How can I find the Lagrangian of a disk rolling in a plane?
Answers
Answered by
0
✔✔stract-A circular disk rolling on a horizontal surface without slip is a common example of
a nonholonomic problem of Lagrangian mechanics. In the literature, very few such problems have
been attacked developing simple and interesting results. Here, it will be shown first that it is possible
✔✔✔to find a steady motion of the system, which is rolling in a circle. Then it will be shown that it
is possible to find the expression for the frequency of small oscillations about this steady motion,
including an approximate expression for the speed required for stabilit,y of that steady motion, which
✔✔✔✔is accurate for circles large in radius compared to the disk. The solution is shown to reduce to the
well-known critical speed for a disk rolling in a straight line. It is then surprising to note that the
critical speed for stability is slower for motion in a circle than for the straight line. This result will
be developed entirely analytically which seems to be quite unusual for such a nonholonomic problem
of this complexity. @ 2002 Elsevier Science Ltd. All rights reserved.
Keywords-Dynamics, Lagrangian, Non-holonomic, Steady motion, Stability, Oscillations.
THE LAGRANGIAN APPROACH
The rolling disk is shown in Figure 1 in order to illustrate the generalized coordinates used here
to establish its position. The five coordinates selected are X, Y, Q,c$, 1+9, where X and Y locate the
center of the disk projected vertically on to the horizontal plane and the three angles describe
the angular position of the rigid body in an Eulerian fashion. The principal axes of moment of
inertia are also identified with x as the axis of symmetry with the other orthogonal pair y and z
with y not rotating with the body but remaining parallel to the horizontal plane. The rotation
rate vectors are also shown on this figure.
For the Lagrangian approach there will be five Lagrange’s equations, one for each coordinate
required and two equations of constraint representing the condition of no slip on the horizontal
plane.
The authors acknowledge the motivation provided by .I. Ferry and E. R. Brandon to bring this work to completion.
The first author also wishes to make special note of the extensive algebraic efforts of the second author that made
and assured the accuracy of the analysis in this work. In addition, the editorial assistance of H. Tada is gratefully
acknowledged.
a nonholonomic problem of Lagrangian mechanics. In the literature, very few such problems have
been attacked developing simple and interesting results. Here, it will be shown first that it is possible
✔✔✔to find a steady motion of the system, which is rolling in a circle. Then it will be shown that it
is possible to find the expression for the frequency of small oscillations about this steady motion,
including an approximate expression for the speed required for stabilit,y of that steady motion, which
✔✔✔✔is accurate for circles large in radius compared to the disk. The solution is shown to reduce to the
well-known critical speed for a disk rolling in a straight line. It is then surprising to note that the
critical speed for stability is slower for motion in a circle than for the straight line. This result will
be developed entirely analytically which seems to be quite unusual for such a nonholonomic problem
of this complexity. @ 2002 Elsevier Science Ltd. All rights reserved.
Keywords-Dynamics, Lagrangian, Non-holonomic, Steady motion, Stability, Oscillations.
THE LAGRANGIAN APPROACH
The rolling disk is shown in Figure 1 in order to illustrate the generalized coordinates used here
to establish its position. The five coordinates selected are X, Y, Q,c$, 1+9, where X and Y locate the
center of the disk projected vertically on to the horizontal plane and the three angles describe
the angular position of the rigid body in an Eulerian fashion. The principal axes of moment of
inertia are also identified with x as the axis of symmetry with the other orthogonal pair y and z
with y not rotating with the body but remaining parallel to the horizontal plane. The rotation
rate vectors are also shown on this figure.
For the Lagrangian approach there will be five Lagrange’s equations, one for each coordinate
required and two equations of constraint representing the condition of no slip on the horizontal
plane.
The authors acknowledge the motivation provided by .I. Ferry and E. R. Brandon to bring this work to completion.
The first author also wishes to make special note of the extensive algebraic efforts of the second author that made
and assured the accuracy of the analysis in this work. In addition, the editorial assistance of H. Tada is gratefully
acknowledged.
Similar questions