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Stiffness of hydraulic systems
A hydraulic motor or cylinder may be used to drive any
kind of mechanism. The speed of the actuatoris determined by the
hydraulic flow that is generated by a hydraulic pump or direction
proportional
valve, or that the hydraulic fluid itself is not compressible.
In this paragraph we will find that this is notreally true.
Hydraulic fluid is compressible. This is expressed in the
formula:
E
P
V
V Δ−=
Δ
With : ΔV = change in volume (because of volume decrease a minus
sign is being
used) [m3]V = original volume [m3]ΔP = pressure increase [N/m2]E
= elasticity of oil [N/m2]
The termV
V Δ in fact is the relative volume change.
The value E for mineral oil fluids is 1E9 tot 1.6E
9 N/m2 = 10.000 – 16.000 bar. , (water has an E value
of 24000 bar.
Another practical expression for the volume change is:
with a pressure increase of 100 bar the volumedecrease is 1%
This effect could also be explained with the use of the figures
as given below. At the left an externalforce is being applied to a
cylinder on which the pressure increases with 200 bar. The
displacement ofthe cylinder than will be 2% or 2 cm. At the
right figure in total 2 dm3 of fluid has to be added (pumped) into
the original volume of 100 dm3to achieve a pressure increase of 200
bar.
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In both situations the tube of the cylinder is pressurized
and will expand to a larger diameter. This ofcourse will have a
damping effect on the oil elasticity. However, the elasticity
module of steel with avalue of 220 x E
9 [N/m2] is nearly 200 times larger then the elasticity of
oil. Therefore the increase of
volume of the cylinder tube can be neglected in relation to the
volume decrease of mineral oil.
Stiffness Co of a cylinder
Because of the elasticity of oil the behaviour of a hydraulic
cylinder may be described like the stiffnessof a mechanical spring.
The stiffness Co of a cylinder can be described with:
Under influence of an external force ΔF the cylinder
piston will show a displacement ΔL. The stiffnessnow may be
defined with:
