P3ax
F (1 + iM)Elli(Z5Z8 Z6Z7)
+Vs cosh pixi) Z6 (T5 sin x, + T6 cos 11 + T7 sinh pix + Ts cosh pixi)]
where the compound beam is loaded at position A. Additional terms are applied to
reduce notation. The constants are defined in Table 4-4, where 321 and I21 are the
same as above.
4.1.3 Extension of the analytical solution for applied couples
This section examines the case of applied couples as shown in Fig 4-2(c) and
Fig 4-2(d). For both systems, the continuity conditions are the same as discussed
above. For excitation at location C, the boundary conditions at position A are
Table 4-1: Notation for force excitation at position A
25 -T212 4T
Z7 -T/3f + T4/3.
Zs = -Vi2/3 + V4/3
F5 = sin 32L2 sinh/32L2 + cosh 32L2 cos/32L2.
F6 = sin 32L2 cosh /32L2 + sinh /32L2 cos 32L2.
F7 = sinh /32L2 cos 32L2 + sin 32L2 cosh /32L2.
Fs = sin 32L2 sinh 32L2 + cosh 32L2 cos 32L2.
Ts sin/31Li1 1F6 + +cos/1Li (F 1) + 21 (F5 + 1).
T6 sin/1L1 (1- F) (Fs + 1) +cos/3iLi 2F6+ ).
T sinh/3iLi ( -F6 ) +cosh/3IL1 (F61 (1 Fs) + (F5 + 1) .
si2312 31 321 21 121/321 2 8 +F6 ).
T6 = sinh 1 2L (-F ) 1) 2 (F5 + 1) + cosh1 AL -( 2 F6 + 2
V5 = sin/1 (2L1 (F8 1) + (F8 + 1) + cos/i3iL( F7 + F~1 17)
V6 sin/31L1 ( I1 F7+ 7) + cos/3iLi ( (8 1) +2 ( + 1)).
V7 = sinh/31L1 (21 (Fs 1) 2 (F + 1)) +cosh/3lL1 (-- 7- 22 F7)
Vs sinh 1L1 ('21 F7 F7) + cosh Li (+2 (1- F8)+ (8 + 1)).
o 0. (4.19)
Due to the free body sign convention, the boundary conditions at position C
2 Vl (L2 )
E2s 2 = Msin wt (4.20a)
0 () 0. (4.20b)
For excitation at position A, the boundary conditions at location C require
?12 (L2) _(L2)
-ax ax (4.21)
while the boundary conditions at location A require
El 21 02V Msinwct, (4.22a)
3v 0. (4.22b)
The system FRFs are obtained using the same procedure as outlined before.
Boundary conditions at the unforced end are applied first, then the continuity
conditions, and then the boundary conditions at the point of excitation.
4.1.4 Comparison of the analytical solution to receptance coupling
This section compares the responses given by the analytical results of sec-
tions 4.1.1 and 4.1.2 to receptance coupling substructure synthesis. Receptance
coupling is an alternative method capable of of predicting the dynamic response of
a stepped beam. The receptance coupling method involves coupling the receptances
for uniform beams obtained analytically or experimentally at the discontinuity
using compatibility and equilibrium conditions. The end result is a 2 by 2 matrix
for each frequency consisting of the primary receptances given by the individual
beam components. The component matrices are written as
hi 1 1 vI vI
R =11 f m ,1 (4.23)
1 P01 01
where the subscripts indicate either direct or cross receptances due to applied
component forces and moments. The individual beam receptances, h, 1, n, and p,
are frequency dependent vectors such that the size of the total matrix is 2 by 2 by
N where N is the length of the frequency vector. The entries into the matrices are
found from a model of the uniform beams given by Bishop .
The analysis that follows uses lowercase h, 1, n, and p to correspond to
component receptances and capital H, L, N, and P for assembly receptances. R
is used to denote the component receptance matrix and G is used to identify the
assembly receptance matrix. The relationship between input forces and output
responses can be written as
vi h 111 f(4.24)
O1 ll P (4.24)
Using reduced notation, the displacement vector (v, 0) can written as the gen-
eralized coordinate u and the force vector (f, m) can written as the generalized
coordinate q such that
u = Rlqi (4.25)
for a direct measurement at location 1. The goal of the process is to determine
relationships between the assembly and component models to obtain the desired
assembly FRF. To examine the solution process, consider the case of solving for
the direct FRF due to force excitation at location 1 as shown in Fig 4-3. Note that
location 1 corresponds to location C in the previous section.
q2b I lq 2 1 Q Q1
A "2b B U2 U
Figure 4-3: Receptance coupling components (a) and assembly (b) models for
excitation at C.
The solution for the case of excitation at location C, given by receptance
coupling [21-23], is obtained first by analyzing the component displacements and
rotations. The component displacements and rotations show the internal reaction
forces due to an applied force or moment on the total system. For body A, the
component displacement and rotations are
Slb Ribib Rlb2b 0 4
I b (4.26)
I' _- R2blb R2b2b q2b
'. -. = R2b2bq2b
Similarly, for Body B
( l R11 R12 q (4.27)
2 R21 R22 92
ul = R11qi + R12q2
U2 R21iq + R22q2
Next, consider the assembly displacements and rotations. The assembly
displacements and rotations show the applied external forces on the total system.
In this case, the applied force is located at position 1. The force at assembly
position 2 is "turned off' such that there are only reaction forces at this location.
The assembly displacements and rotations can be written as
) Gil C1 12 Qi (4.28)
U2 G21 G22 0
U. = GllQ .
The relationship between the components and the assembly is defined by the
compatibility and equilibrium conditions. For the aligned neutral axis model, the
compatibility conditions are
U2 U2b = 0, (4.29a)
tu = Ul, and (4.29b)
u2 U2. (4.29c)
The compatibility conditions state that the coordinates in the component
models are in the same spatial location as the coordinates in the assembly model.
Although true in this application, it is not necessary that the components and
assembly are in the same spatial location for the receptance coupling method.
Finally, consider the equilibrium conditions. The end result for this exercise is
to obtain the solution at position 1 and therefore Q2 is not active. Also, for this
model, there is a rigid connection between components A and B, such that
q2 + 2b 0, (4.30a)
qi = Qi. (4.30b)
At this point, all of the tools necessary to solve for the transfer function are
available. Recall that the goal is to find h11. This is the real v/F direct FRF at
position 1. This is located in G11. Recall
G11 1 (4.31)
Therefore it is necessary to solve for Gil. Recall from the assembly displacements
and rotations, the relationship for Gl is
Ul = GCQi = G11- (4.32)
This result can be refined by utilizing the compatibility conditions and the compo-
nent displacements and rotations for Body B
u = U = Rlql + R12q2 (4.33)
Plugging this result into the expression for GC1 results in
Rilqi + R12q2
Gl l (4.34)
Recall from the equilibrium conditions, Qi = ql, therefore
G = Rl + (4.35)
Because an explicit function for q2/Q1 is not readily available, the ratio is de-
termined as a function of the individual component receptances. The result is
determined from the component displacements and compatibility conditions as
q3 q2b q 2 Q3 Q 21
u3 A "2b B U3 U2
Figure 4-4: Receptance coupling components (a) and assembly (b) models for
excitation at 3.
U2b R2b2bq2b = U2
R2b2bq2b = R21l + R22q2
Recall : q2b= -q2, and qi= Qi
-R2b2bq2 R22q2 R21Q1
q2 -(R22 + R2b2b)-1R21Q
G11 Rl1 R12(R22 + R2b2b)-1R21 ,
where the desired deflection FRF due to an applied force is the first entry in the
matrix for each frequency. The solution for the case of excitation at location A is
found in a similar manner using Fig 4-4. The FRF for excitation at location A (or
position 3 in Fig 4-4) is written as
G33 R33 R32b (R22 + R2b2b)- R2b3
Figure 4-5: Beam dimensions for comparison of the stepped beam analytical solu-
tion to receptance coupling and experiment. Dimensions are in (mm).
Predicted FRFs for Measurement at Location C
200 400 600 800 1000 1200 1400 1600 1800
200 400 600 800 1000 1200 1400 1600 1800
Figure 4-6: FRF comparison between analytical (solid) and receptance coupling
(dashed) methods when forced at position C.
S10-5 Predicted FRFs for Measurement at Location A
0I o- -- ...... ........ -
200 400 600 800 1000 1200 1400 1600 1800
200 400 600 800 1000 1200 1400 1600 1800
Figure 4-7: FRF comparison between analytical (solid) and receptance coupling
(dashed) methods when forced at position A.
To compare the proposed stepped beam analytical solution to the receptance
coupling result, consider the model given in Fig 4-5. The model consists of a
stepped beam with a rectangular cross section and colinear neutral axis. The
material is 7051 aluminum with a density of p = 2; ;[i. .//m3], a Young's modulus
of E = 71 [GPa], and structural damping factor of q = 0.02. Both beam sections
consist of the same material with the same damping factor.
Figures 4-6 and 4-7 show a comparison of the real and imaginary portions of
the FRFs obtained via sections 4.1.1 and 4.1.2 to the receptance coupling solution.
The results show that the solutions are identical. The advantage of the proposed
solution method is in the processing power required to obtain the solutions. As
shown in Eqs. (4.37) and (4.38), the receptance coupling solution requires inverting
the matrices for each frequency of interest. As the frequency vector becomes large,
either due to an increased frequency resolution or larger frequency bandwidth, the
function becomes very costly to perform. The proposed solution, however, requires
far less computing power for large frequency vectors.
Figure 4-8: Experimental setup for FRF testing on a stepped beam.
4.1.5 Experimental verification of the stepped beam solution
This section provides experimental verification for the analyses of sections 4.1.1
and 4.1.2. The experiment consists of a stepped beam of 7051 aluminum with
dimensions given in Fig 4-5. The material has a density of p = 2830 [kg/m3] and a
Young's modulus of E = 71 [GPa]. Structural damping was obtained as a best fit
approximation to the data. For excitation at location C, a damping value T = 0.003
was obtained. For excitation at location A, the damping value was determined to
be r = 0.001. The free-free boundary conditions were obtained in the same manner
as the uniform beam and are shown in Fig. 4-8. Experiments were conducted by
impacting the beam with a modal hammer and obtaining the response with a low
mass accelerometer mounted onto the beam. Figures 4-9 and 4-10 show the results
for the experiments for the first 3 modes for direct FRFs at locations C and A
respectively. The accelerometer mass was measured to be m = 0.7 grams.
S10-3 Mode 1
500 1000 1500
Figure 4-9: Compa
forced at position C
500 1000 1500
rison of experimental (solid) and
260 280 300 320
analytical (dashed) FRF when
The data show experimental modes at 286 Hz, 1159 Hz, and 1759 Hz for
the experimental test measured at location C. Experimental modes for the test
at location A were found to be located at 291 Hz, 1165 Hz, and 1771 Hz. The
differences are due to additional relative inertia of the accelerometer when placed
on the thinner cross section. As the data show, the experimental results are in
excellent agreement with the analytical predictions. As with the uniform beam,
slight errors in magnitude occur due to the use of a structural damping factor
rather than a damping factor per mode.
The predicted results show modes with higher natural frequencies than the
experimental measurements. For both the analytical model as well as the exper-
imental measurements, the test at location C shows almost all of the vibration
occurring in the fundamental mode. The test at location A shows a greater distri-
bution of energy across the higher modes in both the experiment and analytical
x10-4 Total Response 10-4 Mode 1
500 1000 1500
--4 E -4
500 1000 1500 260 280 300 320
Freq (Hz) Freq (Hz)
Figure 4-10: Comparison of experimental (solid) and analytical (dashed) FRF
when forced at position A.
result. The magnitude of the first mode was of the order of 10-3 [m/N] for the test
at location C and 10-4 [m/N] for the test at location A. The smaller motion of the
test at position A is due to larger inertia of the thicker cross section as well as the
additional damping caused by the proximity of position A to the nylon string.
4.2 Receptance Derivation for Discontinuous Beams with Misaligned Neutral Axes
This section develops the receptance functions for a discontinuous beam with
a non-uniform neutral axis and the case of free boundary conditions at the end
locations with one change in cross section as shown in Fig 4-11. The solution is an
expansion of the stepped beam receptance derivation; however this case includes a
coupling of the transverse bending and axial vibrations. The results are compared
to a three sectioned beam used in the l--r removal process by Alcoa.
I I 2 ----
A B C
Figure 4-11: Schematic of a discontinuous notch beam with a misaligned neutral
axis and free boundary conditions at locations A and C.
4.2.1 Discontinuous misaligned beam solution for force excitation at location C
This section develops the frequency response for force excitation at position
C. The continuity conditions of the beam are determined from the free body
diagrams shown in Fig 4-12. In the figures, q represents the transverse shear force,
m represents the bending moment, p represents the axial force, 0 is the slope,
A is the relative displacement between beam sections in the x direction, and e
is the distance between beam neutral axes. The equations are coupled by the
axial deflection and bending moments. From the displacement figure, one can see
that the relationship between the slope and axial displacement between beams is
tan = -. For small angles, it can be shown that tan 0 9 0. Therefore the relative
axial displacement between beams is A = cO, where 0 is the slope written as d.
The moments are related by the axial force such that m = m2 p26.
p (L) 2(0)
m (L ) q m2(O)
Figure 4-12: Free body diagram of (a) forces and (b) displacements for a discontin-
uous notched beam with a misaligned neutral axis.
The continuity conditions are defined as a set of forces and a set of displace-
ments. The displacement continuity conditions state that the transverse deflection,
slope, and axial force are equal for the opposite sides of the joint. The force con-
tinuity conditions state that the bending moment, shear force, and axial force
are equal. The continuity conditions are di-,i'l ,1 in Table 4-2. Note that as the
individual beam sections neutral axes are aligned, the equations become uncoupled
and the continuity conditions would yield the same result as in the aligned case.
Therefore it can be viewed that the aligned neutral axis condition is a special case
of the more general discontinuous notched beam.
Table 4-2: Discontinuous notched beam continuity conditions.
Displacement Continuity Conditions
transverse displacement vi(Li) = v2(0)
slope 01(L1) 02(L1)
axial displacement ui(Li) = u2(0) Oe
Force Continuity Conditions
shear force si(Li) = 82(0)
bending moment mi(Li) = m2(L) p2(0)
axial force pl(Li) = p2(0)
To solve the coupled system, it becomes important to consider the axial
vibration of uniform beams. The equation of motion for the axial vibration of a
uniform Euler-Bernoulli beam is given by reference  as
E2u(x,t) 2u(x,t) (4.39)
S 2 2 (4.39)
where E is the Young's modulus and p is the material density as previously defined.
The solution to Eq. (4.39), when subjected to an input of frequency w (rad/s), can
be separated into a solution in space and time
u(x, t) = Z(x) sin ut .
Substitution of Eq. (4.40) into Eq. (4.39) yields dependence upon the spatial
a2Z(X) 2Z(x) 0, (4.41)
where a = w/~.
The general mode shape solution to Z(x) is
Z(x) Hsin ax+ Jcosax, (4.42)
where H and J are constants determined by suitable boundary conditions. The
axial force, p, can be written as p = AE (. The boundary conditions for
classical conditions listed in Table 4-3.
Table 4-3: Axial vibration boundary conditions for classical beam ends.
u = 0 for a fixed end,
AEa = 0 for a free end
The solution process for system FRFs is the same as for the stepped beam.
The system FRFs are obtained by enforcing boundary conditions at the unforced
end, then the continuity conditions, and then boundary conditions at the point of
excitation. For excitation at location C, the free boundary conditions at A require
a2v1(0) a3V1(0) u1i(0)
x2; 9x9x 03 (4.43)
where the extra boundary condition comes from the axial vibration mode shape.
The continuity conditions are listed in Table 4-2. The forced boundary conditions
at C become
-(L 0 (4.44a)
E2 2 -F sin wt, (4.44b)
2(L2) 0. (4.44c)
The result is a set of 12 equations with 12 unknown constants. Solving these
equations yields a solution of
F (1 + iq)E212(-ZsZ5Z4 + ZsZ6Z2 + Z5Z3Z9 + ZIZ7Z4 Z6ZIZ9 Z3Z7Z2)
(Z7Z2 Z5Z9) (Ti sin 0X2 + T2 cos 0x2 + T3 sinh 02X + T4 cosh 2X2) +
(ZZt8 ZIZ7) (VI sin 32x2 + V2 cos 322 + V3 sinh 32x2 + V4 cosh X2) +
(ZiZ9 Z8Z2) (W1 cos/22 + 2 cosh P2)] ,
where the compound beam is loaded at position C. Additional terms are applied
to reduce notation. The constants are defined in Table 4-4, where /21, 121, and
F, F2, F3, F4 are the same as above. An additional constant is required, E21 z-
The denominator forms the characteristic equation whose roots are the natural
frequencies of the system.
Figure 4-14 shows a comparison of the analytical result for the stepped beam
shown in Fig 4-5 to the analytical result for a similarly dimensioned misaligned
beam shown in Fig 4-13. The distance between the neutral axes, e, is c = (h2 -
h1)/2 6.78 [mm].
As the results show, the shifted and aligned solutions show very similar results.
The first mode has negligible differences. Higher modes are show slight differences
on the order of 2 Hz.
Table 4-4: Notation for FRF with force excitation at position C including a mis-
aligned neutral axis.
T1 1 F4 1 + 3-
T, 1 1, 4 2 21
T2 22E2121312 + 1
T3 2E21 213 F4 + 27 2F3-
T4 1 2 1F,
V2 2E21213 1 + 21
V2 1 E7 | 1 3 -
2E211211 1 2 1 F2
V24 1 4 1
S = F, -sinaiL .
S3 2E21 21 3P21
V4 E21 F4 + F3
i2 -- sin L .1
S2 COS olLL.
Z1 -TI2 sin f2L2 T2 cos f22 + T32L sinh f22 + T432 cosh 2L2
Z2 -Vl30 sin f2L2 V2/3 cos f2L2 + V3/23 sinh f2L2 + V4023 cosh f2L2.
Z3 -Ti23 cos f2L2 + T2 03 sin f32L2 + T303 cosh f2L2 + T4323 sinh f2L2.
Z4 = -Vij3 cos 0f2L2 + V2j3 sin f2L2 + V3f3 cosh f2L2 + V4 3 sinh 32L2.
5 = -Wl23 cos 32L2 + W2/32 cosh 32L2.
Z WI3 sin /32L2 + W2/3 sinh /32L2.
Z7 = a2S COS 1 2L2 a2S2 sin a2L2.
Zs -a2S3 sin c2L2.
Z9 = -2S4 sin a2L2.
4.2.2 Experimental study of the misaligned neutral axis solution
This section provides an experimental study for a three sectioned beam used in
the 1.v-r removal process by Alcoa as shown in Fig 4-15. The experiment consists
of a notched beam of 7051 aluminum with dimensions given. The material has a
density of p = 2830 [kg/m3] and a Young's modulus of E = 71 [GPa]. Structural
damping was obtained as a best fit approximation to the data. A damping value
r = 0.0008 was used in the analysis. As with the other experimental tests, the
free-free boundary conditions were obtained with the schematic shown in Fig 4
8. Figure 4-16 shows the experimental results for the first 2 modes for direct
< 254 >-- 140 -
A B C
Figure 4-13: Dimensions for analytical study of beam with jump discontinuity.
Dimensions are in (mm).
excitation at location D. The accelerometer mass was measured to be m = 0.7
The data shows experimental modes at 162 Hz and 1385 Hz. Analytical
predictions show an error of 13 Hz, or 7 percent, with respect to the first mode.
Results show that analytical predictions are higher than the experimental mea-
surements. Errors are most likely due to neglecting shear and rotary inertia in the
analysis for the individual beam sections. The smaller middle section is much more
flexible than the larger outer sections which causes shear at the discontinuities.
The analysis would be more accurate if the beam sections were modeled with the
more complex Timoshenko beam model  which includes the effects of shear and
rotatory inertia. In this model, transverse and rotational modes are coupled making
it difficult to obtain closed form analytical solutions. Many researchers use finite
element packages to solve the coupled equations [22; 31; 32].
Other possible sources of error include the material properties and beam
dimensions. Material properties are assumed to be equal to the average values
for the particular material. However, these properties can vary slightly between
batches of the same material. In any machining process, dimensions contain a
certain amount of geometric tolerances. The dimensions listed are an average for
the experimental workpiece, however local variations do occur.
200 400 600 800 1000 1200
1400 1600 1800
200 400 600 800 1000 1200 1400 1600 1800
Figure 4-14: Comparison of the analytical FRF with a misaligned neutral axis
(solid) to the analytical FRF with an aligned neutral axis (dashed) when forced at
A B C D
Figure 4-15: Dimensions for experimental study of beam with a jump discontinuity
forced at the end position. Dimensions are in (mm).
200 400 600 800 1000 1200 1400
200 400 600 800 1000 1200 1400
x 10-3 Mode 1
120 140 160 180 200 220
120 140 160 180 200 220
Figure 4-16: Comparison of experimental (solid) and analytical (dashed) FRF for 3
sectioned notch beam with forcing at the end location.
STABILITY OF LAYER REMOVAL PROCESS
This chapter details the stability for machining discontinuous beams assuming
slotting cutting conditions. Stability analyses of machining in literature show
frequency diagrams labeling stable and unstable depths of cut as a function of
various spindle speeds [3-5]. The goal is to obtain a width of cut, or chip width
(b), for which chatter will not occur at any chosen spindle speed. C'! I I r is defined
as a self-excited vibration that will adversely affect the quality of the produced
surface, and may lead to increased tool wear and tool failure. C'!i II 1, causes
vibrations to grow in amplitude at a rapid rate causing the tool to leave the
workpiece. The mechanism for chatter is the regeneration effect. In this model,
a wavy surface is formed on the workpiece due to the motion of the tool during
previous passages. As the tool enters and leaves the cut, the wavy surface is
Stability limits are obtained assuming that chatter occurs due to dynamics of
the spindle holder and tool resonance frequencies. The tool is usually considered
the most flexible part of the dynamic system. However, during milling operations
on the discontinuous beam, the beam will vibrate transversely. As the material
is being removed, the natural frequencies of the workpiece shift and the beam
vibrations will start to grow. After a certain number of passes, the beam becomes
more flexible than the tool and chatter will occur due to regeneration of the beam
5.1 Limiting Chip Width for Machining Process
This section identifies the limiting chip thickness, bum for which the cutting
process is stable. In most high speed applications, the goal is to obtain stability
lobes which detail stable cutting conditions for different spindle speeds. In this
cutting process, the cutting is performed at modest speeds [1350 rpm] which
inhibits the advantages gained by high speed machining. There is no need to obtain
stability lobes for this particular application. Rather, the goal of this section is
to obtain the limiting chip thickness for which the cutting process is stable for all
speeds. The limiting value of chip width is given by T!i-li,- 
blimp (5. 1)
where b is the chip width [mm], Ks is the specific cutting coefficient [N/mm2], and
Re(G)min is the minimum real portion of the frequency response function. The
chip width is albv--, a positive number and therefore Re(G),mi must be the most
negative portion of the FRF for a conservative estimate. The cutting coefficient is
not a material dependent quantity, but is rather process dependent. The limiting
chip width is defined as the most negative portion of the FRF because that
corresponds to a phase shift of 270 degrees between passages. As the phase shift
approaches 270 degrees, there is the largest phase difference between subsequent
passages or the largest amount of amplitude variation per cycle.
The minimum real of the FRF is obtained using the analytical calculations of
the previous section. Consider the fixed-free cutting process shown in Fig 5 1. The
goal is to find the most conservative estimate for bli,. The analysis is performed
with two assumptions in the cutting process; (1) the amplitude of vibrations are
small enough to not pinch the tool during i i 11 iiii:i.- and (2) the most compliant
element of the system is the beam in its final cut configuration. The tool is relieved
such that the first assumption can be held. In this case, the point of largest
vibration occurs at the end of the notch cross section farthest from the fixed end.
This is the point that will be considered the point of excitation for the FRF.
Figure 5-1: Schematic of the clamped-free notched beam during machining. Di-
mensions are given in (mm).
Using the analysis of the previous chapter, the system is modeled as a fixed-
free beam with 3 uniform sections. The force boundary condition is applied to the
shear continuity condition for the second discontinuity. The system is modeled
using 7051 aluminum with dimensions given in Fig 5-1. To obtain a conservative
estimate of the vibrations, it is assumed that the largest vibrations will occur after
machining such that the notch is in its lowest height. The material has a density of
p = 2830[kg/m3] and a Young's modulus of E = 71 [GPa]. Structural damping is
assumed to be ] = 0.02. Figure 5-2 shows a plot of the real response of the notched
beam after machining.
The limiting chip width is calculated using Eq. (5.1). The specific cutting
coefficient is found to be K = 850 [N/mm2] for the tangential component. This
case uses the radial component of the cutting force which is equal to 200 [N/mm2].
The minimum real portion of the FRF is determined to be -1.25 x 10-5 [m/N].
Using these values, the limiting chip width is 0.2 [mm] [.008 in]. As expected, the
structure is so flexible that an extremely small chip width leads to instability. Most
cutting processes have limiting chip widths of orders of magnitude greater than
those recorded here. Chip widths can range from 2 to 30 [mm]. For this particular
test, cuts were performed in steps of 0.254 [mm] which led to chatter.
x 10- Total Response
0 500 1000 1500 2000 2500 3000
x10-5 Mode 1
30 35 40 45 50 55 60
Figure 5 2: Analytical FRF for the notched beam with fixed-free boundary condi-
5.2 Mode Shape Ai, .! ,-i as a Function of the Notch Height
This section compares the measured and theoretical mode shapes as a function
of the notch height with fixed-free boundary conditions. Because all real systems
contain some measure of compliance in the connection, the fixed boundary condi-
tion is modeled in two v--, The first model assumes a perfectly rigid connection.
The second model contains constraints against translation and rotation by use
of translational and rotational springs. The spring connections are applied to
the shear and bending moment boundary conditions respectively. The compliant
boundary conditions are given as 
El I = (5.2a)
where kt and k, and the coefficients of the translation and rotational springs. For
this analysis, the spring constants were obtained as a best fit approximation to the
experimental complaint-free uniform beam data. The results were compared to the
uniform beam configuration because the beam best represents the optimal Euler-
Bernoulli beam before any machining has been completed. Using this method, the
springs constants were found to be kt = 5 x 106 [N/m] and k, = 1 x 105 [Nm/rad].
First Mode Second Mode
x 10 x 10 -5
-10 -1 10
c- -15l -1
_'_j 50 600
100150 200 250 300 0 800 1000 0
Freq (Hz) Notch depth (mm) Freq (Hz) Notch depth (mm)
x 10-5 Measured Real FRF for Uniform and Notch beam
2----------------------- I I II1
100 200 300 400 500 600 700 800
Figure 5-3: Experimental mode shapes as a function of the notch height for Alcoa
Figures 5-3, 5-4, and 5-5 show plots of the measured and theoretical mode
shapes for the FRFs with fixed-free boundary conditions as a function of the
notch height. Results were obtained by taking FRF measurements at the notch in
between l-v-r removal sequences. Each figure shows a plot of the first and second
mode shapes as well as a comparison of the uniform beam and the notched beam
in its final configuration. As the figures show, the notch causes a downward shift of
the natural frequencies. All of the plots show that the second mode shape increases
as a function of the notch height. They also all show the same relative magnitudes
for both modes.
100 200 300 400
0 700 800
Notch depth (mm) Freq (Hz)
Real FRF for Uniform and Notch beam
-1 -i----------------- ------------
100 200 300 400 500
600 700 800 900 1000
Figure 5-4: Analytical mode shapes as a function of the notch height assuming
fixed-free boundary conditions.
The figures show differences in the first mode. The experiment shows a
decrease in the magnitude of the first mode as a function of the notch height. The
theoretical models show a decrease of the mode up to 10 [mm] of notch depth,
and then a sharp rise in the mode size. This result is most likely due to a lack of
sensor response for low frequencies in the experimental measurement. For large
notch heights, the natural frequency shifts to below 40 [Hz]. The sensors used in
the experiment are not accurate below 50 [Hz] and as a result, some of the data
was clipped from the results.
Notch depth (mm)
x 10. ............ ...
. . . xO -. . '
LL --rl----- ^Y
0 .5 ....... ..............
.0 .-0.5 10
60 H0 0 800 0
otch depth (mm) oFreqh (He 800
otch depth (mm) Notch depth (mm)
Real FRF for Uniform and Notch beam
100 200 300 400 500
600 700 800 900 1000
Figure 5-5: Analytical mode shapes as a function of the notch height assuming
compliant-free boundary conditions.
The figures also show differences in the natural frequencies for both modes
for both the fixed-free model and the compliant-free model. The fixed-free model
shows larger errors for the second mode than the first. As the compliant-free model
shows, most of the differences occur in the second mode. In fact, both theoretical
models show very similar results for the first mode. Errors in the natural frequency
are due to two reasons. First, compliant spring constants were obtained as a best
fit approximation. A more accurate model of the clamping force would provide a
better FRF estimate. The second reason is due to the modeling of Euler-Bernoulli
beams. Recall that an underlying assumption for Euler-Bernoulli beams is that
of long, slender beams. For small values of the notch depth, the middle beam
section becomes short and tall as opposed to long and slender. Therefore the Euler-
Bernoulli approximation is best held for large notch depths. However, as discussed
x10 :. "
in the previous chapter, large notch depths also contain shear deformation at the
section boundaries which is neglected in this analysis. The results promotes the use
of more complicated Timoshenko beam models for the experimental system.
Figure 5-6 shows a comparison of the limiting chip thickness as a function of
notch depth for the experiment and both the fixed-free and compliant-free models.
All of the models show a nonlinear relationship between a stable chip thickness
and the notch depth. The general response of each model is an increasing allowable
chip thickness with increasing notch depth until a depth of 12 [mm], followed by a
sharp drop. C'!i I.I r most likely occurs for high notch depths because of the sharp
drop in stable cutting conditions. The fixed-free model over estimates the bum
values relative to the experimental data while the compliant-free model is more
Figure 5: Compason of limiting chip thia ness, bum, as a fu
depth for: (a) experiment, (b) a fixed-free model, and (c) a compliant-free model.
Fixed-free m odel mng chp thcknesso f d l h
035 _0 035
025 0 025
015 0 15
Notch depth (ram) Notch depth (rm)
Figure 5-6: Comparison of limiting chip thickness, bli,, as a function of the notch
depth for: (a) experiment, (b) a fixed-free model, and (c) a compliant-free model.
CONCLUSIONS AND FUTURE WORK
The objective of this work was to examine the dynamic response of discontin-
uous beams. This thesis examined two different discontinuities, one of a stepped
beam, and one of a notched beam. The study analyzed the beams using Euler-
Bernoulli beam theory, neglecting shear and rotary inertia. The study examined
the processes necessary to perform successful modal testing experiments. Various
modal testing complications were discussed including the use of window functions
and mass loading effects due to contact sensors.
The dynamic response of stepped beams was found to be an extension of the
solution for uniform beams. The solution process involved solving a boundary
value problem for the mode shape solution of beam segments using boundary
conditions and continuity conditions. A comparison to receptance coupling has
shown this result to be extremely accurate for beams with long, slender uniform
sections. The advantage of this formulation as compared to receptance coupling
is in the computational time required to obtain solutions. Receptance coupling
requires the use of matrix inversion per frequency of interest in the solution. As
the frequency vector becomes large, either due to increased bandwidth or frequency
resolution, the receptance coupling solution can become very costly. This result is
especially true when considering a structure with many beam segments. Results
were also compared to an experiment of a beam with one step change and free-free
boundary conditions. The results were found to compare very favorably with the
experimental results with minimal errors.
The dynamic response of notched beams was investigated as an extension
of stepped beams. This thesis examined the result as the dynamic coupling
between axial and transverse mode shapes. It was shown that the notched beam
and the stepped beam will show identical results when the distance between
neutral axes is zero. In other words, the stepped beam is a special case of the
more general notched beam case. Results compared to experiment have shown
that this formulation is inaccurate for the applications studied in this work. The
fundamental mode shape showed a 7 percent error between the experimental and
analytical results. It was determined that the errors were due to the beam model
used in the analysis. It was found that the effects of shear at the discontinuities
cannot be ignored.
The stability of beam milling was analyzed to determined the limiting chip
thickness to avoid chatter. It was determined that chatter was being caused by
the flexibility of the workpiece in its notched configuration. The flexibility caused
regeneration of a wavy surface on the workpiece which led to greater vibration
and ultimately the loss of stability. An analysis of the cutting configuration was
performed to determine the analytical and experimental mode shapes as a function
of the notch height. The fixed boundary condition was created with two analytical
models; one of a perfectly rigid boundary condition, and the other of a compliant
boundary. The second mode in the compliant-free model was shown to be more
accurate than that of the fixed-free model. It was also shown that the natural
frequencies shift downward as a result of increased notch height. Qualitatively,
the second mode increased in size with increased notch height. The fundamental
mode was shown to increase in size in the analytical models with the notch height,
however it was shown to decrease in experimental tests. These results were most
likely due to a problem of sensor response in the experimental tests. Some of the
data in the signal was clipped because the accelerometers used in the test were
unable to measure response at frequencies less than 50 Hz. As the fundamental
mode shifted to a lower natural frequency in the experimental test, its response
went below the 50 Hz threshold. A comparison of the limiting chip thickness had
determined that the compliant-free model shows the best qualitative behavior,
however the fixed-free model shows more accurate thickness values.
As discussed in this work, the assumption of the long slender Euler-Bernoulli
does not hold for all values of the notch height. For most of the cutting conditions
in this study, the notched beam section more closely resembles a short, thick
beam. Also, with large values of the notch height, the beam notch becomes much
more flexible than the taller continuous beam sections. As a result, significant
shear deformation is believed to occur at the discontinuities. It is recommended to
rework the individual beam sections considering Timoshenko beam theory which
includes the effects of shear and rotary inertia. Added complications arise in this
model because transverse and rotational modes become coupled, making it difficult
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Michael Koplow was born on Nov. 1, 1980 in Boston, MA. He received a
bachelor's degree with honors in mechanical engineering from the University of
Florida in August, 2004. After receiving his MS in mechanical engineering from the
University of Florida in August, 2005, Michael plans to pursue PhD studies at the
University of California at Berkeley.