Slide TSP302 DinamikaStr PengRekGempa TSP 302 P6 7

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Mata Kuliah
Kode
SKS

: Dinamika Struktur & Pengantar Rekayasa Kegempaan
: TSP – 302
: 3 SKS

Multi Degree of Freedom System
Free Vibration
Pertemuan – 6, 7

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TIU :
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Mahasiswa dapat menjelaskan tentang teori dinamika struktur.
Mahasiswa dapat membuat model matematik dari masalah teknis yang
ada serta mencari solusinya.

TIK :


Mahasiswa mampu mendefinisikan derajat kebebasan, membangun
persamaan gerak sistem MDOF

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Sub Pokok Bahasan :
 Penentuan derajat kebebasan
 Properti matrik kekakuan, massa dan redaman

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Structures cannot always be described by a SDoF
model
 In fact, structures are continuous systems and as such
possess an infinite number of DoF
 There are analytical methods to describe the
dynamic behavior of continuous structures, which
are rather complex and require considerable
mathematical analysis.
 For practical purposes to study Multi Degree of
Freedom System (MDoF), we shall consider the
multistory shear bulding model.

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A shear building may be defined as a structure in which there
is no rotation of a horizontal section at the level of the floors
Assumption for shear building

 The total mass of the structure is concentrated at the
levels of the floors
 The slabs/girders on the floors are infinitely rigid as
compared to the columns
 The deformation of the structure is independent of the
axial forces present in the columns

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u3

m3

F3(t)
k3

F3(t)

k3

m2

F2(t)
k2

m1

F1(t)

u2

m 3ü 3

F2(t)

k3(u3 u2)
k3(u3 u2)
m 2ü 2

F3(t)

k2(u2 u1)
m 1ü 1

k2 u1

k2(u2 u1)

k1u1

k1

k1

Shear Building w/ 3 DoF

Free Body Diagram

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By equating to zero the sum of the forces acting on each
mass :
m1u1  k1u1  k 2 u2  u1   F1 t   0

m2u2  k 2 u2  u1   k3 u3  u2   F2 t   0

m3u3  k3 u3  u2   F3 t   0


M u  K u  F 

(1)

Eq. (1) may conveniently be written in matrix notation as :

(2)

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Where :
m1
M    0
0


And :

0
m2
0

 u1 
u  u2 
u3 

0
0

m3 

k1  k 2
K     k2
 0


 u1 
u  u2 
u3 

 k2
k 2  k3
 k3

0 
 k3 
k3 

 F1 t 

F   F2 t 
 F3 t 

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Natural Frequencies and Normal Modes
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

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M u K u  0

EoM for MDoF system in free vibration is :

u  n qn t   n  An cos nt  Bn sin nt 

(3)

Solution of Eq. (3) is :

K   

M n qn t   0

(4)

The substitution of Eq. (4) into Eq. (3) gives :
n

2
n

(5)

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K   

M n   0

The formulation of Eq (5) is known as an eigenvalue problem.
To indicate the formal solution to Eq. (5), it is rewritten as :
2
n

(6)

Eq (6) has non trivial solution if :

K   n 2 M   0

(7)

Eq.(7) gives a polynomial equation of degree n in n2, known
as characteristic equation of the system
A vibrating system with n DoF has n natural vibration frequencies n, arranged in
sequence from smallest to largest

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The n roots of Eq. (7) determine the n natural frequencies n
of vibration.
These roots of the characteristic equation are also known as
eigenvalues
When a natural frequency n is known, Eq.(6) can be solved
for the corresponding vector n
There are n independent vectors , which are known as
natural modes of vibration, or natural mode shape of
vibration
These vectors are also known as eigenvector.

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E = 200.000 MPa

1  10 ,76

   11


21

 2  38,28

12  0 ,6809  3,0385

1 
 22   1

W10x21

u2

W1 = 11.600 kgf

3m

u1
W10x45

W10x45 I = 10,300 cm4

W10x21

W10x21  I = 4,900 cm4

W2 = 24.000 kgf

W10x45

Example 1
 Determine :
1. The natural frequencies and
corresponding modal shapes

4,5 m

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Modal & Spectral Matrices


The N eigenvalues, N natural frequencies, and n natural
modes can be assembled compactly into matrices
11 12
   ...21 ...22

 n1  n 2

Mode 1
12

2
 




 

2

2 2

... 1n 
...  2 n 
... ... 
...  nn 

DoF 1
2
…..
n

Modal matrix

n





2
n 

Spectral matrix

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Orthogonality of Modes


The natural modes corresponding to different natural
frequencies can be shown to satisfy the following
orthogonality conditions :

n T K r   0

n  M r   0
T

When n ≠ r

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Free Vibration Response : Undamped System
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M u K u  0

EoM for MDoF Undamped Free Vibration is :
(8)

The differential equation (8) to be solved had led to the
matrix eigenvalue problem of Eq.(6).
The general solution of Eq. (8) is given by a superposition of
the response in individual modes given by Eq. (4).

u  1q1  2 q2  3 q3  ...  n qn

Or

u    q
N

n 1

n

n

u   q

(9)
(10)

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Premultiply Eq. (10) with {n}T[M] :

n T M u  n T M  qn 

 n  M 1q1  n  M 2 q2  ....
T

 n  M N qN

(11)

T

T



Regarding the orthogonality conditions :
T
T
n  M u  n  M n qn
 From which :

  M u

  M u
also
q 
q 
  M  
  M  
T

T

n

n

n

T

n

T

n

n

n

n

(11)

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If Eq. (8) is premultiplied by the transpose of the nth modeshape vector {n}T, and using Eq.(10) with its second
derivative it becomes :

  M  q    K  q
T

T

n

n

n

n



n

0

(11)

Kn

Mn
Generalized mass

n

Generalized stiffness

Eq.(11) is an EoM from SDoF Undamped System for mode n,
which has solution :
qn t   qn 0 cos nt 

qn 0 

n

sin nt

(12)

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Solution for Eq. (8) becomes :



qn 0 
u  n qn 0cos nt 
sin nt 
n
n 1


N



(13)

The procedure described above can be used to obtain an
independent SDoF equation for each mode of vibration of
the undamped structure.
 This procedure is called the mode-superposition method

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Example 2
 Based on data from Example 1, and the
following initial condition, for each DoF plot
the time history of displacement, regarding
the 1st, and 2nd mode contribution.
 6  103 
ut 0    8  103 m
10  103 

 0 
ut 0   0 ,20m/s
 0 

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Free Vibration Response : Damped System
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M u  C u  K u  0

EoM for MDoF Damped Free Vibration is :

(14)

M  q  C  q  K  q  0

Using Eq. (10) and its derivative :

  M  q    C  q    K  q  0

(15)

Premultiply Eq. (15) with {}T :
T

T

Mn

Cn

T

Kn

(16)

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Eq.(16) is an EoM from SDoF Damped System for mode n,
which has solution :
qn t   e



 n n t



qn 0    nn qn 0 
sin nDt 
qn 0 cos nDt 

nD



(17)

The displacement response of the system is then obtained by
substituting Eq. (17) in Eq. (10)


q 0    nn qn 0 
u t    n e   t qn 0 cos nDt  n
sin nDt 
nD
n 1


N

n

n

(18)