Spring Displacement in Moving Part: Constant Pressure?

[Andrea Vironda]

hello, I'm new here
according to this scheme:

i would to know, if the pressure is constant, the spring displacement in the moving part in function of time.
the square is without mass, so i thought to use the energy approach, but i don't know how to consider acceleration

 

[jbriggs444]

If there is a force but no mass or an extremely small mass, what happens to acceleration? What does Newton's second law say?

 

[Andrea Vironda]

The acceleration is Very high. But in this case The only mass i have is The spring, which is fixed on one extremity. I don't know how to consider this

 

[A.T.]

The only mass i have is The spring, which is fixed on one extremity.

The acceleration of the centre of mass times the mass is the net force.

 

[jbriggs444]

The acceleration is Very high. But in this case The only mass i have is The spring, which is fixed on one extremity. I don't know how to consider this

One way of considering it is to imagine the spring as a being a chain of ideal massless springs connected via ideal pointlike masses. As you apply force to one end of the spring, the masses near the force end move more and the masses near the fixed end move less.

In principle, you could use Newton's laws, write down an equation for the motion of each of the masses and solve them all.

If one imagines the limit of the process as the length of the component springs get shorter and shorter, the number of component springs get higher and higher and the masses get smaller and smaller, you end up with a continuous spring.

You can apply Newton's laws to obtain a differential equation defining the acceleration of each part of the spring in terms of the local mass density and the local tension gradient. With some simplifying assumptions you can solve this and obtain things like a wave equation.

[Or you could do as @A.T. suggests and imagine the spring as having its mass concentrated in the center. Way easier and probably accurate enough for your purposes].

 

[Andrea Vironda]

The acceleration of the centre of mass times the mass is the net force.

can i assume the spring free end acceleration as the double of the center of mass acceleration?

You can apply Newton's laws to obtain a differential equation defining the acceleration of each part of the spring in terms of the local mass density and the local tension gradient. With some simplifying assumptions you can solve this and obtain things like a wave equation.

i'm curious but I'm not able to implement this differential equation. can you give me a sketch?

 

[A.T.]

can i assume the spring free end acceleration as the double of the center of mass acceleration?

As an approximation, that ignores oscillation within the spring.

 

[Andrea Vironda]

it's good, because i have to modelize a single-effect piston

 

[Andrea Vironda]

As an approximation, that ignores oscillation within the spring.

if i substitute i find i have ~1600g. i think this is the acceleration i receive if i cut the pressure supply outright.
if i have to calculate the acceleration the spring have during the room oil replenishment, how can i do? only for the forward stroke.