A beam is supported by a hinged joint at one end (0) and by a spring at the other end (A) as shown in the figure. The stiffness of the spring is such that the beam end A lies directly under the spring attachment point B when the spring is at its free length (no force). Using Hamilton's principle, develop the Euler-Lagrange equation of motion for the system when subjected to 1. A time-dependent vertical force applied at end A of the beam. 2. A time dependent force at end A of the beam that remains oriented perpendicular to the beam for all angles of the beam. Assume the following for your analysis 1. The beam is rigid. 2. The beam is uniform, i.e. its mass per unit length is constant) 3. The thickness of the beam is small compared to its length 4. The spring stiffness is constant. 5. The spring mass is small and can be ignored. 6. The angular displacements of the beam are not necessarily small.

A beam is supported by a hinged joint at one end (0) and by a spring at the other end (A) as shown in the figure. The stiffness of the spring is such that the beam end A lies directly under the spring attachment point B when the spring is at its free length (no force). Using Hamilton's principle, develop the Euler-Lagrange equation of motion for the system when subjected to 1. A time-dependent vertical force applied at end A of the beam. 2. A time dependent force at end A of the beam that remains oriented perpendicular to the beam for all angles of the beam. Assume the following for your analysis 1. The beam is rigid. 2. The beam is uniform, i.e. its mass per unit length is constant) 3. The thickness of the beam is small compared to its length 4. The spring stiffness is constant. 5. The spring mass is small and can be ignored. 6. The angular displacements of the beam are not necessarily small.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.32P

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