DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS

By

MICHAEL A. KOPLOW

A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2005

Copyright 2005

by

Michael A. Koplow

ACKNOWLEDGMENTS

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

iii

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction to the Problem . . . . . . . . . . . . . . . . . . . . . 11.2 Machining and the Material Removal Process . . . . . . . . . . . . 21.3 Application to Industry . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 EXPERIMENTAL MODAL TESTING . . . . . . . . . . . . . . . . . . . 7

2.1 Dynamic Response of Linear Systems . . . . . . . . . . . . . . . . 72.2 Impact Testing Overview . . . . . . . . . . . . . . . . . . . . . . . 82.3 Contact Sensor Mass Loading Effects . . . . . . . . . . . . . . . . 13

3 DYNAMIC RESPONSE PREDICTION OF CONTINUOUS BEAMS . . 15

3.1 Derivation of the Equation of Motion . . . . . . . . . . . . . . . . 153.2 Dynamic Response Prediction of Uniform Beams . . . . . . . . . . 173.3 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 excitationat location C . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.2 Discontinuous stepped beam solution for force excitationat location A . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.3 Extension of the analytical solution for applied couples . . 294.1.4 Comparison of the analytical solution to receptance coupling 314.1.5 Experimental verification of the stepped beam solution . . 38

iv

4.2 Receptance Derivation for Discontinuous Beams with MisalignedNeutral 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 . . . . . . . . . . . . 495.2 Mode Shape Analysis as a Function of the Notch Height . . . . . 52

6 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . 57

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

v

LIST OF TABLESTable page

3–1 Euler-Bernoulli beam notation. . . . . . . . . . . . . . . . . . . . . . . 16

3–2 Boundary conditions for classical beam ends. . . . . . . . . . . . . . . 18

3–3 Characteristic equations for the free vibration of uniform Euler-Bernoullibeams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

vi

LIST OF FIGURESFigure page

1–1 Alcoa testing procedures. . . . . . . . . . . . . . . . . . . . . . . . . . 3

2–1 Signal processing overview. . . . . . . . . . . . . . . . . . . . . . . . . 9

2–2 Comparison of different modal hammers for: (a) a force measurmentin the time domain and (b) a force amplitude measurement in thefrequency 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 Fand frequency ω, 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 forthe 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 amplitudeF and frequency ω, applied at location C, (b) a force of amplitudeF and frequency ω, applied at location A,(c) a couple of amplitudeM and frequency ω, applied at location C, and (d) a couple of am-plitude M and frequency ω, 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(mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4–6 FRF comparison between analytical (solid) and receptance coupling(dashed) methods when forced at position C. . . . . . . . . . . . . 36

vii

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 whenforced at position C. . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4–10 Comparison of experimental (solid) and analytical (dashed) FRF whenforced at position A. . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4–11 Schematic of a discontinuous notch beam with a misaligned neutralaxis 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) whenforced 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 for3 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 assumingfixed-free boundary conditions. . . . . . . . . . . . . . . . . . . . . 54

5–5 Analytical mode shapes as a function of the notch height assumingcompliant-free boundary conditions. . . . . . . . . . . . . . . . . . 55

5–6 Comparison of limiting chip thickness, blim, as a function of the notchdepth for: (a) experiment, (b) a fixed-free model, and (c) a compliant-free model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

viii

Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS

By

Michael A. Koplow

August 2005

Chair: Brian P. MannMajor 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,

testing, 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 analyze 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

ix

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.

x

CHAPTER 1INTRODUCTION

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

responses.

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 layer 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

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.

Chatter, related to the dynamics of the structure during machining, 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 major aerospace aluminum manufacturer, has implemented a layer

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 layer 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

3

static measurement cycle is repeated several times to determine the residual stress

at each layer of the workpiece material.

Machine

Spindle

Fixed end

Free end

Workpiece

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

process.

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

4

modal superposition techniques [6] and receptance techniques [7]. Modal super-

position requires first solving the eigenvalue problem for the free vibration of the

structure without damping. As the name suggests, 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

loading.

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 [12] 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. Naguleswaran [14; 18; 19] considered the effects of multiple beam spans,

5

a non-symmetrical rigid body at the discontinuity, and applied static axial forces.

Tsukazan [20] 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. Chapter 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. Chapter 3 gives a derivation of the uniform Euler-

Bernoulli beam as well as the procedure to obtain frequency response functions for

6

classical boundary conditions. The anayltical results are verified by experiment.

Chapter 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. Chapter 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, Chapter 6 summarizes the conclusions

and provides recommendations for future work.

CHAPTER 2EXPERIMENTAL 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(τ), 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 τ . The output y(t), for any input

x(t), is given by the convolution integral [24]

y(t) =

∫ ∞

−∞h(τ)x(t− τ)dτ . (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(τ) into the frequency domain frequency response function (FRF). For a

physically realizable system, the frequency response function, given by Bendat and

Piersol [24], is

7

8

H(f) =

∫ ∞

0

h(τ)e−j2πftdτ , (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(τ), 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 = σ + 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 jω 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

9

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 acquistion 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

DATA

INP

UT

OU

TP

UT

ANTI-ALIASING

FILTER

WINDOWING APPLY FFT AVERAGING REAL AND

IMAG FRF

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

10

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

11

soft tip/

large mass

stiff tip/low mass

time

F

(a)

|F| soft tip

stiff tip

frequency

(b)

Figure 2–2: Comparison of different modal hammers for: (a) a force measurment inthe 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 always 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

12

content over a wide range [25]. 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 A(f)F (f)

.

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) = R sin ωt , (2.4)

where R is the amplitude of excitation and ω 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 −ω2F. Converting

the system from accelerance to receptance requires dividing by −ω2. The measured

FRF is the ratio of output over input divided by −ω2,

Y (f)

F (f)=

A(f)/F (f)

−ω2. (2.5)

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

13

imaginary responses or similarly magnitude and phase plots. The final step is

to determine the accuracy of the experiments. The coherence function [24] 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, γ2, is given by Bendat and Piersol [24] as

γ2(f) =|Gxy(f)|2

Gxx(f)Gyy(f), (2.6)

where Gxy is the cross power spectrum between the input and output signals,

Gxx is the input power spectrum, and Gyy 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 [27], is given by

At11 =

Am11

1−MAm11

, (2.7)

14

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 [27]

Am11 =

At11

1 + MAt11

, (2.8)

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

Am12 =

At12

1 + MAt11

, (2.9)

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 −ω2.

CHAPTER 3DYNAMIC 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.

f(x,t)

L

x

dx

v

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.

15

16

f(x,t)

dxQ(x)

Q(x+dx)

M(x+dx)M(x)

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 sectionM = Internal bending momentρ = 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 = ∂M/∂t. 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 θ = ∂v/∂x. In

practicle, 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 analysis presented in this work is most interested in

the fundamental mode vibration because the small amplitude of the higher modes

is less important.

17

Summing the forces on the differential element and using Newton’s second law,

it follows that

∑Fy = ma, ⇒ ∂Q

∂xdx + f(x, t)dx = ρAdx

∂2v(x, t)

∂t2. (3.1)

Application of the first assumption of slender beams to Eq. (3.1) results in

∂2M

∂x2+ f(x, t) = ρA

∂2v(x, t)

∂t2. (3.2)

From bending theory, recall M = EI ∂θ∂x

. Applying the second assumption for

small deflections, it follows that

−∂2

∂x2

(EI

∂2v

∂x2) + f(x, t) = ρA

∂2v(x, t)

∂t2. (3.3)

Assuming only uniform beams, Eq. (3.3) may be rewritten as

ρA∂2v(x, t)

∂t2+ EI

∂4v(x, t)

∂x2= f(x, t) . (3.4)

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 ω (rad/s),

can be separated into a solution in space and time

v = X(x) sin ωt . (3.5)

Substitution of Eq. (3.5) into Eq. (3.4) yields dependence upon the spatial

quantity alone

18

∂4X(x)

∂x4− β4X(x) = 0 , (3.6)

where

β4 =ω2ρA

EI(1 + iη), (3.7)

is the solution to the eigenvalue problem and η is a non-dimensional structural

damping factor. The general mode shape solution to X(x) is

X(x) = a sin βx + b cos βx + c sinh βx + d cosh βx , (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 β is the unknown quantity. Values for β 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 β. The natural frequencies are then

determined solving for ω 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 [29] and are listed in Table 3–3

Table 3–2: Boundary conditions for classical beam ends.

v = ∂v∂x

= 0 for a fixed end,

EI ∂2v∂x2 = EI ∂3v

∂x3 = 0 for a free end,

v = EI ∂2v∂x2 = 0 for a pinned end, and

∂v∂x

= EI ∂3v∂x3 = 0 for a sliding end.

The forced vibration solution [7] is obtained by equating applied forces into

the boundary conditions. Applied forces are equated to the shear force while

19

Table 3–3: Characteristic equations for the free vibration of uniform Euler-Bernoulli beams.

Fixed-Free Beam: cos βL cosh βL + 1 = 0 ,Free-Free Beam: cos βL cosh βL− 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/F2 Translation due to an applied moment v/M3 Bending due to an applied force θ/F4 Bending due to an applied moment θ/M

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 EI∂2v(0)

∂x2= 0 , (3.9a)

EI∂3v(0)

∂x3= 0 , (3.9b)

At x = L EI∂2v(L)

∂x2= 0 , (3.9c)

EI∂3v(L)

∂x3= −F sin ωt . (3.9d)

Applying the boundary conditions given in Eq. (3.9) to Eq. (3.8) results in the

following FRF solution [7]

20

v

F=

sin βL cosh βL− cos βL sinh βL

EI(1 + iη)β3(cos βL cosh βL− 1), (3.10)

where the denominator, cos βL cosh βL = 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 andfrequency ω, 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 string, 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 always 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

ρ = 2830 [kg/m3] and a Young’s modulus of E = 71 [GPa]. Structural damping

was obtained as η = 0.0003 via a best fit approximation to the experimental data.

21

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.

22

500 1000 1500

−1

−0.5

0

0.5

1

x 10−3

Rea

l (m

/N)

Total Response

Freq (Hz)

500 1000 1500

−15

−10

−5

0

5x 10

−4

Imag

(m

/N)

Freq (Hz)

600 620 640 660

−1

−0.5

0

0.5

1

x 10−3

Rea

l (m

/N)

Freq (Hz)

Mode 1

600 620 640 660

−15

−10

−5

0

5x 10

−4

Freq (Hz)

Imag

(m

/N)

Figure 3–5: Comparison of experimental (solid) and analytical (dashed) FRFs forthe uniform beam.

CHAPTER 4RESPONSE 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

given by

23

24

Figure 4–1: Schematic of the stepped beam with aligned neutral axis and freeboundary conditions at locations A and C.

(a) (b)

(c) (d)

Figure 4–2: Schematic of a stepped beam subjected to: (a) a force of amplitude Fand frequency ω, applied at location C, (b) a force of amplitude F and frequencyω, applied at location A,(c) a couple of amplitude M and frequency ω, applied atlocation C, and (d) a couple of amplitude M and frequency ω, applied at locationA.

X1(x1) = c1 sin β1x1 + c2 cos β1x1 + c3 sinh β1x1 + c4 cosh β1x1 , (4.1)

where the subscript 1 refers to the (A-B) beam section. The (A-B) beam sectional

properties are given by E1, I1, ρ1, A1, and β1. As with the uniform beam, β1 is

written as β41 = ω2ρ1A1

E1I1(1+iη). Applying the free boundary condition at location A

requires

∂2v1(0)

∂x21

=∂3v1(0)

∂x31

= 0 . (4.2)

25

Substituting Eq. (4.2) into Eq. (4.1) yields c1 = c3 and c2 = c4. The resulting

expression becomes

X1(x1) = c1 (sin β1x1 − sinh β1x1) + c2 (cos β1x1 − cosh β1x1) . (4.3)

The solution for the second beam section (B-C) is given by

X2(x2) = c5 sin β2x2 + c6 cos β2x2 + c7 sinh β2x2 + c8 cosh β2x2 , (4.4)

where the subscript 2 refers to the (B-C) beam section. The (B-C) beam sectional

properties are given by E2, I2, ρ2, A2, and β2, which is given by β42 = ω2ρ2A2

E2I2(1+iη). It

is understood that β1 and β2 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

conditions are

v1(L1) = v2(0) , (4.5a)

dv1(L1)

dx1

=dv2(0)

dx2

, (4.5b)

E1I1d2v1(L1)

dx21

= E2I2d2v2(0)

dx22

, (4.5c)

E1I1d3v1(L1)

dx31

= E2I2d3v2(0)

dx32

. (4.5d)

Applying the continuity equations yields

26

F1c1 + F3c2

F3c1 − F2c2

F1c1 + F4c2

−F4c1 + F1c2

=

0 1 0 1

β21I21 0 β21I21 0

0 β221I21 0 −β2

21I21

−β321I21 0 β3

21I21 0

c5

c6

c7

c8

, (4.6)

where the undefined terms in the above matrix are

F1 = sin β1L1 + sinh β1L1 , (4.7a)

F2 = sin β1L1 − sinh β1L1 , (4.7b)

F3 = cos β1L1 + cosh β1L1 , (4.7c)

F4 = cos β1L1 − cosh β1L1 , (4.7d)

I21 =E2I2

E1I1

and β21 =β2

β1

. (4.8)

Constants c5, c6, c7, and c8 are eliminated by solving Eq. (4.6). The solution for

X2(x) may now be expressed in terms of the remaining unknown constants c1 and

c2

X2(x2) =c1 (T1 sin β2x2 + T2 cos β2x2 + T3 sinh β2x2 + T4 cosh β2x2)

+ c2 (V1 sin β2x2 + V2 cos β2x2 + V3 sinh β2x2 + V4 cosh β2x2) ,

(4.9)

27

where the undefined terms are

T1 =F4

2I21β321

+F3

2β21

, (4.10a)

T2 =F2

2I21β221

+F1

2, (4.10b)

T3 = − F4

2I21β321

+F3

2β21

, (4.10c)

T4 = − F2

2I21β221

+F1

2, (4.10d)

V1 = − F1

2I21β321

− F2

2β21

, (4.11a)

V2 =F4

2I21β221

+F3

2, (4.11b)

V3 =F1

2I21β321

− F2

2β21

, (4.11c)

V4 = − F4

2I21β221

+F3

2. (4.11d)

Constants c1 and c2 are determined by the boundary conditions at location C.

The boundary conditions at location C require

∂2v2(L2)

∂x22

= 0 , (4.12a)

E2I2∂3v2(L2)

∂x32

= −F sin ωt . (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

28

c1Z1 + c2Z2 = 0 , (4.13a)

c1Z3 + c2Z4 = − F

E2I2

, (4.13b)

where the relationships for Z1 to Z4 are

Z1

Z2

Z3

Z4

=

−T1 β22 −T2 β2

2 T3 β22 T4 β2

2

−V1 β22 −V2 β2

2 V3 β22 V4 β2

2

−T1 β32 T2 β3

2 T3 β32 T4 β3

2

−V1 β32 V2 β3

2 V3 β32 V4 β3

2

sin β2L2

sinh β2L2

cos β2L2

cosh β2L2

(4.14)

Solving Eq. (4.13) yields the frequency response solution

v

F=

1

(1 + iη)E2I2(Z1Z4 − Z2Z3)[Z2 (T1 sin β2x2 + T2 cos β2x2 + T3 sinh β2x2

+T4 cosh β2x2)− Z1 (V1 sin β2x2 + V2 cos β2x2 + V3 sinh β2x2 + V4 cosh β2x2)] ,

(4.15)

where the compound beam is forced at position C, x2 represents the spatial

output location, and η 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

29

∂2v1(0)

∂x22

= 0 , (4.16a)

E2I2∂3v1(0)

∂x32

= F sin ωt . (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

∂2v2(L2)

∂x22

=∂3v2(L2)

∂x32

= 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

solution becomes

v

F=

1

(1 + iη)E1I1(Z5Z8 − Z6Z7)[Z5 (V5 sin β1x1 + V6 cos β1x1 + V7 sinh β1x1

+V8 cosh β1x1)− Z6 (T5 sin β1x1 + T6 cos β1x1 + T7 sinh β1x1 + T8 cosh β1x1)] ,

(4.18)

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 β21 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

30

Table 4–1: Notation for force excitation at position A

Z5 = -T2β21 + T4β

21 .

Z6 = -V2β21 + V4β

21 .

Z7 = -T1β31 + T3β

31 .

Z8 = -V1β31 + V3β

31 .

F5 = sinβ2L2 sinhβ2L2 + coshβ2L2 cos β2L2.F6 = − sinβ2L2 coshβ2L2 + sinhβ2L2 cos β2L2.F7 = sinhβ2L2 cos β2L2 + sin β2L2 coshβ2L2.F8 = − sin β2L2 sinhβ2L2 + coshβ2L2 cos β2L2.

T5 = sinβ1L1

(I21β2

212 F6 + F6

2

)+ cos β1L1

(I21β3

212 (F5 − 1) + β21

2 (F5 + 1)).

T6 = sinβ1L1

(I21β3

212 (1− F5)− β21

2 (F5 + 1))

+ cos β1L1

(I21β2

212 F6 + F6

2

).

T7 = sinh β1L1

(I21β2

212 F6 − F6

2

)+ coshβ1L1

(I21β3

212 (1− F5) + β21

2 (F5 + 1)).

T8 = sinhβ1L1

(I21β3

212 (F5 − 1)− β21

2 (F5 + 1))

+ coshβ1L1

(F62 − I21β2

212 F6

).

V5 = sinβ1L1

(I21β2

212 (F8 − 1) + 1

2 (F8 + 1))

+ cos β1L1

(I21β3

212 F7 + β21

2 F7

).

V6 = − sinβ1L1

(I21β3

212 F7 + β21

2 F7

)+ cos β1L1

(I21β2

212 (F8 − 1) + 1

2 (F8 + 1)).

V7 = sinh β1L1

(I21β2

212 (F8 − 1)− 1

2 (F8 + 1))

+ coshβ1L1

(β21

2 F7 −I21β3

212 F7

).

V8 = sinh β1L1

(I21β3

212 F7 − β21

2 F7

)+ coshβ1L1

(I21β2

212 (1− F8) + 1

2 (F8 + 1)).

∂2v1(0)

∂x21

=∂3v1(0)

∂x31

= 0 . (4.19)

Due to the free body sign convention, the boundary conditions at position C

now become

E2I2∂2v1(L2)

∂x22

= M sin ωt , (4.20a)

∂3v2(L2)

∂x32

= 0 . (4.20b)

For excitation at position A, the boundary conditions at location C require

∂2v2(L2)

∂x22

=∂3v2(L2)

∂x32

= 0 , (4.21)

31

while the boundary conditions at location A require

E1I1∂2v1(0)

∂x21

= −M sin ωt , (4.22a)

∂3v1(0)

∂x31

= 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

R11 =

h1 l1

n1 p1

=

v1

f1

v1

m1

θ1

f1

θ1

m1

, (4.23)

where the subscripts indicate either direct or cross receptances due to applied

component forces and moments. The individual beam receptances, h, l, 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 [7].

The analysis that follows uses lowercase h, l, n, and p to correspond to

component receptances and capital H, L, N , and P for assembly receptances. R

32

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

v1

θ1

=

h11 l11

n11 p11

f1

m1

. (4.24)

Using reduced notation, the displacement vector (v, θ) can written as the gen-

eralized coordinate u and the force vector (f, m) can written as the generalized

coordinate q such that

u1 = R11q1 , (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.

m2

q1

q2

q2b

u1

u2

u2bA B

(a)

Q1

Q2

U1

U2

(b)

Figure 4–3: Receptance coupling components (a) and assembly (b) models forexcitation 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

33

u1b

u2b

=

R1b1b R1b2b

R2b1b R2b2b

0

q2b

, (4.26)

∴ u2b = R2b2bq2b .

Similarly, for Body B

u1

u2

=

R11 R12

R21 R22

q1

q2

(4.27)

∴ u1 = R11q1 + R12q2 ,

u2 = R21q1 + 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

U1

U2

=

G11 G12

G21 G22

Q1

0

, (4.28)

∴ U1 = G11Q1 .

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)

u1 = U1 , and (4.29b)

u2 = U2 . (4.29c)

34

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 + q2b = 0 , (4.30a)

q1 = Q1 . (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 =

h11 l12

n21 p22

(4.31)

Therefore it is necessary to solve for G11. Recall from the assembly displacements

and rotations, the relationship for G11 is

U1 = G11Q1 ⇒ G11 =U1

Q1

(4.32)

This result can be refined by utilizing the compatibility conditions and the compo-

nent displacements and rotations for Body B

u1 = U1 = R11q1 + R12q2 (4.33)

Plugging this result into the expression for G11 results in

35

G11 =R11q1 + R12q2

Q1

(4.34)

Recall from the equilibrium conditions, Q1 = q1, therefore

G11 = R11 +R12q2

Q1

(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

follows

(a)

m2

q2

q2b

q3

u2

u2b

u3 A B

(b)

Q2

U2

Q3

U3

Figure 4–4: Receptance coupling components (a) and assembly (b) models forexcitation at 3.

u2b = R2b2bq2b = u2

R2b2bq2b = R21q1 + R22q2

Recall : q2b = −q2, and q1 = Q1

−R2b2bq2 −R22q2 = R21Q1

q2 = −(R22 + R2b2b)−1R21Q1

Therefore,

G11 = R11 −R12(R22 + R2b2b)−1R21 , (4.37)

36

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)−1 R2b3 . (4.38)

254 140

19.05

5.49

25.4

Figure 4–5: Beam dimensions for comparison of the stepped beam analytical solu-tion to receptance coupling and experiment. Dimensions are in (mm).

200 400 600 800 1000 1200 1400 1600 1800−1

−0.5

0

0.5

1x 10

−3

Freq (Hz)

Rea

l (m

/N)

Predicted FRFs for Measurement at Location C

200 400 600 800 1000 1200 1400 1600 1800−10

−5

0

5x 10

−4

Freq (Hz)

Imag

(m

/N)

Figure 4–6: FRF comparison between analytical (solid) and receptance coupling(dashed) methods when forced at position C.

37

200 400 600 800 1000 1200 1400 1600 1800

−4

−2

0

2

4x 10

−5

Freq (Hz)

Rea

l (m

/N)

Predicted FRFs for Measurement at Location A

200 400 600 800 1000 1200 1400 1600 1800−4

−3

−2

−1

0

1

2x 10

−5

Freq (Hz)

Imag

(m

/N)

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 ρ = 2830[kg/m3], a Young’s modulus

of E = 71 [GPa], and structural damping factor of η = 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

38

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 ρ = 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 η = 0.003

was obtained. For excitation at location A, the damping value was determined to

be η = 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.

39

500 1000 1500−3

−2

−1

0

1

2

3x 10

−3

Rea

l (m

/N)

Total Response

Freq (Hz)

500 1000 1500−6

−5

−4

−3

−2

−1

0

1x 10

−3

Imag

(m

/N)

Freq (Hz)

260 280 300 320−3

−2

−1

0

1

2

3x 10

−3

Rea

l (m

/N)

Freq (Hz)

Mode 1

260 280 300 320−6

−5

−4

−3

−2

−1

0

1x 10

−3

Freq (Hz)

Imag

(m

/N)

Figure 4–9: Comparison of experimental (solid) and analytical (dashed) FRF whenforced at position C.

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

40

500 1000 1500

−3

−2

−1

0

1

2

3

x 10−4

Rea

l (m

/N)

Total Response

Freq (Hz)

500 1000 1500

−6

−5

−4

−3

−2

−1

0

1x 10

−4

Imag

(m

/N)

Freq (Hz)

260 280 300 320

−3

−2

−1

0

1

2

3

x 10−4

Rea

l (m

/N)

Freq (Hz)

Mode 1

260 280 300 320

−6

−5

−4

−3

−2

−1

0

1x 10

−4

Freq (Hz)

Imag

(m

/N)

Figure 4–10: Comparison of experimental (solid) and analytical (dashed) FRFwhen 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 layer removal process by Alcoa.

41

Figure 4–11: Schematic of a discontinuous notch beam with a misaligned neutralaxis 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, θ is the slope,

∆ is the relative displacement between beam sections in the x direction, and ε

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 θ = ∆ε. For small angles, it can be shown that tan θ ≈ θ. Therefore the relative

axial displacement between beams is ∆ = εθ, where θ is the slope written as dvdx

.

The moments are related by the axial force such that m1 = m2 − p2ε.

(a)

m2

1 1

(b)

Figure 4–12: Free body diagram of (a) forces and (b) displacements for a discontin-uous notched beam with a misaligned neutral axis.

42

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 displayed 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 Conditionstransverse displacement v1(L1) = v2(0)

slope θ1(L1) = θ2(L1)axial displacement u1(L1) = u2(0)− θε

Force Continuity Conditionsshear force s1(L1) = s2(0)

bending moment m1(L1) = m2(L1)− p2(0)εaxial force p1(L1) = 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 [30] as

E∂2u(x, t)

∂x2= ρ

∂2u(x, t)

∂t2, (4.39)

where E is the Young’s modulus and ρ is the material density as previously defined.

The solution to Eq. (4.39), when subjected to an input of frequency ω (rad/s), can

be separated into a solution in space and time

u(x, t) = Z(x) sin ωt . (4.40)

43

Substitution of Eq. (4.40) into Eq. (4.39) yields dependence upon the spatial

quantity alone

∂2Z(x)

∂x2− α2Z(x) = 0 , (4.41)

where α = ω√

ρE

.

The general mode shape solution to Z(x) is

Z(x) = H sin αx + J cos αx , (4.42)

where H and J are constants determined by suitable boundary conditions. The

axial force, p, can be written as p = AE ∂u(x,t)∂x

. 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,AE ∂u

∂x= 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

∂2v1(0)

∂x21

=∂3v1(0)

∂x31

=∂u1(0)

∂x1

= 0 , (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

44

∂2v2(L2)

∂x22

= 0 , (4.44a)

E2I2∂3v2(L2)

∂x32

= −F sin ωt , (4.44b)

∂u2(L2)

∂x2

= 0 . (4.44c)

The result is a set of 12 equations with 12 unknown constants. Solving these

equations yields a solution of

v

F=

1

(1 + iη)E2I2(−Z8Z5Z4 + Z8Z6Z2 + Z5Z3Z9 + Z1Z7Z4 − Z6Z1Z9 − Z3Z7Z2)[

(Z7Z2 − Z5Z9) (T1 sin β2x2 + T2 cos β2x2 + T3 sinh β2x2 + T4 cosh β2x2) +

(Z5Z8 − Z1Z7) (V1 sin β2x2 + V2 cos β2x2 + V3 sinh β2x2 + V4 cosh β2x2) +

(Z1Z9 − Z8Z2) (W1 cos β2x2 + W2 cosh β2x2)] ,

(4.45)

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 β21, I21, and

F1, F2, F3, F4 are the same as above. An additional constant is required, E21 = E1

E2.

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, ε, is ε = (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.

45

Table 4–4: Notation for FRF with force excitation at position C including a mis-aligned neutral axis.

T1 = 12E21I21β3

21F4 + 1

2β21F3.

T2 = 12E21I21β2

21F2 + 1

2F1.

T3 = − 12E21I21β3

21F4 + 1

2β21F3.

T4 = − 12E21I21β2

21F2 + 1

2F1.

V1 = − 12E21I21β3

21F1 − 1

2β21F2.

V2 = 12E21I21β2

21F4 + 1

2F3.

V3 = 12E21I21β3

21F1 − 1

2β21F2.

V4 = − 12E21I21β2

21F4 + 1

2F3.

W1 = εA1E21α1

2I2β22

sin α1L1.

W2 = − εA1E21α1

2I2β22

sin α1L1.

S1 = −A1E21α1

A2α2sin α1L1.

S2 = cos α1L1.S3 = εβ1F3.S4 = −εβ1F2.Z1 = −T1β

22 sin β2L2 − T2β

22 cos β2L2 + T3β

22 sinh β2L2 + T4β

22 cosh β2L2

Z2 = −V1β22 sin β2L2 − V2β

22 cos β2L2 + V3β

22 sinh β2L2 + V4β

22 cosh β2L2.

Z3 = −T1β32 cos β2L2 + T2β

32 sin β2L2 + T3β

32 cosh β2L2 + T4β

32 sinh β2L2.

Z4 = −V1β32 cos β2L2 + V2β

32 sin β2L2 + V3β

32 cosh β2L2 + V4β

32 sinh β2L2.

Z5 = −W1β22 cos β2L2 + W2β

22 cosh β2L2.

Z6 = W1β32 sin β2L2 + W2β

32 sinh β2L2.

Z7 = α2S1 cos α2L2 − α2S2 sin α2L2.Z8 = −α2S3 sin α2L2.Z9 = −α2S4 sin α2L2.

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 layer 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 ρ = 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

η = 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

46

254 140

19.05 5.49

25.4

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

grams.

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 [30] 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.

47

200 400 600 800 1000 1200 1400 1600 1800

−4

−3

−2

−1

0

1

2

3x 10

−3

Rea

l (m

/N)

Freq (Hz)

200 400 600 800 1000 1200 1400 1600 1800

−6

−4

−2

0

2

x 10−3

Imag

(m

/N)

Freq (Hz)

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 atposition C.

A B C D

Figure 4–15: Dimensions for experimental study of beam with a jump discontinuityforced at the end position. Dimensions are in (mm).

48

200 400 600 800 1000 1200 1400

−2

−1

0

1

2

x 10−3

Rea

l (m

/N)

Total Response

Freq (Hz)

200 400 600 800 1000 1200 1400

−8

−6

−4

−2

0

2

x 10−3

Imag

(m

/N)

Freq (Hz)

120 140 160 180 200 220−3

−2

−1

0

1

2

3x 10

−3

Rea

l (m

/N)

Freq (Hz)

Mode 1

120 140 160 180 200 220

−8

−6

−4

−2

0

x 10−3

Freq (Hz)

Imag

(m

/N)

Figure 4–16: Comparison of experimental (solid) and analytical (dashed) FRF for 3sectioned notch beam with forcing at the end location.

CHAPTER 5STABILITY 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. Chatter 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. Chatter 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

continually regenerated.

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

vibrations.

5.1 Limiting Chip Width for Machining Process

This section identifies the limiting chip thickness, blim for which the cutting

process is stable. In most high speed applications, the goal is to obtain stability

49

50

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 Tlusty [3]

blim =−1

2KsRe(G)min

, (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 always a positive number and therefore Re(G)min 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 blim. 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 machining, 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.

51

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

ρ = 2830[kg/m3] and a Young’s modulus of E = 71 [GPa]. Structural damping is

assumed to be η = 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 Ks = 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 × 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.

52

0 500 1000 1500 2000 2500 3000−2

−1

0

1

2x 10

−5

Rea

l (m

/N)

Total Response

Freq (Hz)

30 35 40 45 50 55 60−2

−1

0

1

2x 10

−5

Rea

l (m

/N)

Freq (Hz)

Mode 1

Figure 5–2: Analytical FRF for the notched beam with fixed-free boundary condi-tions.

5.2 Mode Shape Analysis 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 ways. 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 [30]

E1I1d2v1(0)

dx2= kr

dv1(0)

dx, (5.2a)

E1I1d3v1(0)

dx3= −ktv1(0) , (5.2b)

53

where kt and kr 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× 106 [N/m] and kr = 1× 105 [Nm/rad].

−10

−5

0100 150 200 250 300

−15

−10

−5

0

5

x 10−6

Notch depth (mm)Freq (Hz)

First Mode

Rea

l FR

F (

m/N

)

−10

−5

0600 800 1000

−1

0

1

x 10−5

Notch depth (mm)

Second Mode

Freq (Hz)

Rea

l FR

F (

m/N

)

100 200 300 400 500 600 700 800−2

−1

0

1

2x 10

−5

Freq (Hz)

Rea

l FR

F (

m/N

)

Measured Real FRF for Uniform and Notch beam

Uniform BeamDiscont Beam

Figure 5–3: Experimental mode shapes as a function of the notch height for Alcoatesting conditions.

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 layer 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

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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.

−10

−5

0100 200 300 400

−5

0

5

10

x 10−6

Notch depth (mm)

First Mode

Freq (Hz)

Rea

l FR

F (

m/N

)

−10

−5

0700 800 900

−5

0

5

x 10−6

Notch depth (mm)

Second Mode

Freq (Hz)

Rea

l FR

F (

m/N

)

100 200 300 400 500 600 700 800 900 1000−2

−1

0

1

2x 10

−5

Freq (Hz)

Rea

l FR

F (

m/N

)

Real FRF for Uniform and Notch beam

Uniform BeamDiscont Beam

Figure 5–4: Analytical mode shapes as a function of the notch height assumingfixed-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.

55

Figure 5–5: Analytical mode shapes as a function of the notch height assumingcompliant-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

56

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. Chatter most likely occurs for high notch depths because of the sharp

drop in stable cutting conditions. The fixed-free model over estimates the blim

values relative to the experimental data while the compliant-free model is more

conservative.

0 5 10 150.1

0.15

0.2

0.25

0.3

0.35

0.4Experimental limiting chip thickness

Notch depth (mm)

blim

(m

m)

(a)

0 2 4 6 8 10 12 140.1

0.15

0.2

0.25

0.3

0.35

0.4Fixed−free model limiting chip thickness

Notch depth (mm)

blim

(m

m)

(b)

0 2 4 6 8 10 12 140.1

0.15

0.2

0.25

0.3

0.35

0.4Compliant−free model limiting chip thickness

Notch depth (mm)

blim

(m

m)

(c)

Figure 5–6: Comparison of limiting chip thickness, blim, as a function of the notchdepth for: (a) experiment, (b) a fixed-free model, and (c) a compliant-free model.

CHAPTER 6CONCLUSIONS 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

57

58

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

59

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

to obtain closed form analytical solutions.

REFERENCES

[1] P. Sooraksa, G. Chen, Mathematical modeling and fuzzy control of a flexible-link robot arm, Mathematical and Computer Modelling 27 (6) (1998) 73–93.

[2] G. Surace, V. Anghel, C. Mares, Coupled bending-bending-torsion vibrationanalysis of rotating pretwisted blades: an integral formulation and numericalexamples, Journal of Sound and Vibration 206 (1997) 473–486.

[3] J. Tlusty, Manufacturing Processes and Equipment, Prentice Hall, UpperSaddle River, NJ, 2000.

[4] B. P. Mann, P. V. Bayly, M. A. Davies, J. E. Halley, Limit cycles, bifurcations,and accuracy of the milling process, Journal of Sound and Vibration 277(2004) 31–48.

[5] T. Insperger, G. Stepan, P. V. Bayly, B. P. Mann, Multiple chatter frequenciesin milling processes, Journal of Sound and Vibration 262 (2003) 333–345.

[6] L. Meirovitch, Elements of Vibration Analysis, 2nd Edition, McGraw-Hill, Inc.,New York, 1986.

[7] R. Bishop, D. Johnson, The Mechanics of Vibration, Cambridge UniversityPress, Cambridge, 1960.

[8] A. Yavari, S. Sarkani, J. N. Reddy, Generalized solutions of beams with jumpdiscontinuities on elastic foundations, Archive of Applied Mechanics 71 (2001)625–639.

[9] A. Yavari, S. Sarkani, E. T. M. Jr, On applications of generalized functionsto beam bending problems, International Journal of Solids and Structures 37(2000) 5675–5705.

[10] B. Biondi, S. Caddemi, Closed form solutions of euler-bernoulli beams withsingularities, International Journal of Solids and Structures 42 (2005) 3027–3044.

[11] S. K. Jang, C. W. Bert, Free vibration of stepped beams: Exact and numericalsolutions, Journal of Sound and Vibration 130 (1989) 342–346.

[12] M. J. Maurizi, P. M. Belles, Free vibration of stepped beams elasticallyrestrained against translation and rotation at one end, Journal of Sound andVibration 163 (1993) 182–187.

60

61

[13] M. A. De Rosa, Free vibrations of stepped beams with elastic ends, Journal ofSound and Vibration 173 (4) (1994) 563–567.

[14] S. Naguleswaran, Natural frequencies, sensitivity, and mode shape details ofan Euler-Bernoulli beam with one-step change in cross-section with ends onclassical supports, Journal of Sound and Vibration 252 (2002) 751–767.

[15] S. K. Jang, C. W. Bert, Free vibration of stepped beams: Higher modefrequencies and effects of steps on frequency, Journal of Sound and Vibration132 (1989) 164–168.

[16] M. A. De Rosa, P. M. Belles, M. J. Maurizi, Free vibrations of stepped beamswith intermediate elastic supports, Journal of Sound and Vibration 181 (1993)905–910.

[17] M. A. De Rosa, M. J. Maurizi, The influence of concentrated masses andpasternak soil on the free vibrations of Euler beams-exact solution, Journal ofSound and Vibration 212 (1998) 573–581.

[18] S. Naguleswaran, Vibration and stability of an Euler-Bernoulli beam with upto three-step changes in cross-section and in axial force, International Journalof Mechanical Sciences 45 (2003) 1563–1579.

[19] S. Naguleswaran, Vibration of an Euler-Bernoulli stepped beam carrying anon-symmetrical rigid body at the step, Journal of Sound and Vibration 271(2004) 1121–1132.

[20] T. Tsukazan, The use of a dynamical basis for computing the models ofa beam system with a discontinuous cross-section, Journal of Sound andVibration 281 (2005) 1175–1185.

[21] T. Schmitz, R. Donaldson, Predicting high-speed machining dynamics bysubstructure analysis, Annals of the CIRP 49 (1) (2000) 303–308.

[22] T. L. Schmitz, G. S. Duncan, Receptance coupling for dynamics prediction ofassemblies with coincident neutral axes, Journal of Sound and Vibration (InPress).

[23] S. S. Park, Y. Altintas, M. Movahhedy, Receptance coupling for end mills,International Journal of Machine Tools and Manufacture 43 (2003) 889–896.

[24] J. S. Bendat, A. G. Piersol, Random Data, Analysis and MeasurementProcedures, 3rd Edition, John Wiley & Sons, Inc., New York, NY, 2000.

[25] D. J. Ewins, Modal Testing: Theory and Practice, John Wiley & Sons, NewYork, 1984.

[26] Agilent Technologies, Fundamentals of Modal Testing (AN 243-3) (May 2000).

62

[27] M. R. Ashory, High quality modal testing methods, Ph.D. thesis, ImperialCollege of Science, Technology and Medicine., London (1999).

[28] J. M. M. Silva, N. M.Maia, A. M. R. Ribeiro, Cancellation of mass loadingeffects of transducers and evaluation of unmeasured frequency responsefunctions, Journal of Sound and Vibration 236 (5) (2000) 761–779.

[29] B. Balachandran, E. B. Magrab, Vibrations, Brooks/Cole, Pacific Grove, CA,2004.

[30] W. Weaver, S. P. Timoshenko, D. H. Young, Vibration Problems in Engineer-ing, 5th Edition, John Wiley and Sons, New York, NY, 1990.

[31] M. Sabuncu, K. Evran, Dynamic stability of a rotating asymmetric cross-section Timoshenko beam subjected to an axial periodic force, Finite Elementsin Analysis and Design 41 (2005) 1011–1026.

[32] H. Antes, M. Schanz, S. Alvermann, Dynamic analyses of plane frames byintegral equations for bars and Timoshenko beams, Journal of Sound andVibration 276 (2004) 807–836.

BIOGRAPHICAL SKETCH

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.

63