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## Spring mechanism

A gas spring is a spring mechanism that, rather than using a physical coil to generate
the spring behaviour, uses the properties of a gas to achieve this behaviour.
Here is a simplified representation of a gas spring:
The force exerted by the gas spring is dictated by the pressure of the gas within it.
As the temperature of the gas remains constant, and no gas escapes, the pressure
is dictated by Boyle’s Law,
p1V1 = p2V2 = k
where p is pressure and V is volume. The force due to the gas is given by
F = pA
where A is the area of the surface the pressure, and thus the force, applies to (in
our case, this area is the cross-sectional area). Note that the air outside the gas
spring will also exert a force, but that the pressure for the air will remain constant.
As such, the total pressure producing the force is the pressure inside the gas spring
minus the air pressure.
a) (*) By considering the forces involved – the force due to pressure in the gas
spring and the force due to gravity, as well as air resistance (linear) – show that
the motion of mass m is described by
m
d2x
dt2 + c
dx
dt −
k
x
+
k
x0
= mg
b) (*) Determine the resting length of the spring, x1, when the mass is attached.
That is, the length for which all forces balance.
c) (**) Using MATLAB’s ode45 function, solve the ODE if m = 0.13, c = 0.27,
k = 0.35, g = 9.81, and x0 = 0.1, with x(0) = 0.03 and x0(0) = 0, for 0  t  5.

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