CN108717490B - Novel method for calculating load-deformation characteristic of diaphragm spring - Google Patents

Novel method for calculating load-deformation characteristic of diaphragm spring Download PDF

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CN108717490B
CN108717490B CN201810475807.9A CN201810475807A CN108717490B CN 108717490 B CN108717490 B CN 108717490B CN 201810475807 A CN201810475807 A CN 201810475807A CN 108717490 B CN108717490 B CN 108717490B
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load
diaphragm spring
coefficient
separation finger
influence
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CN108717490A (en
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刘夫云
赵毅斌
耿立冬
黄宝案
刘文华
赵亮亮
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Guilin University of Electronic Technology
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Abstract

The invention discloses a new method for calculating the load-deformation characteristic of a diaphragm spring, which comprises the steps of firstly solving a load-deformation characteristic curve of the diaphragm spring; replacing diaphragm springs in different window hole forms by a conversion model, and replacing structure parameters of different window holes by three conversion parameters; analyzing the influence rule of the conversion parameters on the load-deformation characteristic of the diaphragm spring; proposing and defining separation finger influence coefficients; obtaining the relation between peak value, inflection point load, valley value and conversion parameter by matrix conversion, and using the relation between each item separation finger influence coefficient and the conversion parameter; obtaining simulation analysis data through an orthogonal test; fitting a multivariate regression analysis method to obtain a function expression of the separation finger influence coefficient and the conversion parameter; and adding the influence coefficient of the separation finger as a correction term of the separation finger part to a calculation formula of a disc spring precision algorithm S-W method to obtain a new calculation method for the load-deformation characteristic of the diaphragm spring. The invention can be used for guiding the design work of the diaphragm spring, shortening the research and development period, reducing the design cost and improving the efficiency.

Description

Novel method for calculating load-deformation characteristic of diaphragm spring
Technical Field
The invention relates to the technical field of characteristic calculation of diaphragm springs, in particular to a novel method for calculating load-deformation characteristics of a diaphragm spring.
Background
The diaphragm spring is a non-linear spring and consists of a disc spring part and a separation finger part. The diaphragm spring has good nonlinear characteristic, and simultaneously has the functions of a compression spring and a release lever, so that the number of parts is reduced, the weight is reduced, the axial distance of the clutch is shortened, and the working reliability is improved. Due to the advantages mentioned above, the diaphragm spring is an important component of the friction clutch.
Clutch manufacturers typically use a disc spring approximation a-L to calculate the load-deflection characteristics of the diaphragm spring. However, the load value calculated by the a-L method has a large error with an actual product, and engineers need to perform design work by using a reverse solution principle. Thus, the product cycle is prolonged, the design cost is increased, and the production efficiency and the economic benefit of the product cannot be ensured.
The error is caused by: theoretical errors exist in the derivation of the A-L method, the influence of the structure of the separation finger is not considered, and the influence of the strengthening treatment is not considered.
In the research on error elimination, the document "research on a correction formula of a diaphragm spring a-L" corrects the a-L method in stages according to the process flow, and a good effect is achieved. This research is leading in China. However, there are several problems with this correction method: 1. the A-L method is corrected, and the theoretical error of the A-L method during derivation cannot be essentially eliminated; 2. the correction formula adopts the ring value ratio c/a as an influence parameter of the separation finger structure to correct the A-L method. The loop ratio is calculated as:
Figure GDA0003446556820000011
where c is the total length of the disk where the separating fingers are attached to the disk and a is the perimeter of the disk where the separating fingers are attached. However, the separation finger portion has other structural parameters besides the ring value ratio c/a, which affect the load deflection characteristics of the diaphragm spring, and this correction method does not sufficiently consider the structural influence of the separation finger.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a novel method for calculating the load-deformation characteristic of the diaphragm spring, which reduces the theoretical error in formula derivation and fully considers the influence of a separation finger part.
The technical scheme for realizing the purpose of the invention is as follows:
a novel method for calculating load-deformation characteristics of a diaphragm spring specifically comprises the following steps:
1) solving to obtain a load-deformation characteristic curve of the diaphragm spring by using finite element simulation software;
2) replacing diaphragm springs in different window hole forms with the conversion model, replacing structural parameters of different window holes with three conversion parameters, and analyzing the load-deformation characteristic of the conversion model according to the method in the step 1) to further obtain the influence rule of the three conversion parameters on the load-deformation characteristic of the diaphragm springs;
3) the influence coefficient of the separation finger is provided and defined, and the relation between the peak value, the inflection point load, the valley value and the conversion parameter is found to be also applicable to the relation between the influence coefficient of the separation finger and the conversion parameter of each time through matrix conversion, so that the influence rule of the influence coefficient of the separation finger on the load-deformation characteristic of the diaphragm spring is obtained;
4) designing an orthogonal experiment according to the discovered influence rule of the separation finger influence coefficient on the load-deformation characteristic of the diaphragm spring, and acquiring a plurality of groups of representative data about the separation finger influence coefficient and three conversion parameters of each item;
5) establishing and simplifying a regression model of each separation finger influence coefficient and three conversion parameters according to the discovered influence rule of the separation finger influence coefficient on the load-deformation characteristic of the diaphragm spring;
6) obtaining a separation finger influence coefficient function formula by utilizing SPSS software regression fitting according to the acquired data about the separation finger influence coefficients of each time item and the three conversion parameters;
7) a disc spring accurate calculation method S-W method is used for replacing an approximate calculation method A-L method, the influence coefficient of a separation finger is used as a correction term of the separation finger part, the correction term is added to a calculation formula of the disc spring accurate calculation method S-W method, a new calculation method of the load-deformation characteristic of the diaphragm spring is obtained, and the load-deformation characteristic of the diaphragm spring is calculated by the new calculation method.
Step 1), solving a load time history curve at a support ring and a displacement time history curve at a pressure plate contact position by using finite element software, and drawing a load-deformation characteristic curve of the diaphragm spring;
step 2), specifically, the diaphragm spring separation finger structures in different window hole forms are converted to obtain a conversion model of the diaphragm spring, and the diaphragm spring separation finger structure parameters in different window hole forms are represented by uniform parameters to obtain three conversion parameters, wherein the three conversion parameters are respectively: the relation between three conversion parameters and the actual structure parameters of the separation finger part is respectively established by the following formulas:
Figure GDA0003446556820000021
Figure GDA0003446556820000031
Figure GDA0003446556820000032
in the above formula, n is an index, r is the inner radius of the disc spring portion, and δ2The width of the window hole groove is delta3For the separation finger contact width with the inner diameter, r' is the window hole fillet radius.
Further, the step 2) is specifically to construct a diaphragm spring conversion geometric model with the structure of the disc spring part unchanged and the structure of the separation finger changed, and perform finite element analysis to obtain a load-deformation characteristic rule of the diaphragm spring by three conversion parameters: along with the increase of the reciprocal 1/n of the exponent, the peak value, the inflection point load and the valley value of the load-deformation curve are increased and are in a forward linear relation, the load-deformation curve can be described by a linear function, the correction loop ratio value c'/a is increased, the peak value, the inflection point load and the valley value of the load-deformation curve are increased, the increasing rate is increased, and the load-deformation curve can be described by a quadratic function. Along with the increase of the ratio i of the radius of the fillet of the window hole, the peak value, the load of the inflection point and the valley value of the load-deformation curve are increased, and the acceleration is accelerated and can be described by a quadratic function.
In the step 3), a separation finger influence coefficient is proposed and defined, and specifically:
the load-deflection curve of the diaphragm spring is a cubic curve through the origin, and the function form is as follows:
F＝a1λ+a2λ2+a3λ3
in the above formula, F is the load, a1、a2、a3Is the coefficient of each order of the diaphragm spring, a1Is the first order coefficient of the diaphragm spring, a2Is the second order coefficient of the diaphragm spring, a3Is the coefficient of the third order of the diaphragm spring, λ is the displacement;
the load-deflection curve of the disc spring is also a cubic curve passing through the origin, and the function form is as follows:
Fdish＝a'1λ+a'2λ2+a'3λ3
In the above formula, a'1、a’2、a’3Denotes each linear coefficient, a 'of the disc spring'1Is the primary coefficient of the disc spring, a'2Is the coefficient of the secondary term of the disc spring, a'3Is the cubic coefficient of the disc spring;
the ratio of each secondary coefficient of the diaphragm spring function to each secondary coefficient of the disc spring function is provided as the influence coefficient of the separation finger, and I is the influence coefficient of the separation finger1Is a first order coefficient of influence, I2Is the coefficient of influence of the quadratic term, I3Is the cubic term influence coefficient, then:
Figure GDA0003446556820000041
in the step 3), the displacement loads of the peak point, the inflection point and the valley point on the load-deformation curve obtained by finite element analysis are substituted into the polynomial function of the diaphragm spring or the disc spring to solve each secondary coefficient of the diaphragm spring or the disc spring, so as to obtain the function form of the load-deformation characteristic, namely, the linear equation is solved as follows:
Figure GDA0003446556820000042
for convenience of description and writing, the linear equation set is expressed by a matrix, so that
Figure GDA0003446556820000043
The matrix form of the system of equations is then expressed as: a is equal to b and a is equal to b,
the matrix Lambda is called coefficient matrix of linear equation system, a is unknown quantity column, b is constant item column, two sides of equation are respectively transposed to obtain aTΛT＝bTAngle of rotationTMove to the right of the equation to get aT＝bT(ΛT)-1，
3 th order matrix ΛTThe sufficient requirement for reversibility is ΛTBeing non-singular, i.e. 3-order matricesTDeterminant | Λ ofT| is not equal to zero, and a 3 rd order matrix Λ is formedTThe determinant of (a) is developed as:
Figure GDA0003446556820000044
in the above equation, the right determinant is a 3 rd order van der Mond determinant due to the peak displacement λ of the load-deformation curve1Displacement of inflection point lambda2Valley point displacement lambda3Are all not zero and are not equal to each other, then |. ΛT| is not equal to zero, ΛTReversible, ΛTReversible matrix of (Λ)T)-1(ii) present;
the coefficients of each secondary term of the diaphragm spring function are respectively the peak value, the inflection point load and the valley value (lambda)T)-1The linear combination of each row of elements has the same rule as the peak value, the inflection point load and the valley value and the conversion parameter, so that the same rule also exists for each secondary coefficient of the diaphragm spring and the conversion parameter, each secondary coefficient constant of one disc spring is removed from each secondary coefficient of the diaphragm spring to obtain each secondary separation finger influence coefficient, and the influence coefficient of each secondary separation finger is removed without influencing the rule, namely, each secondary separation finger influence coefficient and the conversion parameter are considered to have the same rule.
Step 4), specifically, a three-factor five-level is designed according to the approximate range of the distribution of the conversion parameters in the actual productionOrthogonal test of (2), selecting L25(56) The test was performed using a type orthogonal table, as shown in table 1, and a simulation test was performed using finite element software,
TABLE 1 orthogonal Table L25(56)
Figure GDA0003446556820000051
Figure GDA0003446556820000061
Step 5) is specifically to fit a function relation of each secondary coefficient and three conversion parameters by using a ternary quadratic regression model, and replace the reciprocal of the exponent 1/n, the correction ring ratio c'/a and the window hole fillet radius ratio i by x, y and z respectively, then:
Ii(x,y,z)(1≤i≤3)＝a1xy2z2+a2xy2z+a3xy2+a4xyz2+a5xyz+a6xy
+a7xz2+a8xz+a9x+a10y2z2+a11y2z+a12y2
+a13yz2+a14yz+a15y+a16z2+a17z+a18
according to the geometric meaning of the actual existence of the diaphragm spring, the model is simplified, if the number of the separating fingers tends to be infinite, the stress change of the separating finger part is almost absent when the diaphragm spring is loaded, namely the separating finger part does not influence the load characteristic of the diaphragm spring; if the ratio of the correction ring is equal to zero, the membrane spring becomes a disc spring, and a separation finger part does not exist; whether the number of the separating fingers tends to be infinite or the ratio of the correction ring is equal to zero, the separating finger part in the model does not exist, the constant term is equal to 1, the window hole fillet exists depending on the separating fingers, and the radius of the window hole fillet also exists depending on the number of the separating fingers; the model is simplified as follows:
Ii(x,y,z)(1≤i≤3)＝a1xy2z2+a2xy2z+a3xy2+a4xyz2+a5xyz+a6xy+1。
step 6) obtaining a separation finger influence coefficient function by utilizing SPSS software regression fitting, specifically fitting according to a model simplified formula, firstly generating four square interaction terms and two interaction terms of three independent variables contained in the simplified model, and then selecting the variables by adopting a stepwise regression method, wherein the right side of the simplified model equation contains a constant 1, the constant needs to be moved to the left side of the equation to establish a new dependent variable, and the fitting equation does not contain the constant, specifically:
6-1) establishing a catalyst containing x, y, z, I1，I2，I3SPSS datasets of six variables;
6-2) generates a new variable x1y2z2 (representing xy)2z2) X1y2z1 (represents xy)2z), x1y2 (representing xy)2) X1y1z2 (representing xyz2) X1y1z1 (representing xyz), x1y1 (representing xy), I1S1 (representing I)1-1), I2S1 (representing I)2-1), I3S1 (representing I)3-1)；
6-3) entering a linear regression dialog box, selecting I1S1 into a dependent variable box, selecting x1y2z2, x1y2z1, x1y2, x1y1z2, x1y1z1 and x1y1 into an independent variable box, selecting a stepwise regression method, eliminating the option of 'constant is contained in equation', clicking a determination button to obtain a one-time item influence coefficient I1The fit relationship to the three transformation parameters is:
I1(x,y,z)
respectively selecting I2S1 and I3S1 into dependent variable boxes, and solving to obtain quadratic term influence coefficient I2Coefficient of influence of cubic term I3The fit relationship to the three transformation parameters is:
I2(x,y,z)
I3(x,y,z)
in summary, the separation of the fit means that the influence coefficient function is:
Ii(x,y,z)(1≤i≤3)。
the step 7) is specifically that the disc spring accurate calculation method S-W method proposed by professor Schmidt (R.Schmidt) and Wepinner (D.A.Wempner) of the civil engineering department of the university of Arizona in 1959 replaces the approximate calculation method A-L method, and the formula of the S-W method is as follows:
p＝B(A1C+A2C2+A3C3)
Figure GDA0003446556820000071
Figure GDA0003446556820000072
Figure GDA0003446556820000073
Figure GDA0003446556820000074
Figure GDA0003446556820000075
in the formula
Figure GDA0003446556820000076
Figure GDA0003446556820000077
Figure GDA0003446556820000078
Figure GDA0003446556820000079
The formula of the S-W method is expressed as a cubic polynomial in the form:
F＝a1λ+a2λ2+a3λ3
the influence coefficient of the separating fingers is used as a correction term of the separating fingers, and is added to a calculation formula of an S-W method of a disc spring accurate calculation method to obtain a new calculation method of the load-deformation characteristic of the diaphragm spring, wherein the expression is as follows:
F＝a1I1λ+a2I2λ2+a3I3λ3。
the novel method for calculating the load-deformation characteristic of the diaphragm spring eliminates theoretical errors during formula derivation as much as possible and fully considers the influence of the separation finger part. The actual measurement result shows that the error of the calculated peak value of the method is within 3.5 percent, and the error of the calculated valley value is within 9 percent. The error of the peak value calculated by the A-L method is 10 percent, and the error of the valley value is 17.5 percent. The peak-to-valley value error calculated by the new method is smaller, and the actual engineering requirement is met. The new method is applied to the design work of the diaphragm spring, the design period is greatly shortened, the design cost is reduced, and the practical value and the economic value are great.
Drawings
FIG. 1 is a schematic view of a diaphragm spring;
FIG. 2 is a graph of diaphragm spring load versus deflection obtained from finite element analysis;
FIG. 3 is a dimensional view of a diaphragm spring with apertures in the form of oblong, square and trapezoidal shapes, respectively;
FIG. 4 is a conversion model of a diaphragm spring;
FIG. 5 is a graph comparing peak, inflection point loads, and valley values after changing the reciprocal of the exponent by 1/n;
FIG. 6 is a graph comparing peak, inflection load, valley after varying correction ring ratio c'/a;
FIG. 7 is a graph comparing peak, inflection point loads, and valleys after varying the window radius ratio i;
FIG. 8 is a graph showing the variation of the influence coefficient of the separation finger for each time after changing the reciprocal 1/n of the exponent;
FIG. 9 is a graph of the change in the coefficient of influence of the separation finger for each time after changing the correction ring ratio c'/a;
FIG. 10 is a graph showing the variation of the impact coefficients of the respective term separation fingers after varying the aperture radius ratio i;
in the figure, R is the outer radius of the diaphragm spring; r is the inner radius of the disc spring part; h is the height of an inner truncated cone of the disc spring part; t is the thickness of the spring plate; l is the outer bearing radius; l is the inner support radius; r is0Is the small end inner radius; r iseIs the inner radius of the window hole; delta1The width of the small end groove is wide; delta2The window hole groove is wide; delta3Width of contact between the separating finger and the inner diameter; r' is the window radius.
Detailed Description
For the purposes of clarity and completeness, the invention will be derived and described in detail with reference to the drawings and the embodiments.
In the embodiment, a model a oblong hole diaphragm spring is taken as an example, and the structure of the model a oblong hole diaphragm spring is shown in fig. 1. The material properties and structural parameters of model a oblong hole diaphragm springs are shown in tables 2 and 3.
TABLE 2 Material Properties of model A diaphragm spring
Figure GDA0003446556820000091
TABLE 3 structural parameters of type A diaphragm spring
Figure GDA0003446556820000092
A novel method for calculating the load-deformation characteristic of a diaphragm spring is characterized by comprising the following calculation steps of:
1) solving to obtain a load-deformation characteristic curve of the diaphragm spring by using finite element simulation software;
2) replacing diaphragm springs in different window hole forms with the conversion model, replacing structural parameters of different window holes with three conversion parameters, and analyzing the load-deformation characteristic of the conversion model according to the method in the step 1) to further obtain the influence rule of the three conversion parameters on the load-deformation characteristic of the diaphragm springs;
3) the influence coefficient of the separation finger is provided and defined, and the relation between the peak value, the inflection point load, the valley value and the conversion parameter is found to be also applicable to the relation between the influence coefficient of the separation finger and the conversion parameter of each time through matrix conversion, so that the influence rule of the influence coefficient of the separation finger on the load-deformation characteristic of the diaphragm spring is obtained;
4) designing an orthogonal experiment according to the discovered influence rule of the separation finger influence coefficient on the load-deformation characteristic of the diaphragm spring, and acquiring a plurality of groups of representative data about the separation finger influence coefficient and three conversion parameters of each item;
5) establishing and simplifying a regression model of each separation finger influence coefficient and three conversion parameters according to the discovered influence rule of the separation finger influence coefficient on the load-deformation characteristic of the diaphragm spring;
6) obtaining a separation finger influence coefficient function formula by utilizing SPSS software regression fitting according to the acquired data about the separation finger influence coefficients of each time item and the three conversion parameters;
7) a disc spring accurate calculation method S-W method is used for replacing an approximate calculation method A-L method, the influence coefficient of a separation finger is used as a correction term of the separation finger part, the correction term is added to a calculation formula of the disc spring accurate calculation method S-W method, a new calculation method of the load-deformation characteristic of the diaphragm spring is obtained, and the load-deformation characteristic of the diaphragm spring is calculated by the new calculation method.
Specifically, in the step 1), finite element software is used for solving a load time history curve at a supporting ring and a displacement time history curve at a pressure plate contact position, and a load-deformation characteristic curve of the diaphragm spring is drawn, as shown in fig. 2;
specifically, the step 2) classifies the window forms, and the diaphragm springs are divided into oblong-hole diaphragm springs, square-hole diaphragm springs and trapezoidal-hole diaphragm springs, as shown in fig. 3. And (3) converting the diaphragm spring separation finger structures in different window hole forms to obtain a conversion model of the diaphragm spring, as shown in fig. 4. The diaphragm spring separation finger structure parameters of different window hole forms are expressed by uniform parameters, and three conversion parameters are obtained, wherein the three conversion parameters are respectively as follows: the inverse exponent 1/n, the correction ring ratio c'/a, and the window radius ratio i. The following formula establishes the relationship between the three transformation parameters and the actual structural parameters of the separation finger portion.
Figure GDA0003446556820000101
Figure GDA0003446556820000102
Figure GDA0003446556820000103
In the above formulas (1), (2) and (3), n is an index, r is the inner radius of the disc spring portion, and δ is2The width of the window hole groove is delta3For the separation finger contact width with the inner diameter, r' is the window hole fillet radius.
Specifically, step 2) is to construct a transformation geometric model with the unchanged disc spring part and the changed separation finger structure, and perform finite element analysis. Analysis shows that the three conversion parameters have the following rules on the load-deformation characteristic of the diaphragm spring: as the reciprocal 1/n of the exponent is increased, the peak value, the inflection point load and the valley value of the load-deformation curve are increased and have a positive linear relation, and can be described by a linear function, as shown in FIG. 5. The correction ring ratio c'/a increases, the peak value, the inflection point load and the valley value of the load-deformation curve increase, and the increase rate becomes large, which can be described by a quadratic function, as shown in fig. 6. As the ratio i of the fillet radius of the window hole is increased, the peak value, the inflection point load and the valley value of the load-deformation curve are increased, and the acceleration is accelerated, which can be described by a quadratic function, as shown in FIG. 7.
Specifically, the step 3) proposes and defines the separation finger influence coefficient:
the load-deflection curve of the diaphragm spring is a cubic curve through the origin, and the function form is as follows:
F＝a1λ+a2λ2+a3λ3 (4)
in the above formula (4), a1、a2、a3Is the coefficient of each order of the diaphragm spring, a1Is the coefficient of a first order term, a2Is the coefficient of the quadratic term, a3Is the cubic term coefficient.
The load-deflection curve of the disc spring is also a cubic curve passing through the origin, and the function form is as follows:
Fdish＝a'1λ+a'2λ2+a'3λ3 (5)
In the formula (5), a'1、a’2、a’3Each coefficient of the disc spring is expressed.
The ratio of each secondary coefficient of the diaphragm spring function to each secondary coefficient of the disc spring function is provided as the influence coefficient of the separation finger. Let I be the separation finger influence coefficient, I1Is a first order coefficient of influence, I2Is the coefficient of influence of the quadratic term, I3Is the cubic term influence coefficient, then:
Figure GDA0003446556820000111
specifically, in the step 3), displacement loads of a peak point, an inflection point and a valley point on a load-deformation curve obtained by finite element analysis are substituted into a polynomial function of the diaphragm spring or the disc spring to solve each secondary coefficient of the diaphragm spring or the disc spring, so that a functional form of load-deformation characteristics is obtained, namely a linear equation is solved as follows:
Figure GDA0003446556820000112
for convenience of description and writing, the linear equation set is expressed by a matrix, so that
Figure GDA0003446556820000121
The matrix form of the system of equations is then expressed as:
Λa＝b (8)
the matrix Λ is called a coefficient matrix of the system of linear equations, a is the column of unknowns, and b is the column of constant terms.
And (3) respectively transposing two sides of the peer-to-peer equation to obtain:
aTΛT＝bT
handleTMoving to the right of the equation, we get:
aT＝bT(ΛT)-1
3 th order matrix ΛTThe sufficient requirement for reversibility is ΛTBeing non-singular, i.e. 3-order matricesTDeterminant | Λ ofT| is not equal to zero. A 3 rd order matrix lambdaTThe determinant of (a) is developed as:
Figure GDA0003446556820000122
the determinant on the right of the equation is a 3 rd order van der mond determinant. Because of the displacement lambda of peak point, inflection point and valley point of the load-deformation curve1、λ2、λ3Are all not zero and are not equal to each other, then |. ΛT| is not equal to zero, ΛTIt is reversible. LambdaTReversible matrix of (Λ)T)-1Are present.
The coefficients of each secondary term of the diaphragm spring function are respectively the peak value, the inflection point load and the valley value (lambda)T)-1Linear combinations of elements of each column. Because the peak value, the inflection point load and the valley value have the same rule with the conversion parameter, the same rule also exists for each secondary coefficient of the diaphragm spring and the conversion parameter. And removing each secondary coefficient constant of one disc spring from each secondary coefficient of the diaphragm spring to obtain each secondary separation finger influence coefficient. Removing a constant does not affect the rule, i.e. each term separation is considered to mean that the same rule exists between the influence coefficient and the transformation parameter, as shown in fig. 8 to 10.
Specifically, theIn other words, the step 4) designs a three-factor five-level orthogonal test according to the approximate range of the distribution of the conversion parameters in the actual production. Selecting L25(56) The experiments were performed on a type orthogonal table, as shown in table 1. And a simulation test was performed using finite element software.
TABLE 1 orthogonal Table L25(56)
Figure GDA0003446556820000131
Specifically, the step 5) uses a ternary quadratic regression model to fit a functional relation of each secondary coefficient and three conversion parameters. For convenience of writing, the reciprocal exponent 1/n, the correction ring ratio c'/a and the window hole fillet radius ratio i are respectively replaced by x, y and z, and then:
Figure GDA0003446556820000141
in particular, step 5) simplifies the model according to the geometrical significance of the actual existence of the diaphragm spring. If the separation index number tends to be infinite, the separation finger part has almost no stress change when the membrane spring is loaded, namely, the separation finger part does not influence the loading characteristic of the membrane spring. If the correction ring ratio is equal to zero, the diaphragm spring becomes a disc spring and no separation finger exists. Whether the number of fingers tends to infinity or the correction ring ratio is equal to zero, the portion of the fingers in the model should be absent and the constant term equal to 1. The window hole fillet exists depending on the separation index, and the radius of the window hole fillet also exists depending on the separation index. The model is simplified as follows:
Ii(x,y,z)(1≤i≤3)＝a1xy2z2+a2xy2z+a3xy2+a4xyz2+a5xyz+a6xy+1 (11)
specifically, the step 6) is to fit the separation finger influence coefficient function equation by using SPSS software. And fitting according to the model simplified expression, firstly generating four square interaction terms and two interaction terms of three independent variables contained in the simplified model, and then selecting the variables by adopting a stepwise regression method. Since the right side of the simplified model equation contains a constant 1, the constant needs to be moved to the left side of the equation to establish a new dependent variable, and the fitting equation does not contain the constant, specifically:
6-1) establishing a catalyst containing x, y, z, I1，I2，I3SPSS datasets of six variables;
6-2) generates a new variable x1y2z2 (representing xy)2z2) X1y2z1 (represents xy)2z), x1y2 (representing xy)2) X1y1z2 (representing xyz2) X1y1z1 (representing xyz), x1y1 (representing xy), I1S1 (representing I)1-1), I2S1 (representing I)2-1), I3S1 (representing I)3-1)；
6-3) entering a linear regression dialog box, selecting I1S1 into a dependent variable box, selecting x1y2z2, x1y2z1, x1y2, x1y1z2, x1y1z1 and x1y1 into an independent variable box, selecting a stepwise regression method, eliminating the option of 'constant is contained in equation', clicking a determination button to obtain a one-time item influence coefficient I1The fit relationship to the three transformation parameters is:
I1＝-10.248xy2z+9.292xy2+5.02xyz2+7.304xyz+1
the difference is that I2S1 and I3S1 are respectively selected into a dependent variable box, and a quadratic term influence coefficient I is obtained by solving2Coefficient of influence of cubic term I3The fit relationship to the three transformation parameters is:
I2＝-10.011xy2z+8.934xy2+4.707xyz2+7.361xyz+1
I3＝-8.872xy2z+8.76xy2+3.999xyz2+6.497xyz+1
in summary, the separation of the fit means that the influence coefficient function is:
Figure GDA0003446556820000151
specifically, the step 7) replaces the approximate calculation method a-L with the disc spring exact calculation method S-W method proposed by professor schmidt (r.schmidt) and quinuprat sodium (d.a.wempner) of the civil engineering system of arizona university, usa in 1959. The method is an accurate calculation method of the disc spring and has an explicit solution. Compared with the A-L method, the method is closer to the actual curve. The formula of the S-W method is as follows:
p＝B(A1C+A2C2+A3C3) (13)
Figure GDA0003446556820000152
Figure GDA0003446556820000153
Figure GDA0003446556820000154
Figure GDA0003446556820000155
Figure GDA0003446556820000156
in the formula (13)
Figure GDA0003446556820000157
Figure GDA0003446556820000158
Figure GDA0003446556820000159
Figure GDA00034465568200001510
Specifically, the step 7) expresses the calculation formula of the S-W method in the form of a cubic polynomial:
F＝a1λ+a2λ2+a3λ3 (14)
the influence coefficient of the separation finger is used as a correction term of the separation finger part and is added to a calculation formula of an S-W method of a disc spring accurate calculation method to obtain a new calculation method of the diaphragm spring load-deformation characteristic, wherein the expression is as follows:
F＝a1I1λ+a2I2λ2+a3I3λ3 (15)
and writing a detailed formula of the new method into scientific calculation software MATLAB for calculating the load deformation characteristic of the diaphragm spring.
The above examples of the present invention are merely examples for clearly illustrating the method and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (7)

1. A novel method for calculating load-deformation characteristics of a diaphragm spring is characterized by comprising the following steps:
1) solving to obtain a load-deformation characteristic curve of the diaphragm spring by using finite element simulation software;
2) replacing diaphragm springs in different window hole forms with the conversion model, replacing structural parameters of different window holes with three conversion parameters, and analyzing the load-deformation characteristic of the conversion model according to the method in the step 1) to further obtain the influence rule of the three conversion parameters on the load-deformation characteristic of the diaphragm springs;
3) the influence coefficient of the separation finger is provided and defined, and the relation between the peak value, the inflection point load, the valley value and the conversion parameter is found to be also applicable to the relation between the influence coefficient of the separation finger and the conversion parameter of each time through matrix conversion, so that the influence rule of the influence coefficient of the separation finger on the load-deformation characteristic of the diaphragm spring is obtained;
4) designing an orthogonal experiment according to the discovered influence rule of the separation finger influence coefficient on the load-deformation characteristic of the diaphragm spring, and acquiring a plurality of groups of representative data about the separation finger influence coefficient and three conversion parameters of each item;
5) establishing and simplifying a regression model of each separation finger influence coefficient and three conversion parameters according to the discovered influence rule of the separation finger influence coefficient on the load-deformation characteristic of the diaphragm spring;
6) obtaining a separation finger influence coefficient function formula by utilizing SPSS software regression fitting according to the acquired data about the separation finger influence coefficients of each time item and the three conversion parameters;
7) replacing an approximate calculation method A-L method with a disc spring accurate calculation method S-W method, adding a separation finger influence coefficient serving as a correction term about a separation finger part to a calculation formula of the disc spring accurate calculation method S-W method to obtain a new calculation method of the load-deformation characteristic of the diaphragm spring, and calculating the load-deformation characteristic of the diaphragm spring by using the new calculation method;
step 2), specifically, the diaphragm spring separation finger structures in different window hole forms are converted to obtain a conversion model of the diaphragm spring, and the diaphragm spring separation finger structure parameters in different window hole forms are represented by uniform parameters to obtain three conversion parameters, wherein the three conversion parameters are respectively: the relation between three conversion parameters and the actual structure parameters of the separation finger part is respectively established by the following formulas:
Figure FDA0003446556810000011
Figure FDA0003446556810000021
Figure FDA0003446556810000022
in the above formula, n is an index, r is the inner radius of the disc spring portion, and δ2The width of the window hole groove is delta3R' is the aperture radius for the contact width of the separation finger with the inner diameter;
step 2), specifically, a diaphragm spring conversion geometric model with a structure of the disc spring portion unchanged and a structure of the separation finger portion changed is constructed, finite element analysis is performed, and the load-deformation characteristic rule of the diaphragm spring obtained by analysis of three conversion parameters is as follows: along with the increase of 1/n of the reciprocal exponent, the peak value, the inflection point load and the valley value of the load-deformation curve are increased and are in a forward linear relation, the correction loop ratio value c '/a is increased, the peak value, the inflection point load and the valley value of the load-deformation curve are increased, the increase rate is increased, and the correction loop ratio value c'/a is described by a quadratic function; along with the increase of the radius ratio i of the fillet of the window hole, the peak value, the inflection point load and the valley value of a load-deformation curve are increased, and the acceleration is accelerated and is described by a quadratic function;
in the step 3), a separation finger influence coefficient is proposed and defined, and specifically:
the load-deflection curve of the diaphragm spring is a cubic curve through the origin, and the function form is as follows:
F＝a1λ+a2λ2+a3λ3
in the above formula, F is the load, a1、a2、a3Is the coefficient of each order of the diaphragm spring, a1Is the first order coefficient of the diaphragm spring, a2Is the second order coefficient of the diaphragm spring, a3Is the coefficient of the third order of the diaphragm spring, λ is the displacement;
the load-deflection curve of the disc spring is also a cubic curve passing through the origin, and the function form is as follows:
Fdish＝a'1λ+a'2λ2+a'3λ3
In the above formula, a'1、a’2、a’3Denotes each linear coefficient, a 'of the disc spring'1Is the primary coefficient of the disc spring, a'2Is the coefficient of the secondary term of the disc spring, a'3Is the cubic coefficient of the disc spring;
the ratio of each secondary coefficient of the diaphragm spring function to each secondary coefficient of the disc spring function is provided as the influence coefficient of the separation finger, and I is the influence coefficient of the separation finger1Is a first order coefficient of influence, I2Is the coefficient of influence of the quadratic term, I3Is the cubic term influence coefficient, then:
Figure FDA0003446556810000031
2. the method for calculating the load-deformation characteristic of the diaphragm spring as claimed in claim 1, wherein in step 1), a load-deformation characteristic curve of the diaphragm spring is drawn by solving a load time history curve at the supporting ring and a displacement time history curve at the pressure plate contact position by using finite element software.
3. The method for calculating the load-deformation characteristic of the diaphragm spring as claimed in claim 1, wherein in the step 3), the displacement loads of the peak point, the inflection point and the valley point on the load-deformation curve obtained by the finite element analysis are substituted into the polynomial function of the diaphragm spring or the disc spring to solve each polynomial coefficient of the diaphragm spring or the disc spring, so as to obtain the functional form of the load-deformation characteristic, that is, the linear equation is solved as follows:
Figure FDA0003446556810000032
for convenience of description and writing, the linear equation set is expressed by a matrix, so that
Figure FDA0003446556810000033
The matrix form of the system of equations is then expressed as: a is equal to b and a is equal to b,
the matrix Lambda is called coefficient matrix of linear equation system, a is unknown quantity column, b is constant item column, two sides of equation are respectively transposed to obtain aTΛT＝bTAngle of rotationTMove to the right of the equation to get aT＝bT(ΛT)-1，
3 th order matrix ΛTThe sufficient requirement for reversibility is ΛTBeing non-singular, i.e. 3-order matricesTDeterminant | Λ ofT| is not equal to zero, and a 3 rd order matrix Λ is formedTThe determinant of (a) is developed as:
Figure FDA0003446556810000034
in the above equation, the right determinant is a 3 rd order van der Mond determinant due to the peak displacement λ of the load-deformation curve1Displacement of inflection point lambda2Valley point displacement lambda3Are all not zero and are not equal to each other, then |. ΛT| is not equal to zero, ΛTReversible, ΛTReversible matrix of (Λ)T)-1(ii) present;
the coefficients of each secondary term of the diaphragm spring function are respectively the peak value, the inflection point load and the valley value (lambda)T)-1The linear combination of each row of elements has the same rule as the peak value, the inflection point load and the valley value and the conversion parameter, so that the same rule also exists for each secondary coefficient of the diaphragm spring and the conversion parameter, each secondary coefficient constant of one disc spring is removed from each secondary coefficient of the diaphragm spring to obtain each secondary separation finger influence coefficient, and the influence coefficient of each secondary separation finger is removed without influencing the rule, namely, each secondary separation finger influence coefficient and the conversion parameter are considered to have the same rule.
4.The method for calculating the load-deflection characteristics of a diaphragm spring according to claim 1, wherein the step 4) is to design a three-factor five-level orthogonal test according to the approximate range of the distribution of the conversion parameters in actual production, and select L25(56) The test was performed using a type orthogonal table, as shown in table 1, and a simulation test was performed using finite element software,
TABLE 1 orthogonal Table L25(56)
Figure FDA0003446556810000041
Figure FDA0003446556810000051
5. The method for calculating a load-deflection characteristic of a diaphragm spring according to claim 1, wherein in step 5), a ternary quadratic regression model is used to fit a function relation between each secondary coefficient and three transformation parameters, and the reciprocal of the exponent 1/n, the correction ring ratio c'/a and the window hole fillet radius ratio i are respectively replaced by x, y and z, so that:
Figure FDA0003446556810000052
according to the geometric meaning of the actual existence of the diaphragm spring, the model is simplified, if the number of the separating fingers tends to be infinite, the stress change of the separating finger part is almost absent when the diaphragm spring is loaded, namely the separating finger part does not influence the load characteristic of the diaphragm spring; if the ratio of the correction ring is equal to zero, the membrane spring becomes a disc spring, and a separation finger part does not exist; whether the number of the separating fingers tends to be infinite or the ratio of the correction ring is equal to zero, the separating finger part in the model does not exist, the constant term is equal to 1, the window hole fillet exists depending on the separating fingers, and the radius of the window hole fillet also exists depending on the number of the separating fingers; the model is simplified as follows:
Ii(x,y,z)(1≤i≤3)＝a1xy2z2+a2xy2z+a3xy2+a4xyz2+a5xyz+a6xy+1。
6. the method as claimed in claim 1, wherein in step 6), the SPSS software is used to perform regression fitting to obtain the separation finger influence coefficient function, specifically, the fitting is performed according to a model simplified equation, four square interaction terms and two interaction terms of three independent variables included in the simplified model are generated, then a stepwise regression method is used to select the variables, since the right side of the simplified model equation includes the constant 1, the constant needs to be shifted to the left side of the equation to establish a new dependent variable, and the fitting equation does not include the constant, specifically:
6-1) establishing a catalyst containing x, y, z, I1，I2，I3SPSS datasets of six variables;
6-2) generates a new variable x1y2z2 (representing xy)2z2) X1y2z1 (represents xy)2z), x1y2 (representing xy)2) X1y1z2 (representing xyz2) X1y1z1 (representing xyz), x1y1 (representing xy), I1S1 (representing I)1-1), I2S1 (representing I)2-1), I3S1 (representing I)3-1)；
6-3) entering a linear regression dialog box, selecting I1S1 into a dependent variable box, selecting x1y2z2, x1y2z1, x1y2, x1y1z2, x1y1z1 and x1y1 into an independent variable box, selecting a stepwise regression method, eliminating the option of 'constant is contained in equation', clicking a determination button to obtain a one-time item influence coefficient I1The fit relationship to the three transformation parameters is:
I1(x,y,z)
respectively selecting I2S1 and I3S1 into dependent variable boxes, and solving to obtain quadratic term influence coefficient I2Coefficient of influence of cubic term I3The fit relationship to the three transformation parameters is:
I2(x,y,z)
I3(x,y,z)
in summary, the separation of the fit means that the influence coefficient function is:
Ii(x,y,z)(1≤i≤3)。
7. the method for calculating load-deflection characteristics of a diaphragm spring according to claim 1, wherein step 7) is performed by using a disc spring accurate calculation method S-W method instead of an approximate calculation method a-L method, which is proposed by schmidt (r.schmidt) and wiberner (d.a.wempner) of the civil engineering assistant professor schmidt, university of arizona in 1959, and the formula of the S-W method is as follows:
p＝B(A1C+A2C2+A3C3)
Figure FDA0003446556810000071
Figure FDA0003446556810000072
Figure FDA0003446556810000073
Figure FDA0003446556810000079
Figure FDA0003446556810000074
in the formula
Figure FDA0003446556810000075
Figure FDA0003446556810000076
Figure FDA0003446556810000077
Figure FDA0003446556810000078
The formula of the S-W method is expressed as a cubic polynomial in the form:
F＝a1λ+a2λ2+a3λ3
the influence coefficient of the separating fingers is used as a correction term of the separating fingers, and is added to a calculation formula of an S-W method of a disc spring accurate calculation method to obtain a new calculation method of the load-deformation characteristic of the diaphragm spring, wherein the expression is as follows:
F＝a1I1λ+a2I2λ2+a3I3λ3。
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