# Project Report, MAE 4150, University of Colorado Colorado Springs

Authors: Sean Glover, Matt Radel

Editors Note: That isn’t a typo, Colorado is included twice in the school name.

## Abstract

Longswords are often treated as having static vibrational nodes, with implications for sparring. A combination of mathematical analysis, SolidWorks modeling, and practical experiments were used to analyze the motion of a longsword under realistic end conditions. It was determined that not only are the nodes not static with respect to different end conditions, they also vary over time during free vibration after an impulse. Due to the complexity of the vibrations, as well as the very specific design criteria of a sword, it is impractical to attempt to alter the sword to maintain specific node locations. Vibration damping gloves are suggested as a possible remedy to fatigue during sparring.

## Nomenclature

 w Displacement of blade [mm] c Wave speed [m/s] E Elastic modulus [Pa ] a Acceleration of blade [m/s2 ] n Mode number [unitless] τ tension [N] ρ density [kg/m3] v Velocity of blade [m/s] L Blade length [m] t Time after impact [sec] x Linear distance along blade, measured from the hilt [m ]

## Introduction

Longswords have a particularly long effective length. This, arising from the slender nature of a blade and the long blades used on such swords, makes them particularly vulnerable to vibrations. Vibrations are concerning for three principle reasons in a fencing context. First, vibrations transferred to the fencer through the hilt cause numbness and fatigue. Second, vibrations can prevent proper edge alignment. Third, energy going into a vibration is energy not going into the opponent, reducing the effectiveness of the strike.

Among contemporary longsword practitioners, there are several preconceived notions regarding vibrations that this study intends to examine for validity. The first and most common of these is the position of the vibrational node (commonly but incorrectly referred to as the center of percussion) near the end of the blade. This is commonly tested for by lightly gripping the hilt, often with just a finger and thumb, and slapping the pommel (the counterweight at the base of the grip) to provide an impulse and resulting vibrations. The sword will vibrate in a second mode, showing a clear node that is often around 10cm away from the tip. Conventional wisdom indicates that this is where a fencer should strike, as this will produce minimal vibrations. This also indicates second mode motion.

The second most common popular statement made about nodes refers to the positioning of the second node. A competent smith intends to place this at the point in the grip where the middle finger of the lead hand will mostly be positioned. The common test for this is similar to that performed for the other node, in that the grip is lightly held and an impulse to the pommel, moving where the grip is held until minimal vibration is felt in the hand.

These common tests and suggestions make a critical assumption: that the positions of the nodes do not change when the sword is held for actual sparring. This is a highly suspect assumption, as the end conditions of a vibrating beam have significant impact on the vibrational characteristics of the system. Determining the actual effects of hand position on vibrational nodes is the main purpose of this project. Additional information of interest is the actual amount of force lost to vibrations, as well as the force transferred to the hand of the fencer by vibration.

## Model

Two displacement tests were done using SolidWorks dynamic analysis. First, the sword was recreated as a SolidWorks CAD model, shown in Figure 1, as accurately as possible up to the bent tip. This tip was approximated as an elliptical ring at the end of the blade with similar dimensions as the true bend and is shown in Figure 2.

For the first displacement test, a small 0.057 cm2 rectangle was cut out of the blade 85.39 cm up from the guard and a force of 250N was applied to that area for 0.02 seconds. The handle was then fixed on the upper and lower grips and a material of Plain Carbon Steel was applied to the entire sword (see Figure 2). These conditions were used to represent the sword getting struck by another, but no damping was assumed during the test.

The second displacement test model had two small 0.0375 cm2 rectangle cuts 15cm up from the guard and the same force was applied at these areas. There was only one fixed position on the upper grip and 30 points of interest were selected evenly distributed along the side of the blade (see Figure 3). Again, no damping was assumed during the test

## Solution

From the SolidWorks model, the second moments of area were calculated from the section 40cm away from the hilt, as a representative value for the blade:

Iz = 0.5089 cm4 = 5.089*10-9m4
Iy = 0.0040 cm4 = 4.0*10-11m4

The stiffness of a beam is given by:

The elastic modulus, E, for 51CrV4 Carbon Steel is variable based on the heat treatment. The specific heat treatment applicable here is not known. It is assumed the steel is tempered, for a value of 210GPa [1]. Therefore, the estimated stiffness for the sword is 3741 N/m in the blade direction and 29.40 N/m in the flat direction. This large difference between directions validates the assumption that the blade will experience negligible deflection, and therefore negligible vibration, around the blade’s axis. Another favorable feature is the thickness is consistent for most of the blade, and the width only decreases slightly, making the estimated second moment of area relatively accurate.

As the blade is most accurately thought of as a vibrating beam, it will be treated as a distributed-parameter system. The solution will be via the separation of variables method. The boundary conditions will be treated as cantilevered at the handle, and free at the tip. The handle itself will be treated as not vibrating, though this is a simplification.

The governing equation for the vibration of the blade is:

Where n is the mode and the wave speed, c, is:

As we are only dealing with the first two modes, n = 2. For this steel density of 7800 kg/m3 and elastic modulus of 210 GPa, c = 5188 m/s. Therefore, the equation of motion becomes:

The constants are solved for via end conditions. The first end condition is w’’(0,0)=0. This is due to the assumption that there is no acceleration at the hilt at the beginning. For the second end condition, it will be assumed that the impulse provides an initial velocity to the tip, leading to w’(L,0)=20. This assumption is based on the only known data currently available regarding the speed of a sword [2]. From this test, the cutting surface moves at approximately 20m/s, making it a reasonable value for the resultant velocity from the impact. The initial displacement is 0. Therefore, solving for initial displacement provides no information. Taking the derivative of a multivariable function such as this simply entails taking the partial derivative with respect to each variable and adding the results. Therefore, the velocity equation is as follows:

Solving for the velocity produces 1.2884a2-0.3321a1=20. From here, another derivative is needed for the acceleration condition:

Solving for the acceleration condition produces a1=-4a4, allowing for the final constants to be determined and added to the initial equation:

A graphical depiction of these results follows in Figure 4 and Figure 5:

As can be seen in these graphs, there is no defined vibration node when the two modes are combined. As is to be expected with a multiple mode system, there is a fundamental frequency and a higher frequency harmonic of a lower amplitude.

## Results

Practical tests in lieu of campus resources such as accelerometers required a certain amount of ingenuity. As clamped end conditions are not an accurate depiction of the realistic use conditions, tests were conducted in hand. To acquire useable data, a camera on a tripod was used in conjunction with a computer-based stopwatch to provide a stable reference time and position. At the start of testing videos, a grid with known dimensions was placed behind the sword to provide accurate scaling.

Tests were conducted with both one and two hands on the sword. For single-handed tests, the input was a tap with the hand to the pommel. A sample of frames from one of these tests can be found in appendix A. For two-handed tests, the original attempt was a strike with a wooden mallet, but it provided insufficient vibration. After some experimenting with different surfaces, it was found the most effective input is an elastic material with some give. The final tests in this manner used an input of bouncing the blade off a large exercise ball. Given the inherent limits of the equipment at hand, accurate data regarding the input force is unavailable. Unfortunately, this means the force transmissibility was impossible to experimentally verify. In addition, the position of the second vibrational node was also not able to be determined by this method.

After testing, videos were examined in Photoshop to determine amplitudes and vibrational node positions. Difference analysis between frames allowed for depiction of the full amplitude in a single image. This analysis can be found in Figure 6:

The measurements from these tests are given in Table 1:

 Test Amplitude (cm) Node position from tip (cm) 2-handed test 5.13 36.41 1-handed test at clock time 30.6 11.63 49.36 1-handed test at clock time 30.7 3.25 27.15

This is interesting, as it experimentally verifies that the node location not only changes based on hand position, but it also changes in a very short period within the same test, with essentially identical conditions. A difference of more than 20cm across a tenth of a second indicates that the vibration is much more complex than previously thought. The assumption that the node is essentially static is now shown to be incorrect not only for other hand positions, but for different points in time during the same vibrational response to an impact! This also agrees well with the results of the mathematical model.

Determining the single-hand frequency posed a challenge. The vibration was easily visible in the videos, but the frequency was near enough to the frame rate that Nyquist Frequency effects were a possible concern. From visual analysis of the vibration, the frequency was estimated at around 10-20 Hz. Given a frame rate of 30fps, this was concerning, as any vibration above 15 Hz would be impossible to accurately measure via video analysis.

Analysis of the video suggests a frequency of approximately 10 Hz. However, the vibrations visible in the frames are complex and difficult to determine. Difficulty determining the frequency in this manner means the natural frequency of the single hand case cannot be experimentally verified with confidence, given available equipment.

While the single-hand frequency was a large enough amplitude and low enough frequency to detect in the video, the two-hand frequency was much higher. This necessitated a different test method. Audio was taken of the blade being impacted. This audio was analyzed via Spectrum Lab FFT analysis software. The natural frequency was faintly visible, occurring at 1233.9 Hz. Spek frequency analysis software provided a less measurable but more visible example of this analysis, found in Figure 7:

The entire test raster can be found with commentary in appendix B. As a distributed parameter system such as this has a fundamental natural frequency and multiple harmonics, this is presumably not the only frequency of interest. However, it was the only frequency detectable with the available equipment.

This frequency can be converted to 7752.82 radians per second. The theoretical model of the sword suggests a fundamental frequency of 1902 radians per second. This could indicate a significant flaw with the model. However, the measured frequency is approximately 4.08 times the calculated one. This, assuming some level of imprecision with the assumptions made in the mathematical model, suggests the measured frequency is a harmonic of the fundamental frequency. It was near the edge of the microphone’s lower range and was relatively low amplitude.  As harmonic frequencies have lower amplitudes than the fundamental frequency, the next higher harmonic was likely below the noise floor. This also supports the notion that the calculated fundamental frequency is accurate.

## SolidWorks Simulation Results

The first displacement test gave a response corresponding closely, each off by a factor of about 300 when the force was applied, to the first mode of motion shown in appendix E. The only node found using this test was located at the handle. This was expected for the used end conditions of the blade and placement of the force. The maximum displacement for the points of interest occurred 83.4 cm from the guard and the displacement became smaller as the distance from the grip decreased.

The second displacement test gave more variable results for the nodes. The nodes were not stationary along the blade and instead moved over time. The main locations of the nodes were near points 1, 7, 17, 21, 25, and 30 which correspond to 84.72cm, 68.04cm, 45.8cm, 29.12cm, 18cm, and 4.1cm from the tip of the blade, respectively.

These shifting nodes suggest that the vibrations in the sword are made through an overlapping of multiple modes. The first five modes are shown in appendix E, were calculated using SolidWorks vibration analysis, and occur with frequencies of 4.0426 Hz, 20.978 Hz, 50.253 Hz, 56.636 Hz, and 109.82 Hz. The first two modes are shown in Figure 8. These time dependent nodes correlate with the experimental tests done with the real sword.

## Modifications

Modifying swords is a difficult prospect if the sword is to remain useful afterward. The balance, blade cross-section, length, and grip all have many design constraints more important than vibration suppression. For instance, addition of a vibrational absorber such as attached to power lines would obviously be unsuitable for the middle of a longsword’s blade. As the most common sources of vibration are the impulses of hitting either an opponent or an opponent’s blade, limiting the input or adding dampers are likewise not valid options. Two possibilities seem immediately obvious. The grip could be modified to attempt to decrease vibration, or the fencer could adjust their hands to produce favorable vibrational characteristics via tuning of the end conditions.

The first option is the most straightforward, and the most achievable. Either the grip could be wrapped in a damping layer, such as a polymer foam, or sparring gauntlets could incorporate vibration damping in a similar manner. This has two drawbacks. First, it reduces the tactile feedback from the grip, which is valuable information for the fencer. Second, it only mitigates transmissibility to the fencer. It produces no significant effect regarding the location of a vibrational node, nor will it prevent vibrations in the blade causing edge misalignment or reduced force.

The second option relies on being able to maintain very specific end conditions in a complex and stressful situation. This may be possible with training; however difficult it may be. Such a method would allow an experienced fencer to intentionally strike with the vibrational node for the best effect. Combining this with damping at the grip could potentially allow for better performance during sparring. However, as the node positions change more than was originally thought, far more testing would be required before this could be a feasible option. Additionally, such complex requirements would likely interfere with other aspects of sparring.

## Conclusion

The vibrational characteristics of longswords are far more complex than previously believed. Not only do the vibrational nodes move positions based on different end conditions, they also change position during vibrations from an impact. This indicates there are multiple modes at play. Without accelerometers it would be extremely difficult to fully characterize these vibrations. Even upon characterizing these vibrations, it would be extremely difficult to alter the design of a sword to isolate, damp, or otherwise tune the vibrations to a specific purpose. As vibrations transmitted into the hand are fatiguing to fencers, it is possible vibration damping materials such as polymer foams would help mitigate this issue.

It bears noting that the design of many swords, including the one tested here, includes a fuller. This is the groove cut down the middle of the flat of the blade on both sides, and it functions in the same manner as the design of an I-beam. It increases stiffness, which decreases the amplitude of vibrations. In this manner, it could be claimed the sword is already designed to mitigate vibration, in a manner consistent with its application. Many different fuller designs are possible, and it would merit further study of these designs to determine which fullers produce the most favorable result. That is, however, beyond the scope of this paper.

### Appendix A: Sequence of test video frames

In the following images, the final numbers indicate the time of the frame. As the video sampled at 30fps, every tenth of a second had 3 associated frames. The last number is the frame within each tenth of a second.

### Appendix B: Two-handed natural frequency spectrum and commentary

The test had 5 impacts on the sword, shown in the broadband, short duration signals visible. The natural frequency had been previously estimated via matching to test tones as being around 1kHz. The third and fourth tests had sufficient ringing to be detected by the microphone, shown as the longer duration pure tones near 1kHz, as expected. This was then measured with greater detail in Spectrum Lab, providing the natural frequency.

### Appendix C: Sword Displacement Tables

 distance (cm)from tip distance(cm)From guard Time(s) and Point Displacement (mm) .01sec .51sec 1.01sec 1.51sec 2.01sec 2.51sec 3.01sec 84.72 2.78 0.910 0.242 0.347 0.516 0.459 0.644 0.025 81.94 5.56 3.000 0.870 1.220 1.800 1.610 2.250 0.132 79.16 8.34 6.200 2.010 2.720 3.990 3.600 5.020 0.415 76.38 11.12 9.300 3.370 4.400 6.440 5.870 8.130 0.866 73.60 13.90 12.600 5.260 6.560 9.570 8.820 12.200 1.660 70.82 16.68 15.200 7.500 8.810 12.900 12.000 16.500 2.800 68.04 19.46 16.700 9.770 10.700 15.800 14.900 20.400 4.130 65.26 22.24 17.600 12.600 12.600 18.900 18.100 24.600 5.950 62.48 25.02 17.600 15.400 13.900 21.400 20.700 28.100 7.810 59.70 27.80 17.100 18.800 14.700 23.800 23.300 31.400 10.100 56.92 30.58 16.200 22.500 14.800 25.600 25.300 34.100 12.300 54.14 33.36 15.200 25.800 14.300 26.700 26.600 35.700 14.200 51.36 36.14 13.900 29.500 12.900 27.300 27.400 36.700 16.000 48.58 38.92 12.600 32.500 11.100 27.200 27.600 36.800 17.200 45.80 41.70 11.100 35.600 8.500 26.400 27.300 36.100 17.800 43.02 44.48 9.530 37.800 5.460 24.800 26.400 34.600 17.600 40.24 47.26 8.150 39.000 2.550 22.800 25.200 32.600 16.600 37.46 50.04 6.680 39.200 0.847 19.700 23.700 29.900 14.600 34.68 52.82 5.510 38.100 3.760 16.300 22.100 27.200 11.800 31.90 55.60 4.330 35.400 6.800 11.700 20.300 24.000 7.640 29.12 58.38 3.280 31.000 9.420 6.260 18.500 20.900 2.360 26.34 61.16 2.430 25.700 11.300 0.875 16.900 18.200 3.310 23.56 63.94 1.520 18.700 13.100 5.760 15.100 15.400 10.400 20.78 66.72 0.673 10.700 14.400 12.700 13.400 13.000 18.200 18.00 69.50 0.039 2.940 15.200 19.000 11.800 11.100 25.800 15.22 72.28 0.691 6.580 15.700 26.300 9.910 9.300 35.000 12.44 75.06 1.230 15.500 15.800 32.800 8.130 8.030 43.600 9.66 77.84 1.800 25.900 15.800 40.200 6.010 6.900 53.800 6.88 80.62 2.380 36.600 15.500 47.600 3.770 6.020 64.400 4.10 83.40 2.910 46.300 15.100 54.100 1.710 5.370 74.000

*Bold Values correspond to lowest recorded value at that time

 distance (cm)from tip distance(cm)From guard Time(s) and Point Displacement (mm) 3.51sec 4.01sec 4.51sec 5.01sec 5.51sec 6.01sec 6.51sec 84.72 2.78 0.292 0.209 0.226 0.292 0.360 0.975 0.060 81.94 5.56 0.999 0.791 0.705 1.050 1.240 3.370 0.186 79.16 8.34 2.170 1.920 1.340 2.410 2.720 7.410 0.349 76.38 11.12 3.420 3.360 1.820 4.030 4.350 11.900 0.460 73.60 13.90 4.940 5.500 2.060 6.220 6.390 17.400 0.483 70.82 16.68 6.450 8.190 1.830 8.710 8.550 23.300 0.339 68.04 19.46 7.680 11.000 1.110 11.100 10.500 28.400 0.041 65.26 22.24 8.850 14.700 0.324 13.800 12.600 33.900 0.529 62.48 25.02 9.620 18.200 2.130 16.100 14.400 38.300 1.220 59.70 27.80 10.100 22.600 4.750 18.400 16.400 42.600 2.190 56.92 30.58 10.100 27.100 7.900 20.300 18.100 46.200 3.330 54.14 33.36 9.720 31.200 11.000 21.400 19.600 48.700 4.440 51.36 36.14 8.750 35.800 14.800 21.900 20.900 50.900 5.780 48.58 38.92 7.420 39.700 18.300 21.600 21.900 52.100 7.040 45.80 41.70 5.410 43.900 22.200 20.300 22.800 52.800 8.550 43.02 44.48 2.910 47.600 26.000 17.900 23.400 52.700 10.200 40.24 47.26 0.339 50.300 29.100 14.800 23.800 52.200 11.900 37.46 50.04 3.000 52.600 32.300 10.100 24.100 51.000 14.200 34.68 52.82 6.200 53.700 34.700 4.870 24.500 49.600 16.700 31.90 55.60 10.000 53.700 36.800 2.110 25.200 47.700 20.200 29.12 58.38 14.100 52.200 38.300 10.100 26.300 45.500 24.400 26.34 61.16 18.000 49.500 39.100 18.000 27.900 43.500 29.000 23.56 63.94 22.400 45.200 39.400 27.400 30.200 41.100 34.800 20.78 66.72 27.100 39.700 39.300 37.200 33.100 38.800 41.200 18.00 69.50 31.500 33.900 38.800 46.100 36.400 36.700 47.500 15.22 72.28 36.700 26.200 37.700 56.400 40.700 34.400 55.000 12.44 75.06 41.500 18.700 36.500 65.600 45.100 32.400 62.000 9.66 77.84 47.200 9.470 34.700 76.000 50.500 30.100 70.300 6.88 80.62 53.100 0.573 32.600 86.400 56.400 27.800 78.700 4.10 83.40 58.500 9.170 30.700 95.600 61.800 25.700 86.400

*Bold Values correspond to lowest recorded value at that time

 distance (cm)from tip distance(cm)From guard Time(s) and Point Displacement (mm) 7.01sec 7.51sec 8.01sec 8.51sec 9.01sec 9.51sec 9.96sec 84.72 2.78 0.392 0.053 0.350 0.281 0.317 0.667 0.178 81.94 5.56 1.360 0.082 1.240 0.959 1.080 2.390 0.595 79.16 8.34 3.000 0.094 2.800 2.080 2.320 5.470 1.250 76.38 11.12 4.800 0.584 4.610 3.270 3.630 9.110 1.890 73.60 13.90 7.070 1.670 7.040 4.710 5.190 14.100 2.570 70.82 16.68 9.420 3.460 9.800 6.130 6.670 19.700 3.050 68.04 19.46 11.400 5.700 12.500 7.280 7.810 25.200 3.210 65.26 22.24 13.500 9.000 15.700 8.350 8.790 31.600 2.980 62.48 25.02 15.000 12.600 18.800 9.030 9.350 37.400 2.320 59.70 27.80 16.300 17.300 22.500 9.430 9.610 43.600 1.040 56.92 30.58 16.900 22.600 26.500 9.370 9.540 49.200 0.807 54.14 33.36 16.900 27.700 30.200 8.870 9.290 53.600 2.830 51.36 36.14 16.000 33.600 34.600 7.790 8.980 57.600 5.410 48.58 38.92 14.300 38.800 38.700 6.310 8.840 60.200 7.800 45.80 41.70 11.500 44.600 43.400 4.050 9.050 62.000 10.400 43.02 44.48 7.520 49.900 48.000 1.110 9.890 62.700 12.600 40.24 47.26 3.020 54.000 52.000 2.140 11.300 62.500 14.000 37.46 50.04 3.160 57.700 56.000 6.490 13.900 61.500 14.900 34.68 52.82 9.490 60.200 59.000 11.100 17.200 60.100 14.800 31.90 55.60 17.500 61.700 61.400 17.000 22.100 58.100 13.800 29.12 58.38 26.100 62.000 62.600 24.000 28.100 56.100 11.700 26.34 61.16 34.200 61.200 62.600 31.200 34.500 54.400 8.970 23.56 63.94 43.500 59.400 61.600 40.000 42.500 52.600 5.330 20.78 66.72 52.700 56.900 59.700 49.600 51.000 51.200 1.220 18.00 69.50 60.700 54.200 57.300 58.900 59.000 50.100 2.800 15.22 72.28 69.400 50.600 53.800 70.200 68.300 49.200 7.520 12.44 75.06 76.900 47.100 50.300 80.800 76.800 48.600 11.900 9.66 77.84 85.000 42.800 45.900 93.500 86.500 48.200 16.900 6.88 80.62 92.900 38.400 41.200 107.000 96.400 47.900 21.900 4.10 83.40 99.800 34.400 36.800 119.000 105.000 47.800 26.400

*Bold Values correspond to lowest recorded value at that time

### Appendix E: Mode Displacement Table and Graphs

 Distance (cm)from tip Distance(cm)From guard Displacement (mm) Mode 1 4.0426 Hz Mode 2 20.978 Hz Mode 3 50.253 Hz Mode 4 56.636 Hz Mode 5 109.82 Hz 84.72 2.78 0.004989 0.027100 0.007610 0.072300 0.135000 81.94 5.56 0.018310 0.095900 0.018700 0.246000 0.437000 79.16 8.34 0.043150 0.217000 0.038100 0.528000 0.884000 76.38 11.12 0.073780 0.356000 0.062000 0.826000 1.290000 73.60 13.90 0.117600 0.542000 0.096500 1.180000 1.680000 70.82 16.68 0.171200 0.750000 0.139000 1.500000 1.900000 68.04 19.46 0.226600 0.945000 0.184000 1.750000 1.920000 65.26 22.24 0.297700 1.170000 0.243000 1.950000 1.730000 62.48 25.02 0.368200 1.360000 0.303000 2.030000 1.360000 59.70 27.80 0.455700 1.570000 0.378000 2.010000 0.777000 56.92 30.58 0.551300 1.760000 0.462000 1.870000 0.080900 54.14 33.36 0.642900 1.900000 0.544000 1.630000 0.585000 51.36 36.14 0.753000 2.030000 0.645000 1.250000 1.270000 48.58 38.92 0.856900 2.100000 0.742000 0.834000 1.750000 45.80 41.70 0.980200 2.140000 0.860000 0.303000 2.080000 43.02 44.48 1.110000 2.130000 0.987000 0.266000 2.130000 40.24 47.26 1.230000 2.070000 1.110000 0.764000 1.940000 37.46 50.04 1.371000 1.950000 1.250000 1.280000 1.470000 34.68 52.82 1.500000 1.780000 1.390000 1.660000 0.864000 31.90 55.60 1.650000 1.540000 1.540000 1.980000 0.083400 29.12 58.38 1.804000 1.240000 1.710000 2.120000 0.721000 26.34 61.16 1.944000 0.915000 1.870000 2.090000 1.320000 23.56 63.94 2.104000 0.520000 2.050000 1.920000 1.820000 20.78 66.72 2.266000 0.093300 2.240000 1.610000 2.080000 18.00 69.50 2.410000 0.315000 2.410000 1.220000 2.070000 15.22 72.28 2.574000 0.793000 2.610000 0.661000 1.750000 12.44 75.06 2.721000 1.230000 2.780000 0.078500 1.210000 9.66 77.84 2.886000 1.740000 2.990000 0.679000 0.357000 6.88 80.62 3.052000 2.260000 3.200000 1.490000 0.715000 4.10 83.40 3.200000 2.730000 3.380000 2.240000 1.770000

*Bold Values correspond to node points.