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ftf

PURPOSE

Calculates the fatigue failure time distribution.

SYNOPSIS

[time,F]=ftf(e,d,sigma2,sigma_D,number)

DESCRIPTION

``` FTF  Calculates the fatigue failure time distribution.

F(t) = P[ T^f <= t ].

CALL: [t,F] = ftf(e,d,s2,s2D,number);

where

t      = an one column matrix with times  t,
F      = the distribution function F(t),
e      = epsilon, a constant,
d      = the damage intensity,
s2     = the residual variance,
s2D    = the variance of the total damage,
number = plot parameter (optionalinput argument);
if equal to 1 the distribution function will be plotted.

Example:
RFC = tp2rfc(tp);
[t,F] = ftf(5.5e-10,cc2dam(RFC,5)/T,0.06,0.5);

CROSS-REFERENCE INFORMATION

This function calls:
 axis Control axis scaling and appearance. diff Difference and approximate derivative. erf Error function. plot Linear plot. title Graph title. xlabel X-axis label.
This function is called by:
 Chapter4 % CHAPTER4 contains the commands used in Chapter 4 of the tutorial itmkurs_lab1 Script to computer exercises 1

SOURCE CODE

```001 function [time,F]=ftf(e,d,sigma2,sigma_D,number)
002 %FTF  Calculates the fatigue failure time distribution.
003 %
004 %       F(t) = P[ T^f <= t ].
005 %
006 %  CALL: [t,F] = ftf(e,d,s2,s2D,number);
007 %
008 %  where
009 %
010 %        t      = an one column matrix with times  t,
011 %        F      = the distribution function F(t),
012 %        e      = epsilon, a constant,
013 %        d      = the damage intensity,
014 %        s2     = the residual variance,
015 %        s2D    = the variance of the total damage,
016 %        number = plot parameter (optionalinput argument);
017 %                 if equal to 1 the distribution function will be plotted.
018 %
019 % Example:
020 %   RFC = tp2rfc(tp);
021 %   [t,F] = ftf(5.5e-10,cc2dam(RFC,5)/T,0.06,0.5);
022 %
024
025 % Tested on: matlab 5.3
026 % History:
027 % Revised by PJ 10-Jan-2000
028 %   updated for WAFO
029 % Original version from FAT by Mats Frendahl
030 %   Copyright 1993, Mats Frendahl, Dept. of Math. Stat., University of Lund.
031
032 timefailurecenter=1/d/e; number_of_t=99;
033 delta=timefailurecenter/number_of_t;
034 time=.5*timefailurecenter:delta:1.5*timefailurecenter;
035 F=.5+.5*erf(log(d*time.*e)/sqrt(sigma2));
036
037 number_of_x=99; x=-4:8/number_of_x:4; phi_x=phi(x,0,1);
038 I=0;
039 for i=1:length(time)
040     t=log(d*e*time(i)+e*sigma_D*sqrt(time(i))*x)./sqrt(sigma2);
041     y=(.5+.5*erf(t/sqrt(2))).*phi_x;
042     I(i)=trapez(x,y);
043 end
044
045 if nargin==5
046    if number==1
047       plot(time,I)
048       axis([min(time) max(time) -0.1 1.1])
049       title('P[ T^f <= t ]'),xlabel('t')
050       axis;
051    end
052 end
053
054 function p=phi(x,m,v,nr)
055 %  Evalutes the phi-/Phi-function, density/distribution function
056 %  for a Gaussian variable with mean  m  and variance  v.
057 %
058 %  CALL: f = phi(x,m,v,nr)
059 %
060 %  where
061 %
062 %        f  = the density/distribution function,
063 %        x  = a vector of x-values,
064 %        m  = the mean,
065 %        v  = the variance,
066 %        nr = plot parameter  (optional input argument)
067 %
068 %             0 => f = density function,
069 %             1 => f = distribution function.
070
071 %  Copyright 1993, Mats Frendahl, Dept. of Math. Stat., University of Lund.
072
073 if nargin==3, nr=0; end
074
075 p=1/sqrt(2*pi*v)*exp(-0.5*(x-m).^2/v);
076
077 if (nargin==4) & (nr==1)
078   p=(1+erf((x-m)./sqrt(2*v)))./2;
079 end
080
081 function integral=trapez(x,y)
082 %  Calculates an integral according to the trapezodial rule given two
083 %  vectors,  x  and  y,  with  x_k-  and  y_k-values.
084 %
085 %  CALL: I = trapez(x,y)
086 %
087 %  where
088 %
089 %        x = a vector with x_k-values,
090 %        y = a vector with y_k-values.
091
092 %  Copyright 1993, Mats Frendahl, Dept. of Math. Stat., University of Lund.
093
094 integral=.5*(y(2:length(y))+y(1:length(y)-1))*diff(x)';
095```

Mathematical Statistics
Centre for Mathematical Sciences
Lund University with Lund Institute of Technology

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