Spring mass spring mass spring - 2 balls oscillations

In summary, the conversation discusses calculating displacement x(t) for two equal masses attached to two springs with the same spring constant and natural length. The masses are assumed to be in point and gravity is not included. The equations for calculating the displacement are given, with two natural frequencies \sqrt{k/m} and \sqrt{3k/m} appearing when solved. The conversation also mentions the possibility of assuming different values for the masses, spring constants, and natural lengths.
  • #1
mmatras
1
0
http://fatcat.ftj.agh.edu.pl/~i7matras/hej.jpg

Both masses are in point.

I need to count displacement x(t) but I don't know how to write derivative equations? Could someone help? Or at least give me a tip?
 
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  • #2
Are we to include gravity or can we think of this as horizontal? Can we assume that the two masses are equal? Can we assume that the two springs have the same spring constant and natural length?

Assuming all of that, let [itex]x_1[/itex] be the distance from the lower edge to the center of mass of the first ball, [itex]x_2[/itex] be the distance from the lower edge to the center of mass of the second ball and let m be the mass of each ball. Let L be the distance from the lower edge to the upper, let k be the spring constant of each spring and let l be their natural lengths.

"F= ma" of course, so [itex]m d^2x_1/dt^2[/itex] is equal to the total force on the first mass. There is a force from the spring below it equal to the spring contant times its extension: [itex]-k(x_1- l)[/itex]. There is a force from the spring above it, equal to the spring constant times its extension: [itex]k(x_2- x_1- l)[/itex] (positive because if the spring is stretched, it will pull the mass upward).
[tex]m\frac{d^2x_1}{dt^2}= -k(x_1-l)+ k(x_2- x_1- l)= -2kx_1+ kx_2[/tex]

[itex]md^2x_2/dt^2[/itex] is equal to the total force on the second mass. There is a force from the spring below it of [itex]-k(x_2-x_1- l)[/itex], equal and opposite to the force of that spring on the lower mass. There is a force from the spring above it: [itex]k(L- x_2- l[/itex].
[tex]m\frac{d^2 x_2}{dt^2}= -k(x_2-x_1-l)+ k(L- x_2- l)= -2kx_2+ kx_1+ kL[/itex].

Those are your two equations. When you solve them, you should find that you have two "natural frequencies", [itex]\sqrt{k/m}[/itex] and [itex]\sqrt{3k/m}[/itex].-
 

What is the concept behind "Spring mass spring mass spring - 2 balls oscillations"?

The concept behind this is the study of oscillatory motion in a system with multiple masses, connected by springs. This system exhibits complex patterns and behaviors as the masses interact with each other and the springs.

What are the key variables in this system?

The key variables in this system are the masses of the balls, the spring constants of the connecting springs, the initial positions and velocities of the balls, and the length of the springs.

How does the system behave over time?

The system will exhibit a periodic motion, with the balls oscillating back and forth between each other. The amplitudes and frequencies of the oscillations may change depending on the initial conditions and the properties of the system.

What are some real-life applications of this system?

This system can be seen in various mechanical systems, such as pendulums, shock absorbers, and even in musical instruments like the guitar. It also has applications in fields such as acoustics, seismology, and engineering design.

How is this system mathematically modeled?

This system can be modeled using the equations for simple harmonic motion, as well as Newton's laws of motion and Hooke's law. Differential equations can be used to describe the motion of the masses and the forces acting on them.

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