Coupled Vertical Oscillators with Gravity

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SUMMARY

This discussion focuses on solving the problem of coupled vertical oscillators under the influence of gravity. The system consists of two masses, m1 and 2m1, connected by springs with spring constant k. The equations of motion are presented as m1x1'' = -k1x1 + k2(x2-x1) + g and m2x2'' = k2x2 + g. A key insight provided is to shift the coordinate x2 to eliminate the gravitational term g, allowing for a standard solution approach.

PREREQUISITES
  • Understanding of coupled oscillators and their equations of motion
  • Familiarity with spring constants and gravitational effects on oscillatory systems
  • Knowledge of matrix representation in dynamics (MX'' = -Kx)
  • Basic skills in solving differential equations
NEXT STEPS
  • Study the method of shifting coordinates in oscillatory systems
  • Learn about normal modes and frequencies in coupled oscillators
  • Explore the application of linear algebra in mechanical systems
  • Investigate the effects of gravity on oscillatory motion in different configurations
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Students and professionals in physics, mechanical engineering, and applied mathematics who are dealing with oscillatory systems and their dynamics.

mekrob
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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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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
 

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