Mechanical Vibrations

v

i

b

r

a

t

i

o

n

s

.

Brief History

Vibration study started from about 6000 years ago when the 1st musicalinstrument was invented.

v

i

b

r

a

t

i

o

n

s

.

Importance of the study of vibration

Vibration is founded all around us In past, vibration was studied to understand the physical phenomena and derive

a mathematical model to describe it . In recent times, the motivation of studying vibration is the engineering

applications.

v

i

b

r

a

t

i

o

n

s

. Most machines have inherent unbalance

Vibration in engineering :

1. In machines

2. Through structure

Vibration causes fatigue stresses in structures and wear in rotating parts of machine.

Importance of the study of vibration

Resonance is one of the most devastating effects of vibration on machines and structures.

Resonance happen when the natural frequency of the system equals the excitation frequency of the external excitation.

v

i

b

r

a

t

i

o

n

s

.

Tacoma Narrows bridge failure due to wind excitation (July 1,1940 – November

7,1940)

Importance of the study of vibrationv

i

b

r

a

t

i

o

n

s

.

Advantages

Health therapy

Domestic applicatio

ns

Industrial applicatio

ns

Music Disadvantages

Machine failure

Noise Structure failure

Vibration system elementary parts v

i

b

r

a

t

i

o

n

s

.

Potential energy

Kinetic energy

Energy dissipation

Elastic element

Inertial element

Damping element

Examples

Rotational vibration

v

i

b

r

a

t

i

o

n

s

.

Other examples v

i

b

r

a

t

i

o

n

s

.

Translational vibration

Toriosnalvibration

Degree of Freedom (DoF)

Definition

The minimum number of independent coordinates required to

determine completely the positions of all parts of the system at

any instant.

v

i

b

r

a

t

i

o

n

s

.

Degree of Freedom (DoF)

Examples (single DoF)

v

i

b

r

a

t

i

o

n

s

.

Degree of Freedom (DoF)

Examples (Two DoF)

v

i

b

r

a

t

i

o

n

s

.

Degree of Freedom (DoF)

Examples (Three DoF)

v

i

b

r

a

t

i

o

n

s

.

Degree of Freedom (DoF)

Examples (infinite DoF)

v

i

b

r

a

t

i

o

n

s

.

Discrete and continues systems v

i

b

r

a

t

i

o

n

s

.

Vibrating system

Discrete Continues

Finite DoF Infinite DoF

Vibration classifications v

i

b

r

a

t

i

o

n

s

.

Vibration

Excitation

Free

Applying initial

excitation only and leave the system to

vibrate

Forced

Continues repeating

force acting on

the system

Damping

Damped

Energy dissipation

after a period of

time

Un-damped

No energy dissipation

and system

will vibrate for ever

Linearity

Linear

All the basic

components (spring, mass and damper ) all have linear

behaviors

None-linear

If any of the

vibration system

elements act

nonlinearly

Determent

Deterministic

If the magnitude

of excitation is known

at any time

Random

If the magnitude

of excitation can not be predicted at a given

time

Deterministic and random vibrationsv

i

b

r

a

t

i

o

n

s

.

Vibration analysis procedures v

i

b

r

a

t

i

o

n

s

.

Mathematical modeling

Derivation of governing equations

Solution of the governing equations

Interpretation of the results

Vibration system elements v

i

b

r

a

t

i

o

n

s

.

Vibration system

elements

Spring Inertia Damper

Has stiffness (k)

Neglected mass and

damping effect.

Has mass (M,I)

Neglected stiffness and

damping effect.

Has damping effect (C)

Neglected mass and stiffness

Vibration system elements : spring

LoLo

L

F

δ

Applying Load

deflection (δ) = L – Lo

F = K δ where K is thespring stiffness

increasing K makes thespring stiffer.

Stiffer springs needs moreforce to deflect it

Potential energy (U):U = 0.5 k.x2

K=(d4G/8D3n)

S

t

a

t

i

c

s

.

Wire diameter (d)Shear modules (G)

Mean coil diameter (D)Number of turns (n)

Spring special case: rod S

t

a

t

i

c

s

.

l

AEk

Modules of elasticity

Cross sectional area

Spring special case: cantilever beam S

t

a

t

i

c

s

.

Moment of inertia

Modules of elasticity

Cross sectional area

Spring special case: torsional stiffness of shafts S

t

a

t

i

c

s

.

1

4

12

4

12

1

12

3212 l

dDG

l

GJkt

2

4

23

4

23

2

23

3223 l

dDG

l

GJkt

Spring special case: parallel connection S

t

a

t

i

c

s

.keq = k1 + k2

Spring special case: series connection S

t

a

t

i

c

s

.

21

111

kkkeq

ExampleS

t

a

t

i

c

s

.

ExampleS

t

a

t

i

c

s

.

Vibration system elements : mass or inertia S

t

a

t

i

c

s

.

In translational motion systems, we use the mass, M (kg) .

In rotational and torsional vibration systems, we use the mass moment of inertia, I (kg.m2).

Newton’s law of motion:

Translational system:∑forces = Mass * acceleration

Rotational system:∑Moment = mass moment of

inertia *angular acceleration

Case1: translational masses connected to rigid barS

t

a

t

i

c

s

.

3

2

1

32

2

1

21 m

l

lm

l

lmmeq

Case2: translational and rotational masses coupled together S

t

a

t

i

c

s

.

Equivalent translational mass

2R

Jmm o

eq

Equivalent rotational mass

2mRJJ oeq

2.2.

2

1

2

1oJxmT

Damping effect Viscous damping

Dry friction Material or solid or hysteretic damping

S

t

a

t

i

c

s

.

Energy dissipation due to motion of mechanical parts in fluids

Amount of dissipated energy depends on:Size and shapes of vibrating bodies Fluid viscosity Vibration frequency Vibrating body velocity

The damping force is proportional to the velocity of the vibrating body

Harmonic motion S

t

a

t

i

c

s

.

Periodic motion is the motion

that repeats it self after a period

of time

Harmonic motion is the

simplest type of periodic motion

x = Asin(θ) = Asin(ωt)

Velocity:

Acceleration:

tAdt

dx cos

xtAdt

xd 22

2

2

sin

Scotch mechanism

S

t

a

t

i

c

s

.

Vector notation

y=A sin(ωt)

Complex notation

iAeiAbiaX ))sin()(cos(

P

End of chapter

S

t

a

t

i

c

s

.