Two identical uniform rods of length L and mass m are connected at the top by a revolute joint, also of mass m that itself is connected to a fixed support by a vertical spring of natural length L and having the stiffness constant k. An ideal spring of natural length 2L and having stiffness constant also equal to k connects the bottoms of each rod. This is shown in the figure below. The two-rod/two-spring/revolute joint system Assume that gravity acts downward. Assume that throughout the motion, the bottom spring always remains horizontal so that the angle between each rod and the negative vertical is e and that 0 s e s r. In addition, assume an xy coordinate system in the plane of the motion, with its origin at the place where the top spring is at its natural position, with positive x to the right and positive y up. Using the generalized coordinates y (the amount that the top spring is stretched or compressed) and 0 (the angle that each rod makes with the negative vertical), construct expressions for the potential and kinetic energies of this system. Construct the Lagrangian for the system and use this to construct the equations of motion for the coordinates y and 0. ell

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Problem #1
Two identical uniform rods of length L and mass m are connected at the top by a revolute
joint, also of mass m that itself is connected to a fixed support by a vertical spring of
natural length L and having the stiffness constant k. An ideal spring of natural length 2L
and having stiffness constant also equal to k connects the bottoms of each rod. This is
shown in the figure below.
The two-rod/two-spring/revolute joint system
Assume that gravity acts downward.
Assume that throughout the motion, the bottom spring always remains horizontal so that
the angle between each rod and the negative vertical is e and that 0 <e sr. In addition,
assume an xy coordinate system in the plane of the motion, with its origin at the place
where the top spring is at its natural position, with positive x to the right and positive y up.
Using the generalized coordinates y (the amount that the top spring is stretched or
compressed) and e (the angle that each rod makes with the negative vertical),
a.)
b.)
equations of motion for the coordinates y and 0.
construct expressions for the potential and kinetic energies of this system.
Construct the Lagrangian for the system and use this to construct the
Transcribed Image Text:Problem #1 Two identical uniform rods of length L and mass m are connected at the top by a revolute joint, also of mass m that itself is connected to a fixed support by a vertical spring of natural length L and having the stiffness constant k. An ideal spring of natural length 2L and having stiffness constant also equal to k connects the bottoms of each rod. This is shown in the figure below. The two-rod/two-spring/revolute joint system Assume that gravity acts downward. Assume that throughout the motion, the bottom spring always remains horizontal so that the angle between each rod and the negative vertical is e and that 0 <e sr. In addition, assume an xy coordinate system in the plane of the motion, with its origin at the place where the top spring is at its natural position, with positive x to the right and positive y up. Using the generalized coordinates y (the amount that the top spring is stretched or compressed) and e (the angle that each rod makes with the negative vertical), a.) b.) equations of motion for the coordinates y and 0. construct expressions for the potential and kinetic energies of this system. Construct the Lagrangian for the system and use this to construct the
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