Electromagnetic acoustic transducer and methods of determining physical properties of cylindrical bodies using an electromagnetic acoustic transducer

An electromagnetic acoustic transducer for inducing and sensing vibrations in a cylindrical object and methods of using an electromagnetic acoustic transducer to determine resonant frequencies and physical properties of cylindrical objects. The electromagnetic acoustic transducers produce specific modes of vibration in cylindrical objects including axial shear vibrations, torsional vibrations, radial vibrations and plane strain vibrations. The methods of determining physical properties of a cylindrical objects include comparing sensed resonant frequencies of the cylindrical object to known relationships between resonant frequency and the physical properties of interest. The methods can be used to determine the temperature, dimensions, elastic constants, and damping coefficients of cylindrical objects, the magnitude of a load applied to a cylindrical object, or the texture or grain orientation of the material forming a cylindrical object.

BACKGROUND 
A. Field of the Invention 
The invention relates to electromagnetic acoustic transducers usable with 
cylindrical objects, and methods for determining resonant frequencies and 
physical properties of cylindrical objects using electromagnetic acoustic 
transducers. 
B. Summary of the Invention 
The invention is directed to an electromagnetic acoustic transducer 
(hereinafter "EMAT") adapted for use With cylindrical objects, and to 
methods for using an EMAT to determine the resonant frequencies and 
physical properties of a cylindrical object. "Cylindrical object" is used 
to denote a body having a cylindrical shape with an approximately circular 
cross section. The term cylindrical body includes solid cylindrical bodies 
and hollow cylindrical bodies such as pipes and tubes. 
The EMAT of this invention includes a housing having a circular opening, a 
plurality of magnets mounted in the housing at evenly spaced intervals 
around the circular opening, and at least one wire coil mounted in the 
housing adjacent the circular opening and the polar ends of the plurality 
of magnets. Applying an electrical excitation signal to a wire coil of the 
EMAT will excite vibrations in a cylindrical object inserted into the 
housing of the EMAT. When the excitation signal is at a resonant frequency 
of the cylindrical object, the cylindrical object resonates, i.e., the 
forces applied to the cylindrical body by the EMAT constructively 
interfere with the natural vibrations of the cylindrical body, and large 
amplitude vibrations are produced. Once the cylindrical object is 
vibrating, the EMAT of this invention is also able to sense the amplitude 
and frequency of the vibrations in the cylindrical object. 
The EMAT of this invention excites very specific types of vibrational 
motion in the cylindrical object inserted into the circular opening. 
Depending on the orientation of the wire coil relative to the magnets and 
the cylindrical object the EMAT can induce axial shear vibrations, 
torsional vibrations, radial vibrations or plain strain vibrations. 
The EMAT of this invention is also useful in determining the frequencies at 
which a cylindrical object experiences resonant vibrations in each of the 
above identified types of vibrational motion. In addition, various 
physical properties of the cylindrical object, or the loading applied to 
the cylindrical object can be determined by: 
(1) determining the amount of time it takes for resonant vibrations in the 
cylindrical body to decrease to a negligible value; 
(2) comparing resonant frequencies of the cylindrical object to resonant 
frequencies of a standardized cylindrical object made from the same 
material and having approximately the same dimensions; 
(3) determining how the resonant frequencies of several types of 
vibrational motion change as a load is applied to the cylindrical object; 
and 
(4) determining how the amplitude of resonant vibrations change as the 
object is rotated within in the EMAT. 
In particular, this invention provides a non-contact type EMAT usable with 
cylindrical objects. 
This invention also provides an EMAT capable of exciting specific types of 
vibrational motion in the cylindrical object. 
This invention further provides an EMAT capable of exciting axial shear 
vibrations, torsional vibrations, radial vibrations, or plain strain 
vibrations in a cylindrical object. 
This invention provides a method for using an EMAT to determine the 
resonant frequencies of a cylindrical object. 
This invention also provides a method for using an EMAT to determine the 
resonant frequency of a cylindrical object for axial shear vibrations, 
torsional vibrations, radial vibrations, or plain strain mode vibrations. 
This invention further provides methods for using an EMAT to measuring 
physical properties of cylindrical objects. 
This invention further provides a method for using an EMAT to determine the 
radial depth from the exterior surface of a cylindrical object at which 
physical properties of the material of the cylindrical object undergo a 
change. 
This invention also provides a method for using an EMAT to determine the 
magnitude of a load placed on a cylindrical object. 
This invention further provides a method for using an EMAT to determine the 
texture or grain orientation of the material forming a cylindrical object. 
These and other objects and advantages of the present invention will become 
apparent from the following detailed description of preferred embodiments 
when taken in conjunction with the accompanying drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 1 shows a first type of EMAT 100 capable of exciting torsional 
vibrations in a cylindrical object 200. The EMAT 100 comprises a housing 
26 having a circular opening 20. A cylindrical object 200 is inserted into 
the circular opening 20. The cylindrical object 200 shown in FIG. 1 
extends into and out of the page. 
A plurality of magnets 22 are mounted in the housing 26 around the circular 
opening 20 at evenly spaced intervals. The polar ends 22a and 22b of the 
magnets 22 are adjacent the circular opening, and each part of adjacent 
magnets 22 has ends 22a and 22b of opposite polarity. The EMAT 100 also 
has at least one wire coil 24 which is also mounted in the housing 26 
adjacent the circular opening 20. In this first type of EMAT 100, the 
individual wires 24a of the wire coil 24 are mounted adjacent to the polar 
ends 22a and 22b of the plurality of magnets 22, and are arranged in a 
meander pattern wherein the wires 24a extend back and forth through the 
circular opening 20 along the axial direction of the cylindrical object 
200 (i.e. into and out of the page). The wires having an X in the center 
are extending out of the page, and the wires with a dot in the center are 
extending into the page. 
Any type of cylindrical object 200 comprising an electrically conductive 
material can be used with the EMAT 100. As mentioned above, the 
cylindrical object 200 can be solid or hollow. When an electrical 
excitation signal is applied to the wire coil 24 a current begins to flow 
through the wires 24a. Because the wires 24a are located in the magnetic 
fields of the magnets 22, and because the cylindrical object 200 is also 
located adjacent the wire coil 24, eddy currents are induced in the 
cylindrical object 200. If the EMAT 100 is held in a fixed position the 
eddy currents in the cylindrical object 200, which are also in the 
presence of the magnetic field, create Lorentz forces that apply a force 
to the material of the cylindrical object 200 in a certain direction. 
Because the current runs in a first direction through the wires 24a 
adjacent the north ends 22a of the plurality of magnets 22, and the 
current runs in an opposite direction in the wires 24a adjacent the south 
ends 22b of the plurality of magnets 22, the Lorentz forces resulting from 
the application of a voltage to the wire coil 24 will always be oriented 
in the same direction. For the configuration shown in FIG. 1, the Lorentz 
forces will cause a torsional force twisting the cylindrical object 200 in 
a rotational direction. 
If the electrical excitation signal applied to the wire coil 24 is an 
alternating current, the Lorentz forces applied to the cylindrical object 
200 will also alternate. As a result, the cylindrical object 200 will 
first twist in one direction, then twist in the opposite direction as the 
current alternates. If the frequency of the alternating excitation signal 
applied to the wire coil 24 matches a resonant torsional vibrational 
frequency of the cylindrical object 200, the torsional forces applied by 
the EMAT 100 will constructively interfere with the vibrational movement 
of the cylindrical object 200, and the cylindrical object 200 will 
resonate. 
FIG. 2 shows a second type of EMAT 100 capable of exciting axial shear 
vibrations in the cylindrical object 200. As in the first type of EMAT 100 
described above, the wire coil 24 of the second type of EMAT 100 is also 
mounted adjacent the circular opening 20 in the housing 26. In this second 
type of EMAT 100, however, the wire coil 24 is wound around the circular 
opening 20 in the form of a solenoid coil. Applying an electrical 
excitation signal to the wire coil 24 will cause Lorentz forces in the 
cylindrical object 200 that are oriented in the axial direction of the 
cylindrical object 200. In other words, the forces will be directed into 
and out of the page. 
Because the current always runs in the same direction relative to the 
cylindrical object 200, and because the wire coil 24 passes next to 
alternating poles 22a and 22b of the plurality of magnets 22, the current 
causes alternating direction Lorentz in the cylindrical object 200. When 
the current is flowing through the wire coil 24 in a first direction, the 
Lorentz force adjacent a south end 22b of a magnet 22 will be directed out 
of the page and the Lorentz force adjacent a north end 22a of a magnet 22 
will be directed into the page. When the direction of the current flowing 
through the wire 24 is reversed, the Lorentz forces will also reverse. 
Applying an alternating current to the wire coil 24 will create 
alternating axial Lorentz forces in the cylindrical object 200. By 
adjusting the frequency of the electrical excitation signal applied to the 
wire coil 24, the cylindrical object 200 can be induced to vibrate at a 
resonant frequency for axial shear vibrations. 
FIG. 16 shows an axial cross-sectional view of the second type of EMAT 100. 
The EMAT 100 has a hollow cylindrical housing 26, in which the plurality 
of magnets 22 are mounted. The plurality of magnets 22 are mounted at 
evenly spaced intervals around the inside of the cylindrical housing 26 so 
that a cylindrical opening 20 extends through the longitudinal axis of the 
EMAT 100. The EMAT 100 has two wire coils 21 and 29 which are mounted in 
the EMAT 100 adjacent the polar ends 22a and 22b of the plurality of 
magnets 22. The wire coils 21 and 29 form a circle around the cylindrical 
opening 20 in the center of the EMAT 100. The first wire coil 21 has a 
first end 27 and a second end 28. The second wire coil 29 has a first end 
25 and a second end 23. 
FIG. 3 shows a third type of EMAT 100 capable of inducing radial vibrations 
in the cylindrical object 200. In this third type of EMAT 100, the wire 
24a of the wire coil 24 are located between adjacent ends 22a and 22b of 
adjacent magnets 22, and the wire coil 24 is disposed in a meander pattern 
extending into and out of the page as described above for the first type 
of EMAT 100. A voltage applied to the wire coil 24 will create Lorentz 
forces in the cylindrical object 200 which tend to pull the material of 
the cylindrical object 200 outward in a radial direction or push the 
material of the cylindrical object inward in a radial direction. Because a 
current passing through the wire coil 24 will run in a first direction for 
wires having a north pole 22a of a magnet 22 on the left and a south pole 
22b of a magnet 22 on the right and because the current will run in the 
opposite direction for wires having a south pole 22b of a magnet 22 on the 
left and a north pole 22a of a magnet 22 on the right, the Lorentz forces 
applied to the cylindrical object 200 by the wires 24a of the wire coil 24 
will always be oriented in the same radial direction (either into the 
center, or away from the center). Applying a current running in a first 
direction through the wire coil 24 will create Lorentz forces tending to 
pull the cylindrical object 200 outward in a radial direction. Reversing 
the current will cause the Lorentz forces to reverse, tending to push the 
cylindrical object 200 inward toward the central axis of the cylindrical 
object 200. Applying an alternating current to the wire coil 24 will cause 
alternating Lorentz forces which pull outward in the radial direction, 
then push inward in the radial direction. The frequency of an alternating 
current excitation signal applied to the wire coil 24 can be adjusted so 
that the cylindrical object 200 vibrates at a resonant frequency. 
Each of the EMATs 100 described above excite specific types of vibrational 
motion in the cylindrical object 200. In each case, the actual vibrations 
caused by the EMATs 100 are only approximate. In other words, the first 
type of EMAT 100 described above is intended to produce only torsional 
vibrations. In practice the vibrations are not strictly limited to 
torsional motion. The vibrations are, however, close enough to pure 
torsional vibrations that they can be modeled as torsional vibrations for 
the purposes of analyzing the properties of the cylindrical object 200. 
The same is true for axial shear and radial vibrations for the second and 
third types of EMATs 100. 
The term "plane strain vibrations" is used to denote vibrations in a 
cylindrical object which have no axial component. Torsional vibrations and 
radial vibrations are two variants of plane strain vibrations. Other types 
of plane strain vibrations are possible and may be useful for determining 
certain properties of the cylindrical object 200. 
An EMAT 100 as described above can be used to determine the resonant 
frequencies and the physical properties of the cylindrical object 200 
according to the following methods. The EMATs used in the methods 
described below, however, need not be one of the types described above. In 
each of the configurations described above, the wire coil 24 and the 
magnets 22 of the EMAT 100 were mounted in the housing 26 located around 
the exterior of the cylindrical object 200. The methods for determining 
the resonant frequency and the physical properties of the cylindrical 
object 200 described below are equally applicable to other types of EMATs, 
such as those designed to be inserted into the center of a hollow tube. 
The EMATs usable with hollow tubes could have both the wire coil 24 and 
the magnets 22 located inside the tube, or the magnets 22 could be located 
on one side of the tubing wall, and the wire coil 24 on the opposite side. 
The use of the methods described below is not intended to be limited to 
EMATs having any particular configuration. 
FIG. 4 shows a first type of test apparatus for determining a resonant 
frequency of a cylindrical object. The test apparatus comprises an 
impedance analyzer 34 attached to the wire coil of an EMAT 30. The EMAT 30 
is attached to a cylindrical object 32. The EMAT could be one of the 
first, second or third types shown in FIGS. 1, 2 and 3, or any other type 
of EMAT designed to excite a specific type of vibration in the cylindrical 
object 32. 
The impedance analyzer 34 applies an alternating current excitation signal 
to the wire coil of the EMAT 30 to excite a certain type of vibration in 
the cylindrical object 32. The vibrations in the cylindrical object 32, in 
turn, affect the impedance characteristics of the wire coil of the EMAT 
30. The impedance characteristics of the wire coil of the EMAT 30 can be 
used to indicate resonant frequencies of the cylindrical object 32. 
If the frequency of the excitation signal applied to the wire coil is 
gradually increased, as the frequency of the excitation signal passes 
through a resonant frequency of the cylindrical object 32, the amplitude 
of the real and imaginary parts of the impedance of the wire coil will 
experience local extrema. Depending on the electrical characteristics of 
the EMAT 30, the real or imaginary parts of the impedance of the wire coil 
24 could experience a maximum or a minimum at the resonant frequency, or 
the amplitude of the real or imaginary parts of the impedance could 
experience a maximum just below the resonant frequency, followed by a 
minimum just above the resonant frequency, tracing out a Z-shape around 
the resonant frequency. A programmable impedance analyzer 34 can be 
programmed to gradually vary the frequency of an electrical excitation 
signal applied to the wire coil of the EMAT 30 and to note those 
frequencies at which the real or imaginary parts of the impedance 
experience extrema, thus indicating a resonant frequency of the 
cylindrical object 32. 
FIG. 8 shows a plot of the real and imaginary parts of the impedance of the 
wire coil of the EMAT 30 over a portion of the frequency band that 
includes a resonant frequency. AS shown in FIG. 8, the real part of the 
impedance of the wire coil, shown as a solid line, experiences a sudden 
maximum at the resonant frequency. The imaginary part of the impedance, 
shown as a dotted line, trace a Z-shaped pattern around the resonant 
frequency. 
The test apparatus described above can be used with an EMAT 30 having a 
single wire coil, or plural wire coils. If the EMAT 30 has a single wire 
coil, the impedance analyzer 34 applies an electrical excitation signal to 
the wire coil and senses the impedance of the same wire coil. If the EMAT 
30 has first and second wire coils, the impedance analyzer 34 can apply an 
excitation signal to a first wire coil, and determine an impedance 
transfer function between the first and second wire coils. 
FIG. 5 shows a second type of test apparatus for determining the resonant 
frequencies of the cylindrical object 32. In this second type of test 
apparatus, a frequency generator 36 provides a frequency signal to a gated 
amplifier 38 and a receiver 40. The gated amplifier 38 provides an 
alternating current electrical excitation signal to the wire coil of an 
EMAT 30. The receiver 40 is connected to the EMAT 30 and senses an 
electrical response signal induced in wire coil of the EMAT 30 by 
vibrations in the cylindrical object 32. The receiver is also connected to 
a recording device 42 for recording the response signal, and an 
oscilloscope 44 for displaying the response signal. 
In this second type of test apparatus, an alternating current electrical 
excitation signal is applied to wire coil of the EMAT 30 to induce 
vibrations in the cylindrical object 32. After a period of time the 
excitation signal is removed from the wire coil and the cylindrical object 
32 is allowed to freely vibrate. Vibrations in the cylindrical object 32 
will excite an alternating current in wire coil of the EMAT 30 for the 
same reasons the excitation signal produced vibrations in the first place. 
The EMAT 30 is essentially working in reverse. 
After the excitation signal is removed from the wire coil, the receiver 40 
is used to sense any electrical response signal generated in wire coil of 
the EMAT 30 by the vibrations of the cylindrical object 32. When the 
excitation signal applied to the wire coil of the EMAT 30 by the amplifier 
38 is at a non-resonant frequency, the amplitude of the vibrations induced 
in the cylindrical object 32 will be relatively small, and the amplitude 
of the response signal will be correspondingly small. When an excitation 
signal applied to the EMAT 30 by the amplifier 38 is at or near a resonant 
frequency of the cylindrical object 30, the amplitude of the vibrations 
induced in the cylindrical object 32 will be relatively large, and the 
amplitude of the response signal induced in the EMAT 30 by the vibrations 
will be corresponding large. 
By applying a plurality of excitation signals to the wire coil of the EMAT 
30 at different frequencies, and sensing the response signal generated in 
the EMAT 30 by the vibrations occurring after each excitation, the 
resonant frequencies of the cylindrical object 32 can be determined. FIG. 
7 shows a diagram of the amplitude of the response signal generated in the 
wire coil of the EMAT 30 by vibrations in the cylindrical object 32 
following excitation at different frequencies. The amplitude of the 
response signal experiences a sharp maximum spike at the resonant 
frequencies of the cylindrical object. 
As in the first test apparatus, the EMAT 30 used in the second test 
apparatus can have one or more wire coils. If the EMAT 30 has only a 
single wire coil the electrical excitation signal is applied to the single 
wire coil by the gated amplifier 38 for a period of time, then the 
excitation signal is removed and the receiver 40 senses a response signal 
induced in the wire coil. 
Alternately, if the EMAT 30 is provided with first and second wire coils, 
the gated amplifier 38 can be connected to the first wire coil, and the 
receiver 40 can be connected to the second wire coil. In this set up, the 
gated amplifier 38 applies an excitation signal to the first wire coil for 
a period of time to induce vibrations in the cylindrical object 32, then 
the excitation signal is removed. The receiver 40 is used to sense an 
response signal in the second wire coil after the excitation signal is 
removed. 
The first and second wire coils can be physically separated. The only 
requirement is that vibrations excited by the first wire coil travel along 
the cylindrical body 32 and cause an excitation signal at the second wire 
coil. In addition, a first EMAT 30 can be used to excite vibrations in the 
cylindrical body, and a second EMAT 30 can be used to sense vibrations in 
the cylindrical body. Using two EMATs 30, one for exciting and one for 
sensing, is analogous to using a single EMAT having two wire coils. For 
the purposes of this description and the claims, the two methods are 
considered equivalents. 
FIG. 6 shows the results of a first method for determining physical 
properties of a cylindrical object using the EMAT 30. In the first method, 
an excitation signal is applied to the EMAT 30 to excite the cylindrical 
object 32 to vibrate at a resonant frequency. The excitation signal is 
then removed, and the cylindrical object 32 is allowed to vibrate freely. 
The amplitude of an electrical response signal generated in the wire coil 
of the EMAT 30 by the resonant vibrations of the cylindrical object 32 is 
sensed using either of the two test set ups described above. 
The amplitude of the response signal will gradually decrease over a period 
of time as the free vibrations are damped by the material of the 
cylindrical object 32. FIG. 6 shows a diagram of the amplitude of a 
response signal generated in the EMAT 30 by free vibrations of the 
cylindrical object 32 over a period of time. The amount of time taken for 
the amplitude of the response signal to decrease to a negligible value can 
be used to determine the damping coefficient for the cylindrical object 
32. 
The frequencies at which the cylindrical object 32 resonates can also be 
used to determine the temperature of the cylindrical object 32. Changes in 
the temperature affect the frequency at which the cylindrical object 32 
resonates. By determining the frequency of resonant vibrations for a 
particular-sized cylindrical object 32 formed from a particular material 
at a variety of different temperatures, a chart describing the 
relationship between temperature and resonant frequency can be 
constructed. Once the relationship is known, determining the resonant 
frequency will permit the determination of the temperature of the object 
32. 
The diameter of the cylindrical object 32 can also be determined using the 
EMAT 30. As described above for temperature, the diameter of the 
cylindrical object 32 affects its resonant frequency. For similarly-sized 
cylindrical object 32 formed from the same material, a slight change in 
the diameter will result in a slight change in the resonant frequencies. 
By determining the resonant frequencies of different diameter cylindrical 
objects 32 at a particular temperature, the relationship between the 
diameter and the resonant frequency at that temperature cap be determined. 
Once the relationship is known, determining the resonant frequency of the 
cylindrical object 32 allows one to determine the diameter. 
Likewise, the same method can be used to determine the relationship between 
resonant frequency and the wall thickness of tubing or piping. Once the 
relationship is known, determining the resonant frequency allows one to 
determine the wall thickness of the tubing or piping. 
The number of magnets 22 used in the EMAT 30 to produce torsional 
vibrations or radial vibrations does not affect the frequencies at which 
the cylindrical object 32 resonates. These types of EMATs 30 are designed 
so that the forces applied to the cylindrical object 32 are relatively 
uniform around the circumference of the cylindrical object 32. What is 
important for these types of EMATs 30 is that enough magnets 22 are 
provided to apply a uniform force around the circumference of the 
cylindrical object 32. 
However, the number of magnets 22 used in the EMAT 30 to induce axial shear 
vibrations (as described above for FIG. 2) does have an affect on the 
frequencies at which the cylindrical body 32 resonates. For a particular 
axial shear type EMAT 30, several different excitation frequencies will 
produce resonant vibrations in the cylindrical object 32. 
FIG. 9A shows a cross-section of the cylindrical object 32 resonating at a 
first resonant frequency. Each wave peak of the diagram could be located 
adjacent the north pole 22a of a magnet, and each wave trough of the 
diagram could be located adjacent the south pole 22b of a magnet 22. At 
the resonant frequency, the Lorentz forces constructively interfere with 
the natural vibrations of the cylindrical object 32 to produce the 
standing waves shown in FIG. 9A. Changing the number of magnets 22 alters 
the number of peaks and troughs of the standing waves, thus altering the 
frequency of the resonant vibration. 
FIG. 9A represents the wave pattern in the material of the cylindrical 
object 32 when it is resonating at a first resonant frequency. FIG. 9B 
represents the wave pattern of the material of the cylindrical object 32 
at a second, higher, resonant frequency. Likewise, FIGS. 9C and 9D 
represent the wave pattern in the material of the cylindrical object 32 at 
additional higher resonant frequencies. As shown in FIGS. 9A-9D, the 
higher the resonant frequency, the deeper towards the center of the 
cylindrical object 32 the vibrations penetrate. Because the depth of 
penetration of the vibrations varies for different resonant frequencies, 
the axial shear mode resonant frequencies can be used to measure the depth 
at which physical properties of the cylindrical object 32 undergo a 
change. 
As in the method of temperature determination described above, for a 
cylindrical object 32 formed from a particular size and material, changing 
the physical properties of the material will change the resonant 
frequencies of the cylindrical object 32. Because axial shear vibrations 
penetrate to varying radial depths, the change in axial shear frequency of 
the cylindrical object 32 can be exploited to determine the radial depth 
at which the material properties change. 
Methods used to determine the case hardening depth of a cylindrical object 
are described in reference to the FIGS. 10, 11, and 12. 
In a first method, the EMAT 30 was used to determine a single resonant 
frequency for axial shear vibrations for a plurality of different 
cylindrical objects 32. Each of the plurality of cylindrical objects 32 
had similar dimensions and were formed from the same material. The case 
hardening depth, however, was different for each cylindrical object 32. 
FIG. 10 shows the change in the resonant frequency for different case 
hardening depths. Once a chart as in FIG. 10 is constructed, determining 
the resonant frequency of the cylindrical object 32 permits determination 
of the case hardening depth. 
A second method for determining the case hardening depth is described in 
reference to FIGS. 11 and 12. In the second method, the EMAT 30 is used to 
determine a plurality of frequencies at which a first isotropic 
cylindrical object 32 (i.e. an object having homogeneous physical 
properties throughout the object) having no case hardening experiences 
resonant axial shear vibrations. Next, six cylindrical objects formed from 
the same material and having nearly identical dimensions were tested to 
determine the frequencies at which they experienced resonant vibrations. 
Each of the six cylindrical objects had been case hardened to a different 
radial depth. Because the case hardening altered the physical properties 
of each of the cylindrical objects, each cylindrical object 32 had 
different resonant frequencies for axial shear vibrations. 
Next, for each of the cylindrical objects 32, ratios were calculated for 
each resonant axial shear mode. The ratios represented the change in a 
resonant frequency between the isotropic object and the hardened object, 
divided by the resonant frequency of the isotropic object. The ratios were 
then plotted as shown in FIG. 11. Finally, an average was calculated for 
each hardened cylindrical object representing the average of the ratios 
shown in FIG. 11. The average of the ratio values are plotted as shown in 
FIG. 12. As shown in FIG. 12, the average ratio values generally 
correspond to a straight line representing the relationship between case 
hardening depth and the average of the frequency shift ratios. 
The above-described methods can be used to develop charts, as shown in 
FIGS. 10 and 12, for any set of cylindrical objects 32 having a variation 
in the physical properties of the material as a function of the radial 
depth. In addition, the same basic methods could be used to map the value 
of a physical material property as a function of radial penetration depth 
for a single cylindrical object 32. For instance, measuring the various 
axial shear resonant frequencies of the cylindrical object 32 with the 
EMAT 30 permits mapping the elastic constant of the cylindrical object 32 
as a function of radial depth. 
Methods for determining the loading applied to the cylindrical object 32 
using the EMATs 30 are described in reference to FIGS. 13-15. The loading 
could be an axially compressive or tensile load, or in the case of tubes, 
the loading could be a pressure applied to the interior of the tube. 
As in the method for temperature determination, the amount of loading 
applied to the cylindrical object 32, affects the frequency at which it 
experiences resonant vibrations. The simplest way to determine the 
magnitude of loading applied to the cylindrical object 32 is to use the 
EMAT 30 to determine the resonant frequencies of the cylindrical object 32 
at various different magnitudes of loading. First, the EMAT 30 is used to 
determine a resonant frequency of the cylindrical object 30 when no 
loading is applied. A first load is then applied to the cylindrical object 
and the EMAT 30 is used to determine the new resonant axial shear 
frequency. This process is repeated for a number of different loads to 
determine the resonant frequencies at each load. 
The test data is then used to prepare a chart showing the relationship 
between the magnitude of the load and the resonant frequency. FIG. 13 
shows the relationship between a resonant axial shear frequency of the 
cylindrical object 32 and a compressive load applied to the cylindrical 
object 32. FIG. 14 shows the relationship between a resonant torsional 
frequency of the cylindrical object 32 for the same range of compressive 
forces. Once these charts have been constructed one can use the EMAT 30 to 
determine the resonant frequency. The chart is then used to determine the 
magnitude of the load. 
Unfortunately, temperature has a great affect on the resonant frequency of 
the cylindrical object 32. The resonant frequencies plotted in FIGS. 13 
and 14 were measured concurrently by two different EMATs 30 as a 
compressive load applied to the cylindrical object 32 was sequentially 
increased (black dots), then sequentially decreased (hollow squares). The 
sudden jump in the resonant frequencies for both axial shear and torsional 
vibration between 3 and 4 kN that occurred when the forces were being 
increased (the black dots) is the result of a small (a few degrees 
Celsius) temperature change in the testing room between measurements. 
Because of the affect of temperature on the resonant frequencies, the 
charts shown in FIGS. 13 and 14 are only useful for a relatively narrow 
temperature range. To determine the loading with an acceptable level of 
accuracy using a single resonant frequency chart, as shown in FIGS. 13 or 
14, the original testing done to create the charts, and the actual test of 
the loading, must be performed at the same temperature. 
A second method for determining the load, however, accounts for temperature 
variations. In this method, an unloaded cylindrical object is first tested 
to determine a particular resonant axial shear frequency and a particular 
resonant torsional frequency. Next, measurements of the resonant 
frequencies are taken for a number of different magnitudes of load for the 
two different modes of vibrations. As described above, the charts of FIGS. 
13 and 14 will result from the measurements. 
Next, a ratio is calculated, representing the resonant axial shear 
frequency to the resonant torsional frequency for the cylindrical object 
32 with no load applied. Then, a plurality of different ratios are 
calculated representing the resonant axial shear frequency divided by the 
resonant torsional frequency for each of the different compressive forces 
applied to the cylindrical object 32. A plurality of difference 
measurements are then calculated representing the difference between the 
no load ratio, and the each of the loaded ratios. 
The ratios are then plotted as shown in FIG. 15. As shown in FIG. 15, a 
relatively straight line will result showing the relationship between the 
compressive force and the difference between the two ratios. The 
temperature variation occurring between the 3 and 4 kN measurements does 
not affect the difference in the ratios. This method can be used to 
account for temperature variations in determining the magnitude of loading 
on the cylindrical object 32, so long as temperature variations affect 
each type of resonant frequency approximately equally. 
Once a plot as shown in FIG. 15 is produced for a particular cylindrical 
object 32, the amount of a force applied to the cylindrical object 32 can 
be determined by measuring the axial shear and torsional resonant 
frequencies, calculating a ratio between the resonant frequencies, and 
comparing the calculated ratio to the no load ratio. 
As mentioned above, the same method can be used to determine the magnitude 
of a tensile load applied to the cylindrical object 32. If the cylindrical 
object 32 is a hollow tube, this method can also be used to determine a 
pressure within (or without) the tube. 
When a symmetrical EMAT 30 having only two magnets 22 (one on each side of 
the cylindrical object 32) is used to determine the resonant frequency of 
the cylindrical object 32, at some frequencies ranges, instead of getting 
a response signal with a single large amplitude spike at a single resonant 
frequency, two smaller amplitude signals are sensed at two slightly 
separated resonant frequencies. It is believed that the splitting of the 
resonant frequencies is due to the grain orientation of the material 
comprising the cylindrical object 32. This resonant frequency splitting 
can be exploited to determine the grain orientation of the material of the 
cylindrical object 32 using the method described below. 
In a first-step, the cylindrical object 32 is inserted in to the EMAT-30 
having two evenly spaced magnets 22. The cylindrical object 32 is oriented 
at a first rotational position, and the amplitude of a response signal of 
the wire coil of the EMAT 30 is recorded for each of two closely spaced 
resonant frequencies. Next, the cylindrical object 32 is rotated a certain 
number of degrees and the amplitudes of response signals are again 
recorded for each of the two split resonant frequencies. This procedure 
may be repeated for a plurality of different rotational positions of the 
cylindrical object 32. By comparing the amplitudes of the response signals 
at the different rotational positions, the grain orientation of the 
material which forms the cylindrical object 32 can be determined. 
In addition to determining grain orientation, the same basic method can be 
used to determine variations in the thickness of thin-walled tubing around 
the circumference of the tube. If the amplitudes of the split resonant 
frequencies remain relatively constant as the tube is rotated in the EMAT 
30, the wall thickness is relatively constant. Variations in the amplitude 
indicates variations in the wall thickness. 
The use of the EMAT 30 having two magnets 22 in the above described method 
is not essential to the method. All that is required is that the EMAT 30 
produce a split resonant frequency response signal. 
The texture of the cylindrical object 32 can also be determined by 
measuring the elastic constant of the cylindrical object 32 for axial 
shear vibrations and torsional vibrations, and comparing the measurements. 
The comparison of the elastic constants at various rotational positions 
will also provide an indication of the grain orientation of the material 
in the cylindrical object 32. 
Variations in the above-described methods can be used to determine a 
variety of different physical properties of the cylindrical object 32 
using EMATs 30 according to this invention. 
While this invention has been described in conjunction with specific 
embodiments thereof, it is evident that many alternatives, modifications, 
and variations will be apparent to those skilled in the art. Accordingly, 
it is intended to embrace all such alternatives, modifications and 
variations as fall within the spirit and scope of the appended claims.