mod&sim sist fis

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Modelamiento de Sistemas Dinamicos Dr. Jorge A. Olórtegui Yume, Ph.D.

Escuela Académico Profesional de Ingeniería Mecánica

Universidad Privada Cesar Vallejo

MODELAMIENTO DE SISTEMAS FISICOS

Dr. Jorge A. Olortegui Yume, Ph.D.

MOD. SIST. FISICOS Dr. Jorge A. Olortegui Yume, Ph.D.

MODELADO Y SIMULACION DE SISTEMAS FISICOS

For ease in writing linear lumped-parameter differential equations, the D operator (S operator) is introduced

The Laplace transform is used to represent a continuous time domain system, f(t), using a continuous sum of complex exponential functions of the form where s is a complex variable defined as . The complex domain (or s plane as it’s often called) is just a plane with a rectangular x–y coordinate system where is the real part and is the imaginary part


The cause–effect relationship for many systems can be approximated by a linear ordinary differential equation. For example, consider the following second-order dynamic system with one input, , and one output

The transfer function is the ratio of the output variable over the input variable represented as the ratio of two polynomials in the D or s operator

3 step procedure to convert linear ordinary differ. equation to transfer function



3 step procedure to convert linear ordinary differ. equation to transfer function

GAIN



Simulation is the process of solving a block diagram model on a computer

Block diagram models have 2 fundamental objects: signal wires and blocks

Fundamental set of 3 basic blocks that all block diagram languages possess



PROCESO DE DESARROLLO DEL SISTEMA MECATRONICO DE UN MOTOR DC DE POSICIONAMIENTO

Block diagrams are rarely constructed in a standard form, and it is often necessary to reduce them to more efficient or understandable forms


Block diagrams are rarely constructed in a standard form, and it is often necessary to reduce them to more efficient or understandable forms



• R is the input to the BFS • E is the control or error variable • Y is the output

Determination of closed-loop transfer function for the BFS



• R is the input to the BFS • E is the control or error variable • Y is the output



Ejemplo



Ejemplo

The block diagram is to be reduced such that two blocks are present: one in the forward loop and one in the feedback loop. The reduced system will be in BFS form




Most commonly used visual simulations environments are: • MATRIXX/System Build (National Instruments) • MATLAB/Simulink (Mathworks) • LabVIEW (National Instruments) • VisSim (Visual Solutions) • Easy5 (Boeing)



2 approaches for developing block diagram models from system illustrations:

• Direct Method • Modified Analogy Method



Block Diagram Modeling—Direct Method

• Modeling of simple models • Single-discipline models of multidiscipline models with minimal coupling

STARTING POINT: • Set of linear ODE’s • Transfer function • Illustration of the system itself




Transfer Function (or ODE) Conversion to Block Diagram Model

A 6-step process

• Ordinary differential equation (ODE) is a differential equation with all derivatives taken with respect to time

• Time is the independent variable • Set of initial conditions must be specified for each (time) derivative

term

• Transfer function in proper form: order of the numerator polynomial is less than or equal to the order of the denominator polynomial





Ejemplo Given

Step 1 Create the state variable, x(t), by “sliding” the numerator part of the transfer function into a new block located to the right of the denominator part of the transfer function

order of the transfer function





Step 2. From step 1, write the state equation (SE) as the differential equation relating the input, r(t), to the state, x(t).





ny-integrator blocks in series and connect them from left to right beginning with the highest derivative.

In our example

Initial conditions will be added in step 6.



Block Diagram Modeling—Direct Method Transfer Function (or ODE) Conversion to Block Diagram Model

Solve state equation for highest derivative of state variable

Using summing junction, implement previous state equation to block diagram




STATE EQUATIONS TO BLOCK DIAGRAM




Step 5. write the output equation (OE) as the differential equation relating the output, y(t), to the state, x(t), and its derivatives




Step 6. Add the initial conditions, i.e, translate IC’s from output variable, y(t), to state variable, x(t), its derivatives, and input

State: output of an integrator





Solving for the state and its derivatives





Add the initial conditions to the block diagram

BLOCK DIAGRAM WITH INITIAL CONDITIONS




Example Given the transfer function convert it to block diagram model

Solución



Block Diagram Modeling—Direct Method Conversion of Mechanical Illustrations to Block Diagram Models

• For single domain systems (Ex: mechanical translation or rotation • Method uses basic force relationships for : mass, spring ,damper

Example

Given the system illustration is used with input r, output y, and all required initial conditions obtain the block diagram model




Solution Step 1. For each mass in the illustration, write Newton’s 2nd Law equation and solve it for acceleration of the particular mass

Integrate as many times as needed to obtain x(t)




Solution




Example Given the system illustration obtain the block diagram model




Example Given the system illustration obtain the block diagram model

Solution

MOD. SIST. FISICOS Dr. Jorge A. Olortegui Yume, Ph.D. 40

MODELOS DE SISTEMAS BASICOS SISTEMAS MECANICOS

Sistemas Mecánicos: (a) Resorte, (b) Amortiguador, (c) Masa

BLOQUES FUNCIONALES


SISTEMAS MECANICOS

(a) Modelo masa-resorte-amortig., (b) bloque funcional (sistema), (c) DCL

SISTEMAS TRASLACIONALES

MODELOS DE SISTEMAS BASICOS


MODELOS DE SISTEMAS BASICOS SISTEMAS MECANICOS

Masa rotacional en el extremo de un eje (a) situación física (c) modelo

SISTEMAS ROTACIONALES


MODELOS DE SISTEMAS BASICOS SISTEMAS MECANICOS Bloques Funcionales Mecánicos


MODELOS DE SISTEMAS BASICOS SISTEMAS ELECTRICOS

(a) Resistor (b) Inductor (c) Capacitor

BLOQUES FUNCIONALES

iaCapacitanc :

aInductanci :

Eléctrica aResistenci :

Potencial de Diferencia :

Eléctrica Corriente :

C

L

R

v

i


MODELOS DE SISTEMAS BASICOS SISTEMAS ELECTRICOS Bloques Funcionales Eléctricos disipada Potencia :

almacenada Energía :

P

E


MODELOS DE SISTEMAS BASICOS SISTEMAS ELECTRICOS MODELADO

1era LEY DE KIRCHOFF Forma Práctica: Análisis de Nodos

0 salenentranii

0321 iii

321 iii

4321 RR

v

R

v

R

vv AAA

Usando ley de Ohm

R

vi



2da LEY DE KIRCHOFF Forma Práctica: Análisis de Mallas

ivv

21 RiiRiv IIII 1era Malla

2da Malla

2430 RiiRiRi IIIIIII

Sist. De 2 ecuaciones con 2 incognitas

04322

221

RRRiRi

vRiRRi

III

IIIIII ii e despeja Se



1era LEY DE KIRCHOFF Forma Práctica: Análisis de Nodos

321 iii 2da LEY DE KIRCHOFF Forma Práctica: Análisis de Mallas

III ii e despeja Se

II

III

I

ii

iii

ii

3

2

1

:calcula Se


MODELOS DE SISTEMAS BASICOS SISTEMAS ELECTRICOS MODELADO SISTEMA RESISTOR-CAPACITOR

i

2da LEY DE KIRCHOFF

CR vvv

iRvR

dt

dvRCv C

R

dt

dvCi C

CC v

dt

dvRCv

Ecuación lineal de 1er orden

INPUT

“v” OUPUT

“vC”


MODELOS DE SISTEMAS BASICOS SISTEMAS ELECTRICOS MODELADO SISTEMA RESISTOR-INDUCTOR-CAPACITOR

2da LEY DE KIRCHOFF

RLR vvvv

iRvR dt

dvRCv C

R dt

dvCi C

CCC v

dt

vdLC

dt

dvRCv

2

2

Ecuación lineal de 2do orden

INPUT

“v” OUPUT

“vC”

i

2

2

dt

vdLC

dt

dvC

dt

dL

dt

diLv CC

L


MODELOS DE SISTEMAS BASICOS SISTEMAS ELECTRICOS SIMILITUD ENTRE SISTEMAS MECANICOS Y ELECTRICOS

Fuerza (F) Velocidad (v) Const. de Amortiguamiento (c) Amortiguador Resorte Masa

Corriente (i) Diferencia de Potencial (v)

Conductancia (1/R) Resistor Inductor Capacitor


MODELOS DE SISTEMAS BASICOS SISTEMAS ELECTRICOS SIMILITUD ENTRE SISTEMAS MECANICOS Y ELECTRICOS

Amortiguador Resistor

22 1

1

vRR

vP

vRR

vi

2cvP

cvF


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS SIMILITUD ENTRE SISTEMAS ELECTRICOS Y FLUIDICOS

Bloque funcional de un sistema

fluidico

SIST. ELECTRICOS SIST. FLUIDICOS

Corriente Electrica (i) Flujo Volumetrico (q)

Diferencia de Potencial (v)

Diferencia de Presion (p=p1-p2)

Resistencia Electrica (R) Resistencia Hidraulica o Neumatica (R)

Sistemas fluidicos

Hidraulicos

Neumaticos

Fluido no compresible

Fluido compresible

INPUT

“q” OUPUT

“p”


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS SISTEMAS HIDRAULICOS

R

pp

R

pq 21

Resistencia Hidraulica

Oposición al flujo de líquido debido a válvulas o cambios de sección


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS SISTEMAS HIDRAULICOS

dt

dVqqq 21

Capacitancia Hidraulica Energía potencial almacenada por un líquido

g

AC

AhV

dt

dhA

dt

Ahdq

dt

dp

g

A

dt

g

pd

Aq H

H

ghpppp H 21

dt

dpCq H qdt

CpH

1


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS SISTEMAS HIDRAULICOS Inercia Hidraulica

Equivalente de inductancia en un sistema eléctrico

A

LI

LAm

dt

dqIp

pAAppApApFFFneta 212121

dt

dvmmapAFneta

Avq

dt

dqLpA

dt

A

qd

LApA


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS SISTEMAS NEUMATICOS

21 ppR

pm

Resistencia Neumatica

dt

dmm

Oposición al flujo de líquido debido a válvulas o cambios de sección

Definida en función al flujo másico



Capacitancia Neumatica dt

Vdmm

21recipienteen masa de cambio deRazon

dt

dV

dt

dVmm

21

dt

dV

dt

dp

dp

dVmm

21

Para gas ideal:

dt

dp

RTdt

dRT

V

mp

1

dt

dp

RT

V

dt

dp

dp

dVmm 21

dt

dp

RT

V

dp

dVmm 21

dt

dpCCmm 2121

dtmmCC

pp

21

21

21

1dp

dVC 1

RT

VC 2

Capacitancia debida a cambio de Volumen de

recipiente

Capacitancia debida a compresibilidad del gas



Inercia Neumatica

dt

mvdApp 21

Inercia neumatica

LAm Lq

A

qLAmv

Avq

dt

qdLApp

21 qm

dt

md

LApp

21

A

LI

dt

md

Ip

Debida a la caída de presión necesaria para acelerar un bloque de gas


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS

Eléctrico sist. "" Analogos

Neumatico Sist.""

Hidraulico Sist.""

i

m

q

Recordar que:

Eléctrico sist. "" Analogos

Neumatico Sist.""

Hidraulico Sist.""

v

p

p

Eléctrico Sist.

R""

C""

Analogos

E""Disipan Neumatico & Hidraulico Sist.""

E""Almacenan Neumatico & Hidraulico Sist.""

R

C


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS MODELADO SISTEMA HIDRAULICO (EJEMPLO)

Sistema: Recipiente con líquido Entrando/saliendo • Capacitor: Líquido en recipiente • Resistencia: Válvula • Inercia: Desprec. “q´s” lentos

Capacitor:

Resistor: Velocidad salida del fluido igual a velocidad salida por válvula

dt

dpCqqq H 21

221 Rqppp ghpppp H 21R

p

R

ghq H

2

dt

dpC

R

pqq HH

1

g

AC

h

R

g

dt

dhAq

1

Ecuac. Dif. Lineal de 1er Orden


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS MODELADO SISTEMA NEUMATICO (EJEMPLO)

Sistema: Fuelle • Capacitor: El fuelle mismo • Resistencia: Reducción de diámetro entrada • Inercia: Despreciable “dm/dt” lento

Capacitor:

Resistor: Resistencia de constricción

dt

dpCCmm 2

2121

mRqqq 21

dt

dpCCm 2

21

R

qqm 21

dt

dpCC

R

pp 221

21

22

211 pdt

dpCCRp


“Variación de p2 respecto a p1”


MODELOS DE SISTEMAS BASICOS SISTEMAS FLUIDICOS MODELADO SISTEMA NEUMATICO (EJEMPLO-cont.)

Sistema: Fuelle 2

2211 p

dt

dpCCRp

Fuelle forma de resorte

kxFAp2 A

kxp 2

xA

k

dt

dx

A

kCCRp

211


“Cambio en extensión/compresión (x) del fuelle con p1”

Capacitancia neumática debida a cambio de volumen

k

A

dp

k

ApAd

dp

Axd

dp

dVC

2

2

2

22

1

Capacitancia neumática debida a compresibilidad

del aire

RT

Ax

RT

VC 2



DIRECT CURRENT MOTORS

Mathematical Model of a DC Motor




Mathematical Model of a DC Motor

The primary loads on the motor are inertia and friction

The DC servo motor drives a mechanical load which consists of dynamic and static components




Mathematical Model of a DC Motor Example

A permanent magnet (PM) DC gear motor is used to lift a mass, as shown. Develop a mathematical relationship between the voltage applied to the motor and the rotational displacement of the motor shaft which is also a measure of the linear displacement of the mass. Assume that the string is inextensible, and also neglect the friction between the string and the pulleys




Mathematical Model of a DC Motor Solution

Load on the motor considering the gear ratio, G





Relationship between the angular displacement of the motor shaft and gear output shaft





Combining





Combining

both torque constant and voltage constant can be assumed to be equal to k




Mathematical Model of a DC Motor Example

Simulate the response of the system described in Figure for a constant input voltage of 10 V DC using MATLAB. Use the data given for a Shayang gear motor model number IG420049-SY3754





neglecting rotor inertia and damping losses in the motor

applying the Laplace transform

at zero initial condition

at zero initial condition

mod&sim sist fis

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