Depicted above is a network of three strings called a tritar. We are interested in how the eigenvalues of this simple
network vary with changes in the transverse stiffness
k
_{i} of each string. We assume that the longitudinal stiffnesses
σ
_{i} are 1 for each string, and we also assume that the lengths of the strings are all 1 for conveninence.
Letting the vector
${u}_{i}={\left[{u}_{i1}\phantom{\rule{0.222222em}{0ex}}{u}_{i2}\right]}^{T}$ represent the displacements of string
i , we obtain the following system of
differential equations:
As with many systems of differential equations, this one can be solved via the time-honored method of guessing. Noting that
the differential equations of this form equate the second derivative of a function with a constant multiple of itself, wehypothesize that the solution for each component of displacement is some linear combination of sines and cosines:
and have used the above to translate these into
u
_{21} ,
u
_{22} ,
u
_{31} , and
u
_{32} . We need to determine the
coefficients. Applying the boundary condition that
${u}_{1}\left(0\right)=0$ , we get
and by substituting in the desired value of
k and setting this determiniant to zero we can then solve for the eigenvalues
λ of our tritar net. A plot of the first seven eigenvalues as a function of
k is displayed below:
As expected, the eigenvalues increase as
k increases; however for eigenvalues beyond the first, we observe some rather
strange behavior, which suggests that something has gone wrong in the above process. The eigenvalues in the above plot werecomputed using MATLAB's fzero() function at a tolerance of 1e-10. Using a more naive bisection method (which is less likely
to lock onto the wrong root) at the same tolerance, we obtain the following plot:
This seems to have fixed some of the erratic behavior, but neither tightening the tolerance nor increasing the fine-ness of
the mesh along which the determinant is evaluated provides much further improvement. On the other hand, it is apparent thatthe solver's structure and parameters impacts the shape of the plots. Perhaps a better solver of some sort (e.g. Newton's
method, but adapted to search only in a given interval) can fix more of the problem.
Example #2: the quintar
Rather than develop all of the mathematical relations as in the previous example, it should suffice to say that the same procedure is followed. The solutions to the differential equations are still sums of sines and cosines, but more equations have been added to the system. The function obtained by setting the determinant equal to zero is not enlightening and longer than that for the quintar, and so only the final plots will be presented here. By iteratively increasing the angle at the ends of the network, a plot of the angle versus the eigenvalues is obtained in which traces are formed as the eigenvalues change. It is interesting to note how some of the eigenvalues increase in magnitude while others decrease.
A second plot is presented in which the first nine eigenvalues are plotted versus the transverse stiffness parameter for our analytic model. A region of stiffnesses was chosen where the root-finding algorithm is able to successfully lock onto the zeroes of the determinant as they change. It is clear from this plot that for this range of stiffnesses, increasing stiffness results in the modes of vibration increasing in frequency. This result is expected, since in general stiffer members possess higher vibrational frequencies.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.