CN105956270A - Computing method of stress of each of end contact type less-leaf end part enhanced main spring and secondary spring - Google Patents

Abstract

The invention discloses a computing method of stress of each of end contact type less-leaf end part enhanced main spring and secondary spring, and belongs to the technical field of suspension steel plate springs. According to the structure parameter, elasticity modulus of each of the main spring and the secondary spring, the secondary spring action load, and the load born by the main spring and the secondary spring, the stresses of each of the main spring and the secondary spring of the end contact type less-leaf end part enhanced main spring and secondary spring at different positions are computed. Through the practical computation and ANSYS simulation verification, the method can obtain accurate and reliable stress computing values of each of the main spring and the secondary spring at different positions; through the adoption of the method disclosed by the invention, the design level, the product quality, performance and reliability of the end contact type less-leaf end part enhanced main spring and secondary spring are improved, the smoothness and security of the vehicle driving are improved; the quality and cost of the suspension spring are reduced; and meanwhile, the product design expense and testing expense of the product are lowered, and the development design speed of the product is accelerated.

Description

The computational methods of the few sheet reinforcement end each stress of major-minor spring of ends contact formula

Technical field

The present invention relates to few each of the sheet reinforcement end major-minor spring of vehicle suspension leaf spring, particularly ends contact formula should The computational methods of power.

Background technology

Few sheet variable-section steel sheet spring, because having between lightweight, sheet little, the advantage such as noise is little that rubs, is widely used in car In Leaf Spring Suspension System.In order to meet the design requirement of processing technique, stress intensity, rigidity and hanger thickness, in reality During the engineer applied of border, generally few sheet variable-section steel sheet spring is designed as the few sheet reinforcement end major-minor spring of ends contact formula Form.Main spring rigidity and major-minor spring complex stiffness meet suspension performance requirement, and each main spring and auxiliary spring are in various location Stress, should meet life-span and the reliability requirement of leaf spring.But, structure is waited owing to the end flat segments of each main spring is non-, secondary Spring length is less than main spring length, and after load works the contact of load major-minor spring more than auxiliary spring, the deformation of each major-minor spring And end points power has coupling, therefore, extremely difficult in the Stress calculation of various location to each main spring and auxiliary spring.According to being consulted Data understands, current state, the inside and outside each master the most not provided the few sheet reinforcement end major-minor spring of reliable ends contact formula Spring and auxiliary spring are in the calculation method for stress of various location.Sheet reinforcement end change few for ends contact formula the most both at home and abroad cuts Face major-minor spring, is mostly to utilize the finite element emulation software such as ANSYS, by solid modelling to the variable cross-section steel plates bullet of fixed structure Spring carries out stress numerical emulation, although the method can get reliable stress simulation value, but, finite element modeling emulation point Analysis method can only carry out numerical simulation checking to the leaf spring stress of fixed structure and load, it is impossible to provides accurate stress solution Analysis calculating formula, thus can not meet the few sheet reinforcement end variable cross-section major-minor spring modernization design CAD design of ends contact formula and The requirement of software development.Therefore, it is necessary to set up each of the few sheet reinforcement end major-minor spring of a kind of ends contact formula accurate, reliable The main spring of sheet and auxiliary spring, in the computational methods of various location stress, meet the few sheet reinforcement end variable cross-section major-minor of ends contact formula Each main spring of spring and the Stress calculation of auxiliary spring various location and the requirement of strength check, improve few sheet variable-section steel sheet spring Design level, quality, performance, reliability and vehicle ride performance and safety；Meanwhile, product design and test fee are reduced With, accelerate product development and design speed.

Summary of the invention

For defect present in above-mentioned prior art, the technical problem to be solved be to provide a kind of easy, The computational methods of the few sheet reinforcement end each stress of major-minor spring of ends contact formula reliably, calculation flow chart, as shown in Figure 1. The few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is symmetrical structure, and the half symmetrical structure of major-minor spring can be seen as outstanding Arm beam, i.e. symmetrical center line be the fixing end of root, the end stress point of main spring and the contact of auxiliary spring respectively as main spring end points and Auxiliary spring end points, the schematic diagram of half symmetrical structure major-minor spring, as in figure 2 it is shown, wherein, including: main spring 1, root shim 2, auxiliary spring 3, end pad 4；The half symmetrical structure of main spring 1 and auxiliary spring 3 is by root flat segments, parabolic segment, oblique line section, end flat segments Four sections of compositions, booster action is played in variable cross-section end by oblique line section；Main spring 1 and each root flat segments of auxiliary spring 3 and main spring 1 are with secondary Being equipped with root shim 2 between spring 3, the end flat segments of each of main spring 1 is provided with end pad 4, and the material of end pad 4 is carbon Fibrous composite, is used for the frictional noise produced when reducing spring works.The width of main spring 1 and auxiliary spring 3 is b, oblique line section A length of Δ l, a length of l of half of installing space₃, elastic modelling quantity is E.Main reed number is m, the thickness of the root flat segments of main spring Degree is h_2M, the distance of the root of main spring parabolic segment to main spring end points is l_2M=L_M-l₃, the end of each main spring parabolic segment is thick Degree is h_1Mpi, the thickness of parabolic segment compares β_i=h_1Mpi/h_2M, i=1,2 ..., m, the end of parabolic segment to main spring end points away from From l_1Mpi=l_2Mβ_i ²；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, More than the thickness of end flat segments and the length of other each main spring, wherein, the thickness of the end flat segments of each main spring and length Degree is respectively h_1MiAnd l_1Mi=l_1Mpi-Δl；The thickness of each main spring oblique line section compares γ_Mi=h_1Mi/h_1Mpi.Auxiliary spring sheet number is n, secondary The a length of L of half of spring_A, the thickness of each auxiliary spring root flat segments is h_2A, the root of auxiliary spring parabolic segment is to auxiliary spring end points Distance is l_2A=L_A-l₃, the end thickness of each auxiliary spring parabolic segment is h_1Apj, the thickness of auxiliary spring parabolic segment compares β_Aj=h_1Apj/ h_2A, the end of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_1Apj=l_2Aβ_Aj ²；The thickness of the end flat segments of each auxiliary spring and Length is respectively h_1AjAnd l_1Aj=l_1Apj-Δ l, the thickness of auxiliary spring oblique line section compares γ_Aj=h_1Aj/h_1Apj.Auxiliary spring ends points and master The horizontal range of spring end points is l₀, major-minor spring gap delta between auxiliary spring ends points and m sheet main spring end flat segments；Work as load Lotus more than auxiliary spring work load time, in auxiliary spring and main spring end flat segments, certain is put and contacts；After major-minor spring ends contact, Each end stress of major-minor spring differs, and the main spring contacted with auxiliary spring is in addition to by end points power, also holds at contact point Support force by auxiliary spring.Work load and major-minor spring institute at each main spring and the structural parameters of auxiliary spring, elastic modelling quantity, auxiliary spring Bear load given in the case of, each main spring of sheet reinforcement end major-minor spring few to end contact and auxiliary spring are at diverse location The stress at place calculates.

For solving above-mentioned technical problem, few each of the sheet reinforcement end major-minor spring of ends contact formula provided by the present invention should The computational methods of power, it is characterised in that use step calculated below:

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

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

According to width b, the length Δ l of oblique line section of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula, elastic Modulus E；Half length L of main spring_M, the root of parabolic segment is to distance l of spring end points_2M, the root flat segments of each main spring Thickness h_2M, main reed number m, wherein, the thickness of the parabolic segment of i-th main spring compares β_i, the thickness of oblique line section compares γ_Mi, oblique line The root of section is to distance l of main spring end points_1Mpi, the end of oblique line section is to distance l of main spring end points_1Mi, i=1,2 ..., m；To master The half stiffness K of each main spring before auxiliary spring contact_MiCalculate, i.e.

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

In formula, G_X-EiFor the end points deformation coefficient of i-th main spring in the case of end points power effect, i.e.

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

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

According to width b, the length Δ l of oblique line section of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula, elastic Modulus E；Half length L of main spring_M, the thickness h of the root flat segments of each main spring_2M, the root of main spring parabolic segment is to spring Distance l of end points_2M, main reed number m, wherein, the thickness of the parabolic segment of i-th main spring compares β_i, the thickness of oblique line section compares γ_Mi, The root of oblique line section is to distance l of main spring end points_1Mpi, the end of oblique line section is to distance l of main spring end points_1Mi；The half of auxiliary spring is long Degree L_A, the thickness h of the root flat segments of each auxiliary spring_2A, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2A, auxiliary spring sheet Number n, wherein, the thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, the thickness of oblique line section compares γ_Aj, the root of oblique line section is to secondary end Distance l of point_1Apj, the end of oblique line section is to distance l of auxiliary spring end points_1Aj, j=1,2 ..., n；Auxiliary spring contact and main spring end points Horizontal range l₀, the half stiffness K of each main spring after major-minor spring is contacted_MAiCalculate, i.e.

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

In formula, G_X-EiEnd points deformation coefficient for i-th main spring in the case of end points power effect；G_X-EATFor at end points Total end points deformation coefficient of the n sheet superposition auxiliary spring in the case of power effect, G_X-EAjSecondary for the jth sheet in the case of end points power effect The end points deformation coefficient of spring；G_x-DEFor the main spring of m sheet under end points stressing conditions at end flat segments with auxiliary spring contact point Deformation coefficient；G_x-EzmEnd points deformation coefficient for the main spring of m sheet under stressing conditions at major-minor spring contact point；G_x-EzFor The main spring of m sheet under stressing conditions deformation coefficient at end flat segments with auxiliary spring contact point at major-minor spring contact point, i.e.

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

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

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

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

$\begin{array}{c}{G}_{x-DE}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mpm}^{1/2}-l_{2M}^{1/2})(l_{1Mpm}+l_{2M}-3l_0+l_{1Mpm}^{1/2}{l}_{2M}^{1/2})}{Eb}+\\ \frac{2{(l_{1Mm}-l_0)}^2(2l_{1Mm}+l_0)}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3}+\frac{6\mathrm{\Δ}l(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3-2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4+2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}-\\ \frac{24l_{1Mm}{l}_{1Mpm}{\mathrm{\Δl\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}-\frac{6l_0\mathrm{\Δ}l(l_{1Mpm}{\mathrm{\γ}}_{Mm}+l_{1Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3};\end{array}$

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

$\begin{array}{c}{G}_{x-{\mathrm{DE}}_z}=\frac{12l_{2M}^{3/2}(6l_0^2{l}_{2M}^{1/2}+12l_0{l}_{1Mpm}{l}_{2M}^{1/2}-2l_{2M}^{1/2}{l}_{1Mpm}^2-6l_{1Mpm}^{1/2}{l}_0^2-12l_{1Mpm}^{1/2}{l}_0{l}_{2M}+2l_{1Mpm}^{1/2}{l}_{2M}^2)}{3l_{1Mpm}^{1/2}{l}_{2M}^{1/2}Eb}+\frac{4{(l_0-l_{1Mm})}^3}{{\mathrm{Eb\β}}_m^3{\mathrm{\γ}}_{Mm}^3}+\\ \frac{4(L_M-l_{2M})(L_M^2-3L_M{l}_0+L_M{l}_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\frac{6\mathrm{\Δ}l(2l_0^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-l_0^2-2l_0^2{\mathrm{\γ}}_{Mm}^3)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_M^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}+l_0^2{\mathrm{\γ}}_{Mm}^4-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2+3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4+2l_0{l}_{1Mm}-6l_0{l}_{1Mm}{\mathrm{\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}-2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{mm}+2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}-2l_0{l}_{1Mm}{\mathrm{\γ}}_{Mm}^3-6l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^2)}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(6l_0{l}_{1Mm}{\mathrm{\γ}}_{Mm}^2+6l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3-2l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^4+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3-4l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^2{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3};\end{array}$

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

According to the width b of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, springform Amount E；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 parabolic segment is to auxiliary spring end Distance l of point_2A, auxiliary spring sheet number n, wherein, the thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, the thickness of oblique line section compares γ_Aj, tiltedly The root of line segment is to distance l of auxiliary spring end points_1Apj, the end of oblique line section is to distance l of auxiliary spring end points_1Aj, to each auxiliary spring one Half stiffness K_AjCalculate, i.e.

$K_{Aj}=\frac{h_{2A}^3}{G_{x-EAj}};$

In formula,

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

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

I step: end points power P of each main spring_iCalculate:

According to the half the most single-ended point load P that the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is loaded, Auxiliary spring works load p_K, calculated K in I step_Mi, and II step calculates obtained K_MAi, end to each main spring Point power P_iCalculate, i.e.

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

Ii step: end points power P of each auxiliary spring_AjCalculate:

According to the half the most single-ended point load P that the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is loaded, Auxiliary spring works load p_K, main reed number m, the thickness h of the root flat segments of each main spring_2M, auxiliary spring sheet number n, each auxiliary spring The thickness h of root flat segments_2A, calculated K in II step_MAi、G_x-DE、G_x-DEzAnd G_x-EAT, and III step is calculated K_Aj, end points power P to each auxiliary spring_AjCalculate, i.e.

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

(3) calculating of each main spring diverse location stress of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula:

Step A: the calculating of stress at front m-1 sheet main spring diverse location x:

Half length L according to the few sheet main spring of reinforcement end variable cross-section of ends contact formula_M, the length Δ l of oblique line section, each The thickness h of the root flat segments of main spring_2M, the root of main spring parabolic segment is to distance l of main spring end points_2M, main reed number m, its In, end thickness h of the parabolic segment of i-th main spring_1Mpi, the thickness h of the end flat segments of i-th main spring_1Mi, i-th main spring The end of parabolic segment to distance l of main spring end points_1Mpi, length l of the end flat segments of i-th main spring_1Mi；And in i step Calculated P_i, with main spring free end as zero, with main spring end points as zero, few sheet reinforcement end is become and cuts The front main spring of m-1 sheet of face leaf spring stress at diverse location x calculates, i.e.

${\mathrm{\σ}}_{Mi}=\left\{\begin{array}{cc}\frac{6P_{i}x}{{\mathrm{bh}}_{1Mi}^2},& x\∈\[0,l_{1Mi}\]\\ \frac{6P_{i}x}{{\mathrm{bh}}_{2Mi}^2\left(x\right)},& x\∈(l_{1Mi},l_{1Mpi}\]\\ \frac{6P_{i}x}{{\mathrm{bh}}_{2Mpi}^2\left(x\right)},& x\∈(l_{1Mpi},l_{2M}\]\\ \frac{6P_{i}x}{{\mathrm{bh}}_{2M}^2},& x\∈(l_{2M},l_M\]\end{array}\right.,i=1,2,...,m-1;$

In formula, h_2MiX () is i-th main spring oblique line section thickness at x position, h_2MpiX () is i-th main spring parabola Section thickness at x position, i.e.

$h_{2Mi}\left(x\right)=\frac{h_{1Mpi}-h_{1Mi}}{\mathrm{\Δ}l}x+\frac{h_{1Mi}{l}_{1Mpi}-h_{1Mpi}{l}_{1Mi}}{\mathrm{\Δ}l},h_{2Mpi}\left(x\right)=h_{2M}\sqrt{\frac{x}{l_{2M}}};$

Step B: the calculating of stress at m sheet main spring diverse location x:

Half length L according to the few sheet main spring of reinforcement end variable cross-section of ends contact formula_M, the length Δ l of oblique line section, each The thickness h of the root flat segments of main spring_2M, the root of parabolic segment is to distance l of spring end points_2M, auxiliary spring contact and main spring end points Horizontal range l₀；Main reed number m, wherein, end thickness h of the parabolic segment of the main spring of m sheet_1Mpm, the parabolic of the main spring of m sheet The end of line segment is to distance l of main spring end points_1Mpm, length l of the end flat segments of the main spring of m sheet_1MmAnd thickness h_1Mm；And i step Calculated P in Zhou_m, calculated P in ii step_Aj, with main spring end points as zero, few sheet reinforcement end is become The main spring of m sheet of section steel flat spring stress at diverse location x calculates, i.e.

${\mathrm{\σ}}_{Mm}=\left\{\begin{array}{cc}\frac{6P_{m}x}{{\mathrm{bh}}_{1Mm}^2},& x\∈\[0,l_0\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{1Mm}^2},& x\∈(l_0,l_{1Mm}\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2Mm}^2\left(x\right)},& x\∈(l_{1Mm},l_{1Mpm}\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2Mpm}^2\left(x\right)},& x\∈(l_{1Mpm},l_{2M}\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2M}^2},& x\∈(l_{2M},L_M\]\end{array}\right.;$

In formula, h_2MmX () is m sheet main spring oblique line section thickness at x position, h_2MpmX () is m sheet main spring parabola Section thickness at x position, i.e.

$h_{2Mm}\left(x\right)=\frac{h_{1Mpm}-h_{1Mm}}{\mathrm{\Δ}l}x+\frac{h_{1Mm}{l}_{1Mpm}-h_{1Mpm}{l}_{1Mm}}{\mathrm{\Δ}l},h_{2Mpm}\left(x\right)=h_{2M}\sqrt{\frac{x}{l_{2M}}};$

(4) each auxiliary spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is at the meter of diverse location stress Calculate:

Half length L according to few sheet reinforcement end variable cross-section auxiliary spring_A, width b, the thickness of auxiliary spring root flat segments h_2A, the length Δ l of oblique line section, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2A, auxiliary spring sheet number n, wherein, jth sheet The end of the parabolic segment of auxiliary spring is to distance l of auxiliary spring end points_1Apj, the end thickness of auxiliary spring parabolic segment is h_1Apj, end is straight The thickness h of section_1AjWith length l_1Aj；And calculated P in ii step_Aj, j=1,2 ..., n, former with auxiliary spring free end for coordinate Point, calculates, i.e. each auxiliary spring of the few sheet reinforcement end variable-section steel sheet spring stress at diverse location x

${\mathrm{\σ}}_{Aj}=\left\{\begin{array}{cc}\frac{6P_{Aj}x}{{\mathrm{bh}}_{1Aj}^2},& x\∈\[0,l_{1Aj}\]\\ \frac{6P_{Aj}x}{{\mathrm{bh}}_{2Aj}^2\left(x\right)},& x\∈(l_{1Aj},l_{1Apj}\]\\ \frac{6P_{Aj}x}{{\mathrm{bh}}_{2Apj}^2\left(x\right)},& x\∈(l_{1Apj},l_{2A}\]\\ \frac{6P_{Aj}x}{{\mathrm{bh}}_{2A}^2},& x\∈(l_{2A},L_A\]\end{array}\right.,j=1,2,...,n;$

In formula, h_2AjX () is jth sheet auxiliary spring oblique line section thickness at x position, h_2ApjX () is jth sheet auxiliary spring parabola Section thickness at x position, i.e.

$h_{2Aj}\left(x\right)=\frac{h_{1Apj}-h_{1Aj}}{\mathrm{\Δ}l}x+\frac{h_{1Aj}{l}_{1Apj}-h_{1Apj}{l}_{1Aj}}{\mathrm{\Δ}l},h_{2Apj}\left(x\right)=h_{2A}\sqrt{\frac{x}{l_{2A}}}.$

The present invention has the advantage that than prior art

Waiting structure owing to the end flat segments of each main spring is non-, auxiliary spring length is less than main spring length, and when load is more than auxiliary spring After the contact of the load that works major-minor spring, deformation and the end points power of each major-minor spring have coupling, therefore, to each main spring and pair Spring is extremely difficult in the Stress calculation of various location, has not the most provided the few bit end of reliable ends contact formula inside and outside predecessor State Each main spring of portion's reinforced major-minor spring and auxiliary spring are in the calculation method for stress of various location.Both at home and abroad end is connect at present The few sheet reinforcement end variable cross-section major-minor spring of touch, is mostly to utilize the finite element emulation software such as ANSYS, by solid modelling pair Stress numerical emulation is carried out, although the method can get reliable stress simulation to the variable-section steel sheet spring of fixed structure Value, but, owing to finite element modeling simulating analysis can only carry out numerical value to the leaf spring stress of fixed structure and load Simulating, verifying, it is impossible to provide accurate stress analysis calculating formula, becomes so the few sheet reinforcement end of ends contact formula can not be met Cross section major-minor spring modernization design CAD design and the requirement of software development.The present invention can be according to the few bit end of each end contact Each main spring of portion's reinforced variable cross-section major-minor spring and the structural parameters of auxiliary spring, elastic modelling quantity, auxiliary spring work load and major-minor The born load of spring, each main spring of sheet reinforcement end major-minor spring few to end contact and auxiliary spring answering in various location Power carries out accurate Analysis calculating.

By example and ANSYS simulating, verifying, the few sheet end of ends contact formula accurate, reliable that the method can get Each main spring of reinforced major-minor spring and auxiliary spring, in the Stress calculation value of various location, are strengthened for the few sheet end of ends contact formula The calculating of type each stress of major-minor spring provides reliable computational methods, and strong for few sheet variable cross-section reinforcement end major-minor spring Degree is checked and reliable technical foundation has been established in CAD software exploitation.Utilize the method can improve vehicle suspension variable cross-section major-minor spring Design level, product quality, performance and reliability, improve the ride performance of vehicle and safety；Meanwhile, product is also reduced 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 calculation flow chart of the few sheet reinforcement end each stress of major-minor spring of ends contact formula；

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

Fig. 3 is ends contact formula few the 1st main spring of the sheet reinforcement end STRESS VARIATION in various location of embodiment Curve；

Fig. 4 is ends contact formula few the 2nd main spring of the sheet reinforcement end STRESS VARIATION in various location of embodiment Curve；

Fig. 5 is that few 1 auxiliary spring of sheet reinforcement end of ends contact formula of embodiment is bent in the STRESS VARIATION of various location Line；

Fig. 6 is the ANSYS stress simulation cloud atlas of few the 1st main spring of sheet reinforcement end of ends contact formula of embodiment；

Fig. 7 is the ANSYS stress simulation cloud atlas of few the 2nd main spring of sheet reinforcement end of ends contact formula of embodiment；

Fig. 8 is the ANSYS stress simulation cloud atlas of few 1 auxiliary spring of sheet reinforcement end of ends contact formula of embodiment.

Specific embodiments

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

Embodiment: the width b=60mm of the few sheet reinforcement end variable cross-section major-minor spring of certain ends contact formula, installing space Half l₃=55mm, the length Δ l=30mm of oblique line section, elastic modulus E=200GPa.Main reed number m=2, the half of main spring Length L_M=575mm, the thickness h of the root flat segments of each main spring_2M=11mm, the root of main spring parabolic segment is to main spring end points Distance l_2M=L_M-l₃=520mm；End thickness h of the parabolic segment of the 1st main spring_1Mp1=6mm, the thickness ratio of parabolic segment β₁=h_1Mp1/h_2M=0.55, the end of parabolic segment is to distance l of main spring end points_1Mp1=l_2Mβ₁ ²=157.30mm, end is straight The thickness h of section_1M1=7mm, the thickness of oblique line section compares γ_M1=h_1M1/h_1Mp1=1.17, length l of end flat segments_1M1=l_1Mp1- Δ l=127.30mm；End thickness h of the parabolic segment of the 2nd main spring_1Mp2=5mm, the thickness of parabolic segment compares β₂=h_1Mp2/ h_2M=0.45, the end of parabolic segment is to distance l of main spring end points_1Mp2=l_2Mβ₂ ²=105.30mm, the thickness of end flat segments h_1M2=6mm, the thickness of oblique line section compares γ_M2=h_1M2/h_1Mp2=1.20, length l of end flat segments_1M2=l_1Mp2-Δ l= 75.30mm.Half length L of auxiliary spring_A=525mm, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2A=L_A-l₃= 470mm, auxiliary spring sheet number n=1, the thickness h of the root flat segments of this sheet auxiliary spring_2A=14mm, the end of the parabolic segment of auxiliary spring is thick Degree h_1Ap1=7mm, the thickness of parabolic segment compares β_A1=h_1Ap1/h_2A=0.50, the end of parabolic segment is to the distance of auxiliary spring end points l_1Ap1=l_2Aβ_A1 ²=117.50mm, the thickness h of end flat segments_1A1=8mm, the thickness of oblique line section compares γ_A1=h_1A1/h_1Ap1= 1.14, length l of end flat segments_1A1=l_1Ap1-Δ l=87.50mm.Auxiliary spring contact and horizontal range l of main spring end points₀= L_M-L_A=50mm, major-minor spring works load p_K=2404.2N.At the half that major-minor spring is loaded the most single-ended point load P= Each main spring and the stress of auxiliary spring various location in the case of 3040N, to this few sheet reinforcement end variable-section steel sheet spring Calculate.

The computational methods of the few sheet reinforcement end each stress of major-minor spring of the ends contact formula that present example is provided, its Calculation process is as it is shown in figure 1, concrete calculation procedure is as follows:

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

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

Width b=60mm, the length Δ l of oblique line section according to the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula =30mm, elastic modulus E=200GPa；Half length L of main spring_M=575mm, the thickness h of main spring root flat segments_2M= 11mm, the root of main spring parabolic segment is to distance l of main spring end points_2M=520mm；Main reed number m=2, wherein, the 1st main spring The thickness of parabolic segment compare β₁=0.55, the root of oblique line section is to distance l of main spring end points_1Mp1=157.30mm, oblique line section Thickness compares γ_M1=1.17, the end of oblique line section is to distance l of main spring end points_1M1=127.30mm；The parabolic segment of the 2nd main spring Thickness compare β₂=0.45, the thickness of oblique line section compares γ_M2=1.20, the root of oblique line section is to distance l of main spring end points_1Mp2= 105.30mm, the end of oblique line section is to distance l of spring end points_1M2=75.30mm；To auxiliary spring contact before the 1st main spring and The half stiffness K of the 2nd main spring_M1And K_M2It is respectively calculated, i.e.

$K_{M1}=\frac{h_{2M}^3}{G_{x-E1}}=13.29N/mm;$

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

In formula, G_X-E1And G_X-E2The end points deformation being respectively the 1st main spring and the 2nd main spring under end points stressing conditions is Number, i.e.

$\begin{array}{c}{G}_{x-E1}=\frac{4(L_M^3-l_{2M}^3)}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp1}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M1}^3}{{\mathrm{Eb\γ}}_{M1}^3{\mathrm{\β}}_1^3}+\frac{6\mathrm{\Δ}l(4l_{1M1}^2{\mathrm{\γ}}_{M1}-l_{1M1}^2-3l_{1M1}^2{\mathrm{\γ}}_{M1}^2+3l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2-4l_{1Mp1}^2{\mathrm{\γ}}_{M1}^3)}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(-l_{1Mp1}^2{\mathrm{\γ}}_{M1}^4-2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}+2l_{1M1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^3-4l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1})}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}\\ =100.18{\mathrm{mm}}^4/N;\end{array}$

$G_{x-E2}=\frac{4(L_M^3-l_{2M}^3)}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp2}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M2}^3}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}+\frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-$

$\begin{array}{c}\frac{6\mathrm{\Δ}l(-l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}\\ =104.73{\mathrm{mm}}^4/N;\end{array}$

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

Width b=60mm, the length Δ l of oblique line section according to the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula =30mm, elastic modulus E=200GPa.Half length L of main spring_M=575mm, the thickness h of the root flat segments of each main spring_2M =11mm, the root of the parabolic segment of main spring is to distance l of spring end points_2M=520mm, main reed number m=2, wherein, the 1st The thickness of the parabolic segment of main spring compares β₁=0.55, the thickness of oblique line section compares γ_M1=1.17, the root of oblique line section is to main spring end points Distance l_1Mp1=157.30mm, the end of oblique line section is to distance l of main spring end points_1M1=127.30mm；The throwing of the 2nd main spring The thickness of thing line segment compares β₂=0.45, the thickness of oblique line section compares γ_M2=1.20, the root of oblique line section is to the distance of spring end points l_1Mp2=105.30mm, the end of oblique line section is to distance l of spring end points_1M2=75.30mm.Half length L of auxiliary spring_A= 525mm, the thickness h of auxiliary spring root flat segments_2A=14mm, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2A= 470mm, auxiliary spring sheet number n=1, the thickness of the parabolic segment of this sheet auxiliary spring compares β_A1=0.50, the thickness of oblique line section compares γ_A1= 1.14, the root of oblique line section is to distance l of auxiliary spring end points_1Ap1=117.50mm, the end of oblique line section is to the distance of auxiliary spring end points l_1A1=87.50mm；Auxiliary spring contact and horizontal range l of main spring end points₀=50mm.The 1st main spring after major-minor spring is contacted Half stiffness K with the 2nd main spring_MA1And K_MA2It is respectively calculated, i.e.

$K_{MA1}=\frac{h_{2M}^3}{G_{x-E1}}=13.29N/mm;$

$K_{MA2}=\frac{h_{2M}^3(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)}{G_{x-E2}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)-G_{x-E_{z}2}{G}_{x-DE}{h}_{2A}^3}=35.94N/mm;$

In formula,

$\begin{array}{c}{G}_{x-E1}=\frac{4(L_M^3-l_{2M}^3)}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp1}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M1}^3}{{\mathrm{Eb\γ}}_{M1}^3{\mathrm{\β}}_1^3}+\frac{6\mathrm{\Δ}l(4l_{1M1}^2{\mathrm{\γ}}_{M1}-l_{1M1}^2-3l_{1M1}^2{\mathrm{\γ}}_{M1}^2+3l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2-4l_{1Mp1}^2{\mathrm{\γ}}_{M1}^3)}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(-l_{1Mp1}^2{\mathrm{\γ}}_{M1}^4-2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}+2l_{1M1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1Mp1}^2{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1}+2l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^3-4l_{1M1}{l}_{1Mp1}{\mathrm{\γ}}_{M1}^2{\mathrm{ln\γ}}_{M1})}{{\mathrm{Eb\γ}}_{M1}^2{\mathrm{\β}}_1^3{({\mathrm{\γ}}_{M1}-1)}^3}\\ =100.18{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-E2}=\frac{4(L_M^3-l_{2M}^3)}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp2}^{3/2}-l_{2M}^{3/2})}{Eb}+\frac{4l_{1M2}^3}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}+\frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(-l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}\\ =104.73{\mathrm{mm}}^4/N;\end{array}$

$G_{x-EAT}=\frac{1}{\underset{j=1}{\overset{n}{\mathrm{\Σ}}}\frac{1}{G_{x-EAj}}}=77.53{\mathrm{mm}}^4/N;$

$\begin{array}{c}{G}_{x-EA1}=\frac{4(L_A^3-l_{2A}^3)}{Eb}-\frac{8l_{2A}^{3/2}(l_{1Ap1}^{3/2}-l_{2A}^{3/2})}{Eb}+\frac{4l_{1A1}^3}{{\mathrm{Eb\γ}}_{A1}^3{\mathrm{\β}}_{A1}^3}+\frac{6\mathrm{\Δ}l(4l_{1A1}^2{\mathrm{\γ}}_{A1}-l_{1A1}^2-3l_{1A1}^2{\mathrm{\γ}}_{A1}^2+3l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2-4l_{1Ap1}^2{\mathrm{\γ}}_{A1}^3)}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(-l_{1Ap1}^2{\mathrm{\γ}}_{A1}^4-2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}+2l_{1A1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^3-4l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1})}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}\\ =77.53{\mathrm{mm}}^4/N;\end{array}$

$G_{x-DE}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp2}^{1/2}-l_{2M}^{1/2})(l_{1Mp2}+l_{2M}-3l_0+l_{1Mp2}^{1/2}{l}_{2M}^{1/2})}{Eb}+$

$\begin{array}{c}\frac{2{(l_{1M2}-l_0)}^2(2l_{1M2}+l_0)}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}+\frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2+l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\\ \frac{24l_{1M2}{l}_{1Mp2}{\mathrm{\Δl\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\frac{6l_0\mathrm{\Δ}l(l_{1Mp2}{\mathrm{\γ}}_{M2}+l_{1M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3}=86.43{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-E_{z}2}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mp2}^{1/2}-l_{2M}^{1/2})(l_{1Mp2}+l_{2M}-3l_0+l_{1Mp2}^{1/2}{l}_{2M}^{1/2})}{Eb}+\frac{2{(l_{1M2}-l_0)}^2(2l_{1M2}+l_0)}{{\mathrm{Eb\γ}}_{M2}^3{\mathrm{\β}}_2^3}+\\ \frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3+l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}-\\ \frac{6l_0\mathrm{\Δ}l(l_{1Mp2}{\mathrm{\γ}}_{M2}+l_{1M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3}=86.43{\mathrm{mm}}^4/N;\end{array}$

$\begin{array}{c}{G}_{x-{\mathrm{DE}}_z}=\frac{12l_{2M}^{3/2}(6l_0^2{l}_{2M}^{1/2}+12l_0{l}_{1Mp2}{l}_{2M}^{1/2}-2l_{2M}^{1/2}{l}_{1Mp2}^2-6l_{1Mp2}^{1/2}{l}_0^2-12l_{1Mp2}^{1/2}{l}_0{l}_{2M}+2l_{1Mp2}^{1/2}{l}_{2M}^2)}{3l_{1Mp2}^{1/2}{l}_{2M}^{1/2}Eb}+\frac{4{(l_0-l_{1M2})}^3}{{\mathrm{Eb\β}}_2^3{\mathrm{\γ}}_{M2}^3}+\\ \frac{4(L_M-l_{2M})(L_M^2-3L_M{l}_0+L_M{l}_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\frac{6\mathrm{\Δ}l(2l_0^2{\mathrm{\γ}}_{M2}-l_{1M2}^2-l_0^2-2l_0^2{\mathrm{\γ}}_{M2}^3)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_M^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(4l_{1M2}^2{\mathrm{\γ}}_{M2}+l_0^2{\mathrm{\γ}}_{M2}^4-3l_{1M2}^2{\mathrm{\γ}}_{M2}^2+3l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2-4l_{1Mp2}^2{\mathrm{\γ}}_{M2}^3+l_{1Mp2}^2{\mathrm{\γ}}_{M2}^4+2l_0{l}_{1M2}-6l_0{l}_{1M2}{\mathrm{\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}-2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}+2l_{1M2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}+2l_{1Mp2}^2{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2}-2l_0{l}_{1M2}{\mathrm{\γ}}_{M2}^3-6l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2)}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(6l_0{l}_{1M2}{\mathrm{\γ}}_{M2}^2+6l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-2l_0{l}_{1Mp2}{\mathrm{\γ}}_{M2}^4+2l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^3-4l_{1M2}{l}_{1Mp2}{\mathrm{\γ}}_{M2}^2{\mathrm{ln\γ}}_{M2})}{{\mathrm{Eb\γ}}_{M2}^2{\mathrm{\β}}_2^3{({\mathrm{\γ}}_{M2}-1)}^3}\\ =72.75{\mathrm{mm}}^4/N;\end{array}$

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

Width b=60mm, the length Δ l of oblique line section according to the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula =30mm, elastic modulus E=200GPa.Half length L of auxiliary spring_A=525mm, auxiliary spring sheet number n=1, the root of this sheet auxiliary spring The thickness h of flat segments_2A=14mm, the root of parabolic segment is to distance l of auxiliary spring end points_2A=470mm, the end of oblique line section is arrived Distance l of auxiliary spring end points_1A1=87.50mm, the root of oblique line section is to distance l of spring end points_1Ap1=117.50mm；This sheet is secondary The thickness of the parabolic segment of spring compares β_A1=0.50, the thickness of oblique line section compares γ_A1=1.14；Half stiffness K to this sheet auxiliary spring_A1 Calculate, i.e.

$K_{A1}=\frac{h_{2A}^3}{G_{x-EA1}}=35.39N/mm;$

In formula,

$\begin{array}{c}{G}_{x-EA1}=\frac{4(L_A^3-l_{2A}^3)}{Eb}-\frac{8l_{2A}^{3/2}(l_{1Ap1}^{3/2}-l_{2A}^{3/2})}{Eb}+\frac{4l_{1A1}^3}{{\mathrm{Eb\γ}}_{A1}^3{\mathrm{\β}}_{A1}^3}+\frac{6\mathrm{\Δ}l(4l_{1A1}^2{\mathrm{\γ}}_{A1}-l_{1A1}^2-3l_{1A1}^2{\mathrm{\γ}}_{A1}^2+3l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2-4l_{1Ap1}^2{\mathrm{\γ}}_{A1}^3)}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}-\\ \frac{6\mathrm{\Δ}l(-l_{1Ap1}^2{\mathrm{\γ}}_{A1}^4-2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}+2l_{1A1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1Ap1}^2{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1}+2l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^3-4l_{1A1}{l}_{1Ap1}{\mathrm{\γ}}_{A1}^2{\mathrm{ln\γ}}_{A1})}{{\mathrm{Eb\γ}}_{A1}^2{\mathrm{\β}}_{A1}^3{({\mathrm{\γ}}_{A1}-1)}^3}\\ =77.53{\mathrm{mm}}^4/N;\end{array}$

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

I step: end points power P of each main spring_iCalculate:

According to the half the most single-ended point load P=that the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is loaded 3040N, auxiliary spring works load p_K=2404.2N, main reed number m=2；Calculated K in I step_M1=13.29N/mm and K_M2=12.71N/mm, and II step calculate obtained K_MA1=13.29N/mm and K_MA2=35.94N/mm, to the 1st master Spring and end points power P of the 2nd main spring₁And P₂It is respectively calculated, i.e.

$P_1=\frac{K_{M1}{P}_K}{2\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{K}_{Mi}}+\frac{K_{MA1}(2P-P_K)}{2\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{K}_{MAi}}=1110.6N;$

$P_2=\frac{K_{M2}{P}_K}{2\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{K}_{Mi}}+\frac{K_{MA2}(2P-P_K)}{2\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{K}_{MAi}}=1929.4N;$

Ii step: each auxiliary spring end points power P_AjCalculating:

According to the half the most single-ended point load P=that the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is loaded 3040N, auxiliary spring works load p_K=2404.2N, main reed number m=2, the thickness h of the root flat segments of each main spring_2M= 11mm；Auxiliary spring sheet number n=1, the thickness h of the root flat segments of this sheet auxiliary spring_2ACalculated K in=14mm, II step_MA1= 13.29N/mm、K_MA2=35.94N/mm, G_x-DE=86.43mm⁴/N、G_x-DEz=72.75mm⁴/N、G_x-EAT=77.53mm⁴/ N, and Calculated K in III step_A1=35.39N/mm, end points power P to this sheet auxiliary spring_A1Calculate, i.e.

$P_{A1}=\frac{K_{A1}{K}_{MA2}{G}_{x-DE}{h}_{2A}^3(2P-P_K)}{2\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{K}_{Aj}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{K}_{MAi}(G_{x-EAT}{h}_{2M}^3+G_{x-{\mathrm{DE}}_z}{h}_{2A}^3)}=1050.8N;$

(3) each main spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula stress at diverse location x Calculate:

Step A: the 1st main spring Stress calculation at diverse location x:

According to the width b=60mm of the few sheet main spring of reinforcement end variable cross-section of ends contact formula, half length L of main spring_M =575mm, the root of main spring parabolic segment is to distance l of main spring end points_2M=520mm, the thickness of the root flat segments of each main spring Degree h_2M=11mm, main reed number m=2, the thickness of the parabolic segment of the 1st main spring compares β₁The oblique line section of the=0.55, the 1st main spring Root to distance l of spring end points_1Mp1=157.30mm, end thickness h of parabolic segment_1Mp1=6mm, the end of the 1st main spring The thickness h of portion's flat segments_1M1=7mm and length l_1M1=127.30mm；And calculated P in i step₁=1110.6N, with master Spring end points is zero, and the 1st main spring of sheet reinforcement end variable cross-section major-minor spring few to this ends contact formula is at not coordination Put the stress at x to calculate, i.e.

${\mathrm{\σ}}_{M1}=\left\{\begin{array}{cc}\frac{6P_{1}x}{{\mathrm{bh}}_{1M1}^2}=2.27xMPa,& x\∈\[0,127.30\]mm\\ \frac{6P_{1}x}{{\mathrm{bh}}_{2M1}^2\left(x\right)}=\frac{111.06x}{{(-33.33x+11.24)}^2}MPa,& x\∈(127.30,157.30\]mm\\ \frac{6P_{1}x}{{\mathrm{bh}}_{2MP1}^2\left(x\right)}=477.29MPa,& x\∈(157.30,520\]mm\\ \frac{6P_{1}x}{{\mathrm{bh}}_{2M}^2}=0.92xMPa,& x\∈(520,575\]mm\end{array}\right.;$

In formula, h_2M1(x)=-33.33x+11.24,Wherein, the 1st main spring obtained by calculating At the stress changing curve of various location, as shown in Figure 3；

Step B: the 2nd main spring Stress calculation at diverse location x:

According to the width b=60mm of the few sheet main spring of reinforcement end variable cross-section of ends contact formula, half length L of main spring_M =575mm, the thickness h of the root flat segments of each main spring_2M=11mm, the root of parabolic segment is to distance l of main spring end points_2M =520mm, main reed number m=2, wherein, the thickness of the parabolic segment of the 2nd main spring compares β₂=0.45, the root of oblique line section arrives Distance l of spring end points_1Mp2=105.30mm, end thickness h of parabolic segment_1Mp2=5mm, the end flat segments of the 2nd main spring Thickness h_1M2=6mm and length l_1M2=75.30mm；Auxiliary spring contact and horizontal range l of main spring end points₀In=50mm, i step Calculated P₂Calculated P in=1929.4N, ii step_A1=1050.8N, with main spring end points as zero, to this 2nd main spring stress at diverse location x of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula calculates, i.e.

${\mathrm{\σ}}_{M2}=\left\{\begin{array}{cc}\frac{6P_{2}x}{{\mathrm{bh}}_{1M2}^2}=5.36xMPa,& x\∈\[0,50\]mm\\ \frac{6\[P_{2}x-\underset{j=1}{\overset{1}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{1M2}^2}=2.78\×(0.88x+52.54)MPa,& x\∈(50,127.30\]mm\\ \frac{6\[P_{2}x-\underset{j=1}{\overset{1}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2M2}^2\left(x\right)}=100\×\frac{(0.88x+52.54)}{{(-33.33x+8.51)}^2}MPa,& x\∈(127.30,157.30\]mm\\ \frac{6\[P_{2}x-\underset{j=1}{\overset{1}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2Mp2}^2\left(x\right)}=223.47\×\frac{(0.88x+52.54)}{x}Mpa,& x\∈(157.30,520\]mm\\ \frac{6\[P_{2}x-\underset{j=1}{\overset{1}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2M}^2}=0.83\×(0.88x+52.54)MPa,& x\∈(520,575\]mm\end{array}\right.;$

In formula, h_2M2(x)=-33.33x+8.51,Wherein, the 2nd main spring obtained by calculating Stress changing curve at diverse location x, as shown in Figure 4；

(4) each auxiliary spring Stress calculation of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula:

According to the width b=60mm of the few sheet reinforcement end variable cross-section auxiliary spring of ends contact formula, half length L of auxiliary spring_A =525mm, the thickness h of the root flat segments of auxiliary spring_2A=14mm, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2A =470mm, auxiliary spring sheet number n=1, the thickness of the parabolic segment of this sheet auxiliary spring compares β_A1=0.50, the end of parabolic segment is to auxiliary spring Distance l of end points_1Ap1=117.50mm, the end thickness of parabolic segment is h_1Ap1=7mm, the end flat segments of this sheet auxiliary spring Thickness h_1A1=8mm and length l_1A1=87.50mm；And calculated P in ii step_A1=1053.1N, with auxiliary spring end points for sitting Mark initial point, 1 auxiliary spring of sheet reinforcement end variable cross-section major-minor spring few to this ends contact formula stress at diverse location x enters Row calculates, i.e.

${\mathrm{\σ}}_{A1}=\left\{\begin{array}{cc}\frac{6P_{A1}x}{{\mathrm{bh}}_{1A1}^2}=1.64xMPa,& x\∈\[0,87.50\]\\ \frac{6P_{A1}x}{{\mathrm{bh}}_{2A1}^2\left(x\right)}=\frac{105.08x}{{(-33.33x+10.92)}^2}MPa,& x\∈(87.50,117.50\]\\ \frac{6P_{A1}x}{{\mathrm{bh}}_{2Ap1}^2\left(x\right)}=251.99MPa,& x\∈(117.50,470\]\\ \frac{6P_{A1}x}{{\mathrm{bh}}_{2A}^2}=0.54xMPa,& x\∈(470,525\]\end{array}\right.;$

In formula, h_2A1(x)=-33.33x+10.92,Wherein, 1 auxiliary spring obtained by calculating exists Stress changing curve at diverse location x, as shown in Figure 5.

Utilize ANSYS finite element emulation software, according to the few sheet reinforcement end variable cross-section major-minor spring of this ends contact formula Each main spring and the structural parameters of auxiliary spring and elastic modelling quantity, set up the ANSYS phantom of half symmetrical structure major-minor spring, divides Grid, arranges auxiliary spring end points and contacts with main spring, and at the root applying fixed constraint of phantom, applies collection at major-minor spring end points Middle load F=P-P_K/ 2=1837.9N, the major-minor spring of sheet reinforcement end variable-section steel sheet spring few to this ends contact formula Stress carries out ANSYS emulation, the ANSYS stress simulation cloud atlas of the 1st obtained main spring, as shown in Figure 6；2nd main spring ANSYS stress simulation cloud atlas, as shown in Figure 7；The ANSYS stress simulation cloud atlas of 1 auxiliary spring, as shown in Figure 8, wherein, the 1st master Spring stress σ at parabolic segment with root contact position_MA1=213.86MPa, the 2nd main spring are in parabolic segment and oblique line section Stress σ at contact position_MA2=337.61MPa, the 1 auxiliary spring stress σ at parabolic segment with oblique line section contact position_A1= 253.79MPa。

Understand, in the case of same load, the 1st and the 2nd main spring of this leaf spring and the ANSYS of 1 auxiliary spring stress Simulating, verifying value σ_MA1=213.86MPa, σ_MA2=337.61MPa, σ_A1=253.79MPa, respectively with analytical Calculation value σ_MA1= 213.22MPa、σ_MA2=339.45MPa, σ_A1=251.99MPa matches, relative deviation is respectively 0.30%, 0.55%, 0.71%；Result shows the computational methods of the few sheet reinforcement end each stress of major-minor spring of ends contact formula that this invention is provided Being correct, each main spring and auxiliary spring are accurate, reliable in the Stress calculation value of various location.

Claims (1)

1. the computational methods of the few sheet reinforcement end each stress of major-minor spring of ends contact formula, wherein, the few bit end of ends contact formula The half symmetrical structure of portion's reinforced major-minor spring is made up of root flat segments, parabolic segment, oblique line section and end flat segments 4 sections, Booster action is played in the end of variable cross-section major-minor spring by oblique line section；End flat segments non-the grade structure, i.e. the 1st main spring of each main spring The thickness of end flat segments and length, more than the thickness of end flat segments and the length of other each main spring, meet the 1st main spring The requirement of complicated applied force；It is provided with major-minor spring gap between auxiliary spring contact and main spring end flat segments, works load meeting auxiliary spring The design requirement of lotus；When load works load more than auxiliary spring, when the contact of major-minor spring works together, being subject to of each main spring and auxiliary spring Power and unequal at the stress of various location；Work load at each main spring and the structural parameters of auxiliary spring, elastic modelling quantity, auxiliary spring Lotus, major-minor spring bear load given in the case of, each main spring of sheet reinforcement end variable cross-section major-minor spring few to end contact Calculating at the stress of various location with auxiliary spring, concrete calculation procedure is as follows:

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

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

According to the width b of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elastic modelling quantity E；Half length L of main spring_M, the root of parabolic segment is to distance l of spring end points_2M, the thickness of the root flat segments of each main spring Degree h_2M, main reed number m, wherein, the thickness of the parabolic segment of i-th main spring compares β_i, the thickness of oblique line section compares γ_Mi, oblique line section Root is to distance l of main spring end points_1Mpi, the end of oblique line section is to distance l of main spring end points_1Mi, i=1,2 ..., m；To major-minor spring The half stiffness K of each main spring before contact_MiCalculate, i.e.

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

In formula, G_X-EiFor the end points deformation coefficient of i-th main spring in the case of end points power effect, i.e.

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

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

According to the width b of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elastic modelling quantity E；Half length L of main spring_M, the thickness h of the root flat segments of each main spring_2M, the root of main spring parabolic segment is to spring end points Distance l_2M, main reed number m, wherein, the thickness of the parabolic segment of i-th main spring compares β_i, the thickness of oblique line section compares γ_Mi, oblique line The root of section is to distance l of main spring end points_1Mpi, the end of oblique line section is to distance l of main spring end points_1Mi；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 parabolic segment is to distance l of auxiliary spring end points_2A, auxiliary spring sheet number n, Wherein, the thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, the thickness of oblique line section compares γ_Aj, the root of oblique line section is to secondary end points Distance l_1Apj, the end of oblique line section is to distance l of auxiliary spring end points_1Aj, j=1,2 ..., n；Auxiliary spring contact and the level of main spring end points Distance l₀, the half stiffness K of each main spring after major-minor spring is contacted_MAiCalculate, i.e.

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

In formula, G_X-EiEnd points deformation coefficient for i-th main spring in the case of end points power effect；G_X-EATFor in end points power effect In the case of total end points deformation coefficient of n sheet superposition auxiliary spring, G_X-EAjEnd for the jth sheet auxiliary spring in the case of end points power effect Point deformation coefficient；G_x-DEFor the deformation at end flat segments with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions Coefficient；G_x-EzmEnd points deformation coefficient for the main spring of m sheet under stressing conditions at major-minor spring contact point；G_x-EzFor at major-minor spring The main spring of m sheet under stressing conditions deformation coefficient at end flat segments with auxiliary spring contact point at contact point, i.e.

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

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

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

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

$\begin{array}{c}{G}_{x-DE}=\frac{4L_M^3-6l_0{L}_M^2-4l_{2M}^3+6l_0{l}_{2M}^2}{Eb}-\frac{8l_{2M}^{3/2}(l_{1Mpm}^{1/2}-l_{2M}^{1/2})(l_{1Mpm}+l_{2M}-3l_0+l_{1Mpm}^{1/2}{l}_{2M}^{1/2})}{Eb}+\\ \frac{2{(l_{1Mm}-l_0)}^2\left(2l_{1Mm}+l_0\right)}{{\mathrm{Eb\γ}}_{Mn}^3{\mathrm{\β}}_m^3}+\frac{6\mathrm{\Δ}l(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2-2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4+2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3)}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}-\\ \frac{24l_{1Mm}{l}_{1Mpm}{\mathrm{\Δl\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}-\frac{6l_0\mathrm{\Δ}l(l_{1Mpm}{\mathrm{\γ}}_{Mm}+l_{1Mm})}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3}\end{array};$

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

$\begin{array}{c}{G}_{x-{\mathrm{DE}}_z}=\frac{\mathrm{12}{l}_{2M}^{3/2}({\mathrm{6l}}_0^2{l}_{2M}^{1/2}+12l_0{l}_{1Mpm}{l}_{2M}^{1/2}-2l_{2M}^{1/2}{l}_{1Mpm}^2-6l_{1Mpm}^{1/2}{l}_0^2-12l_{1Mpm}^{1/2}{l}_0{l}_{2M}+2l_{1Mpm}^{1/2}{l}_{2M}^2)}{3l_{1Mpm}^{1/2}{l}_{2M}^{1/2}Eb}+\frac{4{(l_0-l_{1Mm})}^3}{{\mathrm{Eb\β}}_m^3{\mathrm{\γ}}_{Mm}^3}+\\ \frac{4(L_M-l_{2M})(L_M^2-3L_M{l}_0+L_M{l}_{2M}+3l_0^2-3l_0{l}_{2M}+l_{2M}^2)}{Eb}-\frac{6\mathrm{\Δ}l({\mathrm{2l}}_0^2{\mathrm{\γ}}_{Mm}-l_{1Mm}^2-l_0^2-2l_0^2{\mathrm{\γ}}_{Mm}^3)}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_M^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(4l_{1Mm}^2{\mathrm{\γ}}_{Mm}-l_0^2{\mathrm{\γ}}_{Mm}^4-3l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2+3l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3-4l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^3+l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^4+2l_0{l}_{1Mm}-6l_0{l}_{1Mm}{\mathrm{\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}-l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}+2l_{1Mm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}+2l_{1Mpm}^2{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm}-2l_0{l}_{1Mm}{\mathrm{\γ}}_{Mm}^3-6l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^2)}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}+\\ \frac{6\mathrm{\Δ}l(2l_0{l}_{1Mm}{\mathrm{\γ}}_{Mm}^2+6l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3-2l_0{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^4+2l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^3-4l_{1Mm}{l}_{1Mpm}{\mathrm{\γ}}_{Mm}^2{\mathrm{ln\γ}}_{Mm})}{{\mathrm{Eb\γ}}_{Mm}^3{\mathrm{\β}}_m^3{({\mathrm{\γ}}_{Mm}-1)}^3}\end{array};$

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

According to the width b of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elastic modulus E； 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 parabolic segment is to auxiliary spring end points Distance l_2A, auxiliary spring sheet number n, wherein, the thickness of the parabolic segment of jth sheet auxiliary spring compares β_Aj, the thickness of oblique line section compares γ_Aj, oblique line section Root to distance l of auxiliary spring end points_1Apj, the end of oblique line section is to distance l of auxiliary spring end points_1Aj, firm to the half of each auxiliary spring Degree K_AjCalculate, i.e.

$K_{Aj}=\frac{h_{2A}^3}{G_{x-EAj}};$

In formula,

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

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

I step: end points power P of each main spring_iCalculate:

According to the half the most single-ended point load P that the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is loaded, auxiliary spring Work load p_K, calculated K in I step_Mi, and II step calculates obtained K_MAi, end points power to each main spring P_iCalculate, i.e.

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

Ii step: end points power P of each auxiliary spring_AjCalculate:

According to the half the most single-ended point load P that the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is loaded, auxiliary spring Work load p_K, main reed number m, the thickness h of the root flat segments of each main spring_2M, auxiliary spring sheet number n, the root of each auxiliary spring The thickness h of flat segments_2A, calculated K in II step_MAi、G_x-DE、G_x-DEzAnd G_x-EAT, and calculated in III step K_Aj, end points power P to each auxiliary spring_AjCalculate, i.e.

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

(3) calculating of each main spring diverse location stress of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula:

Step A: the calculating of stress at front m-1 sheet main spring diverse location x:

Half length L according to the few sheet main spring of reinforcement end variable cross-section of ends contact formula_M, the length Δ l of oblique line section, each main spring The thickness h of root flat segments_2M, the root of main spring parabolic segment is to distance l of main spring end points_2M, main reed number m, wherein, i-th End thickness h of the parabolic segment of the main spring of sheet_1Mpi, the thickness h of the end flat segments of i-th main spring_1Mi, the parabolic of i-th main spring The end of line segment is to distance l of main spring end points_1Mpi, length l of the end flat segments of i-th main spring_1Mi；And i step calculates The P arrived_i, with main spring free end as zero, with main spring end points as zero, to few sheet reinforcement end variable cross-section steel plates The front main spring of m-1 sheet of spring stress at diverse location x calculates, i.e.

${\mathrm{\σ}}_{Mi}=\left\{\begin{array}{cc}\frac{6P_{i}x}{{\mathrm{bh}}_{1Mi}^2},& x\∈\[0,l_{1Mi}\]\\ \frac{6P_{i}x}{{\mathrm{bh}}_{2Mi}^2\left(x\right)},& x\∈(l_{1Mi},l_{1Mpi}\]\\ \frac{6P_{i}x}{{\mathrm{bh}}_{2Mpi}^2\left(x\right)},& x\∈(l_{1Mpi},l_{2M}\]\\ \frac{6P_{i}x}{{\mathrm{bh}}_{2M}^2},& x\∈(l_{2M},L_M\]\end{array}\right.,i=1,2,...,m-1;$

In formula, h_2MiX () is i-th main spring oblique line section thickness at x position, h_2MpiX () is that i-th main spring parabolic segment is at x The thickness of position, i.e.

$h_{2Mi}\left(x\right)=\frac{h_{1Mpi}-h_{1Mi}}{\mathrm{\Δ}l}x+\frac{h_{1Mi}{l}_{1Mpi}-h_{1Mpi}{l}_{1Mi}}{\mathrm{\Δ}l},h_{2Mpi}\left(x\right)=h_{2M}\sqrt{\frac{x}{l_{2M}}};$

Step B: the calculating of stress at m sheet main spring diverse location x:

Half length L according to the few sheet main spring of reinforcement end variable cross-section of ends contact formula_M, the length Δ l of oblique line section, each main spring The thickness h of root flat segments_2M, the root of parabolic segment is to distance l of spring end points_2M, auxiliary spring contact and the water of main spring end points Flat distance l₀；Main reed number m, wherein, end thickness h of the parabolic segment of the main spring of m sheet_1Mpm, the parabolic segment of the main spring of m sheet End to distance l of main spring end points_1Mpm, length l of the end flat segments of the main spring of m sheet_1MmAnd thickness h_1Mm；And in i step Calculated P_m, calculated P in ii step_Aj, with main spring end points as zero, to few sheet reinforcement end variable cross-section The main spring of m sheet of leaf spring stress at diverse location x calculates, i.e.

${\mathrm{\σ}}_{Mm}=\left\{\begin{array}{cc}\frac{6P_{m}x}{{\mathrm{bh}}_{1Mm}^2},& x\∈\[0,l_0\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{1Mm}^2},& x\∈(l_0,l_{1Mm}\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2Mm}^2\left(x\right)},& x\∈(l_{1Mm},l_{1Mpm}\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2Mpm}^2\left(x\right)},& x\∈(l_{1Mpm},l_{2M}\]\\ \frac{6\[P_{m}x-\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{P}_{Aj}(x-l_0)\]}{{\mathrm{bh}}_{2M}^2},& x\∈(l_{2M},L_M\]\end{array}\right.;$

In formula, h_2MmX () is m sheet main spring oblique line section thickness at x position, h_2MpmX () is that m sheet main spring parabolic segment is at x The thickness of position, i.e.

$h_{2Mm}\left(x\right)=\frac{h_{1Mpm}-h_{1Mm}}{\mathrm{\Δ}l}x+\frac{h_{1Mm}{l}_{1Mpm}-h_{1Mpm}{l}_{1Mm}}{\mathrm{\Δ}l},h_{2Mpm}\left(x\right)=h_{2M}\sqrt{\frac{x}{l_{2M}}};$

(4) each auxiliary spring of the few sheet reinforcement end variable cross-section major-minor spring of ends contact formula is in the calculating of diverse location stress:

Half length L according to few sheet reinforcement end variable cross-section auxiliary spring_A, width b, the thickness h of auxiliary spring root flat segments_2A, tiltedly The length Δ l of line segment, the root of auxiliary spring parabolic segment is to distance l of auxiliary spring end points_2A, auxiliary spring sheet number n, wherein, jth sheet auxiliary spring The end of parabolic segment is to distance l of auxiliary spring end points_1Apj, the end thickness of auxiliary spring parabolic segment is h_1Apj, the thickness of end flat segments Degree h_1AjWith length l_1Aj；And calculated P in ii step_Aj, j=1,2 ..., n, with auxiliary spring free end as zero, to few The each auxiliary spring of sheet reinforcement end variable-section steel sheet spring stress at diverse location x calculates, i.e.

${\mathrm{\σ}}_{Aj}=\left\{\begin{array}{cc}\frac{6P_{Aj}x}{{\mathrm{bh}}_{1Aj}^2},& x\∈\[0,l_{1Aj}\]\\ \frac{6P_{Aj}x}{{\mathrm{bh}}_{2Aj}^2\left(x\right)},& x\∈(l_{1Aj},l_{1Apj}\]\\ \frac{6P_{Aj}x}{{\mathrm{bh}}_{2Apj}^2\left(x\right)},& x\∈(l_{1Apj},l_{2A}\]\\ \frac{6P_{Aj}x}{{\mathrm{bh}}_{2A}^2},& x\∈(l_{2A},L_A\]\end{array}\right.,j=1,2,...,n;$

In formula, h_2AjX () is jth sheet auxiliary spring oblique line section thickness at x position, h_2ApjX () is that jth sheet auxiliary spring parabolic segment is at x The thickness of position, i.e.

$h_{2Aj}\left(x\right)=\frac{h_{1Apj}-h_{1Aj}}{\mathrm{\Δ}l}x+\frac{h_{1Aj}{l}_{1Apj}-h_{1Apj}{l}_{1Aj}}{\mathrm{\Δ}l},h_{2Apj}\left(x\right)=h_{2A}\sqrt{\frac{x}{l_{2A}}}.$