CN105740591A - Method for verifying strength of each leaf of end contact type few-leaf oblique main and auxiliary springs - Google Patents

Abstract

The invention relates to a method for verifying strength of each leaf of end contact type few-leaf oblique main and auxiliary springs and belongs to the technical field of suspension leaf springs. It is possible to verify and calculate stress strength of each of main and auxiliary leaves according to structural parameters, elastic model and allowable stress of all main and auxiliary leaves of the end contact type few-leaf oblique variable-section main and auxiliary springs, acting load of each auxiliary spring and maximum load bearable by a main spring. Experiments and simulation tests show that this method is correct, maximum stress verification calculations of all main and auxiliary leaves are accurate and reliable, the design level, product quality and life of the end contact type few-leaf oblique variable-section main and auxiliary leaf springs as well as vehicle driving smoothness may be improved; meanwhile, it is also possible to reduce designing and testing cost and increase product development speed.

Description

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

Technical field

The present invention relates to vehicle suspension leaf spring, particularly the check method of the few sheet bias type each intensity of major-minor spring of ends contact formula.

Background technology

Few sheet variable-section steel sheet spring is compared with multi-disc superposition leaf spring, specifically save material, alleviate unsprung mass, improve the advantage such as vehicle ride comfort and conevying efficiency, therefore, cause the great attention of domestic and international vehicle expert, and have been carried out promotion and application widely abroad.For few sheet variable-section steel sheet spring, in order to meet the requirement of variation rigidity, generally it is designed to major-minor spring, and by major-minor spring gap, guarantee after the load that works more than auxiliary spring, the contact of major-minor spring and work together, meet vehicle suspension designing requirement to leaf spring rigidity and stress intensity in different loads situation.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 actual the 1st designed main spring, usual big than other each main spring, namely in actual design with produce, few sheet variable-section steel sheet spring of the structure such as mostly adopt end non-.Few sheet variable-section steel sheet spring mainly has two types, and one is parabolic type, and another is bias type, and wherein, Parabolic stress is iso-stress, and suffered by it, stress ratio bias type is more reasonable.But, owing to the processing technique of parabolic type variable cross-section is complicated, it is necessary to complicated, expensive process equipment, and the processing technique of bias type is simple, it is only necessary to simple equipment just can be processed, therefore, under meeting rigidity and stress intensity designing requirement premise, the variable-section steel sheet spring of bias type can be adopted.For few sheet bias type variable cross-section major-minor spring, different auxiliary spring length can be adopted to meet the designing requirement of different composite rigidity and stress intensity, therefore, length difference according to auxiliary spring and the different contact position of major-minor, can be divided into ends contact formula and non-ends contact formula two kinds by few sheet bias type variable cross-section major-minor spring.The complex stiffness of few sheet bias type variable cross-section major-minor spring should meet the designing requirement of vehicle ride comfort, and intensity should meet service life and the security requirement of bearing spring, but, owing to the main spring length of few sheet bias type variable cross-section major-minor leaf spring is unequal with auxiliary spring length, the end flat segments of each main spring is non-waits structure, and after the contact of major-minor spring, deformation and the internal force of main spring and auxiliary spring have coupling, therefore, each main spring of few sheet bias type variable cross-section major-minor spring and the maximum stress of auxiliary spring calculate extremely complex, previously failed to provide simplicity always, accurately, the check method of the few sheet bias type variable cross-section each intensity of major-minor spring of reliable ends contact formula.Therefore, must be set up the check method of the few sheet bias type variable cross-section each intensity of major-minor spring of a kind of ends contact formula accurate, reliable, meet Vehicle Industry fast development and the requirement to suspension Precise Design for Laminated Spring, improve few design level of sheet bias type variable cross-section major-minor spring, product quality and service life；Meanwhile, reduce design and testing expenses, accelerate product development speed.

Summary of the invention

For the defect existed in above-mentioned prior art, the technical problem to be solved is to provide the check method of a kind of simplicity, the few sheet bias type each intensity of major-minor spring of reliable ends contact formula.The few sheet bias type variable cross-section major-minor spring of ends contact formula, it each bias type variable-section steel sheet spring including main spring, root shim, auxiliary spring, end pad, main spring and auxiliary spring is to be made up of root flat segments, oblique line section, end flat segments three sections；Being provided with root shim between each root flat segments of main spring and between each root flat segments of auxiliary spring, be provided with end pad between each end flat segments of main spring, the material of end pad is carbon fibre composite, produces frictional noise during to prevent work.Wherein, the thickness of the root flat segments of each main spring is h_2M, width is b, and half length is L_M, the half l of installing space₃, the root of main spring oblique line section is l to the distance of main spring end points_2M；The end flat segments of each main spring is non-waits structure, the thickness of the end flat segments of the 1st main spring and length, more than the thickness of end flat segments and length, the thickness of the end flat segments of each main spring and the length respectively h of other each main spring_1iAnd l_1i, the thickness of the oblique line section of each main spring is than for β_i=h_1i/h_2M, i=1,2 ..., m, m is main reed number.The width of each auxiliary spring is b, and half length is L_A, the half l of installing space₃, the thickness of the root flat segments of each auxiliary spring is h_2A, the thickness of end flat segments and length respectively h_A1jAnd l_A1j, the thickness of the oblique line section of each auxiliary spring is than for β_Aj=h_A1j/h_2A, j=1,2 ..., n, n is the sheet number of auxiliary spring.The half length L of auxiliary spring_AHalf length L less than main spring_M, level between auxiliary spring contact and main spring end points is from for l₀=L_M-L_A, it is provided with certain major-minor spring gap delta between auxiliary spring contact and main spring end flat segments, when load works load more than auxiliary spring, auxiliary spring contact contacts with certain point in the flat segments of main spring end；After major-minor spring contacts, the end points power of each of major-minor spring is unequal, and the 1 main spring contacted with auxiliary spring is except by end points power, also at contact point place by the effect of auxiliary spring contact support power.Work load to, under stable condition, the few each main spring of sheet bias type major-minor spring of end contact and the stress intensity of auxiliary spring being checked at the few each chip architecture parameter of sheet bias type variable cross-section major-minor spring of ends contact formula, elastic modelling quantity, allowable stress, maximum load and auxiliary spring.

For solving above-mentioned technical problem, the check method of the few sheet bias type each intensity of major-minor spring of ends contact formula provided by the present invention, it is characterised in that check step below adopting:

(1) each main spring of the few sheet bias type variable cross-section major-minor spring of ends contact formula and the half of auxiliary spring clamp Rigidity Calculation:

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 bias type variable cross-section_M, the thickness h of the root flat segments of each main spring_2M, width b, elastic modulus E, the half l of installing space₃, the root of oblique line section is to the distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of i-th main spring compares β_i=h_1i/h_2M, wherein, i=1,2 ..., m, m is main reed number, the half clamping stiffness K of each main spring of bias type variable cross-section before major-minor spring is contacted_MiIt is calculated, namely

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

In formula,

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 bias type variable cross-section_M, the thickness h of the root flat segments of each main spring_2M, width b, elastic modulus E, the half l of installing space₃, the root of oblique line section is to the distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of i-th main spring compares β_i, wherein, i=1,2 ..., m, m is main reed number；The half length L of auxiliary spring_A, the thickness h of the root flat segments of each auxiliary spring_2A, the root of auxiliary spring oblique line section is to the distance l of auxiliary spring end points_2A=L_A-l₃, the horizontal range l of auxiliary spring contact and main spring end points₀, the thickness of the oblique line section of jth sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, n is auxiliary spring sheet number, the half clamping stiffness K of each main spring of bias type variable cross-section after major-minor spring is contacted_MAiIt is calculated, namely

$K_{MAi}=\left\{\begin{array}{cc}\frac{h_{2M}^3}{G_{x-Di}},& i=1,2,...,m-1\\ \frac{h_{2M}^3(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}{G_{x-Dm}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)-G_{x-D_{z}m}{G}_{x-CD}{h}_{2A}^3},& i=m\end{array}\right.;$

In formula,

$G_{x-Di}=\frac{4}{Eb}\[{(L_M-l_3/2)}^3-l_{2M}^3\]+\frac{6l_{2M}^3{({\mathrm{\β}}_i+1)}^2\[3({\mathrm{\β}}_i-1)-2{\mathrm{ln\β}}_i(1+{\mathrm{\β}}_i)\]}{Eb}+\frac{4{\mathrm{\β}}_i^3{l}_{2M}^3}{Eb};$

$G_{x-DAT}=\frac{1}{\underset{j=1}{\overset{n}{\Σ}}\frac{1}{G_{x-DAj}}};$ $G_{x-DAj}=\frac{4}{Eb}{(L_A-l_3/2)}^3-l_{2A}^3\]+\frac{6l_{2A}^3{({\mathrm{\β}}_{Aj}+1)}^2\[3({\mathrm{\β}}_{Aj}-1)-2{\mathrm{ln\β}}_{Aj}(1+{\mathrm{\β}}_{Aj})\]}{Eb}+\frac{4{\mathrm{\β}}_{Aj}^3{l}_{2A}^3}{Eb};$

$\begin{array}{c}{G}_{x-CD}=\frac{4{(L_M-l_3/2)}^3+22l_{2M}^3({\mathrm{\β}}_m^3-1)+6l_{2M}^3\[3{\mathrm{\β}}_m({\mathrm{\β}}_m-1)-2(1+{\mathrm{\β}}_m^3){\mathrm{ln\β}}_m-6{\mathrm{\β}}_m(1+{\mathrm{\β}}_m){\mathrm{ln\β}}_m\]}{Eb}+\\ \frac{2\{l_0^3+3{\mathrm{\β}}_m^2\[l_{2M}^2{\mathrm{\β}}_m^2-{(L_M-l_3/2)}^2{\mathrm{\β}}_m-l_{2M}^2\]l_0\}}{{\mathrm{Eb\β}}_m^3};\end{array}$

$\begin{array}{c}{G}_{x-D_{z}m}=\frac{2l_3\[6{(L_M-l_3/2)}^2-6(L_M-l_3/2)l_3-6(L_M-l_3/2)l_0+2l_3^2+3l_0{l}_3\]}{Eb}+\\ \frac{2{\[l_0-(L_M-l_3/2){\mathrm{\β}}_m^2+l_3{\mathrm{\β}}_m^2\]}^2\[l_0+2(L_M-l_3/2){\mathrm{\β}}_m^2-2l_3{\mathrm{\β}}_m^2\]}{{\mathrm{Eb\β}}_m^3}+\frac{6l_0{(L_M-3l_3/2)}^3({\mathrm{\β}}_m-1){({\mathrm{\β}}_m+1)}^2}{{\mathrm{Eb\β}}_m}+\\ \frac{6{(L_M-3l_3/2)}^3{({\mathrm{\β}}_m+1)}^2(3{\mathrm{\β}}_m-2{\mathrm{ln\β}}_m-2{\mathrm{\β}}_m{\mathrm{ln\β}}_m-3)}{Eb};\end{array}$

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_z}=-\frac{2\[6{(L_M-l_3/2)}^2{l}_0-6(L_M-l_3/2)l_0^2+9l_0^2{l}_{2M}+6l_{2M}^3{\mathrm{ln\β}}_m-2{(L_M-l_3/2)}^3+11l_{2M}^3\]}{Eb}+\\ \frac{2{\mathrm{\β}}_m^3{l}_{2M}^3(11-6{\mathrm{ln\β}}_m)}{Eb}-\frac{2{\mathrm{\β}}_m(3l_0^2{l}_{2M}+9l_{2M}^3+18l_{2M}^3{\mathrm{ln\β}}_m)}{Eb}-\frac{4l_0^3+2{\mathrm{\β}}_m^2(6l_{2M}^2{l}_0-9l_0^2{l}_{2M})-6l_0^2{l}_{2M}{\mathrm{\β}}_m}{{\mathrm{Eb\β}}_m^3}+\\ \frac{2{\mathrm{\β}}_m^2(6l_{2M}^2{l}_0+9l_{2M}^3-18l_{2M}^3{\mathrm{ln\β}}_m)}{Eb},\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_AjCalculate:

Half length L according to few sheet bias type variable cross-section auxiliary spring_A, the thickness h of the root flat segments of each auxiliary spring_2A, width b, elastic modulus E, the half l of installing space₃, the root of auxiliary spring oblique line section is to the distance l of auxiliary spring end points_2A=L_A-l₃, the thickness of the oblique line section of jth sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, n is auxiliary spring sheet number, and the half of each bias type variable cross-section auxiliary spring is clamped stiffness K_AjIt is calculated, namely

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

In formula,

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

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

The half of maximum load and single-ended some maximum load P suffered by few sheet bias type variable cross-section major-minor spring_max, main reed number m, auxiliary spring works load p_K, calculated K in I step_Mi, and II step calculates obtained K_MAi, maximum end points power P to each main spring of bias type variable cross-section_imaxIt is calculated, namely

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

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

The half of maximum load and single-ended some maximum load P suffered by few sheet bias type variable cross-section major-minor spring_max, auxiliary spring works load p_K；Main reed number m, the thickness h of the root flat segments of each main spring_2M；The sheet number n of auxiliary spring, the thickness h of the root flat segments of each auxiliary spring_2A, calculated K in II step_MAi、G_x-CD、G_x-CDzAnd G_x-DAT, and calculated K in III step_Aj, maximum end points power P to each bias type variable cross-section auxiliary spring_AjmaxIt is calculated, namely

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

(3) each main spring of the few sheet bias type variable cross-section major-minor spring of 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 main spring of bias type variable cross-section_M, main reed number m, the thickness h of the root flat segments of each main spring_2M, width b, the half l of installing space₃, calculated P in i step_imax, the maximum stress of the front main spring of m-1 sheet is calculated, namely

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

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

The root of the oblique line section according to few sheet main spring of bias type variable cross-section is to the distance l of main spring end points_2M, the thickness h of the root flat segments of each main spring_2M, width b, the thickness of the oblique line section of the main spring of m sheet compares β_m；The horizontal range l of auxiliary spring sheet number n, auxiliary spring contact and main spring end points₀, calculated P in i step_mmax, calculated P in ii step_Ajmax, the maximum stress of the main spring of m sheet is calculated, namely

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

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

Half length L according to few sheet bias type variable cross-section auxiliary spring_A, auxiliary spring sheet number n, the thickness h of the root flat segments of each auxiliary spring_2A, width b, the half l of installing space₃, calculated P in ii step_Ajmax, the maximum stress of each auxiliary spring is calculated, namely

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

(4) each main spring of the few sheet bias type 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 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, check the stress intensity of each of the front main spring of m-1 sheet of the few sheet bias type variable cross-section major-minor spring of end contact, 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, meet 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, check the stress intensity of the main spring of m sheet of the few sheet bias type variable cross-section major-minor spring of end contact, 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, check the stress intensity of each auxiliary spring of the few sheet bias type variable cross-section major-minor spring of end contact, 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, 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 bias type 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, simultaneously, the main spring of m sheet is except by end points power, the effect of auxiliary spring contact support power also it is subject in end flat segments, therefore, the end points power of each main spring of bias type variable cross-section and auxiliary spring calculates extremely complex, previously fails to provide the check method of the few sheet bias type variable cross-section each stress intensity of major-minor spring of ends contact formula always.The present invention can work the maximum load that load, major-minor spring bear according to each main spring of few sheet bias type variable cross-section major-minor spring and the structural parameters of auxiliary spring, elastic modelling quantity, allowable stress, auxiliary spring, and the few each main spring of sheet bias type variable cross-section major-minor spring of end contact and the stress intensity of each auxiliary spring are carried out calculation and check.By checking example and ANSYS simulating, verifying it can be seen that the strength check methods that the ends contact formula that this invention provides lacks sheet bias type variable cross-section major-minor spring is correct, the maximum stress calculation and check value of each main spring and auxiliary spring is accurately and reliably.Utilize the method can improve the few design level of sheet bias type variable cross-section major-minor leaf spring of ends contact formula, product quality and service 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 accompanying drawing.

Fig. 1 is the flow chart of each stress intensity check of the few sheet bias type variable cross-section major-minor spring of ends contact formula；

Fig. 2 is the half symmetrical structure schematic diagram of the few sheet bias type variable cross-section major-minor spring of 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

As shown in Figure 1, the check method step of the present invention is as follows: for the few sheet bias type variable cross-section major-minor spring of ends contact formula, first each main spring of the few sheet bias type variable cross-section major-minor spring of end contact and the half clamping rigidity of auxiliary spring are calculated, secondly, again the few each main spring of sheet bias type variable cross-section major-minor spring of end contact and the maximum end points power of auxiliary spring are calculated, again, the few each main spring of sheet bias type variable cross-section major-minor spring of end contact and the maximum stress of auxiliary spring are calculated, finally, the few each main spring of sheet bias type variable cross-section major-minor spring of end contact and the stress intensity of auxiliary spring are checked.Above-mentioned contact point mean that under state as shown in Figure 2, the contact point formed when the end of auxiliary spring contacts with the lower surface of main spring, in actual contact process, the arris of auxiliary spring end contacts with the surface of main spring, in the method for designing process of the present invention, it is regarded as point cantact and carries out Rigidity Calculation.As in figure 2 it is shown, the half symmetrical structure schematic diagram of the few sheet bias type variable cross-section major-minor spring of ends contact formula, its main spring 1, root shim 2, each bias type variable-section steel sheet spring of auxiliary spring 3, end pad 4, main spring 1 and auxiliary spring 3 is to be made up of root flat segments, oblique line section, end flat segments three sections；It is provided with root shim 2 between each root flat segments of main spring 1 and between each root flat segments of auxiliary spring 3, it is provided with end pad 4 between each end flat segments of main spring 1, the material of end pad 4 is carbon fibre composite, produces frictional noise during to prevent work.Wherein, the thickness of the root flat segments of each main spring is h_2M, width is b, and half length is L_M, the half l of installing space₃, the root of main spring oblique line section is l to the distance of main spring end points_2M；The end flat segments of each main spring is non-waits structure, the thickness of the end flat segments of the 1st main spring and length, more than the thickness of end flat segments and length, the thickness of the end flat segments of each main spring and the length respectively h of other each main spring_1iAnd l_1i, the thickness of the oblique line section of each main spring is than for β_i=h_1i/h_2M, i=1,2 ..., m, m is main reed number.The width of each auxiliary spring is b, and half length is L_A, the half l of installing space₃, the thickness of the root flat segments of each auxiliary spring is h_2A, the thickness of end flat segments and length respectively h_A1jAnd l_A1j, the thickness of the oblique line section of each auxiliary spring is than for β_Aj=h_A1j/h_2A, j=1,2 ..., n, n is the sheet number of auxiliary spring.The half length L of auxiliary spring_AHalf length L less than main spring_M, level between auxiliary spring contact and main spring end points is from for l₀=L_M-L_A, between auxiliary spring contact and main spring end flat segments, it is provided with certain major-minor spring gap delta.

By the examples below the present invention is described in further detail.

Embodiment: the main reed number m=2 of the few sheet bias type variable cross-section major-minor spring of certain ends contact formula, wherein, the half length L of each main spring_M=575mm, width b=60mm, elastic modulus E=200GPa, the half l of installing space₃=55mm, the thickness h of the root flat segments of each main spring_2M=11mm, the root of oblique line section is to the distance l of main spring end points_2M=L_M-l₃=520mm；The thickness of the end flat segments of the 1st main spring is h₁₁=7mm, the thickness of the oblique line section of the 1st main spring is than for β₁=h₁₁/h_2M=0.64；The thickness of the end flat segments of the 2nd main spring is h₁₂=6mm, the thickness of the oblique line section of the 2nd main spring compares β₂=h₁₂/h_2M=0.55.The sheet number n=1 of auxiliary spring, the half length L of this sheet auxiliary spring_AThe horizontal range l of=525mm, auxiliary spring contact and main spring end points₀=L-L_A=50mm, the root of the oblique line section of auxiliary spring is to the distance l of auxiliary spring end points_2A=L_A-l₃=470mm；The thickness h of the root flat segments of auxiliary spring_2A=14mm, the thickness h of end flat segments_A11=8.0mm, the thickness of the oblique line section of this sheet auxiliary spring compares β_A1=h_A11/h_2A=0.57；When load works load more than auxiliary spring, auxiliary spring contact contacts with certain point in the flat segments of main spring end.The half of maximum load suffered by this few sheet main spring of bias type variable-section steel sheet spring and single-ended some maximum load P_max=3040N, auxiliary spring works load p_K=2470N, allowable stress [the σ]=700MPa of leaf spring, check this few each main spring of sheet bias type variable cross-section major-minor spring and the stress intensity of auxiliary spring.

The check method of the few sheet bias type each intensity of major-minor spring of the ends contact formula that present example provides, 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 bias type variable cross-section major-minor spring of ends contact formula and the half of auxiliary spring clamp Rigidity Calculation:

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 bias type variable cross-section_M=575mm, the thickness h of the root flat segments of each main spring_2M=11mm, width b=60mm, elastic modulus E=200GPa, the half l of installing space₃=55mm, the root of oblique line section is to the distance l of main spring end points_2M=520mm；The thickness h of the end flat segments of the 1st main spring₁₁=7mm, the thickness of the oblique line section of the 1st main spring compares β₁=0.64；The thickness h of the end flat segments of the 2nd main spring₁₂=6mm, the thickness of the oblique line section of the 2nd main spring compares β₂=0.55, the 1st main spring and the half of the 2nd main spring before major-minor spring is contacted step up stiffness K_M1And K_M2It is respectively calculated, namely

$K_{M1}=\frac{h_{2M}^3}{G_{x-D1}}=14.31N/mm;$

$K_{M2}=\frac{h_{2M}^3}{G_{x-D2}}=13.17N/mm;$

In formula,

$G_{x-D1}=\frac{4}{Eb}\[{(L_M-l_3/2)}^3-l_{2M}^3\]+\frac{6l_{2M}^3{({\mathrm{\β}}_1+1)}^2\[3({\mathrm{\β}}_1-1)-2{\mathrm{ln\β}}_1(1+{\mathrm{\β}}_1)\]}{Eb}+\frac{4{\mathrm{\β}}_1^3{l}_{2M}^3}{Eb}=93.02{\mathrm{mm}}^4/N,$

$G_{x-D2}=\frac{4}{Eb}\[{(L_M-l_3/2)}^3-l_{2M}^3\]+\frac{6l_{2M}^3{({\mathrm{\β}}_2+1)}^2\[3({\mathrm{\β}}_2-1)-2{\mathrm{ln\β}}_2(1+{\mathrm{\β}}_2)\]}{Eb}+\frac{4{\mathrm{\β}}_2^3{l}_{2M}^3}{Eb}=101.06{\mathrm{mm}}^4/N;$

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 bias type variable cross-section_M=575mm, the thickness h of the root flat segments of each main spring_2M=11mm, width b=60mm, elastic modulus E=200GPa, the half l of installing space₃=55mm, the root of oblique line section is to the distance l of main spring end points_2M=L_M-l₃=520mm；The thickness of the oblique line section of the 1st main spring compares β₁The thickness of the oblique line section of the=0.64, the 2nd main spring compares β₂=0.55.The sheet number n=1 of auxiliary spring, the half length L of this sheet auxiliary spring_AThe horizontal range l of=525mm, auxiliary spring contact and main spring end points₀=50mm, the thickness h of the root flat segments of auxiliary spring_2A=14mm, the root of auxiliary spring oblique line section is to the distance l of auxiliary spring end points_2A=470mm, the thickness of auxiliary spring oblique line section compares β_A1=0.57, the 1st main spring and the half of the 2nd main spring after major-minor spring is contacted clamp stiffness K_MA1And K_MA2It is respectively calculated, namely

$K_{MA1}=\frac{h_{2M}^3}{G_{x-D1}}=14.31N/mm;$

$K_{MA2}=\frac{h_{2M}^3(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}{G_{x-D2}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)-G_{x-D_{z}2}{G}_{x-CD}{h}_{2A}^3}=37.71N/mm;$

In formula,

$G_{x-D1}=\frac{4}{Eb}\[{(L_M-l_3/2)}^3-l_{2M}^3\]+\frac{6l_{2M}^3{({\mathrm{\β}}_1+1)}^2\[3({\mathrm{\β}}_1-1)-2{\mathrm{ln\β}}_1(1+{\mathrm{\β}}_1)\]}{Eb}+\frac{4{\mathrm{\β}}_1^3{l}_{2M}^3}{Eb}=93.02{\mathrm{mm}}^4/N;$

$G_{x-D2}=\frac{4}{Eb}\[{(L_M-l_3/2)}^3-l_{2M}^3\]+\frac{6l_{2M}^3{({\mathrm{\β}}_2+1)}^2\[3({\mathrm{\β}}_2-1)-2{\mathrm{ln\β}}_2(1+{\mathrm{\β}}_2)\]}{Eb}+\frac{4{\mathrm{\β}}_2^3{l}_{2M}^3}{Eb}=101.06{\mathrm{mm}}^4/N;$

$G_{x-DAT}=\frac{1}{\underset{j=1}{\overset{1}{\Σ}}\frac{1}{G_{x-DAj}}}=73.54{\mathrm{mm}}^4/N;$

$G_{x-DA1}=\frac{4}{Eb}\[{(L_A-l_3/2)}^3-l_{2A}^3\]+\frac{6l_{2A}^3{({\mathrm{\β}}_{A1}+1)}^2\[3({\mathrm{\β}}_{A1}-1)-2{\mathrm{ln\β}}_{A1}(1+{\mathrm{\β}}_{A1})\]}{Eb}+\frac{4{\mathrm{\β}}_{A1}^3{l}_{2A}^3}{Eb}=73.54{\mathrm{mm}}^4/N;$

$G_{x-CD}=\frac{4{(L_M-l_3/2)}^3+22l_{2M}^3({\mathrm{\β}}_2^3-1)+6l_{2M}^3\[3{\mathrm{\β}}_2({\mathrm{\β}}_2-1)-2(1+{\mathrm{\β}}_2^3){\mathrm{ln\β}}_2-6{\mathrm{\β}}_2(1+{\mathrm{\β}}_2){\mathrm{ln\β}}_2\]}{Eb}+$

$\frac{2\{l_0^3+3{\mathrm{\β}}_2^2\[l_{2M}^2{\mathrm{\β}}_2^2-{(L_M-l_3/2)}^2{\mathrm{\β}}_2-l_{2M}^2\]l_0\}}{{\mathrm{Eb\β}}_2^3}=83.31{\mathrm{mm}}^4/N;$

$\begin{array}{c}{G}_{x-D_{z}2}=\frac{2l_3\[6{(L_M-l_3/2)}^2-6(L_M-l_3/2)l_3-6(L_M-l_3/2)l_0+2l_3^2+3l_0{l}_3\]}{Eb}+\\ \frac{2{\[l_0-(L_M-l_3/2){\mathrm{\β}}_2^2+l_3{\mathrm{\β}}_2^2\]}^2\[l_0+2(L_M-l_3/2){\mathrm{\β}}_2^2-2l_3{\mathrm{\β}}_2^2\]}{{\mathrm{Eb\β}}_2^3}+\frac{6l_0{(L_M-3l_3/2)}^3({\mathrm{\β}}_2-1){({\mathrm{\β}}_2+1)}^2}{{\mathrm{Eb\β}}_2}+\\ \frac{6{(L_M-3l_3/2)}^3{({\mathrm{\β}}_2+1)}^2(3{\mathrm{\β}}_2-2{\mathrm{ln\β}}_2-2{\mathrm{\β}}_2{\mathrm{ln\β}}_2-3)}{Eb}=83.31{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_z}=-\frac{2\[6{(L_M-l_3/2)}^2{l}_0-6(L_M-l_3/2)l_0^2+9l_0^2{l}_{2M}+6l_{2M}^3{\mathrm{ln\β}}_2-2{(L_M-l_3/2)}^3+11l_{2M}^3\]}{Eb}+\\ \frac{2{\mathrm{\β}}_2^3{l}_{2M}^3(11-6{\mathrm{ln\β}}_2)}{Eb}-\frac{2{\mathrm{\β}}_2(3l_0^2{l}_{2M}+9l_{2M}^3+18l_{2M}^3{\mathrm{ln\β}}_2)}{Eb}-\frac{4l_0^3+2{\mathrm{\β}}_2^2(6l_{2M}^2{l}_0-9l_0^2{l}_{2M})-6l_0^2{l}_{2M}{\mathrm{\β}}_2}{{\mathrm{Eb\β}}_2^3}+\\ \frac{2{\mathrm{\β}}_2^2(6l_{2M}^2{l}_0+9l_{2M}^3-18l_{2M}^3{\mathrm{ln\β}}_2)}{Eb}=69.87{\mathrm{mm}}^4/N;\end{array}$

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

Half length L according to bias type variable cross-section auxiliary spring_A=525mm, the sheet number n=1 of auxiliary spring, the thickness h of the root flat segments of this sheet auxiliary spring_2A=14mm, width b=60mm, elastic modulus E=200GPa, the half l of installing space₃=55mm, the root of auxiliary spring oblique line section is to the distance l of auxiliary spring end points_2A=470mm, the thickness of oblique line section compares β_A1=0.57, the half of this sheet bias type variable cross-section auxiliary spring is clamped stiffness K_A1It is calculated, namely

$K_{A1}=\frac{h_{2A}^3}{G_{x-DA1}}=37.31N/mm;$

In formula,

$G_{x-DA1}=\frac{4}{Eb}\[{(L_A-l_3/2)}^3-l_{2A}^3\]+\frac{6l_{2A}^3{({\mathrm{\β}}_{A1}+1)}^2\[3({\mathrm{\β}}_{A1}-1)-2{\mathrm{ln\β}}_{A1}(1+{\mathrm{\β}}_{A1})\]}{Eb}+\frac{4{\mathrm{\β}}_{A1}^3{l}_{2A}^3}{Eb}=73.54{\mathrm{mm}}^4/N;$

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

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

The half of maximum load and single-ended some maximum load P suffered by few sheet bias type variable cross-section major-minor spring_max=3040N, main reed number m=2, auxiliary spring works load p_KCalculated K in=2470N, I step_M1=14.31N/mm and K_M2=13.17N/mm, and II step calculates obtained K_MA1=14.31N/mm and K_MA2=37.71N/mm, the maximum end points power P to the 1st main spring and the 2nd main spring_1maxAnd P_2maxIt is respectively calculated, namely

$P_{1max}=\frac{K_{M1}{P}_K}{2\underset{i=1}{\overset{2}{\Σ}}{K}_{Mi}}+\frac{K_{MA1}(2P_{max}-P_K)}{2\underset{i=1}{\overset{2}{\Σ}}{K}_{MAi}}=1139.50N;$

$P_{2max}=\frac{K_{M2}{P}_K}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{Mi}}+\frac{K_{MA2}(2P_{max}-P_K)}{2\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}}=1900.50N;$

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

The half of maximum load and single-ended some maximum load P suffered by few sheet bias type variable cross-section major-minor spring_max=3040N, main reed number m=2, auxiliary spring works load p_K=2470N, the thickness h of the root flat segments of each main spring_2M=11mm, auxiliary spring sheet number n=1, the thickness h of the root flat segments of auxiliary spring_2A=14mm, II step calculates obtained K_MA1=14.31N/mm, K_MA2=37.71N/mm, G_x-CD=83.31mm⁴/N、G_x-CDz=69.87mm⁴/ N and G_x-DAT=73.54mm⁴/ N, and calculated K in III step_A1=37.31N/mm, the maximum end points power P to this sheet auxiliary spring_A1maxIt is calculated, namely

$P_{A1max}=\frac{K_{A1}{K}_{MA2}{G}_{x-CD}{h}_{2A}^3(2P_{\max }-P_K)}{2\underset{j=1}{\overset{n}{\Σ}}{K}_{Aj}\underset{i=1}{\overset{m}{\Σ}}{K}_{MAi}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}=1033.00N;$

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

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

Half length L according to few sheet main spring of bias type variable cross-section_M=575mm, the thickness h of the root flat segments of each main spring_2M=11mm, width b=60mm, the half l of installing space₃=55mm, i step calculates obtained P_1max=1139.50N, is calculated the maximum stress of the 1st main spring of bias type variable cross-section, namely

${\mathrm{\σ}}_{1max}=\frac{6P_{1max}(L_M-l_3/2)}{{\mathrm{bh}}_{2M}^2}=515.62MPa;$

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

The root of the oblique line section according to few sheet main spring of bias type variable cross-section is to the distance l of main spring end points_2M=520mm, the root thickness h of each main spring_2MThe thickness of the oblique line section of=11mm, width b=60mm, a 2nd main spring compares β₂=0.5；The horizontal range l of auxiliary spring sheet number n=1, this sheet auxiliary spring contact and main spring end points₀=50mm, i step calculates obtained P_2max=1900.50N, ii step calculates obtained P_A1max=1033.00N, is calculated the maximum stress of the 2nd main spring, namely

${\mathrm{\σ}}_{2max}=\frac{6\[P_{2max}{\mathrm{\β}}_2^2{l}_{2M}-\underset{j=1}{\overset{n}{\Σ}}{P}_{Ajmax}({\mathrm{\β}}_2^2{l}_{2M}-l_0)\]}{b{\left({\mathrm{\β}}_2{h}_{2M}\right)}^2}=516.26MPa;$

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

Half length L according to few sheet bias type variable cross-section auxiliary spring_A=525mm, the thickness h of the root flat segments of this sheet auxiliary spring_2A=14mm, width b=60mm, the half l of installing space₃=55mm, ii step calculates obtained P_A1max=1033.00N, is calculated the maximum stress of this sheet auxiliary spring, namely

${\mathrm{\σ}}_{A1max}=\frac{6P_{A1max}(L_A-l_3/2)}{{\mathrm{bh}}_{2A}^2}=262.21MPa;$

(4) each main spring of the few sheet bias type 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 the 1st main spring is checked:

Allowable stress [σ]=700MPa according to leaf spring, and the maximum stress σ of calculated 1st main spring in step A_1max=515.62MPa, it is known that σ_1max≤ [σ], namely the 1st main spring disclosure satisfy that stress intensity requirement；

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

Allowable stress [σ]=700MPa according to leaf spring, and the maximum stress σ of calculated 2nd main spring in step B_2max=516.26MPa, it is known that σ_2max≤ [σ], namely the 2nd main spring disclosure satisfy that stress intensity requirement；

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

Allowable stress [σ]=700MPa according to leaf spring, and the maximum stress σ of this sheet auxiliary spring calculated in step C_A1max=262.21MPa, it is known that σ_A1max≤ [σ], namely this sheet auxiliary spring disclosure satisfy that stress intensity requirement.

Utilize ANSYS finite element emulation software, major-minor spring structure parameter according to this few sheet bias type variable-section steel sheet spring and material characteristic parameter, set up the ANSYS phantom of half symmetrical structure major-minor spring, grid division, arrange auxiliary spring end points to contact with main spring, and at the root applying fixed constraint of phantom, apply concentrfated load F=P at main spring end points_max-P_K/ 2=1805N, carries out ANSYS emulation to the stress of this few sheet bias type variable-section steel sheet spring major-minor spring in the clamp state, 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 the 1st auxiliary spring, as it is shown in figure 5, wherein, the 1st main spring is at the maximum stress σ clamping root_1max=214.26MPa, the 2nd main spring is at the maximum stress σ of oblique line section Yu flat segments contact position place, end_2max=262.59MPa, the 1st auxiliary spring is at the maximum stress σ clamping root_A1max=248.72MPa.

It can be seen that in same load situation, the ANSYS simulating, verifying value σ of the 1st and the 2nd main spring of this leaf spring and the 1st auxiliary spring maximum stress_1max=214.26MPa, σ_2max=262.59MPa and σ_A1max=248.72MPa, respectively with analytical Calculation value σ_1max=213.42MPa, σ_2max=261.95MPa and σ_A1max=247.71MPa, matches, relative deviation respectively 0.39%, 0.24%, 0.41%；Result shows that the check method of the few sheet bias type each intensity of major-minor spring of ends contact formula that this invention provides is correct, and the stress intensity check value of each main spring and auxiliary spring is accurately and reliably.

Claims (1)

1. the check method of the few sheet bias type each intensity of major-minor spring of ends contact formula, wherein, symmetry one half structure of few sheet bias type variable-section steel sheet spring is made up of root flat segments, oblique line section, end flat segments three sections；The end flat segments of each main spring is non-waits structure, i.e. the thickness of the end flat segments of the 1st main spring and length, more than thickness and the length of other each main spring；The length of auxiliary spring is less than the length of main spring, and when load works load more than auxiliary spring, auxiliary spring contact contacts with certain point in the flat segments of main spring end, and namely major-minor spring is ends contact formula；When load works load more than auxiliary spring, after the contact of major-minor spring, the end points power of each major-minor spring differs, and the 1 main spring contacted with auxiliary spring is except by end points power, also in the effect by auxiliary spring contact support power of the end flat segments；Work load under stable condition at the few each chip architecture parameter of sheet bias type major-minor spring of ends contact formula, elastic modelling quantity, allowable stress, maximum load, auxiliary spring, the few each main spring of sheet bias type major-minor spring of end contact and the stress intensity of auxiliary spring are checked, and concrete check step is as follows:

(1) each main spring of the few sheet bias type variable cross-section major-minor spring of ends contact formula and the half of auxiliary spring clamp Rigidity Calculation:

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 bias type variable cross-section_M, the thickness h of the root flat segments of each main spring_2M, width b, elastic modulus E, the half l of installing space₃, the root of oblique line section is to the distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of i-th main spring compares β_i=h_1i/h_2M, wherein, i=1,2 ..., m, m is main reed number, the half clamping stiffness K of each main spring of bias type variable cross-section before major-minor spring is contacted_MiIt is calculated, namely

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

In formula,

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 bias type variable cross-section_M, the thickness h of the root flat segments of each main spring_2M, width b, elastic modulus E, the half l of installing space₃, the root of oblique line section is to the distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of i-th main spring compares β_i, wherein, i=1,2 ..., m, m is main reed number；The half length L of auxiliary spring_A, the thickness h of the root flat segments of each auxiliary spring_2A, the root of auxiliary spring oblique line section is to the distance l of auxiliary spring end points_2A=L_A-l₃, the horizontal range l of auxiliary spring contact and main spring end points₀, the thickness of the oblique line section of jth sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, n is auxiliary spring sheet number, the half clamping stiffness K of each main spring of bias type variable cross-section after major-minor spring is contacted_MAiIt is calculated, namely

$K_{MAi}=\left\{\begin{array}{cc}\frac{h_{2M}^3}{G_{x-Di}},& i=1,2,...,m-1\\ \frac{h_{2M}^3(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}{G_{x-Dm}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)-G_{x-D_{z}m}{G}_{x-CD}{h}_{2A}^3},& i=m\end{array}\right.;$

In formula,

$G_{x-Di}=\frac{4}{Eb}\[{(L_M-l_3/2)}^3-l_{2M}^3\]+\frac{6l_{2M}^3{({\mathrm{\β}}_i+1)}^2\[3({\mathrm{\β}}_i-1)-2{\mathrm{ln\β}}_i(1+{\mathrm{\β}}_i)\]}{Eb}+\frac{4{\mathrm{\β}}_i^3{l}_{2M}^3}{Eb};$

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

$G_{x-DAj}=\frac{4}{Eb}\[{(L_A-l_3/2)}^3-l_{2A}^3\]+\frac{6l_{2A}^3{({\mathrm{\β}}_{Aj}+1)}^2\[3({\mathrm{\β}}_{Aj}-1)-2{\mathrm{ln\β}}_{Aj}(1+{\mathrm{\β}}_{Aj})\]}{Eb}+\frac{4{\mathrm{\β}}_{Aj}^3{l}_{2A}^3}{Eb};$

$\begin{array}{c}{G}_{x-CD}=\frac{4{(L_M-l_3/2)}^3+22l_{2M}^3({\mathrm{\β}}_m^3-1)+6l_{2M}^3\[3{\mathrm{\β}}_m({\mathrm{\β}}_m-1)-2(1+{\mathrm{\β}}_m^3){\mathrm{ln\β}}_m-6{\mathrm{\β}}_m(1+{\mathrm{\β}}_m){\mathrm{ln\β}}_m\]}{Eb}+\\ \frac{2\{l_0^3+3{\mathrm{\β}}_m^2\[l_{2M}^2{\mathrm{\β}}_m^2-{(L_M-l_3/2)}^2{\mathrm{\β}}_m-l_{2M}^2\]l_0\}}{{\mathrm{Eb\β}}_m^3};\end{array}$

$\begin{array}{c}{G}_{x-D_{z}m}=\frac{2l_3\[6{(L_M-l_3/2)}^2-6(L_M-l_3/2)l_3-6(L_M-l_3/2)l_0+2l_3^2+3l_0{l}_3\]}{Eb}+\\ \frac{2{\[l_0-(L_M-l_3/2){\mathrm{\β}}_m^2+l_3{\mathrm{\β}}_m^2\]}^2\[l_0+2(L_M-l_3/2){\mathrm{\β}}_m^2-2l_3{\mathrm{\β}}_m^2\]}{{\mathrm{Eb\β}}_m^3}+\frac{6l_0{(L_M-3l_3/2)}^3({\mathrm{\β}}_m-1){({\mathrm{\β}}_m+1)}^2}{{\mathrm{Eb\β}}_m}+\\ \frac{6{(L_M-3l_3/2)}^3{({\mathrm{\β}}_m+1)}^2(3{\mathrm{\β}}_m-2{\mathrm{ln\β}}_m-2{\mathrm{\β}}_m{\mathrm{ln\β}}_m-3)}{Eb};\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_AjCalculate:

Half length L according to few sheet bias type variable cross-section auxiliary spring_A, the thickness h of the root flat segments of each auxiliary spring_2A, width b, elastic modulus E, the half l of installing space₃, the root of auxiliary spring oblique line section is to the distance l of auxiliary spring end points_2A=L_A-l₃, the thickness of the oblique line section of jth sheet auxiliary spring compares β_Aj, wherein, j=1,2 ..., n, n is auxiliary spring sheet number, and the half of each bias type variable cross-section auxiliary spring is clamped stiffness K_AjIt is calculated, namely

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

In formula,

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

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

The half of maximum load and single-ended some maximum load P suffered by few sheet bias type variable cross-section major-minor spring_max, main reed number m, auxiliary spring works load p_K, calculated K in I step_Mi, and II step calculates obtained K_MAi, maximum end points power P to each main spring of bias type variable cross-section_imaxIt is calculated, namely

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

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

The half of maximum load and single-ended some maximum load P suffered by few sheet bias type variable cross-section major-minor spring_max, auxiliary spring works load p_K；Main reed number m, the thickness h of the root flat segments of each main spring_2M；The sheet number n of auxiliary spring, the thickness h of the root flat segments of each auxiliary spring_2A, calculated K in II step_MAi、G_x-CD、G_x-CDzAnd G_x-DAT, and calculated K in III step_Aj, maximum end points power P to each bias type variable cross-section auxiliary spring_AjmaxIt is calculated, namely

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

(3) each main spring of the few sheet bias type variable cross-section major-minor spring of 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 main spring of bias type variable cross-section_M, main reed number m, the thickness h of the root flat segments of each main spring_2M, width b, the half l of installing space₃, calculated P in i step_imax, the maximum stress of the front main spring of m-1 sheet is calculated, namely

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

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

The root of the oblique line section according to few sheet main spring of bias type variable cross-section is to the distance l of main spring end points_2M, the thickness h of the root flat segments of each main spring_2M, width b, the thickness of the oblique line section of the main spring of m sheet compares β_m；The horizontal range l of auxiliary spring sheet number n, auxiliary spring contact and main spring end points₀, calculated P in i step_mmax, calculated P in ii step_Ajmax, the maximum stress of the main spring of m sheet is calculated, namely

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

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

Half length L according to few sheet bias type variable cross-section auxiliary spring_A, auxiliary spring sheet number n, the thickness h of the root flat segments of each auxiliary spring_2A, width b, the half l of installing space₃, calculated P in ii step_Ajmax, the maximum stress of each auxiliary spring is calculated, namely

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

(4) each main spring of the few sheet bias type 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 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, check the stress intensity of each of the front main spring of m-1 sheet of the few sheet bias type variable cross-section major-minor spring of end contact, 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, meet 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, check the stress intensity of the main spring of m sheet of the few sheet bias type variable cross-section major-minor spring of end contact, 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, check the stress intensity of each auxiliary spring of the few sheet bias type variable cross-section major-minor spring of end contact, 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, meet stress intensity requirement, j=1, and 2 ..., n.