1. Physics 2215: 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).

2. Physics 2215: 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.

3. Physics 2215: Analysis of Oscillating Systems How the Euler Method Works x (position) time t=0 x = x 0 v = v 0 true motion t1=tt1=t x 1 = x 0 + v 0  t v 1 = v 0 + a 0  t Euler method assumes constant velocity and acceleration during each time interval. Force can depend on position, velocity, and time.  It changes for each time interval as well. x1x1 x 2 = x 1 + v 1  t v 2 = v 1 + a 1  t t 2 =2  t

4. Physics 2215: Analysis of Oscillating Systems Solution Easy in a Spreadsheet time position velocity force acceleration 0 t1=tt1=t t 2 =  t t 3 =  t etc. xoxo vovo F o (x o, v o, t o ) a o =F o /m x 1 = x 0 + v 0  t v 1 = v 0 + a 0  t F 1 (x 1, v 1, t 1 )a 1 =F 1 /m x 2 = x 1 + v 1  t x 3 = x 2 + v 2  t v 2 = v 1 + a 1  t v 3 = v 2 + a 2  t F 2 (x 2, v 2, t 2 ) F 3 (x 3, v 3, t 3 ) a 2 =F 2 /m a 3 =F 3 /m initial conditions

5. Physics 2215: Analysis of Oscillating Systems How the Improved Euler Method Works x (position) time t=0 x = x 0 v = v 0 true motion t1=tt1=t x 1 = x 0 + v 0.5  t x1x1 t 2 =2  t v 0.5 = v 0 + a 0  t/2 x 2 = x 1 + v 1.5  t v 1.5 = v 0.5 + a 1  t The improved method uses estimated velocity halfway between the points in calculations  Numerical simulation is closer to the true motion.

6. Physics 2215: Analysis of Oscillating Systems Improved Euler Method in a Spreadsheet time position velocity force acceleration 0 t1=tt1=t t 2 =  t etc. xoxo vovo F o (x o, v o, t o ) a o =F o /m x 1 = x 0 + v 0.5  t F 1 (x 1, v 0.5, t 1 ) a 1 =F 1 /m x 2 = x 1 + v 1.5  t F 2 (x 2, v 1.5, t 2 ) a 2 =F 2 /m initial conditions t 3 =  t x 3 = x 2 + v 2.5  t F 3 (x 3, v 2.5, t 3 ) a 3 =F 3 /m velocity at halfpoint V 0.5 = v 0 + a 0  t V 1.5 = v 0.5 + a 1  t V 2.5 = v 1.5 + a 2  t V 3.5 = v 2.5 + a 3  t

7. Physics 2215: Analysis of Oscillating Systems 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:  On the lab website under “Hints/Links” …..  Or on your computer in the folder C:\Physics Lab\Lab Files\Physics1809

8. Physics 2215: Analysis of Oscillating Systems Hooke’s Law Restoring force of a spring: Hanging a mass m at the end of the spring yields a change in the length of the spring (  x).  Determine spring constant k: xx

9. Physics 2215: Analysis of Oscillating Systems From theory:. Case A: Undamped, Undriven Oscillator Force acting on mass: m -xx Rest position +x k

10. Physics 2215: Analysis of Oscillating Systems Hanging the Mass Vertically… m -x x Rest position with mass m k Rest position without mass m mg x shif t Total force on mass:  Simply shift the coordinate system origin to the new equilibrium position and use F total = - kx again (and ignore mg). In the new equilibrium position:

11. Physics 2215: Analysis of Oscillating Systems Open spreadsheet: C:\Physics Lab\Lab Files\Physics 1809\Numerical_Analysis_Undriven_Oscillator.xlsx. Case A: Simulating the Undamped, Undriven Oscillator 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.

12. Physics 2215: 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.

13. Physics 2215: Analysis of Oscillating Systems Printing Graphs Click this tab (PVA) for graphs that you want to print out.

14. Physics 2215: Analysis of Oscillating Systems Selecting PVA Tab  Shows All Graphs + Variables x(t) v(t) a(t)

15. Physics 2215: Analysis of Oscillating Systems Case A: Experimentally Measuring the Undamped, Undriven Oscillator with Data Studio m Mass oscillate around equilibrium point Motion sensor measures x(t) Please: Make sure that the mass does not crash into or fall onto the motion sensor. The motion sensor is easily damaged.

16. Physics 2215: Analysis of Oscillating Systems Case B: Damped, Undriven Oscillator m Tape piece of thick paper/carton (e.g., from a manila folder) at the bottom of the mass for damping. Additional force: Modify b (damping coefficient) in the spread sheet

17. Physics 2215: Analysis of Oscillating Systems Case C: Damped, Driven Oscillator m Additional driving force:  For the simulation spreadsheet use: C:\Physics Lab\Lab Files\Physics 1809 \Numerical_Analysis_Driven_Oscillator.xlsx

18. Physics 2215: 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: For an damped oscillator, the resonance frequency is shifted as follows: If there is too much damping (b too large)  no resonance possible (number under square root < 0).

19. Physics 2215: Analysis of Oscillating Systems If #NUM! appears here, then b is chosen too large  Reduce value of b

20. Physics 2215: 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…

21. Physics 2215: Analysis of Oscillating Systems Case C: Damped, Driven Oscillator - Experiment Driver/Oscillator: powered by 750 Interface amplitude adjustment

22. Physics 2215: 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

23. Physics 2215: Analysis of Oscillating Systems Weights Use these specially made aluminum weights only !! (They have the proper weight needed).

24. Physics 2215: 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.

25. Physics 2215: 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

26. Physics 2215: 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 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)

27. Physics 2215: Analysis of Oscillating Systems Trying 4.0 Volt Works in Our Case… Amplitude keeps growing, no beat pattern observed.

28. Physics 2215: Analysis of Oscillating Systems Case D: Undamped, Driven Oscillator m  For the simulation spreadsheet use again: C:\Physics Lab\Lab Files\Physics 1809 \Numerical_Analysis_Driven_Oscillator.xlsx 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

29. Physics 2215: 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.

30. Physics 2215: Analysis of Oscillating Systems Important Information about Printing from the Excel Spreadsheet today !! The spreadsheet will look like this: There are several tabs at the bottom. To print your graphs, first click on the sheet named PVA(print this sheet). Do not print from the other tabs. Otherwise you will be printing out reams of paper filled with numbers.

31. Physics 2215: Analysis of Oscillating Systems Once you are at the PVA-tab of the spreadsheet it looks like this: Make sure that ONLY the PVA tab is selected and use the “Print Preview” command to make sure you are only printing what you want to have printed. Thank you for helping preserve resources!