What is the gas spring stiffness for a system with given pressures and volumes?

[Fluidman117]

Hey,

Lets say I have the following gas spring system:

https://dl.dropboxusercontent.com/u/47965009/asd.jpg

In which
$$P_a,P_b$$
are the pressures and
$$V_a,V_b$$
the volumes.

I would like to know how to determine the gas spring stiffness in this case?

 

[jfizzix]

First you would need to know whether the piston is insulating or not. If no heat flows in or out, you get a different result than if everything stays at the same temperature.

The idea would be to see how the pressures change when you displace the piston a small distance from equilibrium. From the pressures, you can get the net force on the piston. Once you have how the net force depends on the displacement, that should resemble a spring equation, and the constant of proportionality between the force and the displacement will be your gas spring constant.

 

[Fluidman117]

Thanks for the reply. I have knowledge of the force that the spring is subjected to at a certain displacement. And it is easy to get the spring stiffness from that.

The spring stiffness depends on the pressure,volume and area inside the cylinder. Let's say I would like to increase my spring stiffness to a new value. For this I keep my volume and area constant and assume adiabatic and isothermal process. How do calculate the pressure increase or decrease required inside the cylinder for a different spring stiffness value?

I found a paper which proposed the following formula for spring rate( http://www.eng.ox.ac.uk/cryogenics/publications/papers/High%20Speed%20Compressors%2015-Jun-2012.pdf) :

$$k=A_p^2*(ΔP/ΔV)$$

In which Ap cross section area of piston.
Thus in my case the formula:

$$k=A_p^2\frac{P_{b1}-P_{b0}}{V_{b1}-V_{b0}}$$

And the
$$P_{b0}$$
is the initial pressure in the cylinder at equilibrium? And by increasing or decreasing this pressure I also increase or decrease the spring stiffness of the gas spring system?

 

[jfizzix]

the process would have to be either adiabatic or isothermal.

If it's isothermal, then $PV = const$. If it's adiabatic, then $PV^{\gamma}=const$, where $\gamma = \frac{C_{p}}{C_{v}}$. In either case, increasing the equilibrium pressure will increase the constant, so that displacements with a higher baseline pressure will have higher restoring forces, i.e., a stiffer gas spring constant.

 

[Fluidman117]

Yes, thanks for pointing that out.
I have found another paper that gives a formula for isothermal gas stiffness in cylinder.
$$k=\frac{P_{b0}A_{p}^2}{V_{b0}-A_{p}S}$$

In which S is the stroke. And from this formula it can be seen that the gas spring is actually non-linear as the spring rate changes with different stroke lengths.

 

[serbring]

Yes, thanks for pointing that out.
I have found another paper that gives a formula for isothermal gas stiffness in cylinder.
$$k=\frac{P_{b0}A_{p}^2}{V_{b0}-A_{p}S}$$

In which S is the stroke. And from this formula it can be seen that the gas spring is actually non-linear as the spring rate changes with different stroke lengths.

Hi,

there several approaches for modeling an air spring. You can model it as isothermal or polytropic process and it mostly depends by the excitation frequency, for very low frequency the process is almost isothermal and for higher ones the process is polytropic.

 

[Fluidman117]

Hi,

there several approaches for modeling an air spring. You can model it as isothermal or polytropic process and it mostly depends by the excitation frequency, for very low frequency the process is almost polytropic and for higher ones the process is isothermal.

Can you consider 0.1Hz - 0.2Hz low frequency?

 

[serbring]

I'm sorry I made a mistake in my previous post. You can consider the process as isothermal