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Mekanika Klasik (7 - 10) bagian 1 Presentation Transcript

  • 1. Sesion #07-10
    MekanikaKlasik
    Riser Fahdiran, M.Si
    Umiatin, M.Si
    JurusanFisika
    FakultasMatematikadanIlmuPengetahuanAlam
  • 2. Outline
    Introduction Harmonic Oscillator
    Linear & Non Linear Oscillations
    SHO (simple harmonic oscillator)
    Energy of SHO
    Damping Oscillator
    Energy of Damping Oscillator
    Quality Factor
    Driven Oscillator
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    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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  • 3. HARMONIC OSCILLATOR
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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  • 4. 1. Introduction Harmonic Oscillator
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    Consider a system in static or dynamic stable equilibrium. When such a system is displaced slightly from its equilibrium position, the resulting oscillatory motion is called harmonic motion
    Example :
    elastic springs, bending beams, pendula, vibrating strings, resonance of air cavities, and the motion of charges in certain electrical circuits and cavities
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  • 5. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Based on the characteristic oscillator :
    • 6. Linear Oscillator :
    • 7. When the displacement is smaal
    • 8. Non Linear Oscillator :
    • 9. When the displacement of the system from equilibrium is large
    • 10. Linear Oscillator :
    • 11. Harmonic Oscillator
    • 12. Damped Oscillator
    • 13. Driven Oscillator
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  • 14. 2. Linear & Non Linear Oscillations
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    A particle of mass m moving in an arbitrary conservative force field for which the potential energy V(x) is represented by a curve below :
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  • 15. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The potential may be writen as :
    V(x) = ½ k(x-xo)2
    Expand the potential function using Taylor series about point xo
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  • 16. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Since xo is point of minimum, for stable equilibrium in symmetrical potential, the odd term must be zero :
    • 17. While :
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  • 18. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Define :
    • 19. The potential function may be written as :
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  • 20. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Assume that the origin is located at the equilibrium point ( xo = 0 ) and x' = x, by neglecting the higher-order terms:
    • 21. Furthermore, since the motion of the particle is in a conservative force field, using the definition :
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  • 22. 3. SHO (simple harmonic oscillator)
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • From the Newton’s Second Law
    • 23. ωo : free natural angular frequency
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  • 24. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The general solution of SHO motion is :
    • 25. With :
    • 26. To : Period of oscillation
    • 27. Vo : Frequency of oscillation
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  • 28. 4. Energy of SHO
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    We’ve found that displacement of SHO :
    Kinetic energy :
    Potential energy :
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  • 29. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Thus the total energy E, which is always constant whenever there is a conservative force held, is :
    • 30. Those eq can be solved to find x(t) by :
    • 31. Prove that the solution is also periodic function !
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  • 32. 5. Damping Oscillator
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The existence of resistance Force make there’s impossible SHO in natural system. The net force in damping oscillator :
    • 33. Newton 2nd law’s for those motion :
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  • 34. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The illustration of damping oscillator :
    • 35. The general solution is :
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  • 36. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • There’s 3 interesting cases :
    • 37. Try to analysis each case above to differ its characteristic
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  • 38. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The plot of graph (displacement vs time) of 3 case of damping oscillator :
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  • 39. 6.Energy of Damping Oscillator
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The total energy E(t) of a damped harmonic system at any time t is given by :
    • 40. where E(0) is the total energy at time t = 0 and Wfis the work done by friction in the time interval 0 to t.
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  • 41. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Thus the rate of energy loss by friction may be written as :
    • 42. which is negative and represents the rate at which energy is being dissipated into heat. Since Wf< 0, E, continuously decreases with time and may be calculated in the following manner
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  • 43. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Since we assumed light damping, we may write ω1≈ ωo = k/m; hence this equation takes the form :
    • 44. With :
    • 45. So the energy is :
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  • 46. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Constant decay or characteristic time τ (time needed for E to decrease till 1/e of its initial value). Subtitue E(t) = Eo/e and t = τ
    • 47. If y is very small, τ —>~, and if y is very large,τ —> 0
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  • 48. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The fractional rate of decrease in energy may be write in the eq :
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  • 49. 7. Quality Factor
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The quality factor Q, or simply Q value, is a frequently used term in mechanical oscillatory systems as well as electrical oscillatory systems. Q is a dimensionless quantity and represents the degree of damping of an oscillator.
    • 50. P : power loss or the rate of energy disipated
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  • 51. 8. Driven Oscillator
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • To maintain the oscillations, energy from an external source must be supplied at a rate equal the energy dissipated by the oscillator in the damping medium. Fnet acting on the system is :
    • 52. In which :
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  • 53. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Assume the driven force is:
    • 54. The motion of a driven harmonic oscillator could be write in the eq :
    • 55. Above is inhomogeneous differential 2nd degree eq. The solution contain : particular (inhomogeneous) and complementary solution . (homogeneous)
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  • 56. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The solution is :
    • 57. The homogeneous solution xhis called the transient solution, while the particular solution Xi is the steady-state solution.
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  • 58. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • The general solution x will be independent of the influence of the initial conditions except in the beginning when the transient term is still contributing
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  • 59. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • Two interesting cases in driven oscillator :
    • 60. ω < ω1 driving freq less than natural freq
    • 61. ω > ω1 driving freq more than natural freq
    • 62. As is clear from these plots, the transient solution xh is effective only in the beginning and decays to zero as time passes, while the steadystatesolution remains constant with time. Thus the transient solution effects the general solution only in the beginning.
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  • 63. © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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    • if ω < ω1, the transient term xhcauses distortion of the resulting sinusoidal waveform.
    • 64. if ω > ω1 the transient term xh, instead of causing distortion, has the effect of modulating the oscillations due to the force function.
    • 65. After the transient term has died out, the oscillations are governed by the force function.
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    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
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