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

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Problem 6[edit]

Problem Statement[edit]

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[edit]

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[edit]

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




Problem 7[edit]

Problem Statement[edit]

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

Solution[edit]

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[edit]

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




References[edit]