i) A uniform rigid shelf is supported on symmetrically placed identical elastic supports as shown in Figure 3.1. The shelf is of length 2L, mass m and moment of inertia about its centre Im. The supports have stiffness k and are positioned equally spaced apart a distance 2a about the centre of mass C. Centre of mass C k 2a 0 2L Figure 3.1. A shelf on its elastic supports Determine the two lowest natural frequencies, where the shelf is considered to be rigid and undergoing small rotation , about C, and displacement y, of C, from the horizontal equilibrium position. a) Derive the equations of motions, using Lagrange's equations, in terms of the rotation and the displacement. Clearly state any assumptions made and show the relevant diagrams or relationships used in the derivation.

i) A uniform rigid shelf is supported on symmetrically placed identical elastic supports as shown in Figure 3.1. The shelf is of length 2L, mass m and moment of inertia about its centre Im. The supports have stiffness k and are positioned equally spaced apart a distance 2a about the centre of mass C. Centre of mass C k 2a 0 2L Figure 3.1. A shelf on its elastic supports Determine the two lowest natural frequencies, where the shelf is considered to be rigid and undergoing small rotation , about C, and displacement y, of C, from the horizontal equilibrium position. a) Derive the equations of motions, using Lagrange's equations, in terms of the rotation and the displacement. Clearly state any assumptions made and show the relevant diagrams or relationships used in the derivation.

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

Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provide resistance in changing its state of velocity or motion for coming to a state of rest. In other words, Newton's first law of motion describes the law of inertia of a rigid body.

Question

Q.3.
(i) A uniform rigid shelf is supported on symmetrically placed
identical elastic supports as shown in Figure 3.1. The shelf is of
length 2L, mass m and moment of inertia about its centre Im. The
supports have stiffness k and are positioned equally spaced apart
a distance 2a about the centre of mass C.
Centre of mass C
k
3
2a
y
0
k
2L
Figure 3.1. A shelf on its elastic supports
Determine the two lowest natural frequencies, where the shelf is
considered to be rigid and undergoing small rotation , about C, and
displacement y, of C, from the horizontal equilibrium position.
a) Derive the equations of motions, using Lagrange's equations,
in terms of the rotation and the displacement. Clearly state
any assumptions made and show the relevant diagrams or
relationships used in the derivation.
b) Obtain algebraic expressions for the two undamped natural
frequencies and describe these modes in words.

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