Kuliah 8-10- Pengantar Dinamika Struktur SDOF-1

7/26/2019 Kuliah 8-10- Pengantar Dinamika Struktur SDOF-1

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PENGANTAR

DINAMIKA STRUKTUR

Bayzoni 2016


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DINAMIKA STRUKTUR

Single Degree of Freedom (SDOF)Multi Degree of Fredom (MDOF)


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Single-Degree-of-Freedom System


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Single-Degree-of-Freedom System

Sistem SDOF pada blok kaku dengan massa m, pegas

dengan kekakuan k dan peredam viscous c.

Blok massa hanya dapat bertranslasi pada garis

tuggal.


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Persamaan Gerak

Persamaan gerak diturunkan dengan menyeimbangkan semua gaya yangbekerja . Seperti terlihat pada gambar, gaya yang bekerja p(t) danmenghasilkan tiga gaya akibat gerak : gaya inersia fI, gaya redaman fD,dan gaya pegas, fS

Keseimbangan gaya diberikan dengan:

Setiap gaya-gaya yang diberikan pada sebe;lah kiri persamaan di atas merupakanfungsi dari perpindahan u, atau turunannya:

Prinsip dAlemberts, gaya inertia:

Gaya redaman viscous:

Gay Pegas:


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Persamaan Gerak

untuk Beban Akibat Gerakan Tumpuan

Keseimbangan gaya

Gaya inersia

total perpindahan

substitusi untuk inersia,redaman dan gaya

pegas

Persaman akhir


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Pada persamaan di atas, peff(t) menyatakan gaya

efektif akibat pergerakan tumpuan.

Kesimpulan yang dapat diambil bahwa gerakan

relatif sistem, ur(t), yang ditimbulkan oleh gerakan

tumpuan g(t), akan sama dengan gerakan total

sebuah sistem tumpuan kaku, u(t), yang diberi aksi

dengan gaya sama dengan Peff(t) = mg(t)

Persamaan Gerak

untuk Beban Akibat Gerakan Tumpuan


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Solusi Persamaan Gerak

Apakah permasalahan linier atau non-linier

kasus non-linier umumnya diselesaikan dengan

metode numerik.

Tipe bebanTingkat ketelitian

Untuk menentukan response sistem SDOF, u(t),

persamaan gerak diselesaikan secara analitis atau

numerik. Pemilihan metode tergantung dari

beberapa hal:


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Solusi Persamaan Gerak

As for any linear differential equation, the complete general

solution is the sum of complementary solution uc(t) and the

particular solution up(t):

The complementary solution is the solution of the homogeneous

equation

Characteristic Equation

Using the notation

s is solved:

If c > 2m,s will be real valued,

but if c < 2m, s will be a complex number.


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Undamped Free Vibrations

If the system is undamped, i.e., if c = 0, s becomes,

The response

By utilizing Eulers equation:

The result may be written in the form:

This type of motion is called a simple harmonic motion.


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The quantity is the natural angular velocity of the undamped

system (sometimes also referred to as the natural angular

frequency) and is related to natural frequency f as:

The reciprocal of f is called

the natural period T:

Based on the initial conditions: the displacement u(0) = B and

velocity (0) = At at time t = 0



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The Equation of motion u(t) can be recast into

The response is given by the real part, or horizontal projection,of the two rotating vectors.

The phase angle


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Damped Free Vibrations

If the oscillator is damped, c > 0, three different types of

motion are possible, depending on whetherthe value of the term

under the square root in the expression for s

is zero, negative, or positive,


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The value of c that makes the value of the term under the

square root in Equation 3.16 equal to zero iscalled thecritical

damping, cc,

At critical damping, the value of s :

The response is

By imposing the initial conditions the response is

Critical Damping

It is readily observed from that Equation, that the critically damped response does not involve oscillations about the zero deflection point

and the displacement returns to zero in accordance with the exponential decay term. Critical damping is the smallest amount of damping

that keeps a SDOF system from oscillating during free response.


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If the damping is less than critical, c < 2m, it is customary to

express the damping as a ratio to the critical damping value:

Where is called the damping ratio, the value of s :

By using the notation

D

is called the damped vibration frequency. Note that for

typical structures damping ratios rarely exceed about 10% (

< 0.10), and D differs very little from the undamped natural

frequency.

Underdamped Systems


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The response is written as:

By using Eulers equation, the response can be written in the

form:

The second term in equation above is of the same form as the

simple harmonic motion of the undamped oscillator, except

now at the damped, slightly lower frequency. The first term

indicates exponential attenuation of the oscillations. Constantsof integration A and B are again determined based on the

initial conditions u(0) and u(0) as before.

Underdamped Systems

v t( ) e t vov vo

d

sin d t vo cos d t


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REVIEW SDOF

Undamped Free Vibration

vo 1 vov 0

1

v t( )vov

sin t vo cos t

t 0 0.1 20

0 5 10 15 201

0

1

v t( )

t


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REVIEW SDOF

Underdamped Free Vibrationvo 1 vov 0

8 d 9 0.05

v t( ) e t vov vo

d

sin

d t

v

o cos

d t

t 0 0.1 15

0 5 10 151

0

1

v t( )

t


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Duhamel Integral

In the following, an expression for theresponse to an arbitrary dynamic loading isdeveloped based on Equation 3.80. Theconcept is to first derive, based on Equation3.80, the differential response due to adifferential impulse, acting over aninfinitesimal time interval and then, basedon an assumption of linearity, obtain thetotal response as the summation (integral) ofthe differential responses. For the

differential time interval d, the response is(fort > ):

du(t) represents the differential responsecontribution of the impulse p()d to the totalresponse, which is obtained by integratingEquation 3.81 as:

Equation 3.82, known as the Duhamelintegral,can be used to obtain the responseof an undamped2 SDOF system to any

dynamic loadingp(t).

For a damped system, the derivation isidentical except that the free-vibrationresponse initiated by the differential loadimpulse decays exponentially. The Duhamelintegral for a damped SDOF system is:


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Central Difference Method

Metode ini berdasarkan pendekatan beda hingga

turunan perpindahan terhadap waktu, dengan

mengambil step waktu tetap ti= t, maka

ekspresi untuk kecepatan dan percepatan adalah:


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Newmark Method

Th 1959, Newmark keluarga time-stepping

berdasarkan persamaan berikut:

Parameter b gan g untuk menentukan variasi

kecepatan untuk tiap tingkatan waktu dan untuk

stabilitas serta ketelitian.


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Newmark Method


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Newmark Method


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Newmark Method


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Newmark Method


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Wilson - q


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Wilson - q


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Evaluation of Results

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y(t)

t

Cen Diff

Newmark

Wilson-Q

Theoretica l


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Linear Step by Step Procedure


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Non-Linear Step by Step Procedure


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Non-Linear Step by Step Procedure

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Kuliah 8-10- Pengantar Dinamika Struktur SDOF-1