For small displacements of the pendulums from the equilibrium, the Lagrange equations are given by (m1 + m2)l191 + m2l202 + (m1 + m2)gø1 = 0, hö1 + l202 + gó2 = 0. For simplicity, assume that the pendulums have equal lengths: l1 = l2 = 1. Determine the small ocillations of this system, that is, the characteristic frequencies and relative amplitudes for each frequency.
For small displacements of the pendulums from the equilibrium, the Lagrange equations are given by (m1 + m2)l191 + m2l202 + (m1 + m2)gø1 = 0, hö1 + l202 + gó2 = 0. For simplicity, assume that the pendulums have equal lengths: l1 = l2 = 1. Determine the small ocillations of this system, that is, the characteristic frequencies and relative amplitudes for each frequency.
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Hi please show all work and explanation. A helpful hint is already provided.
Transcribed Image Text:For small displacements of
the pendulums from the equilibrium, the Lagrange equations are given by
( ml + m2)lιφι + m21 φ2 + (m + m2) gφι 0,
höi + l202 + gø2 = 0.
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For simplicity, assume that the pendulums have equal lengths: l1 = l2 = l. Determine the small
oscillations of this system, that is, the characteristic frequencies and relative amplitudes for each
frequency.
Hints: Substitute in the Lagrange equations øa =
tions in a matrix form, and equal its determinant to zero, which will give an equation for two values
of the frequency w. Then, for each value of w, substitute w into one of the resulting equations, which
will give a relation between the amplitudes Aa.
Age?wt for a = 1, 2, write the two resulting equa-
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