CN105893684A - Calibrating method for strengths of non-end contact type few-leaf root-reinforcing main and auxiliary springs - Google Patents

Abstract

The invention relates to a calibrating method for strengths of non-end contact type few-leaf root-reinforcing main and auxiliary springs, and belongs to the technical field of suspension steel plate springs. According to the structure parameters, elastic modulus, allowable stress, auxiliary spring action load, and maximum loads of each of the non-end contact type few-leaf root-reinforcing variable-section main and auxiliary springs, the stress strengths of each main spring and each auxiliary spring are calibrated. After proofing by examples and simulation verification, the calibrating method for the strengths of the non-end contact type few-leaf root-reinforcing main and auxiliary springs has the advantages that the calibrating method is correct; the accurate and reliable maximum stress calibrating values of each main spring and each auxiliary spring can be obtained, and the design levels, product qualities and service lives of the non-end contact type few-leaf root-reinforcing variable-section main and auxiliary springs, and the vehicle driving smoothness are improved; the design and test cost is reduced, and the product development speed is accelerated.

Description

The check method of the few sheet root reinforced major-minor each intensity of spring of non-ends contact formula

Technical field

The present invention relates to vehicle suspension leaf spring, be few each of the sheet root reinforced major-minor spring of non-ends contact formula especially The check method of intensity.

Background technology

Few sheet variable-section steel sheet spring, compared with multi-disc superposition leaf spring, is specifically saved material, is alleviated unsprung mass, carries The advantages such as high vehicle ride comfort and conevying efficiency, cause the great attention of domestic and international vehicle expert, and the most carry out Promotion and application widely.For few sheet variable-section steel sheet spring, generally it is designed to major-minor spring, and by between major-minor spring Gap, it is ensured that after the load that works more than auxiliary spring, major-minor spring contacts and works together, meets vehicle suspension in different loads feelings Design requirement to leaf spring rigidity and stress intensity under condition.Owing to the stress of the 1st main spring is complicated, it is subjected to vertical load Lotus, simultaneously also subject to torsional load and longitudinal loading, therefore, the thickness of the end flat segments of the 1st main spring designed by reality And length, it is typically larger than thickness and the length of the end flat segments of his each main spring, i.e. in actual design and production, mostly adopts With the non-few sheet variable cross-section major-minor spring waiting structure in end.Few sheet variable-section steel sheet spring mainly has two types, and one is parabola Type, another is bias type, and wherein, Parabolic stress is iso-stress, and suffered by it, stress ratio bias type is more reasonable； Meanwhile, in order to strengthen the root intensity of parabolic type variable-section steel sheet spring, can increase between root flat segments and parabolic segment Add an oblique line section, i.e. use root reinforced variable cross-section major-minor spring.Few sheet parabolic type variable cross-section major-minor spring, can use different Auxiliary spring length is to meet the design requirement of different composite rigidity and stress intensity, therefore, according to the length difference i.e. major-minor of auxiliary spring Different contact position, few sheet variable cross-section major-minor spring can be divided into ends contact formula and non-ends contact formula two kinds.For set The few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula of meter, its intensity should meet bearing spring service life and Security requirement, but, due to the few sheet root reinforced variable cross-section major-minor leaf spring of non-ends contact formula main spring length with After the structures such as the end flat segments of unequal, each main spring of auxiliary spring length is non-, and the contact of major-minor spring, the deformation of main spring and auxiliary spring and Internal force has coupling, therefore, and each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the end of auxiliary spring Point power and maximum stress calculate extremely complex, fail to provide simplicity, the few sheet root of non-ends contact formula accurate, reliable the most always The check method of portion's reinforced variable cross-section each intensity of major-minor spring.Therefore, it is necessary to set up a kind of non-ends contact accurate, reliable The check method of the few sheet root reinforced variable cross-section each intensity of major-minor spring of formula, meets Vehicle Industry fast development and to suspension steel The requirement of flat spring careful design, improves non-ends contact the formula few design level of sheet root type variable cross-section major-minor spring, product matter Amount and service life；Meanwhile, reduce design and testing expenses, accelerate product development speed.

Summary of the invention

For defect present in above-mentioned prior art, the technical problem to be solved be to provide a kind of easy, The check method of the few sheet root reinforced major-minor each intensity of spring of reliable non-ends contact formula, it checks flow chart, such as Fig. 1 institute Show.The few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula is symmetrical structure, the half pair of each major-minor leaf spring Claiming structure is to be made up of root flat segments, oblique line section, parabolic segment, end flat segments four sections, and wherein, oblique line section is to leaf spring Root play booster action；The major-minor spring of half symmetrical structure can regard cantilever beam as, symmetrical center line will regard half spring as Fixing end, regard main spring end stress point and auxiliary spring ends points as the main spring of half and the end points of auxiliary spring, its half pair respectively Claim structural representation, as in figure 2 it is shown, include: main spring 1, root shim 2, auxiliary spring 3, end pad 4；Each root of main spring 1 is put down Root shim 2 it is provided with, between each end flat segments of main spring 1 between straight section and between each root flat segments of auxiliary spring 3 Being provided with end pad 4, the material of end pad 4 is carbon fibre composite, produces frictional noise during to prevent work.Each The width of main spring is b, a length of L of half_M, half l of installing space₃, a length of Δ l of oblique line section, the root of oblique line section is to main The distance of spring end points is l_2M, the distance of the end of oblique line section to main spring end points is l_2Mp；The root thickness of each main spring is h_2M, tiltedly The end thickness of line segment is h_2Mp, i.e. the thickness of oblique line section compares γ_M=h_2Mp/h_2M；The non-structure that waits of the end flat segments of each main spring, i.e. The thickness of the end flat segments of the 1st main spring and length, more than the thickness of end flat segments and the length of other each main spring；Respectively Thickness and the length of the end flat segments of the main spring of sheet are respectively h_1iAnd l_1i, the thickness of parabolic segment is than for β_i=h_1i/h_2Mp, i= 1,2 ..., m, m are the sheet number of main spring 1.The thickness of the root flat segments of each auxiliary spring is h_2A, width is b, a length of L of half_A, Half l of installing space₃, a length of Δ l of the oblique line section of each auxiliary spring, the distance of the root of oblique line section to auxiliary spring end points is l_2A, the distance of the end of oblique line section to auxiliary spring end points is l_2Ap；The end thickness of oblique line section is h_2Ap, the thickness of oblique line section compares γ_A =h_2Ap/h_2A；Thickness and the length of the end flat segments of each auxiliary spring are respectively h_A1jAnd l_A1j, the thickness of parabolic segment is than for β_Aj =h_A1j/h_2Ap, j=1,2 ..., n, n are the sheet number of auxiliary spring.Half length L of auxiliary spring_AHalf length L less than main spring_M, auxiliary spring Horizontal range between contact and main spring end points is l₀=L-L_A；Vertical distance between auxiliary spring end points and main spring parabolic segment is Major-minor spring gap is δ, when load works load more than auxiliary spring, and certain point in the parabolic segment of auxiliary spring end points and the main spring of m sheet Contact；After the contact of major-minor spring, each end points stress of major-minor spring is unequal, and the main spring of m sheet is in addition to by end points power, Also acted on by auxiliary spring end points support force.Work at each chip architecture parameter of major-minor spring, elastic modelling quantity, allowable stress, auxiliary spring In the case of load and maximum load are given, each main spring of sheet root reinforced major-minor spring few to non-ends contact formula and auxiliary spring Stress intensity is checked.

For solving above-mentioned technical problem, few each of the sheet root reinforced major-minor spring of non-ends contact formula provided by the present invention The check method of intensity, it is characterised in that the following step of checking of employing:

(1) each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the half clamping of auxiliary spring are firm Degree calculates:

I step: the half clamping stiffness K of each main spring before the contact of major-minor spring_MiCalculate:

Half length L according to few sheet root main spring of reinforced variable cross-section_M, main reed number m, each main spring root flat segments Thickness h_2M, width b, elastic modulus E, half l of installing space₃, oblique line segment length Δ l, the root of parabolic segment is to main spring Distance l of end points_2Mp=L_M-l₃-Δ l, the root of oblique line section is to distance l of main spring end points_2M=L_M-l₃, the oblique line of each main spring The thickness of section compares γ_M, the thickness of the parabolic segment of i-th main spring compares β_i, wherein, i=1,2 ..., m, before contacting major-minor spring Each main spring half clamping stiffness K_MiCalculate, i.e.

$K_{Mi}=\frac{h_{2M}^3}{G_{x-Ei}},i=1,2,...,m;$

In formula,

$\begin{array}{c}{G}_{x-Ei}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_i^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2\mathrm{\Δ}l{\mathrm{ln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3};\end{array}$

II step: the half clamping stiffness K of each main spring after the contact of major-minor spring_MAiCalculate:

Half length L according to few sheet root main spring of reinforced variable cross-section_M, main reed number m, the root of each main spring is straight The thickness h of section_2M, width b, elastic modulus E, half l of installing space₃, the length Δ l of oblique line section, the root of parabolic segment arrives Distance l of main spring end points_2Mp=L_M-l₃-Δ l, the root of oblique line section is to distance l of main spring end points_2M=L_M-l₃, the thickness of oblique line section Degree compares γ_M, the thickness of the parabolic segment of i-th main spring compares β_i, wherein, i=1,2 ..., m；Half length L of auxiliary spring_A, auxiliary spring sheet Number n, the thickness h of the root flat segments of each auxiliary spring_2A, auxiliary spring contact and horizontal range l of main spring end points₀, the root of parabolic segment Portion is to distance l of auxiliary spring end points_2Ap=L_A-l₃-Δ l, the root of oblique line section is to distance l of auxiliary spring end points_2A=L_A-l₃, each pair The thickness of the oblique line section of spring compares γ_A, the thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, to major-minor spring The half clamping stiffness K of each main spring after contact_MAiCalculate, i.e.

$K_{MAi}=\left\{\begin{array}{cc}\frac{h_{2M}^3}{G_{x-Ei}},& i=1,2,\mathrm{...},m-1\\ \frac{h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)}{G_{x-Em}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)-G_{x-E_{p}m}{G}_{x-CD}{h}_{2A}^3},& i=m\end{array}\right.;$

In formula,

$\begin{array}{c}{G}_{x-Ei}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_i^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2\mathrm{\Δ}l{\mathrm{ln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3};\end{array}$

$G_{x-EAT}=\frac{1}{\underset{j=1}{\overset{n}{\Σ}}\frac{1}{G_{x-EAj}}},j=1,2,\mathrm{...},n;$

$\begin{array}{c}{G}_{x-EAj}=\frac{4{(L_A-l_3/2)}^3-l_{2A}^3\]}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{Aj}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(3l_{2A}^2{\mathrm{\γ}}_A^2+l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2\mathrm{\Δ}l{\mathrm{ln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3};\end{array}$

$\begin{array}{c}{G}_{x-CD}=\frac{4{(L_M-l_3/2)}^3-6l_0{(L_M-l_3/2)}^3-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}+\frac{8l_{2Mp}^3+16l_{2Mp}^{3/2}{l}_0^{3/2}-24l_{2Mp}^2{l}_0}{{\mathrm{Eb\γ}}_M^3}-\frac{6l_0\mathrm{\Δ}l(l_{2Mp}+l_{2M}{\mathrm{\γ}}_M)}{{\mathrm{Eb\γ}}_M^2}+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3-4l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M);\end{array}$

$\begin{array}{c}{G}_{x-E_{p}m}=\frac{4{(L_M-l_3/2)}^3-6l_0{(L_M-l_3/2)}^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{12}{Eb}\[\frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{2l_{2M}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^3}-\\ \frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3{\mathrm{\γ}}_M^2}-\frac{2l_{2Mp}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}-\frac{{\mathrm{\Δl}}^3}{{({\mathrm{\γ}}_M-1)}^3}{\mathrm{ln\γ}}_M\]-\frac{24l_0{l}_{2Mp}^2-8{l_{2Mp}}^3-16l_0^{3/2}{l}_{2Mp}^{3/2}}{{\mathrm{Eb\γ}}_M^3}-\\ \frac{6l_0\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{\mathrm{Eb\γ}}_M^2};\end{array}$

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_p}=\frac{4(L_M-l_3/2-l_{2M})\[{(L_M-l_3/2)}^2-3l_0(L_M-l_3/2)+(L_M-l_3/2)l_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\\ \frac{12}{Eb}\[\frac{l_0^2\mathrm{\Δ}l{({\mathrm{\γ}}_M-1)}^2-3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}+\frac{2l_{2Mp}\mathrm{\Δ}l(\mathrm{\Δ}l-l_{2Mp}-l_0{\mathrm{\γ}}_M+l_{2M}{\mathrm{\γ}}_M)}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}+\\ \frac{\mathrm{\Δ}l\[2l_0^3\mathrm{\Δ}l({\mathrm{\γ}}_M-1)(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})+3{(L_M-l_3/2)}^2-2(L_M-l_3/2)l_0-2(L_M-l_3/2)l_{2M}{\mathrm{\γ}}_M\]}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l\left(2\right(L_M-l_3/2)l_0{\mathrm{\γ}}_M-4(L_M-l_3/2)l_{2M}-6(L_M-l_3/2)l_3+2l_0^2{\mathrm{\γ}}_M-6(L_M-l_3/2)\mathrm{\Δ}l-l_0^2{\mathrm{\γ}}_M^2-l_0^2)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2l_0{l}_{2M}{\mathrm{\γ}}_M^2+4l_0{l}_{2M}-6l_0{l}_{2M}{\mathrm{\γ}}_M-2l_0{l}_3{\mathrm{\γ}}_M+2l_0{l}_3-2l_0{\mathrm{\Δl\γ}}_M+2l_0\mathrm{\Δ}l-l_{2M}^2{\mathrm{\γ}}_M^2+4l_{2M}^2{\mathrm{\γ}}_M)}{2{({\mathrm{\γ}}_M-1)}^3}+\end{array}$

$\begin{array}{c}\frac{\mathrm{\Δ}l(2l_{2M}{l}_3{\mathrm{\γ}}_M+4l_{2M}{l}_3+2l_{2M}{\mathrm{\Δl\γ}}_M+4l_{2M}\mathrm{\Δ}l+3l_3^2+6l_3\mathrm{\Δ}l+3{\mathrm{\Δl}}^2)}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{{\mathrm{\Δl}}^3{\mathrm{ln\γ}}_M}{{({\mathrm{\γ}}_M-1)}^3}\]+\\ \frac{12}{{\mathrm{Eb\γ}}_M^3}(\frac{2l_{2p}^2-12l_0{l}_{2Mp}-6l_0^2}{3l_{2Mp}^2}+\frac{16l_0^{3/2}}{3l_{2Mp}^{3/2}});\end{array}$

III step: the half clamping stiffness K of each auxiliary spring_AjCalculate:

Half length L according to few sheet root reinforced variable cross-section auxiliary spring_A, auxiliary spring sheet number n, the root of each auxiliary spring is straight The thickness h of section_2A, width b, half l of installing space₃, the length Δ l of oblique line section, the root of parabolic segment is to auxiliary spring end points Distance l_2Ap=L_A-l₃-Δ l, the root of oblique line section is to distance l of auxiliary spring end points_2A=L_A-l₃, the thickness ratio of auxiliary spring oblique line section γ_A, the thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, the half of each auxiliary spring is clamped stiffness K_Aj Calculate, i.e.

$K_{Aj}=\frac{h_{2A}^3}{G_{x-EAj}},j=1,2,\mathrm{...},n;$

In formula,

$\begin{array}{c}{G}_{x-EAj}=\frac{4\[{(L_A-l_3/2)}^3-l_{2A}^3\]}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{Aj}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(3l_{2A}^2{\mathrm{\γ}}_A^2+l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2\mathrm{\Δ}l{\mathrm{ln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3};\end{array}$

(2) non-ends contact formula lacks each main spring and the maximum end points power of auxiliary spring of sheet root reinforced variable cross-section major-minor spring Calculate:

I step: the maximum end points power of each main spring calculates:

Half according to maximum load suffered by few sheet root reinforced variable-section steel sheet spring major-minor spring is the most single-ended maximum Load p_max, auxiliary spring works load p_K, calculated K in I step_Mi, and II step calculates obtained K_MAi, main reed Number m, maximum end points power P to each main spring_imaxCalculate, i.e.

$P_{i\max }=\frac{K_{Mi}{P}_K}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{Mi}}+\frac{K_{MAi}(2P_{\max }-P_K)}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}},i=1,2,\mathrm{...},m;$

Ii step: the maximum end points power of each auxiliary spring calculates:

Half according to maximum load suffered by few sheet root reinforced variable-section steel sheet spring major-minor spring is the most single-ended maximum Load p_max, auxiliary spring works used load P_K, main reed number m, the thickness h of the root flat segments of each main spring_2M, auxiliary spring sheet number N, the thickness h of the root flat segments of each auxiliary spring_2A, calculated K in II step_MAi、G_x-CD、And G_x-EAT, and III step Calculated K in Zhou_Aj, maximum end points power P to each auxiliary spring_AjmaxCalculate, i.e.

$P_{Aj\max }=\frac{K_{Aj}{K}_{MAm}{G}_{x-CD}{h}_{2A}^3(2P_{\max }-P_K)}{2\underset{j=1}{\overset{n}{\Σ}}{K}_{Aj}\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)},j=1,2,\mathrm{...},n;$

(3) non-ends contact formula lacks each main spring and the maximum stress meter of auxiliary spring of sheet root reinforced variable cross-section major-minor spring Calculate:

Step A: the maximum stress of the front main spring of m-1 sheet calculates:

Half length L according to few sheet root main spring of reinforced variable-section steel sheet spring_M, main reed number m, each main spring The thickness h of root flat segments_2M, width b, half l of installing space₃, calculated P in i step_imax, to the front main spring of m-1 sheet Maximum stress be respectively calculated, i.e.

${\mathrm{\σ}}_{imax}=\frac{6P_{imax}(L_M-l_3/2)}{{\mathrm{bh}}_{2M}^2},i=1,2,\mathrm{...}m-1;$

Step B: the maximum stress of the main spring of m sheet calculates:

According to main reed number m, the thickness h of the root flat segments of each main spring_2M, width b, half l of installing space₃, tiltedly The length Δ l of line segment, the root of parabolic segment is to distance l of main spring end points_2Mp=L_M-l₃-Δ l, the oblique line section of each main spring Thickness compares γ_M, the thickness of the parabolic segment of the main spring of m sheet compares β_m；The horizontal range of auxiliary spring sheet number n, auxiliary spring contact and main spring end points l₀, calculated P in i step_mmax, calculated P in ii step_Ajmax, the maximum stress of spring main to m sheet is counted Calculate, i.e.

${\mathrm{\σ}}_{m\max }=\frac{6\[P_{m\max }{\mathrm{\β}}_m^2{l}_{2Mp}-\underset{j=1}{\overset{n}{\Σ}}{P}_{Aj\max }({\mathrm{\β}}_m^2{l}_{2Mp}-l_0)\]}{b{\left({\mathrm{\β}}_m{\mathrm{\γ}}_M{h}_{2M}\right)}^2};$

Step C: the maximum stress of each auxiliary spring calculates:

Half length L according to few sheet root reinforced variable cross-section auxiliary spring_A, auxiliary spring sheet number n, the root of each auxiliary spring is straight The thickness h of section_2A, width b, half l of installing space₃, calculated P in ii step_Ajmax, maximum stress to each auxiliary spring Calculate, i.e.

${\mathrm{\σ}}_{Ajmax}=\frac{6P_{Ajmax}(L_A-l_3/2)}{{\mathrm{bh}}_{2A}^2},j=1,2,\mathrm{...},n;$

(4) non-ends contact formula lacks each main spring and the stress intensity school of auxiliary spring of sheet root reinforced variable cross-section major-minor spring Core:

1. step: the stress intensity of the front main spring of m-1 sheet is checked:

Allowable stress [σ] according to leaf spring, and the maximum of each of the calculated front main spring of m-1 sheet in step A Stress, the stress intensity of each of the front main spring of m-1 sheet of sheet root reinforced variable-section steel sheet spring few to non-ends contact formula Check, it may be assumed that if σ_imax> [σ], then i-th main spring, it is unsatisfactory for stress intensity requirement；If σ_imax≤ [σ], then i-th Main spring, meets stress intensity requirement, i=1, and 2 ... m-1；

2. step: the stress intensity of the main spring of m sheet is checked:

Allowable stress [σ] according to leaf spring, and the maximum stress of the calculated main spring of m sheet in step B, to non- The stress intensity of the main spring of m sheet of the few reinforced variable-section steel sheet spring of sheet root of ends contact formula is checked, it may be assumed that if σ_mmax> [σ], the then main spring of m sheet, it is unsatisfactory for stress intensity requirement；If σ_mmax≤ [σ], the then main spring of m sheet, meets stress intensity Requirement；

3. step: the stress intensity of each auxiliary spring is checked:

Allowable stress [σ] according to leaf spring, and the maximum stress of calculated each auxiliary spring in step C, to non- The stress intensity of each auxiliary spring of the few reinforced variable-section steel sheet spring of sheet root of ends contact formula is checked, it may be assumed that if σ_Ajmax> [σ], then jth sheet auxiliary spring, it is unsatisfactory for stress intensity requirement；If σ_Ajmax≤ [σ], then jth sheet auxiliary spring, meets stress strong Degree requirement, j=1,2 ..., n.

The present invention has the advantage that than prior art

Owing to the end flat segments of each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula is non-etc. Structure, and the length of auxiliary spring is less than the length of main spring, meanwhile, in the case of maximum load, the main spring of m sheet except by end points power it Outward, also being acted on by auxiliary spring contact support power in parabolic segment, therefore, the end points power of each main spring and auxiliary spring calculates the most multiple Miscellaneous, fail to provide the check method of the few sheet root reinforced major-minor each intensity of spring of non-ends contact formula the most always.The present invention Can be according to each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the structural parameters of each auxiliary spring, bullet Property modulus, allowable stress, auxiliary spring work load and maximum load, sheet root reinforced variable cross-section few to non-ends contact formula Each main spring of major-minor spring and the stress intensity of each auxiliary spring are checked.By example and ANSYS simulating, verifying, this The strength check methods of the few sheet root reinforced variable cross-section major-minor spring of bright provided non-ends contact formula is correct, and utilizing should The available the most each main spring of method and the maximum stress calculation and check value of each auxiliary spring, can improve non-ends contact formula Design level, product quality and service life and the vehicle of few sheet root reinforced variable cross-section major-minor leaf spring travel smooth-going Property；Meanwhile, also can reduce design and testing expenses, accelerate product development speed.

Accompanying drawing explanation

In order to be more fully understood that the present invention, it is described further below in conjunction with the accompanying drawings.

Fig. 1 is the flow chart of each stress intensity check of the few sheet root reinforced major-minor spring of non-ends contact formula；

Fig. 2 is the half symmetrical structure schematic diagram of the few sheet root reinforced major-minor spring of non-ends contact formula；

Fig. 3 is the maximum stress emulation cloud atlas of the 1st main spring of embodiment；

Fig. 4 is the maximum stress emulation cloud atlas of the 2nd main spring of embodiment；

Fig. 5 is the maximum stress emulation cloud atlas of 1 auxiliary spring of embodiment.

Specific embodiments

Below by embodiment, the present invention is described in further detail.

Embodiment: the main reed number m=2 of the few sheet root reinforced variable cross-section major-minor spring of certain non-ends contact formula, wherein, respectively Half length L of the main spring of sheet_M=575mm, width b=60mm, elastic modulus E=200GPa, half l of installing space₃= 55mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2Mp=L_M-l₃-Δ l= 490mm, the root of oblique line section is to distance l of main spring end points_2M=L_M-l₃=520mm；The thickness of the root flat segments of each main spring h_2M=11mm, end thickness h of oblique line section_2Mp=10.23mm, the thickness of the oblique line section of each main spring compares γ_M=h_2Mp/h_2M= 0.93；The thickness h of the end flat segments of the 1st main spring₁₁=7mm, the thickness of the parabolic segment of the 1st main spring compares β₁=h₁₁/h_2Mp =0.69；The thickness h of the end flat segments of the 2nd main spring₁₂=6mm, the thickness of the parabolic segment of the 2nd main spring compares β₂=h₁₂/ h_2Mp=0.59.Auxiliary spring sheet number n=1, half length L of this sheet auxiliary spring_A=525mm, the level of auxiliary spring contact and main spring end points away from From l₀=L_M-L_A=200mm；The thickness h of auxiliary spring root flat segments_2A=14mm, end thickness h of auxiliary spring oblique line section_2Ap=13mm, The thickness of oblique line section compares γ_A=h_2Ap/h_2A=0.93；The thickness h of the end flat segments of this sheet auxiliary spring_A11=8mm, the parabolic of auxiliary spring The thickness of line compares β_A1=h_A11/h_2Ap=0.62.Auxiliary spring works load p_K=2380.80N, when load works load more than auxiliary spring During lotus, auxiliary spring end points contacts with certain point in main spring parabolic segment.Allowable stress [the σ]=700MPa of leaf spring, at major-minor The most single-ended some maximum load P of the half of spring maximum load_maxIn the case of=3040N, sheet root few to this non-ends contact formula is strengthened Each major-minor spring stress intensity of type variable-section steel sheet spring is checked.

The check method of the few sheet root reinforced major-minor each intensity of spring of the non-ends contact formula that present example is provided, It checks flow process as it is shown in figure 1, specifically comprise the following steps that

(1) each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the half clamping of auxiliary spring are firm Degree calculates:

I step: the half clamping stiffness K of each main spring before the contact of major-minor spring_MiCalculate:

Half length L according to few sheet root main spring of reinforced variable cross-section_M=575mm, main reed number m=2, each master The thickness h of the root flat segments of spring_2M=11mm, the width b=60mm of the main spring of main spring, elastic modulus E=200GPa, clipping room Away from half l₃=55mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2Mp= 490mm, the root of oblique line section is to distance l of main spring end points_2M=520mm, the thickness of oblique line section compares γ_M=0.93；1st main spring The thickness of parabolic segment compare β₁The thickness of the parabolic segment of the=0.69, the 2nd main spring compares β₂=0.59；Before auxiliary spring is contacted The 1st main spring and the 2nd main spring half clamping stiffness K_M1And K_M2It is respectively calculated, i.e.

$K_{M1}=\frac{h_{2M}^3}{G_{x-E1}}=13.46N/mm;$

$K_{M2}=\frac{h_{2M}^3}{G_{x-E2}}=12.71N/mm;$

In formula,

$\begin{array}{c}{G}_{x-E1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_1^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\end{array}$

$\frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2\mathrm{\Δ}l{\mathrm{ln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=98.87{\mathrm{mm}}^4/N,$

$\begin{array}{c}{G}_{x-E1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_1^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2\mathrm{\Δ}l{\mathrm{ln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=104.76{\mathrm{mm}}^4/N;\end{array}$

II step: the half clamping stiffness K of each main spring after the contact of major-minor spring_MAiCalculate:

Half length L according to few sheet root main spring of reinforced variable cross-section_M=575mm, main reed number m=2, each main spring The thickness h of root flat segments_2M=11mm, width b=60mm, elastic modulus E=200GPa, half l of installing space₃= 55mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2Mp=490mm, oblique line section Root is to distance l of main spring end points_2M=520mm, the thickness of oblique line section compares γ_M=0.93；The thickness of the parabolic segment of the 1st main spring Degree compares β₁The thickness of the parabolic segment of the=0.69, the 2nd main spring compares β₂=0.59；Auxiliary spring sheet number n=1, the half of this sheet auxiliary spring is long Degree L_A=375mm, auxiliary spring contact is to horizontal range l of main spring end points₀=200mm, root thickness h of this sheet auxiliary spring_2A=14mm, The root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2Ap=290mm, the distance of the root of auxiliary spring oblique line section to auxiliary spring end points l_2A=320mm, the thickness of auxiliary spring oblique line section compares γ_A=0.93；The thickness of auxiliary spring parabolic segment compares β_A1=0.62, major-minor spring is connect The 1st main spring after Chuing and the half clamping stiffness K of the 2nd main spring_MA1And K_MA2It is respectively calculated, i.e.

$K_{MA1}=\frac{h_{2M}^3}{G_{x-E1}}=13.46N/mm;$

$K_{MA2}=\frac{h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)}{G_{x-E2}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)-G_{x-E_{p}2}{G}_{x-CD}{h}_{2A}^3}=23.94N/mm;$

In formula,

$\begin{array}{c}{G}_{x-E1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_1^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2\mathrm{\Δ}l{\mathrm{ln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=98.87{\mathrm{mm}}^4/N,\end{array}$

$\begin{array}{c}{G}_{x-E1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_1^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2\mathrm{\Δ}l{\mathrm{ln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=104.76{\mathrm{mm}}^4/N;\end{array}$

$G_{x-EAT}=\frac{1}{\underset{j=1}{\overset{n}{\Σ}}\frac{1}{G_{x-EAj}}}=24.12{\mathrm{mm}}^4/N;$

$\begin{array}{c}{G}_{x-EA1}=\frac{4\[{(L_A-l_3/2)}^3-l_{2A}^3\]}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{A1}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(3l_{2A}^2{\mathrm{\γ}}_A^2+l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2\mathrm{\Δ}l{\mathrm{ln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}=24.12{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-CD}=\frac{4{(L_M-l_3/2)}^3-6l_0{(L_M-l_3/2)}^3-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}+\frac{8l_{2Mp}^3+16l_{2Mp}^{3/2}{l}_0^{3/2}-24l_{2Mp}^2{l}_0}{{\mathrm{Eb\γ}}_M^3}-\frac{6l_0\mathrm{\Δ}l(l_{2Mp}+l_{2M}{\mathrm{\γ}}_M)}{{\mathrm{Eb\γ}}_M^2}+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3-4l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M)=39.29{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-E_{p}2}=\frac{4{(L_M-l_3/2)}^3-6l_0{(L_M-l_3/2)}^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{12}{Eb}\[\frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{2l_{2M}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^3}-\\ \frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3{\mathrm{\γ}}_M^2}-\frac{2l_{2Mp}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}-\frac{{\mathrm{\Δl}}^3}{{({\mathrm{\γ}}_M-1)}^3}{\mathrm{ln\γ}}_M\]-\frac{24l_0{l}_{2Mp}^2-8{l_{2Mp}}^3-16l_0^{3/2}{l}_{2Mp}^{3/2}}{{\mathrm{Eb\γ}}_M^3}-\\ \frac{6l_0\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{\mathrm{Eb\γ}}_M^2}=39.29{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_p}=\frac{4(L_M-l_3/2-l_{2M})\[{(L_M-l_3/2)}^2-3l_0(L_M-l_3/2)+(L_M-l_3/2)l_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\\ \frac{12}{Eb}\[\frac{l_0^2\mathrm{\Δ}l{({\mathrm{\γ}}_M-1)}^2-3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}+\frac{2l_{2Mp}\mathrm{\Δ}l(\mathrm{\Δ}l-l_{2Mp}-l_0{\mathrm{\γ}}_M+l_{2M}{\mathrm{\γ}}_M)}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}+\\ \frac{\mathrm{\Δ}l\[2l_0^3\mathrm{\Δ}l({\mathrm{\γ}}_M-1)(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})+3{(L_M-l_3/2)}^2-2(L_M-l_3/2)l_0-2(L_M-l_3/2)l_{2M}{\mathrm{\γ}}_M\]}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l\left(2\right(L_M-l_3/2)l_0{\mathrm{\γ}}_M-4(L_M-l_3/2)l_{2M}-6(L_M-l_3/2)l_3+2l_0^2{\mathrm{\γ}}_M-6(L_M-l_3/2)\mathrm{\Δ}l-l_0^2{\mathrm{\γ}}_M^2-l_0^2)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2l_0{l}_{2M}{\mathrm{\γ}}_M^2+4l_0{l}_{2M}-6l_0{l}_{2M}{\mathrm{\γ}}_M-2l_0{l}_3{\mathrm{\γ}}_M+2l_0{l}_3-2l_0{\mathrm{\Δl\γ}}_M+2l_0\mathrm{\Δ}l-l_{2M}^2{\mathrm{\γ}}_M^2+4l_{2M}^2{\mathrm{\γ}}_M)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2l_{2M}{l}_3{\mathrm{\γ}}_M+4l_{2M}{l}_3+2l_{2M}{\mathrm{\Δl\γ}}_M+4l_{2M}\mathrm{\Δ}l+3l_3^2+6l_3\mathrm{\Δ}l+3{\mathrm{\Δl}}^2)}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{{\mathrm{\Δl}}^3{\mathrm{ln\γ}}_M}{{({\mathrm{\γ}}_M-1)}^3}\]+\\ \frac{12}{{\mathrm{Eb\γ}}_M^3}(\frac{2l_{2p}^2-12l_0{l}_{2Mp}-6l_0^2}{3l_{2Mp}^2}+\frac{16l_0^{3/2}}{3l_{2Mp}^{3/2}})=19.70{\mathrm{mm}}^4/N;\end{array}$

III step: the half clamping stiffness K of each auxiliary spring_AjCalculate:

Half length L according to this sheet root reinforced variable cross-section auxiliary spring_A=375mm, auxiliary spring sheet number n=1, this sheet auxiliary spring The thickness h of root flat segments_2A=14mm, the width b=60mm of this sheet auxiliary spring, elastic modulus E=200GPa, installing space Half l₃=55mm, the length Δ l=30mm of auxiliary spring oblique line section, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2Ap =290mm, distance l of the root of auxiliary spring oblique line section to auxiliary spring end points_2A=320mm, the thickness of auxiliary spring oblique line section compares γ_A= 0.93, the thickness of auxiliary spring parabolic segment compares β_A1=0.62, the half of this sheet auxiliary spring is clamped stiffness K_A1Calculate, i.e.

$K_{A1}=\frac{h_{2A}^3}{G_{x-EA1}}=113.77N/mm;$

In formula,

$\begin{array}{c}{G}_{x-EA1}=\frac{4\[{(L_A-l_3/2)}^3-l_{2A}^3\]}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{A1}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(3l_{2A}^2{\mathrm{\γ}}_A^2+l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2\mathrm{\Δ}l{\mathrm{ln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}=24.12{\mathrm{mm}}^4/N;\end{array}$

(2) non-ends contact formula lacks each main spring and the maximum end points power of auxiliary spring of sheet root reinforced variable cross-section major-minor spring Calculate:

I step: the maximum end points power of each main spring calculates:

The most single-ended some maximum load P of half according to maximum load suffered by few sheet root reinforced variable cross-section major-minor spring_max= 3040N, auxiliary spring works used load P_K=2380.80N, main reed number m=2, calculated K in I step_M1= 13.46N/mm and K_M2=12.71N/mm, and II step calculate obtained K_MA1=13.46N/mm and K_MA2=23.94N/ Mm, to the 1st main spring and maximum end points power P of the 2nd main spring_1maxAnd P_2maxCalculate respectively, i.e.

$P_{1max}=\frac{K_{M1}{P}_K}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{Mi}}+\frac{K_{MA1}(2P_{\max }-P_K)}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}}=1278.10N;$

$P_{2max}=\frac{K_{M2}{P}_K}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{Mi}}+\frac{K_{MA2}(2P_{\max }-P_K)}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}}=1761.90N;$

Ii step: the maximum end points power of each auxiliary spring calculates:

The most single-ended some maximum load P of half according to maximum load suffered by few sheet root reinforced variable cross-section major-minor spring_max= 3040N, auxiliary spring works load p_K=2380.80N, main reed number m=2, the thickness h of the root flat segments of each main spring_2M= 11mm, auxiliary spring sheet number n=1, the thickness h of this sheet auxiliary spring root flat segments_2ACalculated K in=14mm, II step_MA1= 13.46N/mm、K_MA2=23.94N/mm, G_x-CD=39.29mm⁴/N、G_x-CDp=19.70mm⁴/ N and G_x-EAT=24.12mm⁴/ N, And calculated K in III step_A1=113.77N/mm, maximum end points power P to this sheet auxiliary spring_A1maxCalculate, i.e.

$P_{A1\max }=\frac{K_{A1}{K}_{MA2}{G}_{x-BC}{h}_{2A}^3(2P_{\max }-P_K)}{2\underset{j=1}{\overset{n}{\Σ}}{K}_{Aj}\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{BC}}_p}{h}_{2A}^3)}=1481.60N;$

(3) non-ends contact formula lacks each main spring and the maximum stress meter of auxiliary spring of sheet root reinforced variable cross-section major-minor spring Calculate:

Step A: the maximum stress of the 1st main spring calculates:

Half length L according to few sheet root main spring of reinforced variable cross-section_M=575mm, the root flat segments of each main spring Thickness h_2M=11mm, width b=60mm, half l of installing space₃Calculated P in=55mm, i step_1max= 1278.10N, calculates, i.e. the maximum stress of the 1st root main spring of reinforced variable cross-section

${\mathrm{\σ}}_{1max}=\frac{6P_{1max}(L_M-l_3/2)}{{\mathrm{bh}}_{2M}^2}=578.31MPa;$

Step B: the maximum stress of the 2nd main spring calculates:

The thickness h of the root flat segments according to each main spring_2M=11mm, width b=60mm, auxiliary spring sheet number n=1, this sheet Auxiliary spring contact and horizontal range l of main spring end points₀=200mm, the root of parabolic segment is to distance l of main spring end points_2Mp= 490mm, the thickness of the oblique line section of each main spring compares γ_M=0.93；The thickness of the parabolic segment of the 2nd main spring compares β₂=0.59, i Calculated P in step_2maxCalculated P in=1761.90N, ii step_A1max=1481.60N, adds the 2nd root The maximum stress of the main spring of strong type variable cross-section calculates, i.e.

${\mathrm{\σ}}_{2max}=\frac{6\[P_{2\max }{\mathrm{\β}}_2^2{l}_{2Mp}-\underset{j=1}{\overset{n}{\Σ}}{P}_{Aj\max }({\mathrm{\β}}_2^2{l}_{2Mp}-l_0)\]}{b{\left({\mathrm{\β}}_2{\mathrm{\γ}}_M{h}_{2M}\right)}^2}=955.13MPa;$

Step C: the maximum stress of each auxiliary spring calculates:

Half length L according to this sheet root reinforced variable cross-section auxiliary spring_A=375mm, half l of installing space₃= 55mm, auxiliary spring sheet number n=1, the thickness h of this sheet auxiliary spring root flat segments_2A=14mm, calculates in width b=60mm, ii step The P arrived_A1max=1481.60N, calculates, i.e. the maximum stress of this sheet root reinforced variable cross-section auxiliary spring

${\mathrm{\σ}}_{A1max}=\frac{6P_{A1max}(L_A-l_3/2)}{{\mathrm{bh}}_{2A}^2}=262.68MPa;$

(4) non-ends contact formula lacks each main spring and the stress intensity school of auxiliary spring of sheet root reinforced variable cross-section major-minor spring Core:

1. step: the stress intensity of the 1st main spring is checked:

The maximum of calculated 1st main spring in allowable stress [σ] according to leaf spring=700MPa, and step A Stress σ_1max=578.31MPa, it is known that σ_1max≤ [σ], i.e. the 1st main spring disclosure satisfy that stress intensity requirement；

2. step: the stress intensity of the 2nd main spring is checked:

The maximum of calculated 2nd main spring in allowable stress [σ] according to leaf spring=700MPa, and step B Stress σ_2max=955.13MPa, it is known that σ_2max> [σ], i.e. the 2nd main spring can not meet stress intensity requirement；

3. step: the stress intensity of 1 auxiliary spring is checked:

In allowable stress [σ] according to leaf spring=700MPa, and step C, the maximum of this sheet auxiliary spring calculated should Power σ_A1max=262.68MPa, it is known that σ_A1max≤ [σ], i.e. this sheet auxiliary spring disclosure satisfy that stress intensity requirement.

Utilize ANSYS finite element emulation software, according to the few sheet root reinforced variable cross-section steel plates bullet of this non-ends contact formula The major-minor spring structure parameter of spring and material characteristic parameter, set up the ANSYS phantom of half symmetrical structure major-minor spring, divides net Lattice, arrange auxiliary spring end points and contact with main spring, and at the root applying fixed constraint of phantom, apply to concentrate load at main spring end points Lotus F=P_max-P_K/ 2=1849.60N, sheet root reinforced variable-section steel sheet spring few to this non-ends contact formula is at clamping shape The stress of the major-minor spring under state carries out ANSYS emulation, and the maximum stress emulation cloud atlas of the 1st obtained main spring, such as Fig. 3 institute Show；The maximum stress emulation cloud atlas of the 2nd main spring, as shown in Figure 4；The maximum stress emulation cloud atlas of the 1st auxiliary spring, such as Fig. 5 institute Showing, wherein, the 1st main spring is at the maximum stress σ clamping root_1max=302.26MPa, the 2nd main spring are in oblique line section and end Maximum stress σ at flat segments contact position_2max=684.74MPa, the 1st auxiliary spring are at the maximum stress σ clamping root_A1max= 261.72MPa。

Understand, in the case of same load, this leaf spring the 1st and the 2nd main spring and 1 auxiliary spring maximum stress ANSYS simulating, verifying value σ_1max=302.26MPa, σ_2max=684.74MPa, σ_A1max=261.72MPa, respectively with maximum stress Analytical Calculation value σ_1max=301.21MPa, σ_2max=682.92MPa, σ_A1max=262.68MPa, matches, and relative deviation divides It is not 0.35%, 0.15%, 0.27%；Result shows the few reinforced major-minor of sheet root of non-ends contact formula that this invention is provided The check method of each intensity of spring is correct, and the stress intensity calculation and check value of each main spring and auxiliary spring is reliable.

Claims (1)

1. The check method of the few sheet root reinforced major-minor each intensity of spring of the most non-ends contact formula, wherein, few sheet root is reinforced The half symmetrical structure of variable-section steel sheet spring is to be made up of root flat segments, oblique line section, parabolic segment, end flat segments 4 sections, Spring tang is played booster action by oblique line section；The end flat segments of each main spring is non-that wait structure, i.e. the end of the 1st main spring is straight The thickness of section and length, more than the thickness of end flat segments and the length of other each main spring；Auxiliary spring length is less than main spring length, When load works load more than auxiliary spring, auxiliary spring end points contacts with certain point in main spring parabolic segment, i.e. major-minor spring is non-end Portion's contact；After major-minor spring contacts, the end points power of each major-minor spring differs, and the 1 main spring contacted with auxiliary spring removes By outside end points power, also acted on by auxiliary spring contact support power；In each chip architecture parameter of major-minor spring, elastic modelling quantity, allowable Stress, maximum load and auxiliary spring work load given in the case of, sheet root reinforced major-minor spring few to non-ends contact formula The stress intensity of each main spring and auxiliary spring is checked, and concrete check step is as follows:

   (1) each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the half of auxiliary spring clamp rigidimeter Calculate:

   I step: the half clamping stiffness K of each main spring before the contact of major-minor spring_MiCalculate:

   Half length L according to few sheet root main spring of reinforced variable cross-section_M, main reed number m, the thickness of each main spring root flat segments Degree h_2M, width b, elastic modulus E, half l of installing space₃, oblique line segment length Δ l, the root of parabolic segment is to main spring end points Distance l_2Mp=L_M-l₃-Δ l, the root of oblique line section is to distance l of main spring end points_2M=L_M-l₃, the oblique line section of each main spring Thickness compares γ_M, the thickness of the parabolic segment of i-th main spring compares β_i, wherein, i=1,2 ..., m, to major-minor spring contact before each The half clamping stiffness K of the main spring of sheet_MiCalculate, i.e.

   $K_{Mi}=\frac{h_{2M}^3}{G_{x-Ei}},i=1,2,...,m;$

   In formula,

   $\begin{array}{c}{G}_{x-Ei}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_i^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{\Δlln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3};\end{array}$

   II step: the half clamping stiffness K of each main spring after the contact of major-minor spring_MAiCalculate:

   Half length L according to few sheet root main spring of reinforced variable cross-section_M, main reed number m, the root flat segments of each main spring Thickness h_2M, width b, elastic modulus E, half l of installing space₃, the length Δ l of oblique line section, the root of parabolic segment is to main spring Distance l of end points_2Mp=L_M-l₃-Δ l, the root of oblique line section is to distance l of main spring end points_2M=L_M-l₃, the thickness ratio of oblique line section γ_M, the thickness of the parabolic segment of i-th main spring compares β_i, wherein, i=1,2 ..., m；Half length L of auxiliary spring_A, auxiliary spring sheet number n, The thickness h of the root flat segments of each auxiliary spring_2A, auxiliary spring contact and horizontal range l of main spring end points₀, the root of parabolic segment arrives Distance l of auxiliary spring end points_2Ap=L_A-l₃-Δ l, the root of oblique line section is to distance l of auxiliary spring end points_2A=L_A-l₃, each auxiliary spring The thickness of oblique line section compares γ_A, the thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, major-minor spring is contacted The half clamping stiffness K of each main spring afterwards_MAiCalculate, i.e.

   $K_{MAi}=\{\begin{array}{cc}\frac{h_{2M}^3}{G_{x-Ei}},& i=1,2,\mathrm{...},m-1\\ \frac{h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)}{G_{x-Em}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)-G_{x-E_{p}m}{G}_{x-CD}{h}_{2A}^3},& i=m\end{array};$

   In formula,

   $G_{x-Ei}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_i^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+$

   $\begin{array}{c}\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(3l_{2M}^2{\mathrm{\γ}}_M^2+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2MP}{l}_{2M}{\mathrm{\γ}}_M+2l_{2MP}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2MP}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2p}{\mathrm{\γ}}_M^2{\mathrm{\Δlln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3};\end{array}$

   $G_{x-EAT}=\frac{1}{\underset{j=1}{\overset{n}{\Σ}}\frac{1}{G_{x-EAj}}},j=1,2,...,n;$

   $\begin{array}{c}{G}_{x-EAj}=\frac{4{(L_A-l_3/2)}^3-l_{2A}^3\]}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{Aj}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(3l_{2A}^2{\mathrm{\γ}}_A^2+l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2{\mathrm{\Δlln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3};\end{array}$

   $\begin{array}{c}{G}_{x-CD}=\frac{4{(L_M-l_3/2)}^3-6l_0{(L_M-l_3/2)}^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}+\frac{8l_{2Mp}^3+16l_{2Mp}^{3/2}{l}_0^{3/2}-24l_{2Mp}^2{l}_0}{{\mathrm{Eb\γ}}_M^3}-\frac{6l_0\mathrm{\Δ}l(l_{2Mp}+l_{2M}{\mathrm{\γ}}_M)}{{\mathrm{Eb\γ}}_M^2}+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3+l_{2M}^2{\mathrm{\γ}}_M^2-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3-4l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M);\end{array}$

   $\begin{array}{c}{G}_{x-E_{p}m}=\frac{4{(L_M-l_3/2)}^3-6l_0{(L_M-l_3/2)}^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{12}{Eb}\[\frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2M})}^2}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{2l_{2M}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2MP})}{{({\mathrm{\γ}}_M-1)}^2}-\\ \frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3{\mathrm{\γ}}_M^2}-\frac{2l_{2Mp}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}-\frac{{\mathrm{\Δl}}^3}{{({\mathrm{\γ}}_M-1)}^3}{\mathrm{ln\γ}}_M\]-\frac{24l_0{l}_{2Mp}^2-2{l_{2Mp}}^3-16l_0^{3/2}/l_{2Mp}^{3/2}}{{\mathrm{Eb\γ}}_M^3}-\\ \frac{6l_0\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{\mathrm{Eb\γ}}_M^2};\end{array}$

   $\begin{array}{c}{G}_{x-{\mathrm{CD}}_p}=\frac{4(L_M-l_3/2-l_{2M})\[{(L_M-l_3/2)}^2-3l_0(L_M-l_3/2)+(L_M-l_3/2)l_{2M}+3l_0^3-3l_0{l}_{2M}+l_{2M}^3)}{Eb}-\\ \frac{12}{Eb}\[\frac{l_0^2\mathrm{\Δ}l{({\mathrm{\γ}}_M-1)}^2-3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}+\frac{2l_{2Mp}\mathrm{\Δ}l(\mathrm{\Δ}l-l_{2Mp}-l_0{\mathrm{\γ}}_M+l_{2M}{\mathrm{\γ}}_M)}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}+\\ \frac{\mathrm{\Δ}l\[2l_0^2\mathrm{\Δ}l({\mathrm{\γ}}_M-1)(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})+3{(L_M-l_3/2)}^2-2(L_M-l_3/2)l_0-2(L_M-l_3/2)l_{2M}{\mathrm{\γ}}_M\]}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l\left(2\right(L_M-l_3/2)l_0{\mathrm{\γ}}_M-4(L_M{l}_3/2)l_{2M}-6(L_M-l_3/2)l_3+2l_0^2{\mathrm{\γ}}_M-6(L_M-l_3/2)\mathrm{\Δ}l-l_0^2{\mathrm{\γ}}_M^2-l_0^2)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2l_0{l}_{2M}{\mathrm{\γ}}_M^2+4l_0{l}_{2M}-6l_0{l}_{2M}{\mathrm{\γ}}_M-2l_0{l}_3{\mathrm{\γ}}_M+2l_0{l}_3-2l_0{\mathrm{\Δl\γ}}_M+2l_0\mathrm{\Δ}l-l_{2M}^2{\mathrm{\γ}}_M^2+4l_{2M}^2{\mathrm{\γ}}_M)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2l_{2M}{l}_3{\mathrm{\γ}}_M+4l_{2M}{l}_3+2l_{2M}{\mathrm{\Δl\γ}}_M+4l_{2M}\mathrm{\Δ}l+3l_3^2+6l_3\mathrm{\Δ}l+3{\mathrm{\Δl}}^2)}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{{\mathrm{\Δl}}^3{\mathrm{ln\γ}}_M}{{({\mathrm{\γ}}_M-1)}^3}\]+\\ \frac{12}{{\mathrm{Eb\γ}}_M^3}(\frac{2l_{2p}^2-12l_0{l}_{2Mp}-6l_0^2}{3l_{2Mp}^2}+\frac{16l_0^{3/2}}{3l_{2Mp}^{3/2}});\end{array}$

   III step: the half clamping stiffness K of each auxiliary spring_AjCalculate:

   Half length L according to few sheet root reinforced variable cross-section auxiliary spring_A, auxiliary spring sheet number n, the root flat segments of each auxiliary spring Thickness h_2A, width b, half l of installing space₃, the length Δ l of oblique line section, the root of parabolic segment is to the distance of auxiliary spring end points l_2Ap=L_A-l₃-Δ l, the root of oblique line section is to distance l of auxiliary spring end points_2A=L_A-l₃, the thickness of auxiliary spring oblique line section compares γ_A, jth The thickness of the parabolic segment of sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, the half of each auxiliary spring is clamped stiffness K_AjCount Calculate, i.e.

   $K_{Aj}=\frac{h_{2A}^3}{G_{x-EAj}},j=1,2,...,n;$

   In formula,

   $\begin{array}{c}{G}_{x-EAj}=\frac{4\[{(L_A-l_3/2)}^3-l_{2A}^3\]}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{Aj}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(3l_{2A}^2{\mathrm{\γ}}_A^2+l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2{\mathrm{\Δlln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3};\end{array}$

   (2) non-ends contact formula lacks each main spring and the maximum end points power meter of auxiliary spring of sheet root reinforced variable cross-section major-minor spring Calculate:

   I step: the maximum end points power of each main spring calculates:

   The most single-ended some maximum load of half according to maximum load suffered by few sheet root reinforced variable-section steel sheet spring major-minor spring P_max, auxiliary spring works load p_K, calculated K in I step_Mi, and II step calculates obtained K_MAi, main reed number m, Maximum end points power P to each main spring_imaxCalculate, i.e.

   $R_{imax}=\frac{K_{Mi}{P}_K}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{Mi}}+\frac{K_{MAi}(2P_{max}-P_K)}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}},i=1,2,...,m;$

   Ii step: the maximum end points power of each auxiliary spring calculates:

   The most single-ended some maximum load of half according to maximum load suffered by few sheet root reinforced variable-section steel sheet spring major-minor spring P_max, auxiliary spring works used load P_K, main reed number m, the thickness h of the root flat segments of each main spring_2M, auxiliary spring sheet number n, respectively The thickness h of the root flat segments of sheet auxiliary spring_2A, calculated K in II step_MAi、G_x-CD、G_x-CDpAnd G_x-EAT, and in III step Calculated K_Aj, maximum end points power P to each auxiliary spring_AjmaxCalculate, i.e.

   $P_{Ajmax}=\frac{K_{Aj}{K}_{MAm}{G}_{x-CD}{h}_{2A}^3(2P_{max}-P_K)}{2\underset{j=1}{\overset{n}{\Σ}}{K}_{Aj}\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)},j=1,2,...,n;$

   (3) each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the maximum stress of auxiliary spring calculate:

   Step A: the maximum stress of the front main spring of m-1 sheet calculates:

   Half length L according to few sheet root main spring of reinforced variable-section steel sheet spring_M, main reed number m, the root of each main spring The thickness h of flat segments_2M, width b, half l of installing space₃, calculated P in i step_imax, to the front main spring of m-1 sheet Big stress is respectively calculated, i.e.

   ${\mathrm{\σ}}_{imax}=\frac{6P_{imax}(L_M-l_3/2)}{{\mathrm{bh}}_{2M}^2},i=1,2,...m-1;$

   Step B: the maximum stress of the main spring of m sheet calculates:

   According to main reed number m, the thickness h of the root flat segments of each main spring_2M, width b, half l of installing space₃, oblique line section Length Δ l, the root of parabolic segment is to distance l of main spring end points_2Mp=L_M-l₃-Δ l, the thickness of the oblique line section of each main spring Compare γ_M, the thickness of the parabolic segment of the main spring of m sheet compares β_m；Horizontal range l of auxiliary spring sheet number n, auxiliary spring contact and main spring end points₀, i Calculated P in step_mmax, calculated P in ii step_Ajmax, the maximum stress of spring main to m sheet calculates, i.e.

   ${\mathrm{\σ}}_{mmax}=\frac{6\[P_{mmax}{\mathrm{\β}}_m^2{l}_{2MP}-\underset{j=1}{\overset{n}{\Σ}}{P}_{Ajmax}({\mathrm{\β}}_m^2{l}_{2MP}-l_0)\]}{b{\left({\mathrm{\β}}_m{\mathrm{\γ}}_M{h}_{2M}\right)}^2};$

   Step C: the maximum stress of each auxiliary spring calculates:

   Half length L according to few sheet root reinforced variable cross-section auxiliary spring_A, auxiliary spring sheet number n, the root flat segments of each auxiliary spring Thickness h_2A, width b, half l of installing space₃, calculated P in ii step_Ajmax, the maximum stress of each auxiliary spring is carried out Calculate, i.e.

   ${\mathrm{\σ}}_{Ajmax}=\frac{6P_{Ajmax}(L_A-l_3/2)}{{\mathrm{bh}}_{2A}^2},j=1,2,...,n;$

   (4) each main spring of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the stress intensity of auxiliary spring are checked:

   1. step: the stress intensity of the front main spring of m-1 sheet is checked:

   Allowable stress [σ] according to leaf spring, and the maximum stress of each of the calculated front main spring of m-1 sheet in step A, The stress intensity of each of the front main spring of m-1 sheet of sheet root reinforced variable-section steel sheet spring few to non-ends contact formula carries out school Core, it may be assumed that if σ_imax> [σ], then i-th main spring, it is unsatisfactory for stress intensity requirement；If σ_imax≤ [σ], then i-th main spring are full Foot stress intensity requirement, i=1,2 ... m-1；

   2. step: the stress intensity of the main spring of m sheet is checked:

   Allowable stress [σ] according to leaf spring, and the maximum stress of the calculated main spring of m sheet in step B, to non-end The stress intensity of the main spring of m sheet of the few reinforced variable-section steel sheet spring of sheet root of contact is checked, it may be assumed that if σ_mmax> [σ], the then main spring of m sheet, be unsatisfactory for stress intensity requirement；If σ_mmax≤ [σ], the then main spring of m sheet, meets stress intensity and wants Ask；

   3. step: the stress intensity of each auxiliary spring is checked:

   Allowable stress [σ] according to leaf spring, and the maximum stress of calculated each auxiliary spring in step C, to non-end The stress intensity of each auxiliary spring of the few reinforced variable-section steel sheet spring of sheet root of contact is checked, it may be assumed that if σ_Ajmax> [σ], then jth sheet auxiliary spring, be unsatisfactory for stress intensity requirement；If σ_Ajmax≤ [σ], then jth sheet auxiliary spring, meet stress intensity and want Ask, j=1,2 ..., n.