CN105912756A - Method for checking strength of each of end contact type few-leaf end enhanced master and slave springs - Google Patents

Abstract

The invention relates to a method for checking the strength of each of end contact type few-leaf end enhanced master and slave springs, and belongs to the technical field of suspension steel plate springs. The method can check and calculate the stress strength of each of master springs and slave springs of the end contact type few-leaf end enhanced section-variable master and slave springs according to the structure size, elasticity modulus, allowable stress, slave spring work load and the maximum load of each of the master springs and the slave springs of the end contact type few-leaf end enhanced section-variable master and slave springs. The embodiment and ANSYS simulation verification show that the method for checking the strength of each of end contact type few-leaf end enhanced master and slave springs is correct. The method can obtain accurate and reliable a check value of stress strength of each of the master springs and the slave springs, can further improve the design level, the product quality and the service life of vehicle suspension few-leaf section-variable master and slave springs, improves the transport efficiency and the travel safety of a vehicle, decreases design and testing expenses, and accelerates the product development speed.

Description

The check method of the few sheet reinforcement end each intensity of major-minor spring of ends contact formula

Technical field

The present invention relates to few each of the sheet reinforcement end major-minor spring of vehicle suspension leaf spring, particularly ends contact formula strong The check method of degree.

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 obtain Popularization and application widely.Generally few sheet variable-section steel sheet spring is designed as major-minor spring, and by major-minor spring gap, it is ensured that Load works more than auxiliary spring after load, and major-minor spring contacts and works together, meets vehicle suspension in the case of different loads The design requirement of spring variation rigidity and stress intensity.Owing to the stress of the 1st main spring is complicated, it is subjected to vertical load, 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, More than the thickness of end flat segments and the length of his each main spring, i.e. in actual design with produce, generally use the end non- Few main spring of sheet variable cross-section of structure.Few sheet variable-section steel sheet spring mainly has two types, and one is parabolic type, and another is Bias type, wherein, Parabolic stress is iso-stress, and suffered by it, stress ratio bias type is more reasonable；Meanwhile, in order to strengthen The end intensity of parabolic type variable-section steel sheet spring, can increase by an oblique line section, i.e. between end flat segments and parabolic segment Use reinforcement end variable cross-section major-minor spring.It addition, the length of few sheet parabolic type variable cross-section auxiliary spring is less than the length of main spring, main The way of contact that auxiliary spring divides has ends contact formula and non-ends contact formula two kinds, at identical auxiliary spring root flat segments depth information Under, the complex stiffness of few sheet variable cross-section major-minor spring of ends contact formula is more than the complex stiffness of non-ends contact formula.For set The few sheet reinforcement end variable cross-section major-minor spring of the ends contact formula of meter, not only meets the design requirement of complex stiffness, also should expire The design requirement of foot stress intensity, but, due to the master of the few sheet reinforcement end variable cross-section major-minor leaf spring of ends contact formula After the structures, and the contact of major-minor spring such as the end flat segments of spring length and unequal, each main spring of auxiliary spring length is non-, main spring and auxiliary spring Deformation and internal force have coupling, therefore, each main spring of the few fragment portion of ends contact formula reinforced variable cross-section major-minor spring and pair The calculating of the maximum stress of spring is extremely complex, fails to provide simplicity, the few sheet end of ends contact formula accurate, reliable the most always The check method of reinforced variable cross-section each intensity of major-minor spring.Therefore, it is necessary to it is few to set up a kind of ends contact formula accurate, reliable Each main spring of sheet reinforcement end variable cross-section major-minor spring and the strength check methods of auxiliary spring, meet Vehicle Industry fast-developing and Requirement to suspension variable-section steel sheet spring careful design, improves the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula Design level, product quality 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 reinforcement end each intensity of major-minor spring of ends contact formula, checks flow chart, as shown in Figure 1 reliably. The few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is symmetrical structure, and the half symmetrical structure of major-minor spring can be seen as outstanding Arm beam, i.e. symmetrical center line be the fixing end of root, the end stress point of main spring and the contact of auxiliary spring respectively as main spring end points and Auxiliary spring end points, the half symmetrical structure schematic diagram of major-minor spring, as in figure 2 it is shown, wherein, including: main spring 1, root shim 2, auxiliary spring 3, end pad 4, each root flat segments of main spring 1 and and the root flat segments of auxiliary spring 3 between be provided with root shim 2, main spring 1 The end flat segments of each is provided with end pad 4, and the material of end pad 4 is carbon fibre composite, is used for reducing spring work The frictional noise produced when making；The half symmetrical structure of main spring 1 and auxiliary spring 3 is by root flat segments, parabolic segment, oblique line section, end Portion's flat segments four sections composition；Booster action is played in variable cross-section end by oblique line section.The a length of L of half of each main spring_M, root is straight The thickness of section is h_2M, a length of l of half of installing space₃, the distance of the root of parabolic segment to main spring end points is l_2M=L_M- l₃, the end thickness of each main spring parabolic segment is h_1Mpi, the thickness of the most each main spring parabolic segment compares β_i=h_1Mpi/h_2M, i= 1,2 ..., m, m are main reed number, and the end of each main spring parabolic segment is to distance l of main spring end points_1Mpi=l_2Mβ_i ², oblique line section A length of Δ l；The non-thickness waiting structure, i.e. the end flat segments of the 1st main spring of end flat segments of each main spring and length, greatly The thickness of end flat segments and length, the thickness of each main spring end flat segments and length in other each main spring are respectively h_1MiAnd l_1Mi；The thickness of the oblique line section of each main spring compares γ_Mi=h_1Mi/h_1Mpi.The a length of L of half of each auxiliary spring_A, it is by root Portion's flat segments, parabolic segment, oblique line section, end flat segments four sections composition；Auxiliary spring contact is l with the horizontal range of main spring end points₀, The thickness of the root flat segments of each auxiliary spring is h_2A, a length of l of half of installing space₃, the root of auxiliary spring parabolic segment is to secondary The distance of spring end points is l_2A=L_A-l₃, the end thickness of each auxiliary spring parabolic segment is h_1Apj, the most each auxiliary spring parabolic segment Thickness compares β_Aj=h_1Apj/h_2A, j=1,2 ..., n, n are auxiliary spring sheet number, and the end of each auxiliary spring parabolic segment is to auxiliary spring end points Distance l_1Apj=l_2Aβ_Aj ²；The a length of Δ l of each auxiliary spring oblique line section, the thickness of the end flat segments of auxiliary spring and length are respectively h_1AjAnd l_1Aj=l_1Apj-Δl；The thickness of each auxiliary spring oblique line section compares γ_Aj=h_1Aj/h_1Ap.Auxiliary spring contact is straight with main spring end Section and between be provided with major-minor spring gap delta, when load works load more than auxiliary spring, certain point in auxiliary spring and main spring end flat segments Contact；After major-minor spring ends contact, each end stress of major-minor spring differs, and the main spring contacted with auxiliary spring except Outside end points power, at contact point, also bear the support force of auxiliary spring.At the structural parameters of each major-minor spring, elastic modelling quantity, permitted Work load, maximum load to stable condition, each of sheet reinforcement end major-minor spring few to end contact with stress, auxiliary spring The stress intensity of main spring and auxiliary spring is checked.

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

(1) each main spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula and the half of auxiliary spring clamp rigidity Calculate:

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

Half length L according to few sheet main spring of reinforcement end 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_2M=L_M-l₃；The thickness of the parabolic segment of i-th main spring compares β_i, the thickness of the oblique line section of i-th main spring Compare γ_Mi, the root of oblique line section is to distance l of main spring end points_1Mpi=l_2Mβ_i ², the end of oblique line section is to distance l of main spring end points_1Mi =l_1Mpi-Δ l, wherein, i=1,2 ..., m；The half clamping stiffness K of each main spring before major-minor spring is contacted_MiEnter respectively Row calculates, 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{8l_{2M}^{3/2}(l_{1Mpi}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1Mi}^3}{{\mathrm{Eb\γ}}_{Mi}^3{\mathrm{\β}}_i^3}+\frac{6\mathrm{\Δ}l(4l_{1Mi}^2{\mathrm{\γ}}_{Mi}-l_{1Mi}^2-3l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2+3l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^2)}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{({\mathrm{\γ}}_{Mi}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^3-l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^4-2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}+2l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}+2l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}+2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^3-4l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi})}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{({\mathrm{\γ}}_{Mi}-1)}^3};\end{array}$

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

Half length L according to few sheet main spring of reinforcement end 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_2M=L_M-l₃；The thickness of the parabolic segment of i-th main spring compares β_i, the thickness of the oblique line section of i-th main spring Compare γ_Mi, the root of the oblique line section of i-th main spring is to distance l of main spring end points_1Mpi=l_2Mβ_i ², the oblique line section of i-th main spring End is to distance l of main spring end points_1Mi=l_1Mpi-Δ l, wherein, i=1,2 ..., m；Half length L of auxiliary spring_A, auxiliary spring contact with Horizontal range l of main spring end points₀=L_M-L_A, auxiliary spring sheet number n, the thickness h of the root flat segments of each auxiliary spring_2A, the length of oblique line section Degree Δ l, the root of parabolic segment is to distance l of auxiliary spring end points_2A=L_A-l₃；The thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, The thickness of the oblique line section of jth sheet auxiliary spring compares γ_Aj, the root of oblique line section is to distance l of auxiliary spring end points_1Apj=l_2Aβ_Aj ², oblique line section End is to distance l of auxiliary spring end points_1Aj=l_1Apj-Δ l, wherein, j=1,2 ..., n；Each main spring after major-minor spring is contacted Half clamping stiffness K_MAiIt is respectively calculated, i.e. $K_{MAi}=\left\{\begin{array}{cc}\frac{h_{2M}^3}{G_{x-Ei}},& i=1,2,...,m-1\\ \frac{h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)}{G_{x-Em}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)-G_{x-E_{z}m}{G}_{x-DE}{h}_{2A}^3},& i=m\end{array}\right.;$

In formula,

$\begin{array}{c}{G}_{x-Ei}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mpi}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1Mi}^3}{{\mathrm{Eb\γ}}_{Mi}^3{\mathrm{\β}}_i^3}+\frac{6\mathrm{\Δ}l(4l_{1Mi}^2{\mathrm{\γ}}_{Mi}-l_{1Mi}^2-3l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2+3l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^2)}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{({\mathrm{\γ}}_{Mi}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^3-l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^4-2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}+2l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}+2l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}+2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^3-4l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi})}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{({\mathrm{\γ}}_{Mi}-1)}^3};\end{array}$

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

$\begin{array}{c}{G}_{x-EAj}=\frac{4\[{(L_M-l_3/2)}^3-l_{2A}^3\]}{Eb}-\frac{8l_{2A}^{3/2}(l_{1Apj}^{3/2}-l_{2A}^{3/2})}{Eb}+\frac{4l_{1Aj}^3}{{\mathrm{Eb\γ}}_{Aj}^3{\mathrm{\β}}_{Aj}^3}+\frac{6\mathrm{\Δ}l(4l_{1Aj}^2{\mathrm{\γ}}_{Aj}-l_{1Aj}^2-3l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2+3l_{1Apj}^2{\mathrm{\γ}}_{Aj}^2)}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{({\mathrm{\γ}}_{Aj}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Apj}^2{\mathrm{\γ}}_{Aj}^3-l_{1Apj}^2{\mathrm{\γ}}_{Aj}^4-2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}+2l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}+2l_{1Apj}^2{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}+2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^3-4l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj})}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{({\mathrm{\γ}}_{Aj}-1)}^3};\end{array}$

$\begin{array}{c}{G}_{x-DE}=\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_{2M}^{3/2}(l_{1Mpm}^{1/2}-l_{2M}^{1/2})(l_{1Mpm}+l_{2M}-3l_0+l_{1Mpm}^{1/2}{l}_{2M}^{1/2})}{Eb}+\\ \frac{2{(l_{1Mm}-l_0)}^2(2l_{1Mm}+l_0)}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3}+\frac{6\mathrm{\Δ}l(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3-2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4+2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}-\\ \frac{24l_{1Mm}{l}_{1Mpm}{\mathrm{\Δl\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}-\frac{6l_0\mathrm{\Δ}l(l_{1Mpm}{\mathrm{\γ}}_{Mm}+l_{1Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3};\end{array}$

$\begin{array}{c}{G}_{x-E_{z}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{8l_{2M}^{3/2}(l_{1Mpm}^{1/2}-l_{2M}^{1/2})(l_{1Mpm}+l_{2M}-3l_0+l_{1Mpm}^{1/2}{l}_{2M}^{1/2})}{Eb}+\\ \frac{6\mathrm{\Δ}l(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2+3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4-2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3-4l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{2{(l_{1Mm}-l_0)}^2(2l_{1Mm}+l_0)}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3}-\frac{6l_0\mathrm{\Δ}l(l_{1Mpm}{\mathrm{\γ}}_{Mm}+l_{1Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3};\end{array}$

Wherein, β_mIt it is the thickness ratio of the parabolic segment of the main spring of m sheet；

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

Half length L according to few sheet reinforcement end 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, elastic modulus E, half l of installing space₃, the length Δ l of oblique line section, the root of auxiliary spring parabolic segment Portion is to distance l of auxiliary spring end points_2A=L_A-l₃；The thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, the oblique line section of jth sheet auxiliary spring Thickness compares γ_Aj, distance l of the root of the oblique line section of jth sheet auxiliary spring to auxiliary spring end points_1Apj=l_2Aβ_Aj ², the oblique line section of jth sheet auxiliary spring End to distance l of auxiliary spring end points_1Aj=l_1Apj-Δ l, wherein, j=1,2 ..., n；The half of each auxiliary spring is clamped stiffness K_Aj Calculate, i.e. $K_{Aj}=\frac{h_{2A}^3}{G_{x-EAj}},j=1,2,...,n;$

In formula, $\begin{array}{c}{G}_{x-EAj}=\frac{4\[{(L_M-l_3/2)}^3-l_{2A}^3\]}{Eb}-\frac{8l_{2A}^{3/2}(l_{1Apj}^{3/2}-l_{2A}^{3/2})}{Eb}+\frac{4l_{1Aj}^3}{{\mathrm{Eb\γ}}_{Aj}^3{\mathrm{\β}}_{Aj}^3}+\frac{6\mathrm{\Δ}l(4l_{1Aj}^2{\mathrm{\γ}}_{Aj}-l_{1Aj}^2-3l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2+3l_{1Apj}^2{\mathrm{\γ}}_{Aj}^2)}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{({\mathrm{\γ}}_{Aj}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Apj}^2{\mathrm{\γ}}_{Aj}^3-l_{1Apj}^2{\mathrm{\γ}}_{Aj}^4-2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}+2l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}+2l_{1Apj}^2{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}+2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^3-4l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj})}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{({\mathrm{\γ}}_{Aj}-1)}^3};\end{array}$

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

I step: maximum end points power P of each main spring_imaxCalculate:

The most single-ended point of half according to maximum load suffered by the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is Big load p_max, auxiliary spring works load p_K, main reed number m, calculated K in I step_Mi, and II step calculates obtained K_MAi, wherein, i=1,2 ..., 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: maximum end points power P of each auxiliary spring_AjmaxCalculate:

The most single-ended point of half according to maximum load suffered by the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is Big load p_max, auxiliary spring works load p_K, the thickness h of the root flat segments of each main spring_2M, auxiliary spring sheet number n, each auxiliary spring The thickness h of root flat segments_2A, calculated K in II step_MAi、G_x-DE、G_x-DEzAnd G_x-EAT, and III step is calculated K_Aj, wherein, j=1,2 ..., n, maximum end points power P to each auxiliary spring_AjmaxCalculate, i.e. $P_{Aj\max }=\frac{K_{Aj}{K}_{MAm}{G}_{x-DE}{h}_{2A}^3(2P_{\max }-P_K)}{2\underset{j=1}{\overset{n}{\Σ}}{K}_{Aj}\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)},j=1,2,\mathrm{...},n;$

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

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

Half length L according to few sheet main spring of reinforcement end variable cross-section_M, width b, half l of installing space₃, main reed Number m, the thickness h of the root flat segments of each main spring_2M；And calculated P in i step_imax, wherein, i=1,2 ..., m-1 is right Before few sheet reinforcement end variable-section steel sheet spring, the maximum stress of each of the main spring of m-1 sheet calculates, 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:

Half length L according to few sheet main spring of reinforcement end variable cross-section_M, width b, half l of installing space₃, parabola The root of section is to distance l of main spring end points_2M=L_M-l₃, the thickness of the parabolic segment of the main spring of m sheet compares β_m, the parabolic of the main spring of m sheet The end of line segment is to distance l of main spring end points_1Mpm=l_2Mβ_m ², end thickness h of the parabolic segment of the main spring of m sheet_1Mpm；Auxiliary spring sheet Number n, auxiliary spring contact point and horizontal range l of main spring end points₀, the maximum end points power of the calculated main spring of m sheet in i step P_mmax, calculated P in ii step_Ajmax, wherein, j=1,2 ..., n, the maximum stress of spring main to m sheet calculates, i.e. ${\mathrm{\σ}}_{mmax}=\frac{6\[P_{mmax}{l}_{1Mpm}-\underset{j=1}{\overset{n}{\Σ}}{P}_{Ajmax}(l_{1Mpm}-l_0)\]}{{\mathrm{bh}}_{1Mpm}^2};$

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

Half length L according to few sheet reinforcement end variable cross-section auxiliary spring_A, width b, half l of installing space₃, auxiliary spring sheet Number n, the thickness h of the root flat segments of each auxiliary spring_2A；And calculated P in ii step_Ajmax, few sheet reinforcement end is become The maximum stress of each auxiliary spring of section steel flat spring calculates, i.e. ${\mathrm{\σ}}_{Ajmax}=\frac{6P_{Ajmax}(L_A-l_3/2)}{{\mathrm{bh}}_{2A}^2},j=1,2,...,n;$

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

1. step: the stress intensity of each 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 reinforcement end variable cross-section major-minor spring few to end contact is carried out Check, it may be assumed that if σ_imax> [σ], then it is unsatisfactory for stress intensity requirement；If σ_imax≤ [σ], then meet 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, opposite end in step B The stress intensity of the main spring of m sheet of the few sheet reinforcement end variable cross-section major-minor spring of portion's contact is checked, it may be assumed that if σ_mmax> [σ], then be unsatisfactory for stress intensity requirement；If σ_mmax≤ [σ], then meet 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, opposite end in step C The stress intensity of each auxiliary spring of the few sheet reinforcement end variable cross-section major-minor spring of portion's contact is checked, it may be assumed that if σ_Ajmax> [σ], then be unsatisfactory for stress intensity requirement；If σ_Ajmax≤ [σ], then meet stress intensity requirement, j=1, and 2 ..., n.

The present invention has the advantage that than prior art

Structure is waited owing to the end flat segments of each main spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is non-, 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 is in addition to by end points power, also Being acted on by auxiliary spring contact support power in end flat segments, therefore, each main spring and the end points power of auxiliary spring and maximum stress calculate Extremely complex, fail to provide each main spring and the auxiliary spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula the most always Stress intensity check method.The present invention can be according to each main spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula Work load and maximum load with the structural parameters of each auxiliary spring, elastic modelling quantity, allowable stress, auxiliary spring, to end contact Few each main spring of sheet reinforcement end variable cross-section major-minor spring and the stress intensity of each auxiliary spring are checked.By example and ANSYS simulating, verifying understands, the intensity school of the few sheet reinforcement end variable cross-section major-minor spring of the ends contact formula that this invention is provided Kernel method is correct, utilizes the available the most each main spring of the method and the maximum stress check value of each auxiliary spring, Design level, product quality and the use longevity of the few sheet reinforcement end variable cross-section major-minor leaf spring of ends contact formula can be improved Life and vehicle ride performance；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 strength check flow chart of few each of the sheet reinforcement end major-minor spring of ends contact formula；

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

Fig. 3 is the maximum stress emulation of the 1st main spring of the few sheet reinforcement end major-minor spring of ends contact formula of embodiment Cloud atlas；

Fig. 4 is the maximum stress emulation of the 2nd main spring of the few sheet reinforcement end major-minor spring of ends contact formula of embodiment Cloud atlas；

Fig. 5 is the maximum stress emulation cloud of 1 auxiliary spring of the few sheet reinforcement end major-minor spring of ends contact formula of embodiment Figure.

Specific embodiments

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

Embodiment: the main reed number m=2, auxiliary spring sheet number n of the few sheet reinforcement end variable cross-section major-minor spring of certain ends contact formula =1, wherein, half length L of each main spring_M=575mm, width b=60mm, elastic modulus E=200GPa, each main spring The thickness h of root flat segments_2M=11mm, half l of installing space₃=55mm, the length Δ l=30mm of oblique line section, main spring is thrown The root of thing line segment is to distance l of main spring end points_2M=L_M-l₃=520mm；End thickness h of the parabolic segment of the 1st main spring_1Mp1 The thickness of=6mm, i.e. parabolic segment compares β₁=h_1Mp1/h_2M=0.55, the end of parabolic segment is to distance l of main spring end points_1Mp1= l_2Mβ₁ ²=157.30mm, the thickness h of end flat segments_1M1=7mm, the i.e. thickness of oblique line section compare γ_M1=h_1M1/h_1Mp1=1.17, Length l of end flat segments_1M1=l_1Mp1-Δ l=127.30mm；End thickness h of the parabolic segment of the 2nd main spring_1Mp2= The thickness of 5mm, i.e. parabolic segment compares β₂=h_1Mp2/h_2M=0.45, the end of parabolic segment is to distance l of main spring end points_1Mp2= l_2Mβ₂ ²=105.30mm, the thickness h of end flat segments_1M2=6mm, the i.e. thickness of oblique line section compare γ_M2=h_1M2/h_1Mp2=1.20, Length l of end flat segments_1M2=l_1Mp2-Δ l=75.30mm.Half length L of this sheet auxiliary spring_A=525mm, width b= 60mm, auxiliary spring end points is to horizontal range l of main spring end points₀=L-L_A=50mm, half length l of installing space₃=55mm, tiltedly The length Δ l=30mm of line segment, the root of parabolic segment is to distance l of auxiliary spring end points_2A=L_A-l₃=470mm, this sheet auxiliary spring The thickness h of root flat segments_2A=14mm, end thickness h of the parabolic segment of auxiliary spring_1Ap1=7mm, the thickness of the parabolic segment of auxiliary spring Degree compares β_A1=h_1Ap1/h_2A=0.50, the end of the parabolic segment of auxiliary spring is to distance l of auxiliary spring end points_1Ap1=l_2Aβ_A1 ²= 117.50mm, the thickness h of the end flat segments of auxiliary spring_1A1=8mm, the thickness of the oblique line section of auxiliary spring compares γ_A1=h_1A1/h_1Ap1= 1.14, length l of the end flat segments of auxiliary spring_1A1=l_1Ap1-Δ l=87.50mm.Major-minor spring works half P of load_K= 1325.5N, when load works load more than auxiliary spring, auxiliary spring contacts with certain point in the flat segments of main spring end, leaf spring Allowable stress [σ]=700MPa.In the most single-ended some maximum load P=3040N feelings of the half of maximum load suffered by this major-minor spring Under condition, each slice main spring of sheet reinforcement end variable-section steel sheet spring few to this ends contact formula and the stress intensity of auxiliary spring are carried out Check.

The check method of the few sheet reinforcement end each intensity of major-minor spring of the ends contact formula that present example is provided, its Check flow process as it is shown in figure 1, concrete check step is as follows:

(1) each main spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula and the half of auxiliary spring clamp rigidity 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 main spring of reinforcement end variable cross-section_M=575mm, width b=60mm, elastic modulus E =200GPa, half l of installing space₃=55mm, main reed number m=2, the thickness h of the root flat segments of each main spring_2M= 11mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2M=520mm；1st main spring The thickness of parabolic segment compare β₁The thickness of the oblique line section of the=0.55, the 1st main spring compares γ_M1The oblique line of the=1.17, the 1st main spring The root of section is to distance l of main spring end points_1Mp1=157.30mm；The thickness of the parabolic segment of the 2nd main spring compares β₂=0.45；2nd The thickness of the oblique line section of the main spring of sheet compares γ_M2=1.20；The root of the oblique line section of the 2nd main spring is to distance l of main spring end points_1Mp2= 105.30mm；The end of the oblique line section of the 1st main spring is to distance l of main spring end points_1M1=127.30mm, the oblique line of the 2nd main spring The end of section is to distance l of main spring end points_1M2=75.30mm；The 1st main spring before major-minor spring is contacted and the 2nd main spring Half clamping stiffness K_M1And K_M2Calculate, be respectively

$K_{M1}=\frac{h_{2M}^3}{G_{x-E1}}=14.54N/mm;$

$K_{M2}=\frac{h_{2M}^3}{G_{x-E2}}=13.85N/mm;$

In formula,

$\begin{array}{c}{G}_{x-E1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp1}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M1}^3}{{\mathrm{Eb\γ}}_{M1}^3{\mathrm{\β}}_1^3}+\frac{6\mathrm{\Δ}l(4l_{1M1}^2{\mathrm{\γ}}_{M1}-l_{1M1}^2-3l_{1M1}^2{\mathrm{\γ}}_{M1}^2+3l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2)}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Mp1}^2{\mathrm{\γ}}_{M1}^3-l_{1Mp1}^2{\mathrm{\γ}}_{M1}^4-2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}+2l_{1M1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^3-4l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1})}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}\\ =91.51{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-E2}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp2}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M2}^3}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}+\frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3-l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}\\ =96.07{\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 main spring of reinforcement end variable cross-section_M=575mm, width b=60mm, elastic modulus E= 200GPa, half l of installing space₃=55mm, main reed number m=2, the thickness h of the root flat segments of each main spring_2M=11mm, The length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2M=520mm；The parabola of the 1st main spring The thickness of section compares β₁The thickness of the oblique line section of the=0.55, the 1st main spring compares γ_M1The root of the oblique line section of the=1.17, the 1st main spring arrives Distance l of main spring end points_1Mp1=157.30m, the end of the oblique line section of the 1st main spring is to distance l of main spring end points_1M1= 127.30mm；The thickness of the parabolic segment of the 2nd main spring compares β₂The thickness of the oblique line section of the=0.45, the 2nd main spring compares γ_M2= 1.20；The root of the oblique line section of the 2nd main spring is to distance l of main spring end points_1Mp2=105.30mm；The oblique line section of the 2nd main spring End is to distance l of main spring end points_1M2=75.30mm.Auxiliary spring sheet number n=1, half length L of this sheet auxiliary spring_A=525mm, auxiliary spring Contact and horizontal range l of main spring end points₀=50mm, width b=60mm, half l of installing space₃=55mm, the root of this sheet auxiliary spring The thickness h of portion's flat segments_2A=14mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of auxiliary spring end points_2A= 470mm；The thickness of the parabolic segment of this sheet auxiliary spring compares β_A1=0.50, the thickness of oblique line section compares γ_A1=1.14, the root of oblique line section Distance l to auxiliary spring end points_1Ap1=117.50mm, the end of oblique line section is to distance l of auxiliary spring end points_1A1=87.50mm.To major-minor The 1st main spring after spring contact and the half clamping stiffness K of the 2nd main spring_MA1And K_MA2It is respectively calculated, i.e.

$K_{MA1}=\frac{h_{2M}^3}{G_{x-E1}}=14.54N/mm;$

$K_{MA2}=\frac{h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)}{G_{x-E2}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)-G_{x-E_{z}2}{G}_{x-DE}{h}_{2A}^3}=41.36N/mm;$

In formula,

$\begin{array}{c}{G}_{x-E1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp1}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M1}^3}{{\mathrm{Eb\γ}}_{M1}^3{\mathrm{\β}}_1^3}+\frac{6\mathrm{\Δ}l(4l_{1M1}^2{\mathrm{\γ}}_{M1}-l_{1M1}^2-3l_{1M1}^2{\mathrm{\γ}}_{M1}^2+3l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2)}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Mp1}^2{\mathrm{\γ}}_{M1}^3-l_{1Mp1}^2{\mathrm{\γ}}_{M1}^4-2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}+2l_{1M1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^3-4l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1})}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}\\ =91.51{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-E2}=\frac{4\[{(L_M-l_3/2)}^3-l_{2M}^3\]}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp2}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M2}^3}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}+\frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3-l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}\\ =96.07{\mathrm{mm}}^4/N;\end{array}$

$G_{x-EAT}=\frac{1}{\underset{j=1}{\overset{1}{\Σ}}\frac{1}{G_{x-EAj}}}=63.90{\mathrm{mm}}^4/N;$

$\begin{array}{c}{G}_{x-EA1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2A}^3\]}{Eb}-\frac{8l_{2A}^{3/2}(l_{1Ap1}^{3/2}-l_{2A}^{3/2})}{Eb}+\frac{4l_{1A1}^3}{{\mathrm{Eb\γ}}_{A1}^3{\mathrm{\β}}_{A1}^3}+\frac{6\mathrm{\Δ}l(4l_{1A1}^2{\mathrm{\γ}}_{A1}-l_{1A1}^2-3l_{1A1}^2{\mathrm{\γ}}_{A1}^2+3l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2)}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Ap1}^2{\mathrm{\γ}}_{A1}^3-l_{1Ap1}^2{\mathrm{\γ}}_{A1}^4-2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}+2l_{1A1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^3-4l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1})}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}\\ =63.90{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-DE}=\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_{2M}^{3/2}(l_{1Mp2}^{1/2}-l_{2M}^{1/2})(l_{1Mp2}+l_{2M}-3l_0+l_{1Mp2}^{1/2}{l}_{2M}^{1/2})}{Eb}+\\ \frac{2{(l_{1M2}-l_0)}^2(2l_{1M2}+l_0)}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}+\frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2+l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\\ \frac{24l_{1M2}{l}_{1Mp2}{\mathrm{\Δl\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\frac{6l_0\mathrm{\Δ}l(l_{1Mp2}{\mathrm{\γ}}_{M2}+l_{1M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3}=78.54{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-E_{z}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{8l_{2M}^{3/2}(l_{1Mp2}^{1/2}-l_{2M}^{1/2})(l_{1Mp2}+l_{2M}-3l_0+l_{1Mp2}^{1/2}{l}_{2M}^{1/2})}{Eb}+\\ \frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3+l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{2{(l_{1M2}-l_0)}^2(2l_{1M2}+l_0)}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}-\frac{6l_0\mathrm{\Δ}l(l_{1Mp2}{\mathrm{\γ}}_{M2}+l_{1M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3}=78.54{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-{\mathrm{DE}}_z}=\frac{12l_{2M}^{3/2}(6l_0^2{l}_{2M}^{1/2}+12l_0{l}_{1Mp2}{l}_{2M}^{1/2}-2l_{2M}^{1/2}{l}_{1Mp2}^2-6l_{1Mp2}^{1/2}{l}_0^2-12l_{1Mp2}^{1/2}{l}_0{l}_{2M}+2l_{1Mp2}^{1/2}{l}_{2M}^2)}{3l_{1Mp2}^{1/2}{l}_{2M}^{1/2}Eb}+\frac{4{(l_0-l_{1M2})}^3}{{\mathrm{Eb\β}}_2^3{\mathrm{\γ}}_{M2}^3}+\\ \frac{4\[(L_M-l_3/2-l_{2M})\[{(L_M-l_3/2)}^2-3(L_M-l_3/2)l_0+(L_M-l_3/2)l_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2\]}{Eb}+\\ \frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}+l_0^2{\mathrm{\γ}}_{M2}^4-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3+l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4+2l_0{l}_{1M2}-6l_0{l}_{1M2}{\mathrm{\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}-2l_0{l}_{1M2}{\mathrm{\γ}}_{M2}^3-6l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(6l_0{l}_{1M2}{\mathrm{\γ}}_{M2}^2+6l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-2l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}^4+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\end{array}$

$\frac{6\mathrm{\Δ}l(2l_0^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-l_0^2-2l_0^2{\mathrm{\γ}}_{M2}^3)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_M^3{({\mathrm{\γ}}_{M2}-1)}^3}=65.56{\mathrm{mm}}^4/N;$

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

Half length L according to auxiliary spring_A=525mm, width b=60mm, elastic modulus E=200GPa, the half of installing space l₃=55mm, auxiliary spring sheet number n=1, the thickness h of the root flat segments of this sheet auxiliary spring_2A=14mm, the length Δ l=of auxiliary spring oblique line section 30mm, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2A=470mm；The thickness of the parabolic segment of this sheet auxiliary spring compares β_A1 =0.50, the thickness of oblique line section compares γ_A1=1.14, distance l of the root of auxiliary spring oblique line section to auxiliary spring end points_1Ap1=117.50mm, secondary The end of spring oblique line section is to distance l of auxiliary spring end points_1A1=87.50mm；The half of this sheet auxiliary spring is clamped stiffness K_A1Calculate, I.e.

$K_{A1}=\frac{h_{2A}^3}{G_{x-EA1}}=42.94N/mm;$

In formula,

$\begin{array}{c}{G}_{x-EA1}=\frac{4\[{(L_M-l_3/2)}^3-l_{2A}^3\]}{Eb}-\frac{8l_{2A}^{3/2}(l_{1Ap1}^{3/2}-l_{2A}^{3/2})}{Eb}+\frac{4l_{1A1}^3}{{\mathrm{Eb\γ}}_{A1}^3{\mathrm{\β}}_{A1}^3}+\frac{6\mathrm{\Δ}l(4l_{1A1}^2{\mathrm{\γ}}_{A1}-l_{1A1}^2-3l_{1A1}^2{\mathrm{\γ}}_{A1}^2+3l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2)}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(4l_{1Ap1}^2{\mathrm{\γ}}_{A1}^3-l_{1Ap1}^2{\mathrm{\γ}}_{A1}^4-2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}+2l_{1A1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^3-4l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1})}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}\\ =63.90{\mathrm{mm}}^4/N;\end{array}$

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

I step: maximum end points power P of each main spring_imaxCalculate:

The most single-ended point of half according to maximum load suffered by the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is Big load p_max=3040N, auxiliary spring works load p_K=2651N, main reed number m=2, calculated K in I step_M1= 14.54N/mm and K_M2=13.85N/mm, and II step calculate obtained K_MA1=14.54N/mm and K_MA2=41.36N/mm is right 1st main spring and maximum end points power P of the 2nd main spring_1maxAnd P_2maxIt is respectively calculated, i.e.

$P_{1\max }=\frac{K_{M1}{P}_K}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{Mi}}+\frac{K_{MA1}(2P_{\max }-P_K)}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}}=1124.8N;$

$P_{2\max }=\frac{K_{M2}{P}_K}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{Mi}}+\frac{K_{MA2}(2P_{\max }-P_K)}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}}=1915.2N;$

Ii step: maximum end points power P of each auxiliary spring_AjmaxCalculate:

The most single-ended point of half according to maximum load suffered by the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is Big load p_max=3040N, auxiliary spring works load p_K=2651N；Main reed number m=2, the thickness of the root flat segments of each main spring Degree h_2M=11mm, auxiliary spring sheet number n=1, the thickness h of the root flat segments of this sheet auxiliary spring_2A=14mm；In II step calculated K_MA1=14.54N/mm、K_MA2=41.36N/mm、G_x-DE=78.54mm⁴/N、G_x-DEz=78.54mm⁴/N、G_x-EAT=63.80mm⁴/ N, and Calculated K in III step_A1=42.94N/mm, maximum end points power P to this sheet auxiliary spring_A1maxCalculate, i.e.

$P_{A1max}=\frac{K_{A1}{K}_{MA2}{G}_{x-DE}{h}_{2A}^3(2P_{\max }-P_K)}{2\underset{j=1}{\overset{n}{\Σ}}{K}_{Aj}\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)}=1031.8N;$

(3) ends contact formula lacks each main spring and the maximum stress meter of auxiliary spring of sheet reinforcement end 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 main spring of reinforcement end variable cross-section_M=575mm, width b=60mm, installing space Half l₃=55mm, the thickness h of the root flat segments of each main spring_2M=11mm；And calculated P in i step_1max= 1124.8N, calculates, i.e. the maximum stress of the 1st main spring

${\mathrm{\σ}}_{1max}=\frac{6P_{1max}(L_M-l_3/2)}{{\mathrm{bh}}_{2M}^2}=508.97MPa;$

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

According to the width b=60mm of few sheet main spring of reinforcement end variable cross-section, the root of the oblique line section of the 2nd main spring is to main Distance l of spring end points_1Mp2=105.30mm, end thickness h of the parabolic segment of the 2nd main spring_1Mp2=5mm；Auxiliary spring sheet number n=1, secondary Spring contact and horizontal range l of main spring end points₀Calculated P in=50mm, i step_2max=1915.2N, ii step calculates The P arrived_A1max=1031.8N, calculates, i.e. the maximum stress of the 2nd main spring

${\mathrm{\σ}}_{2max}=\frac{6\[P_{2max}{l}_{1Mp2}-\underset{j=1}{\overset{n}{\Σ}}{P}_{Ajmax}(l_{1Mp2}-l_0)\]}{{\mathrm{bh}}_{1Mp2}^2}=578.43MPa;$

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

Half length L according to this sheet auxiliary spring_A=525mm, width b=60mm, half l of installing space₃=55mm, auxiliary spring The thickness 14mm of root flat segments；And calculated P in ii step_A1max=1031.8N, to this sheet reinforcement end variable cross-section The maximum stress of auxiliary spring calculates, i.e.

${\mathrm{\σ}}_{A1max}=\frac{6P_{A1max}(L_A-l_3/2)}{{\mathrm{bh}}_{2A}^2}=261.90MPa;$

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

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

In allowable stress [σ] according to leaf spring=700MPa, and step A, the maximum of calculated 1st main spring should Power σ_1max=508.97MPa, 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:

In allowable stress [σ] according to leaf spring=700MPa, and step B, the maximum of calculated 2nd main spring should Power σ_2max=578.43MPa, it is known that σ_2max≤ [σ], i.e. the 2nd main spring disclosure satisfy that stress intensity requirement；

3. step: the stress intensity of this sheet 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=261.90MPa, 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 reinforcement end variable cross-section major-minor spring of this ends contact formula Each main spring and the structural parameters of auxiliary spring and material characteristic parameter, set up the ANSYS phantom of half symmetrical structure major-minor spring, Grid division, arranges auxiliary spring end points and contacts with main spring, and at the root applying fixed constraint of phantom, executes at main spring free end Add concentrfated load F=P_max-P_K/ 2=1714.4N, sheet reinforcement end variable cross-section major-minor few to this ends contact formula is at clamped condition Under the stress of major-minor spring carry out ANSYS emulation, the maximum stress emulation cloud atlas of the 1st obtained main spring, as shown in Figure 3； The maximum stress emulation cloud atlas of the 2nd main spring, as shown in Figure 4；The maximum stress emulation cloud atlas of 1 auxiliary spring, as it is shown in figure 5, its In, the 1st main spring is at the maximum stress σ clamping root_1max=201.22MPa, the 2nd main spring connect with oblique line section in parabolic segment Touch the maximum stress σ of position_2max=305.19MPa, 1 auxiliary spring are at the maximum stress σ clamping root_A1max=262.72MPa。

Understand, in the case of same load, the 1st of the few sheet reinforcement end variable cross-section major-minor spring of this this ends contact formula ANSYS simulating, verifying value σ with the 2nd main spring and the maximum stress of 1 auxiliary spring_1max=201.22MPa、σ_2max=305.19MPa、 σ_A1max=262.72MPa, respectively with analytical Calculation value σ_1max=201.77MPa、σ_2max=306.05MPa、σ_A1max=261.90MPa, Matching, relative deviation is respectively 0.27%, 0.28%, 0.31%；Result shows the few bit end of ends contact formula that this invention is provided The check method of portion's reinforced major-minor each intensity of spring is correct, and the stress intensity calculation and check value of each main spring and auxiliary spring is Accurately and reliably.

Claims (1)

1. the check method of the few sheet reinforcement end each intensity of major-minor spring of ends contact formula, wherein, the few bit end of ends contact formula The half symmetrical structure of portion's reinforced major-minor spring is made up of root flat segments, parabolic segment, oblique line section and end flat segments 4 sections, Booster action is played in the end of major-minor spring by the oblique line section added；The end flat segments of each main spring is non-waits structure, i.e. the 1st main spring The thickness of end flat segments and length, more than the thickness of end flat segments and the length of other each main spring, to meet the 1st The requirement of main spring complicated applied force；Auxiliary spring length is less than main spring length, is provided with certain between auxiliary spring contact and main spring end flat segments Major-minor spring gap, work the design requirement of load meeting auxiliary spring；When load works load more than auxiliary spring, auxiliary spring touches Point spring end main with main spring flat segments point contacts, and the end points power of each major-minor spring is unequal, and with the 1 of auxiliary spring contact The main spring of sheet is also acted on by auxiliary spring contact support power；At the structural parameters of each of major-minor spring, elastic modelling quantity, allowable stress, Big load, auxiliary spring work load given in the case of, sheet reinforcement end variable-section steel sheet spring each slice few to end contact The stress intensity of main spring and auxiliary spring is checked, and concrete check step is as follows:

(1) each main spring of the few sheet reinforcement end variable cross-section major-minor spring of 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_MiCalculating:

Half length L according to few sheet main spring of reinforcement end 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_2M=L_M-l₃；The thickness of the parabolic segment of i-th main spring compares β_i, the thickness ratio of the oblique line section of i-th main spring γ_Mi, the root of oblique line section is to distance l of main spring end points_1Mpi=l_2Mβ_i ², the end of oblique line section is to distance l of main spring end points_1Mi= l_1Mpi-Δ l, wherein, i=1,2 ..., m；The half clamping stiffness K of each main spring before major-minor spring is contacted_MiCarry out respectively Calculate, 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\[{\left(L_M-l_3/2\right)}^3-l_{2M}^3\]}{Eb}-\frac{8l_{2M}^{3/2}\left(l_{1Mpi}^{3/2}-l_{2M}^{3/2}\right)}{Eb}+\frac{4l_{1Mi}^3}{{\mathrm{Eb\γ}}_{Mi}^3{\mathrm{\β}}_i^3}+\frac{6\mathrm{\Δ}l\left(4l_{1Mi}^2{\mathrm{\γ}}_{Mi}-l_{1Mi}^2-3l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2+3l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^2\right)}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{\left({\mathrm{\γ}}_{Mi}-1\right)}^3}-\\ \frac{6\mathrm{\Δ}l\left(4l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^3-l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^4-2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}+2l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}+2l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^4{\mathrm{ln\γ}}_{Mi}+2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^3-4l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}\right)}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{\left({\mathrm{\γ}}_{Mi}-1\right)}^3};\end{array}$

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

Half length L according to few sheet main spring of reinforcement end 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_2M=L_M-l₃；The thickness of the parabolic segment of i-th main spring compares β_i, the thickness ratio of the oblique line section of i-th main spring γ_Mi, the root of the oblique line section of i-th main spring is to distance l of main spring end points_1Mpi=l_2Mβ_i ², the end of the oblique line section of i-th main spring Portion is to distance l of main spring end points_1Mi=l_1Mpi-Δ l, wherein, i=1,2 ..., m；Half length L of auxiliary spring_A, auxiliary spring contact and master Horizontal range l of spring end points₀=L_M-L_A, auxiliary spring sheet number n, the thickness h of the root flat segments of each auxiliary spring_2A, the length of oblique line section Δ l, the root of parabolic segment is to distance l of auxiliary spring end points_2A=L_A-l₃；The thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, jth The thickness of the oblique line section of sheet auxiliary spring compares γ_Aj, the root of oblique line section is to distance l of auxiliary spring end points_1Apj=l_2Aβ_Aj ², the end of oblique line section Portion is to distance l of auxiliary spring end points_1Aj=l_1Apj-Δ l, wherein, j=1,2 ..., n；Each main spring after major-minor spring is contacted Half clamping stiffness K_MAiIt is respectively calculated, i.e.

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

In formula,

$G_{x-Ei}=\frac{4\[{\left(L_M-l_3/2\right)}^3-l_{2M}^3\]}{Eb}-\frac{8l_{2M}^{3/2}\left(l_{1Mpi}^{3/2}-l_{2M}^{3/2}\right)}{Eb}+\frac{4l_{1Mi}^3}{{\mathrm{Eb\γ}}_{Mi}^3{\mathrm{\β}}_i^3}+\frac{6\mathrm{\Δ}l\left(4l_{1Mi}^2{\mathrm{\γ}}_{Mi}-l_{1Mi}^2-3l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2+3l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^2\right)}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{\left({\mathrm{\γ}}_{Mi}-1\right)}^3}-$

$\frac{6\mathrm{\Δ}l\left(4l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^3-l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^4-2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}+2l_{1Mi}^2{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}+2l_{1Mpi}^2{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}+2l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^3-4l_{1Mi}{l}_{1Mpi}{\mathrm{\γ}}_{Mi}^2{\mathrm{ln\γ}}_{Mi}\right)}{{\mathrm{Eb\γ}}_{Mi}^2{\mathrm{\β}}_i^3{\left({\mathrm{\γ}}_{Mi}-1\right)}^3};$

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

$\begin{array}{c}{G}_{x-EAj}=\frac{4\[{\left(L_M-l_3/2\right)}^3-l_{2A}^3\]}{Eb}-\frac{8l_{2A}^{3/2}\left(l_{1Apj}^{3/2}-l_{2A}^{3/2}\right)}{Eb}+\frac{4l_{1Aj}^3}{{\mathrm{Eb\γ}}_{Aj}^3{\mathrm{\β}}_{Aj}^3}+\frac{6\mathrm{\Δ}l\left(4l_{1Aj}^2{\mathrm{\γ}}_{Aj}-l_{1Aj}^2-3l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2+3l_{1Apj}^2{\mathrm{\γ}}_{Aj}^2\right)}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{\left({\mathrm{\γ}}_{Aj}-1\right)}^3}-\\ \frac{6\mathrm{\Δ}l\left(4l_{1Apj}^2{\mathrm{\γ}}_{Aj}^3-l_{1Apj}^2{\mathrm{\γ}}_{Aj}^4-2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}+2l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}+2l_{1Apj}^2{\mathrm{\γ}}_{Aj}^4{\mathrm{ln\γ}}_{Aj}+2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^3-4l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}\right)}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{\left({\mathrm{\γ}}_{Aj}-1\right)}^3};\end{array}$

$\begin{array}{c}{G}_{x-DE}=\frac{4{\left(L_M-l_3/2\right)}^3-6l_0{\left(L_M-l_3/2\right)}^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{3l_{2M}^{3/2}\left(l_{1Mpm}^{1/2}-l_{3M}^{1/2}\right)\left(l_{1Mpm}+l_{2M}-3l_0+l_{1Mpm}^{1/2}{l}_{2M}^{1/2}\right)}{Eb}+\\ \frac{2{\left(l_{1Mm}-l_0\right)}^2\left(2l_{1Mm}+l_0\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3}+\frac{6\mathrm{\Δ}l\left(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3-2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{\left({\mathrm{\γ}}_{Mm}-1\right)}^3}+\\ \frac{6\mathrm{\Δ}l\left(3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4+2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{\left({\mathrm{\γ}}_{Mm}-1\right)}^3}-\\ \frac{24l_{1Mm}{l}_{1Mpm}{\mathrm{\Δl\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{\left({\mathrm{\γ}}_{Mm}-1\right)}^3}-\frac{6l_0\mathrm{\Δ}l\left(l_{1Mpm}{\mathrm{\γ}}_{Mm}+l_{1Mm}\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3};\end{array}$

$\begin{array}{c}{G}_{x-E_{z}m}=\frac{4{\left(L_M-l_3/2\right)}^3-6l_0{\left(L_M-l_3/2\right)}^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{3l_{2M}^{3/2}\left(l_{1Mpm}^{1/2}-l_{3M}^{1/2}\right)\left(l_{1Mpm}+l_{2M}-3l_0+l_{1Mpm}^{1/2}{l}_{2M}^{1/2}\right)}{Eb}+\\ \frac{6\mathrm{\Δ}l\left(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2+3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4-2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{\left({\mathrm{\γ}}_{Mm}-1\right)}^3}+\\ \frac{6\mathrm{\Δ}l\left(2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3-4l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{\left({\mathrm{\γ}}_{Mm}-1\right)}^3}+\\ \frac{2{\left(l_{1Mm}-l_0\right)}^2\left(2l_{1Mm}+l_0\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3}-\frac{6l_0\mathrm{\Δ}l\left(l_{1Mpm}{\mathrm{\γ}}_{Mm}+l_{1Mm}\right)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3};\end{array}$

Wherein, β_mIt it is the thickness ratio of the parabolic segment of the main spring of m sheet；

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

Half length L according to few sheet reinforcement end variable cross-section auxiliary spring_A, auxiliary spring sheet number n, the root flat segments of each auxiliary spring Thickness h_2A, width b, elastic modulus E, half l of installing space₃, the length Δ l of oblique line section, the root of auxiliary spring parabolic segment arrives Distance l of auxiliary spring end points_2A=L_A-l₃；The thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, the thickness of the oblique line section of jth sheet auxiliary spring Compare γ_Aj, distance l of the root of the oblique line section of jth sheet auxiliary spring to auxiliary spring end points_1Apj=l_2Aβ_Aj ², the oblique line section of jth sheet auxiliary spring End is to distance l of auxiliary spring end points_1Aj=l_1Apj-Δ l, wherein, j=1,2 ..., n；The half of each auxiliary spring is clamped stiffness K_Aj Calculate, i.e.

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

In formula,

$\begin{array}{c}{G}_{x-EAj}=\frac{4\[{\left(L_M-l_3/2\right)}^3-l_{2A}^3\]}{Eb}-\frac{8l_{2A}^{3/2}\left(l_{1Apj}^{3/2}-l_{2A}^{3/2}\right)}{Eb}+\frac{4l_{1Aj}^3}{{\mathrm{Eb\γ}}_{Aj}^3{\mathrm{\β}}_{Aj}^3}+\frac{6\mathrm{\Δ}l\left(4l_{1Aj}^2{\mathrm{\γ}}_{Aj}-l_{1Aj}^2-3l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2+3l_{1Apj}^2{\mathrm{\γ}}_{Aj}^2\right)}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{\left({\mathrm{\γ}}_{Aj}-1\right)}^3}-\\ \frac{6\mathrm{\Δ}l\left(4l_{1Apj}^2{\mathrm{\γ}}_{Aj}^3-l_{1Apj}^2{\mathrm{\γ}}_{Aj}^4-2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}+2l_{1Aj}^2{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}+2l_{1Apj}^2{\mathrm{\γ}}_{Aj}^4{\mathrm{ln\γ}}_{Aj}+2l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^3-4l_{1Aj}{l}_{1Apj}{\mathrm{\γ}}_{Aj}^2{\mathrm{ln\γ}}_{Aj}\right)}{{\mathrm{Eb\γ}}_{Aj}^2{\mathrm{\β}}_{Aj}^3{\left({\mathrm{\γ}}_{Aj}-1\right)}^3};\end{array}$

(2) each main spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula and the maximum end points power of auxiliary spring calculate:

I step: maximum end points power P of each main spring_imaxCalculate:

The most single-ended maximum load of half according to maximum load suffered by the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula Lotus P_max, auxiliary spring works load p_K, main reed number m, calculated K in I step_Mi, and in II step obtained by calculating K_MAi, wherein, i=1,2 ..., m, maximum end points power P to each main spring_imaxCalculate, i.e.

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

Ii step: maximum end points power P of each auxiliary spring_AjmaxCalculate:

The most single-ended maximum load of half according to maximum load suffered by the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula Lotus P_max, auxiliary spring works load p_K, the thickness h of the root flat segments of each main spring_2M, auxiliary spring sheet number n, the root of each auxiliary spring The thickness h of flat segments_2A, calculated K in II step_MAi、G_x-DE、G_x-DEzAnd G_x-EAT, and calculated in III step K_Aj, wherein, j=1,2 ..., n, maximum end points power P to each auxiliary spring_AjmaxCalculate, i.e.

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

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

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

Half length L according to few sheet main spring of reinforcement end variable cross-section_M, width b, half l of installing space₃, main reed number m, The thickness h of the root flat segments of each main spring_2M；And calculated P in i step_imax, wherein, i=1,2 ..., m-1, to few Before sheet reinforcement end variable-section steel sheet spring, the maximum stress of each of the main spring of m-1 sheet calculates, 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:

Half length L according to few sheet main spring of reinforcement end variable cross-section_M, width b, half l of installing space₃, parabolic segment Root is to distance l of main spring end points_2M=L_M-l₃, the thickness of the parabolic segment of the main spring of m sheet compares β_m, the parabola of the main spring of m sheet The end of section is to distance l of main spring end points_1Mpm=l_2Mβ_m ², end thickness h of the parabolic segment of the main spring of m sheet_1Mpm；Auxiliary spring sheet number Horizontal range l of n, auxiliary spring contact point and main spring end points₀, maximum end points power P of the calculated main spring of m sheet in i step_mmax, Calculated P in ii step_Ajmax, wherein, j=1,2 ..., n, the maximum stress of spring main to m sheet calculates, i.e.

${\mathrm{\σ}}_{mmax}=\frac{6\[P_{mmax}{l}_{1Mpm}-\underset{j=1}{\overset{n}{\Σ}}{P}_{Ajmax}(l_{1Mpm}-l_0)\]}{{\mathrm{bh}}_{1Mpm}^2};$

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

Half length L according to few sheet reinforcement end variable cross-section auxiliary spring_A, width b, half l of installing space₃, auxiliary spring sheet number n, The thickness h of the root flat segments of each auxiliary spring_2A；And calculated P in ii step_Ajmax, to few sheet reinforcement end variable cross-section The maximum stress of each auxiliary spring of leaf spring calculates, 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 reinforcement end variable cross-section major-minor spring of ends contact formula and the stress intensity of auxiliary spring are checked:

1. step: the stress intensity of each 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 reinforcement end variable cross-section major-minor spring few to end contact is checked, If that is: σ_imax> [σ], then it is unsatisfactory for stress intensity requirement；If σ_imax≤ [σ], then meet 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, connect end The stress intensity of the main spring of m sheet of the few sheet reinforcement end variable cross-section major-minor spring of touch is checked, it may be assumed that if σ_mmax> [σ], Then it is unsatisfactory for stress intensity requirement；If σ_mmax≤ [σ], then meet 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, connect end The stress intensity of each auxiliary spring of the few sheet reinforcement end variable cross-section major-minor spring of touch is checked, it may be assumed that if σ_Ajmax> [σ], then It is unsatisfactory for stress intensity requirement；If σ_Ajmax≤ [σ], then meet stress intensity requirement, j=1, and 2 ..., n.