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
, i.e., find the values for the parameters k, c, m.
Recall the equation of motion for the spring dashpot mass system:
![{\displaystyle m(y''_{k}+{\frac {k}{c}}y'_{k})+ky_{k}=f(t)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c076c354d253a192b0749638ea75d729d1a83727)
Dividing the entire equation by m:
![{\displaystyle y''_{k}+{\frac {k}{cm}}y'_{k}+{\frac {k}{m}}y_{k}=f(t)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/472513072c687d7e9dac2a790a5107c1cbcc1a29)
The characteristic equation for the double root :
is:
![{\displaystyle (\lambda +3)^{2}=\lambda ^{2}+6\lambda +9=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73bac01c5c7974a6e5c21791e42b6bdd466d5891)
The corresponding L2-ODE-CC (with excitation) is:
![{\displaystyle y''+6y'+9=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c57ddcc70bcf25432f65f1344f5c871fdb849bac)
Matching the coefficients:
![{\displaystyle y''\Rightarrow 1=1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08017a9a778d4e757c7e4c1f3d47930bf9afe3eb)
![{\displaystyle y'\Rightarrow {\frac {k}{cm}}=6\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24498ac21c29108ddc3d69a435d530d725cabb94)
![{\displaystyle y\Rightarrow {\frac {k}{m}}=9\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92b6df15b15466b9c9f5ba935f9dbd25ed25e039)
After algebraic manipulation it is found that the following are the possible values for k, c, and m:
![{\displaystyle k=18\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0dadbcca8f933e5c02f3c95f8810e81a9fa0679)
![{\displaystyle c={\frac {3}{2}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03575c3e2c81f18efc1c01decb677d4cedc7376c)
![{\displaystyle m=2\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f31758d7fe392cb22147bb55b11d49f703f0f42c)
Solved and typed by - Egm4313.s12.team4.Lorenzo 20:04, 6 February 2012 (UTC)
Reviewed By -
Edited by -
Develop the MacLaurin series (Taylor series at t=0) for:
![{\displaystyle e^{t}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe81434bcb23be7b6e89d6f38099a847db960be)
![{\displaystyle \cos t\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87155c5a49c1b2e762974ae5adcd44cc26527257)
![{\displaystyle \sin t\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/253b6801ee2dce33b0d01c3d9c0375a532e322f3)
Recalling Euler's Formula:
![{\displaystyle e^{i\omega x}=\cos \omega x+i\sin \omega x\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3876e7fdba345a4dc6cc2683298d38ab1c49526e)
Recall the Taylor Series for
at :
(also called the MacLaurin series)
![{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/860229906c54633d755aadd1917102016d0e6124)
By replacing x with t, the Taylor series for
can be found:
![{\displaystyle e^{t}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c10bdf060ee5516c0d082092f2e36f62b1eec4ef)
even powers:
![{\displaystyle i^{2k}=(i^{2})^{k}=(-1)^{k}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecc0dc5b56904763ed9b92012f1c491658c358e)
odd powers:
![{\displaystyle i^{2k+1}=(i^{2})^{k}i=(-1)^{k}i\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad12ed6e7110b18f60f6304c3dd08a0985f2183)
If we let
:
![{\displaystyle 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)!}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a68a5ed02ea5fc4cb336ad65d74457580493f34f)
Using the two previous equations:
![{\displaystyle e^{it}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}t^{2k}}{(2k)!}}+\sum _{k=0}^{\infty }{\frac {(-1)^{k}t^{2k+1}}{(2k+1)!}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b589cb89fa7a40842c3bae7f79fed6607506b9d6)
![{\displaystyle \Rightarrow e^{it}=\cos t+i\sin t\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94baefc5d25157e8e1f58ee8846445c62bcd3a11)
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:
![{\displaystyle \cos t=\sum _{k=0}^{\infty }{\frac {(-1)^{k}t^{2k}}{(2k)!}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9cd3dc0d5039fed1a4a63c38af28fdb0e58f21)
![{\displaystyle \sin t=\sum _{k=0}^{\infty }{\frac {(-1)^{k}t^{2k+1}}{(2k+1)!}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/774e782fb8a9206694373647f6c116241430eac6)
Solved and typed by - Egm4313.s12.team4.Lorenzo 20:05, 6 February 2012 (UTC)
Reviewed By -
Edited by -