Explicit iteration MCS for random vibration of nonlinear...

Explicit iteration MCS for random vibration

of nonlinear systems

Cheng Su, Huan Huang, Haitao Ma

School of Civil Engineering and Transportation

South China University of Technology

Guangzhou 510640, China

December 12, 2014

The 6th Kwan-Hua Forum, Innovations and Implementations in Earthquake

Engineering Research, December 12-14, 2014, Shanghai, China


Contents

1 Introduction

2 Explicit expressions of dynamic responses : linear systems

3 Explicit iteration MCS : nonlinear systems

4 Numerical examples

5 Concluding remarks


Method

Comparision from different points of view

1 2 3 4 5

Stationary Non-

stationary

Wide

band

Narrow

band

White

noise

Non-white

noise Gaussian

Non-

Gaussian

Small

number of

DOFs

Large

number of

DOFs

FPK equation

method √ × √ × √ × √ × √ ×

Stochastic

average

method

√ × √ × √ √ √ √ √ ×

Moment

equation

method

√ √ √ × √ × √ √ √ ×

Stochastic

perturbation

method

√ × √ √ √ √ √ √ √ ×

Equivalent

linearization

method

√ √ √ √ √ √ √ × √ √

Equivalent

nonlinear

system method

√ × √ × √ × √ × √ ×

Nonlinear random vibration analytical methods

1 Introduction


Contents

1 Introduction






1 1

where

t

t X t

( )

, ( ) ( )

0 I 0,

V HV P

YV P W

Y

H WM K M C E

State equation

Recurrence formula

1 1 1 2

2 1

1

2 1

2

1 2

where

e

i i i i

t

X X

i l

t

t

( , , , )

(I ) /

( I) /

H

V TV Q Q

T

Q T H W TH W

Q T H W H W

,0 0 ,1 1 ,

1,0 1

1,1 2

2,0 1,0

2,1 2 1

2,2 1,1

,0 1,0

,1 1,1

( 1, 2, , )

where

( 1)

( 2)

i i i i i i

i i

i i

X X X

i l

i

i

V A A A

A Q

A Q

A TA

A TQ Q

A A

A TA

A TA

, 1, 1

(3 )

(2 )i j i j

i l

j i

A A

Explicit expressions of dynamic

responses

( )MY CY KY MEX t

Motion Equation

( )

0( ) e (0) e ( )d

tt tt

H HV V P

Solution to state equation

Can also be obtained using

the other step-by-step

integration approaches

2 Explicit expressions of dynamic responses: linear systems

Ground acceleration

process


X

t1 t2 t3 t4

tt0

1

tl…A3,2

A4,2

A l,2

A2,2

Explicit expressions of dynamic responses

,0 0 ,1 1 , ( 1,2, , )i i i i i iX X X i l V A A A

The responses due to a

unit impulse at time t0 Vi=Ai,0 (i=1,2,3,…,l)


unit impulse at time t2 Vi=Ai,2 (i=2,3,…,l)

X

t1 t2 t3 t4

tt0

1

tl…A1,0

A2,0

A3,0

A4,0

A l,0

X

t1 t2 t3 t4

tt0

1

tl…A2,1

A3,1

A4,1

A l,1

A1,1

A3,1

A l-1,1A1,1

A2,1

X 0 X 1 X 2 X 3 … X l- 2 X l- 1 X l

t 1 A 1,0 A 1,1

t 2 A 2,0 A 2,1 A 2,2

t 3 A 3,0 A 3,1 A 3,2 A 3,3

t 4 A 4,0 A 4,1 A 4,2 A 4,3

… … … … … …

t l -2 A l -2,0 A l -2,1 A l -2,2 A l -2,3 … A l -2,l -2

t l -1 A l -1,0 A l -1,1 A l -1,2 A l -1,3 … A l -1,l -2 A l -1,l -1

t l A l ,0 A l ,1 A l ,2 A l ,3… A l ,l -2 A l ,l -1 A l ,l

时刻系数向量

Time Coefficient matrices


unit impulse at time t1 Vi=Ai,1 (i=1,2,3,…,l)


系数向量时刻

0X 1X 2X 3X 2lX 1lX lX

1t 1,0A 1,1A

2t 2,0A 2,1A 1,1A

3t 3,0A 3,1A 2,1A 1,1A

2lt 2,0lA 2,1lA 3,1lA 4,1lA 1,1A

1lt 1,0lA 1,1lA 2,1lA 3,1lA 2,1A 1,1A

lt ,0lA ,1lA 1,1lA 2,1lA 3,1A 2,1A 1,1A

The same as those matrices in the second column



■Computational cost=Two deterministic time-history response analyses

■ Storage capacity=Storage of the first two columns of matrices

Time Coefficient matrices

Obtained by a deterministic time-history response analysis with a unit impulse at time t0

Obtained by a deterministic time-history response analysis with a unit impulse at time t1


C. Su et al., Acta Mech. Sinica, 2010, 42(3): 512-520

C. Su et al., Struct. Eng. Mech., 2014, 52 (2): 239-260


Extract the responses at time t1 ,

t2 , …, tl to form the coefficient

matrices A1,0， A2,0， … ， Al,0

Time-history response analysis

with a unit impulse at time t1

Finite element models of PKPM, SAP2000, ETABS, MIDAS

Time-history response analysis

with a unit impulse at time t0

Extract the responses at time t1 ,

t2 , …, tl to form the coefficient

matrices A1,1 , A2,1 , … , Al,1

X

t1 t2 t3 tl-1

tt0

1

tl…

X

t1 t2 t3

tt0

1

tl… tl-1





1 Introduction





Contents


Dimension-

reduced

iteration

Quasi-linear motion equation

= ( )

( )= ( )- ( , )

MU CU KU F

F F f U U

t

t t

Explicit iteration for dynamic responses

(0)

-1

( ) ( 1)

,0 0 ,1 1 , -1 -1 ,

Initial values ( )= ( )

= + [ - ( )]

( =1,2, , ; =1,2, )

i i

j j

i i i i i i i i i i

i l j

-

：f V f V

V A F A F A F A F f V

( , ) ( ) MU CU KU f U U F t

Nonlinear motion equation

,0 0 ,1 1 , -1 -1 ,

=1,2, ,

i i i i i i i i i

i l

V A F A F A F A F

（）

Explicit expressions of dynamic

responses (in form)

1 (0) 1

-1

1 ( ) 1 1 1

,0 0 ,1 1 , -1 -1

1 1 ( -1)

,

2 2 2 2

,0 0 ,1 1 , -1 -1

2 1

,


=

+ [ - ( )]

=

+ [ - ( )]

i i

j

i i i i i i

j

i i i i

i i i i i i

i i i i

：f V f V

V A F A F A F

A F f V

V A F A F A F

A F f V

( =1,2, , ; =1,2, )i l j

Dimension-reduced iteration process

DOFs

associated with

nonlinear terms

DOFs

non-associated

with nonlinear

terms



DOFs

associated with

nonlinear terms

( ) ( =1,2, , )Fk t k N

N samples of non-stationary

random excitation

( , ) ( )

( =1,2, , )

MU CU KU f U U Fk k k k k k t

k N

Nonlinear motion equation

1 (0) 1

, , -1

1 ( ) 1 1 1 1 1 ( -1)

, ,0 ,0 ,1 ,1 , -1 , -1 , , ,

2 2 2 2 2 1

, ,0 ,0 ,1 ,1 , -1 , -1 , , ,


= + [ - ( )]

= + [ - ( )]

k i k i

j j

k i i k i k i i k i i i k i k i

k i i k i k i i k i i i k i k i

：f V f V



( =1,2, , ; =1,2, ; =1,2, , )i l j k N

Dimension-reduced iteration for the kth sample analysis

Calculated only once and remain the

same throughout the solution process

Dimension-

reduced iteration

DOFs

non-associated

with nonlinear

terms



1 Introduction





Contents


ETABS FE model: 103 storeys, a total height of 432m, 19,813 nodes, 18,597 beam

elements, 14,074 shell elements, and a total number of DOFs of 154,583

Maximum value of response spectrum αmax=0.08, ground predominant period Tg=0.35s,

the first three natural periods T1=7.29s (y-direction), T2=7.21s (x-direction), T3=2.67s

(torsion)

4 Numerical examples (1) : Linear system


Time elapsed by the present method

Total number

of DOFs

Elapsed time of

two time-history

response analyses

（ETABS）

Number of

responses

Number of

samples

Elapsed time of

random

simulation

154,583 21 min 10 sec 207 1,000 4 min 4 sec



4 Numerical examples (2) : Duffing system

1 2 50

51 52 100

7

1 2 50

7

51 52 100

1 50 51 100

= =2500kg,

= = = =3000kg;

7.5 10 kN/m,

8 10 kN/m;

533.3, 500.0;

Damping ration 0.05 for the initial linear system

m m m

m m m

k k k

k k k

=

Structural parameters

31 21 1 1 1

3 32 32 2 2 1 1 1 2

3 33 43 3 3 2 2 2 3

3 399 10099 99 99 98 98 98 99

3 3100100 100 100 99 99 99 100

f ，

u uk y y

u uk y k y y

u uk y k y y

u uk y k y y

uk y k y y

Nonlinear restoring forces

100-DOF Duffing system

1u

2u

99u

100u

South China University of Technology C. Su et al., J. Vib. Eng., 2014, 27(2): 1-7



1u

2u

99u

100u

4 2 2 2

02 2 2 2 2 2

2 3

0

4 ( )

( ) 4

where

15.708rad/s, 0.6, 0.05m / s

g g g

x

g g g

g g

S S

S

Kanai-Tajimi Spectrum for ground

acceleration

1 1

1 2

( )

2 3

1 2 3

( / ) 0

( ) 1

where

6.0s 18.0s 30.0s 0.18

2

2

-a t -t

t t t t

g t t t t

e t t t

t t t a

，，，

Uniform modulation function for non-

stationary ground motion

South China University of Technology C. Su et al., J. Vib. Eng., 2014, 27(2): 1-7


Displacement response u1



An excitation sample

Number of iteration in each time step


Standard deviation of displacement u1


Evolutionary PDF of displacement u1



Time elapsed by analysis of 100-DOF system

Time elapsed by analysis of 1,000-DOF system



First-order dynamic eigenvalue

Second-order dynamic eigenvalue

Third-order dynamic eigenvalue

Dynamic eigenvalues at t=18.55s (s-2)





with local nonlinearities

1 2 50 51 52 100

7

1 2 50

7

51 52 100

90 100

= =2500kg, = = = =3000kg;

7.5 10 kN/m,

8 10 kN/m;

1000.0;


m m m m m m

k k k

k k k

=


Nonlinear restoring forces

1 1 2

2 2 3

3

90 90 90 90 90 91

391 9290 90 90 91

3100100100 100 100

0

0

y u u

y u u

k y y u u

u uk y y

uyk y

f ，

Coefficients

of

nonlinear

restoring

forces

( )g

X t

4 Numerical examples (3) : Duffing system with local nonlinearities

DOFs

associated with

nonlinear terms


( )g

X t

4 2 2 2

02 2 2 2 2 2

2 3

0

4 ( )

( ) 4

where

15.708rad/s, 0.6, 0.05m / s

g g g

x

g g g

g g

S S

S


acceleration

1 1

1 2

( )

2 3

1 2 3

( / ) 0

( ) 1

where

6.0s 18.0s 30.0s 0.18

2

2

-a t -t

t t t t

g t t t t

e t t t

t t t a

，，，




with local nonlinearities


Coefficients

of

nonlinear

restoring

forces

DOFs

associated with

nonlinear terms

Dynamic eigenvalues at t=18.55s (s-2)






1 2 50

51 52 100

7

1 2 50

7

51 52 100

= =6000kg,

= = = =5000kg;

8 10 kN/m,

7.5 10 kN/m;


m m m

m m m

k k k

k k k

=


Hysteretic restoring forces

1

1 1

( , ) (1 )( 1,2, ,100)

| | | | | |

where

0.2 1 400m =300m

1 ( 1,2, ,100)

i i

i i i i i i i i i

i i i i i i i i i i

i i i i

i

f y z k y k zi

z A y y z z y z

A

i

，，，，

An n-storey shear structure

( )g

X t

4 Numerical examples (4) : hysteretic system

South China University of Technology C. Su et al., Earthq. Struct., 2014, 7(2): 119-139

( )g

X t


4 2 2 2

02 2 2 2 2 2

3 2 3

0

4 ( )

( ) 4

where

15.708rad/s, 0.6, 1.574 10 m / s

g g g

x

g g g

g g

S S

S


acceleration

1 1

1 2

( )

2 3

1 2 3

( / ) 0

( ) 1

where

6.0s 18.0s 30.0s 0.18

2

2

-a t -t

t t t t

g t t t t

e t t t

t t t a

，，，



South China University of Technology C. Su et al., Earthq. Struct., 2014, 7(2): 119-139

An n-storey shear structure

Restoring force of the first storey ( 0.8)i

Restoring force of the first storey ( 0.2)i






Dynamic eigenvalues at t=18.48s (s-2 )



1 Introduction





Contents


The non-stationary random vibration analysis of nonlinear systems with multiple degrees of freedom is one of the most difficult topics in the field of nonlinear random vibration. A new approach to this highly challenging problem is developed in the present study.

Two explicit iteration schemes based on precise integration and Newmark-β integration are proposed for fast MCS of non-stationary random vibration of nonlinear systems, including Duffing systems and hysteretic systems.

The coefficient matrices used for the solution need to be calculated just once and remain unchanged for different time steps and different samples. Therefore, the solution efficiency can be improved greatly, effectively breaking the bottleneck in MCS.

The proposed method provides a solid foundation for the application of random vibration analysis to large-scale nonlinear engineering problems.

5 Concluding Remarks


Explicit iteration MCS for random vibration of nonlinear...

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