Pendula Nonlinear system




illustration of pendulum



linearizations of pendulum


a classic, extensively studied nonlinear problem dynamics of pendulum under influence of gravity. using lagrangian mechanics, may shown motion of pendulum can described dimensionless nonlinear equation











d

2


θ


d

t

2





+
sin

(
θ
)
=
0


{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\sin(\theta )=0}



where gravity points downwards ,



θ


{\displaystyle \theta }

angle pendulum forms rest position, shown in figure @ right. 1 approach solving equation use



d
θ

/

d
t


{\displaystyle d\theta /dt}

integrating factor, yield











d
θ



c

0


+
2
cos

(
θ
)



=
t
+

c

1




{\displaystyle \int {\frac {d\theta }{\sqrt {c_{0}+2\cos(\theta )}}}=t+c_{1}}



which implicit solution involving elliptic integral. solution not have many uses because of nature of solution hidden in nonelementary integral (nonelementary unless




c

0


=
2


{\displaystyle c_{0}=2}

).


another way approach problem linearize nonlinearities (the sine function term in case) @ various points of interest through taylor expansions. example, linearization @



θ
=
0


{\displaystyle \theta =0}

, called small angle approximation, is











d

2


θ


d

t

2





+
θ
=
0


{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\theta =0}



since



sin

(
θ
)

θ


{\displaystyle \sin(\theta )\approx \theta }





θ

0


{\displaystyle \theta \approx 0}

. simple harmonic oscillator corresponding oscillations of pendulum near bottom of path. linearization @



θ
=
π


{\displaystyle \theta =\pi }

, corresponding pendulum being straight up:











d

2


θ


d

t

2





+
π

θ
=
0


{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\pi -\theta =0}



since



sin

(
θ
)

π

θ


{\displaystyle \sin(\theta )\approx \pi -\theta }





θ

π


{\displaystyle \theta \approx \pi }

. solution problem involves hyperbolic sinusoids, , note unlike small angle approximation, approximation unstable, meaning




|

θ

|



{\displaystyle |\theta |}

grow without limit, though bounded solutions possible. corresponds difficulty of balancing pendulum upright, literally unstable state.


one more interesting linearization possible around



θ
=
π

/

2


{\displaystyle \theta =\pi /2}

, around



sin

(
θ
)

1


{\displaystyle \sin(\theta )\approx 1}

:











d

2


θ


d

t

2





+
1
=
0.


{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+1=0.}



this corresponds free fall problem. useful qualitative picture of pendulum s dynamics may obtained piecing such linearizations, seen in figure @ right. other techniques may used find (exact) phase portraits , approximate periods.








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