Analogous Physical Systems

1
Analogous Physical Systems

• BIOE 4200

2
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

3
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

4
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

5
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)
6
Physical Variables

• Force is a through variable because of Newtons
  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

7
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

8
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

9
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
Fluid Resistance Rf (pipe) Q P/Rf P2/Rf
Thermal Resistance Rt (insulation) q T/Rt T/Rt
10
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
Fluid Fluid storage Cf (balloon) Q Cf dP/dt CfP2/2
Thermal Thermal storage Ct q Ct dT/dt CtT
11
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 ??? ??? ???