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Mass-spring-damper system

  1. Jan 9, 2008 #1
    1. The problem statement, all variables and given/known data

    [​IMG]
    The mass-spring-damper-system is consist of a rotating body (Jo), a flat spring (E, I), a damper (b) and a connecting rod. Only the mass of the rotating body is to be considered. It is assumed that there's only a small angular travels due to the oscillation/vibration.

    Find the differential equation φ(t) for the oscillation/vibration of the rotating body.

    Values given: Jo = 0.3 kg/m² ; b = 200 kg/s ; a = 25cm ; L = 20 cm

    2. Relevant equations
    [tex]F_{D} = [/tex] [tex]b . a . \dot{\varphi}[/tex]

    [tex]F_{F} = [/tex] [tex]c .a . \varphi[/tex]

    3. The attempt at a solution

    i have came up with two approaches.. but i don't know which one is correct

    Solution 1:

    [tex]J_{o}\ddot{\varphi} = -F_{F} . a - F_{D} . a[/tex]

    [tex]J_{o}\ddot{\varphi} + b . a^{2} . \dot{\varphi} + c . a^{2} . \varphi = 0[/tex]

    [tex]\ddot{\varphi} + \frac{b . a^{2}}{J_{o}} . \dot{\varphi} + \frac{c . a^{2}}{J_{o}} . \varphi = 0[/tex]

    [tex]with[/tex]
    [tex] 2\delta = \frac{b . a^{2}}{J_{o}} ; \omega{o}^{2} = \frac{c . a^{2}}{J_{o}}[/tex]

    Solution 2:

    [tex]m . a . \ddot{\varphi} = -F_{F} - F_{D} [/tex]

    [tex]m . a . \ddot{\varphi} + b . a \dot{\varphi} + c . a . \varphi = 0 [/tex]

    [tex]\ddot {\varphi} + \frac{ba}{ma} \dot{\varphi} + \frac {ca}{ma}\varphi = 0[/tex]

    [tex]with[/tex]
    [tex] 2\delta = \frac{b}{m} ; \omega_{o}^{2} = \frac {c}{m} [/tex]

    both would give different answers for calculating other unknowns.. so i wonder which one is correct ?
     
  2. jcsd
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