The mathematics Parametric oscillator

1 mathematics

1.1 transformation of equation
1.2 solution of transformed equation
1.3 intuitive derivation of parametric excitation

the mathematics









d

2


x


d

t

2





+
β
(
t
)



d
x


d
t



+

ω

2


(
t
)
x
=
0


{\displaystyle {\frac {d^{2}x}{dt^{2}}}+\beta (t){\frac {dx}{dt}}+\omega ^{2}(t)x=0}

this equation linear in



x
(
t
)


{\displaystyle x(t)}

. assumption, parameters




ω

2




{\displaystyle \omega ^{2}}

,



β


{\displaystyle \beta }

depend on time , not depend on state of oscillator. in general,



β
(
t
)


{\displaystyle \beta (t)}

and/or




ω

2


(
t
)


{\displaystyle \omega ^{2}(t)}

assumed vary periodically, same period



t


{\displaystyle t}

.

if parameters vary @ twice natural frequency of oscillator (defined below), oscillator phase-locks parametric variation , absorbs energy @ rate proportional energy has. without compensating energy-loss mechanism provided



β


{\displaystyle \beta }

, oscillation amplitude grows exponentially. (this phenomenon called parametric excitation, parametric resonance or parametric pumping.) however, if initial amplitude zero, remain so; distinguishes non-parametric resonance of driven simple harmonic oscillators, in amplitude grows linearly in time regardless of initial state.

a familiar experience of both parametric , driven oscillation playing on swing. rocking , forth pumps swing driven harmonic oscillator, once moving, swing can parametrically driven alternately standing , squatting @ key points in swing arc. changes moment of inertia of swing , hence resonance frequency, , children can reach large amplitudes provided have amplitude start (e.g., push). standing , squatting @ rest, however, leads nowhere.

transformation of equation

we begin making change of variables

q
(
t
)
 




=



d
e
f





 

e

d
(
t
)


x
(
t
)


{\displaystyle q(t)\ {\stackrel {\mathrm {def} }{=}}\ e^{d(t)}x(t)}

where



d
(
t
)


{\displaystyle d(t)}

time integral of damping

d
(
t
)
 




=



d
e
f





 


1
2



∫

t


d
τ
 
β
(
τ
)
.


{\displaystyle d(t)\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{2}}\int ^{t}d\tau \ \beta (\tau ).}

this change of variables eliminates damping term

d

2


q


d

t

2





+

Ω

2


(
t
)
q
=
0


{\displaystyle {\frac {d^{2}q}{dt^{2}}}+\omega ^{2}(t)q=0}

where transformed frequency defined

Ω

2


(
t
)
=

ω

2


(
t
)
−


1
2



(



d
β


d
t



)

−


1
4



β

2


.


{\displaystyle \omega ^{2}(t)=\omega ^{2}(t)-{\frac {1}{2}}\left({\frac {d\beta }{dt}}\right)-{\frac {1}{4}}\beta ^{2}.}

in general, variations in damping , frequency relatively small perturbations

β
(
t
)
=

ω

0



[
b
+
g
(
t
)
]



{\displaystyle \beta (t)=\omega _{0}\left[b+g(t)\right]}









ω

2


(
t
)
=

ω

0


2



[
1
+
h
(
t
)
]



{\displaystyle \omega ^{2}(t)=\omega _{0}^{2}\left[1+h(t)\right]}

where




ω

0




{\displaystyle \omega _{0}}

,



b


{\displaystyle b}

constants, namely, time-averaged oscillator frequency , damping, respectively. transformed frequency can written in similar way:

Ω

2


(
t
)
=

ω

n


2



[
1
+
f
(
t
)
]

,


{\displaystyle \omega ^{2}(t)=\omega _{n}^{2}\left[1+f(t)\right],}

where




ω

n




{\displaystyle \omega _{n}}

natural frequency of damped harmonic oscillator

ω

n


2


 




=



d
e
f





 

ω

0


2



(
1
−



b

2


4


)



{\displaystyle \omega _{n}^{2}\ {\stackrel {\mathrm {def} }{=}}\ \omega _{0}^{2}\left(1-{\frac {b^{2}}{4}}\right)}

and

ω

n


2


f
(
t
)
 




=



d
e
f





 

ω

0


2



{
h
(
t
)
−


1

2

ω

0






(



d
g


d
t



)

−


b
2


g
(
t
)
−


1
4



g

2


(
t
)
}

.


{\displaystyle \omega _{n}^{2}f(t)\ {\stackrel {\mathrm {def} }{=}}\ \omega _{0}^{2}\left\{h(t)-{\frac {1}{2\omega _{0}}}\left({\frac {dg}{dt}}\right)-{\frac {b}{2}}g(t)-{\frac {1}{4}}g^{2}(t)\right\}.}

thus, our transformed equation can written

d

2


q


d

t

2





+

ω

n


2



[
1
+
f
(
t
)
]

q
=
0.


{\displaystyle {\frac {d^{2}q}{dt^{2}}}+\omega _{n}^{2}\left[1+f(t)\right]q=0.}

the independent variations



g
(
t
)


{\displaystyle g(t)}

,



h
(
t
)


{\displaystyle h(t)}

in oscillator damping , resonance frequency, respectively, can combined single pumping function



f
(
t
)


{\displaystyle f(t)}

. converse conclusion form of parametric excitation can accomplished varying either resonance frequency or damping, or both.

solution of transformed equation

let assume



f
(
t
)


{\displaystyle f(t)}

sinusoidal, specifically

f
(
t
)
=

f

0


sin
⁡
2

ω

p


t


{\displaystyle f(t)=f_{0}\sin 2\omega _{p}t}

where pumping frequency




ω

p


≈

ω

n




{\displaystyle \omega _{p}\approx \omega _{n}}

need not equal




ω

n




{\displaystyle \omega _{n}}

exactly. solution



q
(
t
)


{\displaystyle q(t)}

of our transformed equation may written

q
(
t
)
=
a
(
t
)
cos
⁡

ω

p


t
+
b
(
t
)
sin
⁡

ω

p


t


{\displaystyle q(t)=a(t)\cos \omega _{p}t+b(t)\sin \omega _{p}t}

where have factored out rapidly varying components (



cos
⁡

ω

p


t


{\displaystyle \cos \omega _{p}t}

,



sin
⁡

ω

p


t


{\displaystyle \sin \omega _{p}t}

) isolate varying amplitudes



a
(
t
)


{\displaystyle a(t)}

,



b
(
t
)


{\displaystyle b(t)}

. corresponds laplace s variation of parameters method.

substituting solution transformed equation , retaining terms first-order in




f

0


≪
1


{\displaystyle f_{0}\ll 1}

yields 2 coupled equations

2

ω

p





d
a


d
t



=

(



f

0


2


)


ω

n


2


a
−

(

ω

p


2


−

ω

n


2


)

b


{\displaystyle 2\omega _{p}{\frac {da}{dt}}=\left({\frac {f_{0}}{2}}\right)\omega _{n}^{2}a-\left(\omega _{p}^{2}-\omega _{n}^{2}\right)b}








2

ω

p





d
b


d
t



=
−

(



f

0


2


)


ω

n


2


b
+

(

ω

p


2


−

ω

n


2


)

a


{\displaystyle 2\omega _{p}{\frac {db}{dt}}=-\left({\frac {f_{0}}{2}}\right)\omega _{n}^{2}b+\left(\omega _{p}^{2}-\omega _{n}^{2}\right)a}

we may decouple , solve these equations making change of variables

a
(
t
)
 




=



d
e
f





 
r
(
t
)
cos
⁡
θ
(
t
)


{\displaystyle a(t)\ {\stackrel {\mathrm {def} }{=}}\ r(t)\cos \theta (t)}








b
(
t
)
 




=



d
e
f





 
r
(
t
)
sin
⁡
θ
(
t
)


{\displaystyle b(t)\ {\stackrel {\mathrm {def} }{=}}\ r(t)\sin \theta (t)}

which yields equations

d
r


d
t



=

(

α


m
a
x



cos
⁡
2
θ
)

r


{\displaystyle {\frac {dr}{dt}}=\left(\alpha _{\mathrm {max} }\cos 2\theta \right)r}











d
θ


d
t



=
−

α


m
a
x




[
sin
⁡
2
θ
−
sin
⁡
2

θ


e
q



]



{\displaystyle {\frac {d\theta }{dt}}=-\alpha _{\mathrm {max} }\left[\sin 2\theta -\sin 2\theta _{\mathrm {eq} }\right]}

where have defined brevity

α


m
a
x



 




=



d
e
f





 




f

0



ω

n


2




4

ω

p







{\displaystyle \alpha _{\mathrm {max} }\ {\stackrel {\mathrm {def} }{=}}\ {\frac {f_{0}\omega _{n}^{2}}{4\omega _{p}}}}








sin
⁡
2

θ


e
q



 




=



d
e
f





 

(


2

f

0




)

ϵ


{\displaystyle \sin 2\theta _{\mathrm {eq} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {2}{f_{0}}}\right)\epsilon }

and detuning

ϵ
 




=



d
e
f





 




ω

p


2


−

ω

n


2




ω

n


2






{\displaystyle \epsilon \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\omega _{p}^{2}-\omega _{n}^{2}}{\omega _{n}^{2}}}}

the



θ


{\displaystyle \theta }

equation not depend on



r


{\displaystyle r}

, , linearization near equilibrium position




θ


e
q





{\displaystyle \theta _{\mathrm {eq} }}

shows



θ


{\displaystyle \theta }

decays exponentially equilibrium

θ
(
t
)
=

θ


e
q



+

(

θ

0


−

θ


e
q



)


e

−
2
α
t




{\displaystyle \theta (t)=\theta _{\mathrm {eq} }+\left(\theta _{0}-\theta _{\mathrm {eq} }\right)e^{-2\alpha t}}

where decay constant

α
 




=



d
e
f





 

α


m
a
x



cos
⁡
2

θ


e
q





{\displaystyle \alpha \ {\stackrel {\mathrm {def} }{=}}\ \alpha _{\mathrm {max} }\cos 2\theta _{\mathrm {eq} }}

.

in other words, parametric oscillator phase-locks pumping signal



f
(
t
)


{\displaystyle f(t)}

.

taking



θ
(
t
)
=

θ


e
q





{\displaystyle \theta (t)=\theta _{\mathrm {eq} }}

(i.e., assuming phase has locked),



r


{\displaystyle r}

equation becomes

d
r


d
t



=
α
r


{\displaystyle {\frac {dr}{dt}}=\alpha r}

whose solution



r
(
t
)
=

r

0



e

α
t




{\displaystyle r(t)=r_{0}e^{\alpha t}}

; amplitude of



q
(
t
)


{\displaystyle q(t)}

oscillation diverges exponentially. however, corresponding amplitude



r
(
t
)


{\displaystyle r(t)}

of untransformed variable



x
 




=



d
e
f





 
q

e

−
d
(
t
)




{\displaystyle x\ {\stackrel {\mathrm {def} }{=}}\ qe^{-d(t)}}

need not diverge

r
(
t
)
=
r
(
t
)

e

−
d
(
t
)


=

r

0



e

α
t
−
d
(
t
)




{\displaystyle r(t)=r(t)e^{-d(t)}=r_{0}e^{\alpha t-d(t)}}

the amplitude



r
(
t
)


{\displaystyle r(t)}

diverges, decays or stays constant, depending on whether



α
t


{\displaystyle \alpha t}

greater than, less than, or equal



d
(
t
)


{\displaystyle d(t)}

, respectively.

the maximum growth rate of amplitude occurs when




ω

p


=

ω

n




{\displaystyle \omega _{p}=\omega _{n}}

. @ frequency, equilibrium phase




θ


e
q





{\displaystyle \theta _{\mathrm {eq} }}

zero, implying



cos
⁡
2

θ


e
q



=
1


{\displaystyle \cos 2\theta _{\mathrm {eq} }=1}

,



α
=

α


m
a
x





{\displaystyle \alpha =\alpha _{\mathrm {max} }}

.




ω

p




{\displaystyle \omega _{p}}

varied




ω

n




{\displaystyle \omega _{n}}

,




θ


e
q





{\displaystyle \theta _{\mathrm {eq} }}

moves away 0 ,



α
<

α


m
a
x





{\displaystyle \alpha <\alpha _{\mathrm {max} }}

, i.e., amplitude grows more slowly. sufficiently large deviations of




ω

p




{\displaystyle \omega _{p}}

, decay constant



α


{\displaystyle \alpha }

can become purely imaginary since

α
=

α


m
a
x





1
−


(


2

f

0




)


2



ϵ

2






{\displaystyle \alpha =\alpha _{\mathrm {max} }{\sqrt {1-\left({\frac {2}{f_{0}}}\right)^{2}\epsilon ^{2}}}}

if detuning



ϵ


{\displaystyle \epsilon }

exceeds




f

0



/

2


{\displaystyle f_{0}/2}

,



α


{\displaystyle \alpha }

becomes purely imaginary ,



q
(
t
)


{\displaystyle q(t)}

varies sinusoidally. using definition of detuning



ϵ


{\displaystyle \epsilon }

, pumping frequency



2

ω

p




{\displaystyle 2\omega _{p}}

must lie between



2

ω

n




1
−



f

0


2






{\displaystyle 2\omega _{n}{\sqrt {1-{\frac {f_{0}}{2}}}}}

,



2

ω

n




1
+



f

0


2






{\displaystyle 2\omega _{n}{\sqrt {1+{\frac {f_{0}}{2}}}}}

in order achieve exponential growth in



q


{\displaystyle q}

. expanding square roots in binomial series shows spread in pumping frequencies result in exponentially growing



q


{\displaystyle q}

approximately




ω

n



f

0




{\displaystyle \omega _{n}f_{0}}

.

intuitive derivation of parametric excitation

the above derivation may seem mathematical sleight-of-hand, may helpful give intuitive derivation.



q


{\displaystyle q}

equation may written in form

d

2


q


d

t

2





+

ω

n


2


q
=
−

ω

n


2


f
(
t
)
q


{\displaystyle {\frac {d^{2}q}{dt^{2}}}+\omega _{n}^{2}q=-\omega _{n}^{2}f(t)q}

which represents simple harmonic oscillator (or, alternatively, bandpass filter) being driven signal



−

ω

n


2


f
(
t
)
q


{\displaystyle -\omega _{n}^{2}f(t)q}

proportional response



q


{\displaystyle q}

.

assume



q
(
t
)
=
a
cos
⁡

ω

p


t


{\displaystyle q(t)=a\cos \omega _{p}t}

has oscillation @ frequency




ω

p




{\displaystyle \omega _{p}}

, pumping



f
(
t
)
=

f

0


sin
⁡
2

ω

p


t


{\displaystyle f(t)=f_{0}\sin 2\omega _{p}t}

has double frequency , small amplitude




f

0


≪
1


{\displaystyle f_{0}\ll 1}

. applying trigonometric identity products of sinusoids, product



q
(
t
)
f
(
t
)


{\displaystyle q(t)f(t)}

produces 2 driving signals, 1 @ frequency




ω

p




{\displaystyle \omega _{p}}

, other @ frequency



3

ω

p




{\displaystyle 3\omega _{p}}

f
(
t
)
q
(
t
)
=



f

0


2


a

(
sin
⁡

ω

p


t
+
sin
⁡
3

ω

p


t
)



{\displaystyle f(t)q(t)={\frac {f_{0}}{2}}a\left(\sin \omega _{p}t+\sin 3\omega _{p}t\right)}

being off-resonance,



3

ω

p




{\displaystyle 3\omega _{p}}

signal attentuated , can neglected initially. contrast,




ω

p




{\displaystyle \omega _{p}}

signal on resonance, serves amplify



q


{\displaystyle q}

, proportional amplitude



a


{\displaystyle a}

. hence, amplitude of



q


{\displaystyle q}

grows exponentially unless zero.

expressed in fourier space, multiplication



f
(
t
)
q
(
t
)


{\displaystyle f(t)q(t)}

convolution of fourier transforms






f
~



(
ω
)


{\displaystyle {\tilde {f}}(\omega )}

,






q
~



(
ω
)


{\displaystyle {\tilde {q}}(\omega )}

. positive feedback arises because



+
2

ω

p




{\displaystyle +2\omega _{p}}

component of



f
(
t
)


{\displaystyle f(t)}

converts



−

ω

p




{\displaystyle -\omega _{p}}

component of



q
(
t
)


{\displaystyle q(t)}

driving signal at

+

ω

p




{\displaystyle +\omega _{p}}

, , vice versa (reverse signs). explains why pumping frequency must near



2

ω

n




{\displaystyle 2\omega _{n}}

, twice natural frequency of oscillator. pumping @ grossly different frequency not couple (i.e., provide mutual positive feedback) between



−

ω

p




{\displaystyle -\omega _{p}}

,



+

ω

p




{\displaystyle +\omega _{p}}

components of



q
(
t
)


{\displaystyle q(t)}

.