chapter 3

DYNAMIC OF STRUCTURES

CHAPTER 3RESPONSE OF SDOF SYSTEMS

Department of Civil Engineering, University of North Sumatera

DANIEL RUMBI TERUNA, IP-U

RESPONSE OF SDOF SYSTEM

The equation of motion can be rewritten as

q Undamped System Subjected to Harmonic Force

The complementary solution of equation (2) is the free vibration response

(1)

Setting gives the diffrential equation governing forced vibration of the system without damping )0( =c

tptp o sin)( =

)()()()( tptkutuctum =++ &&&

tptkutum o sin)()( =+&& (2)The complete solution is the sum of the complementary and particular solutions is

)()()( tututu pc += (3)


(4)tBtAtu nnc sincos)( +=

(6)

The particular solution of equation (2) is

tCtu p sin)( = (5)

Diffrentiating this twice gives

Substituting eq.(5) and eq. (6) in the eq.(2) leads to a solution for

( )20

/1

1

nkp

C

= (7)

tCtu p sin)(2=&&

C


0=t

(10)

Substituting eq.(4) and eq. (5) in the eq.(3) gives

( )t

kp

tBtAtun

onn

sin/1

1sincos)( 2

++=

( )t

kp

tBtAtun

onnnn

cos/1

cossin)( 2++=&

(8)

(9)

Substituting inital condition at leads to solution for

20

)/(1/)0(

,)0(n

n

n kpu

BuA

==

&

BandA

Substituting eq.(10 in the eq.(8) gives the final result


(11)

( )

( )t

kp

tkpu

tutu

n

o

n

n

n

nn

sin/1

1

sin/1

/)0(cos)0()(

2

20

+

+=

&

transient

Steady state

2.0/ =n

Fig. 1. (a) harmonic force; (b) response of undamped system to harmonic force: 2.0/ =n

kpuu n /)0(,0)0( 0== &


( )t

kp

tun

o

sin/1

1)( 2=

The steady state dynamic response, a sinusoidal oscillation at the forcing frequency, maybe expressed as

(12)

or ( )( )

tutun

st sin

/1

1)( 20 =

Where is the maximum value of the static deformation( )kp

u st0

0 =

(13)

Ignoring yhe dynamic effect gives static deformation at each instant

tkp

tu st sin)( 0= (14)

Equation (13) is rewritten in term of amplitude of the vibratory displacement and phase angle

0utu


( ) ( ) ( ) == tRututu dst sinsin)( 00where

( ) ( )200

/1

1

nstd u

uR

==

and

> n tsin

Fig. 2 deformation response factor and phase angle for an undamped system excited by harmonic force


The diffrential equation governing the response of SDOF system to harmonicforce with damping is

q Harmonic vibration with viscous damping

The particular solution of this diffrential equation is

(1)tptkutuctum sin)()()( 0=++ &&&

0)0()0( === tatuuuu && (2)

where

tDtCtu p cossin)( += (3)

This equation is to be solved subject to the initial conditions

( )[ ] ( )[ ]2222

/2/1

)/(1

nn

no

kp

C

+

= (4)


The complete solution of equation (1) is

(5)

(6)

transient

(7)

( )[ ] ( )[ ]222 /2/1/2

nn

no

kp

D

+

=

The complementary solution of equation (1) is the free vibration response given as

( )tBtAetu DDtc n sincos)( +=

( ) tDtCtBtAetu DDtn cossinsincos)( +++=

Steady state

Where the constants A and B can be deetrmined by standard procedures


Fig. 3. response of damped system to harmonic force: 05.0,2.0/ == nkpuu n /)0(,0)0( 0== &


For lightly damped systems the sinussoidal term in Eq. (8) is small and

; thus

BandA

nD

q Response for 1/ =n

For .Eq.(4) gives ; for and

zero initial condition , the constants in Eq.(5) can be determined:

and . With these solution for

, Eq.(7) becomes

1/ =n 2/)(0 0stuDandC == 1/ =n

2/)( 0stuA=2

0 12/)( = stuA

DandCBA ,,,

( )

+= ttteutu nDD

tst

n

cossin1

cos21

.)(20

........(8)

Fig. 4. response of damped system to harmonic force with 05.0, == n0)0(,0)0( == uu &

( ) [ ] teutu ntst n cos1

21

.)( 0 (9)

Envelope function

q Maximum Deformation and Phase Lag

tDtCtu cossin)( +=

The steady state response due to harmonic force described by Eqs.(3), (4) and (5) can be rewritten

( )[ ] ( )[ ]222 /2/1/2

nn

no

kp

D

+

=

( )[ ] ( )[ ]2222

/2/1

)/(1

nn

no

kp

C

+

=

Equation (9) can be written in other form as

)sin()sin()( 00 == tRkp

tutu d

(9)

(10)

(11)

(12)


( )CDandDCu /tan 1220 =+= Where . Substituting for C and D gives

( )( )2

1

/1

/2tan

n

n

=

( )[ ] ( )[ ]22200

/2/1

1)(

nnstd u

uR

+== (13)

(14)


1. If the frequency ratio

Fig.5 . response of damped system to harmonic force with 2.0,5.0/ == n

This result implies that the dynamic response is essentially the same as the static deformation and is controlled by the stiffness of the system

2. If the frequency ratio >>1 (i.e., the force is rapidly varying),

tend to zero as increases and is essentially unaffected by

damping . For large value of , the is dominant in Eq.(13),

which can be approximated by

n / dR

n /

n /4)/( n

( )2

02

2

00

mp

uu nst = (16)

This result implies that the response is controlled by the mass of the system

3. If the frequency ratio (i.e., the forcing frequency is close to the

natural frequency of the system), is very sensitive to damping and, for the

smaller damping value, can be several time larger than 1. If Eq.(13)

becomes

1/ n

dR 1/ =n

( )n

st

cpu

u

000 2

= (17)

This result implies that the response is controlled by damping of the system

dR

1. If 1 (i.e., the force is rapidly varying), is close to and

the displacement is essentially out of phase relative to applied force. When the

force acts to the right, the system would also be displaced to the left

n / 0180

3. If (i.e., the forcing frequency is equal to the natural frequency),

; for all value of , and the displacement attains its peaks when the

force passes through zero

090=

dR

Fig.6 Deformation response factor and phase angle for a damped system

q Dynamic response Factor

The steady state displacement as described by Eqs.(12), is repeated for convinience:

)sin(/)(

)sin()(0

0 == tRkptu

ortRkp

tu dd (18)

(19)


Diffrerentiating Eq.(18) gives an equation for the velocity response:

)cos(/

)(

0

= tRkmp

tuv

&

Where the velocity response factor is related to byvR dR

dn

v RR

= (20)

(21)

(22)


Diffrerentiating Eq.(19) gives an equation for the acceleration response:

)sin(/)(

0

= tRmptu

a

&&

Where the acceleration response factor is related to byaR dR

dn

a RR2

=

(23)

The simple relation among the dynamic response factors

dn

vn

a RRR

==

/

Fig.7 Deformation response factor for a damped system excited by harmonic force

Fig. 8 Velocity response factor for a damped system excited by harmonic force

Fig. 9 Acceleration response factor for a damped system excited by harmonic force

A resonat frequency is defined as the forcing frequency at which thelargest response amplitude occurs. These resonant frequencies can bedetermined by setting to zero the first derivative of withrespect to ; for gives

q Resonant Frequencies and Resonant Responses


avd RandRR ,,n / 2/1

If are the forcing frequencies on either side of the resonantfrequencies at which the amplitude times the resonantamplitude.

q Half Power Bandwidth

ba and210 isu

2= ba


(25)

(26)

(27)

Fig. 10 The definition of Half Power Bandwidth

chapter 3

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