Finding frequencies of normal modes with the systematic method

In summary, the conversation is about understanding the systematic method for determining frequencies and amplitude ratios in systems with multiple degrees of freedom. The problem statement involves finding the two coupled equations of motion using either the slinky or small-oscillations approximation, and then using the systematic method to find the normal modes. However, there is confusion in setting up the equations of motion, which are later corrected to the correct form.
  • #1
jmm5872
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I am having problems understanding the "systematic method" for determining the frequencies and amplitude ratios of normal modes when a system has more than one degree of freedom. I think I initially have problems setting up the differential equation that describes the motion. Here is the problem statement:

Using either the slinky approximation or the small-oscillations approximation, find the two coupled equations of motion for the transverse displacements Ya and Yb (shown in attached picture). Use the systematic method to find the frequencies and amplitude ratios for the two normal modes.

(Description): The displacement from the relaxed position along the x-axis is given by Ya and Yb. The two normal modes are also shown but I don't think I needed to draw them. !FORGOT TO MENTION...they masses are attached to springs!

So my initial problem is my confusion in setting up the two coupled equations of motion. (They are set up so that gravity does not affect the motion). Here is what I assumed for the two equations:

MYa" = -TYa + T(Yb - Ya)
MYb" = -T(Yb - Yb) - TYb

Let me first see if this is even correct before I move on.
 

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  • #2
Any help is greatly appreciated!No, the equations of motion you have written down are incorrect. The equations of motion should be written as follows:M_a a'' = -T (Y_a - Y_b) - k_a Y_aM_b b'' = T (Y_a - Y_b) - k_b Y_bwhere M_a and M_b are the masses, T is the tension in the spring connecting them, k_a and k_b are the spring constants for each mass and Y_a and Y_b are the displacements of each mass from the relaxed position along the x-axis.
 

1. What is the systematic method for finding frequencies of normal modes?

The systematic method for finding frequencies of normal modes involves setting up a matrix equation using the equations of motion for a system, and then solving for the eigenvalues of the matrix. These eigenvalues correspond to the frequencies of the normal modes of the system.

2. Why is it important to find the frequencies of normal modes?

Finding the frequencies of normal modes is important because it allows us to understand the behavior and stability of a system. By knowing the frequencies, we can predict how a system will respond to different inputs and determine if it is in a stable or unstable state.

3. Can the systematic method be applied to any type of system?

Yes, the systematic method can be applied to any type of system that can be described using equations of motion. This includes mechanical, electrical, and thermal systems, among others.

4. Are there any limitations to the systematic method for finding frequencies of normal modes?

One limitation of the systematic method is that it can become computationally expensive for larger and more complex systems, as the size of the matrix equation increases. Additionally, it assumes that the system is linear and does not take into account any nonlinear effects.

5. How does the systematic method differ from other methods for finding frequencies of normal modes?

The systematic method is a more general and systematic approach compared to other methods, such as the modal analysis method. It does not require any prior knowledge about the system and can be applied to a wide range of systems. However, it can be more time-consuming and computationally intensive compared to other methods.

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