∫ ACCELERATION GENERATING CORRESPONDING TO THE PROPOSED RESPONSE SPECTRA

(1)

ACCELERATION GENERATING

CORRESPONDING TO THE PROPOSED RESPONSE SPECTRA

Jan Dupal

^1a

1University of West Bohemia in Pilsen, Faculty of Applied Sciences, Dept. of Mechanics e-mail: ^adupal@kme.zcu.cz

Abstract

Nuclear devices are checked also from the seismic safety point of view. The response spectra method is one of the most important approaches to obtain an upper estimation of some output variables like displacements, velocities, accelerations, stresses etc. The proposed response spectra of the base especially accelerations serves as an input for extreme estimations of mentioned output quantities depending on frequency [1]. The alternative approach for functionality assessment can be used the base acceleration as an input for time integration of the mathematical model and the output quantities can be obtained in the direct integration way of the model. The main requirement on the input acceleration is posed on the desired response spectrum satisfaction. The paper deals with an approach to generation of the base acceleration in such a way to be satisfied the properties of the proposed prescribed response spectrum of the acceleration.

GENEROVÁNÍ AKCELEROGRAMU NA ZÁKLADĚ NÁVRHOVÉHO SPEKTRA ODEZVY

Summary

Bezpečnost jaderných zařízení je kontrolována mimo jiné také ze seizmického hlediska. Metoda spekter odezvy patří mezi nejdůležitější metody pro určení horního odhadu výstupních veličin, za které považujeme posuvy, rychlosti, zrychlení, napětí atd. Presentovaný přístup pro posouzení funkčnosti využívá zrychlení základu odpovídající návrhovému spektru odezvy jako kinematické buzení matematického modelu a výstupní veličiny lze získat přímou integrací.

1. INTRODUCTION

What is it the response spectrum? It can be defined as a maximum of absolute value of one DOF vibrating system response to a prescribed excitation. The corresponding mathematical model has form

( ) ^t ^b ^q ( ) ^t ^kq ( ) ^t ^f ( ) ^t ^, q

m & & + & + =

₍₁₎

where

f ( ) t

is the mentioned excitation. The response spectrum of such system is expressed as

( ) ^max ¹ ( )

⁽ ⁾

^sin ( ) ^,

max

max 0

,

0

_Ω ∫ ^Ω ⁻

=

Ω

⁻ ^Ω ⁻

∈

T

D t

D

T D

q t

f e t d

S m τ

^τ

τ τ

(2)

where

Ω = k / m

is natural frequency of the un-damped system,

Ω

_D

= Ω 1 − D

² and

(2)

km D b

2 =

is damping ratio which is supposed to be input constant parameter. When the system is excited by the base movement, the excitation can be expressed as

f ( ) t = − m u & & ( ) t

and the response spectrum of displacement has form

( ) ^max ¹ ( )

⁽ ⁾

^sin ( ) ^.

max

max 0

,

0

⁻ _Ω ∫ ^Ω ⁻

=

Ω

⁻ ^Ω ⁻

∈

T

D t

D

T D

q t

u e t d

S ^& ^& τ

^τ

τ τ

(3)

Between response spectrum of displacement and response spectrum of acceleration holds simple relation

( ) Ω = Ω

² _q

( ) Ω ^.

q

S

&& (4)

2. SOLUTION OF THE RESPONSE

The relation (3) expresses a maximum of the absolute value of the system described by eq. of motion

( ) t ² D q ( ) t

²

q ( ) t u ( ) t ^.

q & & + Ω & + Ω = − & &

⁽⁵⁾

As a start point of the presented method is replacement of the kinematic excitation by periodic excitation with frequency step

∆ ω ,

it means by Fourier series with the basic frequency

∆ ω

. In case the frequency step is variable the substitutive excitation will be poly-harmonic. The example of the proposal response spectrum is depicted in Fig. 1

Fig. 1 Example of the proposal response spectrum

Having constant frequency step

∆ ω

we can the frequency values chose in form

, ,

1 ..., , 1 , 0

,

_min

min

ω ω ω

ω

_k

= + k ∆ k = N − = ∆

(6)

it means N values. It means that

ω

_max

= ω

_min

+ ( N − ¹ ) ∆ ω ^.

In case the step

∆ ω

is variable the frequency values of poly-harmonic excitation

ω

_k are chosen from the set

{

₁

^,

₂

^, ^... ^,

_N

} ^,

k

ω ω ω

ω ∈

(7)

where

ω

₁

= ω

_min

, ω

_max

= ω

_N

.

In both cases the eq. (5) takes a form

( ) ² ( ) ( ) ( ^cos ^sin ) ^.

1

2

∑

=

+

−

= Ω + Ω +

N

k

k k k

k

b

a t

q t

q D t

q ^& ^& ω τ ω τ

&

(8)

(3)

The goal is to find such values

a

_k

, b

_k

, k = 1 , 2 , ..., N ,

to achieve the result acceleration response spectrum of the substitutive excitation be identical with the proposal response spectrum in the defined frequency points

ω

_k^{. As the}

experience shown the calculation of convolution integrals in (3) and (4) is very time consuming and for this reason the analytical solution by means of Laplace transform of the eq. (8) was taken which respects, despite of Fourier transform, the zeros initial conditions. The Laplace transform of (8) can be expressed as

( ) ( ) ( ) ∑

=

 

 





+ +

− +

= Ω

+ Ω

+

N

k k

k k k

k

b p

p a p p

Q p

pQ D p Q p

1

2 2 2

2 2

2

2 ω ω

ω

_, ₍₉₎

with solution

( )

1

( )( ² ) ( )( ² ) ^,

2 2

2 2 2

2 2

∑

2

=

 

 





Ω + Ω + + +

Ω + Ω +

− +

=

N

k k

k

b p p D p

p D p p

a p p

Q ω

ω

₍₁₀₎

where

L { q ( ) t } = Q ( ) p ^.

To obtain an inverse Laplace transform of (10) it is necessary to decompose the relation (10) into so called partial fractions according to

( )

( ) ∑

=

−

=

n

j j

j

p p

p c Z

p P

1

, (n is number of roots of the Eq.

Z ( ) p = ⁰

⁾ ⁽¹¹⁾

where

( ) ( )

j

j j

p dp Z

d p c = P

. (12)

It can be written for both fractions in (10)

( ) ^p ( ^p

_k

)( ^p ^D ^p ) ^p ⁱ

_k

^p ^D ⁱ

_D

Z =

²

+ ω

² ²

+ 2 Ω + Ω

²

= 0 ⇒

₁_,₂

= ± ω ,

₃_,₄

= − Ω ± Ω

(13)

( ) p = p ( p + D Ω p + Ω ) ( + p + ) ( p + D Ω )

dp Z d

k

2 2

2

² ² ²

ω

²

. (14)

Now we can come to the coefficients of the first fraction in (10)

( ² ) ^,

2

1

2 , 2

1

= − + Ω + Ω

k k

k

i D

c ω ω ² ( ² ) ^,

1

, 2 1 , 2

2 k

k k

k

c

D

c i =

Ω + Ω

−

= −

ω

₍₁₅₎

( ² ) ^,

2

² ² ² ²

, 3

k D D

D

k

i D i D

i c D

ω + Ω

− Ω Ω

Ω + Ω

= −

(

D D k

)

^k

D

k

c

D i D i

i

c

₄_, ₂ ₂

D

₂ ₂ ₃_,

2 2 =

+ Ω

− Ω Ω + Ω Ω

−

Ω

− Ω

= −

ω

(16) and for the second fraction in (10)

( ² ) ^,

2

1

, 2 1

2

* ,

1 k

k k

k

ic

D i

c i = −

Ω + Ω +

= −

ω

ω ² ( ² ) ^,

1

_*

, 2 1 2

* ,

2 k

k k

k

c

D i

c i =

Ω + Ω

−

= −

ω

(17)

( ² ) ^,

2

² ² ² ²

* , 3

k D D

D

k

i D i D

c ω

ω

+ Ω

− Ω Ω

− Ω

= Ω

( ² ) ^.

2

* , 2 3 2 2

2

* ,

4 k

k D D

D

k

c

D i D

c i =

+ Ω

− Ω Ω + Ω Ω

= −

ω ω

(18)

(4)

t p

j

e

j

p

p =

 



 



−

−1

1 L

⁽¹⁹⁾

we can obtain the total solution of Eq. (8) (inverse Laplace transform of (10)) in form

( ) { [

⁽ ⁾ ⁽ ⁾

]

( ) ( )

[ ]} ^.

,

* , 3

* , 1

* , 1 1

, 3 ,

3 ,

1 ,

1

t i D k t i D k t i k t i k k N

k

t i D k t i D k t i k t i k k

D D

k k

D D

k k

e c e

c e c e c b

e c e

c e c e c a t

q

Ω

− Ω

+ Ω

−

=

Ω

− Ω

+ Ω

−

+ +

−

=

Ω ∑

ω ω

(20)

From the analytical solution (20) it can be simply determined response spectrum of displacement and acceleration

( ) ( )

[ ]

{

( ) ( )

[ ]}

, max

, max ,

* , 3

* , 1

* , 1 1

, 3 ,

3 ,

1 ,

0 ,

0 _max _max

t i D k t i D k t i k t i k k N

k

t i D k t i D k t i k t i k k T

t T

q t

D D

k k

D D

k k

e c e

c e

c e c b

e c e

c e

c e c a t

q S

Ω

− Ω

+ Ω

− −

=

Ω

− Ω

+ Ω

−

∈

+ +

=

Ω ∑

ω ω

p p

(21)

( ) ( )

[ ]

{

( ) ( )

[ ] }

. max

, ,

* , 3

* , 1

* , 1 1

, 3 ,

3 ,

1 ,

0 2 2

max D i t

k t i D k t i k t i k k N

k

t i D k t i D k t i k t i k k T

q t q

D D

k k

D D

k k

e c e

c e

c e c b

e c e

c e

c e c a S

S

Ω

− Ω

+ Ω

−

=

Ω

− Ω

+ Ω

−

∈

+ +

Ω

= Ω Ω

=

Ω ∑

ω ω

p

&

p

&

(22) The terms in (20) are complex conjugated pairs and for this reason the response (20) is of course real. In relations (21) and (22) was used a vector symbol p which means

, ,

,

²

1

2 1

2 N

N N

b b b

a a a

R b

R a

b R

p a ∈

 





 





=

∈

 





 





=

 ∈



 



= 

M

⁽²³⁾

finding of which will be goal of optimization.

3. FORMULATION OF THE OPTIMIZATION PROBLEM [2]

The goal of optimization process will be the minimization of square difference sum proposal response spectrum

( ) Ω

*

S

q&& and current spectrum

S

&_q&

( Ω , p )

. Both spectra will be expressed in points

ω

k

∈ { ω

1

^, ω

2

^, ^... ^, ω

N

} ^.

Firstly we assemble the discrete values of both spectra into the vectors

( )

( ) ( ) ( )

. ,

, , ,

,

² ^,¹

1

1 ,

* 2

* 1

*

* 1 2 ,

1

N

N N

N q q q

q N

N q

q q

q

S S S

Ω R Ω R

S R p p p Ω p

S ∈



 









 







=

∈

 





 





=

∈

 





 





=

ω ω ω

M M M

&

(24) The cost function can be written down in form

( ) p = [ S

&_q&

( Ω ^, p ) − S

^*&_q&

( ) Ω ]

^T

W [ S

&_q&

( Ω ^, p ) − S

^*_q&&

( ) Ω ] ^/ ( S

^*_q&&^T

( ) Ω WS

_q^*&&

( ) Ω ) ^,

ψ

(25)

(5)

where

W ∈ R

^{N ,}^N is diagonal matrix of weighting parameters which will be chosen as identity matrix. In the opposite case this matrix enables the preference of representative frequencies for better coincidence of both spectra in these frequencies. Let us introduce the functional vectors

( )

( ) ( )

, ,

3 3

1 1

32 32

12 12

31 31

11 11

 







 







+ +

+

+ +

+

+ +

+

= Ω

Ω

− Ω

+ Ω

−

Ω

− Ω

+ Ω

−

Ω

− Ω

+ Ω

−

t i D N t i D N t i N t i N

t i D t

i D t

i t

i

t i D t

i D t

i t

i

D D

k k

D D

k k

D D

k k

e c e

c e

c e c

e c e

c e

c e c

e c e

c e

c e c t

ω ω

a

M f

(26)

( )

( ) ( )

   





 





+ +

+

+ +

+

+ +

+

= Ω

Ω

− Ω

+ Ω

− −

Ω

− Ω

+ Ω

−

Ω

− Ω

+ Ω

−

t i D N t i D N t i N t i N

t i D t

i D t

i t

i

t i D t

i D t

i t

i

D D

k k

D D

k k

D D

k k

e c e

c e

c e c

e c e

c e

c e c

e c e

c e

c e c t

* , 3

* , 1

* 2 , 3

* 2 , 1

* 1 , 3

* 1 , 1

,

ω ω

b

M f

(27)

and the total functional vector

( ) ( )

( ^, ) ^.

, , 



 



 Ω

= Ω

Ω t

t t

b a

f f f

(28)

Then we can rewrite the relation (22) into form

( ^, ) ^max ( ^, ) ( ^, ) ^max ( ^, ) ^.

max

max 0,

2 ,

0

2

− Ω − Ω = Ω − Ω

Ω

= Ω

∈

t t t

S

^T

T t T

T T

q&&

p

t

a f

_a

b f

_b

p f

(29)

The proposal response spectrum of acceleration of the nuclear base for

D = 0 . 02

is depicted in Fig. 2 (blue color).

Fig. 2 Response spectra of acceleration

(6)

The generated acceleration corresponding to the achieved response spectrum (Fig. 2-green line) is depicted in Fig. 3

Fig. 3 Generated acceleration

4. CONCLUSION

The presented approach will be used for safety assessment of the nuclear power station Temelin especially its reactor and pipeline system. The requirement of three acceleration functions for kinematic excitation of base in three directions (horizontal x, z and vertical y) was posed by Nuclear Research Institute in Rez. Only one of these three accelerations was presented in this paper for the reason of space lack.

This contribution was supported by the Europe- an Regional Development Fund (ERDF), project

“NTIS - New Technologies for Information Society”, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.

REFERENCES

1. Dimarogonas A.: Vibration for engineers. Prentice Hall inc., 1996.

2. Dupal J.: Generating of the synthetical accelerogram from the proposal acceleration response spectrum, report 52 220-01-12, Univerzity of West Bohemia in Pilsen, 2012. (In Czech)

∫ ACCELERATION GENERATING CORRESPONDING TO THE PROPOSED RESPONSE SPECTRA