1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Modeling mass-spring-dashpot system

  1. Apr 14, 2015 #1
    1. The problem statement, all variables and given/known data
    Suppose that the mass in a mass-spring-dashpot system with m = 25, c = 10, and k = 226 is set in motion with x(0) = 2 and x'(0) = 4. Find the smallest time t>0 for which x(t) = 0

    2. Relevant equations
    mx'' + cx' + kx = 0

    3. The attempt at a solution
    I rushed through this a bit due to lack of time today so there may be mistakes, I just need to be sure I'm doing this correctly.

    The differential equation is: 25x'' + 10x' + 226x = 0
    Which has the characteristic equation: 25r2 + 10r + 226 = 0
    with roots: r = -(1/5) +/- 3i

    Giving the general solution:
    x(t) = e-(1/5)t(Acos(3t) + Bsin(3t))

    Plugging in the initial conditions:
    x(0) = 2 = A

    x'(t) = -(1/5)Ae-(1/5)tcos(3t) -3A e-(1/5)tsin(3t) - (1/5)e-(1/5)tsin(3t) + 3Be-(1/5)tcos(3t)

    x'(0) = 4 = -(1/5)(2) + 3B
    B = 22/15

    Finding combined solution:
    x(t) = Ce-(1/5)tcos(3t-a)

    C = (A2 + B2)1/2 = 3(346)1/2/15

    a = tan-1(B/A) = 0.6327 radians

    x(t) = (3(346)1/2/15)e-(1/5)tcos(3t - 0.6327)

    If this is right (it seems too messy, might have messed up), how would I solve it when equal to zero?
     
  2. jcsd
  3. Apr 15, 2015 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    For which you find A and B correctly A correctly.The first factor is > 0, so basically all you want to solve is
    Acos(3t) + Bsin(3t) = 0​
    And that's not so hard.

    And if you want the other form, same thing: solving cos(3t - atan(B/A)) = 0 should be enough -- and give the same result :rolleyes:

    Something's wrong with your general solution, though, so fix that first !!
     
    Last edited: Apr 15, 2015
  4. Apr 15, 2015 #3
    I checked again, and with Wolfram, and I did get A and B right. C should be 2(346)^0.5 /15 though. When I solve cos(3t - 0.6327) = 0, I get t = 0.7345 which isn't right. This answer x2 is right though, and I'm not sure why.

    Edit: misunderstanding, my answer is right.
     
    Last edited: Apr 15, 2015
  5. Apr 16, 2015 #4

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    Yeah, B was 62/45. Well done!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted