Question

Problem 4. Consider the spring-mass system shown in the figure. The displacement of the mass m as a function of time is as follows: x = Xocoswt) + cos(Wnt) ωη where xo is the initial displacement equals to 0.1 m, čo is the initial velocity equals to 1 m/s, and Wr is the natural frequency of the system equals to 4 rad/s. Calculate the acceleration (second time derivative of displacement) of the mass after 1 s with a time step size of 0.1 s and 0.2 s using the centered finite-divided difference. Determine the true percent relative error and compare the remainder term of the Taylor series expansion for both time steps. Comment on the accuracy of your results.

Answer 1

Given data

the displacement of the mass m,

as a function of time is as follows
Answer sporing - man system show in figure. wn n = no cos(watt no cos (onth for not onl, nool, what this canals. = 0.1Cos Cut) + 0.25 Cos Cut) = 0.35 Coscut That is, 1. 35 Cosutis the olisplacement at any point of time The Ituves velocity is a' (t): ocity is alt)=-1.ulincut and Lion is ā (H) = -6.6Coscut) acceleration divided difference wing second-order central to cire a the approximation to the sire the thod with step second onder derivatine is. (lih-ralt) talt-b). let)

ton tal ho. I we have. , Itu (11) O103 5665, 21 210.9 0 313865a1t) = -022995 -0.10 75665-0.313865 -2.(-0. 2.287715 a" (1) a. - (0.12 ? 3. 618s And so That is. a (1) The actual alllevation and so the error is "L) = 7.6604ou is l. 6185-3.660404 - - 0.048 554. 3.6 11 86-7.bloudy 3.660404 and percent error is! - r.326016 similarly, being tao. 2 we have. À (1.2). 21 (1) th10.8) 3.4693 2" (1) 2 - 10.2g? arrow is 3. u693- 3. 6604 2.0 1911 u absolute Harm this canals. in percentage 34693-3-6604 - 1-224. 3-6604

Thus we find that a larger step in slatine error. leads to larger 544 Lo The Ninainder toris alt) = when het)=0-35 Coscut) = 1lt i o. 35 x 4 89.6 Thus the error terre is bounded 89.6.b 12 - 89.6.20 10 ~ ano o.ooots in the bound ton. b=0.1 y 89.6.20 10 12 89.6.20.2 tor h= 0.2 => 89 0.01191 In the bound. 12 The actual error terus in our case cave much. larger than thil Nenainder terms bounde. > the results have a high degree of error margin especially in the case b=0.2. where the erron terus goes as large as sy o .