A beam is supported by a hinged joint at one end (O) 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.

A beam is supported by a hinged joint at one end (O) 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.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Generally, there are three types of vibration: longitudinal vibration, transverse vibration, and torsional vibration.

Question

A beam is supported by a hinged joint at one end (O) 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.
D
F(t)
F(t)

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