U.S. patent application number 09/875608 was filed with the patent office on 2003-02-27 for apparatus and method for characterizing physical properties of a test piece.
Invention is credited to Hage, Richard Todd.
Application Number | 20030037619 09/875608 |
Document ID | / |
Family ID | 25366072 |
Filed Date | 2003-02-27 |
United States Patent
Application |
20030037619 |
Kind Code |
A1 |
Hage, Richard Todd |
February 27, 2003 |
APPARATUS AND METHOD FOR CHARACTERIZING PHYSICAL PROPERTIES OF A
TEST PIECE
Abstract
An apparatus and method for non-destructively determining
non-musical physical characteristics such as the Young's modulus,
stiffness coefficient, and damping coefficient of a test piece of
arbitrary shape. The apparatus consists of a frame having a natural
vibrational frequency different from the free vibrational frequency
of the test piece, a mechanism for fixing the test piece to the
frame, a mechanism for initiating free vibrations in the test
piece, a mechanism to measure the displacement of the test piece
during the free vibrations, and a mechanism to measure time during
the free vibrations. The amplitude, frequency, and decay
characteristics of the free vibrations are analyzed mathematically
to determine the physical characteristics. The apparatus may
include a mechanism to control the initial deflection applied to
the test piece. The apparatus may include a mechanism to control
the initial load applied to the test piece. The apparatus may
include a device such as a computer to collect, process, record,
and display the information obtained.
Inventors: |
Hage, Richard Todd;
(Champlin, MN) |
Correspondence
Address: |
MERCHANT & GOULD PC
P.O. BOX 2903
MINNEAPOLIS
MN
55402-0903
US
|
Family ID: |
25366072 |
Appl. No.: |
09/875608 |
Filed: |
June 6, 2001 |
Current U.S.
Class: |
73/778 |
Current CPC
Class: |
G01N 3/32 20130101; G01N
2203/0005 20130101; G01N 2203/0658 20130101; G01N 2203/0075
20130101 |
Class at
Publication: |
73/778 |
International
Class: |
G01L 001/00 |
Claims
We claim:
1. An apparatus for measuring at least one non-musical physical
property of a test piece, comprising: a frame having a natural
vibrational frequency different from a free vibration of the test
piece; fixing means adapted to fix at least one fixing point of the
test piece; initiating means adapted to initiate the free vibration
in the test piece resulting in a vibrating test piece; displacement
measuring means adapted to obtain measurements of a displacement of
said vibrating test piece; time measuring means in communication
with said displacement measuring means, said time measuring means
being adapted to obtain a measurement of time.
2. The apparatus as claimed in claim 1, wherein said initiating
means are adapted to apply a deflection to the test piece, and
wherein said initiating means are adapted to release said
deflection whereby the free vibrations are initiated in the test
piece.
3. The apparatus as claimed in claim 1, wherein the free vibrations
are transverse longitudinal vibrations, wherein the displacement is
transverse to a longitudinal axis of the test piece.
4. The apparatus as claimed in claim 1, further comprising
collecting means adapted for collecting said measurements of
displacement and time.
5. The apparatus as claimed in claim 4, wherein said collecting
means comprise a computer.
6. The apparatus as claimed in claim 1, further comprising
processing means adapted for processing said measurements of
displacement and time.
7. The apparatus as claimed in claim 6, wherein said processing
means comprise a computer.
8. The apparatus as claimed in claim 1, further comprising
recording means adapted for recording said measurements of
displacement and time.
9. The apparatus as claimed in claim 8, wherein said recording
means comprise a computer.
10. The apparatus as claimed in claim 1, further comprising display
means adapted for displaying said measurements of displacement and
time.
11. The apparatus as claimed in claim 10, wherein said display
means comprise a computer monitor.
12. The apparatus as claimed in claim 1, wherein said displacement
measuring means and said time measuring means are adapted to obtain
measurements at least 1000 times per second.
13. The apparatus as claimed in claim 1, wherein said fixing means
are adapted to fix a generally longitudinal test piece at a first
end of the test piece.
14. The apparatus as claimed in claim 13, wherein said fixing means
are adapted to fix the test piece at the first end and a second end
of the test piece.
15. The apparatus as claimed in claim 1, wherein said fixing means
are adapted to fix a generally longitudinal test piece at a center
point generally coinciding with a center of mass of the test
piece.
16. The apparatus as claimed in claim 1, wherein said fixing means
are adapted to releasably fix said at least one fixing point of the
test piece.
17. The apparatus as claimed in claim 1, wherein said fixing means
comprise a mechanical clamp.
18. The apparatus as claimed in claim 1, wherein said fixing means
comprise a water bladder clamp.
19. The apparatus as claimed in claim 1, wherein said fixing means
comprise an adhesive.
20. The apparatus as claimed in claim 2, wherein said initiating
means are adapted to be actuatable between a first position wherein
said deflection of the test piece is not enabled, and a second
position wherein said deflection of the test piece is enabled.
21. The apparatus as claimed in claim 20, wherein said initiating
means comprises: a pin releasably connected to the test piece, said
pin being generally perpendicular to a length of the test piece,
said pin being adapted to apply said deflection the test piece,
said deflection being generally transverse to said length of the
test piece; and a rotatable member connected to said frame, said
rotatable member being rotatable about an axis generally
perpendicular to said pin, said rotatable member being adapted to
engage said pin so as to enable said pin to apply said deflection
to the test piece, said rotatable member defining an aperture
therein sized so as to receive said pin in a non-interfering fit;
wherein when said initiating means is in said first position said
rotatable member is oriented such that said aperture receives said
pin, whereby said pin is not enabled to apply said deflection to
the test piece; and wherein when said initiating means is in said
second position said rotatable member is oriented such that said
aperture does not receive said pin, whereby said pin is enabled to
apply said deflection to the test piece.
22. The apparatus as claimed in claim 1, wherein said displacement
measuring means are non-contact measuring means.
23. The apparatus as claimed in claim 1, wherein said displacement
measuring means comprise a linear variable displacement
transducer.
24. The apparatus as claimed in claim 23, wherein said linear
variable displacement transducer comprises a core, core mounting
means for releasably mounting said core to the test piece, and a
body connected to said frame, said body being suitable for
accepting said core therein.
25. The apparatus as claimed in claim 2, further comprising load
measuring means connected to said initiating means, said load
measuring means being adapted to obtain measurements of a load
applied to the test piece during said deflection of the test
piece.
26. The apparatus as claimed in claim 25, wherein said load
measuring means comprise a load cell.
27. The apparatus as claimed in claim 2, further comprising
deflection control means connected to said initiating means, said
deflection control means being adapted to control said deflection
of the test piece.
28. The apparatus as claimed in claim 25, wherein said deflection
control means comprise a micrometer.
29. The apparatus as claimed in claim 1, wherein said time
measuring means comprise a digital timing circuit.
30. A method for measuring at least one non-musical physical
property of a test piece, comprising the steps of: fixing the test
piece to a frame at at least one fixing point, said frame having a
natural vibrational frequency different from a free vibration of
the test piece; initiating the free vibration in the test piece;
measuring a time-varying displacement of the test piece during the
free vibration; and calculating at least one value for said at
least one physical property from the time-varying displacement.
31. The method according to claim 30, wherein a plurality of values
are calculated for said at least one physical property of the test
piece from the time-varying displacement.
32. The method according to claim 30, further comprising the step
of displaying said at least one value.
33. The method according to claim 32, wherein said at least one
value is displayed graphically.
34. The method according to claim 30, wherein said at least one
value is calculated using a computer.
35. The method according to claim 32, wherein said at least one
value is displayed using a computer.
36. The method according to claim 30, wherein said at least one
physical property is a Young's Modulus of the test piece.
37. The method according to claim 30, wherein said at least one
physical property is a damping coefficient of the test piece.
38. The method according to claim 30, wherein said at least one
physical property is a stiffness coefficient of the test piece.
39. The method according to claim 36, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein the at least one value
of the Young's Modulus is calculated according to 38 E = 4 2 L 3 (
kM sensor + k ' M test ) k " T d 2 I ( 1 - ( ln ( y 0 y ( 0 + n ) )
2 n ) 2 ) wherein E is the Young's Modulus of the test piece L is a
length between said at least one fixing point and said displacement
measuring point k is a geometric coefficient corresponding to a
position of said displacement measuring point M.sub.sensor is a
mass of said sensor at said displacement measuring point k' is a
geometric coefficient corresponding to a position of said at least
one fixing point on the test piece M.sub.test is a mass of the test
piece k" is a geometric coefficient corresponding to a an
orientation of the test sample relative to said at least one fixing
point T.sub.d is a damped natural period of the free vibrations I
is a moment of inertia of the test piece about said at least one
fixing point y.sub.0 is the displacement of the test piece after
zero free vibrations y.sub.(0+n) is the displacement of the test
piece after zero plus n free vibrations n is a number of free
vibrations.
40. The method according to claim 36, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein: the test piece is
fixed to said frame at a first end of the test piece; said
displacement measuring point is at a second end of the test piece
opposite the first end; the free vibrations are initiated at a
point generally coincidental with said displacement measuring
point; and the at least one value of the Young's Modulus is
calculated according to 39 E = 4 2 L 3 ( M sensor + ( 0.2357 ) M
test ) 3 T d 2 I ( 1 - ( ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) wherein E
is the Young's Modulus of the test piece L is a length between said
at least one fixing point and said displacement measuring point
M.sub.sensor is a mass of said sensor M.sub.test is a mass of the
test piece T.sub.d is a damped natural period of the free
vibrations I is a moment of inertia of the test piece about said at
least one fixing point y.sub.0 is the displacement of the test
piece after zero free vibrations y.sub.(0+n) is the displacement of
the test piece after zero plus n free vibrations n is a number of
free vibrations.
41. The method according to claim 36, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein the test piece is fixed
to said frame at a first end of the test piece and at a second end
of the test piece opposite the first end; said displacement
measuring point generally coincides with a center of mass of the
test piece; the free vibrations are initiated at a point generally
coincidental with said displacement measuring point; and the at
least one value of the Young's Modulus is calculated according to
40 E = 4 2 L 3 ( M sensor + ( 0.3610 ) M test ) 192 T d 2 I ( 1 - (
ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) wherein E is the Young's Modulus
of the test piece L is a length between said at least one fixing
point and said displacement measuring point M.sub.sensor is a mass
of said sensor M.sub.test is a mass of the test piece T.sub.d is a
damped natural period of the free vibrations I is a moment of
inertia of the test piece about said at least one fixing point
y.sub.0 is the displacement of the test piece after zero free
vibrations y.sub.(0+n) is the displacement of the test piece after
zero plus n free vibrations n is a number of free vibrations.
42. The method according to claim 37, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein the at least one value
of the damping coefficient is calculated according to 41 C = ( 4 (
kM sensor + k ' M test ) T d ) ( ln ( y 0 y ( 0 + n ) ) 2 n 1 - (
ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) wherein C is the damping
coefficient of the test piece k is a geometric coefficient
corresponding to a position of said displacement measuring point
M.sub.sensor is a mass of said sensor k' is a geometric coefficient
corresponding to a position of said at least one fixing point on
the test piece M.sub.test is a mass of the test piece T.sub.d is a
damped natural period of the free vibrations y.sub.0 is the
displacement of the test piece after zero free vibrations
y.sub.(0+n) is the displacement of the test piece after zero plus n
free vibrations n is a number of free vibrations.
43. The method according to claim 37, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein: the test piece is
fixed to said frame at a first end of the test piece; said
displacement measuring point is at a second end of the test piece
opposite the first end; the free vibrations are initiated at a
point generally coincidental with said displacement measuring
point; and the at least one value of the damping coefficient is
calculated according to 42 C = ( 4 ( M sensor + ( 0.2357 ) M test )
T d ) ( ln ( y 0 y ( 0 + n ) ) 2 n 1 - ( ln ( y 0 y ( 0 + n ) ) 2 n
) 2 ) wherein C is the damping coefficient of the test piece
M.sub.sensor is a mass of said sensor M.sub.test is a mass of the
test piece T.sub.d is a damped natural period of the free
vibrations y.sub.o is the displacement of the test piece after zero
free vibrations y.sub.(0+n) is the displacement of the test piece
after zero plus n free vibrations n is a number of free
vibrations.
44. The method according to claim 37, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein the test piece is fixed
to said frame at a first end of the test piece and at a second end
of the test piece opposite the first end; said displacement
measuring point generally coincides with a center of mass of the
test piece; the free vibrations are initiated at a point generally
coincidental with said displacement measuring point; and the at
least one value of the damping coefficient is calculated according
to 43 C = ( 4 ( M sensor + ( 0.3610 ) M test ) T d ) ( ln ( y 0 y (
0 + n ) ) 2 n 1 - ( ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) wherein C is
the damping coefficient of the test piece M.sub.sensor is a mass of
said sensor M.sub.test is a mass of the test piece T.sub.d is a
damped natural period of the free vibrations y.sub.0 is the
displacement of the test piece after zero free vibrations
y.sub.(0+n) is the displacement of the test piece after zero plus n
free vibrations n is a number of free vibrations.
45. The method according to claim 38, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein the at least one value
of the stiffness coefficient is calculated according to 44 K L = 4
2 ( kM sensor + k ' M test ) T d 2 ( 1 - ( ln ( y 0 y ( 0 + n ) ) 2
n ) 2 ) wherein K.sub.L is the stiffness coefficient of the test
piece k is a geometric coefficient corresponding to a position of
said displacement measuring point M.sub.sensor is a mass of said
sensor k' is a geometric coefficient corresponding to a position of
said at least one fixing point on the test piece M.sub.test is a
mass of the test piece T.sub.d is a damped natural period of the
free vibrations y.sub.0 is the displacement of the test piece after
zero free vibrations y.sub.(0+n) is the displacement of the test
piece after zero plus n free vibrations n is a number of free
vibrations.
46. The method according to claim 38, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein the test piece is fixed
to said frame at a first end of the test piece; said displacement
measuring point is at a second end of the test piece opposite the
first end; the free vibrations are initiated at a point generally
coincidental with said displacement measuring point; and the at
least one value of the stiffness coefficient is calculated
according to 45 K L = 4 2 ( M sensor + ( 0.2357 ) M test ) T d 2 (
1 - ( ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) wherein K.sub.L is the
stiffness coefficient of the test piece M.sub.sensor is a mass of
said sensor M.sub.test is a mass of the test piece T.sub.d is a
damped natural period of the free vibrations y.sub.0 is the
displacement of the test piece after zero free vibrations
y.sub.(0+n) is the displacement of the test piece after zero plus n
free vibrations n is a number of free vibrations.
47. The method according to claim 38, further comprising the step
of mounting a sensor to the test piece at a displacement measuring
point, said sensor being adapted for measuring the time-varying
displacement of the test piece, and wherein the test piece is fixed
to said frame at a first end of the test piece and at a second end
of the test piece opposite the first end; said displacement
measuring point generally coincides with a center of mass of the
test piece; the free vibrations are initiated at a point generally
coincidental with said displacement measuring point; and the at
least one value of the stiffness coefficient is calculated
according to 46 K L = 4 2 ( M sensor + ( 0.3610 ) M test ) T d 2 (
1 - ( ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) wherein K.sub.L is the
stiffness coefficient of the test piece M.sub.sensor is a mass of
said sensor M.sub.test is a mass of the test piece T.sub.d is a
damped natural period of the free vibrations y.sub.0 is the
displacement of the test piece after zero free vibrations
y.sub.(0+n) is the displacement of the test piece after zero plus n
free vibrations n is a number of free vibrations.
48. The method according to claim 30, wherein said method is
non-destructive to the sample.
49. The method according to claim 30, wherein the test piece is
generally longitudinal in shape.
50. The method according to claim 30, wherein the test piece has a
cross-section of arbitrary geometry.
51. The method according to claim 30, wherein the test piece has a
cross-section with a geometry that varies along a length of the
test piece.
52. The method according to claim 30, wherein the test piece is
comprised of isotropic material.
53. The method according to claim 30, wherein the test piece is
comprised of orthotropic material.
54. The method according to claim 31, wherein said plurality of
values is calculated from the free vibrations resulting from one
initiation of free vibrations in the test piece.
55. The method according to claim 30, wherein a test period
beginning at initiation of the free vibrations and ending after
calculation of said at least one value for said at least one
physical property is not greater than 5 seconds.
56. The method according to claim 31, wherein said plurality of
values of said at least one physical property is distributed over a
dynamic range of said at least one physical property.
57. The method according to claim 30, further comprising the step
of providing feedback based on said at least one value of said at
least one physical property, said feedback comprising instructions
for adjusting a manufacturing process.
58. A method for controlling a process for manufacturing a product,
comprising the steps of: producing a test piece of the product;
fixing the test piece to a frame at at least one fixing point, said
frame having a natural vibrational frequency different from a
frequency of free vibrations of the test piece; initiating the free
vibrations in the test piece; measuring a time-varying displacement
of the test piece during the free vibrations; calculating at least
one value for at least one physical property of the test piece from
the time-varying displacement; adjusting said process for
manufacturing the product based on said at least one value for said
at least one physical property.
Description
BACKGROUND OF THE INVENTION
[0001] This invention relates to an apparatus for characterizing
physical properties of a test piece, and related methods.
[0002] In many circumstances, it is desirable to determine the
physical properties of an object or structure. For example, in a
manufacturing environment, it may be advantageous to determine
selected physical properties of a product so as to verify that the
product meets the minimum standards for its intended purpose.
Similarly, it may be advantageous to determine the physical
properties of a product in order to adjust a manufacturing process,
so that a consistent and high-quality product may be produced.
Furthermore, by evaluating the physical properties of a particular
type of product while adjusting various process parameters, it is
possible to optimize the process parameters for a particular
product or process. For example, for the manufacture of a drawn
polymer material, such parameters as the temperature of the molten
material, the amount of draw-down, the through-put, and the
viscosity of the base material may be determined and optimized
through proper evaluation of physical properties of the finished
product.
[0003] A wide variety of physical properties may be measured. One
useful property is the elastic modulus. The elastic modulus is a
ratio of the stress applied to an object or structure over strain
exhibited by that object or structure.
[0004] Stress is defined as the applied force divided by the
resisting area of the object. It will be appreciated that an object
may be subjected to many sorts of stress, such as shear,
compression, torsion, and tensile. It will also be appreciated that
the resisting area of an object or structure depends on both the
shape of the object and the type or direction of stress applied to
it.
[0005] Strain is a relative deflection of the object in question in
response to the applied stress. An object may similarly exhibit
various sorts of strains depending on the stresses applied.
[0006] Regardless of the type of stress and strain, however, the
elastic modulus may be understood to be an indication of an
object's response to applied forces.
[0007] One particularly useful elastic modulus is the Young's
modulus. The Young's modulus is a ratio of the applied tensile
stress .delta. over the exhibited tensile strain .epsilon..
Although a wide variety of other elastic moduli may of course be
measured, it is enlightening to consider as a specific example the
Young's modulus and devices known for measuring it.
[0008] Several devices are known for measuring Young's modulus.
Conventional devices include destructive ("test until failure")
systems and dynamic mechanical analyzers, commonly known as DMAs.
Neither type of device is entirely satisfactory, however.
[0009] Conventionally, the Young's modulus of an object is
determined destructively by measuring the deflection of the object
while applying a gradually increasing force to the object until it
"fails", that is, deforms plastically or fractures. The
stress-strain relationship is then determined for the sample. The
stress-strain relationship is plotted as a curve, and a
straight-line fit of the curve is approximated. The slope of the
line may be used as an approximation of Young's modulus.
[0010] This approach has numerous disadvantages. For example, a
useful approximation of Young's modulus can only be determined if
the stress-strain relationship is linear or nearly linear over a
broad range. That is, the material must have a large and generally
uniform range of elastic deformation. Although some materials have
such properties, many others do not. In particular, many composite
materials do not exhibit linear stress-strain relationships.
[0011] Also, the destruction of the object being tested is inherent
in the test method. In order to obtain sufficient useful data for
the straight-line fit, the sample must be tested across essentially
its entire range of elastic deformation. Therefore, it is necessary
to increase the applied force until the sample either deforms
plastically or fractures. In either event, the sample is destroyed,
and is unavailable for sale, further testing, or other
purposes.
[0012] Because conventional destructive methods require a
substantial range of stress-strain data to determine a useful value
of Young's modulus, destructive testing is generally not suitable
for determining Young's modulus at a particular stress, or for a
narrow band of stresses. Conventional destructive testing typically
produces only an average value for the entire range of applied
stress.
[0013] In addition, destructive tests are generally limited to
samples of standard size and shape. This is the case for several
reasons. First, in order to calculate the stress on a sample, the
resisting area of the sample must be known. Thus, in order to
determine the Young's modulus of an arbitrarily-shaped product
using conventional destructive methods, the geometry of the product
must be carefully measured, and the area calculated. For complex
shapes, this is a considerable difficulty.
[0014] Second, the Young's modulus of an object depends not only on
its material and its resisting area but also on its shape. For
example, a U-shaped beam generally exhibits a higher Young's
modulus than a flat strip, even if the strip and the beam have the
same resisting area and are made from the same material. In many
cases, especially for complex shapes, it is impractical or
impossible to calculate in advance the effect of a particular shape
on the Young's modulus. Thus, in order to compare test subjects
without performing complex corrections due to varying geometry, it
is generally necessary to test a sample of standard size and shape
instead of an actual product.
[0015] Although this simplification is convenient, the use of
samples as opposed to actual products produces difficulties. For
example, tests on samples tend to be inaccurate for orthotropic
materials, that is, materials that have a directionally non-uniform
structure. One common example is wood, which has a grain that is
stronger in some directions than in others. Other orthotropic or
partially orthotropic materials include composites, laminates, etc.
Orthotropic materials pose difficulties for conventional tests for
several reasons.
[0016] First, the stress and strain of orthotropic materials do not
always vary linearly with increasing dimension. That is, doubling
the cross-sectional area of a sample of an orthotropic material may
not double the stress required to achieve a given strain, even if
the material composition and shape are kept exactly the same.
[0017] Second, orientation is important for orthotropic materials.
For example, an object composed of many laminated layers will have
a very different Young's modulus if the layers are oriented
parallel to the direction of stress than if they are perpendicular
to the direction of stress. Because it may be inconvenient or
impossible to produce a test sample that is representative of the
orientation in which the actual product will be used, the accuracy
of the test becomes questionable at best.
[0018] Dynamic Mechanical Analyzers operate according to a
different principle. As previously described, the Young's modulus
is a measure of material stiffness. The Young's Modulus of an
object, along with the object's geometry, determine the stiffness
coefficient of that object. In turn, the stiffness coefficient of
an object determines the frequency at which it will vibrate. By
proper analysis, it is therefore possible to determine the Young's
modulus of an object if the geometry and a frequency of vibration
of the object are known.
[0019] In order to obtain data, the object must of course be made
to vibrate. However, the vibration must be at the free vibrational
frequency for the object, also known as its natural frequency. The
free vibrational frequency of an object is the frequency at which
it will vibrate if it is initially disturbed but not subsequently
subjected to additional forces. Such vibrations are known as free
vibrations.
[0020] If an object is driven at its free vibrational frequency,
also known as its harmonic frequency, the object's natural
vibrations and the driving vibrations will combine additively, and
the object is said to be vibrating harmonically. Harmonic vibration
is generally detectable as an apparent increase in the amplitude of
the object's vibrations.
[0021] DMAs exploit this natural phenomenon. Driving vibrations are
applied to an object to be tested. The frequency of the vibrations
is slowly adjusted until the object undergoes harmonic vibration at
its free vibrational frequency. At this point the vibrations are
maintained at a constant, steady-state frequency, and the driving
frequency, which at this point is known to be equal to the free
vibrational frequency, of the object is measured. Young's modulus
is then calculated from the free vibrational frequency of the
object.
[0022] However, DMAs also suffer from serious limitations.
[0023] First, as described above, the free vibrational frequency of
an object depends not only on its Young's modulus but also on its
geometry. As described previously with respect to destructive
testing, determining the geometry of an arbitrary object may be
inconvenient or impossible. Therefore, for similar reasons, DMA
testing is conventionally performed on samples of standard size and
shape.
[0024] However, because of this, the problems inherent in testing
standard samples as opposed to actual objects, as outlined above
with respect to destructive testing, also apply to DMA testing.
[0025] In addition, DMA testing is very time-consuming. In order to
obtain data, the test sample must undergo steady-state harmonic
vibration. Thus, DMA testing requires that the driving frequency be
changed at a rate that permits the harmonic vibration to reach a
steady state and be recognized. In practice, this means that the
frequency of the driving vibrations must be changed very slowly.
Because the range of possible natural frequencies for a test piece
is quite broad, considerable time is required to perform even a
single DMA test.
[0026] Also, because of the need to precisely match an initially
unknown frequency for every test sample, DMA testing requires a
relatively high degree of skill and training to perform reliably.
The need for a highly skilled operator limits the utility of DMA
testing for controlling a manufacturing line from a production
floor.
[0027] Furthermore, DMA testing is severely limited in terms of the
amount of data generated. The stress-strain curve of any object is
never perfectly linear. That is, the Young's modulus of any object
is different for different strains. For a vibration, the strain
corresponds to the amplitude of the vibration. This means that the
free vibrational frequency of an object varies at least slightly
with the amplitude of the vibration. Because the harmonic
vibrations necessary for DMA testing are steady-state, they permit
measurement of only a single point of data on the stress-strain
curve, and hence only a single value for Young's modulus, per test.
Varying the amplitude of the vibrations to determine a value for
Young's modulus at a different strain requires readjustment of the
driving frequency to match the free vibrational frequency of the
object for that different strain. Therefore, in order to generate a
useful stress-strain curve, rather than a single approximating
value, the test must be repeated many times. This can be
inconvenient, especially in view of the relatively long time
necessary for each individual test. In particular, the long time
required renders DMA testing unsuitable for many process control
applications, as it may not be possible to produce data quickly
enough to permit timely adjustment of the process.
SUMMARY OF THE INVENTION
[0028] In contrast, the claimed invention is based on the
observation of free vibrations in a system. Measurements of the
free vibrations of the system may be used to conveniently determine
physical properties of an object of an object of largely arbitrary
geometry, without destroying the object being tested.
[0029] In order to appreciate the structure and function of the
claimed invention, it is illustrative to consider certain
properties of free vibration, and certain exemplary systems.
[0030] An object or system that is struck, deflected or otherwise
disturbed and then left undisturbed undergoes free vibration. Free
vibrations in an object may be represented as a single degree of
freedom mass-spring-damper system 10, as illustrated in FIG. 1.
[0031] An ideal mass-spring-damper system 10 includes a mass 12, a
spring 14 connecting the mass 12 to an effectively immobile base
16, and a damping element 18 also connecting the mass 12 to the
base 16. In an equilibrium state, the spring 14 is neither
compressed nor extended, and thus applies no net force to the mass
12. If the mass 12 is displaced by some initial distance, the
spring 14 is either extended or compressed, and in either case a
net force is a then applied to the mass 12. The mass 12 will then
oscillate back and forth as the spring 14 alternately expands and
contracts. At this point, the system 10 is undergoing free
vibration.
[0032] For purposes of mathematical analysis, the physical
properties of the system are defined as follows. The mass 12 has a
possesses a mass M. The spring 14 has a stiffness coefficient K,
also known as a spring coefficient. The damping element 16 has a
damping coefficient C.
[0033] It is noted that the system illustrated in FIG. 1 is a
one-dimensional system, wherein both the spring 14 and the damping
element 16 are effectively massless, and wherein the mass 12 is a
zero-dimensional point mass. For such a system, the entire mass M
and only mass M have any effect on the system. Thus, the effective
mass of the system is exactly equal to the mass of the mass 12.
However, it will be appreciated by those knowledgeable in the art
that for a real system in two or more dimensions, such as for
example a longitudinal beam undergoing transverse longitudinal free
vibrations, the mass M of a system as it relates to the following
discussion would be an effective mass M.sub.eff, rather than simply
an ideal point mass. For the sake of accuracy in the following
discussion, the mass of the system will be referred to as the
effective mass M.sub.eff. This matter is analyzed in greater detail
below.
[0034] The stiffness coefficient K relates the force necessary to
displace the mass M by a given distance, according to the
equation:
F=Ky (Equation 1)
[0035] wherein
[0036] F is the force applied to the system;
[0037] K is the stiffness coefficient; and
[0038] y is the displacement of the mass.
[0039] Alternatively, Equation 1 may be rearranged to solve for K:
1 K = F y ( Equation 2 )
[0040] The displacement is also referred to as the deflection of
the system, in particular for cases wherein the spring force is
applied to bend or deflect a solid object such as a beam. It will
be appreciated by those knowledgeable in the art that when an
initial displacement is used to initiate free vibrations, the
amplitude of the initial displacement is the same as the initial
amplitude of the free vibrations.
[0041] The damping coefficient C is a measure of the forces
opposing free vibrations in the system. In the simple case wherein
the system is completely undamped, that is, the damping coefficient
C is 0, the system vibrates according to the function: 2 n = K M
eff ( Equation 3 )
[0042] wherein
[0043] .omega..sub.n is the natural frequency of vibrations of the
system;
[0044] K is the stiffness coefficient of the spring; and
[0045] M.sub.eff is the effective mass.
[0046] For an undamped system the free vibration would continue
indefinitely, at an amplitude exactly equal to the original
displacement, and at a frequency determined by the effective mass
and the stiffness coefficient according to Equation 3.
[0047] However, in any real system, energy is lost due to various
damping factors, such as friction, etc. Regardless of the precise
source the damping, energy is gradually lost from any freely
vibrating system. Thus, the vibrations gradually decay until the
system reaches static equilibrium and the free vibrations cease
altogether.
[0048] The motion of a damped system is considerably more complex
than that of an undamped system. For a spring-mass-damper system as
illustrated in FIG. 1, the governing equation of motion may be
written: 3 M eff 2 y t 2 + C y t + Ky = f ( t ) ( Equation 4 )
[0049] wherein
[0050] M.sub.eff is the effective mass;
[0051] C is the damping coefficient;
[0052] K is the stiffness coefficient; and
[0053] y is the displacement of the mass.
[0054] It will be appreciated by those knowledgeable in the art
that both the form of Equation 4 and the spring-mass-damper system
illustrated in FIG. 1 describe linear systems, and that such
systems are exemplary only. A wide variety of other damped free
vibrational systems are possible, including but not limited to
rotational spring-mass-damper systems. It will be further
appreciated that the form of Equation 4 and the spring-mass-damper
system illustrated in FIG. 1 represent one-dimensional systems, and
that such systems are likewise exemplary only. Non-linear systems
and systems undergoing free vibrations in two or more dimensions
may be represented by equations similar to Equation 4. The
mathematical analysis disclosed herein is similarly applicable to
such alternative systems, and the discussion herein disclosed with
respect to the exemplary system of FIG. 1 likewise applies equally
to them.
[0055] Returning to the exemplary case of FIG. 1 and Equation 4, it
will be appreciated that the rate at which the vibrations damp
towards an amplitude of zero depends on the value of the damping
coefficient C, that is, on the amount of damping present in the
system. For any system, there is a quantity of damping for which
the vibrations decay towards static equilibrium at the most rapid
rate possible. This value of the damping coefficient is referred to
as critical damping C.sub.c. If the actual damping value is
substantially lower than C.sub.c the actual rate at which the
vibrations approach an amplitude of zero is lower than it would be
if the damping were C.sub.c. If the actual damping value is
substantially higher than C.sub.c the system does not truly vibrate
at all, but rather moves more or less steadily toward
equilibrium.
[0056] A physical system may be characterized in terms of how
closely the actual damping matches the critical damping value for
that system. The damping ratio is thus defined as 4 C C c (
Equation 5 )
[0057] It will be appreciated that for a critically damped system,
the damping ratio .zeta. is equal to 1.
[0058] It will further be appreciated that the value C.sub.c for
any spring-mass-damper system must depend on the effective mass
M.sub.eff and the stiffness coefficient K. In mathematical terms,
this dependence follows the equation:
C.sub.c=2{square root}{square root over (M.sub.effK)} (Equation
6)
[0059] Combining Equations 5 and 6 yields the following relation: 5
= C 2 M eff K ( Equation 7 )
[0060] Equation 7 may also be solved for the damping coefficient C
as:
C=2.zeta.{square root}{square root over (M.sub.effK)} (Equation
8)
[0061] Returning now to the governing equation of motion as
formulated in Equation 4, dividing all terms by K yields: 6 M eff K
2 y t 2 + C K y t + y = f ( t ) ( Equation 9 )
[0062] As may be seen, by manipulation of terms and substitution of
Equations 3 and 7 the following relation may be determined:
[0063] (Equation 10) 7 C K = ( 2 2 ) ( C ( K K ) ) ( M eff M eff )
= 2 ( M eff K ) ( C 2 M eff K ) = 2 n ( Equation 10 )
[0064] In turn, substituting Equations 3 and 8 into Equation 9
yields: 8 1 n 2 2 y t 2 + 2 n y t + y = f ( t ) ( Equation 11 )
[0065] wherein
[0066] .omega..sub.n is the natural frequency of the system
[0067] y is the displacement of the system;
[0068] t is the time elapsed since the initiation of free
vibrations; and
[0069] .zeta. is the damping ratio of the system.
[0070] Equation 11 may be usefully solved for y in the form:
[0071] y=e.sup.(-.zeta..omega..sup..sub.n.sup.t)(c.sub.1
sin(.omega..sub.n{square root}{square root over
(1-.zeta..sup.1t)})+c.sub- .2 cos(.omega..sub.n{square root}{square
root over (1-.zeta..sup.2t)}) (Equation 12)
[0072] wherein
[0073] c.sub.1 and c.sub.2 are constants reflecting initial system
conditions.
[0074] It is noted that c.sub.1 and c.sub.2 are not components of
or otherwise directly related to the damping coefficient C.
[0075] It will be appreciated by those knowledgeable in the art
that for physical reasons the frequency of free vibrations of a
mass-spring-damper system where the damping coefficient (and hence
the damping ratio) is non-zero is different from the frequency that
would be observed for the same system if the damping coefficient
were zero. The continuing loss of energy due to damping reduces the
frequency of free vibration.
[0076] The actual frequency of free vibration for a
spring-mass-damper system may be determined by inspection of
Equation 12. For such an equation, the frequency is the expression
associated with "t" upon which the trigonometric functions operate.
Considering the case wherein the damping coefficient C and
therefore the damping ratio .zeta. is zero, the expression {square
root}{square root over (1-.zeta..sup.2)} equals exactly 1. Hence,
Equation 12 for that special case could be written
y=e.sup.(-.zeta..omega..sup..sub.n.sup.t)(c.sub.1
sin(.omega..sub.n,t)+c.s- ub.2 cos(.omega..sub.nt)) (Equation
13)
[0077] In such a case, the frequency is .omega..sub.n, as was
previously stated. Similarly, in Equation 12, the frequency with a
non-zero damping coefficient may be extracted as:
.omega..sub.d=.omega..sub.n{square root}{square root over
(1-.zeta..sup.2)} (Equation 14)
[0078] wherein .omega..sub.d is the damped natural frequency of the
system.
[0079] Alternately, Equation 14 may be solved in terms of
.omega..sub.n as 9 n = d 1 - 2 ( Equation 15 )
[0080] It will be appreciated that the period associated with the
damped natural frequency .omega..sub.d of the system is:
[0081] (Equation 16) 10 T d = 2 d ( Equation 16 )
[0082] wherein
[0083] T.sub.d is the damped natural period of the system.
[0084] Returning to Equation 12, by substitution of Equation 14
therein, it can be written as:
y=e.sup.(-.zeta..sup..sub.n.sup.t)(c.sub.1
sin(.omega..sub.dt)+c.sub.2 cos(.omega..sub.dt)) (Equation 17)
[0085] Alternatively, the solution of Equation 12 may be written in
a different but equivalent form:
y=c.sub.3e.sup.(-.zeta..omega..sup..sub.n.sup.t)sin(.omega..sub.dt+.phi.)
(Equation 18)
[0086] wherein
[0087] c.sub.3 is a constant reflecting initial system
conditions;
[0088] .phi. is a phase constant reflecting initial system
conditions.
[0089] It is noted that c.sub.3, like c.sub.1 and c.sub.2, is not a
component of or otherwise directly related to the damping
coefficient C.
[0090] As previously noted, once free vibrations are initiated in a
mass-spring-damper system, energy is gradually lost to the damper
and the vibrations decay. The relative amplitude of the peaks of
two vibrations may be calculated by a simple ratio of Equation 18
for two values of n. It will be appreciated that the ratio of peak
values is exemplary only. Although it is convenient in certain
applications to compare one peak to another, comparing waves at
other phases is also mathematically possible. Such alternative wave
comparisons may be equally suitable for certain applications, and
may be handled similarly.
[0091] It is convenient to consider the case where the first value
of n is zero, and the second remains an arbitrary value n . It will
be appreciated that in such a case, t for the peak at n=0 will be
0, and t for the peak at n will be T.sub.dn. Therefore: 11 y 0 y 0
+ n = c 3 e ( - n t ) sin ( d 0 + ) c 3 e ( - n t + T d n ) sin ( d
T d n + ) = e - n t e - n t + T d n = e n T d n ( Equation 19 )
[0092] Taking the natural log of both sides yields the expression:
12 l ( y 0 y 0 + n ) = le n T d n = n T d n ( Equation 20 )
[0093] Equation 20 can be solved in terms of .zeta. as: 13 = l ( y
0 y 0 + n ) n T d n ( Equation 21 )
[0094] With substitution from Equations 14 and 16, Equation 21 may
in turn be expressed as: 14 = l ( y 0 y 0 + n ) n T d n = l ( y 0 y
0 + n ) n ( 2 d ) n = l ( y 0 y 0 + n ) n 2 ( n 1 - 2 ) n = 1 - 2 l
( y 0 y 0 + n ) 2 n ( Equation 22 )
[0095] Equation 22 represents an exact and general solution for any
value of the damping ratio .zeta.. However, in practice, many
systems of interest have a damping ratio .zeta. that is
substantially less than 1. It will be appreciated by those
knowledgeable in the art that if the damping ratio .zeta. is
substantially less than 1, the value of the expression {square
root}{square root over (1-.zeta..sup.2)} closely approximates
1.
[0096] Using this approximation, Equation 22 may be simplified to
the form: 15 = l ( y 0 y 0 + n ) 2 n ( Equation 23 )
[0097] Based on the above mathematical derivations, it is possible
to modify and substitute various of the preceding equations into
Equation 6 such that the damping coefficient C is expressed in
terms of directly measurable variables. First, Equation 3 is
substituted into Equation 8: 16 C = 2 M eff K = 2 M K M eff = 2 M
eff n ( Equation 24 )
[0098] Then Equation 24 is rearranged into equivalent forms: 17 C =
2 M eff n = 2 M eff 1 n = 4 M eff 2 n = ( 4 M eff ( 2 n ) ) 1 1 - 2
1 - 2 = ( 4 M eff ( 2 n 1 - 2 ) ) ( 1 - ( Equation 25 )
[0099] Next, Equation 14 is substituted into Equation 25: 18 C = (
4 M eff ( 2 n 1 - 2 ) ) ( 1 - 2 ) = ( 4 M eff ( 2 d ) ) ( 1 - 2 ) (
Equation 26 )
[0100] Then, Equation 16 is substituted into Equation 26: 19 C = (
4 M eff ( 2 d ) ) ( 1 - 2 ) = 4 M eff T d ( 1 - 2 ) ( Equation 27
)
[0101] Finally, Equation 23 is substituted into Equation 28: 20 C =
4 M eff T d ( 1 - 2 ) = ( 4 M eff T d ) ( ln ( y 0 y 0 + n ) 2 n 1
- ( ln ( y 0 y 0 + n ) 2 n ) 2 ) ( Equation 29 )
[0102] C is the damping coefficient;
[0103] M.sub.eff is the effective mass;
[0104] T.sub.d is the damped natural period of the system;
[0105] y.sub.o is the peak displacement of the system after zero
free vibrations;
[0106] n is the number of free vibrations; and
[0107] y.sub.0+n is the peak displacement of the system after n
free vibrations.
[0108] It is similarly possible to calculate a value of the
stiffness coefficient K for a system undergoing free vibrations
without depending on the geometry of the system. Returning to
Equation 3, and solving it in terms of K yields the relation:
K=.omega..sub.n.sup.2M (Equation 30)
[0109] wherein
[0110] K is the stiffness coefficient of the spring
[0111] .omega..sub.n.sup.2 is the natural frequency of vibrations;
and
[0112] M.sub.eff is the effective mass.
[0113] Substitution of the expression for .omega..sub.n from
Equation 15 results in the relation: 21 K = ( d 1 - 2 ) 2 M eff = d
2 M eff 1 - 2 ( Equation 31 )
[0114] Equation 16 may be rearranged to solve for the damped
natural frequency .omega..sub.d: 22 d = 2 T d ( Equation 32 )
[0115] Substituting Equation 32 into Equation 31 results in the
relation: 23 K = ( 2 T d ) 2 M eff` 1 - 2 = 4 2 M eff T d 2 ( 1 - 2
) ( Equation 33 )
[0116] Substitution of the expression for .zeta. from Equation 22
yields: 24 K = 4 2 M eff T d 2 ( 1 - l ( y 0 y 0 + n ) 2 n ) 2 (
Equation 34 )
[0117] wherein
[0118] K is the stiffness coefficient;
[0119] M.sub.eff is the effective mass;
[0120] T.sub.d is the damped natural period of the system;
[0121] y.sub.o is the peak displacement of the system after zero
free vibrations;
[0122] y.sub.o+n is the peak displacement of the system after n
free vibrations; and
[0123] n is the number of free vibrations.
[0124] As was pointed out with respect to Equation 1, the force
necessary to displace any spring-mass-damper system is a function
of the stiffness coefficient and the displacement. In addition, in
many useful cases, the force necessary to displace or deflect a
system of a particular type is known or may be calculated based on
the structure of that system.
[0125] It is useful to consider the exemplary case of a
spring-mass-damper system having a beam undergoing a transverse
longitudinal deflection. Two exemplary systems of this type may be
seen in FIGS. 2-5.
[0126] It will be appreciated by those knowledgeable in the art
that for a beam system as described above, the force necessary to
cause a deflection at a particular point along the length of the
beam depends in part on the location of the point or points at
which the beam is secured, and likewise on the location of the
point for which the deflection of the beam is to be established.
FIG. 2 illustrates an exemplary system 20 having a longitudinal
beam 22, fixed at a first end 24. The beam 22 has a center of mass
26. FIG. 3 illustrates the beam 22 with a second end 28 displaced.
The force that must be applied to the second end 28 to establish a
deflection of the beam 22 at the second end 28 may be determined
according to the relation: 25 F = 3 EI L 3 y ( Equation 35 )
[0127] wherein
[0128] F is the applied force;
[0129] E is the elastic modulus or Young's modulus of the beam
[0130] I is moment of inertia of the beam;
[0131] L is the distance between the fixed point and the point at
which deflection is determined; and
[0132] y is the distance of the beam's deflection at the second
end.
[0133] It will be appreciated by those knowledgeable in the art
that for a beam system as illustrated in FIGS. 2 and 3, L is equal
to the length of the beam. That is, the first end 24 is fixed, and
the second end 28 is deflected, so that the L is the distance
between them, the fall length of the beam. Such a configuration is
convenient for certain applications. However, it will be
appreciated that this configuration is exemplary only, and that the
displacement may be measured at essentially any point along the
length of the beam. Consequently, L is not necessarily equal to the
length of the beam for all suitable systems.
[0134] FIG. 4 illustrates a system 30 including a longitudinal beam
32, fixed at a first end 34 and at a second end 38. The beam has a
center of mass 36. FIG. 5 illustrates the beam 32 with the center
of mass 36 of the beam 32 displaced. The force that must be applied
to the center of mass 36 to establish a deflection at the center of
mass 36 may be determined according to the relation: 26 F = 192 EI
L 3 y ( Equation 36 )
[0135] wherein
[0136] y is the distance of the beam's deflection at the center of
mass.
[0137] As noted previously, L is not necessarily equal to the
length of the beam for all suitable systems. For a system wherein
there are two or more fixed points, L is determined from the
distance to the nearest fixed point. It will be appreciated by
those knowledgeable in the art that for a beam system as
illustrated in FIGS. 4 and 5, L is equal to half the length of the
beam.
[0138] The values 3 and 192 in Equations 35 and 36 are geometric
coefficients that correspond to the arrangement of the beam within
the system. It will be appreciated by those knowledgeable in the
art that the choice of a point on the beam at which to establish
the displacement of the beam may be essentially arbitrary. The end
points and the center of mass are often used as a matter of
convenience, but other locations on the beam may be equally
suitable. For other locations or for other systems based on a
longitudinal beam, coefficients other than those in Equations 35
and 36 are applicable. In general, for a longitudinal beam, the
equation of force will be of the form: 27 F = k " EI L 3 y (
Equation 37 )
[0139] wherein
[0140] k" is a geometric coefficient corresponding to the
arrangement of the beam. It is noted that although k" is dependent
on the arrangement of the beam as a whole within the test system,
k" is not dependent on the internal geometry of the beam.
[0141] In physical terms, the value of k" depends on the amount of
deflection that is enabled for a given system at a given force. It
will be appreciated by those knowledgeable in the art that a beam
fixed at both ends requires substantially more force to deflect
than a beam that is fixed at one end. The differing values of k"
reflect this physical difference.
[0142] It will be appreciated by those knowledgeable in the art
that the choice of a point on the beam at which to establish the
displacement of the beam may be essentially arbitrary. The end
points and the center of mass are often used as a matter of
convenience, but other locations on the beam may be equally
suitable.
[0143] It is also noted that k" is not a component of or otherwise
directly related to the stiffness coefficient K.
[0144] Methods for calculating the applicable coefficient k" are
well known, and are not described further herein. Similarly,
methods for calculating the necessary force in general for a given
displacement in systems other than beams is also well-known, and
are not described further herein.
[0145] The moment of inertia I of any structure is dependent on the
distribution of mass and on the geometry of the point about which
the object is to be moved. It will be appreciated by those
knowledgeable in the art that the internal structure is relevant to
the moment of inertia only in so far as it affects the mass
distribution. Thus, so long as the density is constant with respect
to the motion, the internal structure of the system has no direct
effect on the moment of inertia.
[0146] For example, for a beam as in FIGS. 2-5, so long as the
distribution of mass of the beam is constant along the beam's
length, it is not necessary to know the cross-sectional structure
of the beam. A solid cylindrical beam and a hollow square beam of
equal length and equal mass per unit length have exactly the same
moment of inertia about their end points. Thus, for beams of
constant linear density, the internal structure need not even be
determined in order to calculate the moment of inertia.
[0147] Combination of Equations 1 and 37 yields the following
relation: 28 Ky = k " EI L 3 y ( Equation 38 )
[0148] Dividing both sides of Equation 38 by y and solving for E
results in: 29 E = KL 3 k " I ( Equation 39 )
[0149] Substituting the expression for K from Equation 34 into
Equation 39 gives: 30 E = ( 4 2 M eff T d 2 ( 1 - l ( y 0 y 0 + n )
2 n ) 2 ) L 3 k " I ( Equation 40 )
[0150] Equation 40 may be simplified into the form: 31 E = 4 2 M
eff L 3 k " T d 2 I ( 1 - l ( y 0 y 0 + n ) 2 n ) 2 ( Equation 41
)
[0151] wherein
[0152] E is the Young's modulus of the beam;
[0153] L is the distance between the fixed point and the point at
which deflection is determined;
[0154] k" is a geometric coefficient corresponding to the
arrangement of the beam;
[0155] M.sub.eff is the effective mass;
[0156] T.sub.d is the damped natural period of the system;
[0157] I is moment of inertia of the beam;
[0158] y.sub.o is the peak displacement of the system after zero
free vibrations;
[0159] y.sub.0+n is the peak displacement of the system after n
free vibrations; and
[0160] n is the number of free vibrations.
[0161] As previously noted, for a real system as opposed to a
zero-dimensional ideal mass-damper-system, the effective mass
M.sub.eff will not necessarily be equal to the simple total mass of
the system. For a beam system fixed at one end and deflected at the
other end as illustrated in FIGS. 2 and 3, the effective mass
M.sub.eff may be described according to the relation:
M.sub.eff=0.2357M.sub.test (Equation 42)
[0162] wherein
[0163] M.sub.test is the measured mass of the beam.
[0164] Similarly, for a beam system fixed at both ends and
deflected at the center of mass as illustrated in FIGS. 4 and 5,
the effective mass M.sub.eff may be described according to the
relation:
M.sub.eff=0.3610M.sub.test (Equation 43)
[0165] The values 0.2357 and 0.3610 in Equations 42 and 43 are
geometric coefficients that correspond to the relative amount of
motion of the beam along its length, which depends in turn on the
relative location of the fixing point or points along the length of
the beam. In general, for a longitudinal beam, the effective mass
will be of the form:
M.sub.eff=k'M.sub.test (Equation 44)
[0166] wherein
[0167] k' is a geometric coefficient corresponding to the location
along the length of the beam of the point at which force is
applied.
[0168] In physical terms, the value of k' depends on the aggregate
deflection of the beam when force is applied at a given point. It
will be appreciated by those knowledgeable in the art that when a
beam that is fixed at a point or points is deflected, the fixed
points of the beam do not deflect at all. Thus, a deflection of
amplitude y at any particular point on the beam does not imply that
the entire beam has moved a distance y. This variation in the
deflection of various parts of the beam may be accounted for by use
of the coefficient k'. In effect, k' changes the effective mass of
the beam. Although physically the mass of the beam is constant
while the relative displacement during free vibration varies with
position along the length of the beam, it is mathematically
convenient and functionally equivalent to treat the system as
though the displacement of the beam is uniform along its length
while effectively altering the mass of the beam.
[0169] The relative motion of the system as a whole depends in part
on the number and location of the fixed points. This physical
difference is reflected accounted for with differing values of
k'.
[0170] It is noted that k' is not a component of or otherwise
directly related to the stiffness coefficient K.
[0171] Methods for calculating the value of the coefficient k' are
well known, and are not described further herein.
[0172] It will be appreciated by those knowledgeable in the art
that 0.2357 and 0.3610, the exemplary values of k' in Equations 42
and 43, are approximations. However, the value of k' may be
calculated to any arbitrary precision.
[0173] It will also be appreciated by those knowledgeable in the
art that as a practical matter, when observing an actual
spring-mass-damper system, it may be convenient to attach a sensor
to the system in order to facilitate measurement of the
displacement during free vibration. For example, in the case of the
beam systems shown in FIGS. 2-5, it may be convenient to attach a
sensor to the beam. It will be appreciated that the additional mass
of a sensor will change the effective mass of the system. This may
be accounted for by adding a term to Equation 44:
M.sub.eff=kM.sub.sensor+k'M.sub.test (Equation 45)
[0174] wherein
[0175] k is a geometric coefficient corresponding to the location
of the sensor's center of mass along the length of the beam;
and
[0176] M.sub.sensor is the mass of the sensor.
[0177] It will be appreciated that a sensor placed at a fixed point
of a system would not contribute any effective mass to the system,
since it would not move. In such a case, the value of k would be 0.
Contrariwise, a sensor placed at the point at which displacement is
a maximum would contribute its entire mass to the system as
effective mass. In such a case, the value of k would be 1.
[0178] Although the actual mass of a sensor does not depend on its
location, the system may be treated and analyzed as though this
were the case by the use of the coefficient k to adjust the
contribution of the sensor to the system's total effective
mass.
[0179] It is noted that k is not a component of or otherwise
directly related to the stiffness coefficient K.
[0180] It will be appreciated by those knowledgeable in the art
that for certain systems, including but not limited to the beam
systems illustrated in FIGS. 2-5, it is convenient to locate the
sensor such that k has a value of 1. For a beam system fixed at one
end as illustrated in FIGS. 2-3, the sensor would be located at the
second end 28. For a beam system fixed at both ends as illustrated
in FIGS. 4-5, the sensor would be located at the beam's center of
mass 36. However, although such arrangements may be convenient for
certain applications, it will be appreciated that they are
exemplary only, and that the sensor could be placed in other
locations. The value of k may be calculated for any arbitrary
location of the sensor's center of mass.
[0181] Methods for calculating the value of the coefficient k are
well known, and are not described further herein.
[0182] Substitution of Equation 45 into Equations 29, 34, and 41
yields the following relations: 32 C = ( 4 ( kM sensor + k ' M test
) T d ) ( ln ( y 0 y ( 0 + n ) ) 2 n 1 - ( ln ( y 0 y ( 0 + n ) ) 2
n ) 2 ) ( Equation 46 )
[0183] wherein
[0184] C is the damping coefficient;
[0185] k is a geometric coefficient corresponding to sensor
location;
[0186] M.sub.sensor is the mass of the sensor;
[0187] k' is a geometric coefficient corresponding to the location
along the length of the beam of the point at which force is
applied;
[0188] M.sub.test is the mass of the beam;
[0189] T.sub.d is the damped natural period of the system;
[0190] y.sub.o is the peak displacement of the system after zero
free vibrations;
[0191] n is the number of free vibrations; and
[0192] y.sub.0+n is the peak displacement of the system after n
free vibrations. 33 K = 4 ( kM sensor + k ' M test ) T d 2 ( 1 - (
ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) ( Equation 47 )
[0193] K is the stiffness coefficient;
[0194] k is a geometric coefficient corresponding to sensor
location;
[0195] M.sub.sensor is the mass of the sensor;
[0196] k' is a geometric coefficient corresponding to the location
along the length of the beam of the point at which force is
applied;
[0197] M.sub.test is the mass of the beam;
[0198] T.sub.d is the damped natural period of the system;
[0199] y.sub.0 is the peak displacement of the system after zero
free vibrations;
[0200] y.sub.0+n is the peak displacement of the system after n
free vibrations; and
[0201] n is the number of free vibrations. 34 E = 4 2 L 3 ( kM
sensor + k ' M test ) k " T d 2 I ( 1 - ( ln ( y 0 y ( 0 + n ) ) 2
n ) 2 ) ( Equation 48 )
[0202] E is the Young's modulus of the beam;
[0203] L is the distance between the fixed point and the point at
which deflection is determined;
[0204] k is a geometric coefficient corresponding to sensor
location;
[0205] M.sub.sensor is the mass of the sensor;
[0206] k' is a geometric coefficient corresponding to the location
along the length of the beam of the point at which force is
applied;
[0207] M.sub.test is the mass of the beam;
[0208] k" is a geometric coefficient corresponding to the
arrangement of the beam;
[0209] T.sub.d is the damped natural period of the system;
[0210] I is moment of inertia of the beam;
[0211] y.sub.0 is the peak displacement of the system after zero
free vibrations;
[0212] y.sub.0+n is the peak displacement of the system after n
free vibrations; and
[0213] n is the number of free vibrations.
[0214] It will be appreciated by those knowledgeable in the art
that although the beam systems illustrated in FIGS. 2-5 are
analyzed mathematically herein, they are exemplary only. A wide
variety of other systems exhibiting free vibrations may be equally
suitable.
[0215] For example, for certain applications it may be advantageous
to initiate transverse longitudinal free vibrations in a beam fixed
at its center of mass. Such a system is illustrated in FIGS. 10 and
11. FIG. 10 illustrates an exemplary system 50 having a
longitudinal beam 52, fixed at its center of mass 56. The first and
second ends 54 and 58 are free to move. FIG. 11 illustrates the
beam 52 with the first and second ends 54 and 58 displaced.
[0216] Such a system may be analyzed mathematically in a fashion
similar to that disclosed herein with respect to the systems
illustrated in FIGS. 2-5. However, for the sake of brevity, the
system of FIGS. 10 and 11 is not analyzed mathematically herein.
Similarly, systems with test pieces other than longitudinal beams
are not analyzed mathematically herein. However, it will be
appreciated that in addition to those systems specifically analyzed
herein, other systems utilizing longitudinal beams as well as
systems utilizing test pieces with configurations other than
longitudinal beams may be equally suitable.
[0217] It will be appreciated by those knowledgeable in the art
that for a real system, as opposed to the one-dimensional ideal
system illustrated in FIG. 1, free vibration is possible in
multiple directions and orientations, or modes. With regard to
multiple modes of free vibration, it is again illustrative to
consider the exemplary beam system of FIGS. 2 and 3.
[0218] As previously described, FIGS. 2 and 3 illustrate an
exemplary system 20 including a beam 22 fixed at a first end. FIG.
2 illustrates the beam 22 in an undeflected or neutral position.
FIG. 3 illustrates the beam 22 with a transverse longitudinal
deflection, as would be observed during transverse longitudinal
free vibration.
[0219] FIGS. 6 and 7 illustrate the same system 20 from the
perspective of the second end 28. FIG. 6 illustrates the beam 22 in
a neutral position. FIG. 7 illustrates the beam 22 with a torsional
deflection, as would be observed during torsional free vibration.
As shown, the second end 28 of the beam 22 is twisting about an
axis 40 running longitudinally through the beam 22.
[0220] It will be appreciated by those knowledgeable in the art
that the vibratory modes illustrated in FIGS. 2, 3, 6 and 7 are
exemplary only, and that additional vibratory modes beyond
transverse longitudinal and torsional, including but not limited to
compressional, are possible. In addition, although the vibratory
modes illustrated in FIGS. 2, 3, 6, and 7 are shown individually
for purposes of clarity, it will be appreciated that it is possible
for a single system to vibrate simultaneously in multiple
modes.
[0221] It will also be appreciated that, although the longitudinal
beam illustrated in FIGS. 2, 3, 6, and 7 has a simple rectangular
cross-section, this configuration is exemplary only. The principles
of the claimed invention are equally applicable to beams of
essentially arbitrary cross-section, and to objects other than
longitudinal beams, as is elsewhere noted herein.
[0222] It will further be appreciated that a given system will not
necessarily have the same free vibrational properties in all
vibratory modes.
[0223] For example, it will be appreciated by those knowledgeable
in the art that the effective mass of a system will not necessarily
be the same for different vibratory modes. This is because the
relative amount of motion of different parts of the system varies
with the mode of vibration. For example, referring again to FIGS.
2, 3, 6, and 7 it will be appreciated that the effective mass of
the beam 42 for the transverse longitudinal free vibrations
illustrated in FIGS. 2 and 3 will not necessarily be the same as
the effective mass of the beam 42 for the torsional free vibrations
illustrated in FIGS. 6 and 7.
[0224] Similarly, the damping coefficient, stiffness coefficient,
and Young's modulus of a given system will not necessarily be equal
for different modes of vibration.
[0225] For a real system, therefore, Equations 29, 34, and 41
provide values of C, K, and E that are particular to a single mode
of vibration. For example, for a beam system as those illustrated
in FIGS. 2-5, the mode of vibration under consideration is
transverse longitudinal vibration. Thus, Equations 29, 34, and 41
may be more specifically written as 35 C L = ( 4 2 ( kM sensor + k
' M test ) T d ) ( ln ( y 0 y ( 0 + n ) ) 2 n 1 - ( ln ( y 0 y ( 0
+ n ) ) 2 n ) 2 ) wherein ( Equation 49 )
[0226] C.sub.L is the transverse longitudinal damping coefficient.
36 K L = 4 2 ( kM sensor + k ' M test ) T d 2 ( 1 - ( ln ( y 0 y (
0 + n ) ) 2 n ) 2 ) ( Equation 50 )
[0227] K.sub.L is the transverse longitudinal stiffness
coefficient. 37 E L = 4 2 L 3 ( kM sensor + k ' M test ) k " T d 2
I ( 1 - ( ln ( y 0 y ( 0 + n ) ) 2 n ) 2 ) ( Equation 51 )
[0228] E.sub.L is the transverse longitudinal Young's modulus of
the beam;
[0229] It will be appreciated by those knowledgeable in the art
that the values of k, k', k", T.sub.d, and I will also be
particular to each mode of vibration of the system. For example,
for the transverse longitudinal vibration of the systems
illustrated in FIGS. 2-5 these factors could be identified as
k.sub.L, k'.sub.L, k".sub.L, T.sub.dL, and I.sub.L. However, to
avoid unnecessarily complicating the notation, when solving for E,
C, or K in a particular mode of vibration, the factors k, k', k",
T.sub.d, and I are assumed herein to be the appropriate factors for
that mode.
[0230] It is noted that Equation 49 permits calculation of the
damping coefficient from a mass, a measured period, and two
measured peak displacements. In particular, it is noted that
Equation 49 does not depend on the shape or geometry of the test
piece.
[0231] Likewise, it is noted that Equation 50 permits calculation
of the stiffness coefficient from a measured mass, a measured
period, and two measured peak displacements. In particular, it is
noted that Equation 50 does not depend on the shape or geometry of
the test piece.
[0232] It is noted that Equation 51 permits calculation of the
stiffness coefficient from a measured mass, a measured period, a
measured length, and two measured peak displacements. However,
Equation 51 does not depend on the shape or geometry of the beam,
so long as the density of the beam is constant along its
length.
[0233] In view of the preceding, it is the purpose of the claimed
invention to overcome the deficiencies of existing apparatuses and
methods for determining non-musical physical properties of
objects.
[0234] According to the principles of the claimed invention, free
vibrations are initiated in a test piece and are observed as they
decay in amplitude, without continued driving. Data from the free
vibrations as they decay is then analyzed to determine one or more
physical properties of the test piece. Testing according to the
principles of the claimed invention therefore provides data
regarding physical properties across a large dynamic range with
only a single test. Testing is quick, simple, and non-destructive
to the test piece. Furthermore, a sample of arbitrary geometry may
be tested without a need for complex geometric analysis. Because of
this, it is possible to test a piece of an actual product rather
than a standard sample, and the inaccuracies inherent in testing
samples are avoided.
[0235] It is also the purpose of the claimed invention to provide
an apparatus for measuring at least one non-musical physical
property of a test piece.
[0236] An embodiment of a test apparatus in accordance with the
principles of the claimed invention includes a frame with a natural
vibrational frequency that is different from the anticipated free
vibrational frequency of the test piece. It also includes fixing
means for fixing the test piece to the apparatus at at least one
fixing point, so that the test piece may undergo free vibrations.
The apparatus also includes initiating means to initiate vibrations
within the test piece. The apparatus further includes displacement
measuring means to measure the displacement of the test piece and
time measuring means in communication with the displacement
measuring means so as to measure the time-varying displacement of
the test piece as it undergoes free vibration.
[0237] Another embodiment of a test apparatus in accordance with
the principles of the claimed invention includes a mechanism for
collecting measurements of displacement and time. The collecting
mechanism may include a computer.
[0238] Another embodiment of a test apparatus in accordance with
the principles of the claimed invention includes a mechanism for
processing measurements of displacement and time so as to determine
the physical properties of the test piece. The processing mechanism
may include a computer.
[0239] Another embodiment of a test apparatus in accordance with
the principles of the claimed invention includes a mechanism for
recording measurements of displacement and time for later
reference. The recording mechanism may include a computer.
[0240] Another embodiment of a test apparatus in accordance with
the principles of the claimed invention includes a mechanism for
displaying measurements of displacement and time. The measurements
may be displayed in graphic form. The display mechanism may include
a computer.
[0241] It is also the purpose of the claimed invention to provide a
test method for measuring at least one non-musical physical
property of a test piece.
[0242] An embodiment of a test method in accordance with the
principles of the claimed invention includes the steps of fixing a
test piece to a frame, initiating free vibrations in the test
piece, measuring the time-varying displacement of the test piece as
it undergoes free vibration, and calculating at least one value of
the physical property of the test piece.
[0243] Another embodiment of a test method in accordance with the
principles of the claimed invention further includes the step of
displaying the values of the physical property. The values may be
displayed on a computer.
[0244] Another embodiment of a test method in accordance with the
principles of the claimed invention further includes the step of
supplying feedback for adjusting a manufacturing process based on
the values of the physical property.
[0245] It is also the purpose of the claimed invention to provide a
method for controlling a manufacturing process based on measured
values of a physical property of a test piece produced by the
manufacturing process.
[0246] An embodiment of a process control method in accordance with
the principles of the claimed invention includes the steps of
fixing a test piece produced by the process to a frame, initiating
free vibrations in the test piece, measuring the time-varying
displacement of the test piece as it undergoes free vibration,
calculating at least one value of the physical property of the test
piece, and adjusting the process based on the value of the physical
property.
BRIEF DESCRIPTION OF THE DRAWINGS
[0247] Like reference numbers generally indicate corresponding
elements in the figures.
[0248] FIG. 1 is a schematic representation of a single degree of
freedom mass-spring-damper system.
[0249] FIG. 2 is a side perspective view of a system with a
longitudinal beam fixed at one end.
[0250] FIG. 3 is a side perspective view of the system of FIG. 2,
shown with the beam undergoing a transverse longitudinal
deflection.
[0251] FIG. 4 is a side perspective view of a system with a
longitudinal beam fixed at both ends.
[0252] FIG. 5 is a side perspective view of the system of FIG. 4,
shown with the beam undergoing a transverse longitudinal
deflection.
[0253] FIG. 6 is an end perspective view of the system of FIG.
2.
[0254] FIG. 7 is an end perspective view of the system of FIG. 2,
shown with the beam undergoing a torsional deflection.
[0255] FIG. 8 is a perspective view of an embodiment of a test
apparatus in accordance with the principles of the claimed
invention, with a test piece therein.
[0256] FIG. 9 is a perspective view of a load column of the
embodiment shown in FIG. 8.
[0257] FIG. 10 is a side perspective view of a system with a
longitudinal beam fixed at its center of mass.
[0258] FIG. 11 is a side perspective view of the system of FIG. 10,
shown with the beam undergoing a transverse longitudinal
deflection.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
[0259] Referring to FIGS. 8 and 9, an apparatus in accordance with
the principles of the claimed invention includes a vibration decay
evaluator (VDE) tester 100. A test piece 102 is fitted into the VDE
tester 100.
[0260] The VDE tester 100 includes a frame 104. The frame 104 is
substantially rigid, so as to avoid applying a significant damping
force to the test piece 102. The frame 104 also has a natural
vibrational frequency that is different from the anticipated
natural vibrational frequency of any test piece 102 that is to be
tested by the VDE tester 100. This is necessary because, if the
natural vibrational frequency of the frame 104 is similar to that
of a test piece 102, the vibrating test piece would transfer energy
to the frame 104, thereby causing the amplitude of the vibrations
in the test piece 102 to decrease more rapidly than they otherwise
would.
[0261] The frame may be made of any sufficiently rigid material,
including but not limited to metal, plastic, or wood.
[0262] The VDE tester 100 also includes fixing means 106 adapted to
fix the test piece 102 to the frame 104 at at least one fixing
point 108. The fixing means 106 hold the test piece 102 such that
the fixing point 108 is prevented from moving when the test piece
102 undergoes free vibration.
[0263] As illustrated, the VDE tester 100 is adapted to utilize
transverse longitudinal free vibrations in a test piece 102 in the
shape of a substantially longitudinal beam. However, it will be
appreciated by those knowledgeable in the art that this
configuration is exemplary only, and that other configurations of
VDE tester 100 and test piece 102 may be equally suitable, and may
in particular be suitable for utilizing free vibrations other than
transverse longitudinal free vibration, including but not limited
to compressional free vibrations and torsional free vibrations.
[0264] A wide variety of fixing means 106 may be used with the
claimed invention. In a preferred embodiment, the fixing means 106
include a mechanical clamp. In another preferred embodiment, the
fixing means 106 include a water bladder clamp. However, it will be
appreciated by those knowledgeable in the art that these are
exemplary only, and that other fixing means 106 may be equally
suitable.
[0265] As illustrated in FIG. 8, the fixing means 106 fix the test
piece 102 at a first end 110. However, it will be appreciated by
those knowledgeable in the art that this arrangement is exemplary
only, and that other configurations may be equally suitable.
Suitable configurations include but are not limited to fixing the
first and second ends 110 and 112 of the test piece, and fixing the
center of mass 114 of the test piece.
[0266] The VDE tester 100 also includes vibration initiating means
116 adapted to initiate free vibrations in the test piece 102.
[0267] An exemplary embodiment of vibration initiating means 116 is
illustrated in FIG. 9. As shown therein, the vibration initiating
means 116 include a pin 118 connected to the second end 112 of the
test piece 102 by means of a clamp 120. The pin 118 and the clamp
120 are rigidly affixed to the test piece 102 so that moving the
pin 118 likewise moves the second end 112 of the test piece
102.
[0268] The exemplary vibration initiation means 116 also include a
pin release mechanism 122. As illustrated in FIG. 9, an exemplary
pin release mechanism 122 includes a block 124 that is rigidly
mounted to the frame 104, with a rotatable member 126 rotatably
disposed within the block 124. The block 124 defines a first
aperture 128 therein sized to freely accept the pin 118. The
rotatable member 126 similarly defines a second aperture 130
therethrough sized to freely accept the pin 118. The rotatable
member 126 is rotatable between a first position wherein the second
aperture 130 is aligned with the first aperture 128 such that the
first and second apertures 128 and 130 cooperate to accept the pin
118, and a second position wherein the second aperture 130 is not
aligned with the first aperture 128 and does not accept the pin 118
therein.
[0269] The pin release mechanism 122 is positioned such that when
the rotatable member 126 is in the second position, the pin 118
abuts against the rotatable member 126, whereby the second end 112
of the test piece 102 is deflected from an equilibrium position.
When the rotatable member 126 moves from the second position to the
first position, the deflection is released, whereupon transverse
longitudinal free vibrations are initiated in the test piece
102.
[0270] As shown, the block 124 is mounted indirectly to the frame
104 via other elements. However, it will be appreciated by those
knowledgeable in the art that this configuration is exemplary only,
and that the block 124 could be mounted directly to the frame.
[0271] Similarly, it will be appreciated by those knowledgeable in
the art that the aforementioned vibration initiating means 116 are
exemplary only, and that other initiating means may be equally
suitable. In particular, vibration initiating means that are not
mounted to the frame may be equally suitable.
[0272] The VDE tester 100 also includes displacement measuring
means 132 adapted to measure the displacement of the test piece
102.
[0273] An exemplary embodiment of the displacement measuring means
132 is illustrated in FIG. 9. As shown therein, the displacement
measuring means 132 includes a linear voltage displacement
transducer (LVDT) 134. LVDTs measure the position of a core moving
through a body by detecting changes in the electrical properties of
the core with respect to the body, and produce an electrical signal
corresponding to the position of the core within the body. An LVDT
does not require contact between the core and the body, and hence
do not apply extraneous frictional forces to the apparatus. LVDTs
are well known, and are not described further herein. The LVDT 134
includes a core 136 and a body 138. As illustrated in the exemplary
embodiment of FIG. 9, the core 136 is rigidly connected to the pin
118 and moveable therewith, and the body 138 is rigidly mounted to
the frame 104.
[0274] The body 138 defines a core aperture 140 therethrough sized
so as to freely accept the core 136 therein. As the test piece 102
undergoes transverse longitudinal free vibration, the core 136
moves vertically within the core aperture 140 of the body 138.
Because the core 136 is rigidly connected to the pin 118 which is
in turn rigidly connected to the second end 112 of the test piece
102, the motion of the core 136 and consequently the signal output
of the LVDT 134 corresponds to the motion of the second end 112 of
the test piece 102.
[0275] It will be appreciated by those knowledgeable in the art
that, for a system as illustrated in FIG. 8, wherein the test piece
102 is in the form of a longitudinal beam fixed at a first end 110,
any additional mass affixed at the second end 112 will contribute
its entire mass to M.sub.sensor. That is, as noted previously with
respect to Equation 45 and subsequently, k is 1. The nature of the
individual component or components that contribute the mass is not
relevant to the vibration of the test piece 102. Thus, for purposes
of calculation, in the exemplary embodiment shown in FIG. 9 the
clamp 120, the pin 118, and the LVDT core 136 may all be
collectively considered to be the sensor 142, and the combined mass
thereof is M.sub.sensor. It will be appreciated by those
knowledgeable in the art that the embodiment shown in FIG. 9 is
exemplary only, and that the components that contribute to
M.sub.sensor for any alternate displacement measuring means 132
will depend on the structure and nature of the particular
displacement measuring means 132. It will furthermore be
appreciated that for certain displacement measuring means 132,
including but not limited to ultrasonic or laser rangefinding
mechanisms, no additional components will necessarily be affixed to
the test piece 102, and that in such a case M.sub.sensor will be
zero.
[0276] As shown in FIG. 9, the body 138 is mounted indirectly to
the frame 104 via a column spine 144 and a column mount 146.
However, it will be appreciated by those knowledgeable in the art
that this configuration is exemplary only, and that the body 138
could be mounted directly to the frame.
[0277] Similarly, it will be appreciated by those knowledgeable in
the art that the aforementioned displacement measuring means 132
are exemplary only, and that other initiating means may be equally
suitable. In particular, displacement measuring means that are not
mounted to the frame may be equally suitable.
[0278] In a preferred embodiment of the VDE tester 100, the
displacement measuring means 132 are adapted to obtain measurements
of displacement at least 1,000 times per second. In a more
preferred embodiment of the VDE tester 100, the displacement
measuring means 132 are adapted to obtain measurements of
displacement at least 10,000 times per second. In a still more
preferred embodiment of the VDE tester 100, the displacement
measuring means 132 are adapted to obtain measurements of
displacement at least 100,000 times per second.
[0279] An exemplary embodiment of the VDE tester 100 includes
initiation control means 148 adapted for controlling the initial
amplitude of the free vibrations. As illustrated in FIG. 9, the
initiation control means 148 include an adjustment aperture 150
defined in the block 124 of the vibration initiating means 116 and
an adjustment screw 152 rotatably disposed at least partially
within the adjustment aperture 150. The adjustment aperture 150 and
the adjustment screw 152 are threaded so as to cooperate with one
another, such that rotation of the adjustment screw 152 causes a
vertical displacement of the block 124. A vertical displacement of
the block 124 in turn causes a change in the position occupied by
the pin 118 when the rotatable member 126 is in the second
position, that is, when the second aperture 130 is not aligned with
the first aperture 128 and does not accept the pin 118 therein.
Because the pin 118 is rigidly affixed to the test piece 102, a
change in the position of the pin 118 results in a change in the
initial deflection of the test piece 102. Thus, by rotating the
adjustment screw 152, the deflection of the test piece 102 at the
start of free vibration may be controlled.
[0280] As shown in FIG. 9, the adjustment screw 152 is mounted to
the frame 104 via other elements. However, it will be appreciated
by those knowledgeable in the art that this configuration is
exemplary only, and that the adjustment screw 152 could be mounted
directly to the frame.
[0281] An exemplary embodiment of the initiation control means 148
includes deflection measuring means 154 adapted for measuring an
initial deflection of the test piece 102. As illustrated in FIG. 9,
the deflection measuring means 154 include a micrometer
mechanically engaged with the adjustment screw 152, so as to enable
convenient and accurate measurement of the vertical displacement of
the block 124, and thus also the initial deflection of the test
piece 102.
[0282] It will be appreciated by those knowledgeable in the art
that the aforementioned initiation control means 148 are exemplary
only, and that other initiation control means may be equally
suitable. In particular, it will be appreciated that deflection
measuring means 154 other than a micrometer may be equally suitable
for measuring initial deflection, or that the deflection measuring
means 154 may be omitted entirely. Furthermore, it will be
appreciated that for certain embodiments of the VDE tester,
including but not limited to embodiments adapted only for a fixed
initial deflection, initiation control means 148 will not be
included.
[0283] An exemplary embodiment of the VDE tester 100 includes load
measuring means 156 adapted to measure the load applied to the test
piece 102 during initial deflection. As illustrated in FIG. 9, the
load measuring means 156 include a load cell 158. Load cells are
well known, and are not described further herein. The load cell 158
is rigidly affixed to a load cell mount 160, which is in turn
rigidly affixed to the column spine 144. As previously noted, in
the embodiment of FIG. 9 the column spine 144 is rigidly affixed to
the column mount 146, which is rigidly affixed to the frame 104.
Thus, the load cell 158 is rigidly affixed to the frame 104. The
load cell 158 is engaged with the initiating means 116, such that
the force between the initiating means 116 and the frame 104, and
hence between the test piece 102 and the frame 104 during initial
displacement, may be measured by the load cell 158.
[0284] As shown, the load cell 158 is mounted to the frame via
other elements. However, it will be appreciated by those
knowledgeable in the art that this configuration is exemplary only,
and that the load cell 158 could be mounted directly to the frame.
Likewise, as shown the load cell 158 is engaged with the initiating
means 116 via the initiation control means 148. It will be
appreciated that the load cell 158 could be engaged with the
initiating means directly, or via other components.
[0285] Similarly, it will be appreciated by those knowledgeable in
the art that the aforementioned load measuring means 156 are
exemplary only, and that other load measuring means may be equally
suitable. Furthermore, it will be appreciated that certain
embodiments of the VDE tester 100, including but not limited to
embodiments adapted for applying a fixed load at initial
deflection, will not include load measuring means 156.
[0286] Additionally, it will be appreciated that certain
embodiments of the VDE tester 100, including but not limited to
embodiments adapted for initiating free vibrations of a fixed
initial amplitude, will not include initiation control means
148.
[0287] The VDE tester 100 also includes time measuring means 162
adapted to measure time during free vibration. The time measuring
means 162 are in communication with the displacement measuring
means 132, such that the displacement of the test piece 102 may be
determined for a particular time.
[0288] As illustrated, the time measuring means 162 are mounted
directly to the frame 104. However, it will be appreciated by those
knowledgeable in the art that this configuration is exemplary only,
and that the time measuring means 162 may be mounted differently,
or may not be mounted at all. The time measuring means 162 may be
physically remote from the frame 104, so long as they are in
communication with the displacement measuring means 132.
[0289] In a preferred embodiment of the VDE tester 100, the time
measuring means 162 are adapted to obtain measurements of
displacement at least 1,000 times per second. In a more preferred
embodiment of the VDE tester 100, the time measuring means 162 are
adapted to obtain measurements of displacement at least 10,000
times per second. In a still more preferred embodiment of the VDE
tester 100, the time measuring means 162 are adapted to obtain
measurements of displacement at least 100,000 times per second.
[0290] A wide variety of time-measuring means 162 may be used with
the claimed invention. In a preferred embodiment, the time
measuring means 162 include a digital timing circuit. This is
convenient, in that digital timing circuits are compact,
inexpensive, and durable. However, it will be appreciated by those
knowledgeable in the art that this configuration is exemplary only,
and that other time measuring means 162 may be equally suitable.
Time measuring means are well known, and are not described further
herein.
[0291] An exemplary embodiment of the VDE tester 100 includes
collecting means 164 adapted to collect measurements of
displacement and time from the displacement measuring means 132 and
the time measuring means 162. As illustrated in FIG. 8, the
collecting means 164 are part of a computer 172. In a preferred
embodiment of the claimed invention, the collecting means 164
include a plug-in data acquisition card installed in a computer
172. This is advantageous, in that such cards are reliable, simple
to use, and widely available. However, it will be appreciated by
those knowledgeable in the art that this configuration is exemplary
only, and that other collecting means 164 may be equally suitable.
Collecting means are well known, and are not described further
herein.
[0292] It will be appreciated by those knowledgeable in the art
that the measurements of displacement and time may be collected in
a variety of forms, including but not limited to digital and analog
electrical output signals from the displacement measuring means 132
and the time measuring means 162.
[0293] An exemplary embodiment of the VDE tester 100 includes
processing means 166 adapted to process measurements of
displacement and time in order to determine other information
therefrom. For example, the processing means 166 are advantageously
adapted to generate at least one value of at least one physical
property of the test piece 102, including but not limited to
C.sub.L, K.sub.L, and E.sub.L. Advantageously, the processing means
166 are adapted to generate a plurality of values of at least one
physical property of the test piece.
[0294] As illustrated in FIG. 8, the processing means 166 are part
of a computer 172. In a preferred embodiment of the claimed
invention, the processing means 166 include the CPU of a computer
172. In another preferred embodiment of the claimed invention, the
processing means 166 further include a software application adapted
to run on the CPU of a computer 172. However, it will be
appreciated by those knowledgeable in the art that these
configurations are exemplary only, and that other processing means
166 may be equally suitable. Processing means are well known, and
are not described further herein.
[0295] An exemplary embodiment of the VDE tester 100 includes
recording means 164 adapted to record measurements of displacement
and time from the displacement measuring means 132 and the time
measuring means 162. Advantageously, the recording means 168 are
adapted to record processed information from the processing means
166 as well. As illustrated in FIG. 8, the recording means 168 are
part of a computer 172. In a preferred embodiment of the claimed
invention, the recording means 168 include a hard drive of a
computer 172. This is advantageous, in that hard drives are widely
available. However, it will be appreciated by those knowledgeable
in the art that this configuration is exemplary only, and that
other recording means 168 may be equally suitable, including but
not limited to floppy drives, CD drives, and printers. Recording
means are well known, and are not described further herein.
[0296] An exemplary embodiment of the VDE tester 100 includes
display means 170 adapted to display measurements of displacement
and time from the displacement measuring means 132 and the time
measuring means 162. Advantageously, the display means 170 are
adapted to display processed information from the processing means
166 as well, and/or to display recorded information from the
recording means 168. As illustrated in FIG. 8, the display means
170 are part of a computer 172. In a preferred embodiment of the
claimed invention, the display means 170 include a monitor of a
computer 172. This is advantageous, in that computer monitors are
widely available. However, it will be appreciated by those
knowledgeable in the art that this configuration is exemplary only,
and that other display means 170 may be equally suitable, including
but not limited to printers, plotters, LED displays, and indicator
lights. Display means are well known, and are not described further
herein.
[0297] In a preferred embodiment of the VDE tester, the collecting
means 164, processing means 166, recording means 168, and display
means 170 are all parts of a single computer 172. In another
preferred embodiment of the claimed invention, the computer 172 is
a laptop computer. These configurations are advantageous, in that a
laptop computer that includes all of the collecting means 164,
processing means 166, recording means 168, and display means 170 is
convenient and easily portable. However, it will be appreciated by
those knowledgeable in the art that this configuration is exemplary
only. Other computers 172 may be equally suitable for use with the
claimed invention. Furthermore, the collecting means 164,
processing means 166, recording means 168, and display means 170
need not be part of a single computer, and may for example be parts
of several different computers connected via a network, or may be
stand-alone devices not part of any computer.
[0298] As shown, the collecting means 164, processing means 166,
recording means 168, and display means 170 are separate from the
frame 104. However, it will be appreciated that this configuration
is exemplary only, and that some or all of the collecting means
164, processing means 166, recording means 168, and display means
170 may be attached to the frame. Furthermore, it will be
appreciated that some or all of the collecting means 164,
processing means 166, recording means 168, and display means 170
may be remote from the frame 104. For example, it may be
advantageous for certain applications to transmit time and
displacement measurements some distance, i.e. via an internet
connection, prior to collecting, processing, recording, and
displaying them. Additionally, it will be appreciated that a
plurality of collecting means 164, processing means 166, recording
means 168, and display means 170 may be used to collect, process,
record, and display the same measurements of time and displacement,
either simultaneously or at different times.
[0299] It will be appreciated by those knowledgeable in the art
that although collecting means 164, processing means 166, recording
means 168, and display means 170 may be advantageous for certain
exemplary embodiments of the VDE tester 100, some or all of the
collecting means 164, processing means 166, recording means 168,
and display means 170 may be omitted entirely from certain other
exemplary embodiments. For example, it would be possible to
determine values of physical properties from the measurements
produced by the displacement measuring means 132 and the time
measuring means 162 manually, without additional mechanisms.
[0300] In order to determine one or more non-musical properties of
the test piece 102, the test piece 102 is placed in the VDE tester
100, and free vibrations are initiated in the test piece 102 via
the initiating means 116. The displacement of the test piece 102
over time is measured with the displacement measuring means 132 and
the time measuring means 162. Physical properties, including but
not limited to the damping coefficient C, the stiffness coefficient
K, and the Young's Modulus E, may then be calculated from the
displacement and time measurements. For example, values of C.sub.L,
K.sub.L, and E.sub.L for a test piece 102 substantially in the form
of longitudinal beam may be calculated according to Equations 49,
50, and 51. It will be appreciated that, although Equations 49, 50,
and 51 are specific to a longitudinal beam, this is exemplary only.
Values of physical properties for test pieces of other
configurations may be calculated similarly using analogous
equations.
[0301] As has been previously noted, many systems of interest have
a damping ratio .zeta. that is substantially less than 1. It will
be appreciated by those knowledgeable in the art that in such a
case, energy is lost from the system slowly, and therefore a single
decay cycle of free vibrations typically consists of many
individual vibrations. As a result, useful values of one or more
physical properties can be generated from a single decay cycle of
free vibrations. In practice, it is therefore possible to obtain
values for physical properties by initiating free vibrations only
once in a particular test piece. It will also be appreciated,
however, that as initiating free vibrations is non-destructive,
free vibrations could just as well be initiated any number of times
in a particular test piece, rather than only once.
[0302] Furthermore, as previously described, as free vibrations
decay their amplitude decreases. Thus, the peak displacement
decreases over the course of a decay cycle. It will be appreciated
by those knowledgeable in the art that a plurality of values of one
or more physical properties may be calculated from a single
initiation of free vibrations in a test piece. Furthermore, it will
be appreciated that by calculating a values of physical properties
over a range of amplitudes as the vibrations decay, a dynamic range
of values of the properties over a range of displacements may be
determined. For example, K.sub.L may be determined over a range of
strains, rather than at a particular point. However, it will also
be appreciated that it is not necessary to calculate a dynamic
range of values for a physical property, and that for certain
applications it may be advantageous to calculate only a few or even
only a single value.
[0303] For many common materials, free vibrations occur with a
frequency of on the order of hundreds of cycles per second.
Therefore, it is possible to gather substantial data within a
relatively short time, a matter of few seconds. In addition, given
the speed of known processing means, it is possible to determine at
least one value of at least one physical property from measurements
of time and displacement in on the order of one second or less.
Thus, according to the claimed invention, for common materials it
is possible to obtain values of physical properties no more than 5
seconds after initiation of free vibrations in the test piece
102.
[0304] Advantageously, the VDE tester 100 may be used to provide
feedback for a manufacturing process. It will be appreciated by
those knowledgeable in the art that, once information is derived by
use of the VDE tester 100 regarding the test piece 102, this
information may be used to adjust a manufacturing process used to
produce the test piece. In this way, manufacturing processes may be
controlled so as to consistently produce optimum product.
[0305] It will also be appreciated that, because the VDE tester 100
is operates on principles that do not depend on the internal
geometry of the test piece, the VDE tester 100 is insensitive to
the shape and configuration of the test piece 102. Thus, the normal
output of a manufacturing process may be tested on the VDE tester
100.
[0306] In a preferred embodiment of the VDE tester 100, the
processing means 166 are adapted to generate feedback automatically
based on calculated values of physical properties. This is
advantageous, in that it enables persons without extensive training
in the effects of adjusting process parameters to successfully
adjust a manufacturing process. However, it will be appreciated
that this is exemplary only, and that for certain applications it
may be desirable to provide feedback from other sources, or to omit
feedback altogether.
[0307] The above specification, examples and data provide a
complete description of the manufacture and use of the composition
of the invention. Since many embodiments of the invention can be
made without departing from the spirit and scope of the invention,
the invention resides in the claims hereinafter appended.
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