Modeling and Simulation Revision IV - Philadelphia University · 2017-07-17 · Modeling Modeling is the process of representing the behavior of a real system by a collection of mathematical

D R . T A R E K A . T U T U N J I

P H I L A D E L P H I A U N I V E R S I T Y , J O R D A N

2 0 1 7

Modeling and Simulation Revision IV

Modeling

Modeling is the process of representing the behavior of a real system by a collection of mathematical equations and logic.

Models are cause-and-effect structures—they accept external information and process it with their logic and equations to produce one or more outputs. Parameter is a fixed-value unit of information

Signal is a changing-unit of information

Models can be text-based programming or block diagrams

Math Modelling Categories

Static vs. dynamic

Linear vs. nonlinear

Time-invariant vs. time-variant SISO vs. MIMO Continuous vs. discrete Deterministic vs. stochastic

Static vs. Dynamic

Models can be static or dynamic Static models produce no motion, fluid flow, or any other

changes.

Example: Battery connected to resistor

Dynamic models have energy transfer which results in power flow. This causes motion, or other phenomena that change in time.

Example: Battery connected to resistor, inductor, and

capacitor

4

v = 𝒊𝑹

idtC

1

dt

diLRiv

Linear vs. Nonlinear

Linear models follow the superposition principle The summation outputs from individual inputs will be equal to the

output of the combined inputs

Most systems are nonlinear in nature, but linear models can be used to approximate the nonlinear models at certain point.

Linear vs. Nonlinear Models

Linear Systems Nonlinear Systems

Time-invariant vs. Time-variant

The model parameters do not change in time-invariant models

The model parameters change in time-variant models Example: Mass in rockets vary with time as the fuel is

consumed.

If the system parameters change with time, the system is time varying.

Time-invariant vs. variant

Time-invariant

F(t) = ma(t) – g

Where m is the mass, a is the acceleration, and g is the gravity

Time-variant

F = m(t) a(t) – g

Here, the mass varies with time. Therefore the model is time-varying

Linear Time-Invariant (LTI)

LTI models are of great use in representing systems in many engineering applications.

The appeal is its simplicity and mathematical structure.

Although most actual systems are nonlinear and time varying

Linear models are used to approximate around an operating point the nonlinear behavior

Time-invariant models are used to approximate in short segments the system’s time-varying behavior.

SISO vs. MIMO

Single-Input Single-Output (SISO) models are somewhat easy to use. Transfer functions can be used to relate input to output.

Multiple-Input Multiple-Output (MIMO) models involve combinations of inputs and outputs and are difficult to represent using transfer functions. MIMO models use State-Space equations

System States

Transfer functions

Concentrates on the input-output relationship only.

Relates output-input to one-output only SISO

It hides the details of the inner workings.

State-Space Models

States are introduced to get better insight into the systems’ behavior. These states are a collection of variables that summarize the present and past of a system

Models can be used for MIMO models

SISO vs. MIMO Systems

Transfer Function U(t) Y(t)

Continuous vs. discrete

Continuous models have continuous-time as the dependent variable and therefore inputs-outputs take all possible values in a range

Discrete models have discrete-time as the dependent variable and therefore inputs-outputs take on values at specified times only in a range

Continuous vs. discrete

Continuous Models

Differential equations

Integration

Laplace transforms

Discrete Models

Difference equations

Summation

Z-transforms

Deterministic vs. Stochastic

Deterministic models are uniquely described by mathematical equations. Therefore, all past, present, and future values of the outputs are known precisely

Stochastic models cannot be described mathematically with a high degree of accuracy. These models are based on the theory of probability

Block Diagrams

Block diagram models consist of two fundamental objects: signal blocks and wires.

A block is a processing element which operates on input signals and parameters to produce output signals

A wire is to transmits a signal from its origination point (usually a block) to its termination point (usually another block).

Block diagrams are suitable to represent multi-disciplinary models that represent a physical phenomenon.

Dr. Tarek A. Tutunji

Block Diagram Example

[Ref.] Prof. Shetty

Block Diagrams Manipulation

Block Diagrams Manipulation

Block Diagrams: Direct Method Example

Consider the transfer function:

We can introduce s state variable, x(t), in order to separate the polynomials

State Equation

The differential equation is:

Put the needed integrator blocks:

Add the required multipliers to obtain the state equation:

Output Equation

Repeat the same procedure for the output equation:

Connect the two sub-blocks

Block Diagram Modeling: Analogy Approach

Physical laws are used to predict the behavior (both static and dynamic) of systems. Electrical engineering relies on Ohm’s and Kirchoff’s laws Mechanical engineering on Newton’s law Electromagnetics on Faradays and Lenz’s laws Fluids on continuity and Bernoulli’s law

Based on electrical analogies, we can derive the fundamental equations of systems in five disciplines of engineering: Electrical, Mechanical, Electromagnetic, Fluid, and Thermal.

By using this analogy method to first derive the fundamental relationships in a system, the equations then can be represented in block diagram form, allowing secondary and nonlinear effects to be added. This two-step approach is especially useful when modeling large coupled

systems using block diagrams.

Power and Energy Variables: Effort & Flow

[Ref.] Raul Longoria

Transitional Mechanical Systems

Mechanical movements in a straight line (i.e. linear motion) are called “transitional”

Basic Blocks are: Dampers, Masses, and Springs

Springs represent the stiffness of the system

Dampers (or dashpots) represent the forces opposing to the motion (i.e. friction)

Masses represent the inertia

maF

cvF

kxF

Force

displacement

velocity

acceleration mass


Transitional Mechanical Systems

Equations for mechanical systems are based on Newton Laws

Free body diagram

dt

dxc-kx-Fma

Mass Force

Spring

Damper


Example: Mass-Spring-Damper

Note: D is Differentiation 1/D is Integration

Example: Two-Mass Mechanical System

[Ref.] Prof. Shetty

Example: Two-Mass Mechanical System

Example: Mechanical Model

Consider a two carriage train system

22122

12111

xc-)x-k(xxm

xc-)x-k(x-fxm

Mass 1 Force

k(x1-x2)

c v1

x1

Mass 2 k(x1-x2)

c v2

x2

Example continued

Taking the Laplace transform of the equations gives

(s)csX-(s))X-(s)k(X(s)Xsm

(s)csX-(s))X-(s)k(X-F(s)(s)Xsm

22122

2

12112

1

Note: Laplace transforms the time domain problem into s-domain (i.e. frequency)

sX(s)(t)xL

x(t)dteX(s)x(t)L0

st

Example continued

Manipulating the previous two equations, gives the following transfer function (with F as input and V1 as output)

2kcsckmkmsmmcsmm

kcssm

F(s)

(s)V2

212

213

21

221

Note: Transfer function is a frequency domain equation that gives the relationship between a specific input to a specific output

Example continued

Simulation using MATLAB

m1= 5; m2=0.7; k=0.8; c=0.05;

num=[m2 c k];

den=[m1*m2 c*m1+c*m2 k*m1+k*m2+c*c 2*k*c];

sys=tf(num,den); % constructs the transfer function

impulse(sys); % plots the impulse response

step(sys); % plots the step response

bode(sys); % plots the Bode plot

Example continued: Impulse response


Example continued: Step response


Example continued: Bode Plot


Example: Motion of Aircraft

Rotational Mechanical Systems

Consider a mechanical system that involves rotation

•The torque, T, replaces the force, F

•The angle, q, replaces the displacement x

•The angular velocity, w , replaces velocity v

•The angular acceleration, a, replaces the acceleration a

•The moment of inertia J, replaces the mass m

Torque, T

Shaft

Side View

T

q

w dq/dt

J dw/dt kq cw

Top View


Rotational Mechanical Systems

The mechanics equation becomes

2

2

dt

θdJ

dt

dθckθT

JαcωkθT

angular acceleration

moment of inertia

angular velocity

Torque

angle

damper coefficient

spring coefficient


Example: Rotational-Transitional System

Consider a rack-and-pinion system. The rotational motion of the pinion is transformed into transitional motion of the rack

ωcdt

dωJTT 1outin

For simplicity, the spring effects are ignored

Tin

w

v

r

F

pinion

rack


Example continued

The rotational equation is

The transitional equation is dt

dvmvcF 2

ωcdt

dωJTT 1outin

v/rω

rFTout

Using the equations

And manipulating the rotational and transitional equations with the input torque, Tin, as inputs and velocity, v, as output, we get

dt

dvmr

rJvrc

rc

T 21

in


Example continued

Let us take a look at the state space equations

In general, DuBxy

CuAxx

where x is the states vector, y is the output vector, and u is the input vector

In our example, we will

use the states: w and v,

the inputs: Tin and F

the output: v

Manipulating the equations in the previous slide, we get

v

ω10v

F

T

m10

Jr

J1

v

ω

mc

0

0J

c

dtdv

dtdω

in

2

1

Conversion: Transitional and Rotational

Gear Trains

Gear Trains

where

Electrical Systems: Basic Equations

Resistor

Ohm’s Law

Inductor

Capacitor

dt

dVCi

idtC

1V

dt

diLV

RiV

Voltage

current

Resistance

Inductance

Capacitance

Power = Voltage x Current


Kirchoff Laws

Equations for electrical systems are based on Kirchoff’s Laws

1. Kirchoff current law: Sum of Input currents at node = Sum of output currents

2. Kirchoff voltage law: Summation of voltage in closed loop equals zero

Example: RLC circuit

idtC

1

dt

diLRiV cV

dt

diLRiV

dt

dVCi c c2

c2

c Vdt

VdLC

dt

dVRCV

R

V + -

L C

+ VR - + VL - + Vc -

i

Or

since Then

A second order differential equation

Using Kirchoff voltage law

RLC MATLAB Code

R=1000000; % R = 1MW

L=0.001; % L=1 mH

C=0.000001; % C= 1mF

num=1; den=[L*C R*C 1];

sys=tf(num,den);

bode(sys)

Impulse(sys)

Step(sys)

RLC Simulation: Bode Plot

At high frequency, Capacitor is short => Voltage = 0

At DC (i.e. frequency = 0), Capacitor is open => Voltage gain is 0dB (i.e. 1 V/V)


RLC Simulation: Impulse Response

Input voltage is pulse => Capacitor stores energy

And then releases the energy


RLC Simulation: Step Response

At about 2.3 seconds, the capacitor Voltage becomes 90% of the 1 Volts


Op Amps

PM-DC Motor Modeling

The electrical equation is emfin Vdt

diLRiV

The mechanical equation is loadTbωdt

dωJT

Vin

R L i

+ - Motor

T

w

where Vemf (Back electromagnetic voltage) = k1w

where T = k2i

DC Motor Model: Block Diagram

1/(Ls+R) 1/(Js+b) k2

k1

+ + Vin

Tload

w

-

-

T i

Simulation Result

Fluid Systems

Fluid systems can be divided into two categories: Hydraulic: fluid is a liquid and incompressible

Pneumatic: fluid is gas and can be compressed

The volumetric rate of flow, q, is equivalent to the current

The pressure difference, P1-P2, is equivalent to voltage

The basic building blocks for hydraulic systems are: Hydraulic resistance, capacitance, and inertance

Hydraulic resistance

Hydraulic resistance is the resistance to the fluid flow which occurs as a result of valves or pipe diameter changes

The relationship between the volume rate of flow, q, and pressure difference, p1-p2 ,is given by Ohm’s law

Rqpp 21

p2 p1

pipe

p1 p2

valve

Hydraulic Capacitance

Potential energy stored in a liquid such as height of a liquid in a container

dt

dhAqqAhV

dt

dVqq

21

21

Volumetric rate of change

Volume Change

Cross sectional Area

height

q1

q2

p1

p2

h A

Hydraulic Capacitance

hgρppp 21

dt

dp

gρ

A

dt

gρp

d

Aq-qdt

dhAqq 2121

gρ

AC

dt

dpCqq 21

density

gravity height pressure

By letting the hydraulic capacitance be

We get

hgpA/VgpA/mgA/Fp Note that

Hydraulic Inertance

Equivalent to inductance in electrical systems

To accelerate a fluid and increase its velocity a force is required

Avq

ALρm

dt

dvmmaFF 21

mass

Length

Cross sectional Area

F2 = p2 A F1 = p1 A

AppFF 2121

dt

dqIpp 21

A

LρI

using

Then Where the Inertance is

Hydraulic Example Modeling: an interactive 2-tank system

h1(t) A1

A2

h2(t) q2(t)

q1(t)

qin(t)

R2

R1

222

1211

2212

11in1

(t)/Rh(t)q

/R(t)h(t)h(t)q

/A(t)q(t)qdt

dh

/A(t)q(t)qdt

dh

Hydraulic Example Modeling: Block Diagram

1/A1

Sum Integrate Sum 1/R1 1/A2 Sum Integrate

1/R2 1/A2 1/A1

Input: qin

Output: q2

q1

- -

-

h1 h2 dh1/dt


Hydraulic Example: Simulation

Input, qin, is a step Output, q2, is taken to a virtual scope

Here, we assume all the Cross sectional areas and the resistances equals 1

Hydraulic Example: Simulation

Another Form of Analogies Potential and Flow Variables

When systems are in motion, the energy can be Increased by an energy-producing source outside the system

Redistributed between components within the system

Decreased by energy loss through components out of the system.

Therefore, a coupled system becomes synonymous with energy transfer between systems.

Potential Variable = PV

Flow Variable = FV

Analogies: FV and PV

Flow Variable

(FV)

Potential Variable

(PV)

Electrical Current Voltage

Mechanical Transitional Force Velocity

Mechanical Rotational Torque Angular Velocity

Hydraulic Volumetric Flow Rate Pressure

Pneumatic Mass Flow Rate Pressure

Thermal Heat Flow Rate Temperature

Which Analogies to use?

Force-Voltage makes more physical sense

Graphical Representation: Bond Graphs

Force-Current makes mathematical sense

Sum of Currents= Zero and Sum of Forces = Zero

Graphical Representation: Linear Graphs

Conclusion

Mathematical Modeling of physical systems is an essential step in the design process

Simulation should follow the modeling in order to investigate the system response

Mechatronic systems involve different disciplines and therefore an appropriate modeling technique to use is block diagrams

Analogies among disciplines can be used to simplify the

understanding of different dynamic behaviors


Documents

Modeling and Simulation Revision IV - Philadelphia University · 2017-07-17 · Modeling Modeling is the process of representing the behavior of a real system by a collection of mathematical