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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