Week 1:

Fundamental concepts in vibration

Fundamentals of Vibrations

• Definition of vibration

time

Motion

• Displacement• Velocity• Acceleration

Amplitude

period

= Frequency

Unit : second

(Hz)

1.4 Basic Concepts of Vibration

1Period

Fundamentals of Vibrations

Physical Interpretation of Vibration

• Force equilibrium

Dynamic force equilibrium among system elements

x

Equilibrium

Elastic force

Inertia force

Friction force

Fundamentals of Vibrations

Physical Interpretation of Vibration

• Force equilibrium

Dynamic force equilibrium among system elements

Elastic forceInertia force

Friction

Fundamentals of Vibrations

Physical Interpretation of Vibration

• Energy transform Energy transform among system elements

Elastic force

Inertia force

Friction force

Potential energy

Kinetic energy

Energy loss

Fundamentals of Vibrations

Physical Interpretation of Vibration

• Energy transform Energy transform among system elements

: Potential energy (max)UmaxUmax

Tmax: Kinetic energy (max)

Fundamentals of Vibrations

1.4.3 Degree of Freedom

x x2

DOF=1 DOF=2

• Definition A minimum number of variables required to describe the position

of a system

Fundamentals of Vibrations

Spring Stiffness

• Tension/Compression

L

E, A

F

d

EA

FL=d

=

L

EA

F

L

EAk

• Bending

L

E, I

F

d EI

FL

3

3

=d

=

3

3

L

EI

F3

3

L

EIk

• Torsion

Mtq

GJ

LMt=q

=

L

GJ

Mt

L

GJk

Fundamentals of Vibrations

Combination of Springs

• Springs in parallel

k1 k2

W

k1dst k2dstdst

keq

dst

W

st2st1 kkW dd=

k1>>k2

1eq kk

keqdst

Equivalentspring

Equaldeflection

steqkW d=

21eq kkk =st21 )kk( d= st21 )kk(W d=

Fundamentals of Vibrations

• Springs in serial

dst = dst1 + dst2

k1

k2

W

dst1

2st1stst dd=d

keq

dst

W

k2

k1

W

k2 dst2

k1 dst1

Equivalentspring

2st2kW d=

2st21st1 kk d=d

21 kk 2kkeq

Combination of Springs

steq

Wk

d=

21 k

W

k

W= 21 k

1

k

11

=21

21

kk

kk

=

Fundamentals of Vibrations

W

WW

Spring 1

Spring 2

L

EAk2 =

31L

EI3k =

Serial

1.7 Spring Elements

Example 1

Fundamentals of Vibrations

Serial

Parallel Stiffness increase

Stiffness decrease

1.7 Spring Elements

Example 2

Fundamentals of Vibrations

• Mass ..x

..mx

F

..x

Force vs. Acceleration

T

.x

Energy vs. Velocity

m..

xmF =

2.xm

21

T =

1.8 Mass Elements

Fundamentals of Vibrations

Equivalent Mass

22

2

1

2

1 .

G

.

JxmT q=

No slip Translation

Rotation

22

2

2

1

2

1

2

1 ..

mrxm q

=

No slip

2

22

2

1

2

1

2

1

=

..

r

xmrxm

2

2

3

2

1 .

xm

=

.q

G

.x

O

Equivalent

mass

EqualKinetic energy

Fundamentals of Vibrations

.x

.F(x)

• Damping

- Important in vibration

- Very difficult to understand.

1.9 Damping Elements

Fundamentals of Vibrations

Damping Types

• Viscous damping

.x

h

F (viscous force)

F

.x

Force vs. velocity

c xcF =

AF = Adydu

= A

hx

=

x

hA

c

Viscous fluid

Fundamentals of Vibrations

Damping Types

• Coulomb damping

F

(Coulomb force)

F

.x

mgF

• Magnitude: constant

• Direction: opposite to the velocity

Dry friction

• Viscous force (Wet friction)

.x

Fundamentals of Vibrations

• Hysteresis damping

Stress (s)

Strain (e)

F

e=s E

ee=s 'EE

Energyduring loading

Energy recoveredduring unloading

Energy dissipated

Material damping:

• Viscoelastic material Dissipated Energy >> Stored Energy

Rubber, Polymer

• Elastic material Dissipated Energy << Stored Energy

Metal

Damping Types