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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
Recommended