## Gas Spring Stiffness

 (B.1) Consider a thermally insulated gas container that is equipped with a sealed moving piston. Without loss of generality, we can assume that the internal and external pressures of the container are initially equal and, hence, the piston is at resting position. The thermal insulation provides an adiabatic boundary condition for the contained gas. Therefore, at any given temperature, the gas pressure, p, and its volume, V, will follow a trajectory defined by

PVY = C

Where y = cp/cv is the ratio of the gas specific heat at constant pressure to the gas specific

 (B.2) Heat at constant volume and C represents a constant value. A small force, dF, will cause a small displacement, dx, and the gas pressure and volume will change according to the following expressions DV = — AP dx

(B.3)

Where AP is the cross sectional area of the piston.

By differentiating (B.1) and appropriate substitutions from (B.2) and (B.3) we have,

DF

VY— YpAp VY-i dx = 0 (B.4)

AP

Which yields to the gas spring stiffness, KG, as derived below,

KG = dF = — pp

Dx V

Similarly, if gas compression is considered an isothermal process, we have,

PV = C (B.6)

Which yields to the gas spring stiffness, KG, as below,

Kg = A (B.7)