A method for calculating flow-induced stress and fatigue life of a bellows

Abstract

This invention discloses a method for calculating the flow-induced stress and fatigue life of a bellows, comprising: based on the structural parameters and medium parameters of the bellows, sequentially calculating the modal frequencies, excitation velocities of each modality, the first-order radial acoustic modal frequency, and the medium velocity of the bellows; further, determining the excitation mode and the excitation velocity of the excitation mode; based on this, calculating the stress of the bellows; further, correcting the calculated bellows stress by considering a certain degree of uncertainty based on the acoustic resonance results; finally, calculating the fatigue damage of the bellows under each excitation mode based on the corrected bellows stress. The method of this invention can determine the excitation mode based on the structural dimensions and medium parameters of the bellows, and then estimate the flow-induced stress, thereby assessing the resulting bellows fatigue damage, solving the problems of vibration and fatigue caused by fluid scouring that cannot be considered in the prior art.

Description

Technical Field

[0001] This invention belongs to the field of corrugated pipe testing technology, and particularly relates to a method for calculating the flow-induced stress and fatigue life of corrugated pipes. Background Technology

[0002] Bellows are widely used as compensating elements in the pressurization and delivery systems of launch vehicles. When a medium flows inside the bellows, pressure pulsations may occur. When the frequency of these pressure pulsations couples with the frequency of the bellows structure, it can cause resonance, generating flow-induced stress and leading to fatigue failure. Assessing the flow-induced stress and fatigue life of bellows under flowing medium conditions is crucial to addressing the issue of flow-induced vibration fatigue caused by medium erosion in bellows.

[0003] Currently, the commonly used method for assessing bellows fatigue damage is as follows: The stress distribution of the bellows under a given boundary displacement is calculated using finite element method (FEM) software; the stress distribution is then determined using the SN curve to determine if the stress meets the requirements; and the damage under each working condition and the total damage are calculated based on the stress and cycle count for four operating conditions. Bellows selection and matching are performed during the early design and development phase of the engine exhaust system. However, this approach only allows for fatigue damage calculations under static cyclic conditions and does not account for bellows vibration fatigue under media scouring. Summary of the Invention

[0004] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a method for calculating the flow-induced stress and fatigue life of a bellows. The method can determine the excitation mode based on the structural dimensions and medium parameters of the bellows, and then estimate the flow-induced stress to evaluate the resulting bellows fatigue damage. This solves the problems of vibration and fatigue caused by fluid scouring that cannot be considered in the prior art.

[0005] To address the aforementioned technical problems, this invention discloses a method for calculating the flow-induced stress and fatigue life of bellows, comprising:

[0006] Step 1: Calculate the modal frequencies of the bellows based on its inner diameter, wave height, number of layers, single-layer wall thickness, and wave pitch; wherein, each modal frequency includes: each axial modal frequency and local bending modal frequency.

[0007] Step 2: Based on the modal frequencies of the bellows calculated in Step 1, and combined with the Stuhlhal number and wave pitch, calculate the excitation velocities of the bellows in each mode.

[0008] Step 3: Calculate the flow velocity of the medium based on the density, mass flow rate, and inner diameter cross-sectional area of ​​the medium inside the bellows.

[0009] Step 4: Calculate the first-order radial acoustic mode frequency of the bellows based on its inner diameter, wave height, average fillet radius, and the sound velocity of the medium.

[0010] Step 5: Compare the flow velocity of the medium calculated in Step 3 with the excitation flow velocity of each mode of the bellows calculated in Step 2; if the flow velocity of the medium reaches or exceeds the excitation flow velocity of a certain mode of the bellows, then it is determined that the certain mode is excited, and combined with Step 1, the frequency of the excited mode is obtained.

[0011] Step 6: Compare the frequency of the excitation mode obtained in Step 5 with the frequency of the first-order radial acoustic mode of the bellows calculated in Step 4 to obtain the acoustic resonance result; if the frequency of the excitation mode reaches or exceeds the frequency of the first-order radial acoustic mode of the bellows, the acoustic resonance result is: acoustic resonance exists; otherwise, the acoustic resonance result is: acoustic resonance does not exist.

[0012] Step 7: Based on the frequency of the excitation mode obtained in Step 5, and combined with Step 2, obtain the excitation velocity of the excitation mode; based on the obtained excitation velocity of the excitation mode, and combined with the density of the medium, calculate the pulsating pressure.

[0013] Step 8: Calculate the cross-sectional area of ​​the corrugated pipe based on its outer and inner diameters.

[0014] Step 9: Calculate the force exerted by the medium on the bellows based on the pulsating pressure calculated in Step 7 and the cross-sectional area of ​​the bellows calculated in Step 8.

[0015] Step 10: Based on the force of the medium on the bellows calculated in Step 9, and combined with the single-wave stiffness of the bellows, calculate the single-wave displacement of the bellows.

[0016] Step 11: Calculate the stress of the bellows based on the single-wave displacement of the bellows obtained in Step 10.

[0017] Step 12: Based on the acoustic resonance results obtained in Step 6, and considering a certain degree of uncertainty, the stress of the bellows calculated in Step 11 is corrected to obtain the corrected bellows stress.

[0018] Step 13: Based on the corrected bellows stress calculated in Step 12, and combined with the SN curve of the bellows material, calculate the fatigue damage of the bellows under each excitation mode according to constant frequency vibration.

[0019] In the above method for calculating the flow-induced stress and fatigue life of bellows, the calculation formulas for the excitation velocities of each mode of the bellows in step 2 are as follows:

[0020] The excitation velocity of each mode of the bellows = the modal frequency of the bellows × the wave pitch / Stuhal number.

[0021] In the above method for calculating the flow-induced stress and fatigue life of bellows, the formula for calculating the flow velocity of the medium in step 3 is as follows:

[0022] The flow velocity of the medium = the mass flow rate of the medium / the density of the medium / the inner diameter and cross-sectional area of ​​the bellows.

[0023] In the above method for calculating the flow-induced stress and fatigue life of bellows, the formula for calculating the pulsating pressure in step 7 is as follows:

[0024] Pulsating pressure = 0.5 × density of medium × excitation velocity of excitation mode 2 .

[0025] In the above method for calculating the flow-induced stress and fatigue life of bellows, the formula for calculating the cross-sectional area of ​​the bellows in step 8 is as follows:

[0026] The cross-sectional area of ​​the bellows = π × (outer diameter of the bellows) 2 - Inner diameter of the bellows 2 ) / 4.

[0027] In the above method for calculating the flow-induced stress and fatigue life of bellows, the formula for calculating the force exerted by the medium on the bellows in step 9 is as follows:

[0028] The force exerted by the medium on the bellows = pulsating pressure × cross-sectional area of ​​the bellows.

[0029] In the above method for calculating the flow-induced stress and fatigue life of bellows, the formula for calculating the single-wave displacement of the bellows in step 10 is as follows:

[0030] The single-wave displacement of a bellows = the force exerted by the medium on the bellows / the single-wave stiffness of the bellows.

[0031] In the above method for calculating the flow-induced stress and fatigue life of the bellows, in step 12, the stress of the bellows calculated in step 11 is corrected using the following formula:

[0032] The corrected stress of the bellows = stress of the bellows × magnification factor × uncertainty.

[0033] In the above method for calculating the flow-induced stress and fatigue life of the bellows, in step 12, if acoustic resonance is determined to exist based on the acoustic resonance result obtained in step 6, then the amplification factor is not equal to 1.

[0034] In the above method for calculating the flow-induced stress and fatigue life of the bellows, in step 12, if it is determined that there is no acoustic resonance based on the acoustic resonance result obtained in step 6, then the amplification factor is equal to 1.

[0035] The present invention has the following advantages:

[0036] (1) This invention discloses a method for calculating the flow-induced stress and fatigue life of a corrugated pipe. Based on the flow-induced vibration phenomenon and mechanism of the corrugated pipe, a theoretical calculation method for the flow-induced stress and fatigue life is established.

[0037] (2) This invention discloses a method for calculating the flow-induced stress and fatigue life of a bellows, which takes into account the axial modes and local bending modes of the bellows and provides a more comprehensive analysis of the possible excitation modes.

[0038] (3) This invention discloses a method for calculating the flow-induced stress and fatigue life of a bellows, and proposes a method for calculating the excitation flow velocity of each mode of the bellows, as well as an excitation determination method.

[0039] (4) This invention discloses a method for calculating the flow-induced stress and fatigue life of a bellows, and proposes a method for calculating the resultant force of a single wave of fluid on the bellows by using the dynamic pressure of the bellows medium as the pulsating pressure and the pulsating pressure and the structural dimensions of the bellows.

[0040] (5) This invention discloses a method for calculating the flow-induced stress and fatigue life of a corrugated pipe. It adopts a method of calculating the single-wave displacement and structural stress by applying force to a single wave of the corrugated pipe, and obtains the structural stress of the corrugated pipe under flow-induced vibration.

[0041] (6) This invention discloses a method for calculating the flow-induced stress and fatigue life of a bellows, which takes into account the amplification effect of the acoustic resonance of the bellows on the mechanical vibration through the first-order radial acoustic mode of the bellows.

[0042] (7) This invention discloses a method for calculating the flow-induced stress and fatigue life of a corrugated pipe. By using the material SN curve and the method of equating flow-induced vibration to single-frequency vibration, the fatigue life of flow-induced vibration is evaluated. Attached Figure Description

[0043] Figure 1 This is a flowchart of a method for calculating flow-induced stress and fatigue life of a bellows in an embodiment of the present invention. Detailed Implementation

[0044] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments disclosed in the present invention will be described in further detail below with reference to the accompanying drawings.

[0045] Reference Figure 1 In this embodiment, the method for calculating the flow-induced stress and fatigue life of the bellows includes:

[0046] Step 1: Calculate the modal frequencies of the bellows based on its inner diameter, wave height, number of layers, single-layer wall thickness, and wave pitch.

[0047] In this embodiment, the modal frequencies include: axial modal frequencies and local bending modal frequencies.

[0048] Step 2: Based on the modal frequencies of the bellows obtained in Step 1, and combined with the Stuhlhal number and wave pitch, calculate the excitation velocities of the bellows in each mode.

[0049] In this embodiment, the calculation formulas for the excitation velocities of each mode of the bellows are as follows:

[0050] The excitation velocity of each mode of the bellows = the modal frequency of the bellows × the wave pitch / Stuhal number

[0051] Step 3: Calculate the flow velocity of the medium based on the density, mass flow rate, and inner diameter cross-sectional area of ​​the medium inside the bellows.

[0052] In this embodiment, the formula for calculating the flow velocity of the medium is as follows:

[0053] Medium flow velocity = mass flow rate of medium / density of medium / inner diameter and cross-sectional area of ​​bellows

[0054] Step 4: Calculate the first-order radial acoustic mode frequency of the bellows based on its inner diameter, wave height, average fillet radius, and the sound velocity of the medium.

[0055] Step 5: Compare the flow velocity of the medium calculated in Step 3 with the excitation flow velocity of each mode of the bellows calculated in Step 2; if the flow velocity of the medium reaches or exceeds the excitation flow velocity of a certain mode of the bellows, then it is determined that the certain mode is excited, and combined with Step 1, the frequency of the excited mode is obtained.

[0056] Step 6: Compare the frequency of the excitation mode obtained in Step 5 with the frequency of the first-order radial acoustic mode of the bellows calculated in Step 4 to obtain the acoustic resonance result.

[0057] In this embodiment, if the frequency of the excitation mode reaches or exceeds the first-order radial acoustic mode frequency of the bellows, the acoustic resonance result is: acoustic resonance exists; otherwise, the acoustic resonance result is: acoustic resonance does not exist.

[0058] Step 7: Based on the frequency of the excitation mode obtained in Step 5, and combined with Step 2, obtain the excitation velocity of the excitation mode; based on the obtained excitation velocity of the excitation mode, and combined with the density of the medium, calculate the pulsating pressure.

[0059] In this embodiment, the formula for calculating pulsating pressure is as follows:

[0060] Pulsating pressure = 0.5 × density of medium × excitation velocity of excitation mode 2

[0061] Step 8: Calculate the cross-sectional area of ​​the corrugated pipe based on its outer and inner diameters.

[0062] In this embodiment, the formula for calculating the cross-sectional area of ​​the bellows is as follows:

[0063] The cross-sectional area of ​​the bellows = π × (outer diameter of the bellows) 2 - Inner diameter of the bellows 2 ) / 4

[0064] Step 9: Calculate the force exerted by the medium on the bellows based on the pulsating pressure obtained in Step 7 and the cross-sectional area of ​​the bellows obtained in Step 8.

[0065] In this embodiment, the formula for calculating the force exerted by the medium on the bellows is as follows:

[0066] The force exerted by the medium on the bellows = pulsating pressure × cross-sectional area of ​​the bellows

[0067] Step 10: Based on the force of the medium on the bellows calculated in Step 9, and combined with the single-wave stiffness of the bellows, calculate the single-wave displacement of the bellows.

[0068] In this embodiment, the formula for calculating the single-wave displacement of the bellows is as follows:

[0069] Single-wave displacement of a bellows = Force exerted by the medium on the bellows / Single-wave stiffness of the bellows

[0070] Step 11: Calculate the stress of the bellows based on the single-wave displacement of the bellows obtained in Step 10.

[0071] Step 12: Based on the acoustic resonance results obtained in Step 6, and considering a certain degree of uncertainty, the stress of the bellows calculated in Step 11 is corrected to obtain the corrected bellows stress.

[0072] In this embodiment, the stress of the bellows calculated in step 11 is corrected using the following formula:

[0073] The corrected stress of the bellows = stress of the bellows × magnification factor × uncertainty

[0074] If the acoustic resonance result obtained in step 6 determines that an acoustic resonance exists, then the amplification factor is not equal to 1; if the acoustic resonance result obtained in step 6 determines that an acoustic resonance does not exist, then the amplification factor is equal to 1.

[0075] Step 13: Based on the corrected bellows stress calculated in Step 12, and combined with the SN curve of the bellows material, calculate the fatigue damage of the bellows under each excitation mode according to constant frequency vibration.

[0076] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention by utilizing the methods and techniques disclosed above without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

[0077] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for calculating flow-induced stress and fatigue life of a bellows, characterized in that, include: Step 1: Calculate the modal frequencies of the bellows based on its inner diameter, wave height, number of layers, single-layer wall thickness, and wave pitch; wherein, each modal frequency includes: each axial modal frequency and local bending modal frequency. Step 2: Based on the modal frequencies of the bellows calculated in Step 1, and combined with the Stuhlhal number and wave pitch, calculate the excitation velocities of the bellows in each mode. Step 3: Calculate the flow velocity of the medium based on the density, mass flow rate, and inner diameter cross-sectional area of ​​the medium inside the bellows. Step 4: Calculate the first-order radial acoustic mode frequency of the bellows based on its inner diameter, wave height, average fillet radius, and the sound velocity of the medium. Step 5: Compare the flow velocity of the medium calculated in Step 3 with the excitation flow velocity of each mode of the bellows calculated in Step 2; if the flow velocity of the medium reaches or exceeds the excitation flow velocity of a certain mode of the bellows, then it is determined that the certain mode is excited, and combined with Step 1, the frequency of the excited mode is obtained. Step 6: Compare the frequency of the excitation mode obtained in Step 5 with the frequency of the first-order radial acoustic mode of the bellows calculated in Step 4 to obtain the acoustic resonance result; if the frequency of the excitation mode reaches or exceeds the frequency of the first-order radial acoustic mode of the bellows, the acoustic resonance result is: acoustic resonance exists; otherwise, the acoustic resonance result is: acoustic resonance does not exist. Step 7: Based on the frequency of the excitation mode obtained in Step 5, and combined with Step 2, obtain the excitation velocity of the excitation mode; based on the obtained excitation velocity of the excitation mode, and combined with the density of the medium, calculate the pulsating pressure. Step 8: Calculate the cross-sectional area of ​​the corrugated pipe based on its outer and inner diameters. Step 9: Calculate the force exerted by the medium on the bellows based on the pulsating pressure calculated in Step 7 and the cross-sectional area of ​​the bellows calculated in Step 8. Step 10: Based on the force of the medium on the bellows calculated in Step 9, and combined with the single-wave stiffness of the bellows, calculate the single-wave displacement of the bellows. Step 11: Calculate the stress of the bellows based on the single-wave displacement of the bellows obtained in Step 10. Step 12: Based on the acoustic resonance results obtained in Step 6, and considering a certain degree of uncertainty, the stress of the bellows calculated in Step 11 is corrected to obtain the corrected bellows stress. Step 13: Based on the corrected bellows stress calculated in Step 12, and combined with the SN curve of the bellows material, calculate the fatigue damage of the bellows under each excitation mode according to constant frequency vibration.

2. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 1, characterized in that, In step 2, the calculation formulas for the excitation velocities of each mode of the bellows are as follows: The excitation velocity of each mode of the bellows = the modal frequency of the bellows × the wave pitch / Stuhal number.

3. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 1, characterized in that, In step 3, the formula for calculating the flow velocity of the medium is as follows: The flow velocity of the medium = the mass flow rate of the medium / the density of the medium / the inner diameter and cross-sectional area of ​​the bellows.

4. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 1, characterized in that, In step 7, the formula for calculating the pulsating pressure is as follows: Pulsating pressure = 0.5 × density of medium × excitation velocity of excitation mode 2 .

5. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 1, characterized in that, In step 8, the formula for calculating the cross-sectional area of ​​the bellows is as follows: The cross-sectional area of ​​the bellows = π × (outer diameter of the bellows) 2 - Inner diameter of the bellows 2 ) / 4.

6. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 1, characterized in that, In step 9, the formula for calculating the force exerted by the medium on the bellows is as follows: The force exerted by the medium on the bellows = pulsating pressure × cross-sectional area of ​​the bellows.

7. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 1, characterized in that, In step 10, the formula for calculating the single-wave displacement of the bellows is as follows: The single-wave displacement of a bellows = the force exerted by the medium on the bellows / the single-wave stiffness of the bellows.

8. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 1, characterized in that, In step 12, the stress of the bellows calculated in step 11 is corrected using the following formula: The corrected stress of the bellows = stress of the bellows × magnification factor × uncertainty.

9. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 8, characterized in that, In step 12, if acoustic resonance is determined to exist based on the acoustic resonance result obtained in step 6, then the amplification factor is not equal to 1.

10. The method for calculating flow-induced stress and fatigue life of a corrugated pipe according to claim 8, characterized in that, In step 12, if it is determined that there is no acoustic resonance based on the acoustic resonance result obtained in step 6, then the amplification factor is equal to 1.

Images