Harmonic oscillator
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This document discusses solving the harmonic oscillator equation to model different types of vibrations. It covers undamped free vibrations which exhibit simple harmonic motion. Damped free vibrations are also examined, where damping causes the amplitude to decay over time. Forced vibrations, including cases of beats and resonance, are explored. The document suggests the beam vibration can be modeled as a harmonic oscillator. It shows how to write the second order differential equation as a first order system for numerical solution in Matlab. Finally, it notes that the solution depends on ratios of m, c, and k, not their individual values, which is important for solving the inverse problem.
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Phase,
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Last Time…
Our model:
2
frictionless surface
(spring equilibrium position)
The Phase Constant
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frictionless surface
(spring equilibrium position)
A
phase
phase constant
The Phase Constant
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but now we need a phase constant
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Note: This thing moves when you run the power point, the x location at t = 0 is no longer A
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5
What does your
calculator say?
Which one is it? We
need a way to figure
this out.
T
T = 4 s
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To determine the sign of the phase constant, we’ll turn to uniform circular motion
Now what if the object doesn’t start on the x-axis?
6
The x-component of the object’s position vector is:
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Example Problem (15-4)
The position vs. time graph for a SHO is shown below. What is the phase constant ?
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T
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+
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Venus? (LC)
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Pivot
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Harmonic oscillator
- 1. CRSC/SAMSI, May 22 2006 Sava Dediu Solving the Harmonic Oscillator
- 2. ContentsContents 1. Simple illustrative example: Spring-mass system 2. Free Vibrations: Undamped 3. Free Vibrations: Damped 4. Forced Vibrations: Beats and Resonance 5. Will this work for the beam? 6. Writing as a First Order System 7. Summary & References
- 3. 1. Spring-mass system What is a spring-mass system and why it is important? (Hooke’s Law)
- 4. 1. Spring-mass system Dynamic problem: What is motion of the mass when acted by an external force or is initially displaced?
- 5. 1. Spring-mass system Forces acting on the mass Net force acting on the mass
- 6. 1. Spring-mass system Newton’s Second Law of Motion the acceleration of an object due to an applied force is in the direction of the force and given by: For our spring-mass system
- 7. 2. Undamped Free Vibrations no damping no external force (general solution) (particular solutions) A and B are arbitary constants determined from initial conditions
- 8. 2. Undamped Free Vibrations Periodic, simple harmonic motion of the mass
- 9. 2. Undamped Free Vibrations Period of motion Natural frequency of the vibration Amplitude (constant in time) Phase or phase angle
- 10. 3. Damped Free Vibrations no external force Assume an exponential solution Then and substituting in equation above, we have (characteristic equation)
- 11. 3. Damped Free Vibrations Solutions to characteristic equation: The solution y decays as t goes to infinity regardless the values of A and B Damping gradually dissipates energy! overdamped critically damped underdamped
- 12. 3. Damped Free Vibrations The most interesting case is underdamping, i.e:
- 13. 3. Damped Free Vibrations: Small Damping
- 14. 4. Forced Vibrations no damping Periodic external force: Case 1
- 15. 4. Forced Vibrations: Beats
- 16. 4. Forced Vibrations: Beats Rapidly oscillatingSlowly oscillating amplitude
- 17. 4. Forced Vibrations: Resonance unbounded as Case 2
- 18. 5. Will this work for the beam? The beam seems to fit the harmonic conditions • Force is zero when displacement is zero • Restoring force increases with displacement • Vibration appears periodic The key assumptions are • Restoring force is linear in displacement • Friction is linear in velocity
- 19. 6. Writing as a First Order System Matlab does not work with second order equations However, we can always rewrite a second order ODE as a system of first order equations �We can then have Matlab find a numerical solution to this system
- 20. 6. Writing as a First Order System Given the second order ODE with the initial conditions
- 21. 6. Writing as a First Order System Now we can rewrite the equation in matrix-vector form This is a format Matlab can handle.
- 22. 6. Constants are not independent Notice that in all our solutions we never have c, m, or k alone. We always have c/m or k/m. �The solution for y(t) given (m,c,k) is the same as y(t) given (αm, αc, αk). important for the inverse problem
- 23. 7. Summary We can use Matlab to generate solutions to the harmonic oscillator �At first glance, it seems reasonable to model a vibrating beam �We don’t know the values of m, c, or k Need to solve the inverse problem
- 24. 8. References W. Boyce and R.C. DiPrima: Elementary Differential Equations and Boundary Value Problems