Transcript Purpose

Physics 1809: Analysis of Oscillating Systems
Purpose
• Use the “Improved Euler Method” – you learned this
method of solving problems numerically in the
homework.
• Compare measurements and numerical simulations
of oscillating systems (spring-mass system).
Physics 1809: Analysis of Oscillating Systems
The Euler Method Applied to Motion
• Uses the position , velocity, and acceleration of the
system at one point in time to estimate the condition
of that system at the next point in time.
• In general, the larger the time increments are, the
more the estimation deviates from reality.
Physics 1809: Analysis of Oscillating Systems
How the Euler Method Works
x1 = x0 + v0 Dt
v1 = v0 + a0 Dt
x (position)
a1 
F1 v1 , x1 , t1 
m
x2 = x1 + v1 Dt
v2 = v1 + a1 Dt
a2 
F2 v2 , x2 , t 2 
m
x1
true motion
x = x0
v = v0
ao 
Fo vo , xo , to 
m
t=0
Force can depend
on position,
velocity, and time.
 It changes for
each time interval
as well.
t1=Dt
Euler method assumes
constant velocity and
acceleration during
each time interval.
t2=2Dt
time
Physics 1809: Analysis of Oscillating Systems
Solution Easy in a Spreadsheet
initial conditions
time
0
position
xo
velocity
vo
force
acceleration
Fo(xo, vo , to)
ao=Fo/m
a1=F1/m
t1=Dt
x1 = x0 + v0 Dt
v1 = v0 + a0 Dt
F1(x1, v1 , t1)
t2= 2Dt
x2 = x1 + v1 Dt
v2 = v1 + a1 Dt
F2(x2, v2 , t2)
a2=F2/m
t3= 3Dt
x3 = x2 + v2 Dt
v3 = v2 + a2 Dt
F3(x3, v3 , t3)
a3=F3/m
etc.
Physics 1809: Analysis of Oscillating Systems
How the Improved Euler Method Works
x1 = x0 + v0.5 Dt
x (position)
a1 
F1 v0.5 , x1 , t1 
m
v0.5 = v0 + a0 Dt/2
x2 = x1 + v1.5 Dt
a2 
F2 v1.5 , x2 , t 2 
m
v1.5 = v0.5 + a1 Dt
true motion
x1
x = x0
v = v0
ao 
Fo vo , xo , to 
m
t=0
t1=Dt
t2=2Dt
time
The improved method uses estimated velocity halfway between
the points in calculations
 Numerical simulation is closer to the true motion.
Physics 1809: Analysis of Oscillating Systems
Improved Euler Method in a Spreadsheet
initial conditions
time
position
0
xo
velocity
vo
force
Fo(xo, vo , to)
acceleration
velocity at
halfpoint
ao=Fo/m
V0.5 = v0 + a0 Dt
a1=F1/m
V1.5 = v0.5 + a1 Dt
t1=Dt
x1 = x0 + v0.5 Dt
F1(x1, v0.5 , t1)
t2= 2Dt
x2 = x1 + v1.5 Dt
F2(x2, v1.5 , t2)
a2=F2/m
V2.5 = v1.5 + a2 Dt
t3= 3Dt
x3 = x2 + v2.5 Dt
F3(x3, v2.5 , t3)
a3=F3/m
V3.5 = v2.5 + a3 Dt
etc.
Physics 1809: Analysis of Oscillating Systems
Oscillating Systems
You will simulate numerically and measure experimentally:
A.
B.
C.
D.
Undamped, undriven oscillator
Damped, undriven oscillator
Damped, driven oscillator
Undamped, driven oscillator
The spreadsheets for these numerical simulations have already been created.
You can find the two Excel spreadsheets here:
On the lab website under “Hints/Links” …..
Or on your computer in the folder C:\Physics Lab\Lab Files\Physics1809
Physics 1809: Analysis of Oscillating Systems
Hooke’s Law
• Restoring force of a spring:
F  k x
• Hanging a mass m at the end of the spring yields a change in
the length of the spring (Dx).
 Determine spring constant k:
F
mg
k

Dx Dx
Dx
Physics 1809: Analysis of Oscillating Systems
Case A: Undamped, Undriven Oscillator
k
m
-x
x
+x
Rest position
Force acting on mass:
F  k x
From theory: x(t )  A cos(t   )

k
m
.
Physics 1809: Analysis of Oscillating Systems
Hanging the Mass Vertically…
In the new equilibrium position: kxshift
 mg
k
Rest position without mass m
-x
F   k xshift
m
x
xshif
t
Rest position with mass m
mg
Total force on mass: Ftotal  Fspring  mg  k ( xshift  x)  mg  kx
Simply shift the coordinate system origin to the new equilibrium
position and use Ftotal = - kx again (and ignore mg).
Physics 1809: Analysis of Oscillating Systems
Case A: Simulating the Undamped, Undriven Oscillator
Open spreadsheet:
C:\Physics Lab\Lab Files\Physics 1809\Numerical_Analysis_Undriven_Oscillator.xlsx
Enter the mass and
spring constant of
your system. The
damping constant
b should be 0 for
undamped motion.
Here you can also
change the initial
conditions (xo,vo)
and the time
increment of the
Euler method.
More pages with graphs:
 Select here.
.
Physics 1809: Analysis of Oscillating Systems
Here are the
Improved
Euler Method
calculations,
in case you
want to see
how they are
implemented
in a spread
sheet.
Physics 1809: Analysis of Oscillating Systems
Printing Graphs
Click this tab (PVA) for graphs that
you want to print out.
Physics 1809: Analysis of Oscillating Systems
Selecting PVA Tab  Shows All Graphs + Variables
x(t)
v(t)
a(t)
Physics 1809: Analysis of Oscillating Systems
Case A: Experimentally Measuring the Undamped,
Undriven Oscillator with Data Studio
Please: Make sure that the mass does not
crash into or fall onto the motion sensor.
The motion sensor is easily damaged.
m
Mass oscillate around equilibrium point
Motion sensor measures x(t)
Physics 1809: Analysis of Oscillating Systems
Case B: Damped, Undriven Oscillator
Additional force: Fdamp   b v
Ftotal   k x  b v
m
Modify b (damping coefficient)
in the spread sheet
Tape piece of thick paper/carton (e.g.,
from a manila folder) at the bottom of
the mass for damping.
Physics 1809: Analysis of Oscillating Systems
Case C: Damped, Driven Oscillator

Additional driving force: Fdrive  D sin(t )
Ftotal   k x  b v  D sin(t )
m
For the simulation spreadsheet use:
C:\Physics Lab\Lab Files\Physics 1809
\Numerical_Analysis_Driven_Oscillator.xlsx
Physics 1809: Analysis of Oscillating Systems
Resonance
For an undamped oscillator, the most effective frequency with which to
drive (push/pull) it to get it to oscillate with large amplitude is it’s natural
oscillation frequency. That frequency is called “resonant frequency”.
(Like pushing a child on a swing with just the right frequency).
 For undamped oscillator: res 
k
m
For an damped oscillator, the resonance frequency is shifted as follows:
 res
k  b 



m  2m 
2
If there is too much damping (b too large)
 no resonance possible
(number under square root < 0).
Physics 1809: Analysis of Oscillating Systems
If #NUM! appears here, then b is chosen too large
 Reduce value of b
Physics 1809: Analysis of Oscillating Systems
After choosing m, k, b …..
…you can read off the automatically
calculated resonance frequency here …..
…and if you want to see how the system
behaves if driven at the resonance
frequency, you can enter that value
up here as the driving frequency…
Physics 1809: Analysis of Oscillating Systems
Case C: Damped, Driven Oscillator - Experiment
Driver/Oscillator:
powered by 750 Interface
amplitude adjustment
Physics 1809: Analysis of Oscillating Systems
Amplitude Adjustment …
Amplitude: If amplitude is
too large, oscillator may not
rotate (too much torque due
to weight).
Reduce amplitude if necessary
Physics 1809: Analysis of Oscillating Systems
Weights
Use these specially made
aluminum weights only !!
(They have the proper weight
needed).
Physics 1809: Analysis of Oscillating Systems
Running the Driver/Oscillator from Data Studio
Start button will activate
driver and motion sensor.
DC voltage determines
the driving frequency.
DC voltage adjustable to
fine tune driving frequency.
Physics 1809: Analysis of Oscillating Systems
Improper Driver Frequency  Beat Patter is Observed
Beat period:
Here approx. 12s
Beat frequency
=1/12s=.08 Hz
Our driving
Frequency is off
by 0.08 Hz (either
too low or too
high)
Change DC
voltage
Physics 1809: Analysis of Oscillating Systems
How Much Adjustment in DC Voltage ???
Rule of thumb:
A change of 1 Volt changes the driver frequency by 0.2Hz
For a beat frequency of 0.08Hz we need to change the DC
voltage by
0.08
DV   1 Volt
  0.4 Volt
0.2
Before, we had: 3.6 Volts  Try 3.2 Volts or 4.0 Volts
(one will make beat frequency greater, the other will make it disappear)
Physics 1809: Analysis of Oscillating Systems
Trying 4.0 Volt Works in Our Case…
Amplitude keeps growing, no beat pattern observed.
Physics 1809: Analysis of Oscillating Systems
Case D: Undamped, Driven Oscillator

Ftotal   k x  D sin(t )
Careful: Without damping the amplitudes
at resonance can get HUGE. Don’t let the
mass slam into the motion sensor!!!!
m
No more cardboard to dampen motion
For the simulation spreadsheet use again:
C:\Physics Lab\Lab Files\Physics 1809
\Numerical_Analysis_Driven_Oscillator.xlsx
Physics 1809: Analysis of Oscillating Systems
Correction:
Due to some still unfixed software bugs in Capstone, we will use Data
Studio activities in this lab instead of Capstone activities.
Load Data Studio activities from:
C:\Physics Labs\Lab Files\Physics 1809\Data Studio Activities\...
The files are:
Numerical Analysis 1.ds
Numerical Analysis 2.ds
If you need help how to use Data Studio, you can look at the Data Studio
Tutorial that is on the lab website:
On the website click on the link “Manuals” and then look for Data Studio
Toturial. Look how the “smart tool” works in Data Studio.
Other than that, you use “Start” instead of “Record” in Data Studio.