Physical Modelling:

1. Out there world inside here2. Modelling and Design Cycle3. Practical example: Passive Dynamic Walkers4. Base systems and concepts5. Ideals, assumptions and real life6. Similarities in systems and responses

Design Cycle:

https://stillwater.sharepoint.okstate.edu/ENGR1113/default.aspx

Similarities in systems and responses:1. Similarities in systems

– why is a spring like a water tank like a capacitor???2. First derivative time response3. Second derivative time responses (note there is more than one!)4. Damping5. Parallel verses Serial connection of components

Electronic components: Resistors:

• Voltage is proportional to current, Ohms law V = RI

Voltage - ImpedanceCurrent IV

tRitv R

tv

ti

http://ecee.colorado.edu/~mathys/ecen1400/labs/resistors.html

Electronic components: Capacitors:• Voltage is proportional to integral of current

Voltage - Current

tv

ti

tiC

tv1

http://ecee.colorado.edu/~mathys/ecen1400/labs/capacitors.html

Electronic components: Inductors:• Voltage is proportional to differential of current

Voltage - Current

dt

tdiLtv

tv

ti

http://electronics.stackexchange.com/

Force - ImpedanceDistance

Spring

Damper

Mass

Source Nise 2004

tKxtf

dt

tdxDtf

2

2

dt

txdMtf

tf

tx

tf

tx

tf

tx

sXsF

K

Ds

2Ms

Mechanical: Electrical:Capacitor

Resistor

Inductor

Source Nise 2004

Voltage - ImpedanceCurrent

tRitv

dt

tdiLtv

tv

ti

sIsV

R

Ls

tv

ti

tv

ti

tiC

tv1

sC

11

Modelling: Dynamic Systems

www.pbase.comwww.millhouse.nl

Consider dynamic systems: these change with timeAs an example consider water system with two tanks

Water will flow from first tank to second[Assume I stays constant due to nature of Dams]

Plot time response of system….?

Modelling: Dynamic Systems

IF

L

Water flows because of pressure difference [Ignore atmospheric pressure – approx. equal at both ends of pipe]

If have water at one end - what is its pressure? [Tanks with constant cross sectional area A]

Pressure is force per unit area, = F / A, Force (F) is mass of water times gravity g

Mass of water (M) is volume of water * density M = V *

Volume (V) is height of water, h, times its area A: V = h * A

Combining: pressure is

Dynamic Systems

g**hA

g**A*h

FPressure

Pressure

For first tank, pressure is pf = I * * gFor second tank, pressure is ps = L * * g

Thus flow is proportional to the difference in pressures: driving (effort) variable

Flow ∝ pf – ps ∝ (I-L) * * g

as well as on the pipe (its restrictance, R)

Here R is the constant of proportionality… Does flow Increase or decrease as the constant R is changed in different systems?

Flow changes volume of tanks: Volume change = A * rate of change in height (L) = Flow

g**R

L-I Flow

Flow changes volume of tanks: Volume change = A * rate of change in height (L) = Flow

We write ‘rate of change in height L’ as dL/dt = Flow / A

A tank has a capacitance - constants collected together in C =

Thus rate of change in height L: =

Flow stops, and there is no change in height when I = L

A

1*g**

R

L-I dt

dL

g*

A

C

C*R

L-I

dt

dL

Level change – not instantaneous• Initially: Large height difference Large flow L up a lot

• Then: Height difference less Less flow L increases, but by less

• Later: Height difference ‘lesser’ Less flow L up, but by less, etc

Graphically we can thus argue

the variation of level L and flow F is:

Dynamic Flow

F

tTT

L

t

I

Any system of the form:

Rate of change of output variable in an instant = Input variable – Output variable Has a time response (depending on step input):

Time Response of System

Time

OutputExponential

k

O-I

dt

dO

Time responses:• Proportional components

Or

Ratio governed by constant of proportionality

x

f

Time

x

t

Input

Output

Time

x

t

Input

Output

Time responses:• Proportional to derivative components (+ previous)

Or

Ratio governed by gain constantTime of response governed by time constant

dx/dt

f

Time

x

t

Input

Output

Time

x

t

Input

Output

Time responses:• Proportional to second derivative components (+ previous)

Or

Or

Ratio governed by gain constantTime of response governed by time constantsOvershoot governed by damping constant.

d2x/dt2

f

Time

x

t

Input

Output

Time

x

t

Input

Output

Time

x

t

Input

Output

Serial connection of components:• Opposite to parallel connections

• What is equivalent spring?

Draw a free body diagram of a spring

Write down individual equations: Ft = ktxt

Consider laws to combine them: xt = x1 + x2

Consider what does not change: Force must be equal on each spring Ft = F1 = F2

Ft = ktxt = kt (x1 + x2) = kt (F1 / k1 + F2 / k2 ) cancel forces: kt = (1 / k1 + 1 / k2 ) -1

Can extend method to any number of springs in serieshttps://notendur.hi.is/eme1/skoli/edl_h05/masteringphysics/13/springinseries.htm

Parallel connection of components:• Opposite to serial connections

Draw a free body diagram of a spring

Write down individual equations: Ft = ktxt

Consider laws to combine them: Ft = F1 + F2 Consider what does not change: xt = x1 = x2

Ft = ktxt F1 + F2 = kt xt kt xt = k1 x1+ k2x2

cancel distances: kt = k1 + k2

Can extend method to any number of springs in series

Example test questions for PM1. In the Researching phase of the engineering design cycle: state and describe at least five

(5) steps when defining the problem.2. Given a system has the following instantaneous (dynamic) relationships, sketch their

characteristic graphs on appropriate axes:i) y ∝ x ii) y d∝ x/dt iii) y d∝ 2x/dt2

Initially a system starts with a component with the relationship given in (i) sketch its time response to a step change in the effort variable. Plot both the input and output on the same axes (Hint: time is the independent variable).An additional component with the relationship given in (ii) is added; add the new time response clearly labelling the graph. Finally, a component described by (iii) is added; plot and discuss the possible outputs.

3. Given two resistors are in parallel in a connected circuit with a unit voltage effort driving the current flow, draw the diagram labelling important components, variables and constants.

Calculate the equivalent resistor value for the circuit.• Help session available in Mon, Wed AM103, @5 PM, with Howard