Load calculation method for civil aircraft flap needle bearing

Abstract

The invention belongs to the field of uncontained bearing load calculation, and discloses a load calculation method for a civil aircraft flap needle roller bearing. On the basis of the Hertz contact theory, a bearing outer ring is regarded as a flexible outer ring, the influence of bending elastic deformation of the bearing outer ring under an external load on load distribution is considered, a comprehensive deformation calculation formula between a roller pin and a roller path is established, then the roller pin is subjected to slicing discretization treatment along the axis of the needle roller bearing, and according to the obtained deformation, the bearing outer ring is subjected to deformation calculation; the contact load generated between the slice unit and the inner and outer raceways is calculated, then a static equation set of the flap needle bearing is established, the deformation amount of each position of the needle bearing is solved through a Levenberg-Marquardt algorithm, and the load distribution of the needle bearing can be calculated. On the basis, a relational expression among the maximum contact stress, the load and the displacement is established, and the maximum contact stress distribution of the needle bearing is obtained through a numerical iteration method.

Description

Technical Field

[0001] The invention belongs to the field of non-contained bearing load calculation and relates to a load calculation method for a civil aircraft flap needle roller bearing. Background Art

[0002] The flap mechanism of civil aircraft determines the aircraft's safety during takeoff, landing, and low-speed flight. Needle roller bearings, as key and vulnerable components in the flap mechanism, play a vital role. Unlike conventional needle roller bearings, those in flap mechanisms typically function as rollers. Specifically, the inner ring of the bearing is typically connected to the slide frame, while the outer ring rolls on the rails, providing support for the slide frame's movement.

[0003] Existing load analysis models based on rigid ring theory treat the bearing ring as rigid and ignore its structural deformation, accounting only for the Hertzian contact deformation between the rolling elements and the raceway. Flap needle roller bearings are non-contained bearings with an external cylindrical shape. Their outer rings are not contained by the bearing housing, gearbox, or other components, and therefore cannot be considered rigid rings. Under load, the outer rings undergo elastic bending deformation, changing from a circular shape to an elliptical shape. This ring deformation directly affects the bearing's load distribution and load-bearing performance.

[0004] Currently, domestic and foreign scholars have not yet provided a specific solution for the load distribution of non-contained bearings. This paper proposes a load calculation method for non-contained needle roller bearings that takes into account the bending elastic deformation of the outer ring, providing a theoretical basis for the design and application of cylindrical non-contained bearings. Summary of the Invention

[0005] The purpose of the present invention is to provide a load calculation method for an outer cylindrical non-enclosed needle roller bearing, and to analyze and calculate the load distribution of a civil aircraft flap needle roller bearing under the condition of comprehensively considering the bending elastic deformation of the bearing outer ring and the contact deformation between the rolling element and the raceway.

[0006] The technical solution of the present invention:

[0007] A method for calculating the load of a civil aircraft flap needle roller bearing, wherein the needle roller bearing is used as a roller in a civil aircraft flap motion mechanism, wherein the inner ring of the needle roller bearing is connected to a slide rail frame, and the outer ring of the needle roller bearing passively rotates on the rail; the method comprises the following steps:

[0008] Step 1: Establish a global coordinate system on the inner ring of the needle roller bearing, with the X-axis coinciding with the axis of the needle roller bearing and the Y-axis consistent with the direction of the radial load applied to the needle roller bearing; establish a local coordinate system at the center of each needle roller, with the X-axis along the axis of the needle roller bearing and the Y-axis pointing to the center of the inner ring of the needle roller bearing; assuming that the inner ring of the needle roller bearing is fixed and the outer ring of the needle roller bearing moves radially under the action of the load, the radial displacement of the center of the outer ring of the needle roller bearing in the global coordinate system is δ OR, the radial displacement of the needle roller in the local coordinate system is δ yj When the bending deformation of the outer ring of the needle roller bearing is not considered, the Hertzian elastic contact deformation between the needle roller and the inner and outer ring raceways is:

[0009]

[0010] Wherein, j is the needle roller number, j=1, 2, ..., Z, Z is the number of needle rollers; is the needle roller azimuth angle, the maximum load needle roller azimuth angle

[0011] Step 2: Calculate the bending elastic deformation of the outer ring of the needle roller bearing. The outer ring of a non-contained needle roller bearing undergoes bending elastic deformation under the action of load. This bending elastic deformation will significantly affect the load distribution inside the needle roller bearing. In order to accurately calculate the load distribution inside the needle roller bearing, the deformation of each point on the outer ring of the needle roller bearing under the action of load must be considered. The outer ring of the needle roller bearing is regarded as a flexible ring. Assume that the structural elastic deformation of the flexible ring only occurs in the radial plane. The flexible ring will only bend and will not produce circumferential stretching or compression. According to the theory of elastic mechanics, under the action of a single load Q, the deformation calculation formula of the outer ring of the needle roller bearing is:

[0012]

[0013] Among them, K r is the stiffness coefficient, r is the subscript of the stiffness coefficient, r=1, 2,…,∞; θ is the arbitrary azimuth angle of the outer ring of the needle roller bearing;

[0014] External radial load F on needle roller bearings r and the needle roller contact load Q j Under the combined effect of the two, the calculation formula for the bending deformation of the outer ring of the needle roller bearing is:

[0015]

[0016] in, is the needle rolling azimuth. In order to facilitate the calculation, the first-order approximation is performed on the series part on the right side of the above formula:

[0017]

[0018] Among them, R z is the neutral circle radius of the outer ring of the needle roller bearing, I is the cross-sectional inertia moment of the outer ring of the needle roller bearing, t is the thickness of the outer ring of the needle roller bearing, and E is the elastic modulus of the outer ring of the needle roller bearing. The first-order approximate formula can meet the engineering calculation requirements. When a more accurate solution is required, the stiffness coefficient K can be obtained by establishing a finite element model of the outer ring. r .

[0019] Step three, calculate the comprehensive deformation of the needle and the inner and outer raceway

[0020] Along the axis of the needle bearing, the needle is sliced and discretized, and the number of slices is m. In order to avoid stress concentration, the arc slope is used to process the generatrix of the needle, and the center of the circle is on the center line of the needle. The modification amount of the kth slice is calculated according to the following formula:

[0021]

[0022] For the full convex needle, the modification amount of the kth slice can be calculated by the following formula:

[0023]

[0024] Where Δc is the convexity, l c is the generatrix modification length, R c is the radius of the arc, l is the effective length of the needle, l1 is the length of the straight generatrix part in the middle of the needle, l k is the length of the kth slice of the needle from the end of the needle, g k is the gap between the kth needle slice and the raceway, that is, the modification amount.

[0025] The Hertz elastic contact deformation δ ko (j) and the outer race bending elastic deformation δ(θ) are considered comprehensively ki (j), the needle modification amount g k , and the needle bearing radial clearance e to obtain the comprehensive deformation of the needle and the inner and outer raceways:

[0026]

[0027] Where k is the serial number of the needle slice, k = 1, 2, …, m. In order to ensure that the needle and the raceway are always in contact and avoid entering the non-bearing area, the following conditions must be met:

[0028]

[0029] Step four, establish the quasi-static equation of the flap needle bearing:

[0030] According to the Hertz contact theory, the contact load between the needle and the inner and outer raceways is calculated by the load displacement relationship:

[0031]

[0032] Where C L is the contact stiffness coefficient:

[0033]

[0034] Where E1 and E2 are the elastic moduli of the needle roller and the ring respectively; v1 and v2 are the Poisson's ratios of the needle roller and the ring respectively; l is the effective length of the needle roller.

[0035] External radial load F on the bearing outer ring r Contact load Q with needle roller jo It is in equilibrium state, and the quasi-static equilibrium equation can be established:

[0036]

[0037] Among them, F c is the centrifugal force on the needle roller, and its expression is:

[0038]

[0039] Where ρ is the density of the needle roller material; D w is the needle roller diameter; n b is the bearing revolution speed; d m is the bearing pitch diameter; l is the effective length of the needle roller.

[0040] Step 5: Solve the pseudo-static equations of the flap needle roller bearing

[0041] Equation (11) is based on δ OR and δ yj The equation with unknown quantity is solved iteratively by using Levenberg-Marquardt algorithm, and the result is δ OR and δ yj Substitute the value of into formula (7) to calculate the comprehensive deformation between the needle roller and the raceway; then substitute the comprehensive deformation into the load-displacement relationship formula (9) to calculate the contact load of each needle roller, and finally obtain the load distribution of the flap needle roller bearing.

[0042] Step 6: Calculate the maximum contact stress of the needle roller

[0043] The contact between the needle roller and the raceway in the flap needle roller bearing is a finite length line contact problem, and its maximum contact stress p 0k With contact load Q j and comprehensive deformation The relationship is as follows:

[0044]

[0045] in, a k h is the half width of the contact area between the slice and the raceway, k is the thickness of the needle slice, D ik is the flexibility coefficient (derived from Hertz's line contact theory); g i is the modification amount of the i-th needle roller slice.

[0046] Equation (13) is an n+1 order linear equation system. The contact load Q of the needle roller obtained in step 5 is j and modification amount g i The maximum stress p of the contact between the slice unit and the inner and outer rings can be obtained by numerical iteration method. 0k .

[0047] Beneficial effects of the present invention:

[0048] (1) For the outer ring non-enclosed needle roller bearing, based on the Hertz contact theory, the influence of factors such as the bending elastic deformation of the bearing outer ring, radial clearance and oil film thickness are considered, and the calculation formula for the comprehensive deformation of the outer ring non-enclosed needle roller bearing is derived;

[0049] (2) Considering the contact deformation between the needle roller and the inner and outer rings, the bending elastic deformation of the outer ring, the needle roller profile modification, and the radial clearance, the static equations of the flap needle roller bearing are established;

[0050] (3) Based on the load distribution of the needle roller bearing, a corresponding calculation formula for the maximum contact stress between the needle roller and the raceway was established, which can provide theoretical support for the derivation of the rated static load of this type of bearing. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 Schematic diagram of the force on the flap needle roller bearing.

[0052] Figure 2 Schematic diagram of the bearing coordinate system, where (a) is the needle roller local coordinate system and (b) is the bearing global coordinate system.

[0053] Figure 3 Pseudostatic analysis roadmap for flap needle roller bearings.

[0054] Figure 4 The bending deformation of the outer ring under different external loads.

[0055] Figure 5 The bearing load distribution under different external loads. DETAILED DESCRIPTION

[0056] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0057] like Figure 1 As shown in the figure, the structural parameters of the needle roller bearing on a certain type of flap motion mechanism are: initial radial clearance e = 0.03mm, outer diameter of the bearing outer ring D = 60mm, thickness of the bearing outer ring t = 9.06mm, inner diameter of the bearing inner ring d = 20mm, thickness of the bearing inner ring t n=8.56mm, number of needle rollers Z = 52, diameter of needle roller D w = 2.38mm, effective length of needle roller l = 11mm, number of needle roller slices m = 135. The radial load F on the bearing r =60kN.

[0058] Step 1: If Figure 2 As shown in the figure, a global coordinate system is established on the inner ring of the needle roller bearing, with the X-axis coinciding with the axis of the needle roller bearing and the Y-axis aligned with the direction of the radial load applied to the needle roller bearing. A local coordinate system is established at the center of each needle roller, with the X-axis along the axis of the needle roller bearing and the Y-axis pointing to the center of the inner ring of the needle roller bearing. Assuming that the inner ring of the needle roller bearing is fixed and the outer ring of the needle roller bearing moves radially under the action of the load, the radial displacement of the center of the outer ring of the needle roller bearing in the global coordinate system is δ OR , the radial displacement of the needle roller in the local coordinate system is δ yj .

[0059] Step 2: Calculate the bending elastic deformation of the needle roller bearing outer ring

[0060] The outer ring of a non-contained needle roller bearing undergoes bending elastic deformation under load, which significantly affects the load distribution inside the bearing. In order to accurately calculate the load distribution inside a civil aircraft flap needle roller bearing, the deformation of each point on the outer ring under load must be considered. r and the needle roller contact load Q j Under the combined effect of the two, the calculation formula for the bending deformation of the outer ring of the needle roller bearing is:

[0061]

[0062] in, is the needle roller azimuth angle; θ is the arbitrary azimuth angle of the needle roller bearing outer ring;

[0063] Step 3: Calculate the combined deformation of the needle roller and the inner and outer ring raceways

[0064] The needle roller adopts the arc slope trimming method and slices it. The center of the circle is on the center line of the needle roller. The trimming amount of the kth slice is calculated as follows:

[0065]

[0066] Where Δc is the convexity, l c is the busbar trimming length, R c is the arc radius, l is the effective length of the needle roller, l1 is the length of the straight generatrix in the middle of the needle roller, l k g is the length from the center of the kth needle roller to the end of the needle roller, k is the gap between the kth needle roller and the raceway, i.e. the trimming amount.

[0067] The Hertz elastic contact deformation δ is comprehensively considered ko (j), δ ki (j), the outer ring bending elastic deformation δ(θ), the needle shape correction g k The needle bearing radial clearance e obtains the comprehensive deformation of the needle and the inner and outer ring raceways:

[0068]

[0069] Wherein, k is the serial number of the needle slice, k = 1, 2, …, m. In order to ensure that the needle and the raceway are always in contact, and avoid entering the non-bearing area, it is necessary to satisfy:

[0070]

[0071] Step four: establish the statics equation set of the flap needle bearing

[0072] According to the Hertz contact theory, the contact load between the needle and the inner and outer raceways is calculated respectively according to the load displacement relationship:

[0073]

[0074] Wherein, C L is the contact stiffness coefficient:

[0075]

[0076] Wherein, E1, E2 are the elastic modulus of the needle and the ring respectively; v1, v2 are the poisson's ratio of the needle and the ring respectively; l is the effective length of the needle.

[0077] The bearing outer ring is in equilibrium state under the external radial load F r And the needle contact load Q jo The quasi-static equilibrium equation can be established:

[0078]

[0079] Wherein, F c Is the centrifugal force received by the needle, and its expression is:

[0080]

[0081] Wherein, ρ is the density of the needle material; D w Is the diameter of the needle; n b Is the bearing revolution speed; d m Is the pitch diameter of the bearing; l is the effective length of the needle.

[0082] Step five: solve the quasi-static equation set of the flap needle bearing

[0083] Figure 3 As shown, first input the bearing structure, working conditions, material and other parameters, assume that the bending deformation of the bearing outer ring δ(θ) = 0, that is, the outer ring is a rigid ring, substitute the Hertz rigid ring theory to solve, and obtain the bearing load distribution Q under the rigid ring assumption j , substitute it into formula (1) to calculate the bending deformation of the bearing outer ring, then calculate the comprehensive deformation of the needle roller and the inner and outer ring raceways, and obtain the needle roller contact load through the load-displacement relationship; use the Levenberg-Marquardt algorithm to iteratively solve the equation group and obtain the load distribution of the flap needle roller bearing under the condition of outer ring bending deformation. Figure 4 The bending deformation of the bearing outer ring under different external loads shows that the outer ring deforms under load. The greater the external load, the more significant the ring deformation. At 30kN, 50kN and 60kN, the maximum deformation of the ring is -27.72μm, -46.68μm and -56.18μm respectively. Figure 5 Figure 2 shows the load distribution of the bearing under different external loads. The "flattening" of the rings redistributes the forces between the needle rollers and the rings within the bearing. The ring radius decreases at the center of the load-bearing zone, at the very bottom of the ring. This causes the needle rollers there to bear a greater load, while the needle rollers at the edge of the load-bearing zone bear less load, resulting in a steeper overall load distribution curve. Under the rigid ring assumption, the maximum contact loads between the needle rollers and the outer ring under different external loads are 3466N and 6132N, respectively. When considering outer ring bending deformation, the maximum contact loads are 4127N and 7934N, respectively, representing increases of 19% and 29% compared to the rigid ring model.

[0084] Step 6: Calculate the maximum contact stress of the needle roller

[0085] like Figure 3 As shown in the figure, under the condition that the load distribution of the flap needle roller bearing is known, the following formula is numerically iterated based on the flexibility coefficient to obtain the contact stress distribution between the needle roller and the raceway.

[0086]

[0087] in, a k h is the half width of the contact area between the slice and the raceway, k is the thickness of the needle slice, D ik is the flexibility coefficient (derived from Hertz's line contact theory); g i is the modification amount of the i-th needle roller slice.

[0088] Comparison reveals that as the outer ring bending deformation increases, the maximum load inside the bearing increases rapidly. The maximum contact stress also exhibits the same trend as the maximum contact load. Higher contact stress directly shortens the bearing fatigue life and increases the risk of failures such as plastic deformation, so it should be avoided as much as possible.

Claims

1. A load calculation method for a civil aircraft flap needle roller bearing, characterized in that: Needle roller bearings are used as rollers in the flap mechanism of civil aircraft. The inner ring of the needle roller bearing is connected to the slide rail frame, and the outer ring of the needle roller bearing passively rotates on the rail. The following steps are included: Step 1: Establish a global coordinate system on the inner ring of the needle roller bearing, with the X-axis coinciding with the axis of the needle roller bearing and the Y-axis consistent with the direction of the radial load applied to the needle roller bearing; establish a local coordinate system at the center of each needle roller, with the X-axis along the axis of the needle roller bearing and the Y-axis pointing to the center of the inner ring of the needle roller bearing; assuming that the inner ring of the needle roller bearing is fixed and the outer ring of the needle roller bearing moves radially under the action of the load, the radial displacement of the center of the outer ring of the needle roller bearing in the global coordinate system is δ OR , the radial displacement of the needle roller in the local coordinate system is δ yj When the bending deformation of the outer ring of the needle roller bearing is not considered, the Hertzian elastic contact deformation between the needle roller and the inner and outer ring raceways is: Wherein, j is the needle roller number, j = 1, 2, ..., Z, Z is the number of needle rollers; is the needle roller azimuth angle, the maximum load needle roller azimuth angle Step 2: Calculate the bending elastic deformation of the outer ring of the needle roller bearing; The outer ring of a non-contained needle roller bearing undergoes bending elastic deformation under load, which significantly affects the load distribution within the needle roller bearing. To accurately calculate the load distribution within the needle roller bearing, the deformation at each point on the outer ring of the needle roller bearing under load must be considered. The outer ring of the needle roller bearing is considered as a flexible ring, and it is assumed that the structural elastic deformation of the flexible ring occurs only in the radial plane. The flexible ring only bends and does not produce circumferential tension or compression. According to the theory of elastic mechanics, the deformation calculation formula for the outer ring of the needle roller bearing under a single load Q is: Among them, K k is the stiffness coefficient, r is the subscript of the stiffness coefficient, r=1,2,…,∞; θ is the arbitrary azimuth angle of the outer ring of the needle roller bearing; External radial load F on needle roller bearings r and the needle roller contact load Q j Under the combined effect of the two, the calculation formula for the bending deformation of the outer ring of the needle roller bearing is: in, is the needle roller azimuth; Perform first-order approximation on the right-hand side of the above equation: Among them, R z is the neutral circle radius of the needle roller bearing outer ring, I is the cross-sectional inertia moment of the needle roller bearing outer ring, t is the thickness of the needle roller bearing outer ring, and E is the elastic modulus of the needle roller bearing outer ring; Step 3: Calculate the combined deformation of the needle roller and the inner and outer ring raceways; Along the axis of the needle roller bearing, the needle roller is sliced ​​and discretized, and the number of needle roller slices is m. In order to avoid stress concentration, the arc slope trimming method is used to process the needle roller generatrix, with the center of the circle on the needle roller center line. The trimming amount of the kth slice is calculated according to the following formula: For a fully convex needle roller, the modification amount of the kth needle roller slice is as follows: Where Δc is the convexity, l c is the busbar trimming length, R c is the arc radius, l is the effective length of the needle roller, l1 is the length of the straight generatrix in the middle of the needle roller, l k g is the length from the center of the kth needle roller to the end of the needle roller, k is the gap between the kth needle roller slice and the raceway, i.e. the trimming amount; Comprehensive consideration of Hertz elastic contact deformation δ ko (j), δ ki (j), outer ring bending elastic deformation δ(θ), needle roller profile modification g k , the radial clearance e of the needle roller bearing is obtained as the comprehensive deformation of the needle roller and the inner and outer ring raceways: In order to ensure that the needle roller is always in contact with the raceway and avoid entering the non-load-bearing area, the following conditions must be met: Step 4: Establish the pseudo-static equations of the flap needle roller bearing; According to Hertz contact theory, the contact loads between the needle roller and the inner and outer raceways are calculated based on the load-displacement relationship: Among them, C L is the contact stiffness coefficient: Where, E1 and E2 are the elastic moduli of the needle roller and the ring respectively; v1 and v2 are the Poisson's ratios of the needle roller and the ring respectively; l is the effective length of the needle roller; External radial load F on the bearing outer ring r Contact load Q with needle roller jo Under the equilibrium state, the quasi-static equilibrium equation is established: Among them, F c is the centrifugal force on the needle roller, and its expression is: Where ρ is the density of the needle roller material; D w is the needle roller diameter; n b is the bearing revolution speed; d m is the bearing pitch diameter; Step 5, solving the pseudo-static equations of the flap needle roller bearing; Equation (11) is based on δ OR and δ yj The equation with unknown quantity is solved iteratively by using Levenberg-Marquardt algorithm, and the result is δ OR and δ yj Substitute the value of into formula (7) to calculate the comprehensive deformation between the needle roller and the raceway; then substitute the comprehensive deformation into the load-displacement relationship formula (9) to calculate the contact load of each needle roller, and finally obtain the load distribution of the flap needle roller bearing; Step 6: Calculate the maximum contact stress of the needle roller; The contact between the needle roller and the raceway in the needle roller bearing is a finite length line contact problem, and its maximum contact stress p 0k With contact load Q j and comprehensive deformation The relationship is as follows: in, a k h is the half width of the contact area between the slice and the raceway, k is the thickness of the needle slice, D ik is the flexibility coefficient; g i is the modification amount of the i-th needle roller slice; Equation (13) is an n+1 order linear equation system. The contact load Q of the needle roller obtained in step 5 is j and modification amount g i The maximum stress p of the contact between the slice unit and the inner and outer rings can be obtained by numerical iteration method. 0k .

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