Comparison of CPU time for

## 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

where,

The Chebyschev inequality can also be written as

After substituting

where

or

If it is assumed that

Large numbers of samples of random variables

## 3. Dynamic analysis of finite element

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

where

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

1. The initial calculation

The matrices

The initial values

After selecting step

The stiffness matrix is defined as

The stiffness matrix inversion

2. Calculation of each step time

At time

At time

At time

Vectors

## 4. Analysis of mechanical vibration based on CG

Eq.11 can be expressed as

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

2. Calculate the first residual vector

and vector

where,

3. For

The process can be stopped only if

Vectors

The mean of

The variance of

Similarly, the mean and variance of the vector

At time

and

where,

Substituting Eq.25 into Eq.26, the stress for element

Substituting..samples of random variables

The mean of

The variance of

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

The partial derivative of Eq.14 with respect to

where

After

The partial derivative of Eq.30 with respect to

where

After

The displacement is expanded at the mean value point

where,

The variance of

The partial derivative of

The partial derivative of

The partial derivative of Eq.36 with respect to

The partial derivative of Eq.37 with respect to

The mean value and variance of the displacement are obtained at time

The partial derivative of Eq.27 with respect to

The partial derivative of Eq.40 with respect to

The stress is expanded at mean value point

where,

The variance of

## 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.

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,

DSFEM | TSFEM | CG |

19 seconds | 4 seconds | 14 seconds |

DSFEM | TSFEM | CG |

3 hours 8 minutes 17 seconds | 1 hour 45 minutes 10 seconds | 40 minutes 24 seconds |

## 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.