Load spectra of aero-engines usually consist of a variety of load parameter matrices. Each of the load parameter matrices corresponds to certain fatigue damage. The load parameter matrices are often obtained by mixing profile load parameter matrices. When the load parameter matrices are obtained by processing the flight profile in an environmental task, they are called environmental mission profile of the load parameter matrices . The collection of various environmental mission profile load parameter matrices is the environmental task load parameter matrices. The environmental task load parameter matrices include cyclic matrix, distribution matrix and correlation matrix. For the cyclic matrix, it includes power cyclic matrix, speed cyclic matrix, torque cyclic matrix, pressure cyclic matrix, overload cyclic matrix, etc. . Cyclic matrix profile parameters represent load cycles, which are used to calculate the low cycle fatigue damage of aero-engines. Measured data used in this paper are the speed cyclic matrix. By extracting peak values, counting parameter cycles and removing invalid cyclic amplitudes of the flight profile, an environmental task parameter cyclic matrix is obtained.
The remainder of this paper is structured as follows. Section 2 reviews several deterministic fatigue life prediction methods that would be used in the following sections and then introduces the procedure of the fatigue life analysis. In Section 3, by analyzing the measured speed cyclic matrix of an aero-engine, the number of the main work cycles of the aero-engine is obtained. In Section 4, based on the cracks in turbine disk mainly caused by low cycle fatigue failures, the loading stress of the turbine disk is analyzed. In Section 5, by fitting the material S–N curve, the S–N curve of the turbine disk is obtained. In Section 6, by using the S–N curve of the turbine disk and Miner’s rule, the life of the turbine disk is predicted. In the last section, the conclusions are drawn.
2 Fatigue life prediction methods
According to the irreversibility and randomness of fatigue damages, fatigue life prediction methods can be classified into two groups: deterministic fatigue life prediction methods and probabilistic fatigue life prediction methods [3, 10–12]. In this section, the most commonly used deterministic fatigue life prediction methods are reviewed.
2.1 Stress-based fatigue life prediction methods
Stress-based fatigue life prediction method is one of the earliest developed methods used for fatigue life prediction. The S–N curve life prediction is a classic example [4, 9]. Basquin formula is a common fatigue formula used in engineering, which can be expressed as follows : (1)where is the fatigue strength coefficient, is the fatigue life; is the fatigue strength exponent, is the modulus, is the elastic strain range and is the stress amplitude.
Due to the stress concentration, the stress-based fatigue life prediction method can be further divided into nominal stress method, hot spot stress method and notch stress method.
In this paper, the nominal stress method is used to calculate the low cycle fatigue life of the aero-engines. Material S–N curves are derived from material data and determined by the stress concentration factors. Also, it should be noted that different stress concentration factors correspond to different S–N curves .
2.2 Strain-based fatigue life prediction methods
For a heavy service load, during the operation of a component, it is better to use strain–life curve to predict the life of the component. In the 1950s, a second common and more recent approach based on a local strain was established. Manson–Coffin’s formula is one of the most commonly used fatigue life prediction methods in engineering. Manson–Coffin’s formula describes a relationship between the plastic strain and the fatigue life as follows: (2)where is the elastic strain range, is the fatigue ductility coefficient, is the fatigue life and is the fatigue ductility exponent. From eq. (2), it is obvious that determines the value of .
Mason and Hirschberg  put forward an improved formula to express the fatigue life by considering the total strain range: (3)where is the total strain range, is the fatigue strength coefficient, is the modulus, is the fatigue life, is the fatigue ductility coefficient, is the fatigue strength exponent and is the fatigue ductility exponent.
2.3 Miner’s rule
Suppose that there are L levels of stress in the load spectra [7, 8]. They are denoted as , , , . The number of cycles of each level is denoted as , , , , and the numbers of the destruction cycles of each level are denoted as , , , . Miner’s rule assumes that the fatigue damage can be obtained by corresponding circulation ratios. The equation is given as follows: (4)where L is the number of stress levels, is the number of cycles at a given stress level, which can be got from load spectra, is the number of cycles to failure at a given stress level.
A procedure of the fatigue life analysis used in this paper is shown in Figure 1.
First, the S–N curve of the structure should be known. In this paper, the S–N curve of the structure is obtained by fitting the material data. Second, based on the structure analysis, the nominal stress of each cycle can be represented by , , and . Then the equivalent damage corresponding to each cycle can be calculated. Finally Miner’s rule is used to get the fatigue damage and life of the turbine disk.
3 Analysis of speed cyclic matrix data
The measured speed cyclic matrix of an aero-engine of an aircraft is shown in Table 1. The main work cycles can be calculated by analyzing the data in Table 1. Because the data are counted in 2,000 hours many times through the same flight mission, the number of cycles may not be integer. From Table 1, the number of speed cycles of the engines is obtained. Peaks can be defined as four states: take off (>97%), rated take off (94.5–97%), valleys are zero and the ground idle. The numbers of cycles of “0-take off-0,” “ground idle-take off-ground idle,” “0-rated take off-0” and “ground idle-rated take off-ground idle” are 500.1, 304.8, 221.2 and 420.4. The total number of cycles “0-97%” and “94.5-97%-0” is 721.3, which reflects the flight subjects starting from switch off to take off. The number of cycles of “ground idle-take off state-ground idle” is 725.2, which reflects the flight subjects of landing (but not switching off) to take off again. The number of cycles of “flight idle-take off state-flight idle” is high, which reflects the low frequency using this cycle. Then, the number of the main work cycles of the aero-engine can be obtained and are shown in Table 2.
4 Analysis of turbine disk’s loading stress
In this section, the first-stage turbine disk is regarded as objects and their fatigue life is estimated. According to actual turbine disk fracture and related metallographic analysis, the cracks in turbine disk are mainly caused by low cycle fatigue failures.
As shown in Table 1, the centrifugal forces are the main load, which are considered in this paper.
Assuming that the blade’s centrifugal forces are evenly distributed on the contact surface of the mortises of the turbine disk, the centrifugal forces are expressed as follows: (5)where is the mass of the blades, is the radius of the center of mass of the blades and is the angular velocity of the turbine disk. It means that the loading stress of the turbine disk is proportional to the square of the angular velocity.
5 The fitting of S–N curve of the first-stage turbine disk
In this paper, the material S–N curve is used to estimate the low cycle fatigue life of the turbine disk for a rough calculation to verify the feasibility of this method. The material used in the test is GH4698 super alloy. The mechanical and fatigue properties of the tested material under the room temperature are listed in Tables 3 and 4, respectively. Considering the influence of stress concentration on turbine disk, the notched fatigue specimen and notched radius are chosen as follows: , for high cycle fatigue; and , for low cycle fatigue.
According to the material’s fatigue limit Pa, the cycle in the experiment will not cause damage when the stress is less than or equal to 118 MPa. Through the curve fitting, , . Then, the S–N curve can be expressed as follows: (6)The S–N curve is shown in Figure 2 .
6 Fatigue life estimation of the first-stage turbine disk
6.1 Estimation of cyclic stress
6.2 Estimation of symmetric cyclic stress
After getting the cyclic stress, from the cyclic matrix we can obtain the cyclic stress amplitude as shown in Table 5. After applying the modified Goodman diagram and the S–N curve to the experiment data, the stress can be converted into symmetric cyclic stress , the modified equation can be expressed as (8)where is the symmetric cyclic stress, is the amplitude of stress, is the mean stress, is the ultimate strength of the material. To simplify the calculation, we define that and the maximum stress of first-stage turbine disk is . Then, we can get the corresponding symmetrical cyclic stress, as shown in of Table 5.
6.3 Estimation of cumulative damage
The life curve is usually described by the following equation, which is called Goodman diagram: (9)where is the stress amplitude, is the stress amplitude when the mean is 0, is the mean stress and is the mean stress when the amplitude is 0.
The stress cycles of each level can be represented by or . Generally, is accepted in engineering and can be shorted by . According to Goodman diagram, , which corresponds to and called symmetrical cyclic stress, can be calculated by the following equation: (10)Then S–N curve and eq. (10) are used to calculate the fatigue life which corresponds to , the fatigue life is (11)If is known, the corresponding fatigue life can be calculated by (12)Substituting parameters into eq. (12) with , , (13)The corresponding cyclic life by eq. (13) is shown in Table 5.
According to the calculation, the cumulative fatigue damage is 0.018471 after 2,000 flight hours. It indicates that although the main failure mechanisms of the turbine disc are low cycle fatigue, the total damage of the turbine disk is small which is caused by the centrifugal forces in 2,000 flight hours. It should be noted that, in the operation of the turbine disk, it also endures high thermal stresses. But because of the lack of experimental data, in this paper, only the centrifugal forces are used to estimate the life of the turbine disk. So we will do further study about the life of the turbine disk in the future.
In this paper, based on the actual measured load spectra speed cyclic matrix, the fatigue life of the first-stage turbine disk was estimated. The experimental results demonstrated that the centrifugal forces are the main load of the turbine disk. Also, it is clear that the S–N curve and Miner’s rule can be used to estimate the fatigue life of the turbine disk. Similarly, this method can be used to estimate the fatigue life of the other components of the aero-engine. But it should be noted that, when just considering the centrifugal forces of the loading stress, the total damage of the turbine disk is small. In order to have a high precision of fatigue life estimation of the turbine disk, more experimental data about the loading stress are need. Also, the life estimation method can be further improved if there are more accurate parameters.
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About the article
Published Online: 2015-03-10
Published in Print: 2016-04-01
Funding: The authors would like to acknowledge the partial support provided by the National Natural Science Foundation of China under contract number 11272082 and the Fundamental Research Funds for the Central Universities under contract number E022050205.