Lecture 02 Introduction

7/25/2019 Lecture 02 Introduction

1/17

e

c

V

b

a

o

e

c

V

b

a

o

Lecture 2:

Spring and Stiffness

Mass or Inertia Elements Damping

Introduction -1

With many figures and models fromMechanical Vibrations, S. S. Rao


2/17

e

c

V

b

a

o

e

c

V

b

a

o

Spring Elements

Combination of Springs:1) Springs inparallel if we have n spring

constants k1, k2, , kn inparallel, then the

equivalent spring constant keq is:

neq kkkk ...21

Introduction -2


3/17

e

c

V

b

a

o

e

c

V

b

a

o

Spring Elements

Combination of Springs:2) Springs in series if we have n spring

constants k1, k2, , kn in series, then the

equivalent spring constant keq is:

Introduction -3

neq kkkk

1...

111

21


4/17

e

c

V

b

a

o

e

c

V

b

a

o

It is the rigid element of the vibratory system that lose

or gain kinetic energy depending on its velocity

Using mathematical model to represent the actual

vibrating system

E.g. In figure below, the mass and damping of the beam can bedisregarded; the system can thus be modeled as a spring-mass

system as shown.

Mass or Inertia Elements

Mx(t)

x(t)

3

3

l

EIk

Actual system Single degree offreedom model Introduction -4


5/17

e

c

V

b

a

o

e

c

V

b

a

o


Combination of MassesE.g. Assume that the mass

of the frame is negligible

compared to the masses of

the floors. The masses ofvarious floor levels represent

the mass elements, and the

elasticities of the vertical

members denote the spring

elements.

Introduction -5


6/17

e

c

V

b

a

o

e

c

V

b

a

o

Masses Connected by a Rigid Bar

Velocities of masses can be expressed as:

1

1

3

31

1

2

2 x

l

lxx

l

lx


Introduction -6


7/17

e

c

V

b

a

o

e

c

V

b

a

o


Introduction -7

Masses Connected by a Rigid Bar

By equating the kinetic energy of the system:

1xx

eq

2

eqeq

2

33

2

22

2

11 2

1

2

1

2

1

2

1xmxmxmxm

3

2

1

32

2

1

21eq m

l

lm

l

lmm


8/17

e

c

V

b

a

o

e

c

V

b

a

o

Example P1.53

Find theequivalent mass

of this system?

Introduction -8


9/17

e

c

V

b

a

o

e

c

V

b

a

o

Example P1.53

When mass m isdisplaced by x, the

bell crank lever

rotates by the

angle:

This makes thecenter of the

sphere displace by

(xs):

11

tanl

x

l

xbb

2lx

bs

x

xs

Introduction -9


10/17

e

c

V

b

a

o

e

c

V

b

a

o

Example P1.53

The kinetic energy of the system can beexpressed as:

For a sphere:

Equating this to:

11

tanl

x

l

xbb

2

5

2sss

rmJ

2

1

2

2

1

222

2

1

2

2

1

5

2

2

1

2

1

2

1

l

lxm

rl

lxrm

l

xJxmT

s

s

sso

T =1

2meqx

2

meq =m+Jo1

l12+7

5ms

l2

2

l12

Introduction -10


11/17

e

c

V

b

a

o

e

c

V

b

a

o

Damping Elements

Damperis the element in vibratory system thatassumed to have neither mass nor elasticity and

responsible for energy loss in a typical

mechanical system.

Viscous Damping

Energy is dissipated bythe Resistance of a fluid

Friction Damping

Energy is dissipated bythe friction between

rubbing surfaces

Hysteretic Damping

Energy is stored anddissipated by the

material as deformation

takes place

Introduction -11


12/17

e

c

V

b

a

o

e

c

V

b

a

o

Viscous Damping

All real systems dissipate energy when they vibrate. To

account for this we must consider damping. The most simple

form of damping (from a mathematical point of view) is called

viscous damping. A viscous damper (or dashpot) produces a

force that is proportional to velocity.

Damper (c)

x

fc

Mostly a mathematically motivated form, allowinga solution to the resulting equations of motion that predicts

reasonable (observed) amounts of energy dissipation.

Introduction -12


13/17

e

c

V

b

a

o

e

c

V

b

a

o

Viscous Damping

fc is a resistance force and it is always in anopposite direction relative to the body motion.

Introduction -13

x

y

x> y

x

y

x< y

body body


14/17

e

c

V

b

a

o

e

c

V

b

a

o

Example 1.74

Find an expression for the equivalent translationaldamping constant of the system shown below so

that the force F can be expressed as F= ceqv,

where vis the velocity of the rigid bar A.

Introduction -14


15/17

e

c

V

b

a

o

e

c

V

b

a

o

Example 1.74

ceq1 ceq2

Thus the system can be replaced by the two equivalent dampers in parallel:

So that;

Introduction -15


16/17

e

c

V

b

a

o

e

c

V

b

a

o

Example 1.2

Suggest a simple mathematical model byconsidering the elasticity, mass, and damping

of the seat, human body, and restraints for a

vibration analysis of the system.

Introduction -16


17/17

e

c

V

b

a

o

e

c

V

b

a

o

Example 1.2

Introduction -17

Documents

Lecture 02 Introduction