Mechanical Vibration 13-Batch Lec04

7/23/2019 Mechanical Vibration 13-Batch Lec04

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MEHRAN UNIVERSITYOF ENGINEERING & TECHNOLOGY, JAMSHORO

MECHANICAL ENGINEERING, DEPARTMENT

13-BATCH

MECHANICAL VIBRATION

Lecture # 04

Topic

Simple harmonic motion

(cont..)


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SIMPLE HARMONIC MOTION

REVISION

If the motion is repeated after equal intervals of time, it is calledperiodic motion.

The simplest type of periodic motion is known assimple harmonic motion.

C

In Figure-A: a mass hanging on a light spring is pulled and released.

In Figure-B:a simple pendulum is formed by connecting a mass to one end of a

light string, and fixing the other end to the ceiling.

In Figure-C:a U-shaped tube with water is shaken so that water rises and falls in

each of its branches. The higher water rises in one branch, the lower it falls in the

other branch, and the more pressure difference is generated to push it back.

A B


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Harmonic motion, A specific type of periodic motion, simple harmonic motion, has a

rather straightforward mathematical representation:

where A is the amplitude of motion, is the angular frequency, t is the elapsed time,

and is a phase factor identifying at what point in the cycle we chose t = 0. The

angular frequency is related to the period through the relationship:

If an object undergoes motion as described above, the objects velocity and

acceleration as functions of time will be,

Acceleration can be written in form ofxwill be:


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LINEAR MOTION (MASS and SPRING):

As discussed that a spring will be stretched a

distance proportional to the applied force.

Mathematically this is stated as:

F = k * stretch

The negative sign shows that the force exerted

by the spring is always in the opposite

rec on o e s re c o e spr ng.

The net force acting on the mass would be the extra force

due to displacement x. Newtons second law would yield:

Where and

Therefore


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LINEAR MOTION (MASS and SPRING):

Acceleration in Simple Harmonic Motion

1. The acceleration is not constant.

2. More mass means less acceleration.

3. The acceleration is always in the opposite

direction of the displacement.

4. Acceleration is proportional to displacement,

meaning it is greatest when x is greatest ( at

the ends) and zero at x(o).

5. Acceleration is proportional to the spring

constant. The stronger the spring, the stronger

the acceleration.


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SIMPLE PENDULUM:

Applying Newtons 2nd law to the object in the

direction of motion gives:

For small angles, sin()= s/l,

where s

is the displacement of the mass fromequilibrium.

The acceleration a yields:

Where and

Therefore

1. The period of the pendulum does not depend

on the mass of the bob.

2. (in small angles) , the period does not depend

on the beginning displacement. The pendulum

moves faster through long arcs, slower throughsmall arcs.

3. The period is proportional to the square root

of the length. Doubling the length will not double

the period.

4. The period depends on gravity (inverse

square root). Thus pendulums will have different

periods at different locations on the earth.


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PROBLEMS:

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PROBLEMS:

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Mechanical Vibration 13-Batch Lec04