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SM212 Lecture 20

Assignments. Due:Problems 4.5, p. 192,

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SM212 Lecture 20

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    1. SM212 Lecture 20/21 Spring 2006 §4.8/5.6 Modeling unforced, Spring-Mass systems and undriven LRC circuits: unforced, damped harmonic motion

    2. Assignments Due: Problems §4.5, p. 192, #2; §6.3, p.337, #7 Due: Monday, 13 February 2006 Pending: Problems §4.8, p. 219, #1, 2, 7 Due: Tuesday, 14 February 2006 Problems §4.8, p. 219, #3, 8, 9 Due: Wednesday, 15 February 2006 Assign: Problems §4.9, p. 227, #1, 7 Due Friday, 17 February 2006 Exam 2: Wednesday, 22 February 2006

    3. Points to Control Modeling spring-mass systems using 2nd-order ODE’s The complete scenario Variables Forces Governing equation Initial condition Particular scenarios The unforced, undamped oscillator The unforced, damped oscillator

    4. Points to Control Modeling series LRC circuits using 2nd-order ODE’s Recall Kirchoff’s Laws (§3.5) Voltage drop across a R, L, C Voltage drop around the loop Result: 2nd-order LODE: compare to Unforced, damped spring-mass system Forced, damped, undamped spring-mass system

    5. ODE models The design paradigm:

    6. The unforced, damped oscillator Example §4.8, p. 219 #2 (append) Spring-mass system, m = 2 kg, spring: k = 50 nt/m, x(0) = -1/4 m, v(0) = -1 m/s Damping force: -24 v(t) nt (damping coefficient 24 nt/(m/s)) Seek: motion x(t); information about the motion Example §5.6, p. 291 #2 (amend) LRC series circuit L = ź h, C = 1/13 f, R = 2 O; i(0) = 0 amp, q(0) = 3.5 coul Seek: charge q(t), current i(t); information about dynamics Ed note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequencyEd note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequency

    7. The spring-mass system SP211/221: Unforced, damped On Board: Compile the problem: Variables: Displacement from natural state Forces: gravity, spring, damping medium Governing equation: F = mA Auxiliary equations: Refinement of the model: Variables: from equilibrium positionOn Board: Compile the problem: Variables: Displacement from natural state Forces: gravity, spring, damping medium Governing equation: F = mA Auxiliary equations: Refinement of the model: Variables: from equilibrium position

    8. (LRC) circuit equations Basis for model: Kirchhoff, p. 119f

    9. LRC vs Spring-Mass Model equation Simulation: Because of the link, Any phenomenon a spring mass system might display Under-damped oscillation Over-damped motion Transient and steady-state behavior under forcing Bounded/unbounded resonance. Will have a counter-part as a series LRC electric circuit phenomenon. Simulation: Because of the link, Any phenomenon a spring mass system might display Under-damped oscillation Over-damped motion Transient and steady-state behavior under forcing Bounded/unbounded resonance. Will have a counter-part as a series LRC electric circuit phenomenon.

    10. The unforced, damped oscillator The compiled problem (IVP) Normal form for IVP: damping parameter; natural frequency Ed note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequencyEd note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequency

    11. The unforced, damped oscillator: formal solution Normal form for IVP: 2nd-order, linear, homog Solution for IVP: §4.3 Auxiliary equation; quadratic formula: 3 cases (4th case: ? = 0):

    12. Flow of ideas damped harmonic motion Idea: roots for

    13. The unforced, undamped oscillator: formal solution Normal form for IVP: 2nd-order, linear, homog Auxiliary equation; quadratic formula: General solution for ODE: §4.2/4.3

    14. The unforced, un-damped oscillator: solution Example p. 219 #2 (undamped) Take b = 0 nt-s/m a) m = 2 kg, spring: k = 50 nt/m, x(0) = -1/4 m, v(0) = -1 m/s General Solution: IC’s; solution to the IVP Ed note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequencyEd note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequency

    15. The un-damped oscillator: solution in AM format Issue: extract information Trig identity: Ed note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequencyEd note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequency

    16. The un-damped oscillator: solution in AM format Example p. 219 #2 (append) a) m = 2 kg, spring: k = 50 nt/m, x(0) = -1/4 m, v(0) = -1 m/s INFORMATION: signal in AM format (p. 211) AM format: Ed note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequencyEd note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequency

    17. Example: (undamped) Simple Harmonic Oscillator SEE the motion: Undamped oscillation Attributes: Period, frequency, angular frequency Amplitude and phase

    18. Flow of ideas under damped SHM Idea: roots for

    19. Unforced, under damped harmonic motion Example: p. 219, #2’, m =2 kg; b = 4 n/(m/s) ; k = 50 n/m Normal form: Case III: (under damped)

    20. Under damped Harmonic Motion: Formal solution Auxiliary equation; quadratic formula: Case III: Under damped: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    21. Under damped Harmonic Motion: Example Example, p. 219, #2’: m = 2 kg, b = 4 nt-s/m, k = 50 nt/m general Solution: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    22. Under damped Harmonic Motion: Example Example, p. 219, #2’: m = 2 kg, b = 4 nt-s/m, k = 50 nt/m IC “Simult”: Soln to IVP: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    23. The under damped oscillator: solution in AM format Issue: extract information Trig identity: Ed note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequencyEd note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequency

    24. The under-damped oscillator: solution in AM format Example p. 219 #2 (append) m = 2 kg, b = 4 nt-s/m, k = 50 nt/m, x(0) = -1/4 m, v(0) = -1 m/s INFORMATION: signal in AM format (p. 211) AM format: Ed note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequencyEd note: units of normalized constants 2*lambda: [F/(L/T)/M] = [1/T] W0^2: [(F/L)/M] = [1/s^2] natural (angular) frequency

    25. Example: (under damped) SHO SEE the motion: Attributes: Period, frequency, angular frequency Amplitude and phase

    26. Flow of ideas over damped SHM Idea: roots for

    27. Unforced, over damped harmonic motion Example: p. 219, #2’, m =2 kg; b = 24 n/(m/s) ; k = 50 n/m Normal form: Case I: (over damped)

    28. Over damped Harmonic Motion: Formal solution Auxiliary equation; quadratic formula: Case I: Over damped: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    29. Over damped Harmonic Motion: Example Example, p. 219, #2’: m = 2, b = 24, k = 50 general Solution: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    30. Over damped Harmonic Motion: Example Example, p. 219, #2’: m = 2, b = 24, k = 50 IC “Simult”: Soln to IVP: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    31. Example: damped SHO SEE the motion: (m =1, b = 24, k = 25; x(0) = -1/4, x’(0) = -1) Attributes: Overdamped Convergence to equilibrium Period, frequency, angular frequency? Amplitude and phase?

    32. Flow of ideas critically damped SHM Idea: roots for

    33. Unforced, critically damped harmonic motion Example: p. 219, #2’, m =2 kg; b = 20 n/(m/s) ; k = 50 n/m Normal form: Case II: (critically damped)

    34. Critically damped Harmonic Motion: Formal solution Auxiliary equation; quadratic formula: Case II: Critically damped: Roots: General solution: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    35. Critically damped Harmonic Motion: Example Example, p. 219, #2’: m = 2, b = 20, k = 50 general Solution: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    36. Critically damped Harmonic Motion: Example Example, p. 219, #2’: m = 2, b = 20, k = 50 IC “Simult”: Soln to IVP: Cases: Undamped: simple harmonic motion Underdamped: decaying harmonic motionCases: Undamped: simple harmonic motion Underdamped: decaying harmonic motion

    37. Example: critically damped SHO SEE the motion: (m =1, b = 20, k = 25; x(0) = -1/4, x’(0) = -1) Attributes: Critically damped Convergence to equilibrium Period, frequency, angular frequency? Amplitude and phase?

    38. Flow of ideas unforced SHM Idea: roots for

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