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Set up the differential equation of motion for the double pendulum shown in the figure below, using the coordinates given assuming small amplitudes. Find the natural frequencies and the two modes of vibration.WWW
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Mathematical Model: The Double Pendulum The double pendulum is a classic example of a nonlinear dynamical system exhibiting deterministic chaos. To model its motion, we use Lagrangian Mechanics, focusing on energy states rather than force vectors. 1. Generalized Coordinates The system is defined by two angles, θ₁ and θ₂, measured from the vertical. The positions of the masses (m₁, m₂) are: • Mass 1: x₁ = L₁ sin(θ₁), y₁ = -L₁ cos(θ₁) • Mass 2: x₂ = x₁ L₂ sin(θ₂), y₂ = y₁ - L₂ cos(θ₂) 2. Kinetic (
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