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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