Answered: Figure P3.40 illustrates a pendulum…

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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Question

Figure P3.40 illustrates a pendulum with a base that moves horizontally. This
is a simple model of an overhead crane carrying a suspended load with cables.
The load mass is m, the cable length is L, and the base acceleration is a(t).
Assuming that the cable acts like a rigid rod, derive the equation of motion in
terms of ? with a(t) as the input.

Expert Solution

Step 1

To derive the equation of motion for the pendulum with a base that moves horizontally, we can use the Lagrangian approach.

Let $θ$ be the angle between the cable and the vertical, and let x be the horizontal displacement of the base. The kinetic and potential energies of the system can be expressed as:

Kinetic Energy: T = ((1/2) $\times$ m $\times$ (L $\times$ $\dot{θ}$ ²)) + ((1/2) $\times$ m $\times$ $\dot{x}$ ²)

Potential Energy: V = -mgL cos( $θ$ )

where $\dot{θ}$ and $\dot{x}$ are the time derivatives of $θ$ and x, respectively.

The Lagrangian of the system is given by:

L = T - V = ((1/2) $\times$ m $\times$ (L $\dot{θ}$ ²)) + ((1/2) $\times$ m $\times$ $\dot{x}$ ²)+ mgL cos( $θ$ )

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