# Derivation of DEL - Equation

Hello there,

I would be very grateful if I know the answer to these questions:

1- Can i know the transition between equation (27) and equation (28) in Mlife theory?
2- Is there a soure for the equation (2) or it has beed developed by NREL?
3- Is there a way to draw SN- Curve of my component using Mlife depending on the parameter i put?

Best regards,

Dear @Hussam.Al.Halabi,

Here are my responses:

1. To derive equation (28) from (27), solve (27) for DEL and equation (25) for Neq:
DEL = 2*(Lult-LMF)/(Neq^(1/m))
Neq = neq/D
From equations (1) and (2), we know:
D = 1/(2*(Lult-LMF))^m * SUM(n_i * LRF_i^m,i)
Combining and eliminating Neq and D yields (28):
DEL = ((SUM(n_i * LRF_i^m,i)/neq)^(1/m)

2. Equation (2) is the definition of the S-N curve in terms of LRF_i (S) and N_i (N).

3. MLife doesn’t plot the S-N curve, but you can plot Equation (2) directly given user-specified values of Lult, LMF, and m.

Best regards,

Dear @Jason.Jonkman ,

Many thanks for the answer. I have another question about the resulting number of equivalent cycles. If I change the Weibull scale, the number of equivalent cycles also changes. Actually, I still don’t understand the effect of the Weibull parameter on the number of equivalent cycles in Mlife. Wind speed and load distribution were calculated using software such as OpenFast. In the Mlife theory, I found that the Weibull parameters are only used to bin the load cycles to organize the data.

Best regards,

Dear @Jason.Jonkman,

In fact, I have something to add to my previous question.
When choosing the equivalent frequency, I chose that my wind turbine can withstand 10^9 cycles in 20 years. I get equivalent cycles that are smaller than what I chose to determine the equivalent cycle. Why this? Did the Weibull parameter affect my assumed cycle to failure? In this case, wasn’t DEL calculated for 10^9?

Best regards,

Dear @Hussam.Al.Halabi,

I’m not sure I understand your second question. For your first question, the Weibull paremeters determine the probability distribution of mean wind speed.

For typical calculation of operational fatigue (IEC load case 1.2), you’ll bin the operational wind speed range into mean wind speed bins and run a number of simulations in OpenFAST (with different turbulence seeds) within each bin. A typical discretization is 11 2-m/s bins with mean wind speeds of 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24 m/s, which for 6 seeds per bin is 66 total simulations. The Weibull parameters determine the probability that the wind turbine will operate in each bin. For example, Weibull parameters representing a lower mean wind speed means that the wind turbine will operate for more of its time in the lower wind speed bins, so, these bins will have more weight in the fatigue calculation than the higher wind speed bins.

Best regards,

Dear @Jason.Jonkman ,

In this case the DEL is highest at v_ave, which has the maximum frequency. At other wind speeds, the value of DEL is considerably lower. Why do I have to calculate the DEL at other wind speeds?

Best regards,

DEL -Fx at the 5 MW NREL wind turbine with v_ave=8.5 m/s.
X-axis is the tower height. Y-axis is DEL. Color is for the wind speed. Wind speeds are associated with turbulence.

Dear @Hussam.Al.Halabi,

I’m not really understanding your figure. The hub-height of the NREL 5-MW baseline wind turbine is 90 m, so, what do you mean by different tower heights? And presumably the DEL is a short-term DEL.

I would normally expect the short-term DELs to increase with mean wind speed, because wind speed standard deviations (important for fatigue) tends to increase with mean wind speed.

Best regards,

Dear @Jason.Jonkman,

You’re right. The diagram needs further explanation. The marked DEL is valid for life. I calculated DEL at 10 positions of the tower.

Best regards,

Dear @Hussam.Al.Halabi,

I’m not sure I understand what you mean when you say “the marked DEL is valid for life” What do you mean by a lifetime DEL that depends on wind speed?

Best regards,

Dear @Jason.Jonkman,

I found that DEL is calculated by v_ave at the same point that the equivalent frequency is determined. At other wind speeds, the number of equivalent cycles is lower. This means that the DEL does not reach the number of equivalent cycles because it has less weight. I saw the same meaning in your comment: “Weibull parameters representing a lower mean wind speed means that the wind turbine will operate for more of its time in the lower wind speed bins, so, these bins will have more weight in the fatigue calculation than the higher wind speed bins.” After calculating the DEL along the tower I got this chart as an example showing that the DEL is higher but at the same time has a much higher number of equivalent cycles which is not shown in this chart, but it is written in the output files.

In another words, if the program have the highest equivalent frequency at v_ave and other speeds have a lower equivalent frequency, it’s normal to end up with a lower DEL and this can cause confusion when evaluating the DEL at other wind speeds, which tends to have a lower value.

Best regards,

Without the wind speed near to v_ave the results will look like this.

Dear @Jason.Jonkman,

I am very sorry for confusing you. I found out that i have included the wrong value of 9 m/s. The chart looks like this:

Best regards,