Reducing Second Order ODE system to First Order

In summary, the conversation discusses the modeling of a 3-storey building as a system of coupled masses and springs, and how it can be represented by the equation Mx'' + Cx' + Kx=0. The goal is to rewrite this equation in the form x' = Ax, where A is in terms of matrices M, C, and K. The solution involves reducing a 3x3 matrix of second order ODEs to a 6x6 matrix of first order ODEs. The attempt at a solution involves writing a matrix equation, but the individual is uncertain on how to express A in terms of M, C, and K.
  • #1
makeez
4
0

Homework Statement


A 3-storey building can be modeled as a system of coupled masses and springs as showen in attached document. Where mi is the mass of each floor, ki is the spring constant, xi is the displacement of each floor, and ci is the damping coeffcient.

Homework Equations


I understand the equation can be written as:
Mx'' + Cx' + Kx=0

Where:
[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21M%20%3D%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7Bcc%7D%20m_1%20%26%200%20%26%200%20%20%5C%5C%200%20%26%20m_2%20%26%200%20%20%5C%5C%200%20%26%200%20%26%20m_3%20%20%5C%5C%20%5Cend%7Barray%7D%20%7D%20%5Cright%5D%2C%20C%20%3D%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7Bcc%7D%20c_1%20%26%200%20%26%200%20%20%5C%5C%200%20%26%20c_2%20%26%200%20%20%5C%5C%200%20%26%200%20%26%20c_3%20%20%5C%5C%20%5Cend%7Barray%7D%20%7D%20%5Cright%5D%2C%20K%20%3D%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7Bcc%7D%20k_1%20%26%20-k_1%20%26%200%20%20%5C%5C%20-k_1%20%26%20k_1%20%2B%20k_2%20%26%20-k_2%20%20%5C%5C%200%20%26%20-k_2%20%26%20k_2%20%2B%20k_3%20%20%5C%5C%20%5Cend%7Barray%7D%20%7D%20%5Cright%5D.gif

However the form x' = Ax is desired!
Where the matrix A is in terms of the matrices M, C, and K,

This is to be achieved by allowing x4=x'1, x5=x'2, x6=x'3. So reducing a 3x3 matrix of second order ODEs to a 6x6 matrix of first order ODEs.

The Attempt at a Solution


I wrote up the following matix equation based on intuition but I am unsure how to express A in terms of M, C and K.

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7Bcc%7D%201%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%200%20%26%201%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%200%20%26%200%20%26%201%20%26%200%20%26%200%20%26%200%20%5C%5C%200%20%26%200%20%26%200%20%26%20m_1%20%26%200%20%26%200%20%5C%5C%200%20%26%200%20%26%200%20%26%200%20%26%20m_2%20%26%200%20%5C%5C%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%20m_3%20%5C%5C%20%5Cend%7Barray%7D%20%7D%20%5Cright%5D%5Cvec%7Bx%5E%27%7D%3D%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7Bcc%7D%200%20%26%200%20%26%200%20%26%201%20%26%200%20%26%200%20%5C%5C%200%20%26%200%20%26%200%20%26%200%20%26%201%20%26%200%20%5C%5C%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%201%20%5C%5C%20-k_1%20%26%20k_1%20%26%200%20%26%20-c_1%20%26%200%20%26%200%20%5C%5C%20k_1%20%26%20-k_1-k_2%20%26%20k_2%20%26%200%20%26%20-c_2%20%26%200%20%5C%5C%200%20%26%20k_2%20%26%20-k_2-k_3%20%26%200%20%26%200%20%26%20-c_3%20%5C%5C%20%5Cend%7Barray%7D%20%7D%20%5Cright%5D%5Cvec%7Bx%7D.gif
 

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  • #2
I have gone through my notes multiple times but I am not sure how to go about this problem. Any advice or help would be greatly appreciated.
 

1. What is the motivation for reducing a second order ODE system to a first order?

Reducing a second order ODE system to a first order can make the problem more manageable and easier to solve. It also allows for a more efficient use of computational resources.

2. What is the general process for reducing a second order ODE system to a first order?

The general process involves introducing new variables and rewriting the second order equations as a system of first order equations. This usually involves defining new variables for the derivatives of the original variables.

3. Are there any limitations to reducing a second order ODE system to a first order?

There are no inherent limitations to reducing a second order ODE system to a first order. However, depending on the complexity of the system, the resulting first order system may still be difficult to solve.

4. How does reducing a second order ODE system to a first order affect the solutions?

Reducing a second order ODE system to a first order does not change the solutions to the original problem. It simply rewrites the equations in a different form.

5. Are there any real-world applications of reducing a second order ODE system to a first order?

Reducing a second order ODE system to a first order is a common technique used in many fields of science and engineering, such as physics, chemistry, and biology. It is often used to simplify complex systems and make them easier to analyze and solve.

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