VIBRATIONS – Basic Definitions

Dr. S. K. PrasadProfessor of Civil EngineeringS. J. College of EngineeringMysuru 570006skprasad@sjce.ac.inPh: +91-94496-21994

What is vibration?

• Vibrations are oscillations of a system about an equilbrium position.

VibrationIt is also an everyday phenomenon we meet on everyday life

Useful effect of Vibration

Harmful effect of vibration

Noise

Destruction

Compressor

Ultrasonic cleaning

Testing

Wear

Fatigue

Vibration

Vibration in our Lives

• Our heart beats

• Our lungs oscillate

• We hear because our ear drums vibrate

• Vibration makes us snore

• Light waves permit us to see

• Sound waves allow us to hear

• We move because of oscillation of legs

• We can not utter ‘vibration’ without the oscillation of larynges and vocal cords

• We limit our discussion to Mechanical Vibration

• Vibration of dynamic system of a structure

• It is the oscillations of a system that has mass and elasticity

Vibration in our Lives

Vibration – Friend or Foe

Vibration in

Machinery

Vibration in

Recreation

Vibration in

Defense

Vibration in

Transportation

Vibration in

Aerospace

Vibration in

Automobile

Vibration in

Health Care

Vibration in

Structures

Vibration during

Disasters

TYPES OF FORCES

Static or Monotonic

Dynamic

Periodic

Harmonic

Steady State

Transient

Non-Harmonic

Impulse Type

Random

Basic Definitions

Periodic Motion: A motion that repeats itselfafter equal interval of time.

Time Period: Time taken for one completecycle.

Simple Harmonic Motion: Motion of particlewith time that moves round a circle withuniform angular velocity. Trigonometricfunctions can be used to represent suchmotion.

Basic Definitions

Amplitude (Z or 2Z): The maximum displacement of avibrating body from its mean position. The amplitude caneither be single amplitude (Z) when the distance from meanposition to maximum displacement is measured or doubleamplitude (2Z) when the distance from negative maximum topositive maximum displacement (motion) is measured.

Frequency: It is the number of cycles per unit time.Frequency and time period are inversely proportional to eachother. A vibratory motion can have either a very highfrequency or a very low frequency. Frequency can beexpressed either as angular (circular) frequency (ω) oroscillatory frequency (f). ω is expressed in radians per secondand f is expressed in cycles per second or Hertz.

Basic Definitions

Free Vibration: Vibration of a systembecause of its own elastic property. Noexternal force is required for this vibration andonly initiation of vibration may be necessary.

Forced Vibration: A system that vibratesunder an external force at the samefrequency as that of external force.

Basic Definitions

Natural frequency: It is the frequency of freevibration of a system. It is constant for a system. Infact, it is an inherent property of a system. Itdepends on the elastic properties, mass andstiffness of the system.

Resonance: Vibration of a system when thefrequency of external force is equal to the naturalfrequency of the system. The amplitude of vibrationat resonance becomes excessive. Duringresonance, with minimum input, there will be amaximum output. Hence both displacement and thestresses in the vibrating body become very high.

Basic Definitions

Damping: It is the resistance to motion. It is also thesluggishness. Hence it is the delay in response to anyaction. Damping is observed only under fast loading, andnot during static loading.

Degree of freedom: The number of independentcoordinate systems required to specify a motion. If themotion is in one direction due to the vibration of a singlespring, then it is a Single degree of freedom system. If aparticle is likely to vibrate in space, it will have sixdegrees of freedom, namely three translations and threerotations along three axis. A continuum can have infinitedegrees of freedom.

Basic Definitions

Phase difference : The angle between two rotatingvectors representing Simple Harmonic Motion, In timedomain, it can be represented as the delay in one motioncompared with the other.

Wave : It is the vibratory motion of a body or a particlerepresented in time domain or space domain. Forrepresenting a one dimensional wave mathematically,the partial differential equation is given by,

2

22

2

2

x

uv

t

u

Basic Definitions

fT

12

vT

f

v

k

2

T = Time period (in sec)ω = Angular or Circular velocity (in rad/sec)f = Frequency of oscillations (in cycles/sec or Hz)λ = Wave length (in m)k = Wave number = ω/v (in rad/m)v = Wave velocity (in m/sec)

Basic Definitions

TYPES OF LOADING

CYCLIC OR REPETITIVELOADING

SLOWLOADING

RAPID OR TRANSIENT LOADING

STATIC LOADING

TIME

FO

RC

E

TYPES OF LOADING

CYCLIC OR REPETITIVELOADING

SLOW STATIC LOADING

RAPID OR TRANSIENT LOADING

STATIC LOADING

TIME

FO

RC

E

OSCILLATORY LOAD

Large Period Small Period

LOAD

Single Impulse Multiple Impulse

WHAT IS DYNAMIC FORCE ?

Time Time

Time Time

Actual Impulse

Typical Seismogram

• Random• Time Dependent• Cyclic

Start of PrimaryWaves

Start of SecondaryWaves

Start of Surface Waves

TraceAmplitude

Strong Motion

Time

SA

Acceleration

• PGA• Predominant Frequency• Duration of Strong Motion

No two earthquake motions are similar

Free vibration

Equilibrium pos.

• When a system is initially disturbed by a displacement, velocity or acceleration, the system begins to vibrate with a constant amplitude and frequency depending on its stiffness and mass.

• This frequency is called as natural frequency, and the form of the vibration is called as mode shapes

Forced VibrationIf an external force is applied to a system, the system will follow the force with the same frequency.

However, when the forcingfrequency is increased to the system’s natural frequency, amplitudes will dangerously increase in this region. This phenomenon called as“Resonance”

’

Vibration parameters

All mechanical systems can be modeled by containing three basic components:

spring, damper, mass

When these components are subjected to constant force, they react with a constant

displacement, velocity and acceleration

Newton’s Laws of Motion&

Earthquake Engineering

Newton’s First Law of Motion

Lesson: Wear your Seat BeltsLaw of Inertia

Every object continues to remain in its initial status unless acted upon by external force.

Newton’s Second Law of Motion

Lesson: Do not disturb Bad persons

Everyone unconsciously knows the second law that heavier objects require more force to move the same distance as lighter objects

F = m.a

Newton’s Third Law of Motion

Rockets Action: Push down on ground with powerful engine.

Reaction: Ground pushes the rocket upwards with equal force.

Lesson: If you hit some body, expect the same reaction.

For every action there is an equal and opposite reaction

∑FA = 0Statics

Dynamics∑FA - FI = 0

Dynamics is dangerous & action packed. But interesting

FI = m.a

Inertia ???

Effects of Earthquake

ACCELERATIONACCELERATION

DECELERATIONDECELERATION

Inertia Force F = m a

Building at RestBuilding at Rest Ground Accelerates to LeftGround Accelerates to Left

Ground Accelerates to RightGround Accelerates to Right Ground & Building at RestGround & Building at Rest

Period of Vibration

DAMPING AND RESONANCE

Effect of Damping

Effect of Resonance

Spring in vibration

F = C.V

c

Damper - Dashpot

Vibration System