CN105956255A - Method for checking composite stiffness of non-end contact type taper leaf root enhanced main and auxiliary leaf springs - Google Patents

Abstract

The invention relates to a method for checking the composite stiffness of non-end contact type taper leaf root enhanced main and auxiliary leaf springs, and belongs to the technical field of suspension leaf springs. The composite stiffness of the main and auxiliary leaf springs can be checked according to the structural parameter and elastic modulus of each main leaf spring and auxiliary leaf spring of the non-end contact type taper leaf root enhanced variable cross-section main and auxiliary leaf springs. An example and ANSYS simulation verification show that, the method for checking the composite stiffness of the non-end contact type taper leaf root enhanced variable cross-section main and auxiliary leaf springs provided by the invention is correct, and the method can obtain an accurate and reliable composite stiffness checking value of the main and auxiliary leaf springs, so that the design level, the product quality and performance and the vehicle ride comfort of the non-end contact type taper leaf root enhanced variable cross-section main and auxiliary leaf springs can be improved; and meanwhile, the design and testing expenses can be reduced, and the product development speed can be accelerated.

Description

The Method for Checking of the few sheet root reinforced major-minor spring complex stiffness of non-ends contact formula

Technical field

The present invention relates to vehicle suspension leaf spring, be the few sheet root reinforced major-minor spring complex stiffness of non-ends contact formula especially Method for Checking.

Background technology

In order to meet vehicle suspension variation rigidity design requirement under different loads, generally few sheet variable-section steel sheet spring is designed as Major-minor spring, wherein, main spring is designed with certain gap at the contact that connects with auxiliary spring, it is ensured that when load works load more than auxiliary spring After lotus, major-minor spring contacts and cooperatively works.Owing to the stress of the 1st main spring of variable cross-section is complicated, it is subjected to vertical load Lotus, simultaneously also subject to torsional load and longitudinal loading, therefore, the thickness of the end flat segments of the 1st main spring designed by reality With thickness and the length that length is more than his each main spring, the most mostly use the non-few main spring of sheet variable cross-section waiting structure in end；Meanwhile, for Strengthen the stress intensity of few sheet parabolic type variable cross-section major-minor spring, generally between root flat segments and parabolic segment, set up one oblique Line segment, i.e. uses few sheet variable cross-section major-minor spring that root is reinforced.It addition, for the design meeting major-minor spring different composite rigidity Requirement, generally uses the auxiliary spring of different length, and the most main spring is the most different from the contact position of auxiliary spring, and therefore, major-minor spring can be divided into end Portion's contact and non-ends contact formula, wherein, in the case of identical auxiliary spring root flat segments thickness and sheet number, non-ends contact formula The complex stiffness of major-minor spring is less than the complex stiffness of ends contact formula.The complex stiffness of major-minor spring has weight to vehicle ride performance Affect, then, wait owing to main spring end flat segments is non-structure, auxiliary spring length to strengthen with oblique line less than main spring length and root After section, and the contact of major-minor spring, having internal force and Coupling Deformation between main spring and auxiliary spring, therefore, non-ends contact formula root adds The deformation that the few sheet variable-section steel sheet spring of strong type is located at an arbitrary position calculates extremely complex, fails to provide non-ends contact the most always The complex stiffness Method for Checking of the few sheet root reinforced major-minor spring of formula.At present, despite people's once few sheet variable cross-section reinforced to root The deformation of main spring, uses ANSYS modeling and simulating method, but the method can only be to the few sheet variable cross-section providing actual design structure The deformation of leaf spring and complex stiffness carry out simulating, verifying, it is impossible to provide accurate analytical design method formula, solve it is thus impossible to meet Analysis design and the requirement of suspension leaf spring modernization CAD design software development.

Therefore, it is necessary to set up the few sheet root reinforced variable cross-section major-minor spring complex stiffness of a kind of non-ends contact formula accurate, reliable Method for Checking, meet that Vehicle Industry is fast-developing and requirement to few sheet variable cross-section major-minor Precise Design for Laminated Spring, improve and become The design level of section steel flat spring, product quality and performances and vehicle ride performance；Meanwhile, design and test fee are reduced With, accelerate product development speed.

Summary of the invention

For defect present in above-mentioned prior art, the technical problem to be solved is to provide a kind of easy, reliably The Method for Checking of the few sheet root reinforced major-minor spring complex stiffness of non-ends contact formula, it checks flow chart, as shown in Figure 1；Non- The few sheet root reinforced major-minor spring of ends contact formula is symmetrical structure, can regard symmetrical half spring as cantilever beam, will be symmetrical Centrage regards the fixing root of half spring as, regards main spring end stress point and auxiliary spring ends contact stress point as half major and minor The end points of spring, one hemihedrism structural representation, as in figure 2 it is shown, include: main spring 1, root shim 2, auxiliary spring 3, end Each of portion's pad 4, main spring 1 and auxiliary spring 3 is by root flat segments, oblique line section, parabolic segment, four sections of structures of end flat segments Becoming, wherein, leaf spring is played booster action by oblique line section.Between each root of main spring 1 and with auxiliary spring 3 and each root thereof Between be provided with root shim 2, be provided with end pad 4, the material of end pad 4 between each end flat segments of main spring 1 Material for carbon fibre composite, produces frictional noise during to prevent work.Wherein, the few reinforced master of sheet root of non-ends contact formula The width of auxiliary spring is b, half l of installing space₃, a length of Δ l of oblique line section, elastic modelling quantity is E；Main reed number is m, The a length of L of half of main spring_M, the distance of the root of oblique line section to main spring end points is l_2M, the end of oblique line section is to main spring end points Distance is l_2Mp；The thickness of the root flat segments of each main spring is h_2M, the end thickness of oblique line section is h_2Mp, the thickness of oblique line section Degree compares γ_M=h_2Mp/h_2M；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 Degree, more than the thickness of end flat segments and the length of other each main spring；The thickness of the end flat segments of each main spring is h_1i, The thickness of the parabolic segment of each main spring is than for β_i=h_1i/h_2Mp, the length of the end flat segments of each main spring I=1,2 ..., m.The a length of L of half of auxiliary spring_A, auxiliary spring contact is l with the horizontal range of main spring end points₀=L_M-L_A, auxiliary spring oblique line The root of section is l to the distance of auxiliary spring end points_2A, the distance of the end of auxiliary spring oblique line section to auxiliary spring end points is l_2Ap；Auxiliary spring sheet number is N, the thickness of the root flat segments of each auxiliary spring is h_2A, the end thickness of auxiliary spring oblique line section is h_2Ap, the thickness of each oblique line section Compare γ_A=h_2Ap/h_2A；The thickness of the end flat segments of each auxiliary spring is h_A1j, the thickness ratio of the parabolic segment of each auxiliary spring is β_Aj=h_A1j/h_2Ap, the length of the end flat segments of each auxiliary springI=1,2 ..., n.Auxiliary spring contact and main spring parabolic Between line segment, vertical distance is major-minor spring gap delta, when load works load more than auxiliary spring, and auxiliary spring contact and the main spring of m sheet Parabolic segment in certain point contact；After major-minor spring contacts, the end points stress of each main spring and auxiliary spring is unequal, and m The main spring of sheet is in addition to by end points power, also by the end points support force of auxiliary spring；Complex stiffness after the contact of major-minor spring is by major-minor spring The structural parameters of each are determined, and should meet the design required value of complex stiffness.Structure at each main spring and auxiliary spring is joined In the case of number and elastic modelling quantity give, the complex stiffness of sheet root reinforced major-minor spring few to non-ends contact formula checks.

For solving above-mentioned technical problem, the few sheet root reinforced major-minor spring complex stiffness of non-ends contact formula provided by the present invention Method for Checking, it is characterised in that the following step that checks of employing:

(1) the end points deformation coefficient G of each root main spring of reinforced variable cross-section under end points stressing conditions_x-EiCalculate:

According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of main spring_M, the root of main spring parabolic segment is to distance l of main spring end points_2Mp, the root of main spring oblique line section is to main Distance l of spring end points_2M, the thickness of main spring oblique line section compares γ_M, main reed number m, wherein, the parabolic segment of i-th main spring Thickness compare β_i, i=1,2 ..., m, the end points deformation coefficient G to the reinforced main spring of each root under end points stressing conditions_x-Ei Calculate, i.e.

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

(2) the deformation coefficient G at parabolic segment with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions_x-CDCalculate:

According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of main spring_M, the root of main spring parabolic segment is to distance l of main spring end points_2Mp, the root of main spring oblique line section section arrives Distance l of main spring end points_2M, the thickness of main spring oblique line section compares γ_M；Auxiliary spring contact and horizontal range l of main spring end points₀, main reed Number m, to the deformation coefficient G at parabolic segment with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions_x-CDCount Calculate, i.e.

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

(3) the end points deformation coefficient G of the main spring of m sheet under major-minor spring contact point stressing conditions_x-EpmCalculate:

According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of main spring_M, the root of main spring parabolic segment is to distance l of main spring end points_2MpL, the root of main spring oblique line section is to main Distance l of spring end points_2M, the thickness of main spring oblique line section compares γ_M；Auxiliary spring contact and horizontal range l of main spring end points₀, main reed number M, the end points deformation coefficient G to the main spring of m sheet under major-minor spring contact point stressing conditions_x-EpmCalculate, i.e.

$\begin{array}{c}{G}_{x-E_{p}m}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{12}{Eb}\[\frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{{21}_{2M}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^2}-\\ \frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3{\mathrm{\γ}}_M^2}-\frac{2l_{2Mp}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}-\frac{{\mathrm{\Δl}}^3}{{({\mathrm{\γ}}_M-1)}^3}{\mathrm{ln\γ}}_M\]-\frac{24l_0{l}_{2Mp}^2-8{l_{2Mp}}^3-16l_0^{3/2}{l}_{2Mp}^{3/2}}{{\mathrm{Eb\γ}}_M^3}-\\ \frac{6l_0\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{\mathrm{Eb\γ}}_M^2};\end{array}$

(4) the deformation coefficient G at parabolic segment with auxiliary spring contact point of the main spring of m sheet under major-minor spring contact point stressing conditions_x-CDpMeter Calculate:

According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, the one of installing space Half l₃=55mm, elastic modulus E；Half length L of main spring_M, the root of main spring parabolic segment is to the distance of main spring end points l_2Mp, distance l of the root of main spring oblique line section to main spring end points_2M, the thickness of main spring oblique line section compares γ_M；Auxiliary spring contact and main spring Horizontal range l of end points₀, main reed number m, to the main spring of m sheet under major-minor spring contact point stressing conditions in parabolic segment with secondary Deformation coefficient G at spring contact point_x-CDpCalculate, i.e.

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_p}=\frac{4(L_M-l_{2M})(L_M^2-3l_0{L}_M+L_M{l}_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\frac{12}{Eb}\[\frac{l_0^2\mathrm{\Δ}l{({\mathrm{\γ}}_M-1)}^2-3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{2l_{2Mp}\mathrm{\Δ}l(\mathrm{\Δ}l-l_{2Mp}-l_0{\mathrm{\γ}}_M+l_{2M}{\mathrm{\γ}}_M)}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}+\frac{\mathrm{\Δ}l\[2l_0^3\mathrm{\Δ}l({\mathrm{\γ}}_M-1)(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})+3L_M^2-2L_M{l}_0-2L_M{l}_{2M}{\mathrm{\γ}}_M\]}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2L_M{l}_0{\mathrm{\γ}}_M-4L_M{l}_{2M}-6L_M{l}_3-6L_M\mathrm{\Δ}l-l_0^2{\mathrm{\γ}}_M^2+2l_0^2{\mathrm{\γ}}_M-l_0^2+2l_0{l}_{2M}{\mathrm{\γ}}_M^2-6l_0{l}_{2M}{\mathrm{\γ}}_M-2l_0{l}_3{\mathrm{\γ}}_M)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(4l_0{l}_{2M}+2l_0{l}_3-2l_0{\mathrm{\Δl\γ}}_M+2l_0\mathrm{\Δ}l-l_{2M}^2{\mathrm{\γ}}_M^2+4l_{2M}^2{\mathrm{\γ}}_M+2l_{2M}{l}_3{\mathrm{\γ}}_M+4l_{2M}{l}_3+2l_{2M}{\mathrm{\Δl\γ}}_M+4l_{2M}\mathrm{\Δ}l)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(3l_3^2+6l_3\mathrm{\Δ}l+3{\mathrm{\Δl}}^2)}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{{\mathrm{\Δl}}^3{\mathrm{ln\γ}}_M}{{({\mathrm{\γ}}_M-1)}^3}\]+\frac{12}{{\mathrm{Eb\γ}}_M^3}(\frac{2l_{2p}^2-12l_0{l}_{2Mp}-6l_0^2}{3l_{2Mp}^2}+\frac{16l_0^{3/2}}{3l_{2Mp}^{3/2}});\end{array}$

(5) the end points deformation coefficient G of each root reinforced variable cross-section auxiliary spring_x-EAjAnd total end points deformation coefficient of n sheet superposition auxiliary spring G_x-EATCalculate:

According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of auxiliary spring_A, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2Ap, the root of auxiliary spring oblique line section is to auxiliary spring Distance l of end points_2A, the thickness of auxiliary spring oblique line section compares γ_A；Auxiliary spring sheet number n, wherein, the thickness of the parabolic segment of jth sheet auxiliary spring Compare β_Aj, j=1,2 ..., n, the end points deformation coefficient G to each auxiliary spring under end points stressing conditions_x-EAjCalculate, i.e.

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

According to auxiliary spring sheet number n, the end points deformation coefficient G of each auxiliary spring_x-EAj, total end points deformation coefficient G to n sheet auxiliary spring_x-EATEnter Row calculates, i.e.

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

(6) the complex stiffness K of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula_MATChecking computations:

According to main reed number m, the thickness h of the root flat segments of each main spring_2M, the thickness h of the root flat segments of each auxiliary spring_2A, Calculated G in step (1)_x-Ei, calculated G in step (2)_x-CD, calculated G in step (3)_x-Epm, step (4) calculated G in_x-CDp, and calculated G in step (5)_x-EAT, can be reinforced to non-ends contact formula sheet root less The complex stiffness K of variable cross-section major-minor spring_MATCheck, i.e.

$K_{MAT}=\underset{i=1}{\overset{m-1}{\Σ}}\frac{2h_{2M}^3}{G_{x-Ei}}+\frac{2h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)}{G_{x-Em}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)-G_{x-E_{p}m}{G}_{x-CD}{h}_{2A}^3}.$

The present invention has the advantage that than prior art

Wait structure owing to the end flat segments of each main spring is non-, and the length of auxiliary spring is less than the length of main spring, meanwhile, when major-minor spring connects After Chuing, the main spring of m sheet, in addition to by end points power, is also acted on by auxiliary spring contact support power in parabolic segment, therefore, first Before fail to provide the Method for Checking of the few sheet root reinforced variable cross-section major-minor spring complex stiffness of non-ends contact formula always.The present invention can The each main spring according to the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula and the structural parameters of auxiliary spring, elastic modelling quantity, The complex stiffness of sheet root reinforced variable cross-section major-minor spring few to non-ends contact formula checks.Imitated by example and ANSYS True checking understands, the checking computations side of the few sheet root reinforced variable cross-section major-minor spring complex stiffness of the non-ends contact formula that this invention is provided Method is correct, utilizes answering of the available few sheet root reinforced variable cross-section major-minor spring of the most non-ends contact formula of the method Close rigidity checking value, thus non-ends contact the formula few design level of sheet root reinforced variable cross-section major-minor spring, product matter can be improved Amount and performance 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 checking computations flow chart of the few sheet root reinforced major-minor spring complex stiffness of non-ends contact formula；

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

Fig. 3 is the ANSYS deformation simulation cloud atlas of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula of embodiment one；

Fig. 4 is the ANSYS deformation simulation cloud atlas of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula of embodiment two.

Specific embodiments

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

Embodiment one: the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of certain non-ends contact formula, the length of oblique line section Degree Δ l=30mm, elastic modulus E=200GPa.Main reed number m=2, half length L of main spring_M=575mm, installing space Half l₃=55mm, the root of main spring parabolic segment is to distance l of main spring end points_2Mp=L_M-l₃-Δ l=490mm, main spring oblique line The root of section is to distance l of main spring end points_2M=L_M-l₃=520mm；The root flat segments thickness h of each main spring_2M=11mm, main End thickness h of spring oblique line section_2Mp=10.23mm, the thickness of main spring oblique line section compares γ_M=h_2Mp/h_2M=0.93；1st main spring The thickness h of end flat segments₁₁=7mm, the thickness of the parabolic segment of the 1st main spring compares β₁=h₁₁/h_2Mp=0.69；2nd main spring The thickness h of end flat segments₁₂=6mm, the thickness of the parabolic segment of the 2nd main spring compares β₂=h₁₂/h_2Mp=0.59.Auxiliary spring sheet Number n=1, half length L of this sheet auxiliary spring_A=375mm, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2Ap=L_A- l₃-Δ l=290mm, distance l of the root of auxiliary spring oblique line section to auxiliary spring end points_2A=L_A-l₃=320mm；Auxiliary spring root flat segments Thickness h_2A=14mm, end thickness h of auxiliary spring oblique line section_2Ap=13mm, the thickness ratio of auxiliary spring oblique line section γ_A=h_2Ap/h_2A=0.93；The thickness h of the end flat segments of this sheet auxiliary spring_A11=8mm, auxiliary spring parabolical thickness ratio β_A1=h_A11/h_2Ap=0.62；Auxiliary spring contact and horizontal range l of main spring end points₀=L_M-L_A=200mm.According to each main spring and pair The structural parameters of spring and elastic modelling quantity, the complex stiffness of sheet root reinforced variable cross-section major-minor spring few to this non-ends contact formula is carried out Checking computations.

The Method for Checking of the few sheet root reinforced major-minor spring complex stiffness of the non-ends contact formula that present example is provided, its checking computations Flow process is as it is shown in figure 1, concrete checking computations step is as follows:

(1) the end points deformation coefficient G of each root main spring of reinforced variable cross-section under end points stressing conditions_x-EiCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Half length L of main spring_M=575mm, the root of main spring parabolic segment is to main spring Distance l of end points_2Mp=490mm, distance l of the root of main spring oblique line section to main spring end points_2M=520mm, main spring oblique line section Thickness compares γ_M=0.93, main reed number m=2, wherein, the thickness of the parabolic segment of the 1st main spring compares β₁=0.69, the 2nd master The thickness of the parabolic segment of spring compares β₂=0.59, to the 1st under end points stressing conditions and the 2nd root reinforced variable cross-section master The end points deformation coefficient G of spring_x-E1And G_x-E2It is respectively calculated, i.e.

$\begin{array}{c}{G}_{x-E1}=\frac{4(L_M^3-l_{2M}^3)}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_1^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{\Δlln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=107.53{\mathrm{mm}}^4/N,\end{array}$

$\begin{array}{c}{G}_{x-E2}=\frac{4(L_M^3-l_{2M}^3)}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_2^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{\Δlln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=113.42{\mathrm{mm}}^4/N;\end{array}$

(2) the deformation coefficient G at parabolic segment with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions_x-CDCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Half length L of main spring_M=575mm, the root of main spring parabolic segment is to main spring Distance l of end points_2Mp=490mm, distance l of the root of main spring oblique line section to main spring end points_2M=520mm, main spring oblique line section Thickness compares γ_M=0.93, auxiliary spring contact and horizontal range l of main spring end points₀=200mm, main reed number m=2, to end points stress feelings The deformation coefficient G at parabolic segment with auxiliary spring contact point of the 2nd main spring under condition_x-CDCalculate, i.e.

$\begin{array}{c}{G}_{x-CD}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}+\frac{8l_{2Mp}^3+16l_{2Mp}^{3/2}{l}_0^{3/2}-24l_{2Mp}^2{l}_0}{{\mathrm{Eb\γ}}_M^3}-\frac{6l_0\mathrm{\Δ}l(l_{2Mp}+l_{2M}{\mathrm{\γ}}_M)}{{\mathrm{Eb\γ}}_M^2}+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3-4l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M)\\ =44.86{\mathrm{mm}}^4/N;\end{array}$

(3) the end points deformation coefficient G of the m sheet root main spring of reinforced variable cross-section under major-minor spring contact point stressing conditions_x-Ep2Calculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Half length L of main spring_M=575mm, the root of main spring parabolic segment is to main spring Distance l of end points_2Mp=490mm, distance l of the root of main spring oblique line section to main spring end points_2M=520mm, main spring oblique line section Thickness compares γ_M=0.93, auxiliary spring contact and horizontal range l of main spring end points₀=200mm, main reed number m=2, contact major-minor spring The end points deformation coefficient G of the 2nd main spring under some stressing conditions_x-Ep2Calculate, i.e.

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

$\frac{6l_0\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{\mathrm{Eb\γ}}_M^2}=44.86{\mathrm{mm}}^4/N;$

(4) the m sheet root main spring of reinforced variable cross-section under major-minor spring contact point stressing conditions is at parabolic segment with auxiliary spring contact point Deformation coefficient G_x-CDpCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, half l of installing space₃=55mm, elastic modulus E=200GPa；Half length L of main spring_M=575mm, The root of main spring parabolic segment is to distance l of main spring end points_2Mp=490mm, the distance of the root of main spring oblique line section to main spring end points l_2M=520mm, the thickness of main spring oblique line section compares γ_M=0.93, auxiliary spring contact and horizontal range l of main spring end points₀=200mm, main Reed number m=2, to the deformation system at parabolic segment with auxiliary spring contact point of the 2nd main spring under major-minor spring contact point stressing conditions Number G_x-CDpCalculate, i.e.

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_p}=\frac{4(L_M-l_{2M})(L_M^2-3l_0{L}_M+L_M{l}_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\frac{12}{Eb}\[\frac{l_0^2\mathrm{\Δ}l{({\mathrm{\γ}}_M-1)}^2-3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{2l_{2Mp}\mathrm{\Δ}l(\mathrm{\Δ}l-l_{2Mp}-l_0{\mathrm{\γ}}_M+l_{2M}{\mathrm{\γ}}_M)}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}+\frac{\mathrm{\Δ}l\[2l_0^3\mathrm{\Δ}l({\mathrm{\γ}}_M-1)(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})+3L_M^2-2L_M{l}_0-2L_M{l}_{2M}{\mathrm{\γ}}_M\]}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2L_M{l}_0{\mathrm{\γ}}_M-4L_M{l}_{2M}-6L_M{l}_3-6L_M\mathrm{\Δ}l-l_0^2{\mathrm{\γ}}_M^2+2l_0^2{\mathrm{\γ}}_M-l_0^2+2l_0{l}_{2M}{\mathrm{\γ}}_M^2-6l_0{l}_{2M}{\mathrm{\γ}}_M-2l_0{l}_3{\mathrm{\γ}}_M)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(4l_0{l}_{2M}+2l_0{l}_3-2l_0{\mathrm{\Δl\γ}}_M+2l_0\mathrm{\Δ}l-l_{2M}^2{\mathrm{\γ}}_M^2+4l_{2M}^2{\mathrm{\γ}}_M+2l_{2M}{l}_3{\mathrm{\γ}}_M+4l_{2M}{l}_3+2l_{2M}{\mathrm{\Δl\γ}}_M+4l_{2M}\mathrm{\Δ}l)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(3l_3^2+6l_3\mathrm{\Δ}l+3{\mathrm{\Δl}}^2)}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{{\mathrm{\Δl}}^3{\mathrm{ln\γ}}_M}{{({\mathrm{\γ}}_M-1)}^3}\]+\frac{12}{{\mathrm{Eb\γ}}_M^3}(\frac{2l_{2p}^2-12l_0{l}_{2Mp}-6l_0^2}{3l_{2Mp}^2}+\frac{16l_0^{3/2}}{3l_{2Mp}^{3/2}})=23.29{\mathrm{mm}}^4/N;\end{array}$

(5) the end points deformation coefficient G of each root reinforced variable cross-section auxiliary spring_x-EAjAnd total end points deformation coefficient of n sheet superposition auxiliary spring G_x-EATCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Auxiliary spring sheet number n=1, the half length of this sheet root reinforced variable cross-section auxiliary spring L_A=375mm, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2Ap=290mm, the root of auxiliary spring oblique line section is to auxiliary spring Distance l of end points_2A=320mm, the thickness of auxiliary spring oblique line section compares γ_A=0.93, the thickness of auxiliary spring parabolic segment compares β_A1=0.62, End points deformation coefficient G to this sheet root reinforced variable cross-section auxiliary spring under end points stressing conditions_x-EA1Calculate, i.e.

$\begin{array}{c}{G}_{x-EA1}=\frac{4(L_A^3-l_{2A}^3)}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{A1}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2+3l_{2A}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2{\mathrm{\Δlln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}=27.71{\mathrm{mm}}^4/N;\end{array}$

According to auxiliary spring sheet number n=1, the end points deformation coefficient G of this auxiliary spring_x-EA1, total end points deformation coefficient G to this sheet auxiliary spring_x-EAT, Calculate, i.e.

$G_{x-EAT}=\frac{1}{\underset{j=1}{\overset{n}{\Σ}}\frac{1}{G_{x-EAj}}}=G_{x-EA1}=27.71{\mathrm{mm}}^4/N;$

(6) the complex stiffness K of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula_MATChecking computations:

According to main reed number m=2, the thickness h of the root flat segments of each main spring_2M=11mm, the thickness of the root flat segments of this sheet auxiliary spring Degree h_2A=14mm, calculated G in step (1)_x-E1=107.53mm⁴/ N and G_x-E2=113.42mm⁴/ N, step (2) is fallen into a trap The G obtained_x-CD=44.86mm⁴/ N, calculated G in step (3)_x-Ep2=44.86mm⁴/ N, step is calculated in (4) G_x-CDp=23.29mm⁴Calculated G in/N, and step (5)_x-EAT=27.71mm⁴/ N, sheet few to this non-ends contact formula The complex stiffness K of root reinforced variable cross-section major-minor spring_MATCheck, i.e.

$K_{MAT}=\underset{i=1}{\overset{m-1}{\Σ}}\frac{2h_{2M}^3}{G_{x-Ei}}+\frac{2h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)}{G_{x-E2}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)-G_{x-E_{p}2}{G}_{x-CD}{h}_{2A}^3}=70.17N/mm.$

After major-minor spring concurs, in the case of main spring end points applies concentrfated load P=1780N, utilize complex stiffness value of calculation K_MAT=70.17N/mm, the maximum of the half symmetrical structure of sheet root reinforced variable cross-section major-minor spring few to this non-ends contact formula Deformation checks, i.e.

$f_{Dmax}=\frac{2P}{K_{MAT}}=50.73mm.$

Utilize ANSYS finite element emulation software, according to each master of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula Spring and the structural parameters of auxiliary spring and elastic modelling quantity, set up the ANSYS phantom of half symmetrical structure major-minor spring, divides net Lattice, and at the root applying fixed constraint of phantom, apply concentrfated load P=1780N at main spring end points, to this few sheet root The deformation of the major-minor spring of reinforced variable-section steel sheet spring carries out ANSYS emulation, and the ANSYS of this obtained major-minor spring becomes Shape emulation cloud atlas, as it is shown on figure 3, wherein, major-minor spring maximum deformation quantity f at endpoint location_DSmax=50.50mm.

Understand, in the case of same load, ANSYS simulating, verifying value f of this major-minor spring maximum distortion_DSmax=50.50mm, with Maximum distortion f under rigidity checking value_DmaxThe relative deviation of=50.73mm is respectively 0.45%, and result shows that this invention is provided The Method for Checking of the few sheet root reinforced major-minor spring complex stiffness of non-ends contact formula be correct, complex stiffness checking computations value is accurate The most reliably.

Embodiment two: the width of the few sheet root reinforced variable cross-section major-minor spring of certain non-ends contact formula non-ends contact formula B=60mm, the length Δ l=30mm of oblique line section, elastic modulus E=200GPa.Main reed number m=2, the half length of main spring L_M=600mm, half length L of main spring_M=600mm, the root of main spring parabolic segment is to distance l of main spring end points_2Mp=L_M- l₃-Δ l=510mm, distance l of the root of main spring oblique line section to main spring end points_2M=L_M-l₃=540mm；The root of each main spring is put down Straight section thickness h_2M=12mm, end thickness h of main spring oblique line section_2Mp=11mm, the thickness ratio of main spring oblique line section γ_M=h_2Mp/h_2M=0.92；The thickness h of the end flat segments of the 1st main spring₁₁=7mm, the thickness of the parabolic segment of the 1st main spring Degree compares β₁=h₁₁/h_2Mp=0.64；The thickness h of the end flat segments of the 2nd main spring₁₂=6mm, the parabolic segment of the 2nd main spring Thickness compare β₂=h₁₂/h_2Mp=0.55.Auxiliary spring sheet number n=1, half length L of this sheet auxiliary spring_A=410mm, auxiliary spring parabolic segment Root to distance l of auxiliary spring end points_2Ap=L_A-l₃-Δ l=320mm, the distance of the root of auxiliary spring oblique line section to auxiliary spring end points l_2A=L_A-l₃=350mm；The thickness h of the root flat segments of this sheet auxiliary spring_2A=13mm, the end thickness of auxiliary spring oblique line section h_2Ap=12mm, the thickness of auxiliary spring oblique line section compares γ_A=h_2Ap/h_2A=0.92；The thickness h of the end flat segments of auxiliary spring_A11=8mm, The parabolical thickness of auxiliary spring compares β_A1=h_A11/h_2Ap=0.67；Auxiliary spring contact and horizontal range l of main spring end points₀=L_M- L_A=190mm.According to each main spring and the structural parameters of auxiliary spring and elastic modelling quantity, sheet root few to this non-ends contact formula is strengthened The complex stiffness of type variable cross-section major-minor spring checks.

Use the method for designing identical with embodiment one and step, sheet root reinforced variable cross-section steel plates few to this non-ends contact formula The complex stiffness of the major-minor spring of spring checks, and specifically comprises the following steps that

(1) the end points deformation coefficient G of each root main spring of reinforced variable cross-section under end points stressing conditions_x-EiCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Half length L of main spring_M=600mm, the root of main spring parabolic segment is to main spring Distance l of end points_2Mp=510mm, distance l of the root of main spring oblique line section to main spring end points_2M=540mm, main spring oblique line section Thickness compares γ_M=0.92, main reed number m=2, wherein, the thickness of the parabolic segment of the 1st main spring compares β₁=0.64, the 2nd master The thickness of the parabolic segment of spring compares β₂=0.55, the end points of the 1st main spring under end points stressing conditions and the 2nd main spring is deformed Coefficient G_x-E1And G_x-E2It is respectively calculated, i.e.

$G_{x-E1}=\frac{4(L_M^3-l_{2M}^3)}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_1^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+$

$\begin{array}{c}\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{\Δlln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=128.94{\mathrm{mm}}^4/N,\end{array}$

$\begin{array}{c}{G}_{x-E2}=\frac{4(L_M^3-l_{2M}^3)}{Eb}+\frac{4l_{2Mp}^3(2-{\mathrm{\β}}_2^3)}{{\mathrm{Eb\γ}}_M^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M+2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3)-\\ \frac{24l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{\Δlln\γ}}_M}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}=134.42{\mathrm{mm}}^4/N,\end{array}$

(2) deformation coefficient at parabolic segment with auxiliary spring contact point of the m sheet root main spring of reinforced variable cross-section under end points stressing conditions G_x-CDCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Half length L of main spring_M=600mm, the root of main spring parabolic segment is to main spring Distance l of end points_2Mp=510mm, distance l of the root of main spring oblique line section to main spring end points_2M=540mm, main spring oblique line section Thickness compares γ_M=0.92；Auxiliary spring contact and horizontal range l of main spring end points₀=190mm, main reed number m=2, to end points stress feelings The deformation coefficient G at parabolic segment with auxiliary spring contact point of the 2nd main spring under condition_x-CDCalculate, i.e.

$\begin{array}{c}{G}_{x-CD}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}+\frac{8l_{2Mp}^3+16l_{2Mp}^{3/2}{l}_0^{3/2}-24l_{2Mp}^2{l}_0}{{\mathrm{Eb\γ}}_M^3}-\frac{6l_0\mathrm{\Δ}l(l_{2Mp}+l_{2M}{\mathrm{\γ}}_M)}{{\mathrm{Eb\γ}}_M^2}+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(4l_{2Mp}^2{\mathrm{\γ}}_M-l_{2Mp}^2-3l_{2Mp}^2{\mathrm{\γ}}_M^2+3l_{2M}^2{\mathrm{\γ}}_M^2-4l_{2M}^2{\mathrm{\γ}}_M^3+l_{2M}^2{\mathrm{\γ}}_M^4-2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_M^2{({\mathrm{\γ}}_M-1)}^3}(2l_{2Mp}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2M}^2{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M+2l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^3-4l_{2Mp}{l}_{2M}{\mathrm{\γ}}_M^2{\mathrm{ln\γ}}_M)\\ =57.72{\mathrm{mm}}^4/N;\end{array}$

(3) the end points deformation coefficient G of the m sheet root main spring of reinforced variable cross-section under major-minor spring contact point stressing conditions_x-Ep2Calculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Half length L of main spring_M=600mm, the root of main spring parabolic segment is to main spring Distance l of end points_2Mp=510mm, distance l of the root of main spring oblique line section to main spring end points_2M=540mm, main spring oblique line section Thickness compares γ_M=0.92；Auxiliary spring contact and horizontal range l of main spring end points₀=190mm, main reed number m=2, contact major-minor spring The end points deformation coefficient G of the 2nd main spring under some stressing conditions_x-Ep2Calculate, i.e.

$\begin{array}{c}{G}_{x-E_{p}2}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{12}{Eb}\[\frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{2l_{2M}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^2}-\\ \frac{3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3{\mathrm{\γ}}_M^2}-\frac{2l_{2Mp}\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}-\frac{{\mathrm{\Δl}}^3}{{({\mathrm{\γ}}_M-1)}^3}{\mathrm{ln\γ}}_M\]-\frac{24l_0{l}_{2Mp}^2-8{l_{2Mp}}^3-16l_0^{3/2}{l}_{2Mp}^{3/2}}{{\mathrm{Eb\γ}}_M^3}-\\ \frac{6l_0\mathrm{\Δ}l(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}{{\mathrm{Eb\γ}}_M^2}=57.72{\mathrm{mm}}^4/N;\end{array}$

(4) the m sheet root main spring of reinforced variable cross-section under major-minor spring contact point stressing conditions is at parabolic segment with auxiliary spring contact point Deformation coefficient G_x-CDpCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the half of installing space l₃=60mm, the length Δ l=30mm of oblique line section, elastic modulus E=200GPa；Half length L of main spring_M=600mm, main The root of spring parabolic segment is to distance l of main spring end points_2Mp=510mm, the distance of the root of main spring oblique line section to main spring end points l_2M=540mm, the thickness of main spring oblique line section compares γ_M=0.92；Auxiliary spring contact and horizontal range l of main spring end points₀=190mm, main Reed number m=2, to the deformation system at parabolic segment with auxiliary spring contact point of the 2nd main spring under major-minor spring contact point stressing conditions Number G_x-CDpCalculate, i.e.

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_p}=\frac{4(L_M-l_{2M})(L_M^2-3l_0{L}_M+L_M{l}_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\frac{12}{Eb}\[\frac{l_0^2\mathrm{\Δ}l{({\mathrm{\γ}}_M-1)}^2-3\mathrm{\Δ}l{(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})}^2}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{2l_{2Mp}\mathrm{\Δ}l(\mathrm{\Δ}l-l_{2Mp}-l_0{\mathrm{\γ}}_M+l_{2M}{\mathrm{\γ}}_M)}{{({\mathrm{\γ}}_M-1)}^2{\mathrm{\γ}}_M^2}+\frac{\mathrm{\Δ}l\[2l_0^3\mathrm{\Δ}l({\mathrm{\γ}}_M-1)(l_{2M}{\mathrm{\γ}}_M-l_{2Mp})+3L_M^2-2L_M{l}_0-2L_M{l}_{2M}{\mathrm{\γ}}_M\]}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(2L_M{l}_0{\mathrm{\γ}}_M-4L_M{l}_{2M}-6L_M{l}_3-6L_M\mathrm{\Δ}l-l_0^2{\mathrm{\γ}}_M^2+2l_0^2{\mathrm{\γ}}_M-l_0^2+2l_0{l}_{2M}{\mathrm{\γ}}_M^2-6l_0{l}_{2M}{\mathrm{\γ}}_M-2l_0{l}_3{\mathrm{\γ}}_M)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(4l_0{l}_{2M}+2l_0{l}_3-2l_0{\mathrm{\Δl\γ}}_M+2l_0\mathrm{\Δ}l-l_{2M}^2{\mathrm{\γ}}_M^2+4l_{2M}^2{\mathrm{\γ}}_M+2l_{2M}{l}_3{\mathrm{\γ}}_M+4l_{2M}{l}_3+2l_{2M}{\mathrm{\Δl\γ}}_M+4l_{2M}\mathrm{\Δ}l)}{2{({\mathrm{\γ}}_M-1)}^3}+\\ \frac{\mathrm{\Δ}l(3l_3^2+6l_3\mathrm{\Δ}l+3{\mathrm{\Δl}}^2)}{2{({\mathrm{\γ}}_M-1)}^3}-\frac{{\mathrm{\Δl}}^3{\mathrm{ln\γ}}_M}{{({\mathrm{\γ}}_M-1)}^3}\]+\frac{12}{{\mathrm{Eb\γ}}_M^3}(\frac{2l_{2p}^2-12l_0{l}_{2Mp}-6l_0^2}{3l_{2Mp}^2}+\frac{16l_0^{3/2}}{3l_{2Mp}^{3/2}})=31.74{\mathrm{mm}}^4/N;\end{array}$

(5) the end points deformation coefficient G of each root reinforced variable cross-section auxiliary spring_x-EAjAnd total end points deformation coefficient of n sheet superposition auxiliary spring G_x-EATCalculate:

According to the width b=60mm of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；Auxiliary spring sheet number n=1, half length L of this sheet auxiliary spring_A=410mm, auxiliary spring is thrown The root of thing line segment is to distance l of auxiliary spring end points_2Ap=320mm, the distance of the root of auxiliary spring oblique line section to auxiliary spring end points l_2A=350mm, the thickness of auxiliary spring oblique line section compares γ_A=0.92；The thickness of auxiliary spring parabolic segment compares β_A1=0.67, to end points stress feelings The end points deformation coefficient G of this sheet auxiliary spring under condition_x-EA1Calculate, i.e.

$\begin{array}{c}{G}_{x-EA1}=\frac{4(L_A^3-l_{2A}^3)}{Eb}+\frac{4l_{2Ap}^3(2-{\mathrm{\β}}_{A1}^3)}{{\mathrm{Eb\γ}}_A^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(4l_{2Ap}^2{\mathrm{\γ}}_A-l_{2Ap}^2-3l_{2Ap}^2{\mathrm{\γ}}_A^2+3l_{2A}^2{\mathrm{\γ}}_A^2-4l_{2A}^2{\mathrm{\γ}}_A^3)+\\ \frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}(l_{2A}^2{\mathrm{\γ}}_A^4-2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A+2l_{2Ap}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2A}^2{\mathrm{\γ}}_A^2{\mathrm{ln\γ}}_A+2l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^3)-\\ \frac{24l_{2Ap}{l}_{2A}{\mathrm{\γ}}_A^2{\mathrm{\Δlln\γ}}_A}{{\mathrm{Eb\γ}}_A^2{({\mathrm{\γ}}_A-1)}^3}=36.13{\mathrm{mm}}^4/N;\end{array}$

This auxiliary spring sheet number n=1, the end points deformation coefficient G of this sheet root reinforced variable cross-section auxiliary spring_x-EA1, total to n sheet superposition auxiliary spring End points deformation coefficient G_x-EAT, i.e.

$G_{x-EAT}=\frac{1}{\underset{j=1}{\overset{n}{\Σ}}\frac{1}{G_{x-EAj}}}=G_{x-EA1}=36.13{\mathrm{mm}}^4/N;$

(6) the complex stiffness K of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula_MATChecking computations:

According to main reed number m=2, the thickness h of the root flat segments of each main spring_2M=12mm, the thickness of the root flat segments of this sheet auxiliary spring Degree h_2A=13mm, calculated G in step (1)_x-E1=128.94mm⁴/ N and G_x-E2=134.42mm⁴/ N, step (2) is fallen into a trap The G obtained_x-CD=57.72mm⁴/ N, calculated G in step (3)_x-Ep2=57.72mm⁴/ N, step is calculated in (4) G_x-CDp=31.74mm⁴Calculated G in/N, and step (5)_x-EAT=36.13mm⁴/ N, sheet few to this non-ends contact formula The complex stiffness K of root reinforced variable cross-section major-minor spring_MATCheck, i.e.

$K_{MAT}=\underset{i=1}{\overset{m-1}{\Σ}}\frac{2h_{2M}^3}{G_{x-Ei}}+\frac{2h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)}{G_{x-E2}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)-G_{x-E_{p}2}{G}_{x-CD}{h}_{2A}^3}=70.53N/mm.$

After major-minor spring concurs, in the case of the end points of the main spring of half symmetrical structure applies concentrfated load P=1750N, utilize Complex stiffness checking computations value K_MAT=70.53N/mm, the half of sheet root reinforced variable cross-section major-minor spring few to this non-ends contact formula The maximum distortion of symmetrical structure checks, i.e.

$f_{Dmax}=\frac{2P}{K_{MAT}}=49.62mm.$

Utilize ANSYS finite element emulation software, according to each master of the few sheet root reinforced variable cross-section major-minor spring of this non-ends contact formula Spring and the structural parameters of auxiliary spring and elastic modelling quantity, set up the ANSYS phantom of half symmetrical structure major-minor spring, divides net Lattice, and apply fixed constraint at root, applying concentrfated load P=1750N at end points, sheet root few to this non-ends contact formula adds The deformation of strong type variable cross-section major-minor spring carries out ANSYS emulation, the ANSYS deformation simulation cloud atlas of this obtained major-minor spring, As shown in Figure 4, wherein, major-minor spring maximum deformation quantity f at endpoint location_DSmax=49.50mm.

Understand, in the case of same load, ANSYS simulating, verifying value f of this major-minor spring maximum distortion_DSmax=49.50mm, with Maximum distortion f under rigidity checking value_DmaxThe relative deviation of=49.62mm is respectively 0.24%, and result shows that this invention is provided The Method for Checking of the few sheet root reinforced major-minor spring complex stiffness of non-ends contact formula be correct, complex stiffness checking computations value is accurate Really, reliably.

Claims (1)

1. The Method for Checking of the few sheet root reinforced major-minor spring complex stiffness of the most non-ends contact formula, wherein, few reinforced change of sheet root cuts Face major-minor spring is to be made up of root flat segments, oblique line section, parabolic segment and end flat segments 4 sections, and wherein, oblique line section is to spring Play booster action；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 in other each main spring；Auxiliary spring length is less than main spring length, when load acts as more than auxiliary spring Contact with certain point in main spring parabolic segment with auxiliary spring contact during load, the most non-ends contact formula major-minor spring；In each main spring and pair In the case of the structural parameters of spring, elastic modelling quantity are given, the complex stiffness of sheet root reinforced major-minor spring few to non-ends contact formula enters Row checking computations, concrete checking computations step is as follows:

   (1) the end points deformation coefficient G of each root main spring of reinforced variable cross-section under end points stressing conditions_x-EiCalculate:

   According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of main spring_M, the root of main spring parabolic segment is to distance l of main spring end points_2Mp, the root of main spring oblique line section is to main Distance l of spring end points_2M, the thickness of main spring oblique line section compares γ_M, main reed number m, wherein, the parabolic segment of i-th main spring Thickness compare β_i, i=1,2 ..., m, the end points deformation coefficient G to the reinforced main spring of each root under end points stressing conditions_x-Ei Calculate, i.e.

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

   (2) the deformation coefficient G at parabolic segment with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions_x-CDCalculate:

   According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of main spring_M, the root of main spring parabolic segment is to distance l of main spring end points_2Mp, the root of main spring oblique line section section arrives Distance l of main spring end points_2M, the thickness of main spring oblique line section compares γ_M；Auxiliary spring contact and horizontal range l of main spring end points₀, main reed Number m, to the deformation coefficient G at parabolic segment with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions_x-CDCount Calculate, i.e.

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

   (3) the end points deformation coefficient G of the main spring of m sheet under major-minor spring contact point stressing conditions_x-EpmCalculate:

   According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of main spring_M, the root of main spring parabolic segment is to distance l of main spring end points_2MpL, the root of main spring oblique line section is to main Distance l of spring end points_2M, the thickness of main spring oblique line section compares γ_M；Auxiliary spring contact and horizontal range l of main spring end points₀, main reed number M, the end points deformation coefficient G to the main spring of m sheet under major-minor spring contact point stressing conditions_x-EpmCalculate, i.e.

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

   (4) the deformation coefficient G at parabolic segment with auxiliary spring contact point of the main spring of m sheet under major-minor spring contact point stressing conditions_x-CDpMeter Calculate:

   According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, the one of installing space Half l₃=55mm, elastic modulus E；Half length L of main spring_M, the root of main spring parabolic segment is to the distance of main spring end points l_2Mp, distance l of the root of main spring oblique line section to main spring end points_2M, the thickness of main spring oblique line section compares γ_M；Auxiliary spring contact and main spring Horizontal range l of end points₀, main reed number m, to the main spring of m sheet under major-minor spring contact point stressing conditions in parabolic segment with secondary Deformation coefficient G at spring contact point_x-CDpCalculate, i.e.

   $\begin{array}{c}{G}_{x-{\mathrm{CD}}_p}=\frac{4\left(L_M-l_{2M}\right)\left(L_M^2-3l_0{L}_M+L_M{l}_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2\right)}{Eb}-\frac{12}{Eb}\[\frac{l_0^2\mathrm{\Δ}l{\left({\mathrm{\γ}}_M-1\right)}^2-3\mathrm{\Δ}l{\left(l_{2M}{\mathrm{\γ}}_M-l_{2Mp}\right)}^2}{2{\left({\mathrm{\γ}}_M-1\right)}^3}+\\ \frac{2l_{2Mp}\mathrm{\Δ}l\left(\mathrm{\Δ}l-l_{2Mp}-l_0{\mathrm{\γ}}_M+l_{2M}{\mathrm{\γ}}_M\right)}{{\left({\mathrm{\γ}}_M-1\right)}^2{\mathrm{\γ}}_M^2}+\frac{\mathrm{\Δ}l\[2l_0^3\mathrm{\Δ}l\left({\mathrm{\γ}}_M-1\right)\left(l_{2M}{\mathrm{\γ}}_M-l_{2Mp}\right)+3L_M^2-2L_M{l}_0-2L_M{l}_{2M}{\mathrm{\γ}}_M\]}{2{\left({\mathrm{\γ}}_M-1\right)}^3}+\\ \frac{\mathrm{\Δ}l\left(2L_M{l}_0{\mathrm{\γ}}_M-4L_M{l}_{2M}-6L_M{l}_3-6L_M\mathrm{\Δ}l-l_0^2{\mathrm{\γ}}_M^2+2l_0^2{\mathrm{\γ}}_M-l_0^2+2l_0{l}_{2M}{\mathrm{\γ}}_M^2-6l_0{l}_{2M}{\mathrm{\γ}}_M-2l_0{l}_3{\mathrm{\γ}}_M\right)}{2{\left({\mathrm{\γ}}_M-1\right)}^3}+\\ \frac{\mathrm{\Δ}l\left(4l_0{l}_{2M}+2l_0{l}_3-2l_0{\mathrm{\Δl\γ}}_M+2l_0\mathrm{\Δ}l-l_{2M}^2{\mathrm{\γ}}_M^2+4l_{2M}^2{\mathrm{\γ}}_M+2l_{2M}{l}_3{\mathrm{\γ}}_M+4l_{2M}{l}_3+2l_{2M}{\mathrm{\Δl\γ}}_M+4l_{2M}\mathrm{\Δ}l\right)}{2{\left({\mathrm{\γ}}_M-1\right)}^3}+\\ \frac{\mathrm{\Δ}l\left(3l_3^2+6l_3\mathrm{\Δ}l+3{\mathrm{\Δl}}^2\right)}{2{\left({\mathrm{\γ}}_M-1\right)}^3}-\frac{{\mathrm{\Δl}}^3{\mathrm{ln\γ}}_M}{{\left({\mathrm{\γ}}_M-1\right)}^3}\]+\frac{12}{{\mathrm{Eb\γ}}_M^3}\left(\frac{2l_{2p}^2-12l_0{l}_{2Mp}-6l_0^2}{3l_{2Mp}^2}+\frac{16l_0^{3/2}}{3l_{2Mp}^{3/2}}\right);\end{array}$

   (5) the end points deformation coefficient G of each root reinforced variable cross-section auxiliary spring_x-EAjAnd total end points deformation coefficient of n sheet superposition auxiliary spring G_x-EATCalculate:

   According to the width b of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula, the length Δ l of oblique line section, elastic modulus E； Half length L of auxiliary spring_A, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2Ap, the root of auxiliary spring oblique line section is to auxiliary spring Distance l of end points_2A, the thickness of auxiliary spring oblique line section compares γ_A；Auxiliary spring sheet number n, wherein, the thickness of the parabolic segment of jth sheet auxiliary spring Compare β_Aj, j=1,2 ..., n, the end points deformation coefficient G to each auxiliary spring under end points stressing conditions_x-EAjCalculate, i.e.

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

   According to auxiliary spring sheet number n, the end points deformation coefficient G of each auxiliary spring_x-EAj, total end points deformation coefficient G to n sheet auxiliary spring_x-EATEnter Row calculates, i.e.

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

   (6) the complex stiffness K of the few sheet root reinforced variable cross-section major-minor spring of non-ends contact formula_MATChecking computations:

   According to main reed number m, the thickness h of the root flat segments of each main spring_2M, the thickness h of the root flat segments of each auxiliary spring_2A, Calculated G in step (1)_x-Ei, calculated G in step (2)_x-CD, calculated G in step (3)_x-Epm, step (4) calculated G in_x-CDp, and calculated G in step (5)_x-EAT, can be reinforced to non-ends contact formula sheet root less The complex stiffness K of variable cross-section major-minor spring_MATCheck, i.e.

   $K_{MAT}=\underset{i=1}{\overset{m-1}{\Σ}}\frac{2h_{2M}^3}{G_{x-Ei}}+\frac{2h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)}{G_{x-Em}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_p}{h}_{2A}^3)-G_{x-E_{p}m}{G}_{x-CD}{h}_{2A}^3}.$