2 Background Theory

            This project requires a very multi-disciplinary background in order to perform and analyze the experiments.  Some background in electric-magnetism, structural dynamics, fracture mechanics, and digital signal processing (specifically Wavelet transforms) would be needed to understand the project.  To understand the signal from the wire we must know the sources of the signal.  The triboelectric effect provides the basis of our signal but other noise from electric magnetic interference (EMI) could exist.  It is important to know how these phenomena create a signal to provide an accurate analysis of our data.  To verify that the signal is representative of mechanical behavior, we must know the expected frequency of the beam vibration.  We can use this knowledge to predict the natural frequencies of a cantilever beam and then demonstrate that the electrical wires can acquire these frequencies.  Once we establish this fact, we will proceed to acquire the transient signal from a fracture.  Several types of fractures and crack propagation exist, and we must have some knowledge in this field in order to create appropriate tests.  A wavelet analysis will be employed in order to know at what time a certain frequency occurred; something that a Fourier transform can not accomplish. 

2.1 Triboelectric Effect

            The triboelectric effect is an electrical phenomenon in which electrical charge is transferred between materials that are rubbed together.  The amount of charge transferred is dependent on the composition of the material, the forces, the separation rate, and the humidity.  More charge will be transferred if the magnitude of the forces is larger and/or if the rate of separation between the two materials is faster.  For humidity, the drier the air is, the greater the triboelectric effect.  This effect can be seen in cold weather; when the air is generally drier, people get shocked more often [Chapman].  The relationship between material composition and the amount of charge transferred has been tested in the past by various scientists and engineers.  Although no equation or law exists for this relationship, scientists have compiled what is known as the triboelectric series.  The triboelectric series shows which materials will tend to be negatively charged and which materials will tend to be positively charged as a result of the triboelectric effect [Kurtus].

            In most wiring applications, the triboelectric effect is an unwanted behavior.  The rubbing of the wire insulation to the wire causes noise in the signals which may lead to the signal being distorted, and the static shock from the triboelectric effect can damage electronics equipment beyond repair.  In our case, however, the triboelectric effect is the source of our signal.  The vibrations of a structure will cause the insulation to rub against the wire, creating an electrical current.  Part of our project is to confirm that this signal due to the triboelectric effect is indeed related to mechanical behavior of the structure.  We will test the triboelectric effect with the vibration of the cantilever beam. 

 

2.2 Electrodynamics

            The triboelectric effect will not be the only source of electrical current, and we have to pinpoint these sources in order to eliminate them or reduce their impact on our analysis.  Conventional metal wires are susceptible to many kinds of interference, including magnetic and electric fields.  To acquire a signal from the triboelectric effect only, these outside interferences should be eliminated.

  2.2.1 Electromotive Force 

            For electrons to flow in a wire, there must exist some electromotive force (EMF) which “pushes” the electrons to create the current.  The EMF is defined as

                                                                          (1)

where ε is the EMF and f is the force per unit charge.  The sources of the force term stem from electric and magnetic field defined by Lorentz’s force equation

                                                                 (2)

The magnetic field term is usually ignored since charge velocities are normally very small.  We can then plug the electric field in for f in the EMF equation; the result is the potential difference between the endpoints of the path.  Ohm’s Law, V=IR, determines that the current is proportional to the potential difference divided by the resistance.  But should not the force due to the electric field cause an acceleration of charge, not a constant velocity?

            The answer is that Ohm’s law is not a law, but a rule of thumb.  What we are forgetting are the collisions between the electrons as they flow and the inherent velocity due to thermal energy.  In most cases, these two effects cause the electric field to create a constant current in the wire. 

            Another source of EMF is referred to as motional EMF.  Figure 3 will help visualize this effect.

Figure 3:  A Schematic for EMF Induced by a Changing Magnetic Flux [Griffiths]

The gray area is where the magnetic field is applied perpendicular to the plane of the paper.  In the case of motional EMF, the x distance increases or decreases with time, resulting in a magnetic force along the ab path.  The EMF becomes vBh where v is the change of x.  We can relate the EMF to the magnetic flux, Bhx, as

                                                                                (3)

This equation states that an EMF can be induced by a change in magnetic flux [Griffiths].

            2.2.2 Electric and Magnetic Fields

As stated before, the sources of the EMF come from electric and magnetic fields, whose characteristics are defined by Maxwell’s equations:

                                                                          (4)

                                                                             (5)

                                                                                      (6)

                                                     (7)

 

The last of Maxwell’s equations shows that magnetic fields can induce current.  By integrating, employing Stoke’s theorem, and assuming the electric field to be constant, we can rewrite this equation as

                                                                                     (8)

Steady currents produce constant magnetic fields (magnetostatics).  Figure 4 shows a graphical relationship between the magnetic field and the current.

Figure 4:  The Magnetic Field Caused by an Electrical Current [Nave]

A steady current creates this circular magnetic field around the current as determined by the right hand rule.  Therefore, a magnetic field of similar shape can induce a current.  From all of Maxwell’s equations, we can see a strong relationship between the magnetic and electric fields.  Light and all other radiation are simply electromagnetic waves propagating at the speed of light with different frequencies, and while it is almost impossible to block all sources of electric and magnetic radiation, the insulation on the wires does a fair amount of work to block EM fields [Griffiths].  In conclusion, electric and magnetic fields can induce current in several ways.  Figuring out the exact mathematics of the EM fields may not be reasonable, but knowing of their existence is essential when analyzing our signal. 

 

2.3 Structural Mechanics

  2.3.1 Structural Dynamics of a Beam

Our first experiment will test to see if the triboelectric effect can pick up the vibration behavior of a cantilever beam.  The frequency of the electrical signals in the wire will be compared to the frequencies of the beam, but first we need to obtain the natural frequencies of the beam with some basic structural dynamics.  To acquire the natural frequencies, we will represent the structure as a continuous model with infinite degrees of freedom.  Particularly, we will use Bernoulli-Euler Theory for transverse vibrations of linearly elastic beams.  Bernoulli-Euler theory employs five important assumptions [Craig]:

            1.  An axis exists where no extension or contraction occurs

            2.  Cross sections are perpendicular to this axis

3.  Material of the beam is linearly-elastic and homogenous at a single cross section.

            4.  Stress in the axial direction is dominant over the other directions

            5.  The x-y plane is the principal plane.

Figure 5 shows how the beam is modeled.

Figure 5:  Continuous System Model of Beam Undergoing Transverse Deflection [Craig]

To acquire the EOM, the forces and moments are summed, which results in the equation below, a partial differential equation (PDE) with respect to distance and time:

                                            (11)

where E is the modulus of elasticity, I is the moment of inertia, ρ is the density, A is the area, p is the distributed load, and v is the vertical deflection. 

  2.3.2 Natural Frequencies and Modes of Beam

If we assume a free vibration, the load terms disappear.  To solve this PDE, the appropriate boundary conditions of no deflection and no angle of deflection at the base are applied to the EOM.  This process can be quite lengthy, and we assume that the reader can refer to an introductory structural dynamics book to see the actual calculations.  The result is a characteristic equation

                                                      (12)

Although this equation appears similar, no simple expression for the roots is available. So, the solutions must be found numerically.  The resulting equation for natural frequencies is

                                                               (13)

where the values for λrL are solved numerically.  The first three values for λrL  are 1.8751, 4.6941, and 7.8548, and the first three modes are shown in figure 6.

Figure 6:  First 3 Modes of Cantilever Beam Undergoing Transverse Deflection [Craig]

We now can obtain the frequencies expected to occur in a free vibration of the cantilever beam.  In the beam experiment, an accelerometer should retrieve these frequencies, and hopefully, the electrical wires will obtain these frequencies.

 

          2.4 Fractures and Crack Propagation

            The second sets of tests performed this semester involved the acquisition of the signal from fracture and crack propagation.  Previous groups have been able to acquire a transient signals at the point of failure [Callister].  If our tests show a similar transient signal for the fracture, it will further accredit Dr. Stearman’s analysis of the Beechcraft CVR.  We also like to be able to detect the crack before it becomes critical.  An estimated 90% of all structural failure occurs below the yield or tensile strength of the material [Callister].  In these failures, the cracks grow very slowly under cyclic load until it reaches critical length and then propagate very quickly.  If we can hear the crack before it reaches critical then we can use the electrical wire as a cheap health monitoring system. 

            Fractures can be classified into two categories: brittle or ductile.  The difference is that ductile fractures exhibit large plastic deformation while brittle fractures show little plastic deformation.  These characteristics are relative of course and for most material are dependent on temperature.  Above a certain transition temperature for a particular material, the specimen will exhibit ductile fracture while below it will exhibit brittle fracture.  For our purposes, we will only make a distinction between the brittle or ductile fracture by noting the energy needed to fracture the specimen.  A low energy shows a brittle fracture and vice versa for a ductile fracture.  The type of fracture may produce a different type of transient signal.

            Fracture or crack propagation can fall into two processes: stable and unstable.  For our case we would like to simulate fatigue situations to see if the electrical wires can obtain the crack propagation under cyclic load and the failure at critical crack length.  A stable crack is defined as a crack that will not continue to grow without an increase in stress loads.  Once the crack reaches critical, it will propagate at speeds up to 80% the speed of sound of the material; this propagation is the eventual failure stage.  In stable condition, cracks grow slowly under cyclic loads like those experienced in airplanes from take-off to touch-down.  These subcritical crack propagations stem from imperfections in the material or environmental degradation of the material. Our hope is for the electrical wires to sense the growth of the cracks at this stage.  At the critical crack length, we hope the wires will acquire a transient signal to signify the time which failure occurred; this result could be useful to determine the time of failure in aircraft.  

 

          2.5 Cockpit Voice Recorder

Since a flight data recorder was not installed in the Beechcraft 1900C, the cockpit voice recorder (CVR) became the only on-board flight recorder available for analyzing the accident.  The CVR found in Beechcraft 1900C is a “four-track, 30-minute, continuous-loop audio recorder [NTSB-Aircraft].”  CVR's are required on civilian airliners to provide an analog recording of cockpit voice and radio communications [NTSB-Cockpit voice recorder].  Two channels are connected to the pilot and co-pilot’s voice microphone, one channel is dedicated to a cockpit area microphone (CAM), and the last one is connected to what is known as the public address system [NTSB-Aircraft].  The CVR is crucial in the analysis of the accident because it contains valuable information.  It not only recorded the conversation between the pilots and the watchtower, but it also picked up mechanical signals of the plane’s structural behavior.  This information became the source for the wavelet analysis.

          2.6 Fourier Transform (FT) and Wavelet Transform (WT)

            We utilized two kinds of signal processing principles, the Fourier Transform (FT) and the Wavelet Transform (WT), to analyze both the digitized signal of the CVR and the signal obtained from our Impulse Hammer Experiment.  Applying either of the signal processing methods allows us to obtain the frequency contents of a signal.  However, the Wavelet Transform (WT) does have advantages over the Fourier Transform (FT). In this section, we will discuss the theory behind the Fourier Transform (FT) and the Wavelet Transform (WT).

              2.6.1 The Fourier Transform (FT)

            In 1822, the French Mathematician J. Fourier discovered that any function (or signal) could be decomposed into an infinite sum of sine and cosine functions, which can be represented with periodic complex exponential functions [Polikar].  Below are the equations for the Fourier Transform (FT):

                                                                      (14)

                                                         (15)

For equation (14), t is the time, x(t) is the signal, f is a pre-defined frequency at which the signal x(t) has been integrated over all times ().  X(f) is the transformed signal of x(t) to the frequency domain. Equation (15) transforms the signal X(f) back into x(t), which is in the time domain. 

In order to obtain the full frequency spectrum of a signal, we integrate the product of the signal x(t) and the complex exponential function at each value of frequency f, at which the ranges are defined.  If the signal x(t) has a frequency content that matches with the frequency at which the integration takes place, X(f) will give a relatively large value.  Conversely, if the signal x(t) does not have a frequency that matches with the frequency at which the integration takes place, X(f) will be a zero value.  One remark has to be made about the Fourier Transform (FT).  At each frequency, the limits of integration of the product x (t) and  is from negative infinity to positive infinity, i.e. ().  If the signal x(t) is periodic, the Fourier Transform (FT) will find the frequency contents of the signal successfully.  In the condition of periodic signal, the frequency contents are independent of time.  However, if the signal is non-periodic or transient within the whole period (), we would not be able to identify where a particular frequency takes place in terms of time.  Therefore, the Fourier Transform (FT) works well only to analyze the frequency contents of a stationary signal.  Though the Short Term Fourier Transform (STFT) allows us to use the FT to analyze non-periodic signals, STFT still has limitations in refining time and frequency resolutions if the signal is composed of high and low frequencies [Polikar].  In order to identify the time locations of the transient signals of a CVR, we used the Wavelet Transform (WT). 

  2.6.2 The Continuous Wavelet Transform (CWT)

            The Continuous Wavelet Transform (CWT) works in a way similar to the STFT, i.e., STFT includes a window function inside the integral of the Fourier Transform so that the transform computes different segments of the time signal.  However, STFT and CWT have two differences [Polikar]:

1)      The FT does not compute the negative frequencies.

2)      The width of the window of the CWT varies for every spectral component to give the best time and frequency resolution for the analyzed signal.

Equation (16) is the definition of the CWT:

 

                              (16)

 

where                                                                    (17)

There are two variables in the CWT function.  is the translation parameter that corresponds to time and s is the scale parameter which corresponds to , for which f is the predefined frequency.  If the scale parameter s is high, the frequency resolution of a low-frequency signal is better.  This is called a dilation of the mother wavelet, which will be discussed later.  If the parameter value s is low, the time resolution of a high-frequency signal is better. This is called a compression.  CWT has the property to vary the scale parameter in order to obtain the best resolution for different frequency bands of a signal [Polikar].

The last element in the CWT to be discussed is the mother function . “The term mother implies that the functions with different region of support that are used in the transformation process are derived from one main function, or the mother wavelet [Polikar].” The mother wavelet is carefully chosen to serve as a prototype for all process.  The ranges of spectral components inside a signal are obtained by either a compression or a dilation of the mother wavelet.  Figures 7 and 8 are two examples of a mother wavelet function.

Figure 7:  The Mexican hat Function [The MathWorks, Inc.]

Figure 8:  The Morlet Wavelet [Torrence]

    2.6.3 The Discrete Wavelet Transform (DWT) & Haar Wavelet

            The Discrete Wavelet Transform (DWT) is the algorithmically efficient version of the CWT.  DWT reduces the redundancy of reconstructing the signal that “requires a significant amount of computation time and resources” [Polikar].  For our analysis we used a Haar wavelet function to analyze the signal data in our project.  Haar scaling function is the simplest wavelet form that detects the change in the slope of a signal.  In our case, the transient signals we obtained were sharp and high in frequency magnitude relative to the background noise.  Since the transient signals had abrupt changes in the slope of the signal, therefore, the application of Haar wavelet worked particularly well during our signal analyzing process to detect the existence of the transient signal [Boggess & Narcowich].  Figure 9 is a schematic diagram of the Haar wavelet function.

 

 

Figure 9:  The Haar Wavelet            [Boggess & Narcowich]

 

Figure 10 is the equations of the Haar wavelet function and the scaling function.  The Haar wavelet function gives the form and shape of the filter while the scaling function defines the translations and dilations (both in height and width) when processing the data.  For the implementation of the Haar wavelet analysis, we used the Haar wavelet MATLAB code written by Mr. Joshua Foxworth.

Figure 10:  Haar Wavelet and Scaling Function [Boggess & Narcowich]

 

Applying Haar wavelet analysis has both advantages and disadvantages.  The advantage is that Haar wavelet is conceptually simple to understand; thus, it is fast to run on MATLAB code.  However, since Haar wavelet is a discontinuous function, it does not approximate continuous signals very well [Boggess & Narcowich].  Nonetheless, Haar wavelet should be sufficient for our present stage of analysis.