How Do You Model a Mass-Spring-Damper System for a Rotating Body?

[reckk]

Homework Statement

http://img515.imageshack.us/img515/2668/ohyxf6.jpg
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

Homework Equations

$$F_{D} =$$ $$b . a . \dot{\varphi}$$

$$F_{F} =$$ $$c .a . \varphi$$

The Attempt at a Solution

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

Solution 1:

$$J_{o}\ddot{\varphi} = -F_{F} . a - F_{D} . a$$

$$J_{o}\ddot{\varphi} + b . a^{2} . \dot{\varphi} + c . a^{2} . \varphi = 0$$

$$\ddot{\varphi} + \frac{b . a^{2}}{J_{o}} . \dot{\varphi} + \frac{c . a^{2}}{J_{o}} . \varphi = 0$$

$$with$$
$$2\delta = \frac{b . a^{2}}{J_{o}} ; \omega{o}^{2} = \frac{c . a^{2}}{J_{o}}$$

Solution 2:

$$m . a . \ddot{\varphi} = -F_{F} - F_{D}$$

$$m . a . \ddot{\varphi} + b . a \dot{\varphi} + c . a . \varphi = 0$$

$$\ddot {\varphi} + \frac{ba}{ma} \dot{\varphi} + \frac {ca}{ma}\varphi = 0$$

$$with$$
$$2\delta = \frac{b}{m} ; \omega_{o}^{2} = \frac {c}{m}$$

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

 

[kuruman]

It is not possible to decide without a heavy dose of guesswork which of the two equations is correct without a picture of the setup. Alas, the link to the figure is broken.