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11

Dynamics of Mechanical Systems

Lecture 4

CIE 4145 Dynamics and Introduction to Continuum Mechanics

Faculty of Civil Engineering and GeosciencesDelft, The Netherlands

CIE 4145, Module B

Karel N. van Dalen

Section: Structural MechanicsChair: Dynamics of Solids and StructuresRoom: 6.61, Phone: [email protected]://www.tudelft.nl/knvandalen

2

Lecture B4Free Vibration of 1 DOF with Viscous Damping

Content.

Free Damped Vibrations

Equation of Motion

Characteristic Equation and Its Roots

Aperiodic Motion

Critically Damped System

Damped Vibrations

Amplitude of Damped Vibration,

Logarithmic Decrement

Period of Damped Vibrations

Lecture 4



Learningobjectives

Thestudentisableto

derivetheequationofmotionofadampedsingledegreeoffreedom(SDOF)system

derivethegeneralsolutionoftheequationofmotion(freevibrations),forsubcritically,criticallyandsupercriticallydampedSDOFs

explainthephysicalbehaviourinthesedifferentregimes,andhowthisisvisibleinthemathematics

Explainhowthedampingcoefficientofasystemcanbemeasured

Free vibration of skyscrapers after earthquake: http://www.youtube.com/watch?v=EQaHyY9-Fuw

3

Equation of Motion

Lecture 4

Derivation of Equation of Motion

Force mass acceleration

SPRING DASHPOTt t t m F F F x

SPRING

DASHPOT

t k

t c

F x

F x

mx+cx+kx = F t

m

x(t)

F(t)

x



4

Characteristic Equation and Its Roots

Lecture 4

Equation of Free Motion:

0mx cx kx

Parameters:

natural frequency of vibrations

2 damping factor

nk mundamped

c m n

22 0nx nx x

Characteristic Equation:2 22 0ns ns

Characteristic Exponents:2 2 2 2

1 2,n ns n n s n n

The motion of the system depends crucially upon the ratio

crit2n

n c cckm



25

Aperiodic Motion

Lecture 4

Case 1. n > n2 2

1

2 22

1 2

real, negative

real, negative

n

n

s n n

s n n

s s

Particular initial conditions 00 , 0 0x x x

0 1 2 2 11 2

exp expxx t s s t s s ts s

No Vibrations! The motion is called aperiodic

t

x

x0



6

Critically Damped System

Lecture 4

Case 2. n = n1

2

real, negativereal, negative

s ns n

Particular initial conditions 00 , 0 0x x x

0 exp 1x t x nt nt The general solution:

1 2exp expx t X nt X t nt

Still no vibrations!

t

x

x0



7

Damped Vibrations

Lecture 4

Case 3. n < n

2 21

2 22

1 2

complex

complex

Re 0, Re 0

n

n

s n n

s n n

s s

With this notation 1 1exp cos sinx t nt A t B t

2 21

1 1

2 1

New notation:

real, positiven ns n is n i

General initial conditions 0 00 , 0x x x v

0 00 1 11 1

exp cos sinv nxx t nt x t t



8

Damped Vibrations

Lecture 4

0 00 1 11 1

exp cos sinv nxx t nt x t t

0 1 02

2 0 00 0

1 1

0 00

0 1

exp cos

arctan

x t A nt t

v nxA x

v nxx

t

x

x0

A0A0exp(-nt)

-A0exp(-nt)

T1= 2

The motion is completely defined by the amplitude, period and phase



39

Amplitude, Period, Phase

Lecture 4

0 1 0

0

1 1 2 2

0

exp cos

Amplitude: exp2Period: 2

Phase angle: n

x t A nt t

A nt

Tn

t

x

x0

A0A0exp(-nt)

-A0exp(-nt)

T1= 2



10

Amplitude and Frequency in Practice

Lecture 4

1

11

11

1 2 2

Let be a displacement at and a displacement after 1 cycles.Then

exp 1

or log 1

2 logarithmic decrement

s

s

s

n

x tx s

x s nTx

x s nTx

nnTn

Measurements of damping. 0 1 0exp cosx t A nt t

22 2

1 2

Thefrequency of damped vibrationsis normally approximated bythe natural frequency ofthe undamped systemThe reason:

12n n nn

nn



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