Chapter 1-2.doc

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Chapter 1

Modeling: the process of identifying the principal physical

dynamic effects to be considered in analyzing a system, writing

the differential and algebraic equations from the conservation

and property laws, and reducing the equations to a convenient

differential equation form.

System: a system is a set of interacting components connected

together in such a way that the variation or response in the state

of one component affects the state of the others.

Static system: a system that has an output response to an inputthat does not change with time.

Dynamic system: a system that has a response to an input that is

not instantaneously proportional to the input or disturbance and

that may continue after the input is held constant. Dynamic

systems can respond to input signals, disturbance signals, or

initial conditions.

We will study four major types of dynamic systems:

Mechanical systems: systems that possess significant mass,

inertia, and spring and energy dissipation components driven by

forces, torques, displacements, and velocities. Eamples would

include cars, airplane, bicycle suspensions, etc.

Electrical systems: systems that include electrical circuits withresistive, capacitive, and!or inductive components ecited by

voltage or current.

Fluid systems: systems that employ orifices, restrictions, control

valves, accumulators "capacitors#, long tubes "inductors#, and

actuators ecited by pressure or fluid flow. Eamples include

city water systems, hydraulic power systems, pneumatic power

or control systems, etc.

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Mixed systems: a combination of two or three of the above

system types. Eamples include electric motors, microphones

and spea&ers, solenoid actuators, hydraulic pumps, hydraulic or

pneumatic actuators, electronic hydraulic servo control valves,

etc.

Transient response: the transient response of a dynamic system

to an eternal input refers to the behavior of the system as it

ma&es a transition from the initial condition to the final

condition.

Steady state: the state of a dynamic system after all of thetransients have died out.

Settling time: the time it ta&es a dynamic system to reach

'steady state'.

Transfer function: the ratio of the output to the input for a

system with zero initial conditions as determined by the (aplace

transform.

State-space form: ) set of simultaneous first*order differential

equations describing the dynamics of a system.

State variables: the dependent variables of each first*order

differential equation when in state*space form. +he state

variables represent the dynamic response variables of the

system.

System order: the number of independent derivatives in the

dynamic equations of motion for a system.

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Modeling of dynamic systems:

Model formulation

Conservation laws Differential equations

(inear momentum

)ngular momentum

harge

/ass Energy

Engineering properties

0iscous friction

oulomb friction

1nductance

2esistance

apacitance

3


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Conservation of mass

+he net mass flow rate at a location is equal to the rate of

change, with respect to time, of the mass at that location.

[ ]

+== ###dt

dmnet

6r, if we use the concept of a 'fluid capacitor', then the sum of

all mass flow rates is zero:

[ ] -=

#dt

dm net

Conservation of energy

+he 3stlaw of +hermodynamics: +he sum of all power "heat

transfer, mechanical power, and thermal power# in and out of a

system is equal to the rate at which energy is being stored in a

control volume of the system.

cv

neth m$mv

mudt

d$

vhmW"

++=

+++

$$

$$

Property laws:

Mechanical systems:Damping, viscous friction, coulomb friction, spring stiffness,

mechanical inductance "F%ma#

Electrical systems:

2esistance, capacitance, inductance

Fluid systems:

8luid resistance, fluid capacitance, fluid inductance

ngineering systems similarity

9ystems similarity: modeling of many physically different &inds

of dynamic systems result in the same or similar differential

equations and have similar response behaviors.

ffort and flow variables


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1n every discipline, the variables used to write the differential

equations can be classified as effort or flow variables.

Effort variable: a system variable that epresses the effort which

can be placed on a component of a dynamic system.

/echanical effortforce!torque

Electrical effortvoltage

8luid effortpressure

Flow variable: a system variable that epresses the flow, or rate

of change with time, of a system variable.

/echanical flowvelocity

Electrical flowcurrent8luid flowvolumetric flow rate

&mpedance: the ratio of effort to flow variables for a given

energy storage element.

Effort impedancexflow

Dynamic systems elements:

)ll dynamic systems can be represented by elements. +here arethree main types of elements: dissipative, effort storage, and

flow storage.

Dissipative elements: elements that dissipate energy "or provide

a way for energy to be released from a system#. +hey do not

store energy and are described by algebraic equations.

Effort storage elements: store energy by virtue of the effort

variable, and are capacitive.

effortecapacitanc8lowdt

d=

Effort storageelements are capable of storing potential energy.

For mechanical systems:

"position#stiffnessforceor,effortstiffness

3velocity =

=dt

d

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+he capacitive element in mechanical systems is the spring,

which stores potential energy.

For electrical systems:

= voltage

dt

decapacitanccurrent

For fluid systems:

= pressure

dt

decapacitancflow

Flow storageelements: store energy by virtue of the flow

variable. +hese are inductive in nature.

= flowinductanceeffortdt

d

&nductiveelements can store &inetic energy.

For mechanical systems:

=

=

locityangular veinertiatorque

andvelocitymassforce

dtd

dt

d

/ass or inertia is mechanical inductance.

For electrical systems:

= currentinductancevoltagedt

d

For fluid systems:

= rateflowinductancepressuredt

d

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Chapter 1-2.doc