NONLINEAR RANDOM DISPLACEMENT AND FATIGUE ESTIMATES
USING PDF TRANSFORMATIONS
K. A. Sweitzer†‡ and N. S. Ferguson‡
ITT Industries Space Systems Division LLC, Rochester, NY, USA†
Institute of Sound and Vibration Research, University of Southampton, UK‡
Email: [email protected]
INTRODUCTION
The estimation of displacement and fatigue life of nonlinear structures subjected to random
excitation presents several analysis complications that will be discussed. Recent nonlinear
random vibration experiments performed at Wright Patterson Air Force Base (WPAFB) [1]
have generated data that has been analyzed in the time and frequency domains to model this
complicated behavior. This paper will discuss how the probability density function transform
and inverse distribution function methods have been applied.
Probability density function (PDF) transformations have been shown to be useful for
estimating nonlinear random fatigue damage, but preliminary applications were limited [2].
Analysis of the WPAFB data has employed additional normal PDF, peak PDF and nonlinear
peak PDF transformations. These PDF transforms were used to estimate rates of zero
crossings and more importantly, two dimensional rainflow matrices (RFM) of total stress.
The inverse distribution function method has been shown to be very effective for estimating
linear to nonlinear functional relationships. Several functional relationships of linear to
nonlinear displacement and stress are presented, illustrating the method using response and
peak distribution functions.
Standard closed form PDF transformation methods require the linear to nonlinear functional
relationships to be both differentiable and invertible. Numerical PDF transformation methods
have been developed that require only differentiable functional relationships. Examples of the
numerical methods will be presented for a two step serial application of the PDF transform.
Analyses of the WPAFB data and subsequent ODE numerical experiments with Duffing
equation systems have shown that total stress peak PDFs (or 2D RFMs) can be estimated
using the PDF transform method. These estimated RFMs and the resultant time to failure will
be compared to estimates using the raw WPAFB experimental results.
PROBABILITY FUNCTIONS FOR NONLINEAR SYSTEMS
The response PDFs of linear systems to Gaussian (normal) random inputs has been shown to
be normal. If the system is dominated by one mode, the distribution of response peaks can be
assumed (to a first approximation) to have a Rayleigh PDF [3]. A generalized peak PDF for
the response of any linear system has also been developed [3, 4]. If the system is nonlinear,
or the input distribution is not normal, the closed form equations for the response and peak
PDFs are not valid. In these nonlinear cases, the PDF transform method can be used to
estimate the resultant PDFs. To review, the basic transform of a PDF p(x) of a know function
to a PDF p(y) of another function is:
p( y ) =
p( x )
dy dx
(1)
The key to this equation is determining a one-to-one relationship between x and y, i.e.:
y = g ( x)
(2)
where x and y are the linear and nonlinear value respectively. Previous attempts to determine
this functional relationship were based on a power law model using the RMS value of
response at different input levels [2]. Research has uncovered the inverse distribution
function (IDF) method to estimate linear to nonlinear functional relationships using data from
the full range of distribution functions of x and y [5, 6].
Consider the distribution function of x (or F(x)) (also referred to as the cumulative distribution
function (CDF)) which is defined as the probability of occurrence ρ up to a value x:
F ( x ) = Probability ( X ≤ x ) = ρ
The distribution function monotonically increases from F ( −∞ ) = 0 to F ( ∞ ) = 1 .
(3)
The
distribution function is related to the PDF by:
x
F ( x ) = ∫ p (α ) dα
−∞
(4)
The inverse distribution function (IDF) or F −1 ( ρ ) determines the corresponding x value of a
distribution function given a value of probability of occurrence:
F −1 ( ρ ) = F −1 ( F ( x ) ) = x
(5)
The linear to nonlinear functions can be estimated discretely by solving the inverse problem,
given distribution functions for both the linear x and nonlinear y. The IDF method produces
pairs of x and y values that can be used numerically or curve fit to form an equation like (2).
When working with nonlinear data, it is convenient to start with the nonlinear estimates of the
distribution function determined from a histogram or PDF of the data:
FNL ( yi ) = ρi
(6)
At each value of ρi find the point on the linear distribution function with equal probability,
and then use the IDF to determine the corresponding linear function value:
−1
FNL ( yi ) = ρi = FLin ( xi ) ; xi = FLin
( FNL ( yi ) )
(7)
Inverse distribution function of normal response displacement. The response distribution
function for a linear system exposed to normal random excitation is normal, and can be
expressed (for data with zero mean) as [3]:
Fnorm ( x ) =
1
 x 
1 + erf 
 = ρ

2
 σ 2 
where erf is the error function. This normal distribution function can be written as an IDF:
(8)
−1
Fnorm
( ρ ) = σ 2 erf −1 ( 2 ρ − 1) = x
(9)
where erf −1 is the inverse error function.
Estimates of nonlinear displacement response PDFs from the WPAFB data are shown in
Figure 1-a overlaid with a normal distribution function, with the results being normalized by
the standard deviation of the corresponding data set. An example distribution function (CDF)
determined from the 8 g PDF is shown in Figure 1-b overlaid with a normal CDF. The
inverse method can be traced on Figure 2, which shows the positive response portion of the
linear and nonlinear CDFs. Given a nonlinear normalized displacement of 0.8 σ, what linear
displacement will have the same CDF? By inspection, the corresponding linear displacement
is approximately 0.7 σ. Another example is 1.8 σ nonlinear mapping to 2.0 σ linear.
(a)
(b)
0.67 Linear
0.80 Nonlinear
2.00 Linear
1.81 Nonlinear
Figure 1. (a) Normalized displacement response PDFs. (b) Normalized nonlinear
displacement response distribution function (CDF) and linear response CDF.
Figure 2. Linear and nonlinear displacement cumulative distribution functions for 8 g Input.
The IDF method can be repeated at every nonlinear response CDF value. The resulting x and
y pairs are shown in Figure 3-a for the WPAFB data. The positive pairs are shown again in
Figure 3-b on log-log scale. Note that the results from each input level form a set of unique
curves. Curve fits of these functions can be used to estimate σ, kurtosis and rates of zero
crossings.
(a)
(b)
Figure 3. (a) Linear to nonlinear displacement response functions. (b) Log-log plot of linear
to nonlinear displacement response functions for WPAFB data.
Curve fitting the linear to nonlinear functions. In previous attempts [2] a piecewise
power-law was used to curve fit the RMS stress response functions. Research with similar
nonlinear functions [5] suggests that a polynomial function of the nonlinear response is the
preferred method to curve fit “stress stiffening” data:
h ( y) = x
c1 y + c2 y 2 + L + cn y n = x
(10)
where y and x are the nonlinear and linear displacement levels respectively. Note that the
polynomial is now on the left hand side of the equation. A third order polynomial proved to
be a good fit of the displacement data.
Nonlinear peak displacement PDF. The IDF method can also be applied to estimate linear
peak to nonlinear peak functions. In this case, one needs to first estimate nonlinear peak
PDFs and corresponding linear peak PDFs. Peak PDF estimates for the WPAFB data were
generated using Rain Flow Matrices determined using MATLAB functions developed by the
WAFO group [7].
A typical normalized displacement RFM [8] is shown in Figure 4. Positive peak (or maxima)
histograms can be calculated by summing the bin values along the rows of the RFM;
histograms of minima can be calculated by summing the columns. Peak PDFs can then be
estimated from the histograms of peaks.
Figure 4. Normalized peak displacement RFM for 8 g input WPAFB data, σ = 1.09 mm.
Normalized peak displacement PDFs are shown in Figure 5-a for three of the input levels,
with a linear peak PDF, determined from equation (12), for comparison. The data shows
strong nonlinear behavior, with increasing amounts of negative-valued maxima. Further
analysis of peak PDFs requires knowledge of the rates of zero crossings and peaks.
Response rates. The standard method of determining response rates based on spectral
moments [3, 4] cannot be used with nonlinear data [9]. Instead, the rates must be estimated
directly from the time domain data. The WPAFB results files were processed using the
WAFO MATLAB functions to determine the rates of zero crossings. The rates of positive
slope zero crossings Φ 0+  and the rates of displacement peaks Φ  P +  are found by
dividing the total number of positive slope zero level crossings and the number of rainflow
cycles (from a RFM) for each simulation level by the analysis time period. Table 1 gives the
rates of positive slope zero displacement crossings as well as the rates of displacement peaks
for the WPAFB data. Also tabulated is the ratio of displacement zero to peak rates ϖx:
ϖx =
Φ 0+x 
(11)
Φ  Px+ 
Table 1. Rates of displacement zero crossing and peaks for WPAFB experiments.
Input
Φ  0+x  Hz
Φ  Px+  Hz
ϖx
0.5 g
81.4
84.2
0.967
2g
92.0
102
0.903
8g
132
152
0.865
Linear peak probability functions. The normalized (σ = 1) linear peak PDF is a function of
the ratio ϖ [3]:
 k
pPeak ( x ) =  1
 2π
 − x2 
 x 
 − x2  

x
+
exp
ϖ
exp
 2

 1 − Q2   


 2 
 2k1 
 k2  
(12)
The additional terms kn and Qn are also functions of the ratio ϖ:
 x  2
1
2
Qn ( x ) = erfc 
 k1 = 1 − ϖ
2
k
2
 n

k2 = k1 ϖ
(13)
where erfc is the complimentary error function. The normalized linear peak distribution
function (or CDF) is [3]:
 x2 
FPeak ( x) = 1 − Q1 ( x ) − ϖ exp  −  [1 − Q2 x ]
 2
(14)
The linear peak inverse distribution function can be determined from equation (14).
Inverse distribution function method for peak displacement. One major observation from
the WPAFB data is that the ratio ϖ was not constant. This change in the ratio ϖ with the
input level (and degree of non-linearity) requires a decision with regard to how to proceed
with the IDF method. The logical choices were to use one linear peak CDF based on the
linear ratio ϖ, or to allow the ratio ϖ to change and therefore determine a corresponding linear
peak CDF for each input level.
(a)
(b)
Figure 5. (a) Normalized nonlinear peak displacement PDFs and linear peak PDF (b)
Normalized nonlinear peak displacement CDFs (experiment) and linear peak CDFs.
Figure 5-b shows the normalized linear and nonlinear CDFs for the data. Notice that the CDF
for the nonlinear data at the highest input of 8 g crosses the zero displacement level at a
cumulative probability of approximately 0.8 σ, while the CDF for the lowest level input of
0.5 g crosses the zero displacement level of approximately 0.2 σ. The linear peak CDFs
based on variable ratios ϖ cross at about the same level as the nonlinear data.
A more compelling argument for a variable ratio ϖ comes when one inspects the linear to
nonlinear functions determined using the IDF method. Figure 6-a shows the linear to
nonlinear function based on a fixed linear peak CDF, while Figure 6-b shows the function
based on a variable linear peak CDF. Log-Log plots of these functions are shown in
Figure 7-a and Figure 7-b. Notice that in both cases, the data does not all fall along one
common curve. The data for the variable linear peak CDF functions does tend to cross the
zero-zero point, which is considered to be desirable when curve fitting the functions. A curve
fit of the linear to nonlinear function through data values above approximately 1 σ was used.
It is reassuring to note that polynomial curve fits of the data points determined from the fixed
ϖ and variable ϖ linear peak CDF yield similar coefficients as given in Table 2.
(a)
(b)
Figure 6. Linear to nonlinear peak displacement functions for WPAFB data: (a) Fixed ratio
linear peak CDF and (b) variable ratio linear peak CDF.
(a)
(b)
Figure 7. Log-log plot of linear to nonlinear peak displacement functions for WPAFB data:
(a) Fixed ratio linear peak CDF and (b) Variable ratio linear peak CDF.
Table 2. Polynomial coefficients for linear to nonlinear curve fits of WPAFB peak
displacement data.
c1 mm/mm
c2 mm/mm2
c3 mm/mm3
Fixed ϖ CDF
0.918
5.081
212.0
Variable ϖ CDF
0.909
5.637
212.4
PDF TRANSFORMS FOR PEAK RESPONSE
PDF transform for peak displacement. Nonlinear peak PDFs were estimated using the
coefficients in Table 2 using the PDF transform method. The PDF transfer function for the
linear to nonlinear peak displacement “left handed” polynomial (10) is:
pNLpeak ( y ) = dx dy pPeak ( x )
(15)
Taking the derivative of the polynomial in (10) and substituting into (15) yields:
pNLpeak ( y ) = c1 + 2c2 y + 3c3 y 2 pPeak ( xˆ )
(16)
Instead of trying to solve this equation in closed form, the numerical result can be easily
determined by first estimating the linear peak “bin center” values using:
x̂ = c1 y + c2 y 2 + c3 y 3
(17)
where the y values are be based on the nonlinear histogram or PDF bin centers. The linear
peak PDF is given in equation (12).
PDF transform for nonlinear total stress. In this section, the PDF transform method is
used to estimate the nonlinear peak total stress PDF from the nonlinear peak displacement
PDF. The data is from nonlinear SDOF ODE simulations of the WPAFB experiments. The
nonlinear peak displacement to peak total stress function was determined based on a nonlinear
fit [10] of filtered WPAFB data:
z = g ( y)
(18)
z = E ( d1 y + d 2 y 2 + d3 y 3 )
where z and y are nonlinear peak total stress and nonlinear peak displacement respectively.
The material elastic modulus E = 204 GPa was used to convert the strain to stress. The
coefficients di are given in Table 3. Equation (18) is the standard form of the nonlinear
function with the polynomial on the right hand side of the equation.
Table 3. Nonlinear displacement to bottom strain parameters for WPAFB data
Run
d1 µε/mm
d2 µε/mm2
d3 µε/mm3
0.5 g
141.7
40.3
-15.26
2g
141.7
40.3
-14.65
8g
141.7
40.3
-6.10
The PDF transfer function for this nonlinear peak displacement to peak stress function is:
pNLpeak ( z ) =
pNLpeak ( y )
(19)
dz dy
where pNLpeak (y) is the nonlinear PDF of peak displacement. The derivative has two roots:
y=
− d 2 ± d 22 − 3d 2 d3
d1
(20)
The range of results was such that only one root (the minimum) was observed.
The next step in the process of estimating the total stress peak PDF is to assume that each
positive displacement peak is paired with a negative minimum with the same magnitude but
opposite sign. In other words, each point on a positive peak displacement PDF is assumed to
map to a complete displacement cycle with amplitude equal to the positive peak value. This
is considered fair based on the standard assumption that a narrow band response has one
positive peak for each positive zero crossing. The displacement cycle assumption now lets
one calculate the PDF for min and max stresses based on the total stress equation (18).
It is also helpful to observe how the displacement maps to total stress with an example.
Figure 8-a shows six cycles of an increasing displacement function, and Figure 8-b shows the
corresponding total stress based on equation (18). The first three cycles behave “normally”,
but the final three cycles show that the stress function now contains two cycles for one
displacement cycle.
(a)
(b)
Figure 8. (a) Six cycles of an increasing displacement function. (b) Total stress determined
from displacement.
First, define a “primary cycle” that has a stress maxima mapping to the positive displacement
maxima, and a stress minima mapping to the displacement minima. This one to one mapping
works for displacement cycles up to the value determined by equation (20). Note that the
standard PDF transfer function (19) is asymptotic at this point. The primary cycles have a
minimum at roots (equation (20)) of the total stress function. Also, define a “secondary
cycle” that has a minimum stress at this same minimum and a maximum stress at the point
mapping to the negative minimum of the displacement cycle.
The total stress PDF can be determined numerically as follows. First, define a range of
nonlinear peak displacement values at a convenient spacing (e.g. dy = 0.01σ y ) corresponding
to the positive peak displacement range observed in the nonlinear experiment (e.g. 0.1 σy ≤ y
≤ 4 σy). Determine the PDF of nonlinear positive displacement peaks pNLpeak ( y ) based on
equation (15). Also, determine a second nonlinear displacement PDF of negative minima
using the same PDF, but now assigned at the negative values of y.
Figure 9. Nonlinear positive maxima and negative minima displacement PDFs for 8 g ODE
simulation, ζ = 0.003.
Figure 9 is an example of this process for data from and ODE simulation of the 8 g
experiment. Now, given these PDFs, numerically evaluate equation (19). At the lower levels
of input, the mapping of displacement to stress is one to one as shown in Figure 10-a. At
higher input levels, the secondary cycles start to occur resulting in two possibilities for
generating stress maxima and minima. Figure 10-b gives an example of this result. Note that
the total stress PDF has an asymptote at –20 MPa, and that now Figure 10-b shows PDFs of
primary maxima and secondary maxima.
These functions are be plotted in Figure 11-a in a form like the WAFO RFM of ODE stress
shown in Figure 11-b. A visual comparison of the data is very encouraging. These two
functions for the higher inputs are not strictly PDFs because the “two to one” mapping of
stress results in an integrated area greater than one. The correct peak total stress PDF has to
be normalized by the sum of the primary and secondary contributions. The normalization can
be determined as discussed below.
One complication of this numerical PDF transform method is that the total stress functions
determined with equation (18) have uneven spacing. The resulting nonlinear unevenly spaced
PDFs of primary and secondary cycles have different bin centers and widths. They can be
interpolated at evenly spaced points and then normalized by the sum of the two functions.
Another method is to generate histograms of the primary and secondary cycles and then
simply sum the two to get an overall histogram of total cycles. This second method was used
to generate histograms for comparison to those determined from the ODE simulations of total
stress; there was no need to calculate normalized PDFs.
(a)
(b)
Figure 10. Peak total stress PDF determined from transformed positive and negative
displacement PDFs for (a) 0.5 g simulation and (b) 2 g simulation, ζ = 0.003.
(a)
(b)
Figure 11. (a) Nonlinear total stress primary and secondary ranges from displacement PDFs.
(b) Total stress rain flow matrix for nonlinear ODE simulation: Input = 2 g, ζ = 0.003.
Peak stress histograms of minima and maxima are shown in Figure 12. Note, the histograms
are shown with log number of counts in order to allow for better observation of the results
near the tails of the histograms. These figures show that the histogram results determined by
PDF transform compare very well with the rain flow histograms determined from the direct
time domain data of total stress.
(a)
(b)
Figure 12. Nonlinear peak stress histograms of minima and maxima from nonlinear ODE
simulation and transformed displacement PDFs: (a) input=0.5 g, (b) input=2 g, ζ=0.003.
DAMAGE ESTIMATES USING THE PDF TRANSFORM METHOD
Having estimated the PDFs and histograms of nonlinear peak total stress, the next step in the
analysis is to determine time to failure for the transformed displacement results.
Gordon, et al. [1] only mentions that the beam material used in the WPAFB experiments was
“high carbon steel.” The material SAE 1015 (a low carbon steel) was used for the damage
analysis because fatigue properties for different fatigue models were available [11]. The
baseline alternating stress model assumed no mean stress effects, whilst the Morrow (with
true fracture strength) and Walker models included mean stress effects [8, 11].
In this case, damage estimates were determined directly from the primary and secondary
histograms of peak total stress minima and maxima (the RFM were not calculated). Damage
calculations were also made using only the primary histograms of peak total stress minima
and maxima. The calculations based on only the primary cycles were equal to the combined
damage of primary and secondary cycles up to the 5th significant digit or more. The primary
cycle results are given in Table 4. These compare favorably with the time to failure results
from estimates determined from the raw WPAFB Strain data.
Table 4. Time-to-failure estimates for transformed nonlinear displacement PDF primary
cycle data from ODE simulations compared to estimates from WPAFB data.
Baseline (s)
Morrow (s)
Walker (s)
Run
WPAFB
Data
ODE
Sim
WPAFB
Data
ODE
Sim
WPAFB
Data
ODE
Sim
0.5 g
1.6e14
3.7e14
1.5e14
3.6e14
8.9e13
1.9e14
2g
3.8e10
4.1e10
3.4e10
3.4e10
1.7e10
1.3e10
8g
1.9e7
7.0e6
1.3e7
3.1e6
7.2e6
1.5e6
CONCLUSIONS
PDF transforms have been shown to be effective for estimating nonlinear PDFs of response
and peak displacement and total stress. The inverse distribution function method has been
used to estimate polynomial linear to nonlinear functions. A two step process to transform
linear to nonlinear peak displacement to total nonlinear stress PDFs compare well with
measured stress rainflow matrices and subsequent fatigue life estimates.
Acknowledgment. The authors thank the WPAFB Structural Mechanics Branch for
providing the experimental data and ITT Industries for their educational support at the ISVR.
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