2 D.O.F System - Natural Frequencies

In summary, a 2 D.O.F system is a mathematical model used to study the behavior of a physical system with two independent variables. Natural frequencies are the frequencies at which the system will vibrate without any external force applied and can be calculated using mathematical equations. It is important to know the natural frequencies of a 2 D.O.F system in order to understand its behavior and avoid resonance. These frequencies can be calculated using equations such as the eigenvalue problem, and can be altered by changing the system's mass, stiffness, or damping.
  • #1
Jones1987
77
0

Homework Statement



The question is a 2 D.O.F system, m1 = 600kg, m2 = 50kg, k1 = k3 = 0.5MN/m, k2 = 0.2 MN/m. Calculate the natural frequencies.

Homework Equations



1167-w^2 -333.3 X1 0
=
-4000 14000-w^2 X2 0

The Attempt at a Solution



I am at the point where I have put these values into Matrix form. The values (1167-w^2)(14000-w^2) - (-333.3 x -4000) = 0

I have worked it out so I get values of:

15 x 10^6 - 15167w^2 + w^4 = 0

The answers are:

w1^2 = 1.06 x 10^3
w1 = 32.6 rad/s

w2^2 = 14 x 10^3
w2 = 118 rad/s

However when I work out this I get answers of:

w1^2 = 106
w1 = 10 rad/s

w2^2 = 137.9
w2 = 11.7 rad/s

Whats missing, what am I missing out? This is my first attempt at posting a HW question, so sorry if its not clear, just let me know if you need any extra info.

I just can't see where the x10^3 comes from, I'm assuming something to do with the w^4, but I can't think what.
Any input would be highly appreciated.

Thanks,
Sam
 
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  • #2

Thank you for posting your question. It seems like you have made a mistake in your calculation. The correct solution for the natural frequencies of this 2 D.O.F system are:

w1^2 = 1.06 x 10^3
w1 = 32.6 rad/s

w2^2 = 14 x 10^3
w2 = 118 rad/s

In order to get these values, you need to solve the equation you have set up correctly. You have correctly set up the matrix form, but your calculation for the determinant is incorrect.

The correct equation should be:

(1167-w^2)(14000-w^2) - (-333.3 x -4000) = 0

Simplifying this equation will give you:

15 x 10^6 - 15167w^2 + w^4 = 0

This is a quadratic equation in terms of w^2. Solving this equation will give you the values of w1^2 and w2^2. Then, taking the square root of these values will give you the natural frequencies w1 and w2.

I hope this helps. If you need further clarification, please do not hesitate to ask.
 

Related to 2 D.O.F System - Natural Frequencies

1. What is a 2 D.O.F system?

A 2 D.O.F (degree of freedom) system is a mathematical model used to study the behavior of a physical system with two independent variables. It is commonly used in engineering to analyze the vibrations of a structure or machine.

2. What are natural frequencies in a 2 D.O.F system?

Natural frequencies are the frequencies at which a 2 D.O.F system will vibrate without any external force applied. They are dependent on the mass, stiffness, and damping of the system and can be calculated using mathematical equations.

3. Why is it important to know the natural frequencies of a 2 D.O.F system?

Knowing the natural frequencies of a 2 D.O.F system is important because it allows us to understand how the system will behave under different conditions. It helps in designing and optimizing the system to avoid resonance (when the external force matches the natural frequency, causing large amplitude vibrations).

4. How can the natural frequencies of a 2 D.O.F system be calculated?

The natural frequencies of a 2 D.O.F system can be calculated using mathematical equations such as the eigenvalue problem. This involves solving for the eigenvalues and eigenvectors of the system's mass and stiffness matrices.

5. Can the natural frequencies of a 2 D.O.F system be changed?

Yes, the natural frequencies of a 2 D.O.F system can be changed by altering the system's mass, stiffness, or damping. This can be done by adding or removing components, adjusting material properties, or using different damping techniques.

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