Open access peer-reviewed chapter

Stochastic Finite Element Method in Mechanical Vibration

By Mo Wenhui

Submitted: November 13th 2010Reviewed: May 2nd 2011Published: September 9th 2011

DOI: 10.5772/21414

Downloaded: 2299

1. Introduction

Material properties, geometry parameters and applied loads of the structure are assumed to be stochastic. Although the finite element method analysis of complicated structures has become a generally widespread and accepted numerical method in the world, regarding the given factors as constants can not apparently correspond to the reality of a structure.

The direct Monte Carlo simulation of the stochastic finite element method(DSFEM) requires a large number of samples, which requires much calculation time and occupies much computer storage space [1]. Monte Carlo simulation by applying the Neumann expansion (NSFEM) enhances computational efficiency and saves storage in such a way that the NSFEM combined with Monte Carlo simulation enhances the finite element model advantageously [2]. The preconditioned Conjugate Gradient method (PCG) applied in the calculation of stochastic finite elements can also enhance computational accuracy and efficiency [3]. The TSFEM assumes that random variables are dealt with by Taylor expansion around mean values and is obtained by appropriate mathematical treatment [4, 14]. According to first-order or second-order perturbation methods, calculation formulas can be obtained [2, 5, 6,8, 9, 13, 15, 16]. The result is called the PSFEM and has been adopted by many scholars.

The PSFEM is often applied in dynamic analysis of structures and the second- order perturbation technique has been proved to be accurate and efficient. Dynamic reliability of a large frame is calculated by the SFEM and response sensitivity is formulated in the context of stiffness and mass matrix condensation [7]. Nonlinear structural dynamics are developed by the PSFEM. Nonlinearities due to material and geometrical effects have also been included [8]. By forming a new dynamic shape function matrix, dynamic analysis of the spatial frame structure is presented by the PSFEM [9].

It is significant to extend this research to the dynamic state. Considering the influence of random factors, the mechanical vibrations for a linear system are illustrated by using the TSFEM and the CG.

2. Random variable

Material properties, geometry parameters and applied loads of machines are assumed to be independent random variables, and are indicated as a1,a2,,ai,,an1. Their means areμ1,μ2,,μi,,μn1, and their variances are.. σi2,,σn12. When they are subject to normal distributions, the standard method used to simulate them is to take advantage of well-tested computer programs. When they are subject to unknown distributions, the sample of a1,a2,,ai,,an1can be generated from the following method:

P{|xμ|ε}σ2ε2E1

where,xis a random variable,μis the mean, σis the standard deviation,and εis an arbitrary positive number. Eq.1 is called the Chebyschev inequality.

The Chebyschev inequality can also be written as

P{|xμ|ε}1σ2ε2E2

After substitutingε=6σi,x=ai, Eq.2 becomes

P{|aiμi|6σi}0.9722E3

where

ai6σi+μiE4

or

ai6σi+μiE5

If it is assumed that zis a random number within the open interval (0,1), then

ai=6σiz+μiE6
or
ai=6σiz+μiE7

Large numbers of samples of random variables a1,a2,,ai,,an1are produced from Eqs.6 and 7 so as to resolve the stochastic finite element problem through Monte Carlo stimulation.

3. Dynamic analysis of finite element

For a linear system, the dynamic equilibrium equation is given by

[M]{δ¨}+[C]{δ˙}+[K]{δ}={F}E8

where {δ¨},{δ˙},{δ}are the acceleration, velocity and displacement vectors. [M],[K]and [C]are the global mass, stiffness and damping matrices obtained by assembling the element variables in global coordinate system.

In order to program easily, the comprehensive calculation steps of the Newmark method are as follows

1. The initial calculation

The matrices[K], [M]and [C]are formed.

The initial values{δt},{δ˙t},{δ¨t}are given.

After selecting stepΔtand parametersγ,β, the following relevant parameters are calculated:

γ0.50
β0.25(0.5+γ)2E9
b0=1β(Δt)2b1=γβΔt
b2=1βΔtE10
b3=12β1b4=γβ1
b5=Δt2(γβ2)E11
b6=Δt(1γ)
b7=γΔtE12

The stiffness matrix is defined as

[K˜]=[K]+b0[M]+b1[C]E13

The stiffness matrix inversion[K˜]1is solved.

2. Calculation of each step time

At timet+Δt, the load vector is defined as

{F˜t+Δt}={Ft+Δt}+[M](b0{δt}+
b2{δ˙t}+b3{δ¨t})E14
+[C](b1{δt}+b4{δ˙t}+b5{δ¨t})E15

At timet+Δt, the displacement vector is given by

{δt+Δt}=[K˜]1{F˜t+Δt}E16

At timet+Δt, the velocity vector and acceleration vector are obtained as

{δ¨t+Δt}=b0({δt+Δt}{δt})b2{δ˙t}b3{δ¨t}E17
{δ˙t+Δt}={δ˙t}+b6{δ¨t}+b7{δ¨t+Δt}E18

Vectors {δt+i1Δt},{δ˙t+i1Δt},{δ¨t+i1Δt}are solved at timet+i1Δt(i1=2,3,,n3)step-by -step.

4. Analysis of mechanical vibration based on CG

Eq.11 can be expressed as

[K˜]{δt+Δt}={F˜t+Δt}E19

N1samples of random variables a1,a2,,ai,,an1are produced. N1matrices [K˜]and N1Eq.14 are also generated. For a linear vibration, Eq.14 is the system of linear equations. The CG method is an effective method for solving the large system of linear equations according to the following steps:

1. First, select an approximate solution as the initial value

δ(0)t+Δt=(δ(0)(t+Δt)1,δ(0)(t+Δt)2,,δ(0)(t+Δt)N)TE20

2. Calculate the first residual vector

r(0)={F˜t+Δt}[K˜]δ(0)t+ΔtE21

and vector

p(0)=[K˜]Tr(0)E22

where, [K˜]Tis the transposed matrix of [K˜]

3. Fori˜=0,1,2,,n21, iterate step-by-step as follows

αi˜=([K˜]p(i˜),r(i˜))([K˜]p(i˜),[K˜]p(i˜))=(p(i˜),[K˜]Tr(i˜))([K˜]p(i˜),[K˜]p(i˜))=([K˜]Tr(i˜),[K˜]Tr(i˜))([K˜]p(i˜),[K˜]p(i˜))E23
{δt+Δt}(i˜+1)={δt+Δt}(i˜)+αi˜p(i˜)E24
r(i˜+1)=r(i˜)αi˜[K˜]p(i˜)E25
βi˜+1=([K˜]Tr(i˜+1),[K˜]Tr(i˜+1))([K˜]Tr(i˜),[K˜]Tr(i˜))E26
p(i˜+1)=[K˜]Tr(i˜+1)+βi˜+1p(i˜)E27

The process can be stopped only if rn2is small enough.

Vectors{δt+Δt}1,{δt+Δt}2,,{δt+Δt}N1are solutions of N1Eqs.14.

The mean of {δt+Δt}is given by

μ{δt+Δt}={δt+Δt}1+{δt+Δt}2++{δt+Δt}N1N1E28

The variance of {δt+Δt}is given by

Var{δt+Δt}=1N11i=1N1({δt+Δt}iμ{δt+Δt})2E29

Similarly, the mean and variance of the vector {δt+i1Δt}can be solved for at timet+i1Δt(i1=2,3,,n3)step-by-step.

At timet=t+i2Δt(i2=1,2,,n4),the strain and stress vectors for element dare

{ε}=[B]{δtd}E30

and

{σ}=[D]{ε}E31

where,[D]=the material response matrix of element d,[B]=the gradient matrix of elementdand {δtd}=the element dnodal displacement vector at time t.

Substituting Eq.25 into Eq.26, the stress for element dis given by

{σ}=[D][B]{δtd}E32

Substituting..samples of random variables a1,a2,,ai,,an1into Eq.27, the vectors {σ}1,{σ}2,,{σ}N1can be obtained.

The mean of {σ}is given by

μ{σ}={σ}1+{σ}2++{σ}N1N1E33

The variance of {σ}is given by

Var{σ}=1N11i=1N1({σ}iμ{σ})2E34

The CG method belongs to method of iteration with the advantage of quick convergence. For practical purpose, PCG is applied to accelerate the convergence.

5. Analysis of mechanical vibration based on TSFEM

Independent random variables of the system are regarded as a1,a2,,ai,,an1.

The partial derivative of Eq.14 with respect toaiis given by

{δt+Δt}ai=[K˜]1({F˜t+Δt}ai[K˜]ai{δt+Δt})E35

where

{F˜t+Δt}ai={Ft+Δt}ai+[M](b0{δt}ai+b2{δ˙t}ai+b3{δ¨t}ai)+E36
[M]ai(b0{δt}+b2{δ˙t}+b3{δ¨t})E37
+[C](b1{δt}ai+b4{δ˙t}ai+b5{δ¨t}ai)+[C]ai(b1{δt}+b4{δ˙t}+b5{δ¨t})E38

After {δt}ai|t=0=q0,{δ˙t}ai|t=0=q˙0,{δ¨t}ai|t=0=q¨0are given, Eq.31 can be calculated.

The partial derivative of Eq.30 with respect toaiis given by

2{δt+Δt}ai2=[K˜]1
(2{F˜t+Δt}ai22[K˜]ai{δt+Δt}ai2[K˜]ai2{δt+Δt})E39

where

2{F˜t+Δt}ai2=2{Ft+Δt}ai2+2[M]ai2(b0{δt}+b2{δ˙t}+b3{δ¨t})+E40
2[M]ai(b0{δt}ai+b2{δ˙t}ai+b3{δ¨t}ai)
+[M](b02{δt}ai2+b22{δ˙t}ai2+b32{δ¨t}ai2)E41
+2[C]ai2(b1{δt}+b4{δ˙t}+b5{δ¨t})+2[C]ai(b1{δt}ai+b4{δ˙t}ai+b5{δ¨t}ai)+[C](b12{δt}ai2+b42{δ˙t}ai2+b52{δ¨t}ai2)E42

After2{δt}ai2|t=0=r0,2{δ˙t}ai2|t=0=r˙0,2{δ¨t}ai2|t=0=r¨0are given, Eq.33 can be calculated.

The displacement is expanded at the mean value pointa¯=(a¯1,a¯2,,a¯i,,a¯n1)Tby means of a Taylor series. By taking the expectation operator for two sides of above Eq.11, the mean of the displacement is obtained as

μ{δt+Δt}{δt+Δt}|a=a¯+12i=1n12{δt+Δt}ai2|a=a¯σi2E43

where, μ{δt+Δt}expresses the mean value of δt+Δt.

The variance of δt+Δtis given by

Var{δt+Δt}i=1n1({δt+Δt}ai|a=a¯)2σi2E44

The partial derivative ofδ¨t+Δtwith respect toaiis given by

{δ¨t+Δt}ai=b0({δt+Δt}ai{δt}ai)b2{δ˙t}aib3{δ¨t}aiE45

The partial derivative ofδ˙t+Δtwith respect toaiis given by

{δ˙t+Δt}ai={δ˙t}ai+b6{δ¨t}ai+b7{δ¨t+Δt}aiE46

The partial derivative of Eq.36 with respect toaiis given by

2{δ¨t+Δt}ai2=b0(2{δt+Δt}ai22{δt}ai2)
b22{δ˙t}ai2b32{δ¨t}ai2E47

The partial derivative of Eq.37 with respect toaiis given by

2{δ˙t+Δt}ai2=2{δ˙t}ai2+
b62{δ¨t}ai2+b72{δ¨t+Δt}ai2E48

The mean value and variance of the displacement are obtained at time t+i1Δt(i1=2,3,,n3)step-by-step.

The partial derivative of Eq.27 with respect toaiis given by

{σ}ai=[D]ai[B]{δtd}+[D][B]ai{δtd}+[D][B]{δtd}aiE49

The partial derivative of Eq.40 with respect toaiis given by

2{σ}ai2=2[D]ai2[B]{δtd}+
2[D]ai[B]ai{δtd}+2[D]ai[B]{δtd}aiE50
+[D]2[B]ai2{δtd}+2[D][B]ai{δtd}aiE51
+[D][B]2{δtd}ai2E52

The stress is expanded at mean value pointa¯=(a¯1,a¯2,,a¯i,,a¯n1)Tby means of a Taylor series. By taking the expectation operator for two sides of the above Eq.27, the mean of stress is obtained as

μ{σ}{σ}|a=a¯+12i=1n12{σ}ai2|a=a¯σi2E53

where, μ{σ}expresses the mean value ofσ.

The variance ofσis given by

Var{σ}i=1n1({σ}ai|a=a¯)2σi2E54

6. Numerical example

Figure 1 shows a four-bar linkage, or a crank and rocker mechanism. The establishment of differential equation system can be found in literature 10,11,12.The length of bar 1 is 0.075m, the length of bar 2 is 0.176m, the length of bar 3 is 0.29m,and the length of the bar 4 is0. 286m, the diameters of three bars are 0.02m. The torque T is 4Nm, the load F1 is 20sint N. The three bars are made of steel and they are regarded as three elements. Considering the boundary condition, there are 13 unit coordinates. Young’s modulus is regarded as a random variable. For numerical calculation, the means of the Young’s modulus within the three bars are.2×1011. N/m2and the variances of the Young’s modulus are1011N2/m4.

Figure 2 shows the mean of the displacement at unit coordinate 11. Unit coordinate 11 is the deformation of the upper end of bar 3 in the vertical direction. The DSFEM simulates 1000 samples. The TSFEM produces an error of less than 0.5%. The CG produces an error of less than 0.1%. Figure 3 shows the variance of the displacement at unit coordinate 11. TSFEM produces an error of less than 1.0%. CG produces an error of less than 0.4%.Figure 4 shows the mean of stress at the top of bar 3. The TSFEM produces an error of less than 0.85%.The CG produces an error of less than 0.13%.Figure 5 shows the variance of stress at the top of bar 3. The TSFEM produces an error of less than 1%. The CG produces an error of less than 0.3%.The results obtained by the CG method and the TSFEM are very close to that obtained by the DSFEM. Table 1 indicates the comparison of CPU time when the mechanism has operated for six seconds.

Figure 6 shows a cantilever beam. The length, the width, the height, the Poisson’s ratio,the Young’s modulus and the load F are assumed to be random variables. Their means are 1m, 0.1m, 0.05m, 0.2,2×1011N/m2,100N.Their standard deviation are 0.2, 0.1, 0.1, 0.01, 109, 0.1. Load subjected to the cantilever beam is Fsin(100t)N. It is divided into 400 rectangle elements that have 505 nodes. Figure 7 shows the mean of vertical displacement at node 505. DSFEM simulates 100 samples. The result obtained by the TSFEM produces an error of less than 2%. CG produces an error of less than 0.5%. Figure 8 shows the variance of vertical displacement at node 505.The TSFEM produces an error of less than 3.0%. CG produces an error of less than 0.8%.Figure 9 shows the mean of horizontal stress at node 5. The TSFEM produces an error of less than 2.4%. CG produces an error of less than 0.9%. Figure 10 shows the variance of horizontal stress at node 5. The TSFEM produces an error of less than 3.2%. CG produces an error of less than 1.3%. Table 2 indicates the comparison of CPU time when the cantilever beam has operated for six seconds.

Figure 1.

A four-bar linkage

Figure 2.

The mean of displacement at unit coordinate 11 for σ E 2 = 10 11

Figure 3.

The variance of displacement at unit coordinate 11 for σ E 2 = 10 11

Figure 4.

The mean of stress at the top of bar 3 for σ E 2 = 10 11

Figure 5.

The variance of stress at the top of bar 3 for σ E 2 = 10 11

DSFEMTSFEMCG
19 seconds4 seconds14 seconds

Table 1.

Comparison of CPU time for σE2=1011

Figure 6.

A cantilever beam

Figure 7.

The mean of vertical displacement at node 505

Figure 8.

The variance of vertical displacement at node 505

Figure 9.

The mean of horizontal stress at node 5

Figure 10.

The variance of horizontal stress at node 5

DSFEMTSFEMCG
3 hours 8 minutes 17 seconds1 hour 45 minutes 10 seconds40 minutes 24 seconds

Table 2.

Comparison of CPU time

7. Conclusions

Considering the influence of random factors, the mechanical vibration in a linear system is presented by using the TSFEM. Different samples of random variables are simulated. The combination of CG method and Monte Carlo method makes it become an effective method for analyzing the vibration problem with the characteristics of high accuracy and quick convergence.

© 2011 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

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Mo Wenhui (September 9th 2011). Stochastic Finite Element Method in Mechanical Vibration, Recent Advances in Vibrations Analysis, Natalie Baddour, IntechOpen, DOI: 10.5772/21414. Available from:

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