University of Florida/Egm4313/s12.team4.Lorenzo/R2

Problem Statement
Realize spring-dashpot-mass systems in series as shown in Fig. p.1-4 with the similar characteristic as in (3) p.5-5, but with double real root $$\lambda=-3$$, i.e., find the values for the parameters k, c, m.

Solution
Recall the equation of motion for the spring dashpot mass system:
 * $$m(y''_k + \frac{k}{c}y'_k)+ky_k=f(t)

\!$$ Dividing the entire equation by m:
 * $$y''_k + \frac{k}{cm}y'_k+ \frac{k}{m}y_k=f(t)

\!$$ The characteristic equation for the double root :$$\lambda=-3\!$$ is:
 * $$ (\lambda+3)^2 = \lambda^2+6\lambda+9=0

\!$$ The corresponding L2-ODE-CC (with excitation) is:
 * $$y''+6y'+9=0\!$$

Matching the coefficients:
 * $$y''\Rightarrow 1=1\!$$


 * $$y'\Rightarrow \frac{k}{cm}=6\!$$


 * $$y\Rightarrow \frac{k}{m}=9\!$$

After algebraic manipulation it is found that the following are the possible values for k, c, and m:
 * k=18\!




 * m=2\!

Author
Solved and typed by - Egm4313.s12.team4.Lorenzo 20:04, 6 February 2012 (UTC) Reviewed By - Edited by -

Problem Statement
Develop the MacLaurin series (Taylor series at t=0) for:
 * $$e^t\!$$
 * $$\cos t\! $$
 * $$\sin t\!$$

Solution
Recalling Euler's Formula:
 * $$e^{i\omega x}=\cos\omega x+i\sin\omega x\!$$

Recall the Taylor Series for $$e^x$$ at :$$x=0$$ (also called the MacLaurin series)
 * $$e^{x}=\sum_{n=0}^{\infty}\frac{x^n}{n!}\!$$

By replacing x with t, the Taylor series for $$e^t$$ can be found:

even powers:
 * $$i^{2k}=(i^2)^k=(-1)^k\!$$

odd powers:
 * $$i^{2k+1}=(i^2)^k i=(-1)^k i\!$$

If we let $$x=it$$:
 * $$e^{it}=\sum_{n=0}^{\infty}\frac{i^n t^n}{n!}=\sum_{k=0}^{\infty}\frac{i^{2k} t^{2k}}{(2k)!}+\sum_{k=0}^{\infty}\frac{i^{2k+1} t^{2k+1}}{(2k+1)!}\!$$

Using the two previous equations:
 * $$e^{it}=\sum_{k=0}^{\infty}\frac{(-1)^{k} t^{2k}}{(2k)!}+\sum_{k=0}^{\infty}\frac{(-1)^{k} t^{2k+1}}{(2k+1)!}\!$$


 * $$\Rightarrow e^{it}=\cos t+i \sin t\!$$

Therefore, the first part of the equation is equal to the Taylor series for cosine, and the second part is equal to the Taylor series for sine as follows:



Author
Solved and typed by - Egm4313.s12.team4.Lorenzo 20:05, 6 February 2012 (UTC) Reviewed By - Edited by -