DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS

Transcription

1 DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS By MICHAEL A. KOPLOW A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005

2 Copyright 2005 by Michael A. Koplow

3 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

4 TABLE OF CONTENTS page ACKNOWLEDGMENTS iii LIST OF TABLES vi LIST OF FIGURES vii ABSTRACT ix CHAPTER 1 INTRODUCTION Introduction to the Problem Machining and the Material Removal Process Application to Industry Literature Survey EXPERIMENTAL MODAL TESTING Dynamic Response of Linear Systems Impact Testing Overview Contact Sensor Mass Loading Effects DYNAMIC RESPONSE PREDICTION OF CONTINUOUS BEAMS Derivation of the Equation of Motion Dynamic Response Prediction of Uniform Beams Experimental Response of Uniform Beams DYNAMIC RESPONSE PREDICTION OF DISCONTINUOUS BEAMS Receptance Derivation for Discontinuous Beams with Aligned Neutral Axes Discontinuous stepped beam solution for force excitation at location C Discontinuous stepped beam solution for force excitation at location A Extension of the analytical solution for applied couples Comparison of the analytical solution to receptance coupling Experimental verification of the stepped beam solution.. 38 iv

5 4.2 Receptance Derivation for Discontinuous Beams with Misaligned Neutral Axes Discontinuous misaligned beam solution for force excitation at location C Experimental study of the misaligned neutral axis solution 45 5 STABILITY OF LAYER REMOVAL PROCESS Limiting Chip Width for Machining Process Mode Shape Analysis as a Function of the Notch Height CONCLUSIONS AND FUTURE WORK REFERENCES BIOGRAPHICAL SKETCH v

6 Table LIST OF TABLES page 3 1 Euler-Bernoulli beam notation Boundary conditions for classical beam ends Characteristic equations for the free vibration of uniform Euler-Bernoulli beams Beam primary receptances Notation for force excitation at position A Discontinuous notched beam continuity conditions Axial vibration boundary conditions for classical beam ends Notation for FRF with force excitation at position C including a misaligned neutral axis vi

7 Figure LIST OF FIGURES page 1 1 Alcoa testing procedures Signal processing overview Comparison of different modal hammers for: (a) a force measurment in the time domain and (b) a force amplitude measurement in the frequency domain Schematic of a fixed-free forced beam Free body diagram of a beam element Schematic of a uniform beam subjected to a force of amplitude F and frequency ω, applied at x=l Experimental setup for FRF testing on a uniform beam Comparison of experimental (solid) and analytical (dashed) FRFs for the uniform beam Schematic of the stepped beam with aligned neutral axis and free boundary conditions at locations A and C Schematic of a stepped beam subjected to: (a) a force of amplitude F and frequency ω, applied at location C, (b) a force of amplitude F and frequency ω, applied at location A,(c) a couple of amplitude M and frequency ω, applied at location C, and (d) a couple of amplitude M and frequency ω, applied at location A Receptance coupling components (a) and assembly (b) models for excitation at C Receptance coupling components (a) and assembly (b) models for excitation at Beam dimensions for comparison of the stepped beam analytical solution to receptance coupling and experiment. Dimensions are in (mm) FRF comparison between analytical (solid) and receptance coupling (dashed) methods when forced at position C vii

8 4 7 FRF comparison between analytical (solid) and receptance coupling (dashed) methods when forced at position A Experimental setup for FRF testing on a stepped beam Comparison of experimental (solid) and analytical (dashed) FRF when forced at position C Comparison of experimental (solid) and analytical (dashed) FRF when forced at position A Schematic of a discontinuous notch beam with a misaligned neutral axis and free boundary conditions at locations A and C Free body diagram of (a) forces and (b) displacements for a discontinuous notched beam with a misaligned neutral axis Dimensions for analytical study of beam with jump discontinuity. Dimensions are in (mm) Comparison of the analytical FRF with a misaligned neutral axis (solid) to the analytical FRF with an aligned neutral axis (dashed) when forced at position C Dimensions for experimental study of beam with a jump discontinuity forced at the end position. Dimensions are in (mm) Comparison of experimental (solid) and analytical (dashed) FRF for 3 sectioned notch beam with forcing at the end location Schematic of the clamped-free notched beam during machining. Dimensions are given in (mm) Analytical FRF for the notched beam with fixed-free boundary conditions Experimental mode shapes as a function of the notch height for Alcoa testing conditions Analytical mode shapes as a function of the notch height assuming fixed-free boundary conditions Analytical mode shapes as a function of the notch height assuming compliant-free boundary conditions Comparison of limiting chip thickness, b lim, as a function of the notch depth for: (a) experiment, (b) a fixed-free model, and (c) a compliantfree model viii

9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DYNAMIC RESPONSE OF DISCONTINUOUS BEAMS By Michael A. Koplow August 2005 Chair: Brian P. Mann Major Department: Mechanical and Aerospace Engineering The dynamic response of discontinuous structures is often of vital importance in the design of many engineering applications. In many cases, it is preferable to have an analytical model of the system which can reduce the amount of design, 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 individual 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

10 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

11 CHAPTER 1 INTRODUCTION 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 beamlike 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 beamlike 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

12 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

13 3 static measurement cycle is repeated several times to determine the residual stress at each layer of the workpiece material. Machine Spindle Workpiece Free end Fixed end 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

14 4 modal superposition techniques [6] and receptance techniques [7]. Modal superposition 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 individual 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 discontinuities 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 conditions 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 elastic supports including an intermediate support, and the effects of concentrated masses. Naguleswaran [14; 18; 19] considered the effects of multiple beam spans,

15 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 discontinuous beams. Substructuring methods allow the prediction of assembly frequency response functions (FRFs) using FRFs from individual components obtained either 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 discontinuous 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

16 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.

17 CHAPTER 2 EXPERIMENTAL MODAL TESTING This chapter details the basic operations for acquiring experimental frequency response measurements. The goal of this chapter is to provide an overview of experimental modal analysis techniques. Specifically, this chapter provides information 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

18 8 H(f) = 0 h(τ)e j2πft dτ, (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

19 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 response function from time series data. The steps are broken down into a sequence for the input (hammer/force measurement) and the output (accelerometer measurement). 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. INPUT TIME DOMAIN DATA ANTI-ALIASING FILTER WINDOWING APPLY FFT AVERAGING REAL AND IMAG FRF OUTPUT 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

20 10 systems are possible. During the analog to digital conversion process, an antialiasing 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. However, 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

21 11 F stiff tip/low mass F soft tip soft tip/ large mass stiff tip time frequency (a) (b) Figure 2 2: Comparison of different modal hammers for: (a) a force measurment in the time domain and (b) a force amplitude measurement in the frequency domain. of samples taken, this may or may not be the case. To prevent leakage in the data, it is 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

22 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 ω 2 F. Converting the system from accelerance to receptance requires dividing by ω 2. The measured FRF is the ratio of output over input divided by ω 2, Y (f) F (f) = A(f)/F (f) ω 2. (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

23 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, γ 2, is given by Bendat and Piersol [24] as γ 2 (f) = G xy(f) 2 G xx (f)g yy (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 A t 11 = A m 11, (2.7) 1 MA m 11

24 14 where A 11 represents a direct impact accelerance FRF, the super script m represents 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 sensor. 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] A m 11 = A t 11, (2.8) 1 + MA t 11 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 A m 12 = A t 12, (2.9) 1 + MA t 11 where A 12 represents a transfer accelerance FRF measured at one location and forced at another. As discussed previously, the accelerance result can be transformed into receptance by dividing by ω 2.

25 CHAPTER 3 DYNAMIC RESPONSE PREDICTION OF CONTINUOUS BEAMS This chapter develops the equations of motion for continuous, uniform beams and reviews the method to acquire receptance functions for the case of excitation at the boundary conditions. For this analysis, the system will be modeled using the Euler Bernoulli beam theory, neglecting shear and rotary interia. The beams are modeled using receptance techniques whereby external forces are applied directly to the boundary conditions. The discontinuous beam formulation will be shown in later chapters to be an expansion of the uniform beam. 3.1 Derivation of the Equation of Motion This section derives the equation of motion for an Euler-Bernoulli beam. Figure 3 1 shows a typical fixed-free beam with an applied transverse force. Figure 3 2 shows the free body diagram for a differential beam element with a constant cross sectional area, where the beam notation is defined in Table 3 1. v f(x,t) x L dx 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

26 16 f(x,t) M(x) M(x+dx) Q(x) dx Q(x+dx) Figure 3 2: Free body diagram of a beam element. Table 3 1: Euler-Bernoulli beam notation. f(x,t) = Applied transverse force as a function of space (x) and time (t) Q = Shear force acting on the cross section M = Internal bending moment ρ = Mass density (kg/m 3 ) A = Cross sectional area (m 2 ) E = Young s Modulus (P a) I = Area moment of inertia about the neutral axis (m 4 ) 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.

27 Summing the forces on the differential element and using Newton s second law, it follows that 17 Fy = ma, Q x dx + f(x, t)dx = ρadx 2 v(x, t) t 2. (3.1) Application of the first assumption of slender beams to Eq. (3.1) results in 2 M x 2 + f(x, t) = ρa 2 v(x, t) t 2. (3.2) From bending theory, recall M = EI θ. Applying the second assumption for x small deflections, it follows that 2 ( 2 v EI x 2 x ) + f(x, t) = v(x, t) 2 ρa 2. (3.3) t 2 Assuming only uniform beams, Eq. (3.3) may be rewritten as ρa 2 v(x, t) t 2 + EI 4 v(x, t) x 2 = 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

28 18 4 X(x) x 4 β 4 X(x) = 0, (3.6) where β 4 = ω 2 ρa 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 EI 2 v x 2 v v x = v = 0 for a fixed end, x = EI 3 v = 0 for a free end, x 3 = EI 2 v = 0 for a pinned end, and x 2 = EI 3 v = 0 for a sliding end. x 3 The forced vibration solution [7] is obtained by equating applied forces into the boundary conditions. Applied forces are equated to the shear force while

29 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/f 2 Translation due to an applied moment v/m 3 Bending due to an applied force θ/f 4 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 At x = L EI 2 v(0) = 0, x 2 (3.9a) EI 3 v(0) = 0, x 3 (3.9b) EI 2 v(l) = 0, x 2 (3.9c) EI 3 v(l) = F sin ωt. x 3 (3.9d) Applying the boundary conditions given in Eq. (3.9) to Eq. (3.8) results in the following FRF solution [7]

30 20 v F = sin βl cosh βl cos βl sinh βl EI(1 + iη)β 3 (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 and frequency ω, applied at x=l. 3.3 Experimental Response of Uniform Beams This section provides experimental verification for the uniform beam receptance functions. The experiment consists of the uniform beam of aluminum 7051 which is [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 comparison, the total mass of the beam is 0.54 [kg]. The material has a density of ρ = 2830 [kg/m 3 ] and a Young s modulus of E = 71 [GP a]. Structural damping was obtained as η = via a best fit approximation to the experimental data.

31 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.

32 22 x 10 3 Total Response x 10 3 Mode Real (m/n) Real (m/n) Freq (Hz) Freq (Hz) 5 x x Imag (m/n) 5 10 Imag (m/n) Freq (Hz) Freq (Hz) Figure 3 5: Comparison of experimental (solid) and analytical (dashed) FRFs for the uniform beam.

33 CHAPTER 4 RESPONSE PREDICTION OF DISCONTINUOUS BEAMS This chapter develops the receptance functions for the dynamic response of discontinuous beams. For brevity, the derivation will applied to the case of free boundary conditions at the end locations with one change in cross section. The discontinuity is treated by assuming two separate uniform Euler-Bernoulli beams coupled with continuity conditions at the joint between beams. The problem is solved as a boundary value problem with 8 unknown constants. This method can be easily expanded for beams containing different boundary conditions or additional uniform sections. The case of a notched beam with an unaligned neutral axis is treated with a coupling of the transverse bending and axial vibrations. 4.1 Receptance Derivation for Discontinuous Beams with Aligned Neutral Axes This section develops the receptance functions for a discontinuous beam with an aligned neutral axis and the case of free boundary conditions at the end locations with one step change in cross section as shown in Fig 4 1. The solution can be viewed as expansion of the uniform beam receptance derivation whereby the individual sections are modeled as separate beams with continuity conditions applied at the joints. The following section will solve for the cases of force and couple excitation as shown in Fig 4 2. The results are compared to an alternative solution using receptance coupling and to experiment 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

34 24 Figure 4 1: Schematic of the stepped beam with aligned neutral axis and free boundary 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 F and frequency ω, applied at location C, (b) a force of amplitude F and frequency ω, applied at location A,(c) a couple of amplitude M and frequency ω, applied at location C, and (d) a couple of amplitude M and frequency ω, applied at location A. X 1 (x 1 ) = c 1 sin β 1 x 1 + c 2 cos β 1 x 1 + c 3 sinh β 1 x 1 + c 4 cosh β 1 x 1, (4.1) where the subscript 1 refers to the (A-B) beam section. The (A-B) beam sectional properties are given by E 1, I 1, ρ 1, A 1, and β 1. As with the uniform beam, β 1 is written as β1 4 = ω2 ρ 1 A 1 E 1 I 1. Applying the free boundary condition at location A (1+iη) requires 2 v 1 (0) x 2 1 = 3 v 1 (0) x 3 1 = 0. (4.2)

35 Substituting Eq. (4.2) into Eq. (4.1) yields c 1 = c 3 and c 2 = c 4. The resulting expression becomes 25 X 1 (x 1 ) = c 1 (sin β 1 x 1 sinh β 1 x 1 ) + c 2 (cos β 1 x 1 cosh β 1 x 1 ). (4.3) The solution for the second beam section (B-C) is given by X 2 (x 2 ) = c 5 sin β 2 x 2 + c 6 cos β 2 x 2 + c 7 sinh β 2 x 2 + c 8 cosh β 2 x 2, (4.4) where the subscript 2 refers to the (B-C) beam section. The (B-C) beam sectional properties are given by E 2, I 2, ρ 2, A 2, and β 2, which is given by β 4 2 = ω2 ρ 2 A 2 E 2 I 2 (1+iη). It is understood that β 1 and β 2 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 v 1 (L 1 ) = v 2 (0), (4.5a) dv 1 (L 1 ) dx 1 = dv 2(0) dx 2, (4.5b) d 2 v 1 (L 1 ) E 1 I 1 dx 2 1 d 3 v 1 (L 1 ) E 1 I 1 dx 3 1 Applying the continuity equations yields d 2 v 2 (0) = E 2 I 2 dx 2 2 d 3 v 2 (0) = E 2 I 2 dx 3 2, (4.5c). (4.5d)

36 26 F 1 c 1 + F 3 c 2 F 3 c 1 F 2 c 2 F 1 c 1 + F 4 c 2 F 4 c 1 + F 1 c 2 = β 21 I 21 0 β 21 I β21i β21i 2 21 β21i β21i c 5 c 6 c 7 c 8, (4.6) where the undefined terms in the above matrix are F 1 = sin β 1 L 1 + sinh β 1 L 1, F 2 = sin β 1 L 1 sinh β 1 L 1, F 3 = cos β 1 L 1 + cosh β 1 L 1, F 4 = cos β 1 L 1 cosh β 1 L 1, (4.7a) (4.7b) (4.7c) (4.7d) I 21 = E 2I 2 E 1 I 1 and β 21 = β 2 β 1. (4.8) Constants c 5, c 6, c 7, and c 8 are eliminated by solving Eq. (4.6). The solution for X 2 (x) may now be expressed in terms of the remaining unknown constants c 1 and c 2 X 2 (x 2 ) =c 1 (T 1 sin β 2 x 2 + T 2 cos β 2 x 2 + T 3 sinh β 2 x 2 + T 4 cosh β 2 x 2 ) + c 2 (V 1 sin β 2 x 2 + V 2 cos β 2 x 2 + V 3 sinh β 2 x 2 + V 4 cosh β 2 x 2 ), (4.9)

37 27 where the undefined terms are T 1 = F 4 + F 3, 2I 21 β21 3 2β 21 (4.10a) T 2 = F 2 + F 1 2I 21 β21 2 2, (4.10b) T 3 = F 4 + F 3, 2I 21 β21 3 2β 21 (4.10c) T 4 = F 2 + F 1 2I 21 β21 2 2, (4.10d) V 1 = F 1 F 2, 2I 21 β21 3 2β 21 (4.11a) V 2 = F 4 + F 3 2I 21 β21 2 2, (4.11b) V 3 = F 1 F 2, 2I 21 β21 3 2β 21 (4.11c) V 4 = F 4 + F 3 2I 21 β (4.11d) Constants c 1 and c 2 are determined by the boundary conditions at location C. The boundary conditions at location C require 2 v 2 (L 2 ) x v 2 (L 2 ) E 2 I 2 x 3 2 = 0, (4.12a) = 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

38 28 c 1 Z 1 + c 2 Z 2 = 0, c 1 Z 3 + c 2 Z 4 = F E 2 I 2, (4.13a) (4.13b) where the relationships for Z 1 to Z 4 are Z 1 Z 2 Z 3 = T 1 β2 2 T 2 β2 2 T 3 β2 2 T 4 β2 2 V 1 β2 2 V 2 β2 2 V 3 β2 2 V 4 β2 2 T 1 β2 3 T 2 β2 3 T 3 β2 3 T 4 β2 3 sin β 2 L 2 sinh β 2 L 2 cos β 2 L 2 (4.14) Z 4 V 1 β 3 2 V 2 β 3 2 V 3 β 3 2 V 4 β 3 2 cosh β 2 L 2 Solving Eq. (4.13) yields the frequency response solution v F = 1 (1 + iη)e 2 I 2 (Z 1 Z 4 Z 2 Z 3 ) [Z 2 (T 1 sin β 2 x 2 + T 2 cos β 2 x 2 + T 3 sinh β 2 x 2 +T 4 cosh β 2 x 2 ) Z 1 (V 1 sin β 2 x 2 + V 2 cos β 2 x 2 + V 3 sinh β 2 x 2 + V 4 cosh β 2 x 2 )], where the compound beam is forced at position C, x 2 represents the spatial (4.15) output location, and η represents the structural damping factor. The denominator Z 1 Z 4 Z 2 Z 3 = 0 forms the so called frequency equation whose roots are the natural frequencies of the system 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

39 29 2 v 1 (0) x v 1 (0) E 2 I 2 x 3 2 = 0, (4.16a) = 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 2 v 2 (L 2 ) x 2 2 = 3 v 2 (L 2 ) x 3 2 = 0. (4.17) Using the same procedure as outlined before, the response of the compound beam to force excitation is obtained. However, for loading at position A, the order of the procedure is reversed. In this case, the boundary conditions at location C are applied first, then the continuity conditions at location B, and then finally the boundary conditions at location A. Using the method as outlined before, the solution becomes v F = 1 (1 + iη)e 1 I 1 (Z 5 Z 8 Z 6 Z 7 ) [Z 5 (V 5 sin β 1 x 1 + V 6 cos β 1 x 1 + V 7 sinh β 1 x 1 +V 8 cosh β 1 x 1 ) Z 6 (T 5 sin β 1 x 1 + T 6 cos β 1 x 1 + T 7 sinh β 1 x 1 + T 8 cosh β 1 x 1 )], (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 β 21 and I 21 are the same as above 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