Second order ODE solution for this system?

In summary, the conversation discusses finding the analytical solution for a second order ODE system and the use of the mass-spring-damper system to solve it. The characteristic equation is written and the quadratic formula is used to solve for the exponents, which determine the solution. The possibility of different or same exponents is mentioned and the equation is modified for angular motion by using "theta" instead of "x".
  • #1
karamustafa
2
0
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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  • #2


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?
 
  • #3


thanks a lot, that is the answer if the motion is linear, how about the angular motion?
how can i modify this equation.??
 
  • #4


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

Related to Second order ODE solution for this system?

1. What is a second order ODE?

A second order ordinary differential equation (ODE) is a mathematical equation that relates the values of an unknown function to its first and second derivatives. It is commonly used to model physical phenomena in fields such as physics, engineering, and biology.

2. How do you solve a second order ODE?

The general solution to a second order ODE is a linear combination of two linearly independent solutions. These solutions can be found by using various methods such as the substitution method, variation of parameters, or the Laplace transform method. The specific method used depends on the form of the ODE and the initial conditions.

3. What is the difference between a first and second order ODE?

A first order ODE involves only the first derivative of the unknown function, while a second order ODE involves both the first and second derivatives. This means that a second order ODE requires two initial conditions to solve, while a first order ODE only requires one.

4. How is a second order ODE used in science?

Second order ODEs are used in various scientific fields to model and understand physical phenomena. For example, in physics, they are used to describe the motion of objects under the influence of forces, and in engineering, they are used to design and analyze systems such as circuits and control systems.

5. Can a second order ODE have more than one solution?

Yes, a second order ODE can have an infinite number of solutions depending on the initial conditions and the method used to solve it. However, for a specific set of initial conditions, there will only be one unique solution.

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