MECN 4100

History of Vibration

Strings (Music)

Egyptians

Pythagoras: Monochord

Aristotle, Aristoxenus & Euclid

Vitruvius – acoustic properties of theater

Seismograph

Zhang Heng

History of Vibration

Laws of Vibrating String

Galileo – simple pendulum, resonance

Mersenne – “father of acoustics”

Hooke – relation between pitch and frequency

Sauveur – modes shapes and nodes, harmonics

Equation of Motions- Vibrating Body

Newton - Newton‟s Second Law

Taylor – Taylor‟s Theorem

History of Vibration

Principle of Superposition

Bernoulli

Thin Beam Theory

Euler-Bernoulli

Analytical Solution of Vibrating String

Lagrange

Torsional Oscillations

Coulomb

History of Vibration

Theory of Vibrating Plates

E.F.F. Chladni

Sophie Germain

Vibration of Flexible Membrane

Poisson

Clebsh

Thick Beam Theory

Timoshenko

History of Vibration

Thick Plates

Mindlin

Nonlinear

Poincare – pertubation

Lyapunov – stability

Random

Lin and Rice

Crandall and Mark

Finite Element Method

Importance of the Study of

Vibration

Importance of the Study of

Vibration

Importance of the Study of

Vibration

Importance of the Study of

Vibration – on the good side!

Basic Concepts

• Vibration

– Any motion that repeats itself after an interval of time

• Theory of Vibration

– Deals with the study of oscillatory motions of bodies and the forces associated with them

Basic Concepts

• Vibratory System

– Storing potential energy (spring)

– Storing kinetic energy (mass)

– Energy dissipation (damper)

Basic Concepts

• Degree of Freedom

– The minimum number of independent coordinates required to determine completely the position of all parts of a system at any instant of time defines the degree of freedom of the system

• Generalized Coordinates

– Coordinates necessary to describe the motion of a system

Basic Concepts

• Discrete (Lumped) System

– A system that can be describe using a finite number of degree of freedom

• Continuous (distributed)System

– A system that can be describe using a infinite number of degree of freedom

Classification

• Free Vibration

– A system which after an initial disturbance is left to vibrate on its own. No external force acts on the system

• Forced Vibration

– A system subjected to an external force resulting in a vibrating system

Classification

• Undamped

• Damped

• Linear

• Nonlinear

• Deterministic

• Random

Vibration Analysis Procedure

• A vibratory system is a dynamic system for which the variables such as the excitations (inputs) and response (outputs) are time-dependent. The response of a vibrating system generally depends on the initial as well as the external excitations.

• Predict the behavior under specified input conditions

• Consider a simple model of the complex physical model

Vibration Analysis Procedure

• Procedure

–Mathematical Modeling

–Derivation of the governing

equations

–Solution of the equations

–Interpretation of the results

Vibration Analysis Procedure

• Mathematical Modeling– Represent all important features for the

purpose of deriving the mathematical equations governing the system behavior

– Simple as possible

– Linear or Nonlinear

– Great deal of “engineering judgment”

– Sequential: First a crude or elementary model and then a refined model including more components and/or details

Vibration Analysis Procedure

• Derivation of Governing Equations

– Use principle of dynamics and derive the

descriptive equations of a vibration system

– The equation of motion is usually in the form of a

set of ordinary differential equations for a discrete

system and partial differential equations for a

continuous system

– Linear or Nonlinear

– Approaches: Newton‟s second law, D‟Alambert‟s

principle, and principle of conservation of energy

Vibration Analysis Procedure

• Solution of the governing equations– Standard methods of solving DFQs

• Ordinary

• Partial

– Laplace transform methods

– Matrix methods

– Numerical methods

Vibration Analysis Procedure

• Interpretation of the Results

– Displacements

– Velocities

– Accelerations

Vibration Analysis Procedure

• Example– The following figure shows a motorcycle with

a rider. 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 strut, masses of the

wheels, and elasticity , damping, and mass of

the rider

Vibration Analysis Procedure

Vibration Analysis Procedure

Vibration Analysis Procedure

Vibration Analysis Procedure

• Example– A reciprocating engine is mounted on a

foundation as shown in the following figure. The

unbalanced forces and moments developed in

the engine are transmitted to the frame and the

foundation. An elastic pad is placed between the

engine and the foundation block to reduce the

transmission of vibration. Develop two

mathematical models of the system using gradual

refinement of the modeling process.

Vibration Analysis Procedure

Vibration Analysis Procedure

Elements

• Springs

• Mass/Inertial

• Damping

Spring Elements

• A linear spring is a type of mechanical link with

negligible mass and damping

• The spring force is proportional to the amount of

deformation

• The work done in deforming a spring is stored

as strain or potential energy in the spring

kxF

2

21 kxU

Spring Elements

• Actual spring are nonlinear

kxF

Spring Elements

• Linearization process

2

2

2*

** 2

1x

dx

Fdx

dx

dFxFFF

xx

xkF

Spring Elements

• Elastic elements like beams also behave like

spring

EI

Wlst

3

3

3

3

l

EIWk

st

Spring Elements

• Combination of Springs

– Spring in Parallel

stst kkW 21

steqkW

neq kkkk 21

Spring Elements

• Combination of Springs

– Spring in Series

21st

11kW

neq kkkk

1111

21

22kW

steqkW

eqeqkkk 2211

1

1k

k eqeq

2

2k

k eqeq

st

steqsteq

k

k

k

k

21

Spring Elements

• Example– The figure shown the suspension system of a

freight truck with a parallel-spring arrangement.

Find the equivalent spring constant of the

suspension if each of the three helical springs is

made of steel with a shear modulus G = 80 x 109

N/m2 and has five effective turns, mean coil

diameter D = 20 cm, and wire d = 2 cm

nD

Gdk

3

4

8mN /000,40

52.08

10802.03

94

mNkkeq /000,1203

Spring Elements

• Example– Determine the torsional spring constant of the

steel propeller shaft

Spring Elements

12

12

12 l

GJkt

12

4

12

4

12

32l

dDG

232

2.03.01080 449

23

23

23 l

GJkt

23

4

23

4

23

32l

dDG

332

15.025.01080 449

radmN /105255.25 6

radmN /109012.8 6

2312

2312

tt

tt

tkk

kkk

eq

radmN /105997.6 6

Spring Elements

• Find the equivalent

spring constant of the

system shown in the

figure. Assume that

k1, k2, k3 and k4 are

torsional and k5 ,k6

are linear spring

constant.

Spring Elements

;1111

321123 kkkk

Series of spring,

133221

321123

kkkkkk

kkkk

Using energy equivalence,

2

6212

5212

123212

4212

21 RkRkkkkeq

2

6

2

51234 RkRkkkkeq

65

2

133221

3214 kkR

kkkkkk

kkkkkeq

Spring Elements

• Consider two helical springs with the

following characteristic:

– Determine the equivalent spring constant

when (a) spring 2 is placed inside 1 (b) spring

2 is placed on top of 1

material # turns Mean

coil dia.

Wire dia. Free

length

Shear

modulus

Spring 1 steel 10 12 in. 2 in. 15 in. 12 x 106

psi

Spring 2 Aluminum 10 10 in. 1 in. 15 in. 4 x 106

psi

Spring Elements

3

4

64nR

GdkFor helical spring,

inlbk /89.388,161064

210123

25

1

inlbkkka eq /89.438,1: 21

inlbk /00.5051064

11043

25

2

inlbkkk

beq

/26.48111

:21

Mass or Inertia Elements

Assumed to be a rigid

body

Gain or lose kinetic

energy whenever the

velocity changes

The work done on the

mass is stored for in the

form of mass „s kinetic

energy

Mass or Inertia Elements

Combination of masses

Translational Masses Connected by a Rigid

Bar

1

1

22 x

l

lx

1

1

33 x

l

lx

Mass or Inertia Elements

Translational Masses Connected by a Rigid

Bar

Equating the KE of the three-mass system to that of the

equivalent mass system

1xxeq

2

212

33212

22212

1121

eqeqxmxmxmxm

3

2

1

32

2

1

21 m

l

lm

l

lmmeq

Mass or Inertia Elements

Combination of masses

Translational Masses and Rotational Masses

Coupled Together

○ Equivalent translational mass 2

212

21

oJxmT

xxeq

eqeqeq xmT 21

Rx /2

212

212

21

R

xJxmxm oeq

R

Jmm o

eq

Mass Elements

Example

A cam-follower mechanism is used to convert

the rotary motion of a shaft into the oscillating

or reciprocating motion valve. The follower

system consist of a pushrod of mass mp, a

rocker arm of mass mr, and mass moment of

inertia Jr , a valve of mass mv , and a valve

spring of negligible mass. Find the equivalent

mass of this cam-follower system by assuming

the location as (i) point A and (ii) point C

1/ lxr

122 // lxllx rv

133 / lxllx rr

Mass Elements

2

212

212

212

21

rrrrvvpp xmJxmxmT

eqeqeq xmT 21

1

2

l

lxxv

xxp

1l

xr

1

3

l

lxxr

2

1

2

3

2

1

2

2

2

1 l

lm

l

lm

l

Jmm rv

rpeq

Mass Elements

• In the figure find the equivalent mass of the rocker arm assembly with respect to the x coordinate

Mass Elements

• In the figure find the equivalent mass of the rocker arm assembly with respect to the x coordinate

2

0212

2212

11212

21 Jxmxmxmeq

b

x

b

xax1

2

02

2

1

1

bJm

b

ammeq

Damping Elements

The mechanism by which the

vibrational energy is gradually

converted into heat or sound

A damper is assume to have

neither mass nor elasticity, and

exist only if there is relative

velocity between the two ends of

the damper

Damping Elements

Types;

Viscous

○ Resistance offered by the fluid to the moving

body that causes the energy to be dissipated

○ Factors affecting the damping;

Size

Shape

Viscosity

Frequency

Velocity

Damping Elements

○ The damping force is proportional to the velocity

of the vibrating body

○ Typical example;

Fluid film between sliding surfaces

Fluid around a piston in a cylinder

Fluid flow through an orifice

Fluid film around a journal in a bearing

Coulomb/Dry Friction

○ Constant in magnitude but opposite in direction

of motion to the vibrating body

○ Caused by friction between rubbing surfaces

Damping Elements

Material or Solid or Hysteretic

○ Energy is dissipated by the material when

deform. This is due to friction between the

internal planes, which slip or slide

Damping Elements

Construction

dy

du

hdy

du

ch

AAF

h

Ac

Example 1

Develop an expression for the damping

constant of the dashpot shown

Example 1

Using the shear stress and rate of fluid flow,

dydy

dDlDldF

dy

dv

2

2

dy

vdDldyF

2

4

D

Pp

2

4

D

PDdyp

The pressure,

The pressure force,

Example 1

Integrating this equation twice and using the boundary condition v=0 at y=0 and v=0 at y=d

2

24

dy

vdDldydy

P

lD

P

dy

vd22

2 4

d

yvyyd

lD

Pv 1)(

20

2

2

4

002

3

2

1

6

2dv

lD

PdDDdyvQ

The rate of flow,

Example 1

The volume of the liquid flowing through

03

3

4

213

vd

D

dlD

P

D

d

d

lDc

21

4

33

3

2

4

0 DQ

v

Substituting,

Writing P=cv,

Example 2

The force (F) –velocity (x) relationship of

a nonlinear damper is given by

where a and b are constant. Find the

equivalent linear linear damping constant

when the relative velocity is 5 m/s with

a=5 N s/m and b=0.2 N s2/m2

2xbxaF

Example 222 2.05 xxxbxaF

00

0

)( xxxd

dFxFxF

x

,/50 smxat

30252.055)( 0xF

74.055

0

xxd

dF

x

57)5(730)( xxxF

xcxxF eq 7)( msNceq /7

Example 3

The damping constant (c) due to skin friction drag of a rectangular plate moving in a fluid of a viscosity μ is given by

Design a plate-like damper that provide an identical damping constant for the same fluid

dlc 2100

Example 3

dlc 2100

h

Ac

h

lddl 2100

lc

100

1

Harmonic Motion

tAAx sinsin

tAdt

dxcos

xtAdt

xdsin

2

2

Harmonic

Vectorial

Representation

tAy sin

tAx cos

OPvector

Amagnitude

Harmonic Motion

Complex

Number

2/122 baA

a

b1tan

ibaX

sincos iAAX

iAeiAX sincos

Harmonic

Using complex

number

tiAeX

XiAeiAedt

d

dt

Xd titi

XAeAeidt

d

dt

Xd titi

22

2

2

Harmonic Motion

The displacement, velocity, and

acceleration;

tAAe ti cos]Re[ntdisplaceme

tAAei ti sin]Re[velocity

tAAe ti cos]Re[onaccelerati 22

90cos tA

180cos tA

Harmonic Motion

magnitude,2

2

2

21 sincos AAAA

cos

sintan

21

21

AA

Athe angle,

Check -Example 1.11

Harmonic Motion

2

2

1f

Definition and terminology

Cycle

Amplitude

Period of oscillation

Frequency of oscillation

Harmonic Motion

tAx sin11

Definition and terminology

Phase angle tAx sin22

Harmonic Motion

tt

Xtx2

cos2

cos2

Definition and terminology

Natural frequency

Beats

Octave

Harmonic Motion

0

2

0

log20log10X

X

X

XdB

Definition and terminology

Decibel

Harmonic Analysis

In many cases the vibrations of a system are

periodic. Any periodic function of time can be

represented by Fourier series as an infinite

sum of sine and cosine terms

Harmonic Analysis

)sincos(2

2sinsin

2coscos2

)(

1

0

21

210

tnbtnaa

tbtb

tataa

tx

n

n

n

0

/2

00

2txdttxa

tdtntxdttntxan cos2

cos0

/2

0

tdtntxdttntxbn sin2

sin0

/2

0

Harmonic Analysis

1

000

1

0

2222

222)(

n

nntinnntinti

n

tintin

n

tintin

n

i

ibae

ibae

ibae

i

eeb

eea

atx

tin

n

nectx

dtetxc tin

n

0

1

Complex Fourier Series

Harmonic Analysis

Frequency Spectrum

Harmonic Analysis Time and Frequency Domain Representation

Check: Example 1.12/13