Purpose

1. 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. 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. How the Euler Method Works x2 = x1 +v1 Dt v2 = v1 +a1 Dt x1 = x0 +v0 Dt v1 = v0 +a0 Dt x (position) x1 true motion x = x0 v = v0 time t=0 t1=Dt t2=2Dt • Force can depend • on position, • velocity, and time. • It changes for each time interval as well. Euler method assumes constant velocity and acceleration during each time interval.

4. Solution Easy in a Spreadsheet initial conditions velocity force acceleration position time 0 Fo(xo,vo ,to) xo ao=Fo/m vo t1=Dt F1(x1,v1 ,t1) a1=F1/m v1 = v0 +a0 Dt x1 = x0 +v0 Dt t2= 2Dt a2=F2/m v2 = v1 +a1 Dt x2 = x1 +v1 Dt F2(x2,v2 ,t2) F3(x3,v3 ,t3) a3=F3/m v3 = v2 +a2 Dt x3 = x2 +v2 Dt t3= 3Dt etc.

5. How the Improved Euler Method Works x1 = x0 +v0.5 Dt x2 = x1 +v1.5 Dt x (position) v0.5 = v0 +a0 Dt/2 v1.5 = v0.5 +a1 Dt true motion x1 x = x0 v = v0 time t=0 t1=Dt t2=2Dt The improved method uses estimated velocity halfway between the points in calculations  Numerical simulation is closer to the true motion.

6. Improved Euler Method in a Spreadsheet initial conditions velocity at halfpoint force velocity position acceleration time Fo(xo,vo ,to) 0 xo V0.5 = v0 +a0 Dt vo ao=Fo/m t1=Dt F1(x1,v0.5 ,t1) a1=F1/m x1 = x0 +v0.5 Dt V1.5 = v0.5 +a1 Dt t2= 2Dt F2(x2,v1.5 ,t2) x2 = x1 +v1.5 Dt V2.5 = v1.5 +a2 Dt a2=F2/m t3= 3Dt F3(x3,v2.5 ,t3) x3 = x2 +v2.5 Dt V3.5 = v2.5 +a3 Dt a3=F3/m etc.

7. Oscillating Systems • You will simulate numerically and measure experimentally: • 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

8. 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 (Dx).  Determine spring constant k: Dx

9. k m +x -x x Rest position Case A: Undamped, Undriven Oscillator Force acting on mass: From theory: .

10. Hanging the Mass Vertically… In the new equilibrium position: k Rest position without mass m -x xshift m Rest position with mass m x mg Total force on mass: • Simply shift the coordinate system origin to the new equilibrium position and use Ftotal= - kx again (and ignore mg).

11. 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.

12. Here are the Improved Euler Method calculations, in case you want to see how they are implemented in a spread sheet.

13. Printing Graphs Click this tab (PVA) for graphs that you want to print out.

14. Selecting PVA Tab  Shows All Graphs + Variables x(t) v(t) a(t)

15. Case A: Experimentally Measuring the Undamped, Undriven Oscillator with Capstone 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)

16. Case B: Damped, Undriven Oscillator Additional force: 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.

17. Case C: Damped, Driven Oscillator w Additional driving force: m For the simulation spreadsheet use: C:\Physics Lab\Lab Files\Physics 1809 \Numerical_Analysis_Driven_Oscillator.xlsx

18. 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. If #NUM! appears here, then b is chosen too large  Reduce value of b

20. 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. Case C: Damped, Driven Oscillator - Experiment Driver/Oscillator: powered by 750 Interface amplitude adjustment

22. Amplitude Adjustment … Amplitude: If amplitude is too large, oscillator may not rotate (too much torque due to weight). Reduce amplitude if necessary

23. Weights Use these specially made aluminum weights only !! (They have the proper weight needed).

24. Running the Driver/Oscillator from Capstone DC voltage determines the driving frequency. DC voltage adjustable to fine tune driving frequency. Record button will activate driver and motion sensor (not in picture).

25. Improper Driver Frequency  Beat Patter is Observed • Beat period: • Here approx. 10s • Beat frequency =1/10s=.1 Hz • Our driving Frequency is off by 0.1 Hz (either too low or too high) • Change DC voltage

26. 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 0.1Hz we need to change the DC voltage by Before, we had: 4.0 Volts  Try 3.5 Volts or 4.5 Volts (one will make beat frequency greater, the other will make it disappear)

27. Trying 4.5 Volt Works in Our Case… Amplitude keeps growing, no beat pattern observed.

28. Case D: Undamped, Driven Oscillator w 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

29. 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.

30. 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!