Simulation of a Mechatronic Machine Lumped fluid theory, Flow types.

Simulation of a Mechatronic Machine

Lumped fluid theory, Flow types


Summary of previous lecture

The two properties that are essential for modelling hydraulics are viscosity and bulk modulus.

The bulk modulus effects to the stiffness of the hydraulic components and for this reason the flexibility of the whole system.

Properties of liquids

it

i

ctctoe BV

V

BV

V

BV

V

BB

2

2

1

111


Lumped fluid theory


The lumped fluid theory is commonly used modelling approach for hydraulic systems.

The theory can be applied to hydraulic systems in which the effect of acoustic pressure waves is insignificant.

In mobile machines, the pipelines are short and work cycles are relatively slow in which case the pressure waves are of no significance.

Hydraulipiiri


In the lumped fluid theory, the hydraulic circuit is divided into volumes in which the pressure is assumed to be constant.

- Differential equations are formed for the volumes. Equations are solved in order to obtain the pressures of the system.

- The different volumes are assumed to be separated by throttle valves through which the fluid can flow. The flow rates through the throttle valves can be calculated using the pressures.

- The direction, pressure and flow valves as well as long pipelines used in real systems, are replaced by throttles which control the flow rate between the different volumes.


xx ,

Example

111 ,, VBp

222 ,, VBp

Q


Hydraulipiiri

222 ,, AeAA VBp

Idealisointi

111 ,, AeAA VBp pp

111 ,, BeBB VBp Tp


Pressures in the different locations of the hydraulic circuit can be calculated using the flow continuity equation.

iii Qm

ooo Qm is the densityQ is the flow rate

iiQ ooQQ


The mass of the volume can be written as follows:

where V is the volume under consideration.

- The mass of the volume changes if the in and out going flows are not equal:

VM

oi mmMdt

d iiQ ooQQ


iiQ ooQQ

Employing the equation of mass flow one can get:

Which gives us:

Assuming that the in going fluid density is the same as the out coming density above equation can be written in the form:

ooii QQVdt

d

ooii QQdt

dVV

dt

d

oi QQdt

dVV

dt

d

iii Qm


Using the relation in the equation of

bulk modulus leads to

By substituting this result to equation of flow continuity gives us

d

V

dV

dV

dpVBe

eB

dpd

oie

QQdt

dV

B

V

dt

dp


Reorganize of the equation leads to

Where the change of the volume dV/dt means, in practice, the motion of the piston

- As a result, first order differential equation for the unknown pressure is formed. The equation is called the flow continuum equation.

dt

dVQQ

V

B

dt

dpoi

e

oie

QQdt

dV

B

V

dt

dp


Flow types

Simulation of a Mechatronic Machine Lumped fluid theory, Flow types.

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