A collar of mass m slides without friction on a horizontal rigid rod and is restrained by a pair of identical springs with spring constant k. A pendulum consisting of a uniform rigid bar of length L and mass Mis suspended from the collar by a frictionless pivot. L, M 2.1 Select a complete and independent set of generalized coordinates for this system. 2.2 Derive using Lagrange's principle, the equations of motion of the system in the form of a matrix differential equation. 2.3 Develop the state-space model. å 2.4 Derive an eigenvalue problem of the form [K]{a}=¹[M]{a}

A collar of mass m slides without friction on a horizontal rigid rod and is restrained by a pair of identical springs with spring constant k. A pendulum consisting of a uniform rigid bar of length L and mass Mis suspended from the collar by a frictionless pivot. L, M 2.1 Select a complete and independent set of generalized coordinates for this system. 2.2 Derive using Lagrange's principle, the equations of motion of the system in the form of a matrix differential equation. 2.3 Develop the state-space model. å 2.4 Derive an eigenvalue problem of the form [K]{a}=¹[M]{a}

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 Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.

Question

Hint: Assume your own initial conditions

A collar of mass m slides without friction on a horizontal rigid rod and is restrained by a pair of identical
springs with spring constant k. A pendulum consisting of a uniform rigid bar of length L and mass M is
suspended from the collar by a frictionless pivot.
m
g
\L, M
2.1 Select a complete and independent set of generalized coordinates for this system.
2.2 Derive using Lagrange's principle, the equations of motion of the system in the form of a matrix
differential equation.
2.3 Develop the state-space model.
2.4 Derive an eigenvalue problem of the form
[K]{a} = o* [M]{a}
for natural modes of the form
S*_ Ja, sin(or + p)
x l4, sin(ox + p)f
2.5 Solve analytically for the mode shapes {a, a,}' and the eigenvalues of and o. Construct the
modal matrix [0] whose columns are the mode shape vectors.

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