Answer to Question #145943 in Classical Mechanics for sphume

Question #145943
A planar double pendulum consists of two pendula that
swing in a common plane (see sketch). Find the equations
of motion for the angles ϕ1 and ϕ2 using D’Alembert’s
Principle.
1
Expert's answer
2020-11-23T10:32:50-0500

As per the given question,


"x_1=l_1\\sin\\phi_1"

"y_1=-l_1\\cos\\phi_1"

"x_2=l_1\\sin\\phi_1+l_2\\sin\\phi_2"

"y_2=-(l_1\\cos\\phi_1+l_2\\cos\\phi_2)",

Now, taking the differentiation ,

"\\frac{dx_1}{dt}=l_1\\cos\\phi_1 \\frac{d\\phi_1}{dt}"


"\\frac{dy_1}{dt}=l_1\\sin\\phi_1 \\frac{d\\phi_1}{dt}"


"\\frac{y_2}{dt}=l_1\\sin\\phi_1\\frac{d\\phi_1}{dt}+l_2\\sin\\phi_2\\frac{d\\phi_2}{dt}"


"\\frac{x_2}{dt}=(l_1\\cos\\phi_1\\frac{d\\phi_1}{dt}+l_2\\cos\\phi_2 \\frac{d\\phi_2}{dt})"


"L=T-V=\\frac{m_1v_1^2}{2}+\\frac{m_2v_2^2}{2}-V"


"T=\\frac{m_1((\\frac{x_1}{dt})^2+(\\frac{y_1}{dt})^2)}{2}+\\frac{m_1((\\frac{x_2}{dt})^2+(\\frac{y_2}{dt})^2)}{2}"

Hence langrangian equation will be

"L=\\frac{(m_1+m_2)l_1^2}{2}(\\frac{d\\phi_1}{dt})^2+\\frac{(m_2)l_2^2}{2}(\\frac{d\\phi_2}{dt})^2+m_2l_1l_2\\frac{d\\phi_1}{dt}\\frac{d\\phi_2}{dt}\\cos(\\phi_1-\\phi_2)+(m_1+m_2)gl_1\\cos\\phi_1+m_2gl_2\\cos\\phi_1"


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