User:Eml4507.s13.team2/Report4

Honor Pledge
On our honor, this problem was solved on our own with reference to Dr. Vu-Quoc's Fead.s13.sec53b lecture notes located here.

Given: A spring-mass-damper system
Consider the following spring-mass-damper system as given on p.53-13.



Find the eigenvector X2
Find the eigenvector $$ X_{2} $$ corresponding to the eigenvalue $$ \lambda_{2} $$ for the mass-spring-damper system shown.

Plot the mode shape
Plot and comment on the mode shape obtained from the eigenvector.

Verify eigenvectors are orthogonal
Verify that the eigenvectors $$ X_{1} $$ and $$ X_{2} $$ are orthogonal.

Known:

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$$  \displaystyle K=\begin{bmatrix} 3 & -2\\ -2 & 5 \end{bmatrix} $$     (1.0)
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$$  \displaystyle \gamma= 4\mp \sqrt{5} $$     (1.1)
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$$  \displaystyle \gamma_{1}=4-\sqrt{5} $$     (1.2)
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$$  \displaystyle X_{1}=\begin{Bmatrix} \frac{1+\sqrt{5}}{2}\\ 1 \end{Bmatrix} $$     (1.3)
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Find the eigenvector X2

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$$  \displaystyle \gamma_{2}=4+\sqrt{5} $$     (1.4)
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$$  \displaystyle \left [ K-\gamma _{2}I \right ]X=\begin{bmatrix} -1-\sqrt{5} &-2 \\ -2 & 1-\sqrt{5} \end{bmatrix}\begin{Bmatrix} x_{1}\\ x_{2} \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$     (1.5)
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Set the following:
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$$  \displaystyle x_{2}=1 $$     (1.6)
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$$  \displaystyle -2x_{1}+(1-\sqrt{5})(1)=0 $$     (1.7)
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$$  \displaystyle x_{1}=\frac{(1-\sqrt{5})}{2} $$     (1.8)
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$$  \displaystyle X_{2}=\begin{Bmatrix} \frac{(1-\sqrt{5})}{2}\\ 1 \end{Bmatrix} $$     (1.9)
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Plot the mode shape
The plot of the mode shape is given below.



This coincides with the mass $$ m_1 $$ moving 0.618 units to the left and mass $$ m_2 $$ moving 1 unit to the right.



Verify eigenvectors are orthogonal
Two vectors are orthogonal when their dot product is equal to zero.
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$$  \displaystyle X_{1}\cdot X_{2}=0 $$     (1.10)
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$$  \displaystyle \left ( \frac{1+\sqrt{5}}{2} \right )\left ( \frac{1-\sqrt{5}}{2} \right )+1*1=0 $$     (1.11)
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$$  \displaystyle \frac{1-5}{4}+1=0 $$     (1.12)
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$$  \displaystyle 0=0 $$     (1.13)
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Hence, the two eigenvectors are orthogonal.

Honor Pledge
On our honor, this problem was solved on our own, using only the above problem R4.1 for reference.

Given: Eigenvector assumptions
Let the coefficients of eigenvectors:


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$$  \displaystyle \mathbf x_1=\begin{Bmatrix} x_{11} \\ x_{21} \end{Bmatrix}, \ \mathbf x_2=\begin{Bmatrix} x_{12} \\ x_{22} \end{Bmatrix} $$     (2.0)
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...be written as follows:


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$$  \displaystyle \begin{bmatrix} \mathbf x_{1} & \mathbf x_{2} \end{bmatrix} = \begin{bmatrix} x_{ij} \end{bmatrix}=\begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} $$     (2.1)
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Instead of assuming the 2nd row to be:
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$$  \displaystyle x_{21}=x_{22}=1 $$     (2.2)
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Assume the 1st row to be:


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$$  \displaystyle x_{21}=x_{22} = 1 $$     (2.3)
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Find the eigenvectors
Calculate the eigenvectors for the system.

Plot the eigenvectors
Plot the eigenvectors and compare the mode shapes to those obtained with the first assumption above.

Create mode shape animations
Create an animation for each mode shape using the gif format.

Find the eigenvectors
From problem 4.1 we know that:
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$$  \displaystyle \left [ K-\gamma _{1}I \right ]X=\begin{bmatrix} -1+\sqrt{5} &-2 \\ -2 & 1+\sqrt{5} \end{bmatrix}\begin{Bmatrix} x_{1}\\ x_{2} \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$     (2.0) and
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$$  \displaystyle \left [ K-\gamma _{2}I \right ]X=\begin{bmatrix} -1-\sqrt{5} &-2 \\ -2 & 1-\sqrt{5} \end{bmatrix}\begin{Bmatrix} x_{1}\\ x_{2} \end{Bmatrix}=\begin{Bmatrix} 0\\ 0 \end{Bmatrix} $$     (2.1) As stated in the problem statement,we want to assume the first row to be equal to one, so for X1, x1=1
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$$  \displaystyle -2(1)+x_2 (1+\sqrt{5})=0 $$ $$ x_2 =2/(1+\sqrt{5}) $$     (2.2) Thus,
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$$  \displaystyle X_1 = \begin{Bmatrix} 1\\ 2/(1+\sqrt{5}) \end{Bmatrix} $$     (2.3) and
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$$  \displaystyle X_2 = \begin{Bmatrix} 1\\ 2/(1-\sqrt{5}) \end{Bmatrix} $$     (2.4)
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Plot the eigenvectors
For mode 1 we get: and for mode 2 we get:

Create mode shape animations
Create an animation for each mode shape using the gif format.

Honor Pledge
On our honor, this problem was solved on our own with reference to Dr. Vu-Quoc's lecture notes Eml4500.f08.2.djvu located here.

Given: 2 methods to find axial member forces of a two bar truss
Reconcile analytically the following two methods for computing the axial member forces of a two bar system: taking the square root of the sum of squares and using a transformation matrix.

Also, compare the computational efficiency of the two methods using geometry and algebra.



Method 1: Square root of the sum of squares
$$ P^{(e)}_1 = \sqrt{(f^{(e)}_1)^2 + (f^{(e)}_2)^2} $$  <p style="text-align:right"> (3.1)

P = axial member force, f1 = force component in x direction, f2 = force component in y direction

Method 2: Transformation Matrix
$$ P^{(e)} = T^{(e)}f^{(e)} $$  <p style="text-align:right"> (3.2)

Method 1: Square root of the sum of the squares
To begin using this method, we first use simple techniques from statics to solve for the x and y force components at each node in the system. The techniques from statics we can use include the method of joints, or more often the method of sections, in which all of elements coming from a node are “cut” through so as to solve for the nodal component forces. We then use the Pythagorean Theorem to solve for the axial forces in each member, where the two sides of the right triangle are the x and y force components and the hypotenuse is the axial force. Thus, this is why it is called the “square root of the sum of the squares” method, as shown in Eq. It is a much more basic method that can be used fairly well for simpler truss systems. For example, as long as a system is statically determinate and has a relatively small number of elements (less than five or so), we can use this method without an enormous amount of calculations.

Method 2: Transformation Matrix
$$ P^{(e)} = T^{(e)}f^{(e)} $$

$$ T = \left[\begin{array}{cccc} l & m & 0 & 0 \\ 0 & 0 & l & m \\ \end{array} \right] $$ <p style="text-align:right"> (3.3) l = cos(θ), m = sin(θ)

Using the Transformation matrix and the given angles, we can change the forces in the x and y direction of the member into the axial forces of the member. This method is efficient for systems that are statically indeterminant or systems that are complex (lots of elements). If forces in the x and y direction at each node is given we can calculate the axial member force using Eq(3.3).

Conclusion
In conclusion, both of the aforementioned methods can be used to solve for the axial member forces of the given truss system. While the first method would work just fine for the given two-member system, it is very inefficient and extremely time consuming for larger, more complex systems. The second method should be used for these types of systems, as it is much more computationally efficient because we would only have to solve the one matrix equation as opposed to many different equations if we used the first method.

Honor Pledge
On our honor, we referenced problem R3.6 from EML4507.s13.team2/Report3 located here.

Given: Material information for a plane truss
Continuation of R3.6, involving the plane truss shown below.



The mass density of the material is given to be $$ \rho = 5,000 \ kg/m^3 $$

Find the lumped mass matrix
Construct the lumped (diagonal) mass matrix of the system.

Find the 3 lowest eigenpairs
Find the 3 lowest eigenpairs for $$ (w_j,\phi_j) \ for \ j=1,2,3 $$.

Plot the 3 lowest mode shapes
Plot the 3 lowest mode shapes and animate them using gif files.

MATLAB Code
Readout of eigenvalues and eigenvectors

D =

Columns 1 through 6

-3.4909e-008           0            0            0            0            0 0 3.1099e-008            0            0            0            0 0           0  8.9492e-008            0            0            0 0           0            0  4.0921e+006            0            0 0           0            0            0  1.3605e+007            0 0           0            0            0            0  2.4654e+007 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0

Columns 7 through 12

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0  2.8135e+007            0            0            0            0            0 0  4.118e+007            0            0            0            0 0           0  5.3747e+007            0            0            0 0           0            0  6.0392e+007            0            0 0           0            0            0  7.5968e+007            0 0           0            0            0            0  1.0563e+008 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0

Columns 13 through 18

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0  1.3332e+008            0            0            0            0            0 0 1.9204e+008            0            0            0            0 0           0  1.9856e+008            0            0            0 0           0            0  2.0526e+008            0            0 0           0            0            0  2.2958e+008            0 0           0            0            0            0  2.5299e+008 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0

Columns 19 through 24

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0  2.8646e+008            0            0            0            0            0 0 2.9447e+008            0            0            0            0 0           0  3.2176e+008            0            0            0 0           0            0  3.3174e+008            0            0 0           0            0            0  3.6902e+008            0 0           0            0            0            0  3.7378e+008 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0

Columns 25 through 28

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0  3.8857e+008            0            0            0 0 4.1168e+008            0            0 0           0  4.5178e+008            0 0           0            0  4.7778e+008

V =

Columns 1 through 7

0.092063     0.42067      0.27171     -0.24191      0.20606      0.61593      0.27807      0.73897     -0.51572      0.17906      0.74673     -0.51818     -0.14572     -0.41608      0.19592      0.22885       0.4105      0.20048     -0.36981      0.72724     -0.29109      0.73897     -0.51572      0.17906      0.76074     -0.55198     -0.16391     -0.47639     0.092063      0.42067      0.27171     -0.19682     0.096574       0.4535     0.067727      0.63512      -0.3239     0.040281      0.15633      0.33341     0.010399       0.5402      0.19592      0.22885       0.4105      0.19678     -0.34716      0.64656     -0.25424      0.63512      -0.3239     0.040281      0.20202      0.28381    -0.046857      0.54119     0.092063      0.42067      0.27171    -0.091692    -0.030513      0.12531   -0.0096049      0.53127     -0.13209    -0.098503     -0.37374      0.55159      0.17953    -0.026362      0.19592      0.22885       0.4105      0.14761     -0.18966      0.45334    0.0039061      0.53127     -0.13209    -0.098503     -0.33532      0.63251       0.1653      0.17428     0.092063      0.42067      0.27171     0.040395    0.0019793     -0.20383      0.10902      0.42741     0.059727     -0.23729     -0.57153     -0.08336    -0.016936     -0.69164      0.19592      0.22885       0.4105     0.040395   -0.0019793      0.20383      0.10902      0.42741     0.059727     -0.23729     -0.57153      0.08336     0.016936     -0.69164     0.092063      0.42067      0.27171      0.14761      0.18966     -0.45334    0.0039061      0.32356      0.25154     -0.37607     -0.33532     -0.63251      -0.1653      0.17428      0.19592      0.22885       0.4105    -0.091692     0.030513     -0.12531   -0.0096049      0.32356      0.25154     -0.37607     -0.37374     -0.55159     -0.17953    -0.026362     0.092063      0.42067      0.27171      0.19678      0.34716     -0.64656     -0.25424      0.21971      0.44336     -0.51485      0.20202     -0.28381     0.046857      0.54119      0.19592      0.22885       0.4105     -0.19682    -0.096574      -0.4535     0.067727      0.21971      0.44336     -0.51485      0.15633     -0.33341    -0.010399       0.5402     0.092063      0.42067      0.27171      0.20048      0.36981     -0.72724     -0.29109      0.11585      0.63517     -0.65364      0.76074      0.55198      0.16391     -0.47639      0.19592      0.22885       0.4105     -0.24191     -0.20606     -0.61593      0.27807      0.11585      0.63517     -0.65364      0.74673      0.51818      0.14572     -0.41608

Columns 8 through 14

0.42384     0.15054     -0.34098    -0.059514      0.83481      0.39472      0.58017      -0.1569     -0.07994     -0.41932     -0.32303       0.1441      0.11336    0.0068501      -0.1631     -0.25248      0.22739      0.87401      0.22501     -0.86377      0.52241     -0.19259     -0.10544      -0.5758     -0.49083      0.27464      0.28335     0.050438      0.20444     0.029876      -0.5338     -0.38109      0.40489      0.27656     -0.22204      0.46425      0.25781       0.3778      0.36859     -0.25536      -0.3005     -0.38416     -0.13287     -0.19142      0.16559      0.57522      0.11806     -0.34557      0.07095      0.64115       0.4275     0.066927     0.050024      0.16092     0.044478       0.3642       0.2682      0.20058     -0.49071     -0.45559     -0.30141     -0.26072     -0.58511     -0.60658     -0.59257    -0.025052     0.016698     -0.15281       0.1237     0.057759     0.037019     0.030338      0.35549      0.19874     -0.36021      0.34411     -0.56953     -0.40575     -0.61188     -0.27316    -0.096797     0.090029     -0.21839     -0.08089      0.17501     -0.07142     -0.41642     -0.28724     -0.60897     -0.63107      0.13935     0.029703      0.62875      0.18793      0.13372    -0.078668      0.26379      0.30744     -0.17501     -0.07142      0.41642     -0.28724     -0.60897      0.63107     -0.13935    -0.029703      0.62875     -0.18793      0.13372    -0.078668     -0.26379     -0.30744    -0.037019     0.030338     -0.35549      0.19874     -0.36021     -0.34411      0.56953      0.40575     -0.61188      0.27316    -0.096797     0.090029      0.21839      0.08089      -0.2682      0.20058      0.49071     -0.45559     -0.30141      0.26072      0.58511      0.60658     -0.59257     0.025052     0.016698     -0.15281      -0.1237    -0.057759      0.13287     -0.19142     -0.16559      0.57522      0.11806      0.34557     -0.07095     -0.64115       0.4275    -0.066927     0.050024      0.16092    -0.044478      -0.3642     -0.20444     0.029876       0.5338     -0.38109      0.40489     -0.27656      0.22204     -0.46425      0.25781      -0.3778      0.36859     -0.25536       0.3005      0.38416       0.1631     -0.25248     -0.22739      0.87401      0.22501      0.86377     -0.52241      0.19259     -0.10544       0.5758     -0.49083      0.27464     -0.28335    -0.050438     -0.42384      0.15054      0.34098    -0.059514      0.83481     -0.39472     -0.58017       0.1569     -0.07994      0.41932     -0.32303       0.1441     -0.11336   -0.0068501

Columns 15 through 21

-0.36804   -0.029885       0.2574     -0.63744      0.34468    -0.083395     -0.59515   -0.0082754    0.0070421    -0.010763     0.065714     -0.11916    -0.021461      0.16666      0.49337    -0.011226     -0.76821      0.35892       0.4231      0.13759     0.067272    -0.077702     0.092251      0.32517     -0.47469      0.41221     0.066013     -0.37208      0.11128      0.11355       0.1204      0.18868     -0.14467      0.14418      0.74912      0.27334      0.20946       0.5475     -0.56333      0.49371       0.2887      0.21402     0.052545  -0.00085696     0.025426    -0.049686     -0.12231    -0.044732    -0.030132      -0.1968      -0.2944     -0.67527      0.64524     -0.41345     -0.17022    -0.039726      0.44006    -0.051352     -0.26552      0.33385      0.27508       0.3355      0.12963     -0.65173     -0.57274     -0.21933    0.0083882      0.36828      0.61944      0.33657     -0.40985    -0.085054      0.44413     0.079253     -0.58889     -0.13515     0.095554      0.59958      0.62543      0.15132      0.13121     -0.46923      -0.5221     -0.34922     -0.13291    -0.080994     -0.17547     -0.21593      0.12248      0.21572     -0.51433     0.017204      0.79156     0.056774      0.28071    -0.073913      0.72472     -0.12197     -0.13291     0.080994     -0.17547      0.21593      0.12248     -0.21572     -0.51433     0.017204     -0.79156     0.056774     -0.28071    -0.073913     -0.72472     -0.12197     -0.40985     0.085054      0.44413    -0.079253     -0.58889      0.13515     0.095554      0.59958     -0.62543      0.15132     -0.13121     -0.46923       0.5221     -0.34922      0.44006     0.051352     -0.26552     -0.33385      0.27508      -0.3355      0.12963     -0.65173      0.57274     -0.21933   -0.0083882      0.36828     -0.61944      0.33657     0.052545   0.00085696     0.025426     0.049686     -0.12231     0.044732    -0.030132      -0.1968       0.2944     -0.67527     -0.64524     -0.41345      0.17022    -0.039726      0.11128     -0.11355       0.1204     -0.18868     -0.14467     -0.14418      0.74912      0.27334     -0.20946       0.5475      0.56333      0.49371      -0.2887      0.21402      0.49337     0.011226     -0.76821     -0.35892       0.4231     -0.13759     0.067272    -0.077702    -0.092251      0.32517      0.47469      0.41221    -0.066013     -0.37208     -0.36804     0.029885       0.2574      0.63744      0.34468     0.083395     -0.59515   -0.0082754   -0.0070421    -0.010763    -0.065714     -0.11916     0.021461      0.16666

Columns 22 through 28

0.056167    -0.24865     0.055097      0.41508      0.13022     -0.36482       0.3124      0.22032     -0.70663     -0.76713      0.24871      0.51075     -0.37982      0.21327     -0.89652      0.63921     0.036916     -0.33188      0.69784     -0.70038      0.41825     -0.44706       1.0697       1.1248     -0.33224     -0.59907      0.36768     -0.18545     -0.19288      0.22938     -0.41394     -0.66249      0.22382      0.33064     -0.45015     -0.20211    -0.062593     -0.21076    -0.043417     0.092746      0.06165    -0.085966      0.44183     -0.42226    -0.025178      0.24844     -0.59495      0.72349     -0.48099      0.13422       0.1086       0.1923    -0.041278     -0.19366      0.14598    -0.079588       -0.136      -0.1915      0.27435      0.65126     -0.50493    -0.096539      0.52184     0.084389    -0.050219     0.023178     0.012598     -0.13784    -0.011063     0.096382      0.43035      0.29075      0.31703     -0.29148     0.039471      -0.5283      0.51818      0.10684     0.040685      0.13132     0.053448     0.014578     -0.10034     0.081482      0.66948   -0.0036594      0.13832     -0.45892      0.41969     -0.22946     -0.54001      0.23208     0.025416       0.1134      0.04812      0.10057    -0.055857    -0.093242     -0.66948   -0.0036594     -0.13832      0.45892      0.41969      0.22946     -0.54001     -0.23208     0.025416      -0.1134     -0.04812      0.10057     0.055857    -0.093242     -0.43035      0.29075     -0.31703      0.29148     0.039471       0.5283      0.51818     -0.10684     0.040685     -0.13132    -0.053448     0.014578      0.10034     0.081482        0.136      -0.1915     -0.27435     -0.65126     -0.50493     0.096539      0.52184    -0.084389    -0.050219    -0.023178    -0.012598     -0.13784     0.011063     0.096382     -0.44183     -0.42226     0.025178     -0.24844     -0.59495     -0.72349     -0.48099     -0.13422       0.1086      -0.1923     0.041278     -0.19366     -0.14598    -0.079588      0.19288      0.22938      0.41394      0.66249      0.22382     -0.33064     -0.45015      0.20211    -0.062593      0.21076     0.043417     0.092746     -0.06165    -0.085966      0.89652      0.63921    -0.036916      0.33188      0.69784      0.70038      0.41825      0.44706       1.0697      -1.1248      0.33224     -0.59907     -0.36768     -0.18545    -0.056167     -0.24865    -0.055097     -0.41508      0.13022      0.36482       0.3124     -0.22032     -0.70663      0.76713     -0.24871      0.51075      0.37982      0.21327