Infinite-Extent 2D Mass-Spring System vibration and dispersion relation

In summary, the conversation was about finding the equation of motion and solutions for a 2D infinite lumped mass spring system with identical masses and springs and a periodicity of n=a. The equation of motion was given and it was mentioned that it can be solved using Fourier Analysis. The solution for the displacement of each mass was also provided, along with the dispersive relation for finding the frequency of the system.
  • #1
M_Abubakr
10
1
Hi,
I am trying to find equation of motion and its solutions for a 2D infinite lumped mass spring system as depicted in figure. All the masses are identical, All the springs are identical, and even the horizontal and vertical periodicity is the same n=a.
Mass-spring-system.png

I need to try find dispersive relation for such a system. Can anyone tell me how to solve this? I am fairly confident with mass-spring systems in one dimension with MDOF but I don't know how to solve this one and I can't find any similar problem solved over the internet.
 
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  • #2
Thanks!The equation of motion for this system is given by:m*a*ddot{x_n} + k*(x_{n+1} + x_{n-1} - 2*x_n) = 0where m is the mass of each lumped mass, a is the periodicity of the system, k is the spring constant, and x_n is the displacement of the nth mass. Solving this equation is a bit involved, but it can be done using Fourier Analysis. The solution is:x_n = A*cos(w*n*a) + B*sin(w*n*a)where A and B are constants and w is the frequency of the system, which can be found from the dispersive relationw^2 = k/mHope this helps!
 

Related to Infinite-Extent 2D Mass-Spring System vibration and dispersion relation

What is an "Infinite-Extent 2D Mass-Spring System"?

An Infinite-Extent 2D Mass-Spring System is a theoretical model used to study the behavior of a large number of masses connected by springs in a two-dimensional space. It assumes that the system has an infinite number of masses and springs, allowing for a more accurate analysis of the system's vibration and dispersion properties.

What is the purpose of studying the vibration and dispersion relation of an Infinite-Extent 2D Mass-Spring System?

The purpose of studying the vibration and dispersion relation of an Infinite-Extent 2D Mass-Spring System is to understand how the system behaves under different conditions, such as varying mass and spring constants. This knowledge can be applied to real-world systems, such as structures and materials, to predict and improve their performance and stability.

How is the vibration and dispersion relation of an Infinite-Extent 2D Mass-Spring System calculated?

The vibration and dispersion relation of an Infinite-Extent 2D Mass-Spring System is calculated using mathematical equations that take into account the mass and spring constants, as well as the number and arrangement of masses and springs in the system. These equations can be solved numerically or analytically to determine the system's natural frequencies and mode shapes.

What factors affect the vibration and dispersion relation of an Infinite-Extent 2D Mass-Spring System?

The vibration and dispersion relation of an Infinite-Extent 2D Mass-Spring System can be affected by various factors, such as the mass and spring constants, the number and arrangement of masses and springs, and the boundary conditions of the system. Changes in any of these factors can alter the system's natural frequencies and mode shapes, leading to different vibration and dispersion characteristics.

How is the concept of dispersion related to the vibration of an Infinite-Extent 2D Mass-Spring System?

The concept of dispersion in an Infinite-Extent 2D Mass-Spring System refers to the relationship between the system's natural frequencies and the corresponding mode shapes. A system with a high degree of dispersion will have a wide range of natural frequencies, resulting in a more complex vibration pattern, while a system with low dispersion will exhibit simpler vibration behavior.

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