Table of Contents
International Conference on Advances in Nuclear Science and Engineering in Conjunction with LKSTN 2007 (65-69)
Dynamic Transient Analysis Of The Structure
Based On The Finite Element With
The Normal Mode Method
Utaja1, Khairina Ns2*
National Nuclear Energy Agency of Indonesia - PRPN, Serpong, Indonesia,
2
National Nuclear Energy Agency of Indonesia - PPIN, Serpong, Indonesia,
*E-mail: yenny@batan.go.id
1
Abstract
DYNAMIC TRANSIENT ANALYSIS OF THE STRUCTURE BASED ON THE FINITE ELEMENT WITH
THE NORMAL MODE METHOD. The dynamic transient events frequently encounter on turbine and NPP
building or other engineering problems. The dynamic transient was happened on the object or structure that
was affected by the momentary force which is time function. The consequence of this force, the object or the
structure will deform its shape momentary and then back to its original. At the deformed condition, the stress
will appear on the structure. The existing deformity and stress of the structure should be analyzed to know the
structure condition as the result of the momentary force. This events occurred on the structure because of the
short shake force, for example on the turbine shaft and the structure building i.e.: on the NPP and the
transportation. The dynamic transient analysis of the structure has been done by the finite element modeling.
The dynamic transient solution of the finite element modeling will include the vibration solution. If the solution
was done by the direct integration method it will be time consuming, depend on the time step that was used
between two process integration. Another consequence of the direct integration method was a big file needed to
store the result. To overcome these difficulties, the transient dynamic solution was done by the normal mode
method. On the normal mode method, the structure deformation which was expressed on the Cartesian
coordinates, was transferred to the normal mode coordinates. On the next step the eigenvalue and the
eigenvector then be applied on the dynamic transient solution. With this method, the dynamic analysis can be
solved easier and faster.
Keywords: normal mode, finite element, dynamic transient
Introduction
The vibration problems which are exist in
engineering for example on the NPP plant, should
be analyzed carefully. The vibration will create
many problems for example dynamic transient,
shock spectrum, vibration response, and random
vibration.
The vibration will be coming on if the
structure be introduced either by intermittent or
momentary force. The structures response depends
on the material behavior, structures geometrical and
the disturbing force as time function.
This paper describes the dynamic transient
analysis only. The dynamic transient analysis can
be done by finite element method which in the next
step will be solved by direct integration method or
by normal mode method. The direct integration
method can be sought in many references [1,2]. The
principal method of the direct integration is step by
step solution of the matrix equation. The result of
this method depends on the matrix solution
stability, and it will produce a great amount of data
to be stored.
The principal method of the normal mode is
coordinate transformation, where the Cartesian
coordinate should be transformed to the normal
mode coordinate. With this method, the former
couple matrix equation of n x n size will be
transformed to the m second uncouple partial
differential equation which can be solved easier.
For this process, the eigenvalue and the eigenvector
solutions were needed. On the next step the
eigenvalue and the eigenvector then be used for
dynamic transient solution with superposition
process[1 ]
The NISA USERS MANUAL (third
manual) explains the principal method without
describing the solution process, especially how
eigenvalue and eigenvector be extracted The
deformed position U then be constructed by
superposition principal of eigenvector and
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International Conference on Advances in Nuclear Science and Engineering in Conjunction with LKSTN 2007 (65-69)
amplitude variables. This process is done on the
post processing step, where the deformed analyzed
structure will be displayed as one of the vibration
mode.
Beside the less matrix operation, the normal
mode method can be used for another structure
analysis such as for the vibration response analysis,
the shock spectrum analysis and the random
vibration analysis [1].
Theoritical Overview
If the rest structure be disturbed by shock or
intermittent forces, it will deform its shape and then
back to original shape. This process will initiate
vibration which have many frequencies and
amplitudes variation. In the finite element method,
the equilibrium position can be expressed as [1,2,3]:
MU’’+CU’+KU=F(t)
(1)
where: M = global mass matrix;
C = global damping factor;
K = stiffness matrix
U =global position vector (column matrix);
F(t) = vector force (time function)
This matrix equation is matrix couple
equation where each row has relation with the other
row which is expressed by the matrix coefficient
laid in the column. This matrix couple equation
will be changed to the un-couple matrix equation.
In this dynamic transient analysis, the
solution of the equation (1) is divided into three
steps, that are :
1. Eigenvalue and eigenvector solution.
2. Coordinates transformation
3. Superposition process
1. Eigenvalue and eigenvector solution
Both the principal of eigenvalue and
eigenvector solution are determining the natural
frequencies of structure. During this step, the
damping factor and the force will be omitted (the
principal of free vibration analysis), so that the
equation (1) can be expressed as:
MU”+KU=0
(2)
Assuming a solution of U is expressed as [1] : U = ϕ
e (iωt + ψ)
Then the equation (1) will gives :
(K-λM)ϕ=0
where : λ = Eigenvalue = (2πf)2 = ω2 :
f = frequency (Hz)
ϕ = Vector column of displacement U
(3)
For non trivial solution (ϕ ≠ 0) of equation (3), the
matrix (K - λM) must be singular, so that the next
equation should be fulfilled [3,4]:
det(K-λiM)=0
(4)
with λi = are the roots of the equation ( there are n
roots, where n is the nodes amount)
Since both K and M are real symmetric matrix, it
can be proved that all the roots λi and so the ϕi are
real.
Eigenvalue and eigenvector can be
determined by orthogonality principal, where from
orthogonality it can be shown [1,3]:
ϕiTMϕj = 0 for i≠j
(5)
= 1 for i = j
ϕiTKϕj = 0 for i ≠ j
= ω2 for i = j
th
where: ϕi = the i Eigenvector
The solution of equation (5) is done by finite
element and power iteration method.
The results of this solution are n pair of eigenvalue
and eigenvector, but only some of these results are
needed. The results of the equation (5) may be
written with:
ΦTMΦ = I
(6)
ΦTKΦ = diag( ωi2 )
Φ = [ ϕ1, ϕ2, ϕ3….ϕn.]
where: I = identity matrix
To proceed equation (6), the inverse power iteration
and subspace method are used.
2. Coordinates transformation.
The solution of the equation (6) will give the
eigenvalue and eigenvector which are used during
the coordinates transformation process.
The
principal idea in the coordinates transformation is a
theorem of symmetric matrices orthogonality,
which can be written [5]:
“any arbitrary vector can be expressed as linier
combination of the eigenvector of a symmetric
matrix”. In this method, the nodal displacement
response is expressed in term of normal mode
method. So the nodal displacement U can be
expressed by [1]:
(7)
U = Φ q = Σ λi qi
where:
U = Column matrix of nodal displacement, on
Cartesian coordinates;
Φ = matrix ( n x m ), in each column is an
eigenvector;
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International Conference on Advances in Nuclear Science and Engineering in Conjunction with LKSTN 2007 (65-69)
qi = The mode-amplitude coordinates or normal
mode coordinates;
λi = an eigenvalue
m = the amount of the mode vibration that will
exist on the structure;
n = the amount of the nodes on the structure
modeling;
m << n
Introducing equation (7) into equation (1) and then
pre multiplying by ΦT, follow by orthogonality
relationships will give :
q” + ΦTCΦq’ + diag(ωi2)q = ΦTF(t)
(8)
According to orthogonality principal, the triple
production ΦTCΦq’ can be expressed by
ΦTCΦq’ = diag (2ξiωi)
(9)
where : ξi = damping ratio of mode i
Equation (8) is an uncoupled equation which
consists of m one degree of freedom equation. The
r-th uncouple modal equation of equation (8) ( r = 1,
2, 3……,m) becomes [1]:
qr” + 2ξirωr qr’+ ωr2qr = fr(t)
(10)
where:
fr(t) = forcing function for mode r = ϕrTF(t)
qr = amplitude mode r
The solution of equation (10) will give the structure
amplitude of mode r-th. The amount of mode to be
selected depend on the accuracy needed, where five
modes are enough (maximum r = 5).
3. Superposition process.
The results of equation (10) are m position
modes, where on every mode have one pair of
eigenvalue and eigenvector of λi and ϕi. The nodes
deformation U can be constructed from m pairs of
eigenvalue and eigenvector through superposition
process showed at next equation .
m
U = Φ q = Σ λ i ϕi
(11)
1
This equation similar with equation (7), except that
in equation (11) we need m modes only (m< n ).
So the equation (11) can be solved if five (5) modes
needed have been obtained.
To proceed all steps, a computer code has
been developed, especially for eigenvalue and
eigenvector extraction.
Computer Code
The above described process is done by
finite element method, so that the computer code
should be developed. This code consist of three
codes, pre-processing, processing or solver, and
post-processing.
Pre-processing is used for
geometrical discreetitation, data preparation of the
element, material, and loads which is needed in
matrices coeffisient generating. Solver is use for
matrix equation solving and to prepare the result in
a file which will be processed by post-processing
code. In this paper, the solver main task is for
eigenvalue and eigenvector pairs extracting. The
LU decomposition is used for both matrix equation
solution and the combination of inverse power
iteration and sub space method is used for
eigenvalue and eigenvector extraction. The postprocessor is used for result display, especially for
vibration mode display and superposition display.
This developing code has been written on Visual
Basic 5.0 [5]
Results And Discussions
In this paper two supports beam and a
cantilever beam will be observed as samples for
dynamic transient verification. The result will be
compared with the result of NISA II.
The
geometrical of two supports beam is showed at
Figure 1 and a cantilever beam is showed in Figure
7.
For complexity reducing, the analysis is
performed by 2-D element beam. For eigenvalue
and eigenvector extraction, the Gram-Schmidt
iteration and consistent mass matrix are used.
F= 100 lbs
8o inch = 200 cm
Figure 1. A two supported beam
Beam properties are of [1]:
Material
1. Modulus Elasticity (EX) : 3.0 E7 psi (2.04E6
kg/cm2)
2. Poisson’s ratio (NUXY) : 0.30
3. Density : 7.28 E(-4) lbs.sec2/in4
Bar cross section
1. Cross section area: 4 in2 (25.8 cm2)
2. Moment inertia: 1.3333 in4 (55,5 cm4)
Force position: at the middle of the beam Time
function:
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International Conference on Advances in Nuclear Science and Engineering in Conjunction with LKSTN 2007 (65-69)
1
amplitude of Figure 5 is much less then the
amplitude of Figure 4.
0.0024 sec
0.0048se
Amplitud
e
Time
Figure 4. The deformation of 1’st mode
Figure 2. Time function of force
The beam is divided into 30 equal length part (so
there are 30 elements and 31 nodes) in order the
result can be view as continues curve, which is
showed at Figure 3 and Figure 4.
16
1
Node
Figure 5. The deformation of 2’nd mode
31
31
Element
Figure 3. Thirty (30) elements and thirty and one
(31) nodes
The beam is modeled using 2-D beam element.
Translation in X and Y direction are constrained at
both ends of the beam (at node 1 and node 31).
The first three natural frequencies are extracted and
are compared with NISA II and showed in Table 1..
Table 1. Developing
compared to NISA II
Mode
1
2
3
Developing
Code
28,79 Hz
278,33 Hz
Not yet found
code
result
is
NISA II/90
28,766 Hz
115,398 Hz
263,012 Hz
It can be seen that mode 1 eigenvalue of developing
code is close to NISA II, unfortunately the second
eigenvalue is very different. This is caused by the
very high accuration computation need. The third
eigenvalue not yet determined, because there’s still
fault in the second eigenvalue.
Through the mode 1 and mode 2, the
structure deformation result by mode 1 and mode 2
can be shown at Figure 4 and Figure 5. The
deformation result by the time function force as
Figure 2 can be view at Figure 6.
At Figure 4 it can be seen that both at the end of the
beam there are nodes and at the middle of the beam
there is peak. This ship is the main mode of
vibration.
At figure 5 there are three peaks with
nodes at both end of the beam. The amplitude of
Figure 4 and 5 is not in proportional, actually the
Figure 6. Deformation which is caused by
time function force
Figure 6 shows the result of time function force at
the middle of the beam. The first trace (0 to
0.00048 second) is caused by combination of force
and natural vibration, even at the second trace (
more then 0.00048 second) is caused by natural
vibration only. It can be seen that the vibration is
damped.
The cantilever beam can be seen at Figure 7.
Clamped support
Free-end
80 inc
Figure 7. Cantilever beam
Beam properties are of [1]:
Material:
1. Modulus Elasticity (EX) : 3.0 E7 psi (2.04E6
kg/cm2)
2. Poisson’s ratio (NUXY) : 0.30
3. Density : 7.28 E(-4) lbs.sec2/in4
Bar cross section:
1. Cross section area : 4 in2 (25.8 cm2)
2. Moment inertia
: 1.3333 in4 (55,5 cm4)
The beam is divided into 30 equal part length for
continus curve purpose .
The beam is modeled using 2-D beam element.
Translation in X and Y direction are constrained at
left ends of the beam (at node 1 ).
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International Conference on Advances in Nuclear Science and Engineering in Conjunction with LKSTN 2007 (65-69)
The first three natural frequencies are extracted and
are compared with NISA II and showed in
Table 2.
Table 2. Developing code result is compared to
NISA II
Mode
Developing
NISA II/90
Code
1
10,98 Hz
10,246 Hz
2
56,016 Hz
64,218 Hz
3
Not yet found
179,503 Hz
It can be seen that mode 1 eigenvalue of developing
code is close to NISA II, unfortunately the second
eigenvalue is rather different. Once again, this
phenomena is caused by very high accuracy
computation need. The third eigenvalue is quiet
different, there’s still fault in the third eigenvalue.
The first mode of the nodes displacement
can be seen at Figure 8, and the second mode at
Figure 9.
Figure 8. The deformation of 1’st mode
Figure 9. The 2’nd mode deformation
Figure 8 and Figure 9 show that the free-end beam
is a peak, at Figure 9 another peak position is
located at one third length from the clamped node.
If a load as Figure 2 with different time
step is introduced at the free end beam, the beam
deformation will display as Figure 10.
The
vibration effect appear from first process to the end
of the analysis.
CONCLUSIONS
The result of the two samples analysis show that
the combination of the finite element and the
normal mode method can be used for transient
dynamic solving efficiently. Accordance with the
un-accurate result, the developing code should be
completed so that the sufficient eigenvalue and
eigenvector pair can be extracted.
ACKNOWLEDGEMENT
The authors would like to thanks KPTF PRPN
for very useful discussion to enhance this paper
quality.
REFERENCES
1. ANONIM, ” NISA User Manual “, EMRC
(Engineering
Mechanics
Research
Corporation), Michigan USA 1990
2. FRANK.L STASA, “Applied Finite Element
Analysis for Engineers”, CBS College
Publishing, New York, USA 1985
3. KLAUS JURGEN BATHE, ”Finite Element
Procedures”, Prentice Hall International
Inc, New Jersey, USA 1996
4. ALAN JENNINGS, “Matrix Computation for
Engineers and Scientists”, John Wiley
& Sons, New York, USA
5. EVANGELOS PETROUTSOS, “Mastering
Visual Basic 5”,
Sybex, San Francisco,
USA 1997
DISCUSSION
LIEM PENG HONG
1. Do you really need to use quad-precission
intrider to obtain 2-nd or higher order
eigenvalues.
2. In Reactor Physics fields, we use Wieland
method but we do not need quad-precission.
UTAJA
1. Yes, we really need to use great precission or
high accuracy to obtained 2nd and higher eigen
value, because some references suggest it.
2. I would like to try again your method
(wrilenad) in my software. Thank you for your
information.
Figure 10. Deformation caused by time function
force
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