Chapter 14 Monte Carlo Simulation

14.1 Introduction

• Find several parameters

• Parameter follow the specific probability distribution

• Generate parameter following the probability distribution

14.2 Generation of random number

14.3 generation of random numbers following standard uniform

distribution

• The pseudo random number following standard uniform distribution:

mind by the limitation of computer used.

14.2.2 Random variables with nonuniform distribution

• Distribution function:

)(1 uFx x

Niux ii ..,,.........2,1;ln

14.2.3 Generation of discrete random variable

• X can take n+1 distinct values with mass function pX(xi), the cumulative distribution function is given by:

The discrete random number Xi and Xi+1 can be determined by:

i

jjXiX xpxF

0

)()(

14.3 Generation of jointly distributed random variable

14.3.1 independent variable

Density and distribution function

14.3.3 generation of correlated normal random variables

Vector of mean value

Covariance matrix

Express Xi as:

The mean values and standard deviation:

The mean values of Wj can be determined as

XaW 1

Example 14.9 the torque transmitted by a plate clutch with a single pair of friction surfaces, shown in Fig. 14.6, can be expressed as:

pressureuniformfordD

dDFfT

and

wearuniformfordDFf

T

)(4

)(4

22

33

14.4 computation of reliability

14.4.1 Sampling size and error in simulation

14.4.2 Example: reliability analysis of a straight line mechanism