Spherical pendulum

Spherical pendulum -...

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Example: Spherical pendulum http://andreas-bischoff.dyndns.org/lehre/KURSE/PRT001/course_main. .. 1 of 3 10/28/2010 9:16 AM Next: Inertia of rigid bodies Up: LAGRANGE's equations Previous: First formulations Contents Example: Spherical pendulum Let us consider the idealized spherical pendulum presented in figure 5.1 . Figure 5.1: Spherical pendulum This mechanical system consists of a mass , which is attached to a spherical joint via a massless rod. Two angles und describe the position of the mass. Assuming that all initial conditions are known, we have to derive the equations of motion of the mass under the influence of gravity. As a first step we develop the systems Lagrangian . If related to the origin of the pendulum, the position of the mass is given by the vector , which leads to: (5.6) Consequently the kinetic energy is:
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Unformatted text preview: http://andreas-bischoff.dyndns.org/lehre/KURSE/PRT001/course_main. .. 2 of 3 10/28/2010 9:16 AM and for the potential energy we find: where is the gravitational constant. The resulting Lagrangian is: Herein the generalized coordinates are . Substituting the Lagrangian into Lagrange's equation of motion we find the following individual terms: Example: Spherical pendulum http://andreas-bischoff.dyndns.org/lehre/KURSE/PRT001/course_main. .. 3 of 3 10/28/2010 9:16 AM Thus the dynamics of the system are given by: (5.7) Given the initial position and velocity of the center of mass, equation ( 5.7 ) uniquely describes the motion of the pendulum. Next: Inertia of rigid bodies Up: LAGRANGE's equations Previous: First formulations Contents Michael Gerke 2001-01-18...
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Spherical pendulum -...

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