Two Spring System: Steady State Motion

In summary, the problem considers a system consisting of two springs suspended from the ceiling, with different spring constants and masses, connected by a mass and subject to a periodic force. The equations of motion are derived using free body diagrams and taking into account the difference in displacement between the two springs. The final solution is expressed in terms of the displacements of the two masses, x1 and x2, and the angular frequency, w.
  • #1
mwkfun
4
0

Homework Statement


Consider a system consisting of two springs suspended from the ceiling. The first has a spring constant k-1, the second k-2. They are connected by a mass m and the second spring also has a mass m connected at the bottom. A periodic force is applied to the upper mass. What is the steady state motion for each mass?



Homework Equations


Fcoswt
x1 = Acos wt
x2 = Bcos wt


The Attempt at a Solution

 
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  • #2
Welcome to PF!

Hi mwkfun! Welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help. :smile:
 
  • #3
I tried two different ways...I attempted to turn the system into an electrical circuit equivalent and I tried with free body diagrams and the following equations:
mX=mg + k1x1 + k2x2 - Fcoswt
mX=mg + Fcoswt - k1x1 - k2x2

(For these two eqns. the X means X double dot indicating acceleration)

mX2 = mg + k2x2 - Fcoswt
mX2 = mg + Fcoswt - k2x2

(again, X2 indicates x double dot...I don't know how to type it the correct way)
I think I should be solving for x1 and x2, but I am not sure if I have the correct equations. The eqns. reflect the free body diagrams I have drawn. Thanks for any help you can give.
 
  • #4
I am also wondering whether I should use the x1= Acoswt and x2 = Bcoswt as a substitutions or if they are specific solutions to the differential eqns.
 
  • #5
mwkfun said:
… I tried with free body diagrams and the following equations:
mX=mg + k1x1 + k2x2 - Fcoswt
mX=mg + Fcoswt - k1x1 - k2x2

(For these two eqns. the X means X double dot indicating acceleration)

mX2 = mg + k2x2 - Fcoswt
mX2 = mg + Fcoswt - k2x2

Hi mwkfun! :smile:

(on this forum, it's best to use dashes instead of dots: X1'' :wink:)

I don't understand why you've put each of these equations in pairs :confused:

Anyway, you need to take into account that the displacement of the lower mass is not the same as the displacement of the lower spring … it's the displacement of the lower spring minus the displacement of the upper spring :smile:
 
  • #6
Thanks so much for your help. I think I got it!
 

What is a two spring system?

A two spring system is a physical system in which two springs are connected together in series or parallel. This system is commonly used in physics experiments to study the behavior of springs and their properties.

What is steady state motion in a two spring system?

Steady state motion in a two spring system is when the system reaches a state of equilibrium, meaning the forces acting on the springs are balanced and the system is not accelerating. This results in a constant amplitude and frequency of oscillation.

What factors affect the steady state motion of a two spring system?

The stiffness of the springs, the mass attached to the springs, and the damping coefficient of the system are all factors that can affect the steady state motion of a two spring system. Changes in these factors can alter the amplitude and frequency of oscillation.

How is the steady state motion of a two spring system mathematically represented?

The steady state motion of a two spring system can be mathematically represented using Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. This can be expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement.

What are some real-life applications of a two spring system?

A two spring system has many real-life applications, such as in shock absorbers for vehicles, suspension systems for bridges, and musical instruments such as guitars and violins. It is also used in various industries for vibration isolation and noise reduction.

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