Attenuating floor oscillations with a cushion

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In summary, the floor and the cushion oscillate with each other. The cushion attenuates the motion of the floor.
  • #1
LCSphysicist
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Homework Statement
All below.
Relevant Equations
Trying to do one.
1597260466085.png

I am trying to find any relation between the three parameters:
Position of the floor wrt an inertial frame f
Position of the cushion wrt floor c
Position of the man wrt cushion m

But this is really confusing, leaving me to a lot of unnecessary variables
Do you know one smart way to start?
 
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  • #2
The cushion would in general act as a damped spring, where the damping takes the form of a lower spring constant during expansion than during compression. But that's all a bit hard, so just treat it as an ideal spring, constant k.

Let the floor height vary as ##A\sin(\omega t)## and the height of the top of the cushion, relative to the equilibrium position, be y.
Can you write the differential equation? Can you solve it?
 
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  • #3
haruspex said:
The cushion would in general act as a damped spring, where the damping takes the form of a lower spring constant during expansion than during compression. But that's all a bit hard, so just treat it as an ideal spring, constant k.

Let the floor height vary as ##A\sin(\omega t)## and the height of the top of the cushion, relative to the equilibrium position, be y.
Can you write the differential equation? Can you solve it?
I tried to imagine what follows:
The floor is oscillating with the spring, i will make one action by time:
First the spring stretched to y, after the floor will "go up" by ##A\sin(\omega t)## , all this lead me with:

##my'' = -k(y-A\sin(\omega t))##

I think we can ignore the transient solution here, so we end with:

1597407577328.png


the module of the maximum would need to be equal A/100, solving:

k = m156
Well, when we sit on the spring:

m156x = mg,
x = g/156
x ~ 0.064 m

The answer is: "about to 6 cm"

so i think this is right, the only thing let me skeptical is the fact i need to assume the module of the expression above, this means the spring and the floor are out of phase?
 

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  • #4
LCSphysicist said:
I tried to imagine what follows:
The floor is oscillating with the spring, i will make one action by time:
First the spring stretched to y, after the floor will "go up" by ##A\sin(\omega t)## , all this lead me with:

##my'' = -k(y-A\sin(\omega t))##

I think we can ignore the transient solution here, so we end with:

View attachment 267755

the module of the maximum would need to be equal A/100, solving:

k = m156
Well, when we sit on the spring:

m156x = mg,
x = g/156
x ~ 0.064 m

The answer is: "about to 6 cm"

so i think this is right, the only thing let me skeptical is the fact i need to assume the module of the expression above, this means the spring and the floor are out of phase?
That all looks right.

Wrt phase, seems to me the solution says they are in phase. Of course, in the real world, there would be some damping too, though not proportional to velocity.

When I first looked at this problem I was a bit sceptical, so I did two checks:
- if we double the cushion thickness we would expect double the sinking depth and double the attenuation factor
- dimensional analysis gives the only way to combine the data to get a distance as ##g\omega^2##
So this suggests the answer is proportional to ##g\omega^2\times## attenuation, but there could have been some constant factor, like 2 or pi. Turns out it is 1.
 
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1. What are floor oscillations and why do they need to be attenuated?

Floor oscillations are vibrations or movements that occur in a floor structure. They can be caused by various factors such as walking, jumping, or even external forces like wind or earthquakes. These oscillations can be uncomfortable for occupants and can also cause damage to the structure over time.

2. How does a cushion help attenuate floor oscillations?

A cushion, also known as a vibration isolator or shock absorber, helps to reduce floor oscillations by absorbing and dissipating the energy from the vibrations. The cushion acts as a barrier between the floor and the source of the vibration, reducing the transmission of the oscillations to the rest of the structure.

3. What materials are commonly used for cushions to attenuate floor oscillations?

Commonly used materials for cushions include rubber, foam, and springs. Rubber is a popular choice as it is flexible and can effectively absorb vibrations. Foam is also commonly used as it can be easily customized to fit specific needs. Springs are another option, as they can provide a more rigid support while still reducing vibrations.

4. How do you determine the right cushion for a specific floor structure?

The right cushion for a specific floor structure depends on various factors such as the type of floor, the frequency and amplitude of the oscillations, and the weight and movement of the occupants. It is important to consult with a structural engineer to determine the best cushion material and design for a particular floor structure.

5. Can cushions completely eliminate floor oscillations?

Cushions can significantly reduce floor oscillations, but they may not completely eliminate them. The effectiveness of a cushion depends on various factors such as the type and intensity of the vibrations, as well as the quality and design of the cushion. In some cases, additional measures may be needed to completely eliminate floor oscillations.

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