On the Nature of Random System Matrices in Structural Dynamics S. A - - PowerPoint PPT Presentation

SLIDE 1

On the Nature of Random System Matrices in Structural Dynamics

• S. ADHIKARI AND R. S. LANGLEY

Cambridge University Engineering Department Cambridge, U.K.

Nature of Random System Matrices – p.1/20

SLIDE 2

Outline of the Talk

• Introduction
• System randomness: Probabilistic approach
• Parametric and non-parametric modeling
• Maximum entropy principle
• Gaussian Orthogonal Ensembles (GOE)
• Random rod example
• Conclusions

Nature of Random System Matrices – p.2/20

SLIDE 3

Linear Systems

Equations of motion: M

✂✁ ✄✆☎ ✝ ✞

C

✟ ✁ ✄✆☎ ✝ ✞

Ky

✄✆☎ ✝ ✠

p

✄✆☎ ✝

(1) where M, C and K are respectively the mass, damping and stiffness matrices, y

✄✆☎ ✝

is the vector of generalized coordinates and p

✄✆☎ ✝

is the applied forcing function.

Nature of Random System Matrices – p.3/20

SLIDE 4

System Randomness

We consider randomness of the system matrices as M

✠

M

✞

• M

C

✠

C

✞

• C

and K

✠

K

✞

• K

✁

(2) Here,

✄✄✂ ✝

and

• ✄✄✂

✝

denotes the nominal (deterministic) and random parts of

✄✄✂ ✝

respectively.

Nature of Random System Matrices – p.4/20

SLIDE 5

Parametric Modeling

The Stochastic Finite Element Method (SFEM)

• Probability density function
• q

✄

q

✝

• f random

vectors q

✁ ✂ ✄

have to be constructed from the random fields describing the geometry, boundary conditions and constitutive equations by discretization of the fields.

• Mappings q

☎

G

✄✝✆

q

✞

q

✝✟✞ ✂ ✄ ☎ ✂ ✠☛✡ ✠

, where G denotes M

☞

C or K, have to be explicitly

• constructed. For an analytical approach, this step
• ften requires linearization of the functions.
• For Monte-Carlo-Simulation:

Re-assembly of the element matrices is required for each sample.

Nature of Random System Matrices – p.5/20

SLIDE 6

Non-parametric Modeling

• Direct construction of pdf of M

☞

C and K without having to determine the uncertain local parameters of a FE model.

• Soize (2000) has used the maximum entropy

principle for non-parametric modeling of system matrices in structural dynamics. Philosophy of Jayne’s Maximum Entropy Principle (1957):

• Make use of all the information that is given and

scrupulously avoid making assumptions about information that is not available.

Nature of Random System Matrices – p.6/20

SLIDE 7

Entropy

What is entropy? – A measure of uncertainty.

For a continuous random variable

• ✁

✁

, Shannon’s Measure of Entropy (1948):

✂ ✄

• ✄
• ✝

✝ ✠ ✄ ☎

• ✄
• ✝

✆ ✝

• ✄
• ✝

✞

• Constraint:

✟ ☎

• ✄
• ✝

✞

• ✠

✠

. Suppose only the mean is known. Additional constraint:

✟ ☎

• ✄
• ✝

✞

• ✠

✡

.

Nature of Random System Matrices – p.7/20

SLIDE 8

Maximum Entropy Principle

Construct the Lagrangian as

• ✠

✄ ☎

• ✄
• ✝

✆ ✝

• ✄
• ✝

✞

• ✄

✁✄✂ ☎

• ✄
• ✝

✞

• ✄

✠ ✄ ✁✄☎ ☎

• ✄
• ✝

✞

• ✄

✡ ✠ ☎ ✆ ✄

• ✄
• ✝

✝ ✞

• where

✆ ✄

• ✄
• ✝

✝ ✠ ✄

• ✄
• ✝

✆ ✝

• ✄
• ✝

✄ ✁✝✂

• ✄
• ✝

✄ ✁✝☎

• ✄
• ✝

✞ ✁✝✂ ✞ ✡ ✁✝☎

(3)

Nature of Random System Matrices – p.8/20

SLIDE 9

Maximum Entropy Principle

From the calculus of variation, for

• ✠
• it is

required that

✆ ✄

• ✄
• ✝

✝

must satisfy the Euler-Lagrange equation

✁ ✆ ✄

• ✄
• ✝

✝ ✁

• ✄
• ✝

✄ ✁ ✁

• ✁

✆ ✄

• ✄
• ✝

✝ ✁

• ✄
• ✝

✠

• (4)

Substituting

✆ ✄

• ✄
• ✝

✝

from (3), equation (4) results

✄ ✆ ✝

• ✄
• ✝

✄ ✠ ✄ ✁✝✂ ✄ ✁✝☎

• ✞

✁✝☎ ✠

• r
• ✄
• ✝

✠ ✂☎✄ ✆ ✝✟✞ ✠

That is, exponential distribution.

Nature of Random System Matrices – p.9/20

SLIDE 10

Soize Model (2000)

The probability density function of any system matrix (say

• ) is defined as

✂✁ ✄ ☎ ✄✆

• ✝

✝ ✠ ✞✠✟ ✡☞☛ ✌ ✍ ✎ ✄✆

• ✝

✝✑✏ ✄ ✄ ✞ ✒ ✓ ✆

• ✝

✝ ✝✕✔ ✆ ☎ ✖ ✒ ✗ ✘ ✄ ✄ ✙ ✄ ✠ ✞ ✚ ✁ ✄ ✝ ✚ ✛✢✜ ✣✤ ✒ ✄

• ✝

where

✏ ✄ ✠ ✄ ✚✢✥ ✝ ✆ ✠ ✌ ✠ ✆ ☎ ✎✦ ✧ ✙ ✄ ✠ ✞ ✚ ✁ ✄ ✚ ✠ ✌ ✠ ✆ ☎ ★ ✩ ✝✪✔ ✎✦ ✩ ✠ ✄✬✫ ☎ ✭ ✄ ✙ ✄ ✠ ✞ ✚ ✁ ✄ ✝ ✚

Nature of Random System Matrices – p.10/20

SLIDE 11

Soize Model (2000)

The ‘dispersion’ parameter

✁ ✄ ✠ ✠ ✚

• ✩

✄ ✠ ✄

• ✩

✄ ✄ ✙ ✄ ✠ ✝ ✞ ✄ ✛✢✜ ✣✤ ✒ ✆

• ✝

✝ ✩ ✛✢✜ ✣✤ ✒ ✄✆

• ✩

✝ ✝

and

• ✄

✠

• ✁

✆

• ✝

✄ ✆

• ✝

✁ ✁ ✆

• ✝

✁ ☎ ✦ ✩ ✞✠✟ ✡ ☛ ✌ ✍ ✎ ✄✆

• ✝

✝ ✠ ✠

if

✆

• ✝

✁ ★ ✠ ✄ ✂ ✝

• therwise 0. Here

★ ✠ ✄ ✂ ✝

is the subspace of

✠ ✄ ✂ ✝

constituted of all

✙ ✖ ✙

positive definite symmetric real matrices.

Nature of Random System Matrices – p.11/20

SLIDE 12

GOE (Gaussian Orthogonal Ensembles)

• 1. The ensemble (say H) is invariant under every

transformation H

☎

W

• HW where W is any
• rthogonal matrix.
• 2. The various elements

✁✄✂ ☎ ☞ ✆ ✝ ✞

are statistically independent.

• 3. Standard deviation of diagonals are twice that of

the off-diagonal terms,

✟ ✄ ✁ ✂ ✂ ✝ ✠ ✚ ✟ ✄ ✁ ✂ ☎ ✝ ✠ ✟ ☞ ✠ ✞ ✡ ✠ ✆

The probability density function

• H

✄

H

✝ ✠ ✒ ✗ ✘ ☛ ✄ ☞ ✛✢✜ ✣✤ ✒ ✄

H

✩ ✝ ✞ ✌ ✛✢✜ ✣✤ ✒ ✄

H

✝ ✞ ✏ ✍

Nature of Random System Matrices – p.12/20

SLIDE 13

GOE in Structural Dynamics

The equations of motion describing free vibration of a linear undamped system in the state-space Ay

✠

• where A

✁ ✂ ✩ ✠☛✡ ✩ ✠

is the system matrix. Transforming into the modal coordinates u

✠

• where

✁ ✂ ✩ ✠☛✡ ✩ ✠

is a diagonal matrix.

Nature of Random System Matrices – p.13/20

SLIDE 14

GOE in Structural Dynamics

Suppose the system is now subjected to

• constraints
• f the form

✄

C

✄

I

✝

u

☎

u

✩ ✠

• where C

✁ ✂ ✁ ✡ ✌ ✩ ✠ ✆ ✁ ✎

constraint matrix, I is the

• ✖
• identity matrix, u

☎

and u

☎

are partition of u. If the entries of C are independent, then it can be shown (Langley, 2001) that the random part of the sys- tem matrix of the constrained system approaches to GOE.

Nature of Random System Matrices – p.14/20

SLIDE 15

Random Rod

Equations of motion:

✁ ✁

• ✂
• ✄
• ✝

✁

• ✁
• ✠

✡ ✄

• ✝

✁ ✩

• ✁

☎ ✩ ✞

• ✝
• ✝

✁

(5) Boundary condition: fixed-fixed (U(0)=U(L)=0)

✡ ✄

• ✝

✠ ✡ ✂ ✆ ✠ ✞✄✂ ☎ ☎ ☎ ✄

• ✝

✝ ✂

• ✄
• ✝

✠ ✂

• ✂

✆ ✠ ✞✄✂ ✩ ☎ ✩ ✄

• ✝

✝ ☎✝✆ ✄

• ✝

are zero mean random fields. Deterministic mode shapes:

✞ ☎ ✄

• ✝

✠ ☞ ✟ ✠ ✝ ✄ ✆ ✥

• ✡

✁ ✝

where

☞ ✠ ✚ ✡ ✁ ✡ ✂

Nature of Random System Matrices – p.15/20

SLIDE 16

Random Rod

Consider the mass matrix in the deterministic modal coordinates:

✡

• ✂

☎ ✠ ✁ ✂ ✞ ✂ ✄

• ✝

✡ ✂ ✞ ☎ ✄

• ✝

✞

• ✞✄✂

☎ ✁ ✂ ✞ ✂ ✄

• ✝

☎ ☎ ✄

• ✝

✞ ☎ ✄

• ✝

✞

• ✠

✡

• ✂

✂ ✄ ✞✄✂ ☎ ☎ ✡

• ✂

☎

The random part

☎ ✡

• ✂

☎ ✠ ✟ ✁ ✂ ✞ ✂ ✄

• ✝

☎ ☎ ✄

• ✝

✞ ☎ ✄

• ✝

✞

• ✆

☎ ✡

• ✂

☎ ☎ ✡

• ✝✞

✟ ✠ ✁ ✂ ✁ ✂ ✞ ✂ ✄

• ☎

✝ ✞ ☎ ✄

• ☎

✝ ✞ ✝ ✄

• ✩

✝ ✞ ✞ ✄

• ✩

✝ ✠ ✡ ✞ ✄

• ☎

☞

• ✩

✝ ✞

• ☎

✞

• ✩

Nature of Random System Matrices – p.16/20

SLIDE 17

Case 1:

☎ ☎ ✄

• ✝

is

• correlated (white noise):

✠ ✡ ✞ ✄

• ☎

☞

• ✩

✝ ✠

• ☎
• ✄
• ☎

✄

• ✩

✝

Results:

• ✆

☎ ✡

• ✂

✂ ☎ ✡

• ✝

✝ ✟ ✠ ✠ ✁ ☞ ✧

• ☎

✁

,

✞ ✡ ✠ ✂

• ✆

☎ ✡

• ✂

✂ ☎ ✡

• ✂

✂ ✟ ✠ ✄ ☎ ☞ ✧

• ☎

✁

• ✆

☎ ✡

• ☎

✂ ☎ ✡

• ☎

✂ ✟ ✠ ✠ ✁ ☞ ✧

• ☎

✁

,

✆ ✡ ✠ ✞

• ✆

☎ ✡

• ☎

✂ ☎ ✡

• ✝✞

✟ ✠

• ✆

☎ ✡

• ☎

☎ ☎ ✡

• ☎

✝ ✟ ✠

• ,

✆ ✡ ✠ ✂

Nature of Random System Matrices – p.17/20

SLIDE 18

Case 2:

Case 2:

☎ ☎ ✄

• ✝

is fully correlated:

✠ ✡ ✞ ✄

• ☎

☞

• ✩

✝ ✠

• ✩

for

• ☎

☞

• ✩

✁ ✆

• ☞

✁ ✝

Results:

• ✆

☎ ✡

• ✂

✂ ☎ ✡

• ✝

✝ ✟ ✠ ✠ ✁ ☞ ✧

• ✩

✁ ✩

,

✞ ✡ ✠ ✂

• ✆

☎ ✡

• ✂

✂ ☎ ✡

• ✂

✂ ✟ ✠ ✠ ✁ ☞ ✧

• ✩

✁ ✩

• ✆

☎ ✡

• ☎

✂ ☎ ✡

• ☎

✂ ✟ ✠

• ,

✆ ✡ ✠ ✞

• ✆

☎ ✡

• ☎

✂ ☎ ✡

• ✝✞

✟ ✠

• ✆

☎ ✡

• ☎

☎ ☎ ✡

• ☎

✝ ✟ ✠

• ,

✆ ✡ ✠ ✂

Nature of Random System Matrices – p.18/20

SLIDE 19

Conclusions & Future Research

• Although mathematically optimal given

knowledge of only the mean values of the matrices, it is not entirely clear how well the results obtained from Soize model will match the statistical properties of a physical system.

• Analytical works show that GOE may be a

possible model for the random system matrices in the modal coordinates for very large and complex systems.

• The random rod analysis has shown that the

system matrices in the modal coordinates is close to GOE (but not exactly GOE) rather than the Soize model.

Nature of Random System Matrices – p.19/20

SLIDE 20

Conclusions & Future Research

• Future research will address more complicated

systems and explore the possibility of using GOE (or close to that, due to non-negative definiteness) as a model of the random system matrices.

• Such a model would enable us to develop a

general Monte-Carlo simulation technique to be used in conjunction with FE methods.

Nature of Random System Matrices – p.20/20