MECHANICAL

VIBRATIONS

ME 421 12-Sep-14

1

Marks distribution 2

Quizes (10%-15%)

Announced & surprised

Assignments (5%)

2% for attendance (will be taken anytime after 15

mins)

Sessional Exams (30%)

Final Exam (45%-50%)

12-Sep-14

Books 3

Mechanical Vibrations by S. S. Rao 2nd/3rd/4th

Schaum's Outline of Mechanical Vibrations

12-Sep-14

Course Contents

Fundamentals of Vibrations

Free vibrations and Single DOF systems

Harmonically Excited vibrations

Vibration under general forcing conditions

Two DOF systems

Multi DOF systems

Misc.

12-Sep-14

4

Vibration

Any motion which repeats itself after an interval

of time is called vibration or oscillation

Example

Swing, Pendulum, Legs during walking etc

12-Sep-14

5

Vibration in a positive sense

Hearing, Breathing, Speaking

Music

Vibratory conveyer, Hoppers, sieves, washing

machines, electric tooth brush, Clock,

massagers etc

Vibration improves efficiency of some

machining, casting, forging etc.

Earthquakes simulation

Geological research

12-Sep-14

6

12-Sep-14 7

Vibration in negative Sense

Unbalance in engines

In Turbines, unbalance can cause mechanical

failure

Structures

Vibrations cause wear of mechanical parts and

produce noise

Fasteners become loose

Poor surface finish

Resonance!!

12-Sep-14

8

12-Sep-14 9

Vibrating system

A vibrating system generally includes

A means of storing potential energy

Spring or Elasticity

A means of storing kinetic energy

Mass or Inertia

A means by which the energy gradually lost

Dampers

EXAMPLE???

12-Sep-14

10

Vibration of a system

PE is converted into KE and KE is converted

into PE alternatively

Some of the energy is lost in each cycle if the

system is damped

The dissipated energy should be replaced by an

external source in order to maintain the steady

state of vibration

EXAMPLE??

12-Sep-14

11

Example of a Pendulum

Initial angular displacement =

KE = 0, PE = mgh @ 1. move due to gravitational torque

PE = 0 @ 2. Bob will not stop due to KE

KE = 0, PE = mgh @ 3

This process will continue

Oscillatory motion

In reality

Air will offer resistance, and bob will slow down gradually and will stop at the end

12-Sep-14

12

Degree of Freedom

The minimum number of

independent coordinates required

to determine completely the

positions of all parts of a system at

any instant of time defines the

number of degrees of freedom of

the system.

Angle:

Cartisian: x, y

X2+y2=l2 How Many DOF?

12-Sep-14

13

Degree of Freedom

12-Sep-14

14

Degree of Freedom

12-Sep-14

15

Degree of Freedom

12-Sep-14

16

Discrete and Continuous Systems

Discrete Systems

System having finite number of DOF

Continuous Systems

Systems having infinite number of DOF

Example: Cantilever Beam

Infinite no of mass points require infinite no of

coordinates to specify the deflection (Elastic curve)

12-Sep-14

17

Classification of Vibration

Free and Forced Vibration

Un-damped and Damped Vibration

Linear and Nonlinear Vibration

Deterministic and Random Vibration

12-Sep-14

18

Free and Forced Vibration

Free Vibration

Only initial disturbance is applied to the system and it

is left to vibrate on its own. No external force acts on

the system.

Example: oscillation of a simple pendulum

Forced Vibration

The system is subjected to an external force (often, a

repeating type of force).

If the frequency of the external force coincides with

one of the natural frequencies of the system, a

condition known as resonance occurs, and the

system undergoes dangerously large oscillations.

Failures of such structures as buildings, bridges,

turbines, and airplane wings have been associated

with the occurrence of resonance

12-Sep-14

19

Undamped and Damped Vibration

undamped vibration

If no energy is lost or dissipated in friction or other

resistance during oscillation, the vibration is known as

undamped vibration.

damped vibration

If any energy is lost, it is called damped vibration.

12-Sep-14

20

Linear and Nonlinear Vibration

Linear Vibration

If all the basic components of a vibratory system the spring, the mass, and the damper behave linearly, the resulting vibration is known as linear vibration.

Nonlinear Vibration

If any of the basic components behave nonlinearly, the vibration is called nonlinear vibration.

The differential equations that govern the behavior of linear and

nonlinear vibratory systems are linear and nonlinear,

respectively.

If the vibration is linear, the principle of superposition holds, and

the mathematical techniques of analysis are well developed. For

nonlinear vibration, the superposition principle is not valid, and

techniques of analysis are less well known.

12-Sep-14

21

Components of a vibratory system

12-Sep-14

22

Spring

Mass

Damper

Deterministic and Random

Vibration

Deterministic Vibration

If the value or magnitude of the excitation (force or motion) acting on a vibratory

system is known at any given time, the excitation is called deterministic.

The resulting vibration is known as deterministic vibration.

Random Vibration

In Random excitation, the value of the excitation at a given time cannot be

predicted.

In these cases, a large collection of records of the excitation may exhibit some

statistical regularity. It is possible to estimate averages such as the mean and

mean square values of the excitation.

Examples of random excitations are wind velocity, road roughness, and ground

motion during earthquakes.

12-Sep-14

23

https://sites.google.com/site/421vibrations

Web link 24

12-Sep-14

Vibration Analysis Procedure

The response of a vibrating system generally depends on the initial

conditions as well as the external excitations.

Most practical vibrating systems are very complex.

Often the overall behavior of the system can be determined by

considering even a simple model of the complex physical system.

The analysis of a vibrating system usually involves

mathematical modeling

derivation of the governing equations

solution of the equations

interpretation of the results

12-Sep-14

25

Step1: Mathematical Modeling

The purpose of mathematical modeling is to represent all the important features of

the system for the purpose of deriving the mathematical (or analytical) equations

governing the system behavior.

The mathematical model should include enough details to allow describing the

system in terms of equations without making it too complex.

The mathematical model may be linear or nonlinear, depending on the behavior of

the system s components.

Linear models permit quick solutions and are simple to handle; however, nonlinear

models sometimes reveal certain characteristics of the system that cannot be

predicted using linear models.

Sometimes the mathematical model is gradually improved to obtain more accurate

results. In this approach, first a very crude or elementary model is used to get a

quick insight into the overall behavior of the system.

Subsequently, the model is refined by including more components and/or details so

that the behavior of the system can be observed more closely.

12-Sep-14

26

Step1: Mathematical Modeling (cont.)

Forging Hammer

First approximation

Refined approximation

Further refinement ???

12-Sep-14

27

Step 2: Derivation of Governing Equations

Once the mathematical model is available, we use the principles of

dynamics and derive the equations that describe the vibration of the

system.

The equations of motion can be derived conveniently by drawing

the free-body diagrams of all the masses involved.

The free-body diagram of a mass can be obtained by isolating the

mass and indicating all externally applied forces, the reactive

forces, and the inertia forces.

Linear or non linear equations??

depending on the behavior of the components of the system.

Governing equations can be derived using Newton s second law of

motion, DAlembert s principle, and the principle of conservation of energy.

12-Sep-14

28

Step 3: Solution of the Governing

Equations.

The equations of motion must be solved to find the

response of the vibrating system.

Depending on the nature of the problem, the following

techniques can be used

Standard methods of solving differential equations

Laplace transform methods

Matrix methods

Numerical methods.

Nonlinear equations can be solved in closed form.

12-Sep-14

29

Step 4: Interpretation of the Results.

The solution of the governing equations gives the

displacements, velocities, and accelerations of the

various masses of the system.

These results must be interpreted with a clear view of

the purpose of the analysis and the possible design

implications of the results.

12-Sep-14

30

31

12-Sep-14

Develop a sequence of three mathematical models of the system for

investigating vibration in the vertical direction.

Consider the elasticity of the tires, elasticity and damping of the struts

(in the vertical direction), masses of the wheels, and elasticity,

damping, and mass of the rider

Example

Elements of Vibrating system

12-Sep-14

32

Spring Elements

Mass/Inertia Elements

Damping Elements

Spring/Elastic Elements

12-Sep-14

33

A spring is a type of mechanical link, which in most applications is assumed

to have negligible mass and damping.

The most common type of spring is the helical-coil spring used in

retractable pens and pencils, staplers, and suspensions of freight trucks

and other vehicles.

Any elastic or deformable body or member, such as a cable, bar, beam,

shaft or plate, can be considered as a spring

Restoring force is also developed

Potential Energy

Nonlinear Springs 34

12-Sep-14

Nonlinear Springs (More than one linear Springs)

12-Sep-14

35

Linearization of Nonlinear Spring

12-Sep-14

36

In many practical applications we assume that the deflections are

small and make use of the linear relation

Example 37

12-Sep-14

Spring Constant of a cantilever

beam

12-Sep-14

38

Spring Constant of a Rod

12-Sep-14

39

Combination of Springs 40

Springs in Parallel Springs In Series

12-Sep-14

Example (From Book)

12-Sep-14

41

A hoisting drum, carrying a steel wire rope, is mounted at the end of

a cantilever beam as shown in Figure.

Determine the equivalent spring constant of the system when the

suspended length of the wire rope is l. Assume that the net cross-

sectional diameter of the wire rope is d and the Young s modulus of

the beam and the wire rope is E.

42

12-Sep-14

Example (From Book)

12-Sep-14

43

44

12-Sep-14

END 45

12-Sep-14