Physics 1809: Analysis of Oscillating Systems
• 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
• 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 x (position)
x
1 v
1 a
1
= x
= v

F
1
0
+ v
0
D t
0
+ a
0
D t
 v
1
, x
1
, t
1
 m x
2 v
2 a
2
= x
1
+ v
1
D t
= v
1
+ a
1
D t

F
2
 v
2
, m x
2
, t
2
 x
1 true motion a o
 x = x
0 v = v
0
F o
 v o
, x o
, t o
 m t=0
Force can depend on position, velocity, and time.

It changes for each time interval as well.
t
1
=
D t
Euler method assumes constant velocity and acceleration during each time interval.
t
2
=2
D t time

Physics 1809: Analysis of Oscillating Systems
initial conditions time position velocity force acceleration
0 x o v o
F o
(x o
, v o
, t o
) a o
=F o
/m t
1
=
D t x
1
= x
0
+ v
0
D t v
1
= v
0
+ a
0
D t F
1
(x
1
, v
1
, t
1
) a
1
=F
1
/m t
2
=
2D t x
2
= x
1
+ v
1
D t v
2
= v
1
+ a
1
D t F
2
(x
2
, v
2
, t
2
) a
2
=F
2
/m t
3
=
3D t x
3
= x
2
+ v
2
D t v
3
= v
2
+ a
2
D t F
3
(x
3
, v
3
, t
3
) a
3
=F
3
/m etc.

Physics 1809: Analysis of Oscillating Systems
x (position) x
1 a
1
= x
0
+ v
0.5
D t

F
1
 v
0 .
5
, m x
1
, t
1
 v
0.5
= v
0
+ a
0
D t/2 v
1.5
= v
0.5
+ a
1
D t x
2 a
2
= x
1
+ v
1.5
D t

F
2
 v
1 .
5
, m x
2
, t
2
 true motion x
1 a o
 x = x
0 v = v
0
F o
 v o
, x o
, t o
 m t=0 t
1
=
D t t
2
=2
D t 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
initial conditions time position velocity force acceleration velocity at halfpoint
0 x o v o
F o
(x o
, v o
, t o
) a o
=F o
/m
V
0.5
= v
0
+ a
0
D t t
1
=
D t x
1
= x
0
+ v
0.5
D t
F
1
(x
1
, v
0.5
, t
1
) a
1
=F
1
/m V
1.5
= v
0.5
+ a
1
D t t
2
=
2D t x
2
= x
1
+ v
1.5
D t F
2
(x
2
, v
1.5
, t
2
) a
2
=F
2
/m V
2.5
= v
1.5
+ a
2
D t t
3
=
3D t x
3
= x
2
+ v
2.5
D t F
3
(x
3
, v
2.5
, t
3
) a
3
=F
3
/m V
3.5
= v
2.5
+ a
3
D t etc.

Physics 1809: Analysis of Oscillating Systems
You will simulate numerically and measure experimentally:
A. Undamped, undriven oscillator
B. Damped, undriven oscillator
C. Damped, driven oscillator
D. Undamped, driven oscillator
The spreadsheets for these numerical simulations have already been created.
You can find the two Excel spreadsheets here:
 O n 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
D x
• 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 (
D x).
 Determine spring constant k: k

F
D x
 mg
D x

Physics 1809: Analysis of Oscillating Systems
k m
-x x
Rest position
+x
Force acting on mass:
From theory:
F
  k x x ( t )

A cos(
 t
 
)
  k m
.

Physics 1809: Analysis of Oscillating Systems
In the new equilibrium position: kx shift
 mg k
F
  k x shift
Rest position without mass m
-x t x shif
Rest position with mass m x mg
Total force on mass: F total

F spring
 mg
  k ( x shift
 x )
 mg
  kx
 Simply shift the coordinate system origin to the new equilibrium position and use F total
= - 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 (x o
,v o
) 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
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.
Mass oscillate around equilibrium point
Motion sensor measures x(t)

Physics 1809: Analysis of Oscillating Systems
Additional force: F damp
  b v
F total
  k x
 b v
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

Additional driving force: F drive

D sin(
 t )
F total
  k x
 b v

D sin(
 t )
For the simulation spreadsheet use:
C:\Physics Lab\Lab Files\Physics 1809
\ Numerical_Analysis_Driven_Oscillator.xlsx

Physics 1809: Analysis of Oscillating Systems
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 m
 b
2 m
2
If there is too much damping (b too large)
 no resonance possible
(number under square root < 0).

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

Physics 1809: Analysis of Oscillating Systems
Driver/Oscillator: powered by 750 Interface amplitude adjustment

Physics 1809: Analysis of Oscillating Systems
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
Use these specially made aluminum weights only !!
(They have the proper weight needed).

Physics 1809: Analysis of Oscillating Systems
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
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
D
V
 
1 Volt
0 .
08
0 .
2
 
0 .
4 Volt
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
Amplitude keeps growing, no beat pattern observed.

Physics 1809: Analysis of Oscillating Systems

F total
  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!!!!
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
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.