A response spectrum method for rapid evaluation of structural fatigue life

Abstract

The present application relates to a response spectrum method for realizing rapid evaluation of structural fatigue life, including: using the Single Moment Method to evaluate the contribution of each mode to the fatigue damage of the system, separating the vibration mode participation coefficient and the stress vibration mode, thereby establishing a fatigue damage response spectrum that is independent of the load and structural spatial information; for multi-degree-of-freedom systems, considering the influence of vibration mode correlation on fatigue damage, forming the Complete Quadratic Combination (CQC) and Square Root of the Sum of Squares (SRSS) and Modified Square Root of the Sum of Squares (MRS) calculation formats of the fatigue life evaluation response spectrum method. Using the method given in the application, the reliability and efficiency of the method in this paper are verified by taking the L-type test piece as an example.

Description

Technical Field

[0001] The invention relates to a response spectrum method for realizing rapid evaluation of structural fatigue life, and is suitable for rapid evaluation of fatigue life of a mechanical system subjected to basic acceleration. Background Art

[0002] Fatigue failure is one of the main forms of mechanical system failure under complex load environments. Mechanical systems such as the nacelle structure that wraps the engine on an aircraft and the effective load on a launch vehicle are all subject to strong basic acceleration. In recent years, with the increasing attention paid to the dynamic strength design of mechanical systems such as launch structures, the study of rapid fatigue life assessment methods has become an important topic.

[0003] Generally speaking, the frequency domain method of random fatigue uses the power spectral density function and spectral moment of the response to represent its statistical characteristics, and then conducts fatigue life assessment. Its computational cost is mainly reflected in the following two aspects: one is that a large number of numerical integration operations are required, and the other is that the calculation process is complicated, involving the storage of a large number of intermediate variables, and has high requirements on the computer's internal and external memory and transmission performance. Even if the original problem is modally reduced, the efficiency improvement is not significant for large and complex structures with too many local modes. The high computational cost poses a huge challenge to the combination and further development of the fatigue life assessment process with other scientific problems. Therefore, it is of great significance to develop a frequency domain fatigue life assessment method with simple process and low computational cost. Summary of the invention

[0004] In view of the above problems, the present invention provides a response spectrum method for realizing rapid evaluation of structural fatigue life.

[0005] Based on the mode decomposition method, the present invention establishes the concept of equivalent Von-Mises stress distribution vector for multi-degree-of-freedom structural systems, and proposes a description method of the contribution of each order mode to the total damage of the system when evaluating the multi-axial fatigue life of the structure through equivalent Von-Mises stress, from which the mode participation coefficient and the stress mode are separated, and the modal damage response is defined, so as to establish a fatigue damage response spectrum that is independent of the load and structural spatial information. The fatigue damage response spectrum not only has the characteristic of relatively gentle change with the natural frequency, but also reduces the influence of the response bandwidth property. Considering the influence of mode correlation on fatigue damage, the calculation formats of two response spectrum methods, Complete Quadratic Combination (CQC) and Square Root of the Sum of Squares (SRSS), are formed under the conditions of fully considering and completely ignoring the correlation between modes. Combining the advantages of the two methods of CQC and SRSS, by introducing the concepts of modal group and center frequency, the influence of the correlation between modes on the total fatigue damage of the structure is partially considered, and the Modified Square Root of the Sum of Squares (MRS) calculation format is formed. In summary, the response spectrum method for rapid fatigue life assessment proposed in this invention includes three calculation formats: CQC, SRSS and MRS, which avoids the calculation of the power spectrum density function and spectral moment of the structural stress response and has higher calculation efficiency. In addition, since the structural spatial information and frequency information are decoupled from each other, the huge calculation cost caused by re-analysis is effectively reduced.

[0006] A response spectrum method for realizing rapid evaluation of structural fatigue life of the present invention comprises the following steps:

[0007] (1) The calculation of equivalent Von-Mises stress is deeply integrated with the mode decomposition method to establish the equivalent Von-Mises stress distribution vector. The Single Moment method is used to evaluate the contribution of each mode to the fatigue damage of the system.

[0008] (2) The modal participation coefficient and stress modal are separated from the contribution of each modal mode to the fatigue damage of the system, the concept of modal damage response is established, and a fatigue damage response spectrum that is independent of load and structural spatial information is constructed.

[0009] (3) For multi-degree-of-freedom systems, three calculation formats of fatigue life assessment response spectrum methods are formed under three conditions: fully considering, completely ignoring, and partially considering the influence of vibration mode correlation on fatigue damage.

[0010] Step (1) comprises:

[0011] Consider a discretized structural system with n degrees of freedom, whose equation of motion under the action of a uniform base acceleration is:

[0012]

[0013] M, C, K represent the mass matrix, damping matrix and stiffness matrix of the system respectively, u(t), and denote the displacement, velocity and acceleration vectors of the structure, respectively. represents the base acceleration, and E is the inertial force indicator vector, which is composed of elements 0 and 1. Considering the mode decomposition method, the displacement response of the structure is expanded in the mode space as

[0014]

[0015] Where r represents the modal cutoff number, φ i and η i (t) are the i-th order vibration mode vector and modal coordinates respectively. Considering the mass normalized vibration mode and orthogonal damping, the orthogonality of the vibration mode is used to decouple equation (1) into r single degree of freedom system equations, namely:

[0016]

[0017] Among them, i and ω i Represents the i-th order vibration mode damping ratio and natural frequency, vibration mode participation coefficient The solution of equation (3) can be obtained by using the impulse response function h corresponding to the i-th vibration mode: i (t) is expressed as

[0018]

[0019] Modal coordinate η i The expression of (t) consists of two parts. One is the mode participation coefficient γ related to the structural characteristics and load form. i , and the other part is only related to the natural frequency and modal damping ratio, denoted as δ i (t), represents the response of a virtual single degree of freedom system. Under zero-mean stationary broadband random excitation Under the action of i (t) is a zero-mean stationary narrow-band random process. From equation (2), we can know that the response of the system is

[0020]

[0021] For small deformation problems, the stress response and displacement response of the structure are related to each other through linear operators, so the displacement mode φ can be transformed into i Transformed into stress vibration mode φ σi , then the stress response of the structure can be expressed by formula (5):

[0022]

[0023] The power spectral density function matrix of σ(t) is

[0024]

[0025] Denotes the random response δ i (t) and δ j The cross power spectral density function of (t), the superscript * indicates the complex conjugate, Represents the bilateral power spectral density function of the basic acceleration, H i (ω) represents the frequency response function corresponding to the i-th order modal coordinates, Is an imaginary unit.

[0026] The equivalent Von-Mises stress that satisfies the zero-mean stationary Gaussian property is usually expressed as S E，p =tr[QS σ，p ], where tr[] represents the trace of the matrix, It represents the 6 components of the i-th order stress vibration mode at the p-th position, Q is the coefficient matrix, and the specific form is

[0027]

[0028] Substituting equation (7) into the expression of equivalent Von-Mises stress, we have

[0029]

[0030] Among them, the equivalent Von-Mises stress distribution vector Then the kth order spectral moment of the equivalent Von-Mises stress of the structure at the concerned position is

[0031]

[0032] Among them, m k，ij represents the i-th order stress vibration mode φ σi and the jth stress vibration mode φ σj The contribution of the correlation between the stress response and the kth order spectral moment is m. k，ii represents the i-th order stress vibration mode φ σi The contribution to the k-th order spectral moment of the system response when acting alone, m k，ij Can be expressed as

[0033]

[0034] Applying the SingleMoment method to obtain the single stress vibration mode φ σiContribution to the expected damage rate of the structure. Let k = 2 / b, and convert the m calculated by equation (24) into 2 / b，ii Substituting into formula (21), we have

[0035]

[0036] Among them, vector A° a Represents the vector A = [A1 A2…] T All elements in are raised to the power of a, that is

[0037] Step (2) comprises:

[0038] Rewrite equation (12) to separate the mode participation coefficient and stress mode from the contribution of each mode to the fatigue damage of the system, that is,

[0039]

[0040] in, Denote the stress response as δ i (t) is the expected damage rate of the virtual single-degree-of-freedom system. Perform 1 / b power operation on both sides of equation (13),

[0041] Then there is

[0042]

[0043] Among them, the vector Γ i It is called the damage response distribution vector, which represents the i-th order stress vibration mode φ σi The spatial distribution of the expected damage rate of the system is: is called the i-th stress mode φ σi The damage response under the action of Follow i The curve formed when it changes is defined as the fatigue damage response spectrum. The fatigue damage response spectrum is related to the load spectrum, the natural frequency of the structure, the modal damping ratio and the fatigue performance of the material, but has nothing to do with the spatial information reflected by the vibration mode and the stress vibration mode. It shows the change of the damage response of a single stress vibration mode with the natural frequency.

[0044] Step (3) includes:

[0045] From the SingleMoment method, we know that the total expected damage rate of the system is

[0046]

[0047] Introducing the concept of fatigue damage response spectrum into equation (15), and rewriting it using equation (3) in equation (14), we have

[0048]

[0049]

[0050] Formula (16) is called the CQC combination method of the fatigue life assessment response spectrum method.

[0051] When the natural frequencies are well separated, the influence of the correlation between the vibration modes on the structural response is relatively small. Assume that this influence can be ignored, that is, when i≠j, let ρ 2 / b，ij = 0, the SRSS combination method for spectral moment calculation can be obtained, that is,

[0052]

[0053] The error generated in the calculation process of the SRSS method is difficult to determine a priori. In order to reduce the degree of result deviation caused by completely ignoring the modal correlation, this method is considered to be corrected based on the natural frequency.

[0054]

[0055] It measures the strength of the vibration mode correlation. ij When >1, the correlation between vibration modes is relatively weak, so ρ 2 / b，ij =0, then the SRSS method is used for calculation; when α ij When ≤1, the correlation between vibration modes is strong, so ρ 2 / b，ij = 1, the modal group composed of these modes is equivalent to an independent mode, and its center frequency is the mean of the frequencies of each order that make up the modal group. Assuming that there are r-order natural frequencies involved in the calculation, the natural frequency sequence {ω i}, where r sp The order separation is good, and the remaining rr sp The order natural frequency forms R equivalent modes, and the qth equivalent mode is formed by r q The natural frequencies and vibration modes are combined (where ), the damage response of this equivalent mode From the equivalent natural frequency ω q (r q The damage response distribution vector of the equivalent mode is calculated as

[0056]

[0057] Rewrite the natural frequency sequence, the first r sp The first natural frequency is the real natural frequency of the system, and the last R natural frequencies are equivalent natural frequencies, so we have The formula is rewritten as

[0058]

[0059] This calculation method is called the MRS method of fatigue life assessment response spectrum method. Compared with the SRSS method, the MRS method retains the influence of the part with strong vibration mode correlation, which can significantly reduce the degree of error caused by ignoring vibration mode correlation, while avoiding the calculation of correlation coefficients and a large number of intermediate variables, ensuring the efficiency of the calculation.

[0060] The fatigue damage response spectrum established by the response spectrum method of the present invention not only has the characteristic of relatively gentle change with the natural frequency, but also reduces the influence of the response bandwidth property. This method avoids the calculation of the power spectrum density function and spectral moment of the structural stress response, has higher calculation efficiency, realizes the mutual decoupling of structural space information and frequency information, effectively reduces the huge calculation cost brought by re-analysis, and brings convenient conditions for the further development and application of fatigue life assessment. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 A schematic diagram of the process flow of this application is shown.

[0062] Figure 2 Schematic diagram of the L-shaped test piece used to verify the present method.

[0063] Figure 3 is the fatigue damage response spectrum.

[0064] Figure 4 (a) shows the expected fatigue damage rate distribution of the structure calculated by the self-developed Dirlik method.

[0065] Figure 4 (b) shows the expected fatigue damage rate distribution of the Dirlik Method (MSC.Patran / Nastran) structure.

[0066] Figure 4 (c) shows the expected fatigue damage rate distribution of the structure using the Single Moment method.

[0067] Figure 4 (d) shows the expected fatigue damage rate distribution of the response spectrum method (CQC) structure.

[0068] Figure 4 (e) shows the expected fatigue damage rate distribution of the response spectrum method (SRSS) structure.

[0069] Figure 4 (f) shows the expected fatigue damage rate distribution of the response spectrum method (MRS) structure. DETAILED DESCRIPTION

[0070] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0071] Reference example: Fatigue life evaluation of L-type specimen.

[0072] This application relates to a response spectrum method for realizing rapid evaluation of structural fatigue life. Taking an L-type test piece fatigue life evaluation as an example, the method of this application is verified. The finite element model and its dimensional information are as follows: Figure 2 The finite element model has 11211 nodes and 10802 elements. The Young's modulus of the material is E = 70000 MPa, and the density is ρ = 2.67×10 -9 t / mm 3 , Poisson's ratio is μ = 0.33. The relevant parameters of the material SN curve are b = 3, C = (4535.72MPa) b .

[0073] The calculation steps include:

[0074] 1. First, use MSC.Patran\Nastran 2018 to perform real eigenvalue analysis. In the frequency range [10, 2000] Hz, there are 31 natural frequencies. The structure is subjected to lateral foundation acceleration, and its power spectrum is in the form of limited-band white noise f∈[10, 2000] Hz, with an amplitude of W0=0.1g 2 / Hz, there is a conversion relationship between the unilateral power spectral density function with frequency f as the independent variable and the bilateral power spectral density function with circular frequency ω as the independent variable W0(f)=4πS0(ω). Under this condition, when the modal damping ratio is ζ=[0.01, 0.02, 0.03, 0.04, 0.05], the fatigue damage response spectrum of the structure is as follows: Figure 3 shown.

[0075] 2. Using the structural information such as natural frequency and stress mode obtained from the software, calculate the physical quantities required for solution formats such as CQC, SRSS and MRS.

[0076] 3. The modal damping ratio is selected as ζ = 0.02. The fatigue damage response corresponding to each vibration mode is determined through the original natural frequency sequence and the fatigue damage response spectrum obtained by pre-processing calculation, as shown in Table 1.

[0077] Table 1 Fatigue damage response corresponding to each vibration mode

[0078]

[0079] 4. According to the calculation format of the three fatigue life assessment rapid evaluation response spectrum methods, namely CQC, SRSS and MRS, calculate the expected damage rate of the structure and draw the distribution cloud diagram as follows Figure 4The calculation time comparison is shown in Table 2. Overall, the distribution of the expected damage rate calculated by the response spectrum method, Single Moment method and Dirlik method for fatigue life assessment is basically consistent, and the calculation result of the Dirlik method is slightly conservative. The calculation results of the three calculation formats of the response spectrum method are well consistent with the Single Moment method in terms of distribution. In terms of calculation time, the calculation efficiency of the response spectrum method is significantly higher than that of the Dirlik method and the Single Moment method.

[0080] Table 2 Comparison of calculation time

[0081]

[0082] The present invention provides a response spectrum method for realizing rapid evaluation of structural fatigue life, which is applicable to rapid evaluation of fatigue life of mechanical systems under any basic acceleration. The present invention avoids the calculation of power spectrum density function and each order spectral moment of stress response, greatly simplifies the calculation process and reduces the calculation cost. In the whole calculation process, the processing of structural vibration mode information and the processing of fatigue information do not interfere with each other, and the fatigue life of the structure is obtained only at the end by combining the two. This feature effectively reduces the huge calculation cost brought by re-analysis, and brings convenient conditions for the further development and application of fatigue life evaluation.

Claims

1. A response spectrum method for realizing rapid evaluation of structural fatigue life, characterized by the following steps: Step (1) comprises: Consider a discretized structural system with n degrees of freedom, whose equation of motion under the action of a uniform base acceleration is: M, C, K represent the mass matrix, damping matrix and stiffness matrix of the system respectively, u(t), and denote the displacement, velocity and acceleration vectors of the structure, respectively. represents the base acceleration, E is the inertial force indicator vector, which is composed of elements 0 and 1; considering the mode decomposition method, the displacement response of the structure is expanded in the mode space as Where r represents the modal cutoff number, φ i and η i (t) are the i-th order vibration mode vector and modal coordinates respectively; considering the mass normalized vibration mode and orthogonal damping, the orthogonality of the vibration mode is used to decouple equation (1) into r single-degree-of-freedom system equations, namely: Among them, i and ω i Represents the i-th order vibration mode damping ratio and natural frequency, vibration mode participation coefficient The solution of equation (3) uses the impulse response function h corresponding to the i-th vibration mode i (t) is expressed as Modal coordinate η i The expression of (t) consists of two parts. One is the mode participation coefficient γ related to the structural characteristics and load form. i , and the other part is only related to the natural frequency and modal damping ratio, denoted as δ i (t), represents the response of a virtual single-degree-of-freedom system; under zero-mean stationary broadband random excitation Under the action of i (t) is a zero-mean stationary narrow-band random process; then from equation (2), we can know that the response of the system is For small deformation problems, the stress response and displacement response of the structure are related to each other through linear operators, so the displacement mode φ is transformed into i Transformed into stress vibration mode φ σi , then the stress response of the structure is expressed by formula (5): The power spectral density function matrix of σ(t) is Denotes the random response δ i (t) and δ j The cross power spectral density function of (t), the superscript * indicates the complex conjugate, Represents the bilateral power spectral density function of the basic acceleration, H i (ω) represents the frequency response function corresponding to the i-th order modal coordinates, is an imaginary unit; The equivalent Von-Mises stress that satisfies the zero-mean stationary Gaussian property is usually expressed as S E，p =tr[QS σ，p ], where tr[] represents the trace of the matrix, It represents the 6 components of the i-th order stress vibration mode at the p-th position, Q is the coefficient matrix, and the specific form is Substituting equation (7) into the expression of equivalent Von-Mises stress, we have Among them, the equivalent Von-Mises stress distribution vector Then the kth order spectral moment of the equivalent Von-Mises stress of the structure at the concerned position is Among them, m k，ij represents the i-th order stress vibration mode φ σi and the jth stress vibration mode φ σj The contribution of the correlation between the stress response and the kth order spectral moment is m. k，ii represents the i-th order stress vibration mode φ σi The contribution to the k-th order spectral moment of the system response when acting alone, m k，ij Expressed as Applying the SingleMoment method to obtain the single stress vibration mode φ σi Contribution to the expected damage rate of the structure; let k = 2 / b, and convert the m calculated by Eq. 2 / b，ii Substituting into the formula, we have Among them, vector A °a Represents the vector A = [A1 A2…] T All elements in are raised to the power of a, that is Step (2) comprises: Rewrite equation (12) to separate the mode participation coefficient and stress mode from the contribution of each mode to the fatigue damage of the system, that is, in, Denote the stress response as δ i (t) is the expected damage rate of the virtual single-degree-of-freedom system; performing 1 / b power operations on both sides of equation (13), we have Among them, the vector Γ i It is called the damage response distribution vector, which represents the i-th order stress vibration mode φ σi The spatial distribution of the expected damage rate of the system is: is called the i-th stress mode φ σi The damage response under the action of Follow i The curve formed when the load changes is defined as the fatigue damage response spectrum; the fatigue damage response spectrum is related to the load spectrum, the natural frequency of the structure, the modal damping ratio and the fatigue performance of the material, but has nothing to do with the spatial information reflected by the vibration mode and the stress vibration mode, and shows the change of the damage response of a single stress vibration mode with the natural frequency; Step (3) includes: According to the SingleMoment method, the total expected damage rate of the system is Introducing the concept of fatigue damage response spectrum into equation (15), and rewriting it using equation (3) in equation (14), we have Formula (16) is called the CQC combination method of the fatigue life assessment response spectrum method; When the natural frequencies are well separated, the influence of the correlation between vibration modes on the structural response is relatively small. Assume that this influence is negligible, that is, when i≠j, let ρ 2 / b，ij = 0, then the SRSS combination method for spectral moment calculation is obtained, that is, The error generated in the calculation process of the SRSS method is difficult to determine a priori. In order to reduce the degree of result deviation caused by completely ignoring the modal correlation, this method is considered to be corrected based on the natural frequency; the parameter is introduced Measures the strength of the vibration mode correlation; when α ij When >1, the correlation between vibration modes is relatively weak. Let ρ 2 / b，ij =0, then the SRSS method is used for calculation; when α ij When ≤1, the correlation between vibration modes is strong, let ρ 2 / b，ij = 1, the modal group composed of these modes is equivalent to an independent mode, whose center frequency is the mean of the frequencies of each order that constitute the modal group; assuming that there are r-order natural frequencies involved in the calculation, the natural frequency sequence {ω i }, where r sp The order separation is good, and the remaining rr sp The order natural frequency forms R equivalent modes, and the qth equivalent mode is formed by r q The natural frequencies and vibration modes are combined (where ), the damage response of this equivalent mode From the equivalent natural frequency ω q (r q The damage response distribution vector of the equivalent mode is calculated as Rewrite the natural frequency sequence, the first r sp The first natural frequency is the real natural frequency of the system, and the last R natural frequencies are equivalent natural frequencies, so we have The formula is rewritten as

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