Second order ODE solution for this system?

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Discussion Overview

The discussion revolves around finding the analytical solution to a second-order ordinary differential equation (ODE) related to a mechanical system, specifically questioning whether it can be modeled as a mass-spring-damper system. The inquiry also extends to the implications for angular motion.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asks for the analytical solution to a second-order ODE and suggests the possibility of modeling it as a mass-spring-damper system.
  • Another participant provides the general form of the differential equation and outlines the characteristic equation, including the use of the quadratic formula to find solutions.
  • A later reply questions how to adapt the solution for angular motion and seeks modifications to the original equation.
  • There is a suggestion to replace the variable "x" with "theta" to represent angular motion.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the solution for angular motion, and the discussion remains unresolved regarding the modifications needed for the ODE.

Contextual Notes

Participants have not specified assumptions regarding linearity or the nature of the system, and the discussion lacks details on the specific parameters of the ODE.

karamustafa
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second order ODE solution for this system??

hello guys,
I am wondering if what is the analytical solution for this system?
can we solve it as a mass-spring-damper system?
thanks for your helps.
the rectangular part is removed from the disk.

[URL=http://img3.imageshack.us/my.php?image=odev.jpg][PLAIN]http://img3.imageshack.us/img3/3610/odev.th.jpg[/URL][/PLAIN]
 

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So the DE is

ax'' + bx' + cx = 0

Write the characteristic equation...

an^2 + bn + c = 0

Solve for n using the quadratic formula...

n = [-b +- sqrt(b^2 - 4ac)] / 2a

This will give you two (possibly non-unique) exponents. if the exponents are different, say n1 and n2, then the solution is

x(t) = Aexp(n1 t) + Bexp(n2 t)

If the exponents are the same, then

x(t) = Aexp(n t) + B t exp(n t)

Am I missing something, or does this answer your question?
 


thanks a lot, that is the answer if the motion is linear, how about the angular motion?
how can i modify this equation.??
 


To make it angular, rewrite it using "theta" instead of "x".
 

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