Slide CIV 308 DinamikaStr PengRekGempa CIV 308 P5

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Pertemuan - 5

Single Degree of Freedom System

Numerical Evaluation of Dynamic Response

Mata Kuliah

: Dinamika Struktur & Pengantar Rekayasa Kegempaan

Kode

: CIV

–

308

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

TIU :



Mahasiswa dapat menjelaskan tentang teori dinamika struktur.



Mahasiswa dapat membuat model matematik dari masalah teknis yang

ada serta mencari solusinya.



TIK :

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

Sub Pokok Bahasan :



Interpolasi



Central Difference

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Numerical Evaluation of Dynamic Response



Analytical solution of the equation for a SDOF System is

usually

not possible

if the excitation

–

applied force p(t) or

ground acceleration

–

varies arbitrarily with time or if the

system is nonlinear



Methods of Numerical evaluation are :

- interpolation of excitation

- central difference

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Interpolation of Excitation

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Interpolation of Excitation (Example

₁

)

An SDF system has the following properties :



m = 4.500 kg-sec

2

_/m



k = 178.400 kgf/m



T

_n

= 1 sec (

w

_n

= 6,283 rad/sec)



x

= 0,05.



Determine the response u(t) of this system to p(t) = 4,500

sin(

p

t/0,6) kgf, defined by the half-cycle sine pulse by using

piecewise linear interpolation of p(t) with



t = 0,1 sec.

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u

_i

in m

t_i p_i C.p_i D.p_i+1 ůi B.ůi ui A.ui C'.pi D'.pi+1 A'.ui B'.ůi

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Central Difference Method

Central Difference Method

is stable if :

p

1





n

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Central Difference Method

(Example 2)

Solve Example

₁

, by the central difference method using



t =

0,1 sec.

t_i p_i u_i-1 u_i p_i' u_i+1 0 0,00 0,0000 0,0000 0,0000 0,0000 0,1 2250,00 0,0000 0,0000 2250,0000 0,0048 0,2 3897,11 0,0000 0,0048 7395,2181 0,0159 0,3 4500,00 0,0048 0,0159 13884,5060 0,0299 0,4 3897,11 0,0159 0,0299 18538,8179 0,0399 0,5 2250,00 0,0299 0,0399 18033,8488 0,0389 0,6 0,00 0,0399 0,0389 10627,9866 0,0229 0,7 0,00 0,0389 0,0229 -411,7948 -0,0009 0,8 0,00 0,0229 -0,0009 -10620,7730 -0,0229 0,9 0,00 -0,0009 -0,0229 -16125,5428 -0,0347 1 0,00 -0,0229 -0,0347 -15096,8118 -0,0325

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Newmark’s Method

Newmark’s method is stable if :

Newmark’s Method –

Average Acceleration (Example 3)

t

_i

p

_i

ü

_i



p

_i



p

_i

'



u

_i



ů

_i



ü

_i

ů

_i

u

_i

Newmark’s Method –

Linear Acceleration (Example 4)

t

_i

p

_i

ü

_i



p

_i



p

_i

'



u

_i



ů

_i



ü

_i

ů

_i

u

_i

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Assignment

An SDF system has the following properties :



m = 4.500 kg-sec

2

_/m



k = 178.400 kgf/m



T

_n

= 1 sec (

w

_n

= 6,283 rad/sec)



x

= 0,05.

Determine the response u(t) of this system due to El-Centro 1940 N-S,

using :

a.

Interpolation

b.

Central Difference

c.

Newmark Average Acceleration

d.

Newmark Linear Acceleration