CN114547771B - Contact pressure controllable non-circular C-shaped damping ring design method - Google Patents
Contact pressure controllable non-circular C-shaped damping ring design method Download PDFInfo
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Abstract
The invention provides a design method of a non-circular C-shaped damping ring with controllable contact pressure. Based on the design method, the corresponding non-circular C-shaped damping ring free profile can be designed according to different contact pressure requirements, so that the contact pressure not only depends on the rotation speed of the gear, but also depends on the deformation of the profile, the problem that the contact pressure is uncontrollable under the designed rotation speed is solved, and the optimal contact pressure corresponding to the maximum damping efficiency of the damping ring can be obtained under the designed rotation speed. Based on the design method, the contact pressure between the damping ring and the gear is uniformly distributed and the contact rate is high. When the gear is installed in the gear, uniform contact pressure can be generated, the contact rate is high, and the reduction of vibration reduction performance caused by a non-contact area can be avoided.
Description
Technical Field
The invention relates to a damping piece design method.
Background
When excessive vibration energy is accumulated in an aviation high-speed thin web gear due to traveling wave resonance, sudden traveling wave resonance damage is easy to occur, the damage form is represented by that the gear is broken from a web to a gear tooth in a blocking mode, and the fracture property is pitch-diameter type fatigue fracture. Broken tooth masses directly threaten the overall drive train and even engine safety, and this form of failure has caused many major flight failures. Therefore, the aviation high-speed thin-web gear has extremely high risk of traveling wave resonance damage, effective measures are needed to reduce vibration energy during traveling wave resonance so as to avoid failure risks, and the damping ring is a technical means which has both light weight and vibration damping effect.
The C-shaped damping ring can be used for reducing vibration energy during traveling wave resonance and weakening the influence of the resonance on the gear transmission system. However, the existing C-shaped damping ring technology has at least the following disadvantages:
(1) The existing C-shaped damping ring is circular in a natural state and is called as a regular circular C-shaped damping ring. Since the pre-pressure of the damping ring is small, the working contact pressure of the damping ring and the gear is mainly caused by the deformation interference of the damping ring and the gear at the contact surface under the action of high-speed centrifugation, obviously, the working contact pressure depends on the rotation speed of the gear and is positively correlated with the square of the rotation speed, namely the working contact pressure is larger when the rotation speed of the gear is higher, and the rising slope of the contact pressure is larger. However, the working contact pressure of the damping ring is highly dependent on the centrifugal field formed by high rotating speed, the contact pressure is not controllable, and the difficulty of designing the optimal contact pressure required by reaching the maximum friction energy consumption is increased.
(2) Because the diameter of the ring groove is slightly smaller than the maximum size of the damping ring, the damping ring can form pre-pressure after being installed in the ring groove, although the right circular C-shaped damping ring is circular in a natural state, the material is deformed after being installed in the ring groove, the damping ring can only be in local contact with the gear, namely a non-contact area is formed, the contact pressure distribution is uneven, an effective area of friction energy consumption of the damping ring is lost, and the vibration damping capacity of the damping ring is further reduced. Although the non-contact area is gradually reduced as the rotation speed increases, the reduction rate is slowed, and the non-contact area does not completely disappear even if the rotation speed increases to a higher level.
(3) The existing C-shaped damping ring is designed according to an empirical formula and data, but the damping ring has the defects of uncontrollable contact pressure, obvious non-contact area and the like, and an effective improved design method is not available at present.
Disclosure of Invention
The invention aims to provide a method for designing a non-circular C-shaped damping ring with controllable contact pressure, which is characterized by comprising the following steps:
the noncircular C-shaped damping ring is a noncircular ring with a notch in a natural state;
after the non-circular C-shaped damping ring is arranged on a ring groove on the gear, the outer side of the non-circular C-shaped damping ring is deformed by pressure to be in a circular closed state;
the non-circular C-shaped damping ring is designed into a profile in a natural state through the following steps:
(1) According to the parameters of the ring grooves, setting the parameters of the section of the non-circular C-shaped damping ring: axial width w, radial thickness t, compression profile radius r r ；
Further, the design range of the axial width w is: (0,0.99w) g ]，w g For the ring groove width, the design range of radial thickness t is: t is greater than or equal to t d ，t d Depth of ring groove, compression profile radius r r Radius r of ring groove g Equal; the number of teeth of the gear is z;
(2) Calculating an allowable stress value [ sigma ] according to the non-circular C-shaped damping ring material parameters and the safety factor n:
the yield strength of the non-circular C-shaped damping ring material is sigma according to the selected material s Density rho and elastic modulus of the material are E; the value range of the safety coefficient n is more than or equal to 1:
the allowable stress value [ sigma ] is:
(3) Calculating the contact pressure p caused by centrifugation c And deformation causing contact pressure p s ：
Calculating the gear rotation speed (rotation speed at which resonance occurs) ω of the resonance point:
wherein: f is the resonance point frequency, z is the number of teeth;
calculating the radius r of the neutral layer n ：
Contact pressure p caused by centrifugation c ：
p c ＝ρ·t·r n ·ω 2 (4)
Calculating the contact pressure p caused by deformation s ：
p s ＝p d -p c (5)
Wherein: p is a radical of d To design contact pressure;
(4) Calculating the free profile coordinates of the noncircular C-shaped damping ring under a polar coordinate system:
the circle center of the non-circular C-shaped damping ring in the installation state is used as a pole, the radial direction is used as a polar axis, a polar coordinate is established in a cross section perpendicular to the axial direction of the non-circular C-shaped damping ring, and the symmetry axis of the C-shaped damping ring in the non-installation state is positioned on the polar axis of 0 degrees:
when the non-circular C-shaped damping ring is bent and deformed in the installation state, any point B on the ring is positioned at a infinitesimal element
The moment for a section with a polar angle θ is:
wherein the content of the first and second substances,
the angle of any point B in the established polar coordinate system is theta, and the angle of the cross section in the established polar coordinate system is theta;
the total bending moment at the theta angle section is:
the axial force at the θ -angle section is:
N(θ)＝p s ·wr n ·(1+cosθ) (8)
the differential equation of the deflection line of the non-circular C-shaped damping ring is as follows:
in the formula, A is the damping ring cross-sectional area, u, v are damping ring cross-sectional centroid along circumference and radial displacement respectively, s is the direction of passing the cross-sectional centroid but being perpendicular to the cross-section, jz is the mark, for the radial thickness is t, the axial width is the rectangular damping ring cross-section of w, its value is:
for a circumferential displacement u, the boundary condition is
u＝0；θ＝0 (11)
For a radial displacement v, the boundary condition is
Circumferential displacement u and radial displacement v can be solved by combining a differential equation of the deflection line and boundary conditions of each displacement;
combining the coordinates (theta, r) of any point of the neutral layer in the mounting state n ) And obtaining the corresponding external surface profile coordinates and internal surface profile coordinates in a natural state as follows:
outer surface profile coordinates (theta) e ,r e ):
Inner surface profile coordinates (θ) i ,r i ):
(5) Judging whether the designed non-circular C-shaped damping ring structure can pass the strength check;
if the damping force can be passed, taking the axial width w and the radial thickness t set in the step (1), the material selected in the step (2) and the non-circular C-shaped damping ring free profile coordinate calculated in the step (4) as design parameters;
and if the axial width w and/or the radial thickness t cannot pass through the step (1), resetting the axial width w and/or the radial thickness t, and repeating the steps to perform design calculation and strength check.
Further, in the step (5), the maximum stress value sigma of the non-circular C-shaped damping ring is calculated max And intensity checking is performed.
Further, when judging whether the designed non-circular C-shaped damping ring structure can be checked through the strength:
calculating the bending stress sigma of each cross section b The maximum value is distributed along the circumferential direction:
further solving theta partial derivatives of the above formula, and substituting the theta partial derivatives into an expression of axial force and bending moment to obtain
σ b The maximum value of the bending stress of each cross section along the circumferential direction is set to be 0 according to a derivative judgment method of the extreme value, and a function extreme value is obtained to further judge the maximum value. According to the analysis, σ b (θ, t/2) takes a maximum value at θ =0. Therefore, the inner point (0,t/2) of the damping ring is the stress risk point where the maximum stress value is located, and further the maximum stress value sigma of the non-circular C-shaped damping ring max Is composed of
σ max ＝σ b (0,t/2) (19)
Judging the maximum stress value sigma of the non-circular C-shaped damping ring max Magnitude and allowable stress value [ sigma ]]Magnitude relation, i.e. whether the following holds
|σ max |＜[σ] (20)
The invention has the technical effects that:
(1) The contact pressure between the damping ring and the gear is controllable. Based on the design method, the corresponding non-circular C-shaped free profile of the damping ring can be designed according to different contact pressure requirements, so that the contact pressure not only depends on the rotation speed of the gear, but also depends on the deformation of the profile, the problem that the contact pressure is uncontrollable under the designed rotation speed is solved, and the optimal contact pressure corresponding to the maximum damping efficiency of the damping ring can be obtained under the designed rotation speed.
(2) Based on the design method, the contact pressure between the damping ring and the gear is uniformly distributed and the contact rate is high. When the gear is installed in the gear, uniform contact pressure can be generated, the contact rate is high, and the reduction of vibration reduction performance caused by a non-contact area can be avoided.
Drawings
FIG. 1 is a schematic view of a non-circular C-shaped damping ring designed according to the present invention;
FIG. 2 is a schematic illustration of the rim ring grooves required for the thin web bevel gear to attach non-circular C-shaped damping rings;
FIG. 3 is a schematic illustration of the rim ring grooves required for the thin web spur gear with the additional non-circular C-shaped damping rings;
FIG. 4 is a schematic illustration of a non-circular C-shaped damping ring attached to a thin web bevel gear;
FIG. 5 is a schematic illustration of a non-circular C-shaped damping ring attached to a thin web spur gear using a double-sided mounting;
FIG. 6 is a schematic illustration of a non-circular C-shaped damping ring attached to a thin web spur gear using a left side mounting;
FIG. 7 is a schematic illustration of a non-circular C-shaped dampening ring attached to a thin web spur gear using a right side mounting;
FIG. 8 is a flow chart of a method of the present invention for implementing a contact pressure controllable non-circular C-shaped damping ring design;
FIG. 9 is a schematic diagram illustrating the structural parameters of the dimensions of the non-circular C-shaped damping ring and the ring groove in the installed state;
FIG. 10 is a schematic view showing the contact pressure between the non-circular C-shaped damping ring and the gear;
FIG. 11 is a schematic diagram of a mechanical model for solving a natural state profile of a non-circular C-shaped damping ring;
FIG. 12 is a schematic view of an aviation high-speed thin-web bevel gear with the addition of a non-circular C-shaped damping ring for achieving different contact pressures;
FIG. 13 shows an aviation high-speed thin-web spur gear with the addition of a non-circular C-shaped damping ring to achieve different contact pressures.
In the figure: non-circular C-shaped damping ring 1, free profile 101, pre-pressing profile 102, bevel gear 2, ring groove 201, straight gear 3 and ring groove 301.
Detailed Description
The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.
In the specific embodiment, the schematic diagram of the non-circular C-shaped damping ring designed according to each embodiment is shown in fig. 1, and the damping ring is called a non-circular C-shaped damping ring 1 because the free profile 101 is in a non-circular open state in an uninstalled natural state. In the installed state, the non-circular C-shaped damping ring 1 is deformed under pressure, and the pre-pressing profile 102 is in a perfect circular closed state.
The non-circular C-shaped damping ring 1 designed by the invention is used for being attached to an aviation high-speed thin spoke plate gear, the additional object can be a thin spoke plate bevel gear or a thin spoke plate straight gear, but the non-circular C-shaped damping ring 1 is installed by partially fine-adjusting the rim structure of the gear. As shown in fig. 2, it is a schematic diagram of the rim ring groove required for the additional non-circular C-shaped damping ring of the thin-web bevel gear, and for a certain bevel gear 2, a ring groove 201 is required to be opened on the rim for the non-circular C-shaped damping ring 1. As shown in fig. 3, which is a schematic diagram of a rim ring groove required by a thin-web spur gear with an additional non-circular C-shaped damping ring, for a straight gear 3, an open ring groove 301 is required on a rim for installing the non-circular C-shaped damping ring 1.
The non-circular C-shaped damping rings 1 can be installed in a number determined according to actual requirements and structural space limitations of aviation high-speed thin-spoke gears. As shown in fig. 4, a schematic diagram of attaching a non-circular C-shaped damping ring to a thin-web bevel gear shows that, for a certain bevel gear 2, due to the limitation of the rim space, only the large end side is provided with a condition for forming a ring groove, and then only the large end side is attached with the non-circular C-shaped damping ring 1, and at the same time, after the non-circular C-shaped damping ring 1 is installed in the bevel gear 2, the pre-pressing profile 102 of the non-circular C-shaped damping ring is attached to the ring groove 201, that is, the radial dimensions of the non-circular C-shaped damping ring and the ring groove are the same. As shown in fig. 5, the non-circular C-shaped damping ring is attached to the thin-spoke straight gear in a double-side installation manner, for a certain straight gear 3, the left and right sides of the straight gear are provided with annular grooves, and then a double-side installation manner of installing the non-circular C-shaped damping ring 1 on the annular grooves on the left and right sides can be adopted, and meanwhile, after the non-circular C-shaped damping ring 1 is installed in the straight gear 3, the prepressing profile line 102 of the non-circular C-shaped damping ring is attached to the annular groove 301, that is, the radial sizes of the non-circular C-shaped damping ring and the annular groove are the same. According to actual requirements, even if the damping ring is installed on two sides, the damping ring can be installed on one side. As shown in FIG. 6, a non-circular C-shaped damping ring is attached to a thin web spur gear using a left side mounting. As shown in FIG. 7, a non-circular C-shaped damping ring is added to the thin web spur gear in a right side mounting manner.
The prepressing profile line 102 of the non-circular C-shaped damping ring 1 is attached to the Bao Fuban bevel gear ring groove 201 or the thin spoke plate straight gear ring groove 301, so that high contact rate between the damping ring and a gear can be achieved, the contact pressure between the damping ring and the gear can be uniformly distributed by reasonably designing the free profile line 101, and the contact pressure can be adjusted by adjusting the free profile line 101. Therefore, the basic idea of the present invention is to design the non-circular C-shaped damping ring free profile to achieve a high contact ratio between the damping ring and the gear, and a uniform and adjustable contact pressure. As shown in fig. 8, it is a flow chart of the method for designing a non-circular C-shaped damping ring with controllable contact pressure according to the present invention, and includes the following steps (the damping rings not specifically described below refer to non-circular C-shaped damping rings):
in a specific embodiment, the design method of each embodiment includes the following steps and its design principle:
(1) And (3) giving design working condition parameters and target gear parameters, and determining non-circular C-shaped damping ring section parameters. Adding a damping ring to the harmful resonance point of the target gear to damp vibration, wherein the harmful resonance point is the design working condition, further giving the resonance point frequency f (the frequency when resonance occurs), and designating the design contact pressure p d 。
As shown in fig. 9, description is made of not in the mounted stateThe structural parameter schematic diagram of the sizes of the circular C-shaped damping ring and the ring groove. Given the tooth number z of the target gear and the radius r of the ring groove g Width w of ring groove g And depth t of ring groove d . The section parameter requirement of the non-circular C-shaped damping ring is as follows: radius r of compression profile r Radius r of ring groove g The damping rings are equal to the ring grooves so as to ensure that the damping rings are attached to the ring grooves; the axial width w of the damping ring has a value interval of (0, 0.99w) g ]The damping ring has enough axial sliding space, but the increase of the axial width w of the damping ring is beneficial to the generation of more friction energy consumption, so the large value of the axial width w of the damping ring is considered firstly; the radial thickness t of the damping ring is more than or equal to t d The damping ring is convenient to disassemble and assemble, but the radial thickness t of the damping ring is too large, so that the additional weight is increased, the lightweight design requirement of an aviation high-speed gear transmission system is contradicted, the bending resistance of the damping ring is increased, the assembly and disassembly difficulty is further improved, and the radial thickness t of the damping ring is small on the basis of meeting the use requirement.
Therefore, according to the requirements of the section parameters of the non-circular C-shaped damping ring, the axial width w and the radial thickness t of the section parameters of the damping ring are preliminarily determined for subsequent calculation.
(2) And determining the non-circular C-shaped damping ring material, and obtaining the material parameters and allowable stress value [ sigma ]. Because the non-circular C-shaped damping ring needs to depend on deformation, the size of the cross section of the damping ring is far smaller than the perimeter of the damping ring, which is equivalent to a slender curved beam, and large stress can be generated on a symmetrical section in the prepressing process. If the damping ring deforms and is installed, the stress in the ring exceeds the yield strength of the material and plastic deformation occurs, so that the elastic recovery capability of the material is weakened, the damping ring and the gear cannot be tightly attached, and deformation is influenced to cause contact pressure. Therefore, the non-circular C-shaped damping ring has higher requirements on the yield strength of materials, and the non-circular C-shaped damping ring is preferably made of high-yield-strength materials such as spring steel, including but not limited to the spring steel grades 70, 65Mn,60Si2MnA,60Si2CrVA and the like.
After the non-circular C-shaped damping ring material and the specific mark are selected, the yield strength sigma of the corresponding material is determined according to the relevant standards of GB/T1222-2007 spring Steel and the like s . And determining the density rho and the elastic modulus E of the damping ring material. According to the actual conditions of materials and designA safety factor n, and then obtaining allowable stress value [ sigma ]]I.e. by
(3) Calculating the contact pressure p caused by centrifugation c Building up a deformation-induced contact pressure p s . As shown in fig. 10, the contact pressure between the non-circular C-shaped damping ring and the gear is schematically illustrated. <xnotran> C curve1 curve2 curve3 , , curve2 , curve3 . </xnotran> The non-circular C-shaped damping ring is designed for a certain design point, so that the point corresponds to the design contact pressure p at the rotating speed of the gear d The contact pressure p caused by the centrifugation of the point c And deformation causing contact pressure p s And (4) forming. Since the gear mesh excitation causes the design point resonance, the gear rotational speed ω at the design point can be obtained from the given resonance point frequency f and the number of teeth z, that is
The mechanism of the contact pressure between the non-circular C-shaped damping ring and the gear caused by centrifugation is as follows: the annular restriction loss of the damping ring caused by the annular gap exists in the annular direction, so that the damping ring is expanded outwards under the action of high-speed centrifugation, but the expansion is limited by the gear, and further, the contact pressure p is caused by the centrifugation formed on the contact surface c . For solving the contact pressure, the contact pressure caused by other factors is ignored, namely the damping ring is uniformly contacted with the gear only by virtue of centrifugal force, meanwhile, as the damping ring is of a thin-wall structure and the radial thickness of the damping ring is smaller than the diameter, the centrifugal force can be considered to uniformly act on a neutral layer of the damping ring, and the radius r of the neutral layer n Approximately the location of the centroid of the cross-section in a thin-walled structure, i.e.
The contact pressure p caused by centrifugation is known from the stress balance condition c The magnitude of the centrifugal force is equal to the centrifugal force applied to the neutral layer of the damping ring (but the directions of the centrifugal force and the centrifugal force are opposite), namely
p c ＝ρ·t·r n ·ω 2 (4)
The mechanism of the contact pressure between the non-circular C-shaped damping ring and the gear caused by the deformation is as follows: the free profile of the damping ring is pressed and then installed in the gear ring groove, and contacts with the gear under the action of elastic restoring force to form deformation to cause contact pressure p s . Because of the design contact pressure p d Contact pressure p caused by the design point corresponding to the centrifugation c And deformation causing contact pressure p s So that the following relational expression can be constructed
p s ＝p d -p c (5)
(4) And calculating the free profile coordinates of the non-circular C-shaped damping ring in the polar coordinate system. Solving the coordinate of the damping ring free profile should be carried out in the static environment of the gear, namely, only considering the contact pressure p caused by deformation s And (4) acting. As shown in fig. 11, the mechanical model diagram is a schematic diagram for solving the natural state profile of the non-circular C-shaped damping ring, and one half of the damping ring symmetry can be taken as a research object, and a mechanical model with one fixed end and the other free end is established. The circle center of the non-circular C-shaped damping ring in the installation state is taken as a pole, the radial direction is taken as a polar axis, a polar coordinate is established in a cross section perpendicular to the axial direction of the non-circular C-shaped damping ring, and the symmetry axis of the C-shaped damping ring in the non-installation state is on the polar axis of 0 degree. Contact pressure p caused by deformation of static damping ring in installation state s Acting (contact pressure being the main force, ignoring other forces), if on the basis of this a contact pressure p is superimposed which causes deformation s Forces p of equal but opposite magnitude s 'on the one hand, the damping ring is stressed to 0 according to the superposition of force systems, namely, the damping ring is in a natural state, on the other hand, points A, B and the like in the installation state move to A' in the natural state along with the deformation of the damping ring,b', and the contour coordinates of the damping ring can be obtained by combining the contour coordinates of the installation position, the radial displacement and the circumferential displacement. Meanwhile, the non-circular C-shaped damping ring free profile is derived on the basis of full contact and uniform contact pressure, so that uniform contact pressure can be generated and the contact rate is high after the non-circular C-shaped damping ring free profile is installed into a gear.
Because the damping ring is of a thin-wall structure, the radial displacement of each point on the same cross section can be assumed to be the same, and then a neutral layer is taken as an analysis object. Under the action of uniform radial pressure, the damping ring is bent and deformed, and any point B on the ring is located in a infinitesimal element
Moment to a cross section with a polar angle theta of
Wherein, the first and the second end of the pipe are connected with each other,
and the angle of any point B in the established polar coordinate system is shown, and theta is the angle of the cross section in the established polar coordinate system.
According to the section method of material mechanics and the balance relation between internal force and external force, the total bending moment at the theta-angle section is obtained by the integral of each infinitesimal moment in the left arc section of the section
The axial force of the infinitesimal pair theta angle section is analyzed and integrated in the same way, and further the axial force at the theta angle section can be obtained
N(θ)＝p s ·wr n ·(1+cosθ) (8)
The damping ring can be regarded as a thin-wall curved beam, and according to the mechanics of materials, the differential equation of the deflection line of the curved beam after deformation is
Wherein A is the cross-sectional area of the damping ring, u and v are the displacement of the section centroid (the centroid of the thin-wall structure is approximately positioned on the neutral layer) of the damping ring along the circumferential direction and the radial direction respectively, s is the direction passing through the section centroid but perpendicular to the section, jz is a mark, and the value is that for the cross section of the rectangular damping ring with the radial thickness of t and the axial width of w
For a circumferential displacement u, the boundary condition is
u＝0；θ＝0 (11)
For the radial displacement v, the boundary condition is
The circumferential displacement u and the radial displacement v can be solved by combining the differential equation of the deflection line and boundary conditions of each displacement, thereby combining the coordinates (theta, r) of any point of the neutral layer in the installation state n ) The corresponding external surface profile coordinates (theta) in the natural state are obtained e ,r e ) And inner surface profile coordinates (theta) i ,r i ) Are respectively as
(5) Calculating the maximum stress value sigma of the non-circular C-shaped damping ring max And performing intensity check. Theoretically, the principal stress sigma exists at one point in the damping ring 1 ，σ 2 ，σ 3 But the damping ring radial thickness is small relative to the radius of curvature, the ring internal stress being at a bending stress σ b Mainly due to this factOne point within the nylon ring, the Von Mises stress, can be approximated as
For any point in the damping ring (at the off-angle theta, radial distance from the neutral layer l), the bending stress sigma is b (θ, l) calculated by the following equation
From the above formula, the maximum stress value on the cross section of the damping ring is located at l = t/2. Therefore, each cross-sectional bending stress σ b The maximum values are distributed in the circumferential direction as follows
Further solving theta partial derivative of the formula, and substituting the theta partial derivative into the expression of axial force and bending moment to obtain the theta partial derivative
σ b The variation rate of the maximum value of the bending stress of each cross section along the circumferential direction is set to be 0 according to a derivative judgment method of an extreme value, and a function extreme value is obtained to judge the maximum value. According to the analysis, σ b (θ, t/2) takes a maximum value at θ =0. Therefore, the damping ring inner point (0,t/2) is the stress danger point where the maximum stress value is located, and further the non-circular C-shaped damping ring maximum stress value sigma max Is composed of
σ max ＝σ b (0,t/2) (19)
The allowable stress is obtained by the formula (1). From the equation (15), the Von Mises stress at the stress danger point is similar to the corresponding bending stress, i.e. the maximum stress value sigma of the non-circular C-shaped damping ring max 。
(6) Judging whether the designed non-circular C-shaped damping ring structure can pass throughAnd (6) checking the strength. Judging the maximum stress value sigma of the non-circular C-shaped damping ring max Magnitude and allowable stress value [ sigma ]]Magnitude relation, i.e. whether the following holds
|σ max |＜[σ] (20)
If the formula (20) is established, the next step is performed.
If the expression (20) does not hold, reconstructing the damping ring section parameters, namely adjusting the axial width w and the radial thickness t, and then repeating the steps (2) to (5) until the expression (20) holds.
(7) And converting the free profile coordinates of the non-circular C-shaped damping ring into a Cartesian coordinate system, and combining the section parameters to form the final non-circular C-shaped damping ring design. The free profile line coordinates in the cartesian coordinate system are adopted, so that the damping ring engineering drawing is convenient to draw, and the damping ring is convenient to process and manufacture by using a linear cutting method and the like, so that the free profile line in the polar coordinate system obtained by the steps needs to be subjected to coordinate system conversion.
Finally, combining the free profile coordinates, the axial width w and the radial thickness t of the damping ring to finish the appointed design contact pressure p d The following non-circular C-shaped damping ring design. It is apparent that different design contact pressures correspond to different non-circular C-shaped damping profiles, and therefore, the contact pressure can be controlled by designing the non-circular C-shaped damping profiles.
Example 1:
designing a non-circular C-shaped damping ring attached to a high-speed thin-spoke bevel gear (shown in figure 2) of a certain aviation, and designing according to a specific flow of the design method for the non-circular C-shaped damping ring with controllable contact pressure, wherein the method comprises the following steps:
(1) And (3) giving design working condition parameters and target gear parameters, and determining the section parameters of the non-circular C-shaped damping ring. Given the harmful resonance point frequency f =2895Hz of the target gear and the design contact pressure p d =0.8MPa. Given the number of teeth z =86 of the target gear, the radius r of the ring groove g =50.5mm, ring groove width w g =4.0mm, ring groove depth t d =2.0mm. According to the requirements of the section parameters of the non-circular C-shaped damping ring (the range of the axial width w is (0,0.99w) g ]The radial thickness t is in the range of t ≥ t d ) Preliminary determination of damping ring cross-sectionThe axial width of the parameter is 0.975w g I.e. w =3.9mm; radial thickness of 2.35t d I.e. t =4.7mm, for subsequent calculations.
(2) Determining non-circular C-shaped damping ring material, and obtaining material parameters and allowable stress value [ sigma ]]. Spring steel with the mark of 60Si2CrVA is selected as a damping ring material, and the yield strength sigma of the spring steel after heat treatment is known from GB/T1222-2007 spring steel s It should be greater than 1665MPa. Taking the density rho =7860kg/m of the damping ring material 3 Elastic modulus E =210GPa. A safety factor n =1.2 is determined. Obtaining allowable stress value [ sigma ] from formula (1)]＝1387.5MPa。
(3) Calculating the contact pressure p caused by centrifugation c Building up a deformation-induced contact pressure p s . The gear rotation speed omega =211.51rad/s of the design point is obtained from the formula (2), and the centrifugal induced contact pressure p is obtained from the combined type (3) and the formula (4) c =0.08MPa. And the contact pressure p is caused by the deformation obtained by the formula (5) s ＝0.72MPa。
(4) And calculating the free profile coordinates of the non-circular C-shaped damping ring in the polar coordinate system. The joint vertical type (6) - (14) obtains the external surface contour line coordinate (theta) of the non-circular C-shaped damping ring in the natural state e ,r e ) And inner surface profile coordinates (theta) i ,r i )。
(5) Calculating the maximum stress value sigma of the non-circular C-shaped damping ring max And intensity checking is performed. Calculating the maximum stress value sigma of the damping ring according to the formulas (15) to (19) max The size is 923MPa.
(6) And judging whether the designed non-circular C-shaped damping ring structure can pass the strength check. Maximum stress value sigma of damping ring max Size 923MPa, and allowable stress value [ sigma ]]Since 1387.5MPa, equation (20) is satisfied, and the strength verification requirement is satisfied.
(7) And converting the free profile coordinates of the non-circular C-shaped damping ring into a Cartesian coordinate system, and combining the section parameters to form the final non-circular C-shaped damping ring design.
And then, a group of design contact pressure of 1.1MPa is designated to form a design comparison, and the corresponding non-circular C-shaped damping contour line design is carried out according to the same steps (1) to (7). As shown in FIG. 12, the aviation high-speed thin-web bevel gear is additionally provided with a non-circular C-shaped damping ring for realizing different contact pressures. As can be seen from the figure, according to the method for designing the non-circular C-shaped damping contour line of the present invention, different damping contour lines can meet different design contact pressure requirements, i.e., the contact pressure can be controlled by designing the damping contour line, and the non-circular C-shaped damping contour line is derived on the basis of the full contact and uniform contact pressure distribution, so that the non-circular C-shaped damping contour line can generate uniform contact pressure and has high contact rate when installed in a gear.
Example 2:
designing a non-circular C-shaped damping ring attached to a certain aviation high-speed thin-spoke-plate straight gear (as shown in figure 3), and designing according to a specific flow of the design method for the non-circular C-shaped damping ring with controllable contact pressure, which is disclosed by the invention, the steps are as follows:
(1) And (3) giving design working condition parameters and target gear parameters, and determining the section parameters of the non-circular C-shaped damping ring. Given the harmful resonance point frequency f =5744Hz of the target gear and the design contact pressure p d =0.7MPa. The tooth number z =96 of the target gear is given, and the radius r of the ring groove g =65.0mm, ring groove width w g =4.9mm, ring groove depth t d =2.0mm. According to the requirements of the section parameters of the non-circular C-shaped damping ring (the range of the axial width w is (0,0.99w) g ]The radial thickness t is in the range of t ≥ t d ) Preliminarily determining the axial width of the damping ring section parameter to be 0.98w g I.e. w =4.8mm; the radial thickness is 2.6t d I.e. t =5.2mm, for subsequent calculations.
(2) Determining non-circular C-shaped damping ring material, and obtaining material parameters and allowable stress value [ sigma ]]. Spring steel with the mark of 60Si2CrVA is selected as a damping ring material, and the yield strength sigma of the spring steel after heat treatment is known from GB/T1222-2007 spring steel s It should be greater than 1665MPa. Taking the density rho =7860kg/m of the damping ring material 3 The elastic modulus E =210GPa. A safety factor n =1.2 is determined. Obtaining allowable stress value [ sigma ] from formula (1)]＝1387.5MPa。
(3) Calculating the contact pressure p caused by centrifugation c Building up a deformation-induced contact pressure p s . The gear rotation speed ω =375.94rad of the design point is obtained from the formula (2)S, combined vertical (3) and formula (4) obtaining centrifugally induced contact pressure p c =0.36MPa. And the contact pressure p is caused by the deformation obtained by the formula (5) s ＝0.34MPa。
(4) And calculating the free profile coordinates of the non-circular C-shaped damping ring under the polar coordinate system. The joint vertical type (6) - (14) obtains the external surface contour line coordinate (theta) of the non-circular C-shaped damping ring in the natural state e ,r e ) And inner surface profile coordinates (theta) i ,r i )。
(5) Calculating the maximum stress value sigma of the non-circular C-shaped damping ring max And intensity checking is performed. Calculating the maximum stress value sigma of the damping ring according to the formulas (15) to (19) max The size is 595.3MPa.
(6) And judging whether the designed non-circular C-shaped damping ring structure can pass the strength check. Maximum stress value sigma of damping ring max 595.3MPa, and allowable stress value [ sigma ]]=1387.5MPa, equation (20) holds, and the strength check requirement is satisfied.
(7) And converting the free profile coordinates of the non-circular C-shaped damping ring into a Cartesian coordinate system, and combining the section parameters to form the final non-circular C-shaped damping ring design.
And then, a group of design contact pressure of 1.0MPa is designated to form a design comparison, and the corresponding non-circular C-shaped damping contour line design is carried out according to the same steps (1) to (7). As shown in fig. 13, the non-circular C-shaped damping ring is added to an aviation high-speed thin-web spur gear to realize different contact pressures. As can be seen from the figure, according to the method for designing the non-circular C-shaped damping contour line of the present invention, different damping contour lines can meet different design contact pressure requirements, i.e., the contact pressure can be controlled by designing the damping contour line, and the non-circular C-shaped damping contour line is derived on the basis of the full contact and uniform contact pressure distribution, so that the non-circular C-shaped damping contour line can generate uniform contact pressure and has high contact rate when installed in a gear.
Claims (3)
1. A design method of a non-circular C-shaped damping ring with controllable contact pressure is characterized in that:
the noncircular C-shaped damping ring is a noncircular ring with a notch in a natural state;
after the non-circular C-shaped damping ring is arranged on the ring groove on the gear, the outer side of the non-circular C-shaped damping ring is deformed by pressure to be in a perfect circular closed state;
the non-circular C-shaped damping ring is designed into a profile line in a natural state through the following steps:
(1) According to the parameters of the ring grooves, setting the parameters of the section of the non-circular C-shaped damping ring: axial width w, radial thickness t, compression profile radius r r ；
(2) Calculating an allowable stress value [ sigma ] according to the non-circular C-shaped damping ring material parameters and the safety factor n:
the yield strength of the non-circular C-shaped damping ring material is sigma according to the selected material s Density rho and elastic modulus of the material are E; the value range of the safety coefficient n is more than or equal to 1:
the allowable stress value [ sigma ] is:
(3) Calculating the contact pressure p caused by centrifugation c And deformation-induced contact pressure p s ：
Calculating the gear rotating speed omega of the resonance point:
wherein: f is the resonance point frequency, z is the number of teeth;
calculating the radius r of the neutral layer n ：
Centrifugal induced contact pressure p c ：
p c ＝ρ·t·r n ·ω 2 (4)
Calculating the contact pressure p caused by deformation s ：
p s ＝p d -p c (5)
Wherein: p is a radical of d To design contact pressure;
(4) Calculating the free profile coordinates of the non-circular C-shaped damping ring under a polar coordinate system:
taking the circle center of the non-circular C-shaped damping ring in the mounting state as a pole and the radial direction as a polar axis, establishing a polar coordinate in a section perpendicular to the axial direction of the non-circular C-shaped damping ring, and enabling the symmetry axis of the C-shaped damping ring in the non-mounting state to be on the polar axis of 0 degree;
when the non-circular C-shaped damping ring is bent and deformed in the installation state, any point B on the ring is positioned at a infinitesimal element
The moment for a section with a polar angle θ is:
wherein the content of the first and second substances,
for any point B, the angle in the established polar coordinate system is theta, and theta is the angle of the cross section in the established polar coordinate system:
the total bending moment at the theta angle section is:
the axial force at the θ -angle section is:
N(θ)＝p s ·wr n ·(1+cosθ) (8)
the differential equation of the deflection line of the non-circular C-shaped damping ring is as follows:
in the formula, A is damping ring cross-sectional area, u, v are damping ring cross-sectional centroid along circumference and radial displacement respectively, s is the direction of crossing the cross-sectional centroid but perpendicular to the cross-section, jz is the mark, and for radial thickness be t, the axial width is the rectangular damping ring cross-section of w, its value is:
for a circumferential displacement u, the boundary condition is
u＝0；θ＝0 (11)
For the radial displacement v, the boundary condition is
Circumferential displacement u and radial displacement v can be solved by combining a differential equation of the deflection line and boundary conditions of each displacement;
combining the coordinates (theta, r) of any point of the neutral layer in the mounting state n ) And obtaining the corresponding external surface profile coordinates and internal surface profile coordinates in a natural state as follows:
outer surface profile coordinates (theta) e ,r e ):
Inner surface profile coordinates (θ) i ,r i ):
(5) Judging whether the designed non-circular C-shaped damping ring structure can pass the strength check;
if the damping force can be passed, taking the axial width w and the radial thickness t set in the step (1), the material selected in the step (2) and the non-circular C-shaped damping ring free profile coordinate calculated in the step (4) as design parameters;
and (3) if the axial width w and/or the radial thickness t cannot pass through the step (1), resetting the axial width w and/or the radial thickness t, and repeating the steps to perform design calculation and strength check.
2. The method of claim 1, wherein the contact pressure of the non-circular C-shaped damping ring is controlled by:
in the step (5), the maximum stress value sigma of the non-circular C-shaped damping ring is calculated max And intensity checking is performed.
3. The design method of the non-circular C-shaped damping ring with controllable contact pressure according to claim 1 or 2, characterized in that:
when judging whether the designed non-circular C-shaped damping ring structure can be checked through strength:
calculating the bending stress sigma of each cross section b The maximum value is distributed along the circumferential direction:
theta partial derivative is calculated according to the formula, and the theta partial derivative is substituted into the expression of axial force and bending moment to obtain
σ b ' the variation rate of the maximum value of the bending stress of each cross section along the circumferential direction is set to be 0 according to a derivative judgment method of the extreme value, a function extreme value is obtained, and then the maximum value, sigma, is judged b (theta, t/2) takes the maximum value at the position of theta =0, the inner point (0,t/2) of the damping ring is the stress danger point at which the maximum stress value is positioned, and then the maximum stress value sigma of the non-circular C-shaped damping ring max Is composed of
σ max ＝σ b (0,t/2) (19)
Judging the maximum stress value sigma of the non-circular C-shaped damping ring max Magnitude and allowable stress value [ sigma ]]Magnitude relation, i.e. whether the following holds
|σ max |＜[σ] (20)。
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| CN113177273A (en) * | 2021-04-19 | 2021-07-27 | 西安交通大学 | Gas turbine compressor and turbine blade contact rigidity and damping analysis solving method |
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