single degree of freedom systems

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http://WikiCourses.WikiSpaces.com Single Degree of Freedom Systems

Mohammad Tawfik

Introduction to Vibrations of

Structures

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Mohammad Tawfik

References

• M. Bismarck-Nasr, "Structural Dynamics in Aeronautical

Engineering," AIAA Educational Series, 1999

• D. Inman and E. Austin, “Engineering Vibration,” 2nd edition,

Prentice Hall, 2001

• A. A. Shabana, "Vibration of Discrete and Continuous Systems," 2nd

edition, Springer, 1997

• D. Thorby, “Structural Dynamics and Vibration in Practice” Elsevier,

2008

• A. G. Ambekar, “Mechanical Vibrations and Noise Engineering”

Prentice Hall – India, 2006

• Leonard Meirovitch, “Fundamentals of Vibrations,” 1st edition,

McGraw Hill, 2001

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Mohammad Tawfik

Single degree of freedom

systems

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Mohammad Tawfik

Objectives

• Recognize a SDOF system

• Be able to solve the free vibration equation of a SDOF system with and without damping

• Understand the effect of damping on the system vibration

• Apply numerical tools to obtain the time response of a SDOF system

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Mohammad Tawfik

Single degree of freedom systems

• When one variable can describe the

motion of a structure or a system of

bodies, then we may call the system a 1-D

system or a single degree of freedom

(SDOF) system. e.g. x(t), q(t) Z(t), y(x).

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Mohammad Tawfik

Stiffness

• From strength of materials recall:

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Mohammad Tawfik

Newton’s Law

• Newton’s Law:

00 )0(,)0(

0)()(

)()(

vxxx

tkxtxm

tkxtxm

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Mohammad Tawfik

Solving the ODE

• The ODE is

• The proposed

solution:

• Into the ODE you get

the characteristic

equation:

• Giving:

0)()( tkxtxm taetx )(

02 tt aem

kae

m

k2

m

kj

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Mohammad Tawfik

Solving the ODE (cont’d)

• The proposed

solution becomes:

• For simplicity, let’s

define:

• Giving:

tm

kjt

m

kj

eaeatx

21)(

m

k

tjtj eaeatx 21)(

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Mohammad Tawfik

Let’s manipulate the solution!

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Mohammad Tawfik

Recall

ajSinaCose ja

bSinaCosbCosaSinbaSin

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Mohammad Tawfik

Manipulating the solution

• The solution we have:

• Rewriting:

tjtj eaeatx 21)(

tjSintCosa

tjSintCosatx

2

1)(

tSinaajtCosaatx 2121)(

tSinAtCosAtx 21)(

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Mohammad Tawfik

Further manipulation

tSinAtCosAtx 21)(

2

2

2

1 AAA

A

ASin

A

ACos 12 &

tSinCostCosSinAtx )(

tASintx )(

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Mohammad Tawfik

Different forms of the solution

tjtjeaeatx

tCosAtSinAtx

tASintx

21

21

)(

)(

)()(

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NOTE!

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Mohammad Tawfik

Natural Frequency of Oscillation

• In the previously obtained solution:

• The frequency of oscillation is

• It depends only on the characteristics of the oscillating system. That is why it is called the natural frequency of oscillation

tASintx )(

m

kn

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Mohammad Tawfik

Frequency

period theis s 2

Hz2s 2

cycles

rad/cycle 2

rad/s

frequency natural thecalled is rad/sin is

n

nnnn

n

T

f

We often speak of frequency in Hertz or

RPM, but we need rad/s in the arguments

of the trigonometric functions.

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Mohammad Tawfik

Recall: Initial Conditions

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Mohammad Tawfik

Amplitude & Phase from the ICs

Phase

0

01

Amplitude

2

02

0

0

0

tan ,

yields Solving

cos)0cos(

sin)0sin(

v

xvxA

AAv

AAx

n

n

nnn

n

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Mohammad Tawfik

Some useful quantities

peak value A

T

Tdttx

Tx

0

valueaverage = )(1

lim

valuesquaremean root = 2xxrms

valuesquare-mean = )(1

lim0

22

T

Tdttx

Tx

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Mohammad Tawfik

Peak Values

Ax

Ax

Ax

2

max

max

max

:onaccelerati

:velocity

:ntdisplaceme

Maximum or peak (amplitude) values:

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Mohammad Tawfik

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Mohammad Tawfik

Samples of Vibrating Systems

• Deflection of continuum (beams, plates,

bars, etc) such as airplane wings, truck

chassis, disc drives, circuit boards…

• Shaft rotation

• Rolling ships

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Mohammad Tawfik

Wing Vibration

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Ship Vibration

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Effective Stiffness of

Structures

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Bars

• Longitudinal motion

• A is the cross sectional

area (m2)

• E is the elastic modulus

(Pa=N/m2)

• l is the length (m)

• k is the stiffness (N/m) x(t)

m

EAk

l

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Mohammad Tawfik

Rods

• Jp is the polar

moment of inertia of

the rod

• J is the mass

moment of inertia of

the disk

• G is the shear modulus, l is the

length

Jp

J qt)

0

pGJ

k

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Mohammad Tawfik

Helical Spring

2R

x(t)

d = diameter of wire

2R= diameter of turns

n = number of turns

x(t)= end deflection

G= shear modulus of

spring material

3

4

64nR

Gdk

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Mohammad Tawfik

Beams

f

m

x

• Strength of materials

and experiments

yield:

3

3

3

3

m

EI

EIk

n

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Mohammad Tawfik

Equivalent Stiffness

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Summary

• Write down the equation of motion using Newton’s law

• Solve the equation of motion for a SDOF

• Use initial conditions to determine the amplitude and phase of vibration for a SDOF

• Evaluate the effective stiffness of structural members

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Mohammad Tawfik

1. The amplitude of vibration of an undamped system is measured to be 1 mm. the phase shift is measured to be 2 rad and the frequency 5 rad/sec. Calculate the initial conditions.

2. Using the equation: evaluate the constant A1 and A2 in terms of the initial conditions

HW #1

tSinAtCosAtx 21)(

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Mohammad Tawfik

HW #1 (cont’d)

3. An automobile is modeled as 1000 kg

mass supported by a stiffness k=400000

N/m. When it oscillates, the maximum

deflection is 10 cm. when loaded with the

passengers, the mass becomes 1300 kg.

calculate the change in the frequency,

velocity amplitude, and acceleration if the

maximum deflection remain 10 cm.

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Mohammad Tawfik

Adding Damping

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Mohammad Tawfik

Objectives

• Understand the damping as a force

resisting motion

• Adding viscous damping to the equation of

motion of a SDOF

• Understand the difference in the

responses of different systems depending

on the value of the damping

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Mohammad Tawfik

Damping

• Damping is some form of friction!

• In solids, friction between molecules result in damping

• In fluids, viscosity is the form of damping that is most observed

• In this course, we will use the viscous damping model; i.e. damping proportional to velocity

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Mohammad Tawfik

Viscous Damping

• A mathematical form

called a dashpot or

viscous damper

somewhat like a shock

absorber the constant c

has units: Ns/m or kg/s )(txcfc

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Shock Absorbers

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Mohammad Tawfik

Spring-mass-damper systems

• From Newton’s law:

00 )0( ,)0(

0)()()(

)()()(

vxxx

tkxtxctxm

tkxtxctxm

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Mohammad Tawfik

Solution (dates to 1743 by Euler)

0)()(2)( 2 txtxtx nn

km

c

2=

Where the damping Ratio

is given by: (dimensionless)

Divide the equation of motion by m

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Mohammad Tawfik

roots theof nature the

determines ,1nt discrimina theHere

equation quadratic a of roots thefrom

1

:in equation algebraican now iswhich

02

motion of eq. into subsitute & )(Let

2

2

2,1

22

nn

t

n

t

n

t

t

aeeaea

aetx

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Mohammad Tawfik

Three possibilities:

00201

21

,

:conditions initial theUsing

)(

221=

damped critically called

repeated & equal are roots1 )1

xvaxa

teaeatx

mkmcc

n

tt

ncr

nn

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Mohammad Tawfik

Critical damping cont’d

• No oscillation occurs

t

nnetxvxtx

])([)( 000

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Mohammad Tawfik

12

)1(

12

)1( where

)()(

1

:roots realdistinct two-damping-over called ,1 )2

2

0

2

02

2

0

2

0

1

1

2

1

1

2

2,1

22

n

n

n

n

ttt

nn

xva

xva

eaeaetx nnn

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Mohammad Tawfik

The over-damped response

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Mohammad Tawfik

Most interesting Case!

2

2,1 1

:as formcomplex in roots write

pairs conjugate as rootscomplex Two

commonmost -motion dunderdampe called ,1 )3

jnn

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Mohammad Tawfik

Under-damping

00

01

2

0

2

00

2

1

2

1

1

tan

)()(1

frequency natural damped ,1

)sin(

)()(22

xv

x

xxvA

tAe

eaeaetx

n

d

dn

d

nd

d

t

tjtjt

n

nnn

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Mohammad Tawfik

Under-damped-oscillation • Gives an oscillating response with exponential decay

• Most natural systems vibrate with an under-damped

response

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Mohammad Tawfik

Summary

• Modeling viscous damping

• Solving the equation of motion involving

viscous damping

• Recognizing the different types of

response based on the level of damping

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Mohammad Tawfik

1. Use the given data to plot the response of the

SDOF system

2. Solve the equation

And plot the response

HW #1 (cont’d)

8.0,6.0,4.0,2.0,1.0,01.0

/0,1sec,/2 00

smvmmxradn

0,1

0

00

vx

xxx

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Mohammad Tawfik

HW #1 (cont’d)

• Homework is due next week:

26/9/2010

single degree of freedom systems

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