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

Problem 6

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Problem Statement

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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.

Solution

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Recall the equation of motion for the spring dashpot mass system:

 



Dividing the entire equation by m:

 



The characteristic equation for the double root :  is:

 



The corresponding L2-ODE-CC (with excitation) is:

 



Matching the coefficients:

 



 



 



After algebraic manipulation it is found that the following are the possible values for k, c, and m:

 



 



 


Author

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Solved and typed by - Egm4313.s12.team4.Lorenzo 20:04, 6 February 2012 (UTC)
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Problem 7

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Problem Statement

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Develop the MacLaurin series (Taylor series at t=0) for:

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  •  
  •  

Solution

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Recalling Euler's Formula:

 



Recall the Taylor Series for   at :  (also called the MacLaurin series)

 



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

 



even powers:

 


odd powers:

 



If we let  :

 



Using the two previous equations:

 



 



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

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




References

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