Download Analogous physical systems

Analogous Physical Systems
BIOE 4200
Creating Mathematical Models

Break down system into individual
components or processes
 Need to model process outputs as function of
inputs and process states
 Models can be obtained experimentally
– Measure output using a range of inputs

Models can be obtained theoretically
– Derive equations using physical principles

This lecture will focus on physical principles
Theoretical Modeling

Many real physical systems can be modeled
using a combination of ideal elements
 Processes represent “lumped” parameters
– Idealized mechanical systems use point masses,
springs, dampers
– Idealized electrical circuits use resistors,
capacitors, inductors

Assume spatial (3-D) properties do not affect
the results of your model
 Different types of idealized physical systems
are governed by the same equations
Physical Variables

Physical quantities can be categorized in two
groups
 “Through” variables
– Quantities that pass through ideal element
– Value of through variable is same going into and
coming out of ideal elements

“Across” variables
– Quantities are measured across ideal elements
– Values do not make sense unless they are
measured relative to a reference point
Physical Variables
System
Through variable
Across variable
Translation
Force (F)
Velocity (v)
Rotation
Torque (t)
Angular velocity
(w)
Electrical
Current (i)
Voltage (V)
Fluid
Volumetric flow
rate (Q)
Pressure (P)
Thermal
Heat flow rate (q)
Temperature (T)
Physical Variables

Force is a through variable because of
Newton’s 3rd law
– Pulling on one end of spring produces equal and
opposite force on other end

Velocity, pressure and temperature are
across variables because they are relative
– Pressure must be measured across two points
– Temperature difference is relevant in heat transfer
– Velocity is relative – Newtonian frame of reference
Ideal Elements





Categorize ideal elements by how energy is
transferred within the process
Processes generally transfer energy from one
source to another or convert energy from one
form to another
Energy dissipation – energy entering process
is dissipated as heat loss
Capacitive storage – energy entering the
process is accumulated as velocity or charge
Inductive storage – energy entering the
process is stored as a force or electric field
Mathematical Relationships

Energy Dissipation
– Through ~ Across
– Energy dissipated ~ Across2 or Through2

Capacitive Storage
– Through ~ d(Across)/dt
– Energy stored ~ Across2/2

Inductive Storage
– Across ~ d(Through)/dt
– Energy stored ~ Through2/2
Energy Dissipation
System
Element
Equation
Energy
Translation
Damping b
(friction)
F = bv
bv2
Rotation
Damping b
(friction)
t = bw
bw2
Electrical
Resistance R
i = V/R
V2/R
Q = P/Rf
P2/Rf
q = T/Rt
T/Rt
Fluid
Thermal
Resistance Rf
(pipe)
Resistance Rt
(insulation)
Capacitive Storage
System
Element
Equation
Energy
Translation
Mass m
F = m dv/dt
(F = ma)
mv2/2
Rotation
Inertia J
t = J dw/dt
Jw2 /2
Electrical
Capacitor C
i = C dV/dt
CV2/2
Q = Cf dP/dt
CfP2/2
q = Ct dT/dt
CtT
Fluid
Thermal
Fluid storage
Cf (balloon)
Thermal
storage Ct
Inductive Storage
System
Element
Equation
Energy
Translation
Linear
spring k
kv = dF/dt
(F = kx)
F2/2k
Rotation
Torsional
spring k
kw = dt/dt
(t = kq)
t2 /2k
Electrical
Inductor L
(magnet/coil)
V = L di/dt
Li2/2
Fluid
Fluid inertia I
P = I dQ/dt
IQ2/2
Thermal
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