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.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?