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Wind Energy Exercises Week 11

Exercises and solutions week 11

Kursus

Introduktion til vindenergi (46000)

24 Dokumenter

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Danmarks Tekniske Universitet

Akademisk år: 2022/2023

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Calculation exercise / Solution

A fiber-polymer composite spar cap of a wind turbine rotor blade has a fatigue strength coefficient of C=500 MPa and an exponent of m=10 at R=-1. During a blade certification test, the cap is subject to a constant equivalent stress amplitude of σa=150 MPa. Using the Basquin law of Part 2 (slide 9), calculate the lifetime of the cap.

Solution:

Step_1: we isolate N from the Basquin law given by Eq:

𝜎𝑎 = 𝐶 𝑁−1 𝑚⁄ (1)

𝑁 = ( 𝜎𝐶

𝑎

)

𝑚 (2)

Step_2: we calculate the fatigue life by substituting the material parameters and stress amplitude into Eq:

( 500150 )

10 = 169350 (3)

Answer: the lifetime of the component is approx. 1 × 10 5 cycles.

Due to some minor changes in controller setup of the wind turbine, aeroelastic calculations show that the equivalent stress amplitude in the cap is slightly increased by 2% to σa=153 MPa but everything else remains the same. Calculate the effect of the increased amplitude on the fatigue lifetime in % compared to the original controller setup.

Solution:

Step_1: we calculate the new fatigue life by substituting the stress amplitude into Eq.

( 500153 )

10 = 138926 (4)

Step_2: we calculate the reduction of fatigue life:

(1 − 138926 169350

) ∙ 100 ≅ 18%

Answer: A slight increase in stress amplitude of 2% results in a significant fatigue lifetime reduction of 18% showing the effect of the power law.

The manufacturer decides to produce the cap from a novel pultruded composite material with the same strength coefficient of C=500 MPa but an exponent of m=12. The stress range remains at σa= MPa and R=-1. Calculate the increase of the fatigue life compared to the old design. Calculate the potential material saving in terms of thickness reduction of the cap for maintaining the same fatigue lifetime of the old design.

Solution:

Step_1: we calculate the fatigue lifetime of the new design by substituting the exponent into Eq.

(500 153 )

12 = 1438689

Step_2: we calculate the factor 𝜂 of the increased fatigue life as follows:

𝜂 = 1438689 138926 ≈ 10

Answer: A material system offering only a 20% higher fatigue exponent entails a tenfold increase of the fatigue life compared to the old design.

Step_4: We calculate the stress amplitude for the old lifetime N=138926 but with m=12 using Eq:

500 ∙ 138926−1 12⁄ = 186 [MPa]

Step_5: We calculate the reduction of the cap thickness as the ratio of the stress amplitudes for m= and m=12 for N=138926 cycles.

(1 −

153 186) ∙ 100 ≅ 18%

Answer: By switching to the new material system, the manufacturer could reduce the cap thickness by 18% without compromising the fatigue life of the old design.

Redesign of the blade considering the material mass reduction enabled a lighter design where aeroelastic load calculations indicated a decreased cap stress amplitude of σa=135 MPa. In the meantime, a more detailed fatigue characterization of the new material system revealed the presence of a second slope in the SN-curve for 𝑁 ≥ 2 × 10 6 cycles with a strength parameter of C=369 MPa and an exponent of m=16. Show that the second slope can be exploited in this case. Calculate the effect of the second slope on the lifetime of the cap.

Solution:

Step_1: we calculate the fatigue lifetime with the decreased amplitude for C=500 and m=12:

(500 135 )

12 = 6 × 10 6

Answer: since 6 × 10 6 > 2 × 10 6 we can indeed utilize the second slope

Step_2: we calculate the more accurate fatigue life using C=369 MPa and m=16 as follows:

(369 135 )

16 = 9 × 10 6

Answer: Extrapolation of the fatigue life for 𝑁 > 2 × 10 6 using the slope C=500 and m= underestimates the fatigue life by approx. 30% for the prevailing amplitude. However, this difference increases with decreasing stress amplitude.

The figure below shows a strain based CLD where the ordinate represents the strain range in % and the abscissa represents the mean strain in %. The red dot in the CLD shows that the fatigue life of a fiber-polymer composite material for ∆𝜀 = 1 % at R=-1 (i. 𝜀𝑚 = 0) is 𝑁 = 10 6 cycles. Estimating with the unaided eye from the CLD: (a) what is the allowable strain range under a mean tensile strain magnitude of 𝜀𝑚 = 0 % but without compromising on the fatigue life? (b) how much lower is the fatigue life for keeping ∆𝜀 = 1 % and applying a mean strain of 𝜀𝑚 = 0 % according to the nearest contour line.

Answer (b): we find the nearest contour of 𝑁 = 10 5 at the intersection ∆𝜀 = 1 % and 𝜀𝑚 = 0 % and conclude that the reduction of fatigue life is more than one order of magnitude.

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Wind Energy Exercises Week 11

Kursus: Introduktion til vindenergi (46000)

24 Dokumenter

Studerende delte 24 dokumenter i dette kursus

Universitet: Danmarks Tekniske Universitet

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Calculation exercise / Solution

A fiber-polymer composite spar cap of a wind turbine rotor blade has a fatigue strength coefficient of

C=500 MPa and an exponent of m=10 at R=-1. During a blade certification test, the cap is subject to a

constant equivalent stress amplitude of σa=150 MPa. Using the Basquin law of Part 2 (slide 9), calculate

the lifetime of the cap.

Solution:

Step_1: we isolate N from the Basquin law given by Eq.1:

𝜎𝑎= 𝐶 𝑁−1 𝑚

⁄ (1)

𝑁 = (𝐶

𝜎𝑎)𝑚 (2)

Step_2: we calculate the fatigue life by substituting the material parameters and stress amplitude into

Eq.2:

(500

150)10 =169350 (3)

Answer: the lifetime of the component is approx. 1.7×105 cycles.

Due to some minor changes in controller setup of the wind turbine, aeroelastic calculations show that

the equivalent stress amplitude in the cap is slightly increased by 2% to σa=153 MPa but everything

else remains the same. Calculate the effect of the increased amplitude on the fatigue lifetime in %

compared to the original controller setup.

Solution:

Step_1: we calculate the new fatigue life by substituting the stress amplitude into Eq.2

(500

153)10 =138926 (4)

Step_2: we calculate the reduction of fatigue life:

(1−138926

169350)∙100 ≅18%

Answer: A slight increase in stress amplitude of 2% results in a significant fatigue lifetime reduction of

18% showing the effect of the power law.

The manufacturer decides to produce the cap from a novel pultruded composite material with the

same strength coefficient of C=500 MPa but an exponent of m=12. The stress range remains at σa=153

MPa and R=-1. Calculate the increase of the fatigue life compared to the old design. Calculate the

potential material saving in terms of thickness reduction of the cap for maintaining the same fatigue

lifetime of the old design.

Solution:

Step_1: we calculate the fatigue lifetime of the new design by substituting the exponent into Eq.2

(500

153)12 =1438689

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