17.2: Dos Péndulos Acoplados
- Page ID
- 130806
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Tomaremos dos péndulos iguales, acoplados por un resorte ligero. Tomamos que la fuerza restauradora del resorte sea directamente proporcional a la diferencia angular entre los péndulos. (Esto resulta ser una buena aproximación.)
Para pequeños ángulos de oscilación, tomamos el Lagrangiano para ser
\ begin {ecuación}
L=\ frac {1} {2} m\ ell^ {2}\ punto {\ theta} _ {1} ^ {2} +\ frac {1} {2} m\ ell^ {2}\ punto {\ theta} _ {2} ^ {2} -\ frac {1} {2} m g\ ell\ theta_ {1} ^ {2} -\ frac {1} {2} m g\ ell\ theta_ {2} ^ {2} -\ frac {1} {2} C\ izquierda (\ theta_ {1} -\ theta_ {2}\ derecha) ^ {2}
\ end {ecuación}
Denotando la frecuencia de péndulo único por\(\omega_{0}\), las ecuaciones de movimiento son (escribir\(\omega_{0}^{2}=g / \ell, k=C / m \ell^{2}\), entonces\(\left.[k]=T^{-2}\right)\)
\ begin {ecuación}
\ begin {array} {l}
\ ddot {\ theta} _ {1} =-\ omega_ {0} ^ {2}\ theta_ {1} -k\ izquierda (\ theta_ {1} -\ theta_ {2}\ derecha)\
\ ddot {\ theta} _ {2} =-\ omega_ {0} ^ {2}\ theta_ {2} -k\ izquierda (\ theta_ {2} -\ theta_ {1}\ derecha)
\ end {array}
\ end {ecuación}
Buscamos una solución periódica, escribiendo
\ begin {ecuación}
\ theta_ {1} (t) =A_ {1} e^ {i\ omega t},\ quad\ theta_ {2} (t) =A_ {2} e^ {i\ omega t}
\ end {ecuación}
(Las soluciones finales del ángulo físico serán la parte real.)
Las ecuaciones se convierten (en notación matricial):
\ begin {ecuación}
\ omega^ {2}\ left (\ begin {array} {c}
A_ {1}\\
A_ {2}
\ end {array}\ right) =\ left (\ begin {array} {cc}
\ omega_ {0} ^ {2} +k & -k\\
-k &\ omega_ {0} ^ {2} +k
\ end {array}\ derecha)\ izquierda (\ begin {array} {c}
A_ {1}\\
A_ {2}
\ end {array}\ right)
\ end {ecuación}
Denotando el\(2 \times 2 \text { matrix by } \mathbf{M}\)
\ begin {ecuación}
\ mathbf {M}\ vec {A} =\ omega^ {2}\ vec {A},\ quad\ vec {A} =\ left (\ begin {array} {l}
A_ {1}\\
A_ {2}
\ end {array}\ right)
\ end {ecuación}
Esta es una ecuación de vector propio, con\(\omega^{2}\) el valor propio, que se encuentra por el procedimiento estándar:
\ begin {ecuación}
\ operatorname {det}\ left (\ mathbf {M} -\ omega^ {2}\ mathbf {I}\ derecha) =\ izquierda|\ begin {array} {cc}
\ omega_ {0} ^ {2} +k-\ omega^ {2} & -k\\
-k &\ omega_ {0} ^ {2} +k-\ omega^ {2}
\ end {array}\ derecha|=0
\ end {ecuación}
Resolviendo\(\omega^{2}=\omega_{0}^{2}+k \pm k\),, es decir
\ begin {ecuación}
\ omega^ {2} =\ omega_ {0} ^ {2},\ quad\ omega^ {2} =\ omega_ {0} ^ {2} +2 k
\ final {ecuación}
Los vectores propios correspondientes son (1,1) y (1, −1).