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AEROSPACE 305W STRUCTURES & DYNAMICS
LABORATORY
Laboratory Experiment #1
Bending of Selectively Reinforced Beams
March 24, 2009
Stephanie Penatzer
Lab Section 14
Lab Partners:
Christopher BerryDennis Larkin
George Wilson
Kevin Kennedy
Nicholas Kesler
Course Instructor: Dr. Stephen Conlon
Lab TA:Patrick Williams
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Abstract
The objective of this experiment is to investigate the flexural behavior of structural beam
elements through experimentation and analysis. Additionally, the goal of the experiment is to
assess the utility of composite materials as reinforcement components. This will be achieved
by testing three beams; an aluminum beam, an aluminum with aluminum caps, and an
aluminum with graphite/epoxy caps. These beams will be loaded, hanging a load hook with
weights at the end of the beam, in an I-beam orientation and in an H-beam orientation. The
beams are loaded; the strain is measured using a strain gage and the displacement is measured
using the LVDT. The theoretical data, found by utilizing the Euler-Bernoulli beam theory,
supports the idea that composite reinforcements resist deflection and strain. When comparing
the theoretical data to the experimental there is a clear similarity between them. However, the
experimental data has more displacement and strain than the theoretical data due to the error in
the real world experimentation. Although there is considerable error, the experimental results
provide support to these claims. The error comes from assumptions made in the theoretical
calculations and human errors during experimentation. Even with error, the experimental data
supports the conclusion that composites make excellent strengthening materials in structural
design.
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I. Introduction
The goal of this experiment is to explore the flexibility of structural beams through
experimentation and analysis. This can be accomplished through the comparison of the Euler-
Bernoulli Beam Theory with the experimental results. Along with assessing flexibility, the
experiment examines the effectiveness of composite materials as reinforcements to the structural
beams. This type of information is useful in aerospace structural design; specifically in regards to
reduction of weight with an increase in strength. The goal is to design an aircraft with minimal
materials that accomplishes such design goals as low weight, high stability, high strength, and low
cost. In order to determine these objectives, three beams were loaded; each with a different
reinforcement, and the strain was measured and recorded. This experiment used three specimens: an
aluminum irregular T-beam, an aluminum irregular T-beam reinforced with aluminum caps, and an
aluminum irregular T-beam reinforced with graphite/epoxy caps. A schematic of the beams
compositions are featured in figure 1.
Figure 1. Test Beam compositions
This experiment utilizes the theory of beam bending to analytically determine the flexibility of the
beams as well as the utility of composites. Figure 2 provides a pictorial representation of the beams
directional axis. This theory makes four assumptions: stress is related to strain, xx, through Hookes
Law, equation 1, deformations due to stresses in the y-direction and z-direction are much smaller
than those in the x-direction, cross-sections remain planar and normal to the centroid, and the load
passes through the neutral axis. Equation 1 utilizes xx, stress in the x-direction, and E, Youngs
modulus of elasticity.
3
Aluminum Aluminum with Aluminum Caps Aluminum with Graphite/EpoxyCaps
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Figure 2. Beam Axis
The shear force, Sz(x) produced by a transverse load is given by equation 2, where A is the cross-sectional
area and the bending moment about the y-axis, My, is given by equation 3. These equations are used to
determine the transverse displacement, w(x). Equation 4 provides the transverse displacement of a beam
based on its length (as show in figure 3), properties (such as modulus of elasticity), and the force applied
(P). The limit load for each beam can be determined using the yield stress, found in equation 5, which is
half the load value that will produce a yield stress in the beams. Equations 4 and 5 are dependent upon the
lengths measured in the set-up of the experiment as pictured in figure 3. The derivation of the equation for
strain in the x-direction, xx, as well as the derivation of the transverse displacement, w(x), can be found
in the appendix. Equation 6 describes the cross-sectional moment of inertia for a beam where b is the
base of the beam and t is the thickness. Equation 7 is the adjustment known as the parallel axis theorem,
which gives the moment of inertia of an asymmetrical beam. z_bar is the defined by equation 8 where i is
the specific beam configuration, j is an individual beam section and all z_bar measurements are measured
from the vertical center of the beams. Both z_bar and EI values can be found in table 1.
.
(1)
(2)
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(3)
(4)
(5)
(6)
(7)
=jj
jjj
iAE
zAEz (8)
Figure 3. Important Specimen Dimensions
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Table 1. Measured Dimensions, EI values, and z values
L (in)Lf(in)
Ld(in)
Ls(in) z (in)
EI (lbin^2)
Alum. I-beam
9.812
5 9.375
7.062
5
0.687
5
0.238
92 92347
Alum. OnAlum. I-beam 9.75 9.875
7.812
5 0.875
0.283
94 218448
Graphite/Epoxy onAlum. I-beam
9.437
5
9.187
5 6.125 1
0.277
06 341372
Alum. H-beam
9.687
5 9.375
6.687
5 1
0.238
92 92347
Alum. OnAlum. H-beam 9.875
9.562
5
7.562
5 1
0.283
94 218448
Graphite/Epoxy on
Alum. H-beam 9.375 9.1257.312
5 1
0.277
06 341372
The theoretical data described above can be confirmed through the experimental data. The beam
is clamped at one end and loaded using a load hook at the tip of the beam. The displacement is then
recorded using the LVDT, an apparatus which sits on the top of the beam and measures the drop in
the z-direction. The strain is measured by strain gages placed on the top and side of the beam and is
read with the strain gage box. The beam is loaded from 0 to 5 pounds and then unloaded to record
hysteresis; a process where a lag occurs between the application and removal of a force. In a case
such as this the beam will register different strain and displacement at the beginning and end of
loading.
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II. Experimental Procedure
The basic process encompassed by this experiment is to load three different beams to determine
the flexural behavior and also to determine the value of composites as a strengthening tool in beams.
Before beginning the experiment the strain gage box was powered on and zeroed by depressing the
AMP ZERO button and adjusting the AMP ZERO potentiometer to display plus or minus zero. The
box was then left to sit for five minutes and readjusted in the manor previously described.
Each beam was clamped into the fixture as pictured in figure 4; with the filler blocks flush with
the clamp and the desired length the of the beam outside of the clamp. The filler blocks were used to
clamp the beam without damaging it and were kept flush with the fixture to prevent it from effecting
the beams rigidity and strength.
Figure 4. Beam into fixture configuration
The beams had two strain gages, one on the top and one on the side. When testing the beam as an
I-beam, the top strain gage was used and when testing the beam as an H beam, the side strain gage
was used. These strain gages were wired into the strain gage box with the red wire into the P+, the
black wire into the S-, and the white wire into the D (350) as pictured in figure 5.
7
Clamp Configuration
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The load hanger was then suspended from a point near the end of the beam farthest from the
clamp. The LVDT was mounted on the fixture as shown in figure 6 in order to monitor the transverse
displacement of the beam. The distances from the root to all the apparatus were measured as
pictured in figure 3.
Figure 6. LVDT Mount Location and Load Hook Location
The strain gage was zeroed again to account for the load hanger by adjusting the AMP ZERO
potentiometer display to read zero and the recording software (Lab View) was also zeroed by creating a
dummy file and moving the LVDT arm until it recorded the voltage as close to zero as possible. The Load
hanger was then loaded from 0 to 2 pounds at half pound increments and then from 2 to 5 pounds at one
pound increments as well as unloaded from 5 to 0 in the same fashion.
The beam was then turned onto its side and tested as an H beam and the side strain gage was
connected to the strain gage box. After changing the orientation of the beam and zeroing the equipment,
the beam was loaded as before and the strain values were recorded. This procedure was repeated for each
of the three beams in the I and H orientations and all of the strain data was recorded in Lab View.
8
Red Wire-Red Knob (P+)
Black Wire-White Knob (S-)
White Wire-Yellow Knob
(D350)
LVDT
Load Hook
Figure 5. Strain Gage Box Set-Up
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III. Results and Discussion
Analytically speaking, the displacements of the beams decrease with the addition of a stronger
composite. Therefore, the aluminum beam should displace much more than the aluminum cap beam
and the graphite/epoxy cap beam. However, in review of the experimental data, it is important to note
that the displacement of the purely aluminum beam is greater than that of the beam with
graphite/epoxy caps and even greater than the displacement of the beam with aluminum caps. These
results can be seen clearly in figures 7 through 9. The discrepancy between the theoretical data as
compared to the experimental data is great, especially with the graphite/epoxy capped beam. These
differences can be accounted for through various types of error, values presented in table 2, which
occurred during the experiment. The error was calculated by subtracting the experimental data from
the theoretical, dividing that number by the theoretical data and multiplying by one hundred. Sincethe apparatus was being continually changed throughout the experiment the beams were not placed
precisely and the LVDT was not perfectly zeroed because it is so sensitive. This could have thrown
off the reading for the displacements. The addition, the caps also seemed to bring in more error
throughout the experiment. Since the LVDT was placed directly on the caps the bond between the
specimen and the cap could have been loose and the displacement of the cap was measured but not
the total displacement of the beam. Also, the beam could have displaced in the y and z directions as
well which would throw off the displacement measurements in the x-direction.
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Figure 7. Analytical and Experimental Displacement Vs Applied Load for an Aluminum I-
Beam
Figure 8. Analytical and Experimental Displacement Vs Applied Load for an Aluminum I-
Beam with Aluminum Caps
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Figure 9. Analytical and Experimental Displacement Vs Applied Load for an Aluminum I-Beam with Graphite/Epoxy Caps
Table 2. Percent Error of Displacement for I-Beam Specimens
Aluminum I-Beam
Aluminum I-Beam withAluminum Caps
Aluminum I-BeamwithGraphite/EpoxyCaps
0 0 0
45.67025357 48.2650506 359.9499431
57.27324263 48.5603993 371.9198475
59.10159242 48.79011496 372.8536698
59.72569259 48.80652322 379.0508543
54.93506453 48.5603993 378.4566038
52.94849216 49.44644542 373.7025992
51.70380788 50.78535956 382.7182293
56.28435151 51.98152182 392.4851619
60.91236192 52.43274901 409.0180618
68.16423009 55.05807084 408.4662577
70.45845747 53.7125934 417.8469274
70.24749403 59.48830144 442.4658797
70.88038434 65.39527556 486.7799938
0 0 0
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Theoretically, the strain in the beams should have reduced when the caps were applied to the
beams and the reduction of the strain should have coincided with the strength of the caps applied.
Again, the experimental data does not match our theoretical data. However, the experimental strain
more closely adheres to the theoretical strain than the displacements do. This is because the
displacement of the I-beam was more sensitive to the boundary conditions set at the beginning of the
experiment. The graphs in figures 10 through 12 show that there is considerable error, especially inthe graphite/epoxy capped beam. This error can be observed in table 3 and were calculated in the
same manner as the displacement errors. These differences occur from many outside faults. The
strain gages could have been attached with a poor connection to the strain gage box. Also, the beam
could have been clamped incorrectly which produced strain in the y and z directions. The bonding
element between the beam and the caps could have provided more error if the bond was not strong
enough. In addition, more error could have been produced if the strain gage box was not perfectly
zeroed.
Along with errors, the experimental data differs from the actual data due to the assumptions made
when using the beam bending theory as well as the assumptions made to form boundary conditions.
The displacement is based on the assumption that deformations due to stresses in the y-direction and z-
direction are much smaller than those in the x-direction, cross-sections remain planar and normal to the
centroid, and the load passes through the neutral axis. The strain, however, is only bounded by the
assumption that stress is related to strain, xx, through Hookes Law. These assumptions, described
above, can cause inconsistencies in the data attained through experimentation while making the
analytical data easier to work with.
It can be observed that the error in the graphite/epoxy caps is much higher than that of the other
beams. This is due to the fact that the assumed Youngs modulus is incorrect. If the displacement is
graphed linearly and a line is fit to it as displayed in figure 13 the true Youngs modulus (E) can be
determined. In order to solve for E, the equation for slope from the graph is set equal to the equation
of displacement, equation 4 without the applied load, and the equation is rearranged. The resulting E
is equal to 10.2E6 psi.
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Figure 10. Analytical and Experimental Strain Vs Applied Load for an Aluminum I-Beam
Figure 11. Analytical and Experimental Strain Vs Applied Load for an Aluminum I-Beam withAluminum Caps
13
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Figure 12. Analytical and Experimental Strain Vs Applied Load for an Aluminum I-Beam
with Graphite/Epoxy Caps
Figure 13. Linear fit to a graph of Displacment verses Load for an Aluminum Beam with
Graphite/Epoxy caps
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Table 3. Percent Error of Strain for I-Beam Specimens
Aluminum I-Beam
Aluminum I-Beamwith AluminumCaps
Aluminum I-BeamwithGraphite/EpoxyCaps
0 0 0
5.742963732 11.96059886 66.91121529
6.624154725 1.583927675 81.814012046.917885056 0.67349484 86.78161096
3.539984023 3.495270846 113.1098687
2.481489992 5.752689086 98.70383632
2.848652906 6.034867969 94.48138025
2.716474257 5.52694769 88.96735027
1.306567555 4.341801083 100.4424967
1.012836853 4.059624338 97.71031754
2.218197532 3.495270846 84.79456838
7.181176774 0.455214993 92.74272364
3.068950654 1.583927675 111.6195905
12.79249168 8.356195219 96.7168088
0 0 0
The experiment also included the displacement and strain of each specimen turned on its side and
tested as an H-beam. These tests were not accurate and contained many points of error which can be
viewed in table 4 and 5. Most of the error came from the fact that the H-beam was not symmetrical,
meaning that its left edge was longer than its right edge. When loading such a beam, the specimen
tends to twist which throws off the readings. Also, with the beam twisting, the assumption that there
is only strain in the x-direction is no longer valid and the analytical data is not close to the actual
data. These findings are easily observed in figures 14 through 16. It is clear that the experimental
displacement more closely matches the theoretical displacement than the experimental strain to the
theoretical strain. This is best explained by the afore mentioned error and assumptions. Based on the
assumptions made, the strain is the most sensitive to those conditions. Also, the z_bar is small in the
experiment which will give a small strain that is of the same order as ambient noise and therefore
creates error in the measurements.
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Figure 14. Analytical and Experimental Data for an Aluminum H-beam
Figure 15. Analytical and Experimental Data for an Aluminum H-beam with Aluminum
Caps
Figure 16. Analytical and Experimental Data for an Aluminum H-beam with
Graphite/Epoxy Caps
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Table 4. Percent Error of Displacement for H-Beam specimens
Aluminum H-Beam
Aluminum H-Beam withAluminum Caps
Aluminum H-BeamwithGraphite/EpoxyCaps
0 0 0
19.7838562 19.3340872 144.383741
18.62800398 21.82925447 152.2116577
17.27950973 23.02259535 153.1662817
16.48967738 23.18532365 151.1615713
16.97128247 25.84321923 150.9388257
18.87362258 25.87034061 153.5958625
16.27006546 26.55922375 153.3190215
15.00151765 27.87732298 158.512176
15.93101547 30.29112611 162.7125216
14.23576556 32.18962295 167.1037919
14.69810644 36.33015416 172.0041951
11.2305498 39.62088202 181.232227
11.46172024 50.794892 209.2981722
0 0 0
Table 5. Percent Error of Strain for an H-Beam
Aluminum H-Beam
Aluminum H-Beam withAluminumCaps
Aluminum H-Beamwith Graphite/EpoxyCaps
0 0 0
76.23413815 57.29026107 130.034837
87.20299782 83.98384565 133.0383358
92.07804605 86.94980149 130.0348471
89.94521158 83.98385015 133.0383358
90.249902 85.76342036 132.0371746
93.14446328 94.21638971 133.0383358
90.49365526 93.9494544 131.836939391.31631962 99.11021415 130.0348483
90.85928438 97.62723772 129.0336859
87.20299782 104.4489293 125.529619
86.59361545 113.0501985 108.0092899
87.20299782 121.3548695 127.0313684
79.89042317 156.9463126 112.0139348
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0 0 0
Regardless of the error, it is clear that the composite materials do affect the beams resistance to
strain and displacement. This is because the composites are stronger and more rigid than the beam
itself. When a load is applied the beam distributes it along its upper and lower edges. With the
application of the composites the load is carried by the stronger cap. Even adding a cap of the same
material gives the beam more rigidity to resist the applied load. All of the experimental and
theoretical data can be found in tables 6 through 11 in the appendix.
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IV. Conclusions
Based upon the observed and theoretical data presented in this lab it can be concluded that a
beams flexural characteristic can be changed based on its composition. The beam with the thin
aluminum top and bottom pieces will theoretically deflect more than the beam with the extra
aluminum caps. For example, in the experiment, the aluminum beam deflected approximately 0.0144
inches when fully loaded where as the beam with the aluminum caps only deflected 0.0077 inches
under full load. The beam that has caps made of graphite/epoxy should hypothetically deflect even
less because it adds more rigidity and strength to the aluminum beam. When examining the data
attained by using the Euler-Bernoulli beam theory, the displacement and strain in the aluminum
capped beam should have been 0.0051 inches and 58.49 () respectively while the graphite/epoxy
capped beam should have had a displacement of 0.00196 inches and a strain of 33.23 (). Clearly,this shows that the graphite/epoxy capped beam should have less strain and less displacement.
However, the experimental data does not support the Euler-Bernoulli beam theory entirely. In the
experiment, the aluminum cap beam had a displacement of 0.0077 inches and strain a strain of 55.26
() which is less than the graphite/epoxy capped beam with a displacement of 0.0095 inches and a
strain of 62.78 () when fully loaded. In addition, these numbers do not match with the Euler-
Bernoulli beam theory numbers calculated at the beginning of the lab. The discrepancy can be
attributed to error and assumptions. Such error includes the sensitivity of the LVDT when being
zeroed, the extra rigidity given to the beam by the clamping method, the poor connection between the
strain gage wires and the strain gage box, and the swinging of the load hook after loading. The
assumptions made for the Euler-Bernoulli beam theory, strain and displacement only occur in the x
direction, and that the beam does not deform and stays planar are more sources of error. The
experimental data does however support the second objective of the experiment, which states that
composites are effective as reinforcements for structural beams. As expected, the graphite/epoxy cap
resisted deflection more than the un-capped aluminum beam.
This experiment makes the idea of using composites in everyday structural design a plausible
idea. Using a lighter weight material with lower strength will decrease the weight of the component
and then covering the bulk material in a strong composite skin will to distribute the loads held by the
beam.
In order to minimize the amount of error in the experiment it would be better to do calculations
without assumptions; although this would make the calculations much more difficult. Also, the
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5
3
33.07586
7 -0.004527 35.09475939 -0.003047245
4
43.96911
6 -0.006072 46.79301252 -0.004062994
5
55.25848
4 -0.007658 58.49126566 -0.005078742
4
44.76135
3 -0.006175 46.79301252 -0.004062994
3
33.67004
4 -0.004645 35.09475939 -0.003047245
2
22.57873
5 -0.00315 23.39650626 -0.002031497
1.5
17.62725
8 -0.002342 17.5473797 -0.001523623
1
11.88354
5 -0.00162 11.69825313 -0.001015748
0.5 6.337891 -0.00084 5.849126566 -0.000507874
0 0 -7.00E-05 0 0
Table 8. Strain and Displacement Measured and Calculated for an Aluminum I-Beam with
Graphite/Epoxy Caps
Force(lbs)
Strain()
Displacement (in)
TheoreticalStrain ()
TheoreticalDisplacement (in)
0 0 0.000023 0 0
0.5
5.54565
4 -0.000903 3.3225173 -0.0002
1
12.0816
04 -0.001853 6.6450346 -0.00039
1.5
18.6175
54 -0.002785 9.9675519 -0.00059
2
28.3224
49 -0.003762 13.290069 -0.00079
3
39.6118
16 -0.005636 19.935104 -0.00118
4
51.6934
2 -0.00744 26.580138 -0.00157
5
62.7847
29 -0.009477 33.225173 -0.00196
4
53.2778
93 -0.007735 26.580138 -0.00157
339.4137
57 -0.005996 19.935104 -0.00118
2
24.5593
26 -0.003993 13.290069 -0.00079
1.5
19.2117
31 -0.00305 9.9675519 -0.00059
1
14.0621
95 -0.00213 6.6450346 -0.00039
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0.5 6.53595 -0.001152 3.3225173 -0.0002
0
0.59417
7 -0.000203 0 0
Table 9. Strain and Displacement Measured and Calculated for an Aluminum H-Beam
Force(lbs) Strain() Displacement (in) TheoreticalStrain () TheoreticalDisplacement (in)
0
0.19805
9 0.00003 0 0
0.5
2.57476
8 -0.000694 10.833893 -0.00087
1
2.77282
7 -0.001408 21.667786 -0.00173
1.5
2.57476
8 -0.002147 32.501678 -0.0026
2 4.3573 -0.00289 43.335571 -0.00346
3
6.33789
1 -0.00431 65.003357 -0.00519
4
5.94177
2 -0.005615 86.671143 -0.00692
5
10.2990
72 -0.007244 108.33893 -0.00865
4
7.52624
5 -0.005883 86.671143 -0.00692
3
5.94177
2 -0.004364 65.003357 -0.00519
2
5.54565
4 -0.002968 43.335571 -0.00346
1.5 4.3573 -0.002214 32.501678 -0.0026
1
2.77282
7 -0.001536 21.667786 -0.00173
0.5 2.17865 -0.000766 10.833893 -0.00087
0
1.78253
2 -3.50E-05 0 0
Table 10. Strain and Displacement Measured and Calculated for an Aluminum H-Beam
with Aluminum Caps
Force(lbs)
Strain()
Displacement (in)
TheoreticalStrain ()
TheoreticalDisplacement (in)
0
-
1.58447
3 0.000008 0 0
0.5
2.37670
9 -0.00055 5.564794 -0.00046
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1
1.78253
2 -0.001123 11.129588 -0.00092
1.5 2.17865 -0.001701 16.694382 -0.00138
2
3.56506
3 -0.002271 22.259176 -0.00184
3
4.75341
8 -0.00348 33.388764 -0.00277
4
2.57476
8 -0.004641 44.518352 -0.00369
5
3.36700
4 -0.005833 55.64794 -0.00461
4
0.39611
8 -0.004715 44.518352 -0.00369
3
0.79223
6 -0.003603 33.388764 -0.00277
2
-
0.99029
5 -0.002437 22.259176 -0.00184
1.5 -2.17865 -0.001885 16.694382 -0.00138
1
-
2.37670
9 -0.001287 11.129588 -0.00092
0.5
-
3.16894
5 -0.000695 5.564794 -0.00046
0
-
4.15924
1 -6.15E-05 0 0
Table 11. Strain and Displacement Measured and Calculated for an Aluminum H-Beam
with Graphite/Epoxy CapsForce(lbs)
Strain()
Displacement (in)
TheoreticalStrain ()
TheoreticalDisplacement (in)
0
0.59417
7 -0.000028 0 0
0.5
-
0.99029
5 -0.00064 3.2971546 -0.00026
1 -2.17865 -0.001321 6.5943091 -0.00052
1.5
-
2.97088
6 -0.001989 9.8914637 -0.00079
2 -4.3573 -0.002631 13.188618 -0.00105
3
-
6.33789
1 -0.003943 19.782927 -0.00157
4 -8.7146 -0.005313 26.377237 -0.0021
5 -
10.4971
-0.006634 32.971546 -0.00262
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