Coupled Vertical Oscillators with Gravity

In summary, the conversation is about a problem involving two masses attached to a board by springs. The goal is to find the normal frequencies and coordinates in a constant gravitational field. The person is having trouble setting it up, but their best guess is to use coupled oscillators. Another person suggests shifting x_2 to eliminate the gravity term before solving the equations.
  • #1
mekrob
11
0
Hey, I'm just having some trouble getting started with this problem.

-------------
(
)
(
m1
(
)
(
2m1
Crude representation: (The parantheses are supposed to be the springs)

There is a mass (m1) that is attached vertically to a board by a spring of spring constant k and length b. There is a second mass (2m1) attached by an identical spring to the first mass.

I'm supposed to find the normal frequencies in a constant (so it isn't affected by x1 and x2, right?) gravitational field and the normal coordinates. I can do coupled oscillators pretty easily, but I'm just having a hard time setting it up.

Best guess...
m1x1'' = -k1x1 +k2(x2-x1) + g
m2x2'' = k2x2 + g

I guess I'm not exactly sure where g goes into the MX''=-Kx matrix.
 
Last edited:
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  • #2
mekrob said:
Hey, I'm just having some trouble getting started with this problem.

-------------
(
)
(
m1
(
)
(
2m1
Crude representation: (The parantheses are supposed to be the springs)

There is a mass (m1) that is attached vertically to a board by a spring of spring constant k and length b. There is a second mass (2m1) attached by an identical spring to the first mass.

I'm supposed to find the normal frequencies in a constant (so it isn't affected by x1 and x2, right?) gravitational field and the normal coordinates. I can do coupled oscillators pretty easily, but I'm just having a hard time setting it up.

Best guess...
m1x1'' = -k1x1 +k2(x2-x1) + g
m2x2'' = k2x2 + g

I guess I'm not exactly sure where g goes into the MX''=-Kx matrix.

Just shift x_2. It's easy to get a shifted x_2 that will get rid of the g in both equations. Than you can solve the usual way and in the final solution you can go back to the original x_2
 

1. What are coupled vertical oscillators with gravity?

Coupled vertical oscillators with gravity are a type of mechanical system that consists of two or more masses connected by springs and experiencing the influence of gravity. This system can exhibit complex vibrational behavior due to the interaction between the masses and the force of gravity.

2. How do coupled vertical oscillators with gravity behave?

Coupled vertical oscillators with gravity can exhibit a range of behaviors, including periodic motion, chaotic motion, and resonance. This behavior is dependent on the initial conditions and the parameters of the system, such as the masses, spring constants, and the strength of gravity.

3. What are the applications of coupled vertical oscillators with gravity?

Coupled vertical oscillators with gravity have various applications in engineering, physics, and other fields. They are used to model and study the behavior of structures, such as buildings and bridges, subjected to external forces, as well as to understand the dynamics of molecular systems.

4. How do you model coupled vertical oscillators with gravity?

Coupled vertical oscillators with gravity can be modeled using mathematical equations, such as the equations of motion and Hooke's law. These equations can be solved using numerical methods or simulated using computer software, allowing for the analysis of the system's behavior under different conditions.

5. What are the advantages of studying coupled vertical oscillators with gravity?

Studying coupled vertical oscillators with gravity can provide insight into the behavior of complex mechanical systems and help in the design and optimization of structures. It can also aid in understanding the effects of external forces, such as earthquakes, on structures and how to mitigate their impact.

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