Mathworks SIMSCAPE 3 Reference

Simscape™ 3

Reference

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Simscape™ Reference

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Revision History

March 2007 Online only New for Version 1.0 (Release 2007a) September 2007 Online only Revised for Version 2.0 (Release 2007b) March 2008 Online only Revised for Version 2.1 (Release 2008a) October 2008 Online only Revised for Version 3.0 (Release 2008b) March 2009 Online only Revised for Version 3.1 (Release 2009a) September 2009 Online only Revised for Version 3.2 (Release 2009b) March 2010 Online only Revised for Version 3.3 (Release 2010a)

Block Reference

Foundation ....................................... 1-2

Electrical Hydraulic Magnetic Mechanical Physical Signals Pneumatic Thermal

........................................ 1-2

........................................ 1-5

........................................ 1-6

....................................... 1-7

.................................. 1-10

....................................... 1-12

......................................... 1-14

Contents

Utilities

........................................... 1-16

Blocks — Alphabetical List

Function Reference

Language Reference

Simscape Foundation Domains

Domain Types and Directory Structure .............. 5-2

vi Contents

Electrical Domain

Hydraulic Domain

Magnetic Dom ain

Mechanical Rotational Domain

Mechanical Translational Domain

Pneumatic Domain

Thermal Domain

................................. 5-4

................................. 5-5

.................................. 5-7

................................ 5-10

.................................. 5-12

..................... 5-8

.................. 5-9

Configuration Parameters

Simscape Pane: General ............................ 6-2

Simscape Pane Overview Editing Mode Explicit solver used in model containing Physical Networks

blocks

Input f il teri ng used in model containing Physical Networks

blocks Log simulation data Workspace v aria ble name Limit data points Data history (last N steps)

..................................... 6-5

......................................... 6-7

......................................... 6-9

.................................. 6-12

........................... 6-4

............................... 6-10

........................... 6-11

.......................... 6-13

Bibliography

Glossary

Index

vii

viii Contents

Block Reference

Foundation (p. 1-2 )

Utilities (p. 1-16) Essential environment blocks for

Basic hydraulic, pneumatic, mechanical, electrical, magnetic, thermal, and physical signal blocks

creating Physical Networks models

1 Block Reference

Foundation

Electrical (p. 1-2)

Hydraulic (p. 1-5)

Magnetic (p.

Mechanical (p. 1-7) Mechanical elements for rotational

Physical Signals (p. 1-10) Blocks for transmitting physical

Pneumatic (p. 1-12)

Thermal (p. 1-14)

1-6)

Basic electrical diagram blocks, such as inductors, diodes, capacitors, sensors and source s

Basic hydraul such as orif ic and sources,

Basic electromagnetic diagram blocks, such as reluctances, electromagnetic converters, sensors and sources

and translational motion, as well as mechanical sensors and sources

control signals

Basic pneumatic diagram blocks, such as orifices, chambers, sensors and s ou rces, and pneumatic utilities

thermal blocks, such as heat

Basic

sfer blocks, thermal mass,

tran

ors and source s

sens

ic diagram blocks,

es, chambers, sensors

and hydraulic utilities

1-2

Electrical

ctrical E lements (p. 1-3)

Ele

Electrical Sensors (p. 1-4) Current and voltage sensors

Electrical Sources (p. 1-4) Current and voltage sources

Electrical building blocks, such as inductors, diodes, and capacitors

Electrical Elements

Capacitor Simulate linear capacitor in

electrical systems

Foundation

Diode

Electrical Reference Simulate connection to electrical

Gyrator Simulate ideal gyrator in electrical

Ideal Transformer Simulate ideal transformer in

Inductor

Mutual Inductor

Op-Amp Simulate ideal operational amplifier

Resistor

Rotational Electromechanical Converter

Switch Simulate switch controlled by

Translational Electromechanical Converter

Simulate piecewise linear diode in electrical systems

ground

systems

electrical systems

Simulate linear inductor in electrical systems

Simulate mutual inductor in electrical systems

Simulate linear resistor in electrical systems

Provide i nterface between electrical and mechanical rotational domains

external physical signal

Provide i nterface between electrical and mechanical translational domains

Variable Resistor

Simulate linear variable resistor in electrical systems

1-3

1 Block Reference

Electrical Sensors

Current Sensor Simulate current sensor in electrical

systems

Voltage Sensor Simulate v oltage sensor in electrical

systems

Electrical Sources

AC Current Source Simulate ideal sinusoidal current

source

AC Voltage Source Simulate ideal constant voltage

source

Controlled Current Source Simulate ideal current source driven

by input signal

Controlled Voltage Source Simulate ideal voltage source driven

by input signal

1-4

Current-Controlled Current Source Simulate linear current-controlled

current source

Current-Controlled Voltage Source Simulate linear current-controlled

voltage source

DC Current Source Simulate ideal constant current

source

DC Voltage Source Simulate ideal constant voltage

source

Voltage-Controlled Current Source Simulate linear voltage-controlled

current source

Voltage-Controlled Voltage Source Simulate linear voltage-controlled

voltage source

Hydraulic

Foundation

Hydraulic Elements (p. 1-5)

Hydraulic Sen

Hydraulic So

Hydraulic Ut

sors (p. 1-6)

urces (p. 1-6)

ilities (p. 1-6)

Hydraulic Elements

Constant A

Constant Chamber

Fluid Inertia

Hydra

Hydr

rea Hydraulic Orifice

Volume Hydraulic

ulic Piston Chamber

aulic Reference

Hydraulic build such as orifice hydro-mechani

Hydraulic sensors

Hydraulic sources

Basic hydraulic environment blocks, such as custom hydraulic fluid

Simulate h constant

Simulate constan

Simulat tube or fluid v

Simul capac

Simu pres

cross-sectional area

tvolume

e pressure differential across channel due to change in

elocity

ate variable volume hydraulic

ity in cylind ers

late connection to atmospheric

sure

ing blocks,

s, chambers, and

cal converters

ydraulic orifice with

hydraulic capacity of

Hydraulic Resistive Tube

Linear Hydraulic Resistance

Rotational Hydro-Mechanical Converter

Translational Hydro-Mechanical Converter

late hydraulic pipeline which

Simu

ounts for friction losses only

acc

ulate hydraulic pipeline with

Sim

near resistance losses

mulate ideal hydro-mechanical

ansducer as building block for

otary actuators

imulate single chamber of hydraulic

S cylinder as building block for various cylinder models

1-5

1 Block Reference

Variable Area Hydraulic O rifice Simulate hydraulic variable orifice

created by cylindrical spool and sleeve

Variable Hydraulic Chamber Simulate hydraulic capacity of

variable volume with compressible fluid

Hydraulic Sensors

Hydraulic Flow Rate Sensor Simulate ideal flow meter

Hydraulic Pressure Sensor Simulate ideal pressure sensing

device

Hydraulic Sources

Hydraulic Flow Rate Source Simulate ideal source of hydraulic

energy, characterized by flow rate

1-6

Hydraulic Pressure Source Simulate ideal source of hydraulic

energy, characterized by pressure

Hydraulic Utilities

Custom Hydraulic Fluid Set working fluid prop erties by

specifying parameter values

Magnetic

Magnetic Elements (p. 1-7)

Magnetic Sensors (p. 1-7) Flux and mmf sensors

Magnetic Sources (p. 1-7) Flux and mmf sources

Magnetic building blocks, such as reluctances, electromagnetic converters, and actuators

Magnetic Elements

Electromagnetic Converter Simulate lossless electromagnetic

energy conversion device

Magnetic Reference Simulate reference for magnetic

ports

Foundation

Reluctance

Reluctance Force Actuator Simulate magnetomotive device

Variable R eluctance

Simulate magnetic reluctance

based on reluctance force

Simulate variable reluctance

Magnetic Sensors

Flux Sensor Simulate ideal flux s ensor

MMF Sensor Simulate ideal magnetomotive force

sensor

Magnetic Sources

Controlled Flux Source Simulate ideal flux source driven by

input signal

Controlled MMF Source Simulate ideal magnetomotive force

source driven by input signal

Flux Source Simulate ideal flux source

MMF Source Simulate ideal magnetomotive force

source

Mechanical

Mechanical Sensors (p. 1-8)

Mechanical Sources (p. 1-8)

Mechanisms (p. 1-9)

Mechanical sensors and sources

Various simple mechanisms

1-7

1 Block Reference

Rotational E lements (p. 1-9) Mechanical elements for rotational

motion

Translational Elements (p. 1-9) Mechanical elements for

translational motion

Mechanical Sensors

Ideal Force Sensor Simulate force sensor in mechanical

translational systems

Ideal Rotational Motion Sensor Simulate motion sensor in

mechanical rotational systems

Ideal Torque Sensor Simulate torque sensor in

mechanical rotational systems

Ideal Trans lational Motion Sensor Simulate motion sensor in

mechanical translational systems

1-8

Mechanical Sources

Ideal Angular Velocity Source Simulate ideal angular velocity

source in mechanical rotational systems

Ideal Force Source Simulate ideal source of mechanical

energy that generates force proportional to the input signal

Ideal Torque Source Simulate ideal source of mechanical

energy that generates torque proportional to the input signal

Ideal Translational Velocity Source Simulate ideal velocity source in

mechanical translational systems

Mechanisms

Gear Box Simulate gear bo xe s in mechanical

systems

Foundation

Lever

Wheel and Axle Simulate wheel and axle mechanism

Simulate lever in mechanical systems

in mechanical system s

Rotational Elements

Inertia

Mechanical Rotational Reference Simulate reference for mechanical

Rotational Damper

Rotational Friction

Rotational Hard Stop Simulate double-sided rotational

Rotational Spring Simulate ideal spring in mechanical

Simulate inertia in mechanical rotational systems

rotational ports

Simulate viscous damper in mechanical rotational systems

Simulate friction in contact between rotating bodies

hard stop

rotational systems

Translational Elements

Mass

Mechanical Translational Reference Simulate reference for mechanical

Translational Damper

Simulate mass in mechanical translational systems

translational ports

Simulate viscous damper in mechanical translational systems

1-9

1 Block Reference

Translational Friction

Translational Hard Stop Simulate double-sided translational

Translational Spring Simulate ideal spring in mechanical

Simulate friction in contact between moving bodies

hard stop

translational systems

Physical Signals

Functions (p. 1-10) Perform math operations on physical

signals

Linear Operators (p. 1-11) Simulate continuous-time functions

for physical signals

Lookup Tables (p. 1-11) Perform one- and two-dimensional

table lookup to generate physical signals

Nonlinear O perators (p. 1-11) Simulate discontinuities, such as

saturation or dead zone, for physical signals

Sources (p. 1-12) Simulate physical signal sources

1-10

Functions

PS Add Add two physical signal inputs

PSDivide Computesimpledivisionoftwo

input physical signals

PS Gain

PS Math Function Apply mathematical function to

Multiply input physical signal by constant

input physical signal

Foundation

PS Product

PS Subtract Compute simple subtraction of two

Multiply two physical signal inputs

input physical signals

Linear Operators

PS Integrator

Integrate physical signal

Lookup Tables

PS Lookup Table (1D) Approximate one-dimensional

function using specified lookup method

PS Lookup Table (2D) Approximate two-dimensional

function using specified lookup method

Nonlinear Operators

PS Abs Output absolute value of input

physical signal

PS Ceil Output the smallest integer larger

than or equal to input physical signal

PS Dead Zone Provide region of zero output for

physical signals

PS Fix

PS Floor Output the largest integer smaller

PS Max Output maximum of two input

PS Min Output minimum of two input

Round input physical signal toward zero

than or equal to input physical signal

physical signals

1-11

1 Block Reference

PS Saturation Limit range of physical signal

PS Sign Output sign of input physical signal

PS Switch Simulate single-pole double-throw

switch controlled by external physical signal

Sources

PS Constant Generate constant physical signal

Pneumatic

Pneumatic Elements (p. 1-12)

Pneumatic Sensors (p. 1-13)

Pneumatic Sources (p. 1-13)

Pneumatic Utilities (p. 1-14)

Pneumatic building blocks, such as orifices, chambers, and pneumo-mechanical converters

Pneumatic sensors

Pneumatic sources

Basic pneumatic environment blocks, s uch as gas properties

Pneumatic Elements

Adiabatic Cup Simulate thermal element with no

thermal mass and perfect insulation

Constant Area Pneumatic Orifice Simulate sharp-edged orifice in

pneumatic systems

Constant Area Pneumatic Orifice (ISO 6358)

Constant Volume Pneumatic Chamber

Simulate fixed-area pn eumatic orifice complying with ISO 6358 standard

Simulate constant volume pneumatic chamber based on ideal gas law

1-12

Pneumatic A bso lute Reference Simulate reference to zero absolute

pressure and temperature for pneumatic ports

Pneumatic Atmospheric Reference Simulate reference to ambient

pressure and temperature for pneumatic ports

Pneumatic Piston Chamber Simulate translational pneumatic

piston chamber based on ideal gas law

Foundation

Pneumatic Resistive Tube

Rotary Pneumatic Piston Chamber Simulate rotational pneumatic

Rotational Pneumatic-Mechanical Converter

Variable Area Pneumatic Orifice Simulate variable orifice in

Simulate pressure loss and added heatduetoflowresistancein pneumatic pipe

piston chamber based on ideal gas law

Provide interface between pneumatic and mechanical rotational domains

pneumatic systems

Pneumatic Sensors

Pneumatic Mass & H eat Flow Sensor Simulate ideal mass flow and heat

flow sensor

Pneumatic Pressure & Temperature Sensor

Simulate ideal pressure and temperature sensor

Pneumatic Sources

Controlled Pneumatic Flow Rate Source

Simulate ideal compressor with signal-controlled mass flow rate

Controlled Pneumatic Pressure Source

Simulate ideal compressor with signal-controlled pressure difference

1-13

1 Block Reference

Pneumatic Flow Rate Source Simulate ideal compressor with

constant mass flow rate

Pneumatic Pressure Source Simulate ideal compressor with

constant pressure difference

Pneumatic Utilities

Gas Properties Specify pneumatic domain properties

for attached circuit

Thermal

Thermal Elements (p. 1-14)

Thermal Sensors (p. 1-15) Temperature and heat flow sensors

Thermal Sources (p. 1-15) Temperature and heat flow sensors

Thermal building blocks, such as thermal mass and various heat transfer blocks

and sources

Thermal Elements

Conductive Heat Transfer Simulate heat transfer by conduction

Convective Heat Transfer Simulate heat transfer by convection

Radiative Heat Transfer Simulate heat transfer by radiation

Thermal Mass

Thermal Reference Simulate reference for thermal ports

Simulate mass in thermal systems

1-14

Thermal Sensors

Ideal Heat Flow Sensor Simulate ideal heat flow meter

Ideal Temperature Sensor Simulate ideal temperature sensor

Thermal Sources

Ideal Heat Flow Source Simulate ideal source of thermal

energy, characterized by heat flow

Ideal Temperature Source Simulate ideal source of

thermal energy, characterized by temperature

Foundation

1-15

1 Block Reference

Utilities

Connection Port Create P hysical Modeling connector

port for subsystem

PS-Simulink Converter Convert physical signal into

Simulink

Simulink-PS Converter Convert Simulink input signal into

physical signal

Solver Configuration

Two-Way Connection Create two-way connector port for

Represent Physical Networks environment and solver configuration

subsystem

output signal

1-16

Blocks — Alphabetical List

AC Current Source

Purpose Simulate ide al sinusoidal current source

Library Electrical Sources

Description The AC Current Source block represents an ideal current source that

maintains sinusoidal current through it, independent of the voltage

across its terminals.

The output current is defined by the following equation:

II t=+

iisin( )ωϕ

where

Dialog Box a nd Parameters

The p ositive direction of the current flow is indicated by the arro w.

Current

Peak amplitude

Frequency

Phase shift

Time

2-2

AC Current Source

Peak amplitude

Peak current amplitude. The default value is 10*sqrt(2), or

14.1421 A.

Phase shift

Phase shift in angular units. The default value is

Frequency

Current frequency. The default value is

60 Hz.

Ports The block has two electrical conserving ports associated with its

terminals.

See Also Convective Heat Transfer

Radiative Heat Transfer

401

2-11

Connection Port

Purpose Create Physical Modeling connector port for subsystem

Library Utilities

Description The Connection Port block transfers both the conserving and the

physical signal connections to the outside boundary of a subsystem

block. This transfer is similar to the Inport and Outport blocks in

Simulink models. A subsystem needs a Connection Port block for each

physical connection line that crosses its boundary. You can manually

place a Conne ction Port block inside a subsystem, or Simulink can

automatically insert a Connection P ort block when you create a

subsystemwithinanexistingnetwork.

Port Appearance on Subsystem Boundary

The ports on the subsystem boundary change their appearance

depending on the type of port to which the Connection Port block is

connected inside the subsystem.

Connection Port Block Inside a Subsystem Connects to ...

A Conserving port

A Physical Signal inport or outport

Atwo-wayconnectorportofthe Two-Way Connection block

A SimMechanics™ connector port, either:

Round connector port Round connector port

Body coordinate sys tem port

2-12

... and Appears on the Outside Boundary of the Subsystem as ...

A square Conserving port

A triangular Physical Signal

A two-way connector port

A SimMechanics connector port, either:

Body coordinate system port

inport or outport

Connection Port

Port Location and Orientation on Subsystem Boundary

The orientation of the parent subsystem block and your choice of port location determine the Connection Port block port location on the p arent subsystem boundary.

• A subsystem is in its fundamental orientation when its Simulink

signal inports occur on its left side and its Simulink signal outports occur on its right side.

When a subsystem is oriented in this way, the actual port location on the subsystem boundary respects your choice of port location (left or right) for the connector port.

• A subsyste m orientation is reversed, with left and right interchanged,

when its Simulink signal inports occur on i ts right side and its Simulink signal outports occur on its left side.

When a subsystem is oriented in this way, the actual port location on the subsystem boundary reverses your choice of port location. If you choose left, the port appears on the right side. If you choose right, the port appears on the left side.

2-13

Connection Port

Dialog Box a nd Parameters

Port number

Labels the subsystem connector port that this block creates. Each connector port on the boundary of a single subsystem requires a unique number as a label. The default v alue for the first port is

Port location on parent subsystem

Choose here which side of the parent subsystem boundary the port is located. T h e choices are is

Left.

See “Port Location and Orientation on Subsystem Boundary” on page 2-13.

Left or Right. The default choice

See Also In the Simulink User’s Guide, see “Working w ith Block Masks”.

2-14

Constant Area Hydraulic Orifice

⎝

⎠

Purpose Simulate hydraulic orifice with constant cross-sectional area

Library Hydraulic Elements

Description The Constant Area Hydraulic Orifice block models a sharp- e d ged

constant-area orifice. T he model distinguishes between the laminar and turbulent flow regimes by comparing the Reynolds number with its critical value. The flow rate through the orifice is proportional to the pressure differential across the orifice, and is determined according to the follow i ng equations:

⎧ ⎪

⎪

⎨ ⎪ ⎪

⎩

pp p

=−

Re =

where

pA,p

C A p sign p Re Re

CADpReRe

⎛

⎜ ⎜

| | for >=

νρ

()

iν

⎞

⎟ ⎟

for <

Flow rate

Pressure differential

Gauge pressures at the block terminals

2-15

Constant Area Hydraulic Orifice

Basic Assumptions and Limitations

Flow discharge coe fficient

Orifice passage area

Orifice hydraulic diameter

Fluid density

Fluid kinematic viscosity

Reynolds number

Critical Reynolds num ber

The block positive direction is from port A to port B. This means

thattheflowrateispositiveifitflowsfromAtoB,andthepressure

differential is determined as

pp p

=−

The model is based on the following assumptions:

• Fluid inertia is not taken into account.

• The transition between laminar and turbulent regimes is assumed to

be sharp and taking place exactly at

Re=Re

2-16

Dialog Box a nd Parameters

Constant Area Hydraulic Orifice

Orifice area

Orifice passage area. The default value is

Flow discharge coefficient

Semi-empirical parameter for orifice capacity characterization. Its value depends on the geometrical properties of the orifice, and usually is provided in textbooks or manufacturer data sheets. The default value is

Critical Reynolds number

The maximum Reynolds number for laminar flow. The transition from laminar to turbulent re gime is supposed to take place when the Reynolds number reaches this value. The value of the parameter depends on orifice geometrical profile, and the recommendations on the parameter value can be found in hydraulic textbooks. The default value is a round orifice in thin material with sharp edges.

0.7.

1e-4 m^2.

12, which corresponds to

2-17

Constant Area Hydraulic Orifice

Global Parameters

Fluid density

The parameter is determined by the type of working fluid selected for the system under design. Use the Custom Hydraulic Fluid block, or the Hydraulic Fluid block a vailable with SimHydraulics block libraries, to specify the fluid properties.

Fluid kinematic viscosity

The parameter is determined by the type of working fluid selected for the system under design. Use the Custom Hydraulic Fluid block, or the Hydraulic Flui d block available with SimHydraulics block libraries, to specify the fluid properties.

Ports The block has the following ports:

Hydraulic co n serving port a sso ciated with the orifice inlet.

Hydraulic conserving port associated with the orifice outlet.

See Also Variable Area Hydraulic Orifice

2-18

Constant Area Pneumatic Orifice

Purpose Simulate sharp-edged orifice in pneumatic systems

Library Pneumatic Elements

Description The Constant Area Pneumatic Orifice block models the flow rate of an

ideal gas through a sharp-edged orifice.

Theflowratethroughtheorificeisproportion al to the orifice area and the pressure differential across the orifice.

⎤

⎞

⎛

⎜

⎝

⎥

⎟

⎥

⎠

⎥ ⎦

GCAp

ii i

211

−

⎡

⎛

⎢

⎜

⎢

RTpp

⎝

⎢ ⎣

⎞

−

⎟

⎠

where

G Mass flow rate

Discharge coefficient, to account for effective loss of area due to orifice shape

i,po

Orifice cross-sectional area

Absolute pressures at the orifice inlet and outlet, respectively. The inlet and outlet change depending on flow direction. For positive flow (G >0),p

The ratio of specific heats a t constant pressure and constant volume, c

p/cv

Specific gas constant

Absolute gas temperature

= pA,otherwisepi= pB.

The choked flow occurs at the critical pressure ratio defined by

⎛

⎜

⎝

−

⎞

⎟

⎠

2-19

Constant Area Pneumatic Orifice

after which the flow rate depends on the inlet pressure only and is

computed with the expression

γγ1

GCAp

ii i

diicr

The square root relationship has infinite gradient at zero flow, which

can present numerical solver difficulties. Therefore,forverysmall

pressure differences, defined by p

replaced by a linear flow-pressure relationship

GkCAT p p

=−

di io

where k is a constant such that the flow predicted for po/piis the same

as that predicted by the original flow equatio n for p

Theheatflowoutoftheorificeisassumedequaltotheheatflowinto

the orifice, based on the following considerations:

−

05.

()

> 0.999, the flow equation is

o/pi

=0.999.

2-20

• The orifice is square-edged or sharp-edged, and as such is

characterized by an abrupt change of the downstream area. This means that practically all the dynamic pressure is lost in the expansion.

• The lost energy appears in the form of internal energy that rises the

output temperature and makes it very close to the inlet temperature.

Therefore, q

= qo,whereqiand qoare the input and output heat flows,

respectively.

The block positive direction is from port A to port B. This means that

theflowrateispositiveifitflowsfromAtoB.

Constant Area Pneumatic Orifice

Basic Assumptions and Limitations

Dialog Box a nd Parameters

The model is based on the following assumptions:

• The gas is ideal.

• Specific heats at constant pressure and constant volume, c

are constant.

• The process is adiabatic, that is, there is no heat transfer with the

environment.

• Gravitational effects can be neglected.

• The orifice adds no net heat to the flow.

and cv,

Discha

Orifi

rge coefficient, Cd

Semi-e Its val usual The de

ce area

Spec m^2.

mpirical parameter for orifice capacity characterization.

ue depends on the geometrical properties of the orifice, and

ly is provided in textbooks or manufacturer data sheets.

fault value is

ify the orifice cross-sectional area. The default value is

0.82.

Ports The block has the following ports:

1e-5

2-21

Constant Area Pneumatic Orifice

Pneumatic conserving port associated with the orifice inlet for positive flow.

Pneumatic conserving port associated with the orifice outlet for positive flow.

See Also Constant Area Pneumatic Orifice (ISO 6358)

Variable Area Pneumatic Orifice

2-22

Constant Area Pneumatic Orifice (ISO 6358)

Purpose Simulate fixed-area pneumatic orifice complying with ISO 6358

standard

Library Pneumatic Elements

Description The Constant Area Pneumatic Orifice (ISO 6358) block models the flow

rate of an ideal gas through a fixed-area sharp-edged o rifice. The model conforms to the ISO 6358 standard and is based on the following flow equations, originally proposed by Sanville [1]:

⎧

⎪

⎛ ⎜

⎝

⎞

−

1iiif (laminarβ ))

⎟

⎠ ⎪ ⎪ ⎪

⎪

ii iρβ1

⎨

i ref

⎪

ref

⎪ ⎪

ref

⎪

iiρ<=

⎪

i ref

⎪⎪

⎩

−

lam

where

G Mass flow rate

lam

Pressure ratio at laminar flow, a value between 0.999 and

0.995

Critical pressure ratio, that is, the ratio between the outlet pressure p

and inlet pressure piat which the gas velocity

achieves sonic speed

ref

−

⎛

−

⎜ ⎝

sign p p

⎛ ⎜ ⎜ ⎜ ⎜

⎝

lam

−

()

⎞

−

⎟

−

⎟ ⎟

⎠

−

⎞ ⎟

−

⎠

lam

if (pC

if (choked)pC

lam

ssubsonic)

2-23

Constant Area Pneumatic Orifice (ISO 6358)

C Sonic conductance of the component, that is, the ratio

between the mass flow rate and the product of inlet pressure

and the mass density at standard conditions when the flow

is choked

ref

i,po

i,To

Gas de nsity at standard conditions (1.185 kg/m^3 for air)

Absolute pressures at the orifice inlet and outlet, respectively. The inlet and outlet change depending on flow direction. For positive flow (G >0),p

= pA,otherwisepi= pB.

Absolute gas temperatures at the orifice inlet and outlet, respectively

ref

Gas temperature at standard conditions (T

=293.15K)

ref

The equation itself, parameters b and C, and the heuristic on how to measure these parameters experimentally form the basis for the standard ISO 6358 (1989). The values of the critical pressure ratio b and the sonic conductance C depend on a particular design of a component. Typically, they are determined experimentally and are sometimes given on a manufacturer data sheet.

The block can also be parameterized in terms of orifice effective area or flow coefficient, instead of sonic conductance. When doing so, block parameters are co nv erted into an equivalent value for sonic conductance. When specifying effective area, the following formula proposed by Gidlund and de tailed in [2] is used:

C =0.128d

where

C Sonic conductance in dm^3/(s*bar)

Inner diameter of restriction in mm

The effective area (whether specified directly, or calculated when the orifice is parameterized in terms of C

or Kv, as described below) is used

2-24

Constant Area Pneumatic Orifice (ISO 6358)

to determine the inner diameter d in the Gidlund formula, assuming a circular cross section.

Gidlund also gives an approximate formula for the critical pressure ratio in terms of the pneumatic line diameter D,

b = 0.41 + 0.272 d / D

This equation is not used by the block and you must specify the critical pressure ratio directly.

If the orifice is parameterized in terms of the C the C

coefficient is turned into an equivalent effective o rifice area for

[2] coefficient, then

use in the Gidlund formula:

A = 1.6986e – 5 C

By definition, an opening or restriction has a Cvcoefficient of 1 if it passes 1 gpm (gallon per minute) of water at pressure drop of 1 psi.

If the orifice is parameterized in ter ms of the K the K

coefficient is turned into an equivalent effective o rifice area for

[2] coefficient, then

use in the Gidlund formula:

A = 1.1785e – 6 C

Kvis the SI counterpart of Cv. An opening or restriction has a K

coefficient of 1 if it passes 1 lpm (liter per minute) of water at pressure drop of 1 bar.

Theheatflowoutoftheorificeisassumedequaltotheheatflowinto the orifice, based on the following considerations:

• The orifice is square-edged or sharp-edged, and as such is

characterized by an abrupt change of the downstream area. This means that practically all the dynamic pressure is lost in the expansion.

2-25

Constant Area Pneumatic Orifice (ISO 6358)

• The lost energy appears in the form of internal energy that rises the

output temperature and makes it very close to the inlet temperature.

Basic Assumptions and Limitations

Therefore, q respectively.

The block positive direction is from port A to port B. This means that theflowrateispositiveifitflowsfromAtoB.

The model is based on the following assumptions:

• The gas is ideal.

• Specific heats at constant pressure and constant volume, c

are constant.

• The process is adiabatic, that is, there is no heat transfer with the

environment.

• Gravitational effects can be neglected.

• The orifice adds no net heat to the flow.

= qo,whereqiand qoare the input and output heat flows,

and cv,

2-26

Dialog Box a nd Parameters

Constant Area Pneumatic Orifice (ISO 6358)

2-27

Constant Area Pneumatic Orifice (ISO 6358)

2-28

Orifice is s pecified with

Select one of the following model parameterization methods:

Constant Area Pneumatic Orifice (ISO 6358)

• Sonic conductance —Providevalueforthesonicconductance

of the orifice. The values of the sonic conductance and the critical pressure ratio form the basis for the ISO 6358 compliant flow equations for the orifice. This is the default m ethod.

•

Effective area — Provide value for the orifice effective

area. This value is internally converted by the block into an equivalent value for sonic conductance.

•

Cv coefficient (USCU) —Providevaluefortheflow

coefficient specified in US units. This value is internally converted by the block into an equivalent value for the orifice effective area.

•

Kv coefficient (SI) — Provide value for the flow coefficient

specified in SI units. This value is internally converted by the block into an equivalent value for the orifice effective area.

Sonic conductance

Specify the sonic conductance of the orifice, that is, the ratio between the mass flow rate and the product of upstream pressure and the mass density at standard conditions w hen the flow is choked. This value depends on the geometrical properties of the orifice, and usually is provided in textbooks or manufacturer data sheets. The default value is in the dialog box if Orifice is specified with parameter is set to

Sonic conductance.

1.6 l/s/bar. This parameter appears

Effective area

Specify the orifice cross-sectional area. The default value is m^2. This parameter appears in the dialog box if Orifice is specified with parameter is set to

Effective area.

Cv coefficient

Specify the value for the flow coefficient in US units. The default value is is specified with parameter is set to

0.6. This parameter appears in the dialog box if Orifice Cv coefficient (USCU).

1e-5

2-29

Constant Area Pneumatic Orifice (ISO 6358)

Kv coefficient

Specify the value for the flow coefficient in SI units. The default value is

is specified with parameter is set to

Critical pressure ratio

Specify the critical pressure ratio, that is, the ratio between the downstream pressure and the upstream pressure at w hich the gas velocity achieves sonic speed. The default value is

Pressure ratio at laminar flow

Specify the ratio b etween the downstream pressure and the upstream pressure at laminar flow. This value can be in the range between 0.995 and 0.999. The default value is

Temperature at standard conditions

Specify the gas temperature at which the sonic conductance was measured. The default value is

Pressure at standard conditions

Specify the gas pressure at which the sonic conductance was measured. The default value is

8.5. This parameter appears in the dialog box if Orifice Kv coefficient (SI).

293.15 K.

101325 Pa.

0.528.

0.999.

Ports The block has the following ports:

Pneumatic conserving port associated with the orifice inlet for positive flow.

Pneumatic conserving port associated with the orifice outlet for positive flow.

References [1] Sanville, F. E. “A New Method of Specifying the Flow Capacity of

Pneumatic Fluid Power Valves.” Paper D3, p.37-47. BHRA. Second International Fluid Power Symposium, Guildford, England, 1971.

[2] Beater, P. Pneumatic Drives. System Design, Modeling, and Control. New York: Springer, 2007.

2-30

Constant Area Pneumatic Orifice (ISO 6358)

See Also Constant Area Pneumatic Orifice

Variable Area Pneumatic Orifice

2-31

Constant Volume Hydraulic Chamber

Purpose Simulate hydraulic capacity of constant volume

Library Hydraulic Elements

Description The Constant Volume Hydraulic Chamber block models a fixed-volume

chamber with rigid or flexible walls, to be used in hydraulic valves, pumps, manifolds, pipes, hoses, and so on. Use this block in models where you have to account for some form of fluid compressibility. You can select the appropriate representation of fluid compressibility using the block parameters.

Fluid compressibility in its simplest form is simulated according to the following equations:

2-32

where

Flow rate into the chamber

Volume of fluid in the chamber

Geometrical chamber volume

Fluid bulk modulus

Gauge pressure of fluid in the chamber

If pressure in the chamber is likely to fall to negative values and approach cavitation limit, the above equations must be enhanced. In this block, it is done by representing the fluid in the chamb er as a mixture o f liquid and a small amount of entrained, nondissolved gas. The mixture bulk modulus is determined as:

where

Constant Volume Hydraulic Chamber

⎛

⎜

⎝

np p

()

⎞

⎟

⎠

Pure liquid bulk modulus

Atmospheric pressure

Relative gas content at atmospheric pressure, α =

Gas volume at atmospheric pressure

Volume of liquid

Gas-specific heat ratio

G/VL

The main objective of representing fluid as a mixture of liquid and gas is to introduce an approximate model of cavitation, which takes place in a chamber if pressure drops below fluid vapor saturation level. As it is seen in the graph below, the bulk modulus of a mixture decreases

At high pressure,

, thus considerably slowing down further pressure change.

→

, a small amount of nondissolved gas has

practically no effect on the system behavior.

2-33

Constant Volume Hydraulic Chamber

2-34

Cavitation is an inherently thermodynamic process, requiring consideration of multiple-phase fluids, heat transfers, etc., and as such cannot be accurately simulated with Simscape software. But the simplified version implemented in the block is good enough to signal if pressure falls below dangerous level, and to prevent computation failure that normally occurs at negative pressures.

If it is known t hat cavitation is unlikely in the system under design, you can set the relative gas content in the fluid properties to zero, thus increasing the speed of computations. Use the Hydraulic Fluid or the Custom Hydraulic Fluid block to set the fluid properties.

If chamber walls have noticeable compliance , the ab ov e equations must be further enhanced by representing geometrical chamber volume as a function of pressure:

Constant Volume Hydraulic Chamber

VdL

=π24/ i

where

() ()=

+1 τ

Coefficient

Internal diameter of the cylindrical chamber

Length of the cylindrical chamber

Proportionality coefficient (m/Pa)

Time constant

Laplace operator

establishes relationship between press ure and the

internal diameter at steady-state conditions. For metal tubes, the coefficient can be computed as (see [1]):

⎛

dEDd

⎜

−

⎝

⎞

⎟ ⎟

⎠

where

Pipe external diameter

Modulus of elasticity (Young’s modulus) for the pipe material

Poisson’s ratio for the pipe material

For hoses, the coefficient can be provided by the manufacturer.

The process of expansion and contraction in pipes and especially in hosesisacomplexcombinationofnonlinear elastic and viscoelastic deformations. This process is approximated in the block with the

2-35

Constant Volume Hydraulic Chamber

first-order lag, whose time constant is determined empirically (for example, see [2]).

As a result, by selecting appropriate values, you can implement four different models of fluid compressibility with this block:

• Chamber with rigid walls, no entrained gas in the fluid

• Cylindrical chamber with compliant walls, no entrained gas in the

fluid

• Chamber with rigid walls, fluid with entrained gas

• Cylindrical chamber with compliant walls, fluid with entrained g as

The block allows two methods of specifying the chamber size:

• By volume — Use this option for cylindrical or non-cylindrical

chambers w ith rigid walls. You only need to know the volume of the chamber. This chamber type does not account for wall compliance.

• By length and diameter — Use this option for cylindrical chambers

with rigid or compliant walls, such as circular pipes or hoses.

Basic Assumptions and Limitations

2-36

The block has one hydraulic conserving port associated with the chamber inlet. The block positive direction is from its port to the referencepoint. Thismeansthattheflowrateispositiveifitflows into the chamber.

The model is based on the following assumptions:

• No inertia associated with pipe walls is taken into account.

• Chamber with compliant walls is assumed to have a cylindrical

shape. Chamber with rigid wall can ha ve any shape.

Dialog Box a nd Parameters

Constant Volume Hydraulic Chamber

2-37

Constant Volume Hydraulic Chamber

2-38

Constant Volume Hydraulic Chamber

Chamber specification

Theparametercanhaveoneoftwovalues:

length and diameter

recommended if a chamber is formed by a circular pipe. If the parameter i s set to account. The default value of the parameter is

Chamber wall type

Theparametercanhaveoneoftwovalues: If the parameter is set to account, which can improve computational efficiency. The value

Compliant is recommended for hoses and metal pipes, where

.ThevalueBy length and diameter is

By volume, wall compliance is not taken into

Rigid, wall compliance is not taken into

By volume or By

By volume.

Rigid or Compliant.

2-39

Constant Volume Hydraulic Chamber

compliance can affect the system behavior. The default value of the p arameter is

specification parameter is set to

Chamber volume

Volume of fluid in the chamber. The default value is The parameter is used if the Chamber specification parameter is set to

By volume.

Chamber internal diameter

Internal diameter of the cylindrical chamber. The default value is

0.01 m. The parameter is used if the Chamber specification

parameter is set to

Cylindrical chamber length

Length of the cylindrical chamber. The default value is parameter is used if the Chamber specification param eter is set to

By length and diameter.

Static pressure-diameter coefficient

Coefficient the internal diameter at steady-state conditions. The parameter can be determ ined analytically or experimentally. The default value is

1.2e-12 m/Pa. The parameter is used if Chamber wall

type is set to

Rigid. The parameter is used if the Chamber

By length and diameter.

that establishes relationship between pressure and

Compliant.

By length and diameter.

1e-4 m^3.

1 m. The

2-40

Viscoelastic process time constant

Time constant in the transfer function relating pipe internal diameter to pressure variations. With this parameter, the simulated elastic or v iscoelastic process is approximated with the first-order lag. The parameter is determined experimentally or provided by the manufacturer. The default value is parameter is used if Chamber wall type is set to

Compliant.

Specific heat ratio

Gas-specific heat ratio. The default value is

1.4.

Initial pressure

Initial pressure in the chamber. This parameter specifies the initial condition for use in computing the block’s initial state at

0.01 s. The

Constant Volume Hydraulic Chamber

the beginning of a simulation run. For more information, see “Computing Initial Conditions”. The default value is

Restricted Parameters

When your model is in Restricted editing mode, you cannot modify the following parameters:

• Chamber specification

• Chamber wall type

All other block parameters are available for modification. The actual set of modifiable block parameters depends on the values of the Tube cross section type and Chamber wall type parameters at the time the model entered Restricted mode.

Global Parameters

Fluidbulkmodulus

The parameter is determined by the type of working fluid selected for the system under design. Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

Nondissolved gas ratio

Nondissolved gas relative content determined as a ratio of gas volume to the liquid volume. The parameter is determined by the type of working fluid se le cted for the system under design. Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

Ports The block has one hydraulic conserving port associated with the

chamber inlet.

References [1] Meritt, H.E., Hydraulic Control Systems, John Wiley & Sons, N ew

York, 1967

[2] Holcke, Jan, Frequency Response of Hydraulic Hoses, RIT, FTH, Stockholm, 2002

2-41

Constant Volume Hydraulic Chamber

Mathworks SIMSCAPE 3 Reference

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