A gas spring consists of a piston of area A moving in a...

A gas spring consists of a piston of area $A$ moving in a cylinder of gas. As the piston moves, the gas expands and contracts, changing the pressure exerted on the piston. The process occurs adiabatically (without heat transfer), so $$ p=C \rho^\gamma $$ where $p$ is the gas pressure, $\rho$ is the gas density, $\gamma$ is the constant ratio of specific hears, and $C$ is a constant dependent on the initial state. Consider a spring when the initial pressure is $p_0$ and the initial temperature is $T_0$. Ar this pressure, the height of the gas column in the cylinder is $h$. let $F=\rho_0 A+\delta F$ be the pressure force acting on the piston when it has displaced a distance $x$ into the gas from its initial height. (a) Determine the relation between $\delta F$ and $x$. (b) Linearize the relationship of part (a) to approximate the air spring by a linear spring. What is the equivalent stiffness of the spring? (c) What is the required piston area for an air spring $(\gamma=1.4)$ to have a stiffness of $300 \mathrm{~N} \cdot \mathrm{m}$ for a pressure of $150 \mathrm{kPa}$ (absolute) with $b=30 \mathrm{~cm}$.

A gas spring consists of a piston of area $A$ moving in a cylinder of gas. As the piston moves, the gas expands and contracts, changing the pressure exerted on the piston. The process occurs adiabatically (without heat transfer), so
$$
p=C \rho^\gamma
$$
where $p$ is the gas pressure, $\rho$ is the gas density, $\gamma$ is the constant ratio of specific hears, and $C$ is a constant dependent on the initial state. Consider a spring when the initial pressure is $p_0$ and the initial temperature is $T_0$. Ar this pressure, the height of the gas column in the cylinder is $h$. let $F=\rho_0 A+\delta F$ be the pressure force acting on the piston when it has displaced a distance $x$ into the gas from its initial height.
(a) Determine the relation between $\delta F$ and $x$.
(b) Linearize the relationship of part (a) to approximate the air spring by a linear spring. What is the equivalent stiffness of the spring?
(c) What is the required piston area for an air spring $(\gamma=1.4)$ to have a stiffness of $300 \mathrm{~N} \cdot \mathrm{m}$ for a pressure of $150 \mathrm{kPa}$ (absolute) with $b=30 \mathrm{~cm}$.

Key Concepts

Equivalent Stiffness

The concept of equivalent stiffness, or spring constant, quantifies the amount of force required to produce a unit displacement in a system. In systems like gas springs, the effective stiffness is derived by linearizing the nonlinear relationship between pressure and displacement, thus enabling the modeling of the gas spring as a linear spring over small displacements.

Adiabatic Process

This concept describes a thermodynamic process in which no heat is exchanged with the environment. Under adiabatic conditions, changes in a gas’s pressure, volume, and density are related by a power law, where the pressure is proportional to the density raised to a constant power (the adiabatic index). This relation is fundamental when analyzing rapid compressions or expansions where heat transfer is negligible.

Pressure-Volume Relationship

In gas systems, the pressure and volume are interrelated through specific thermodynamic processes such as adiabatic changes. Understanding how pressure changes as the volume (or density) of the gas is altered is crucial for analyzing devices like gas springs, where mechanical motion directly affects the gas’s state.

Linearization

Linearization involves approximating a nonlinear relationship by a linear one near a specific operating point. This technique simplifies analysis by allowing the use of linear system theory to model small deviations around an equilibrium state, making it easier to derive quantities like effective stiffness in mechanical systems.

Force Balance in Fluid Systems

This concept involves analyzing the forces due to pressure acting on surfaces in fluid systems, such as pistons. By applying the principle of force balance, one can relate the change in pressure caused by fluctuations in fluid density or volume to the net force, which is essential for understanding the mechanical response of systems like gas or air springs.

Transcript

00:01 Okay, so we have that the pressure and volume of a gas in a cylinder of length .8 meters.

00:09 So we have a length of 0 .8 meters and a radius of 0 .2 meters.

00:20 It has a movable piston and the pressure and volume are related by pv to the 1 .4 is equal to k.

00:31 Or k is a constant when the piston is fully extended the gas pressure is 2 ,000 kilopascals so 2 ,000 kilopascals means the pressure is equal to 2 times 10 to the 6th pascales so this is newton per meter squared okay so we need to calculate k so for part a, we have pv to the 1 .4 power is equal to k.

01:20 So we have p is 2 times 10 to the 6th, and the volume is pi r squared times the length.

01:39 So the volume is equal to pi r squared l.

01:45 So this is pi times 0 .2 squared squared.

01:54 Times 0 .8.

01:56 So this is equal to 0 .032 pi.

02:05 And this is to the 1 .4 power.

02:09 So when we plug this into a calculator, we get that k is equal to 80 ,213 .9.

02:22 Part b.

02:23 The force on the piston is pa, where a is the piston's area.

02:27 Calculate the force as a function of the length x of the column of gas.

02:35 Okay, so first let's find the area of a piston.

02:38 The area is equal to pi r squared, so this is 0 .04 pi.