Physics 2215 Analysis of Oscillating Systems Purpose Use

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).

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.

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

Physics 2215: 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 a 1=F 1/m t 1=Dt x 1 = x 0 + v 0 Dt v 1 = v 0 + a 0 Dt F 1(x 1, v 1 , t 1) t 2= 2 Dt x 2 = x 1 + v 1 Dt v 2 = v 1 + a 1 Dt F 2(x 2, v 2 , t 2) a 2=F 2/m t 3= 3 Dt x 3 = x 2 + v 2 Dt v 3 = v 2 + a 2 Dt F 3(x 3, v 3 , t 3) a 3=F 3/m etc.

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

Physics 2215: 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 V 0. 5 = v 0 + a 0 Dt a 1=F 1/m V 1. 5 = v 0. 5 + a 1 Dt t 1=Dt x 1 = x 0 + v 0. 5 Dt F 1(x 1, v 0. 5 , t 1) t 2= 2 Dt x 2 = x 1 + v 1. 5 Dt F 2(x 2, v 1. 5 , t 2) a 2=F 2/m V 2. 5 = v 1. 5 + a 2 Dt t 3= 3 Dt x 3 = x 2 + v 2. 5 Dt F 3(x 3, v 2. 5 , t 3) a 3=F 3/m V 3. 5 = v 2. 5 + a 3 Dt etc.

Physics 2215: Analysis of Oscillating Systems You will simulate numerically and measure experimentally: A. B. C. D. Undamped, 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: ØOn the lab website under “Hints/Links” …. . ØOr on your computer in the folder C: Physics LabLab FilesPhysics 1809

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 (Dx). Determine spring constant k: Dx

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

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

Physics 2215: Analysis of Oscillating Systems Case A: Simulating the Undamped, Undriven Oscillator Open spreadsheet: C: Physics LabLab FilesPhysics 1809Numerical_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 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.

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

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

Physics 2215: 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 2215: Analysis of Oscillating Systems 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.

Physics 2215: Analysis of Oscillating Systems Case C: Damped, Driven Oscillator w Additional driving force: m For the simulation spreadsheet use: C: Physics LabLab FilesPhysics 1809 Numerical_Analysis_Driven_Oscillator. xlsx

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).

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

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…

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

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

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

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.

Physics 2215: Analysis of Oscillating Systems Improper Driver Frequency Beat Patter is Observed Beat period: Here approx. 12 s Beat frequency =1/12 s=. 08 Hz Our driving Frequency is off by 0. 08 Hz (either too low or too high) Change DC voltage

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. 2 Hz For a beat frequency of 0. 08 Hz 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)

Physics 2215: Analysis of Oscillating Systems Trying 4. 0 Volt Works in Our Case… Amplitude keeps growing, no beat pattern observed.

Physics 2215: Analysis of Oscillating Systems 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 LabLab FilesPhysics 1809 Numerical_Analysis_Driven_Oscillator. xlsx

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 LabsLab FilesPhysics 1809Data 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.

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.

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!