Modelling and Simulation of Mechatronic Systems

Prof.Ing.Petr Noskievi, CSc.

Department of Automatic Control and Instrumentation

VB Technical University of Ostrava

Hydraulic systems

The modelling od hydraulic systems will be focused on the mathematical description of the properties of the systems, in which the fluid is in the move and in which the compressed fluid works. The creation of the mathematical models is based on the use of the knowledge of fluid mechanics.

The real fluid is compressible, but less than gas and steam.

The creation of the mathematical models is based on the application of the two general laws: conservation of mass and conservation of energy.

The need of the mathematical models of hydraulic systems is by the control of the fluid flow, control of the fluid reserve in the tanks and by the control of the motion of the mechanisms with hydraulic drives.

Conservation of mass Continuity equation

Conservation of the mass

Two types of the mass changes: Local change of the mass dmt is caused by the compressibility of the fluid, Convective change of the mass dms is caused by the difference of the mass flowing into and out of the unit volume.

Continuity equation for one dimensional steady flow:

0=+= ts dmdmdm

( ) 0.. =vSdsd

constvS.vSvS === ..... 222111

const= .. constvSQ ==

Conservation of energy Bernoulli`s equation

Bernoulli`s equation expresses the conservation energy law by the flowing of the ideal fluid in the gravitational field characterized by the gravitational acceleration. Bernoulli`s equation for the ideal incompressible fluid in the gravitational field and for the steady flow has the following form:

..2

2

consthgpv =++

.constWWW hpK =++

2

21 v

mWw KK == hg

p .=

hgmWw Ph .==

consthgp

gv

=++ 2

2

WK Kinetic energy WP Potential energy due to pressure Wh Potential energy - due to gravity

Fluid outlet from the tank

Fluid outlet from the closed (pressurized) tank

Zhgvphgvp .2

.2

222

211 ++=++

h

p1

v , p2 2

S1

S2

S

gvhZ 2.

22=

2211 .. vSvS =1

221

.SvSv =

gvgvphgv

SSp

2..

2...

21 22

2222

2

2

1

21

++=+

+

2

1

2

21

2

1

..2

+

+

=

SS

pphgv

+

+=

+

+

=

2121

..2.11

1

..2 pphg

pphgv

For the small opening area S2 in comparison with the cross-section area of the tank can be obtained:

For ideal fluid we can write: 0=

+=

21..2 pphgvt

Dynamics of the Liquid level in the open tank

h

p , S0 1

p , S0 2v2 ,Q2

v1

hgv ..22 =Outlet velocity:

2211 .. vSvS = dtdhv =1

hgSdtdhS ..2.. 21 =

0.21

2 =+ hgSS

dtdh

The liquid level h by the gravitational outflow from the tank is described using the homogenous nonlinear first order differential equation:

h(t)h01S1

2gS2

Q2h(t)

Block diagram of the mathematical model of the liquid level the systm without inflow

Atmospheric pressure

Outlet from the tank with inlet

h(t)

p ,S0 1

S2p0 v2, Q2

Q1

211. QQdtdhS =

ghSQdtdhS 2.. 211 =

111

2 .1.2. QS

hgSS

dtdh

=+

The steady state for the constant inlet is equal: 1Q

.

0=dtdh

1

1

1

2 .2.SQhg

SS

= 22

21

2gSQh =

h(t)h0Q1

Q2

S2

1S1

2gh(t)

Because of the existence of the steady state the system has the property called self controllability. (For the constant inflow the level of the liquid is constant).

The rate of the change of the fluid level can be expressed by the nonlinear differential equation:

Block diagram of the mathematical model of the liquid level- tank with inlet

Control of the liquid level using the variable inlet

h(t)

S1

S2 Q2

Q1=Q(u)

hW e uReg

huhh

The relay controller with hysteresis can be used for level control. The controller will control the solenoid valve. The flow Q1 is given by the equations for the command value u :

1max11 == uproQQ 001 == uproQ

h(t)h0

Q1hW e u

Q2

S2

1S1

2gh(t)

Q1=Q (u)1u

0eemaxemin

Closed loop level control using the relay controller

Simulation model of the liquid level control

Liquid level control S1=1%liquid surface m2

S2=0.00065%cross section area of the outlet pipe KQ=0.01%flow gain m3/s

g=9.81%m/s2

s 1

level h

S2*sqrt(2*g)

Outlet flow

MATLAB Function

sqrt

KQ

Inlet flow

1/S1

Liquid surface h e

Step Relay

Flow difference

0 10 20 30 40 50 60 70 80 90 1000.75

0.8

0.85

0.9

0.95

1

1.05

1.1

h[m]

t[s]

The course of the liquid level Simulation model in Simulink

Modelling of the hydraulic drives

Hydraulic drives the power transmission is realized by the fluid. Hydrodynamic drives x hydrostatic drives Hydrodynamic drives use the kinetic energy of the fluid. Hydrostatic drives use the potential energy of the fluid due to pressure. The hydraulic circuit is a set of the elements needed for the transformation of the energy and their transmission. The basic elements forming the hydraulic circuit are: hydraulic pump, hydraulic cylinder, flow valves, pressure valves, pipelines, hoses etc.

The transmitted power is given by: pQP = .

Description of the hydraulic circuits

The properties of the hydraulic circuits are described in general by the system of algebraic equations, which we obtain using the generalized Kirchhoff`s laws used in the circuit theory. They have the following form for the hydraulic circuits: I. The sum of the actual flows into and from the node is equal to zero.

II. The sum of the actual pressure drops around any closed loop is equal to zero. The number of the created linear independent equations must be equal to the number of variables. It is not possible to use both laws on each node and each closed loop, but on the independent nodes and independent loops in the same way as by the description of the electric circuits.

=

=n

iiQ

1

0

=

=n

iip

1

0

Hydraulic lumped elements

R, R

H, H

D, D

The basic system elements in an engineering system can be divided into two groups: energy storage elements and energy dissipation elements. Fluid resistor R (fluid resistance) Fluid inertor H (fluid inertance, fluid inductance) Fluid capacitor C, (Fluid capacitance) or resistance against the deformation D

The transmission of the pressure energy is not without losses. The pressure energy of the fluid is changed into the other forms of energy dissipation of the energy, mainly into the thermal energy, kinetic energy and deformation energy.

Hydraulic resistance

nQpR =

Linear hydraulic resistance n=1

QpRL

= QRp L .= [ ] sNmRL 5=

Laminar flow

Turbulent flow Nonlinear resistance n=2

2QpRN

= 2.QRp N=

[ ] 28 = sNmRN

d.vRe =Reynolds number:

2320ReRe = krit

kritReRe

Where v is the velocity of the flow, d is a characteristic dimension (by the pipeline the inner diameter), kinematic viscosity.

Hydraulic resistance

.2v..2

dlp =Laminar flow

2..4vdQ

SQ

== d.v

.64Re64

==Friction coefficient Flow velocity

.Qd

lp 4....128

= 4....128

dlvRL =

4 Re3164,0

=BTurbulent flow Experimental obtained formula - Blasius

252

2

.....8..

21...

2v.. Q

dl

SQ

dl

dlp

=

==52 ....8dlRN

=

Hydraulic Inductance

Sp1 p2 x, v, a

m

The hydraulic inductance can be derived from the motion equation Newton`s law for the piston of the hydraulic cylinder. The piston and load mass is m:

( ) ... 21 Sppdtdvm =

p

S

vp1

V; m; p2

Hydraulic inductance the column of the liquid

Hydraulic Inductance piston and load mass is m dtdv

SS

Smp ..=

dtdQ

Smp .2=

2SmH =

Sl

SSl

SmH ... 22

===

dtdQpH =

Definition of the hydraulic inductance

[ ] 52

4 mNs

mkgH ==

Hydraulic Capacitance

The hydraulic capacitance CH describes the fact that by the change of the pressure - difference p - the fluid changes the volume the difference V.

VDp = . VpD

=

pVCH

=

Resistance against the deformation

Hydraulic capacitance

VVKp = .

[ ] 5mND =

( )=t

dQV0

( )

= t

dQ

pD

0

dtpd

DQ = .1

VCp H = .

K is the bulk modulus.

Mathematical models of the elements of the hydraulic circuits

Hydraulic circuit The hydraulic circuit consists of the bas ic hydraul ic elements: Pump (gear, piston, vane pump), relief valve, control valve proportional valve or servovalve which control the direction and velocity of the motion, hydraulic cylinder, load and moved mass, pipelines, filter and tank.

u

x,v

m

Pump QHG

DifferencialCylinder

Directional Control Valve

Relief Valve p0

Filter

M

Load Force F SB

SA

pA pB

Pump

R = 1GP P

Qt

Qt

QtQs

Qs

Qz

Qz

p

Q

p

Symbol, resistance circuit and static characteristic of the pump

GRp p

=1

pGQRpQQQQ ptp

tzts =

==

Q V nt t=

sin44

22

== ppp

g Dd

ihd

V

Band axis axial piston pump

The dynamics of the positioning system can be described by the second order system using the differential equation or transfer function:

HGHGHGHGHG uKyyTyT =++ !!! 2 12)( 2 ++= sTsTKsG

HGHGHG

HGHG

KyuHG HG

= max

max

Pump

KHGTHGHG

yHGy

KQGQHG

Hydrogenertor

uHG ps

GHG

1/CHsQ

Block diagram of the controlled pump

The linear mathematical model of the pump with the hydraulic capacity CH can be written in the form of the state model:

sHGHG

HG

s

HG

HG

H

HG

H

QG

HG

HG

HGs

HG

HG

QuTK

pvy

CG

CK

TTpvy

+

=

100

0

0

0

021

010

22

!!!

Hydraulic pipeline

C

p i p i+1 Q i

T - element

Hi

R Hi L Hi R Hi L Hi

Q i+1

C C

R Hi L Hi

- element

Hi Hi

Q i p

I p i+1

Q i Q i+1

C

R Hi L Hi p

i p I+1

L - element

Hi

Q i p i

Q i Q i+1

The structures of the different models of the piece of the pipeline.

l fc

nl fc