User:EML4500.f08.RAMROD.E/HW6

Lecture Notes 11/5/08 Meeting 30
To solve continuous Principle of Virtual Work problem, we must use integration by parts. Recall from calculus integration by parts shown below.

Applying this principle to the equation for continuous PVW equation:

$$\int_{0}^{L}{\left\{u(x)[EA\frac{\delta w}{\delta x}] + f - m\ddot{u} \right\}dx}$$

Denoting the first term, $$\displaystyle u(x)(EA)\frac{\delta u}{\delta x}$$, as $$\displaystyle r(x)$$ and $$\displaystyle w(x)$$ as $$\displaystyle s(x)$$, we can plug these terms into the above formula and simplify the continuous PVW equation using integration by parts.

$$\int (r(x)s(x))' = \int_{0}^{L}{\left\{u(x)[EA\frac{\delta w}{\delta x}] + f - m\ddot{u} \right\}dx} = \left[w(EA)\frac{\delta u}{\delta x} \right]_{x=0}^{x=L} - \int_{x=L}^{x=0}{\frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}}dx$$

$$= \left[w(L) (EA)_{L}\frac{\delta u} {\delta x}_{L,t} \right] - \left[w(0) (EA)_{0}\frac{\delta u}{\delta x}_{0,t} \right] - \int \frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}dx$$

Where the first term is denoted as $$\displaystyle N(L,t)$$ and the second term is denoted by $$\displaystyle N(0,t)$$.

<\br>Now, return to the model problem of a beam shown below.



The boundary conditions for this model are:

$$\displaystyle u(0,t) = 0$$ and $$\displaystyle N(L,t) = F(t)$$

At x=0, select the weighting function, $$\displaystyle w(x)$$, such that $$\displaystyle w(0) = 0$$. (i.e. kinematically admissible)

Back to Continuous PVW

By selecting $$\displaystyle w(0) = 0$$, unknown reaction, $$\displaystyle N(0,t)$$ term cancels.

So, continuous PVW simplifies down to:

$$\int_{0}^{L}{w(m\ddot{u})dx} + \int_{0}^{L}{\frac{\delta w}{\delta x}(EA)\frac{\delta u}{\delta x}dx} = w(L)F(t) + \int_{0}^{L}{(wf)dx}$$ for all $$\displaystyle w(x)$$ such that $$\displaystyle w(0)=0$$.

Matlab Portion
In order to construct the lumped mass matrix, a matrix that stores the mass associated with each degree of freedom at each node, the density, 7.8 g/cm^3, is multiplied by the cross sectional area, 4 cm^2, and by half of the length of the associated element. To do this, a length matrix was also created using the x and y coordinates of each node and the Pythagorean theorem. After the lumped mass matrix, M, was created, a simpler mass matrix, m, was made from the diagonals of M. Lastly, assuming K_bar and M_bar are the simplified stiffness and mass matrices using PVW, we can solve for the eigen values and eigen vectors of the generalized eigenvector problem:

$$\displaystyle \bar{K}v=\lambda \bar{M}v$$

In this code, matrix D are the eigenvectors and the diagonal of matrix V are the eigenvalues.

Finally, we isolate only the three smallest eigenvalue/eigenvector pairs and use them to plot the associated three deformations of the structure with respect to the undeformed structure. This is shown in the three figures following the matlab code below. This code is intended to be executed following the above techniques for plotting the deformed shape.

The following are the results and plots from the above Matlab Code:

V =

Columns 1 through 7

-0.0014944    -0.05031     0.038896     0.014198    -0.032696    -0.001106    0.0071974   -0.0094129     0.021545    -0.011552    0.0065082     0.020884    0.0011283   -0.0094644   -0.0026749    0.0029728    -0.063282     0.017282    -0.032582  -0.00085494    0.0066026   -0.0020615     0.009503     -0.01669     0.018202    -0.020147  -0.00055093    0.0079385   -0.0049418    -0.019243   -0.0096861     0.011171    -0.015833  -0.00026065  -0.00036456   -0.0083028      0.01154    0.0040931    0.0074832     0.015165   0.00083623   -0.0067624   -0.0049418    -0.019243   -0.0096861     0.011171    -0.016113  -0.00029973  -0.00025865   -0.0049431    -0.018737      0.05144     0.010434   0.00010155   0.00012578  -0.00032828   -0.0049418    -0.019243   -0.0096861     0.011171    -0.015761  -0.00031107  -0.00042716   -0.0027531    0.0027253  -0.00033855     0.016338    -0.014736  -0.00037307    0.0055664   -0.0073676     -0.01539   -0.0077489    0.0073001  -0.00091268   0.00031983   -0.0065232   -0.0075273     0.010308    0.0034739    0.0087205    0.0092429   0.00055298   -0.0040796   -0.0073676     -0.01539   -0.0077489    0.0073001  -0.00087416   0.00031888   -0.0064171   -0.0055158    0.0071134    0.0018676      0.01193  9.7725e-005   0.00035942  -0.00046686 -0.0073676    -0.01539   -0.0077489    0.0073001  -0.00091143   0.00032318   -0.0065745   -0.0035286     0.003957   0.00028073       0.0151   -0.0088887  -0.00015875    0.0029767   -0.0097911    -0.011541   -0.0058137    0.0034335     0.011117   0.00045368    -0.009729   -0.0067275    0.0090379    0.0028352    0.0099965    0.0040127   0.00040091   -0.0021783   -0.0097911    -0.011541   -0.0058137    0.0034335     0.011098    0.0010523   -0.0096099   -0.0043041    0.0051888       0.0009     0.013863   -0.0038229  8.9657e-005    0.0012635 -0.011003  -0.0096163   -0.0048461    0.0015002     0.014537   0.00035216    -0.011152    -0.007188    0.0097693    0.0032029    0.0092618    0.0040206    0.0005884   -0.0025633    -0.011003   -0.0096163   -0.0048461    0.0015002     0.014342     0.000869    -0.010619   -0.0055158    0.0071134    0.0018676      0.01193  8.9107e-005  -0.00070175  -0.00060388 -0.011003  -0.0096163   -0.0048461    0.0015002     0.014591    0.0014347    -0.011022   -0.0038679    0.0044959   0.00055166     0.014559    -0.003822   0.00025881    0.0017894    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016733    0.0014591    -0.011545   -0.0076242     0.010462    0.0035513    0.0085658    0.0033657    0.0017022   -0.0021194    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016631    0.0014314    -0.011354   -0.0065094    0.0086915    0.0026611     0.010344    0.0030594  -0.00063799   -0.0029699    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016621   0.00044098    -0.011493   -0.0045222    0.0055352    0.0010742     0.013515   -0.0028964   -0.0009389    0.0017509    -0.012215   -0.0076917   -0.0038785  -0.00043309     0.016676   0.00043757    -0.011595   -0.0034316    0.0038031   0.00020332     0.015255   -0.0031524    0.0014629    0.0018065    -0.012988   -0.0064638   -0.0032611   -0.0016665       0.0163    0.0036404   -0.0098054   -0.0071395    0.0096923    0.0031642    0.0093392    0.0036554    0.0003512   -0.0032446    -0.012988   -0.0064638   -0.0032611   -0.0016665     0.016242   -0.0019621    -0.010752   -0.0038921    0.0045344   0.00057101      0.01452   -0.0034357   3.967e-005    0.0023552 -0.014616  -0.0038772   -0.0019607   -0.0042649    0.0050086      0.01472    0.0038994   -0.0084967     0.011848     0.004248    0.0071739   -0.0038492     0.007872    0.0048045    -0.014616   -0.0038772   -0.0019607   -0.0042649    0.0048501     -0.01516   -0.0019165   -0.0025592    0.0024174  -0.00049337     0.016647    0.0040403     0.008784   -0.0022205    -0.016623  -0.00069005  -0.00035828   -0.0074664 -1.7925e-005    0.0063344     0.015577 -0.012859    0.018776    0.0077314   0.00021403    -0.026786    -0.046864    -0.055085    -0.016623  -0.00069005  -0.00035828   -0.0074664  -0.00066765    0.0044429     0.011469     -0.01029     0.014696      0.00568    0.0043126    -0.010806   -0.0066796   -0.0028508    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0010076    0.0035273    0.0095149   -0.0092237     0.013003    0.0048285    0.0060139   -0.0064944    0.0032482    0.0044614    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0012259    0.0028835    0.0082753   -0.0084482     0.011771    0.0042093    0.0072512   -0.0039643    0.0092514    0.0047445    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0015742    0.0018451    0.0073598   -0.0074788     0.010231    0.0034352    0.0087978   -0.0013382     0.014987    0.0044037    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0019923   0.00059961    0.0069186   -0.0062913    0.0083451    0.0024869     0.010692  -0.00020806     0.019347    0.0042055    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0019997   -0.0011384    0.0065969   -0.0047645    0.0059201    0.0012677     0.013128   0.00055104     0.019693    0.0025555    -0.016623  -0.00069005  -0.00035828   -0.0074664    -0.001584    -0.002481    0.0065811   -0.0035528    0.0039955   0.00030008     0.015062    0.0016607      0.01567   0.00083891    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0012435   -0.0036022    0.0071142   -0.0026077    0.0024944  -0.00045466      0.01657    0.0041473      0.01023   -0.0015388    -0.016623  -0.00069005  -0.00035828   -0.0074664   -0.0010142   -0.0043633     0.008214   -0.0018321    0.0012626   -0.0010739     0.017807     0.006589    0.0042216   -0.0034572    -0.016623  -0.00069005  -0.00035828   -0.0074664  -0.00064793    -0.005506     0.010106  -0.00074159   -0.0004695   -0.0019448     0.019547     0.010882   -0.0063904     0.001122    -0.016623  -0.00069005  -0.00035828   -0.0074664   0.00015962   -0.0080805     0.014385     0.001803   -0.0045111   -0.0039768     0.023607     0.027895    -0.053301     0.047945     -0.01718   0.00019526  8.6827e-005   -0.0083557   -0.0032142    -0.001658    0.0057069 -0.01092    0.015697    0.0061832    0.0033073    -0.013746     -0.01368   -0.0095212     -0.01718   0.00019526  8.6827e-005   -0.0083557   -0.0032288      0.00119    0.0056899 -0.00013572  -0.0014318   -0.0024286     0.020514     0.013702    -0.013691     0.005942    -0.017483   0.00067641   0.00032873    -0.008839   -0.0042235   -0.0039517    0.0046073   -0.0098538     0.014003    0.0053317    0.0050086   -0.0089283   -0.0023228   0.00075412    -0.017483   0.00067641   0.00032873    -0.008839   -0.0041914    0.0034874    0.0054104   -0.0011778   0.00022335   -0.0015965     0.018851    0.0090606   -0.0018008   -0.0012768    -0.017665   0.00096509   0.00047387    -0.009129   -0.0046243   -0.0048842    0.0049671   -0.0092237     0.013003    0.0048285    0.0060139   -0.0066272    0.0030477    0.0034716    -0.017665   0.00096509   0.00047387    -0.009129   -0.0041038   -0.0036227    0.0063496    -0.008012     0.011078    0.0038609    0.0079472   -0.0026624     0.012219    0.0047374    -0.017665   0.00096509   0.00047387    -0.009129   -0.0035153    -0.002223    0.0069672   -0.0068487    0.0092304     0.002932    0.0098031  -0.00061368     0.017688    0.0044436    -0.017665   0.00096509   0.00047387    -0.009129   -0.0032428  -0.00042462    0.0074362   -0.0055158    0.0071134    0.0018676      0.01193   0.00018008     0.020205    0.0034991    -0.017665   0.00096509   0.00047387    -0.009129   -0.0034892    0.0014233    0.0075404   -0.0041829    0.0049963   0.00080324     0.014056   0.00095992     0.018184    0.0017389    -0.017665   0.00096509   0.00047387    -0.009129   -0.0040526    0.0029032    0.0073444   -0.0030439    0.0031872  -0.00010632     0.015874    0.0028869     0.013143  -0.00040913    -0.017665   0.00096509   0.00047387    -0.009129   -0.0045661    0.0043338     0.006219   -0.0018321    0.0012626   -0.0010739     0.017807    0.0067302    0.0039641   -0.0024242    -0.020088    0.0048143    0.0024091    -0.012996    -0.012773    -0.023915    0.0046152   -0.0092237     0.013003    0.0048285    0.0060139    -0.006803    0.0026545    0.0036709    -0.020088    0.0048143    0.0024091    -0.012996    -0.012464     0.023128     0.013457   -0.0018321    0.0012626   -0.0010739     0.017807    0.0069032    0.0035928   -0.0028846

Columns 8 through 14

0.0066546    0.018818   -0.0072996     0.021862     0.015029    -0.015567    -0.013693    -0.014606     -0.01366     0.014578    -0.018684     -0.01909     0.026004      0.02282    -0.007662     0.017069    0.0086158     0.024413     0.014404     0.016827      0.02152    -0.015646    0.0099807     0.015714     0.023035     0.017577     0.022624     0.023672   0.00057936    0.0052153   -0.0013058  -0.00028167   -0.0042208   -0.0033513   -0.0055633    -0.012112   -0.0078425     0.011319    -0.008877     -0.01045     0.016385     0.013797  -5.195e-006    0.0062641  5.8781e-005    0.0018738   -0.0032159    0.0019347    0.0019017 -0.0068736 -0.00084155    0.0073392    0.0011035  -0.00035545     0.013768     0.014389  -0.00056303    0.0047573    0.0013334 -4.0125e-006   -0.0043484    0.0065872    0.0073062 -0.012905   0.0050606     0.012051     0.012486    0.0096137     0.014271     0.011743   0.00045534   -0.0076039  -0.00051373     -0.01946    -0.017665    0.0013262   -0.0011838    -0.011249   0.00072992    0.0090261    0.0041759   -0.0011735    0.0074295    0.0048084   0.00031187   -0.0071523  -0.00022058    -0.017611    -0.016037    0.0015853  -0.00048388    -0.010646  -0.00080776    0.0077078    0.0014474  1.6662e-006    0.0018068   -0.0037741 0.00015335  -0.0078603   0.00012241    -0.020033    -0.018305    0.0027887  -0.00053652    -0.012009    -0.002967    0.0099424   -0.0011645   0.00070286    0.0072079    0.0011141   0.00016652   -0.0049453  -0.00037902   -0.0042007   -0.0036596    -0.002754   0.00075953   -0.0098021    0.0042999    0.0059142    0.0055826  -0.00061762  -0.00063858   -0.0050555    0.0012193   -0.0047806  -0.00090763   -0.0042104   -0.0035975   -0.0026283  -0.00019841    -0.010273   -0.0056355    0.0064372   -0.0031484   0.00083897   -0.0027709   -0.0073692   0.00020412    0.0040957  -0.00099802    0.0073957  -0.00031887   -0.0037235  -0.00036719   -0.0083677     0.010547    0.0035844    0.0099322   -0.0013981     -0.00368   -0.0084516   0.00090541    0.0024708   -0.0008352    0.0048302  -0.00022073   -0.0018631   0.00089764    -0.010716  -0.00061777    0.0049711    0.0010323   0.00016098   -0.0068412   -0.0094633    0.0018121    0.0041723   -0.0010859    0.0074942  -0.00032555   0.00064445    0.0041615    -0.009112    -0.011122    0.0044496   -0.0078457    0.0019855   -0.0086869    -0.012305    0.0015331     0.012465   -0.0007433     0.015077    0.0015499    0.0003494  -0.00090943   -0.0057456       0.0148    0.0016457     0.010274   -0.0026705   -0.0032403    -0.010062    0.0015559      0.01156  -0.00076085     0.013719    0.0014912    0.0004219   0.00010342   -0.0099528    0.0079996     0.003304    0.0089868   -0.0024377   -0.0077263   -0.0097806   0.00056878     0.011285   -0.0011587     0.013286    0.0014097   -0.0009701    0.0021191    -0.010692   -0.0087847    0.0044835   -0.0069132    0.0029417     -0.01184    -0.015201   0.00061577     0.011764   -0.0012315     0.013934    0.0014093  -0.00070067    0.0028758   -0.0063287    -0.014351    0.0021133   -0.0082928    0.0035201   -0.0098438    -0.011506    0.0046923     0.015608  -0.00088502     0.013999    0.0030114    0.0050119   -0.0016117   -0.0077337     0.013114    0.0017569     0.011323   -0.0035634   -0.0061465     -0.01006   -0.0029975     0.014531 -3.2679e-005     0.013237    0.0026879   -0.0043136   -0.0011128 -0.0085031    -0.01298    0.0028626   -0.0090424    0.0042917    -0.012174    -0.014322     0.022278     0.013709   -0.0036124    -0.014494    0.0087426     0.020364   -0.0045793    0.0063637     0.015648    -0.003688   -0.0075924    -0.003578    0.0028635   -0.0095753    -0.024356     0.011616    0.0099995    -0.010778    0.0070027    -0.020287    -0.023433    0.0076003    -0.014085   -0.0069026    0.0056614    0.0055495    -0.004477    0.0014002   -0.0045508 -5.7031e-005   -0.0096163   -0.0060887    0.0067607    0.0038396     0.016364 0.038374   -0.010291     0.016216   -0.0023307     0.017948     0.016688     -0.01344  -0.00035847   -0.0020919   -0.0047725   -0.0062583     0.011705     0.012888      0.00285    0.0023535     0.013259    -0.014923   -0.0039882    -0.016594    -0.024673     0.027007    0.0014884   -0.0023001   -0.0034625   -0.0062617     0.011893     0.011306    0.0031097    0.0024398     0.014651    -0.010587   -0.0061214   -0.0097526   -0.0033283   -0.0021663    0.0024289   -0.0027026   -0.0019896   -0.0060187     0.011988    0.0083103    0.0046988    0.0094551     0.021195   -0.0062669    -0.012177   -0.0069277    0.0023437    -0.010074    0.0020794   -0.0050287   -0.0019391   -0.0036606     0.011002    0.0058826    0.0043496     0.016292     0.022583   -0.0029114    -0.014072   -0.0031642    0.0091915    -0.014098   0.00083095   -0.0075368  -0.00055325   -0.0018046     0.010501   0.00060967     0.004508     0.021099     0.011645   -0.0038619   -0.0082723  -0.00021821     0.016225    -0.014995   -0.0012132   -0.0076824    0.0018541   -0.0013014    0.0098783    -0.006317    0.0025992     0.021881   -0.0078466   -0.0063384    0.0034517    0.0029722       0.0111   -0.0034253   -0.0027469    -0.005289    0.0039517   -0.0025361    0.0095021    -0.010008   -0.0016801     0.017834    -0.019778   -0.0080751     0.010463    0.0053227   -0.0010602    0.0067537   -0.0034107    -0.003034    0.0047744   -0.0044908    0.0098783    -0.010375   -0.0046068     0.010829     -0.01892    -0.010311    0.0088316      0.00942   -0.0089837    0.0054424   -0.0026886   -0.0025977    0.0062134   -0.0043238    0.0088284    -0.010947   -0.0058995    0.0030455    -0.012722    -0.011821    0.0030047     0.012374    -0.010863   0.00050371   -0.0011435   -0.0023285     0.007443    -0.004095    0.0078891     -0.01078   -0.0064964    0.0011195    -0.010779    -0.013715   0.00069984     0.016164    -0.018132  -0.00037073    0.0024649   -0.0012776     0.010633   -0.0033872    0.0044267   -0.0074837   -0.0070775     0.027962    0.0048221     0.007436    0.0030068     -0.01068    0.0069168    0.0013799  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Columns 15 through 21

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Columns 22 through 28

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 0.0010764   -0.0031652   -0.0054849    0.0022046    0.0010433     0.014098     0.001295    0.0010855   0.00079062  5.8421e-005   0.00016889    0.0014714  -0.00083413    0.0035194 0.0010104  -0.0012874    -0.002365   0.00065713   0.00041122   0.00090464   0.00098478   -0.0028464   -0.0049751   -0.0039497    0.0027504    0.0066968    0.0061394    0.0090195    0.0019899    0.0006759  -0.00022316  -0.00075254   0.00021565     -0.01119   0.00015844    -0.010125   -0.0036472    0.0050856   0.00048529  -0.00061744      0.00163    0.0039188    0.0042837    0.0022247   0.00015979   -0.0015046    0.0023785    -0.021044    0.0016533   -0.0046442    0.0024157    0.0070413   -0.0027031   -0.0063361    -0.010735    -0.009079    0.0041744    0.0011245  2.1244e-005  -0.00082335    0.0064522    -0.022428    0.0078248 0.022855   0.0048396    -0.017277    0.0025285     0.012019     0.003445   -0.0011898   -0.0025448 -5.3475e-005    0.0060605   -0.0015068    0.0051485    -0.022247    0.0074618 0.02653 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Columns 29 through 35

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Columns 36 through 42

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Columns 43 through 49

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Columns 50 through 56

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Columns 57 through 63

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Columns 64 through 70

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Columns 71 through 77

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Columns 85 through 91

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Column 92

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D =

Columns 1 through 7

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Columns 8 through 14

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Columns 15 through 21

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Columns 22 through 28

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Columns 29 through 35

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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Columns 36 through 42

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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Columns 43 through 49

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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Columns 50 through 56

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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Columns 57 through 63

0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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Columns 64 through 70

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0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            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6.2484e+005            0            0            0 0           0            0            0  6.4629e+005            0            0 0           0            0            0            0  6.7059e+005            0 0           0            0            0            0            0  6.9386e+005 0           0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0            0           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Columns 71 through 77

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Columns 78 through 84

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Columns 85 through 91

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Column 92

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min_evec1 =

-0.0014944  -0.0094129   -0.0026749   -0.0020615   -0.0049418   -0.0083028   -0.0049418   -0.0049431   -0.0049418   -0.0027531   -0.0073676   -0.0075273   -0.0073676   -0.0055158   -0.0073676   -0.0035286   -0.0097911   -0.0067275   -0.0097911   -0.0043041    -0.011003    -0.007188    -0.011003   -0.0055158    -0.011003   -0.0038679    -0.012215   -0.0076242    -0.012215   -0.0065094    -0.012215   -0.0045222    -0.012215   -0.0034316    -0.012988   -0.0071395    -0.012988   -0.0038921    -0.014616   -0.0084967    -0.014616   -0.0025592    -0.016623    -0.012859    -0.016623     -0.01029    -0.016623   -0.0092237    -0.016623   -0.0084482    -0.016623   -0.0074788    -0.016623   -0.0062913    -0.016623   -0.0047645    -0.016623   -0.0035528    -0.016623   -0.0026077    -0.016623   -0.0018321    -0.016623  -0.00074159    -0.016623     0.001803     -0.01718     -0.01092     -0.01718  -0.00013572    -0.017483   -0.0098538    -0.017483   -0.0011778    -0.017665   -0.0092237    -0.017665    -0.008012    -0.017665   -0.0068487    -0.017665   -0.0055158    -0.017665   -0.0041829    -0.017665   -0.0030439    -0.017665   -0.0018321    -0.020088   -0.0092237    -0.020088   -0.0018321

min_evec2 =

-0.05031    0.021545    0.0029728     0.009503    -0.019243      0.01154    -0.019243    -0.018737    -0.019243    0.0027253     -0.01539     0.010308     -0.01539    0.0071134     -0.01539     0.003957    -0.011541    0.0090379    -0.011541    0.0051888   -0.0096163    0.0097693   -0.0096163    0.0071134   -0.0096163    0.0044959   -0.0076917     0.010462   -0.0076917    0.0086915   -0.0076917    0.0055352   -0.0076917    0.0038031   -0.0064638    0.0096923   -0.0064638    0.0045344   -0.0038772     0.011848   -0.0038772    0.0024174  -0.00069005     0.018776  -0.00069005     0.014696  -0.00069005     0.013003  -0.00069005     0.011771  -0.00069005     0.010231  -0.00069005    0.0083451  -0.00069005    0.0059201  -0.00069005    0.0039955  -0.00069005    0.0024944  -0.00069005    0.0012626  -0.00069005   -0.0004695  -0.00069005   -0.0045111   0.00019526     0.015697   0.00019526   -0.0014318   0.00067641     0.014003   0.00067641   0.00022335   0.00096509     0.013003   0.00096509     0.011078   0.00096509    0.0092304   0.00096509    0.0071134   0.00096509    0.0049963   0.00096509    0.0031872   0.00096509    0.0012626    0.0048143     0.013003    0.0048143    0.0012626

min_evec3 =

0.038896   -0.011552    -0.063282     -0.01669   -0.0096861    0.0040931   -0.0096861      0.05144   -0.0096861  -0.00033855   -0.0077489    0.0034739   -0.0077489    0.0018676   -0.0077489   0.00028073   -0.0058137    0.0028352   -0.0058137       0.0009   -0.0048461    0.0032029   -0.0048461    0.0018676   -0.0048461   0.00055166   -0.0038785    0.0035513   -0.0038785    0.0026611   -0.0038785    0.0010742   -0.0038785   0.00020332   -0.0032611    0.0031642   -0.0032611   0.00057101   -0.0019607     0.004248   -0.0019607  -0.00049337  -0.00035828    0.0077314  -0.00035828      0.00568  -0.00035828    0.0048285  -0.00035828    0.0042093  -0.00035828    0.0034352  -0.00035828    0.0024869  -0.00035828    0.0012677  -0.00035828   0.00030008  -0.00035828  -0.00045466  -0.00035828   -0.0010739  -0.00035828   -0.0019448  -0.00035828   -0.0039768  8.6827e-005 0.0061832 8.6827e-005 -0.0024286  0.00032873    0.0053317   0.00032873   -0.0015965   0.00047387    0.0048285   0.00047387    0.0038609   0.00047387     0.002932   0.00047387    0.0018676   0.00047387   0.00080324   0.00047387  -0.00010632   0.00047387   -0.0010739    0.0024091    0.0048285    0.0024091   -0.0010739