forced harmonic oscillators lecture

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  • 7/24/2019 Forced Harmonic Oscillators Lecture

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    Forced Harmonic Oscillator

    DAMPEDUNDAMPED

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    x

    y=y0coswt

    Undamped Forced Oscillator :

    Effect of external force on the oscillator.

    Force applied : End of the spring + Spring moves(y=y0coswt)

    Change in the length of the spring = x - y(Equilibrium Position)

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    )( yxkxm

    Eqn. of motion without friction

    Substituting for y

    tm

    Ft

    m

    kyxx coscos 002

    0

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    t

    m

    Ft

    m

    kyxx coscos 002

    0

    F0cost = Driving force

    F0 = Amplitude = Driving frequency

    Very special form of driving force !!!

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    Examples of such forced oscillators in Nature :

    1. Response of a bound electron to an EM field2. Tidal response of a lake to the periodic force of the

    moon or sun.

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    Guess

    tAx cos

    Why ?

    RHS of eqn. has cos(wt)

    LHS of eqn. must have cos(wt)

    Therefore

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    Substituting tAx cos

    In LHS of eqn.

    tm

    Fxx cos

    02

    0

    2

    mk

    FA o

    22

    1

    o

    o

    m

    FA

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    tm

    Fx

    cos

    122

    0

    0

    The solution

    Correct, Not complete

    No arbitrary

    constants

    Must able to

    specify x0and v0

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    Complete solution

    )cos(cos1

    022

    0

    0

    tBtm

    Fx

    Steady state solution General solution

    0

    2

    0 xx Motion : Free undamped oscillator

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    )cos(cos122

    0

    0

    tBtmFx

    For a damped system

    B : Decreases exponentially

    Steady state solution

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    Amplitude of oscillation vs driving frequency

    220

    0 1

    mFA

    0 20 40 60 80 100

    -0.010

    -0.005

    0.000

    0.005

    0.010

    0

    A

    A finite at0

    Resonance

    0A

    A0

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    0 20 40 60 80 100

    -0.010

    -0.005

    0.000

    0.005

    0.010

    0

    A

    221

    o

    o

    mFA

    00;A 00;A

    0;A

    -ve A ?

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    0

    Displacement 180 degree out of phase with driving force

    Mathematically

    )180cos(cos

    cos

    ttntDisplaceme

    tForce

    NegativeAmplitude

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    0

    Displacement in phase with driving force

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    Phenomenon of resonance :(+) ve and () ve aspects

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    - ve aspects :Avoid motions of large amplitude in the springs of an automobile

    To reduce response at resonance dissipative frictionforce is needed :

    Analysis of the Forced Damped Harmonic Oscillator

    http://en.wikipedia.org/wiki/File:R75-rear-shock.jpg
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    Forced damped harmonic oscillator

    Motion of the oscillator = - bvRetarding force (Viscous)

    FTot= Fspr+ Fvis+ Fdriving

    = -kx - bv + F0cost

    tcosFbv-kxxm 0

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    tmFxxx oo cos

    2

    Willx =A cos t satisfy this differential equ.?No!

    The velocityterm givessin

    t

    tcosFbv-kxxm 0

    tcosmFx

    mkx

    mbx 0

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    t

    m

    Fxxx oo cos2

    How to find the solution?

    Write the above equation in complex form

    ti

    em

    F

    zzz

    02

    0

    Solution will be of the formz = z

    oeit

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    Real part of z = zo

    eitgives the solution toForced damped harmonic oscillator

    Substituting z = zo

    eitin complex equation

    im

    Fz

    em

    Fiez titi

    22

    0

    00

    02

    020

    1

    )(

    tie

    mFzzz

    02

    0

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    i

    1

    m

    Fz

    22

    0

    00

    2222

    0

    2200

    )()(

    i(

    m

    F

    )

    Put zo in cartesian form by multiplying the

    numerator and denominator by the complexconjugate of the denominator

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    In

    polar

    form

    2

    0

    2

    1

    2

    1

    2222

    0

    0*

    00

    0

    tan

    )()(

    1

    Re

    m

    FzzR

    z i

    2222

    0

    22

    000

    )()(

    i)(

    m

    F

    z

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    Real part

    The complete solution isz = zo

    eit

    titii

    ReeRez

    )cos( tRx

    1/22222

    o

    o

    1

    m

    FRA

    22

    1tano

    Phase difference between

    the driving force

    and the displacement

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    1/22222

    o

    o

    1

    m

    FRA

    0d

    dA

    21

    20m 2Q

    11

    At = max

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    For light damping, Ais maximum for = oand the amplitude at resonanceis:

    o

    oo

    m

    FA )(

    Behavior of A and as functions of ,

    depends on the ratio / o

    1/22222oo

    1

    m

    FRA

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    10

    1

    Q 01

    1/22222oo

    1

    m

    FA

    2

    1

    20m 2Q

    11

    1F

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    As increases, the maximum amplitude occurs at a

    frequency less than the resonant frequency

    1

    0

    2

    1

    20m 2Q

    11

    1/22222o

    o

    1

    m

    FA

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    22

    1tano

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    0 20 40 60 80 100

    -0.010

    -0.005

    0.000

    0.005

    0.010

    0

    A

    Undamped FHO Damped FHO

    1/22222

    o

    o

    1

    m

    FA

    22

    1tan

    o

    22

    1

    o

    o

    m

    FA