MAURBHANJ SCHOOL OF ENGINEERING�BARIPADA : 757107 , ODISHA�

Branch : Mechanical Engineering

Semester : 4^th Sem

Subject : THEORY OF MACHINE

Topic : VIBRATION

Faculty : Er. Malabika Nayak

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Vibrations

Periodic Motion

• Periodic motion is one �of the most important kinds �of physical behavior�
• A large number of systems �can be modeled with this idea

Introduction

Periodic Motion and Waves

• Periodic motion can cause disturbances that move through �a medium in the form of a wave
• Many kinds of waves

Big Bang, Inflation, Gravitational Waves

2014

Amplitude / Space

• Amplitude, A
• The amplitude is the maximum � position of the object from �its equilibrium position

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��In the absence of friction, �an object in simple harmonic motion �will oscillate between the positions x = ±A

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Period and Frequency / Time

• The period, T, is the time that it takes for the object to complete one complete cycle of motion

Time from x = A to x = - A and back to x = A� period T, is time, in seconds

• The frequency, ƒ, is the number �of complete cycles or vibrations �per unit time (per one second)

Frequency is the reciprocal of the period, ƒ = 1 / T��f, frequency , is in reciprocal seconds = Hertz = Hz

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Hooke’s Law

• F_s = - k x

F_s is the spring force

k is the spring constant

It is a measure of the stiffness of the spring:

a large k indicates a stiff spring �and a small k indicates a soft spring

x = 0 at the equilibrium position� x is the displacement of the object from � its equilibrium position �

The negative sign indicates that the force �is directed opposite to the displacement

x = 0

x < 0

x

Hooke’s Law Force

• The force acts toward the equilibrium position

It is called the restoring force

Simple Harmonic Motion

• Motion that occurs when the net force along the direction of motion obeys Hooke’s Law�

The force is proportional to the displacement and always directed toward the equilibrium position�

• The motion of a spring-mass system is an example of Simple Harmonic Motion

Period and Frequency �from Circular Motion

• Period

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This gives the time required for an object of mass m attached to a spring of constant k �to complete one cycle

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• Frequency

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Units are (# of cycles)/second = Hertz = Hz

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Simple Pendulum

• The simple pendulum is another example of a system that exhibits simple harmonic motion
• The restoring force is the component of the weight tangent to the path of motion

F_t = - mg sin θ

Period of Simple Pendulum

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• This shows that the period is independent of the amplitude A and the mass m
• The period depends on:� the length of the pendulum L � and the acceleration of gravity g

A simple pendulum is suspended from the ceiling of a stationary elevator, and the period is measured. � If the elevator accelerates upward, the period T

1. increases
2. decreases
3. remains the same

Quick Quiz

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(falls down freely)

Angular Frequency

• The angular frequency is related to the frequency

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• The frequency gives the number of cycles per second�
• The angular frequency gives the number �of radians per second

Simple Harmonic Motion �and Uniform Circular Motion

• A ball is attached to the rim of a turntable of radius A�
• The focus is on the shadow that the ball casts on the screen�
• When the turntable rotates with a constant angular speed, the shadow moves in simple harmonic motion

Wave Motion

• A wave is the motion of a disturbance�
• Mechanical waves require
• Some source of disturbance
• A medium that can be disturbed
• Some physical connection or mechanism though which adjacent portions of the medium influence each other�
• Waves carry energy and momentum

Types of Waves – Traveling Waves

• Flip one end of a long rope that is under tension and fixed at the other end
• The pulse travels to the right with a definite speed
• A disturbance of this type is called �a traveling wave

Types of Waves – Transverse

• In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion

Types of Waves – Longitudinal

• In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave
• A longitudinal wave is also called a compression wave

Amplitude and Wavelength

• Amplitude A is the maximum displacement of string above the equilibrium position�
• Wavelength, λ, is the distance between two successive points that behave identically

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Waveform – A Picture of a Wave

• The brown curve is a “snapshot” of the wave at t = 0 instant in time
• The blue curve is that �at a later in time, t
• The high points are crests of the wave
• The low points are troughs of the wave

Wavelength λ = vT

Speed of a Wave

• v = ƒλ

Is derived from the basic speed equation of �Distance = Speed x Time� λ = v T = v/f T= 1/f

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• This is a general equation that can be applied to many types of waves

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Interference of Waves

• Two traveling waves can meet and pass through each other without being destroyed or even altered
• Waves obey the Superposition Principle
• When two or more traveling waves encounter each other while moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point

only true for waves with small amplitudes

Constructive Interference

• Two waves, a and b, have the same frequency and amplitude
• Are in phase�
• The combined wave, c, has the same frequency and a greater amplitude