CN105912743A - Method of designing gap between end of root-reinforced few-leaf variable-section main spring at end and auxiliary spring - Google Patents

Abstract

The invention relates to a method of designing the gap between the end of a root-reinforced few-leaf variable-section main spring and an auxiliary spring, and belongs to the technical field of suspension leaf springs. First, the endpoint deformation coefficient of each leaf of main spring and the deformation coefficient G(x-DE) of the contact point between the end straight section of an Nth leaf and an auxiliary spring are determined according to the structure size and elastic modulus of each leaf of root-reinforced variable-section main spring; then, the endpoint force F(N) of the Nth leaf of main spring is obtained according to the auxiliary spring working load and the deformation coefficient of the endpoint of each leaf of main spring; and next, the main-auxiliary spring gap between the end straight section of the main spring and the contact of the auxiliary spring is designed according to the root thickness h2 of the Nth leaf of main spring, F(N) and G(x-DE). Simulation verification shows that an accurate main-auxiliary spring gap value can be designed using the method, the design requirements of the auxiliary spring working load are met, and the product design level and performance and the vehicle riding comfort are improved. Moreover, product development is speeded up, and the design and test costs are reduced.

Description

The reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of root

Technical field

The present invention relates to the reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of vehicle suspension leaf spring, particularly root.

Background technology

In order to meet vehicle suspension variation rigidity design requirement under different loads, few sheet variable-section steel sheet spring is generally designed as major and minor spring, and wherein, main spring is designed with certain gap at the contact that connects with auxiliary spring, guaranteeing after more than certain load, major and minor spring contacts and cooperatively works.Owing to the 1st its stress of few main spring of sheet variable cross-section is complicated, it is subjected to vertical load, simultaneously also subject to torsional load and longitudinal loading, therefore, the end thickness of the 1st leaf spring designed by reality, generally than other each the thickest, the most mostly use few sheet variable-section steel sheet spring of end structure such as non-grade；Meanwhile, in order to strengthen the stress intensity of few sheet variable-section steel sheet spring, between main spring root flat segments and parabolic segment, generally set up an oblique line section, i.e. use few main spring of sheet variable cross-section that root is reinforced.Further, since for the design requirement meeting major-minor spring different composite rigidity, generally use the auxiliary spring of different length, the most main spring is the most different from the contact position of auxiliary spring, therefore, end flat segments contact can be divided into contact with parabolic segment.Then, the deformation located at an arbitrary position due to the reinforced few sheet variable-section steel sheet spring of root calculates extremely complex, therefore, fails to provide the method for designing in the reinforced main spring of sheet less of root major and minor spring gap at end flat segments with auxiliary spring contact point the most always.

Although previously, once someone gave the method for designing of few sheet variable-section steel sheet spring, such as, Peng Mo, high army is once in " automobile engineering ", (volume 14) the 3rd phase in 1992, propose the design and calculation method of Varied section leaf spring, the method is primarily directed to few sheet parabolic type variable-section steel sheet spring of the structures such as end and is designed, its weak point is to meet the design requirement of the non-few sheet variable-section steel sheet spring waiting structure of root, more can not meet the design in the reinforced few main spring of sheet variable cross-section of root major and minor spring gap at end flat segments with auxiliary spring contact point.At present, deformation despite people's once few sheet variable cross-section main spring reinforced to root, use ANSYS modeling and simulating method, but the method can only carry out simulating, verifying to the deformation of the few sheet variable-section steel sheet spring providing actual design structure, it is not provided that accurate analytical design method formula, to meet the requirement of analytical design method, vehicle fast development and the requirement to the modernization CAD design software development of suspension leaf spring more can not be met.

Therefore, must be set up the method for designing in the reinforced few main spring of sheet of a kind of root accurate, reliable major and minor spring gap at end flat segments with auxiliary spring contact point, meet Vehicle Industry fast development and the requirement to few major and minor Precise Design for Laminated Spring of sheet variable cross-section, improve the design level of variable-section steel sheet spring, product quality and performances, improve vehicle ride performance；Meanwhile, reduce design and testing expenses, accelerate product development speed.

Summary of the invention

For defect present in above-mentioned prior art, the technical problem to be solved is to provide a kind of simplicity, the reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of reliable root, its design cycleFigure such as figure 1Shown in.The reinforced few sheet variable cross-section major-minor spring of root is symmetrical structure, the half spring of symmetrical structure can see cantilever beam as, will symmetrical center line as the fixing end of root of half spring, using the end stress point of main spring and the contact of auxiliary spring as main spring end points and auxiliary spring end points.The half symmetrical structure signal of the reinforced few sheet variable cross-section major-minor spring of rootFigure,Such as figure 2Shown in, including, main spring 1, root shim 2, auxiliary spring 3, end pad 4；Main spring 1 is made up of N sheet, and the half of each total length is L, is to be made up of root flat segments, oblique line section, parabolic segment and end flat segments 4 sections, and wherein, tapered spring is played booster action by oblique line section；The thickness of the root flat segments of every main spring is h₂, the half of installing space is l₃, a length of Δ l of oblique line section, the root thickness of parabolic segment is h_2p, i.e. the thickness of oblique line section is than γ=h_2p/h₂；1 each end flat segments of main spring is non-structure such as grade, and the thickness of i.e. the 1st main spring end flat segments and length, more than other thickness of each and length, thickness and the length of each end flat segments are respectively h_1iAnd l_1i；The thickness of each parabolic segment compares β_i=h_1i/h_2p, i=1,2 ..., N；Each root flat segments of main spring 1 and and the root flat segments of auxiliary spring 3 between be provided with root shim 2, be provided with end pad 4 between the end flat segments of main spring 1, the material of end pad carbon fibre composite, in order to reduce frictional noise during spring works；The a length of L of half of auxiliary spring 3_A, i.e. auxiliary spring end points is to horizontal range l of main spring end points₀=L-L_A, it is provided with certain major and minor spring gap delta between end flat segments and the ends points of auxiliary spring 3 of the N sheet of main spring 1, when load works load more than auxiliary spring, auxiliary spring contacts with certain point in the flat segments of main spring end.Each chip architecture parameter of main spring, material characteristic parameter, auxiliary spring length, auxiliary spring work load given in the case of, few sheet variable cross-section main spring major-minor spring gap end flat segments and auxiliary spring contact between reinforced to root is designed.

For solving above-mentioned technical problem, the reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of root provided by the present invention, it is characterised in that use following design procedure:

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

According to half length L of few sheet root main spring of reinforced variable cross-section, width b, half l of installing space₃, the length Δ l of oblique line section, the root of parabolic segment is to distance l of main spring end points_2p=L-l₃-Δ l, the root of oblique line section is to distance l of main spring end points₂=L-l₃, elastic modulus E, the thickness of oblique line section compares β than γ, the thickness of the parabolic segment of i-th main spring_i, wherein, i=1,2 ..., N, N are main reed number, the end points deformation coefficient G to each main spring_x-EiCalculate, i.e.

(2) the main springs of N sheet are at the deformation coefficient G of end flat segments Yu auxiliary spring contact point_x-DECalculate:

According to half length L of few sheet root main spring of reinforced variable cross-section, width b, the length Δ l of oblique line section, the root of parabolic segment is to distance l of main spring end points_2p, the root of oblique line section is to distance l of main spring end points₂, elastic modulus E, the thickness of oblique line section compares β than γ, the thickness of the parabolic segment of the main spring of N sheet_N, auxiliary spring contact and horizontal range l of main spring end points₀, the spring main to N sheet deformation coefficient G at end flat segments with auxiliary spring contact point_x-DECalculate, i.e.

(3) auxiliary spring works end points power F of the main spring of N sheet under load_NCalculate:

I step: according to the thickness h of the root flat segments of few sheet root main spring of reinforced variable cross-section₂, and the end points deformation coefficient G of calculated each main spring in step (1)_x-Ei, determine the half stiffness K of each main spring_Mi, i.e.

II step: the half the most single-ended point load P of the load that works according to auxiliary spring, and the half stiffness K of each main spring determined by I step_Mi, end points power F of the main spring of N sheet under load that auxiliary spring is worked_NCalculate, i.e.

In formula, K_MNIt is the half rigidity of the main spring of N sheet.

(4) the root reinforced few main spring of sheet variable cross-section major-minor spring gap delta design between end flat segments and auxiliary spring contact:

The thickness h of the root flat segments according to main spring₂, in II step, calculated auxiliary spring works end points power F of the main spring of N sheet under load_N, and the G obtained by the middle calculating of step (2)_x-DE, few sheet variable cross-section main spring major-minor spring gap delta end flat segments and auxiliary spring contact between reinforced to root is designed, i.e.

The present invention has the advantage that than prior art

It is extremely complex that the reinforced few sheet variable-section steel sheet spring of root deforms calculating at an arbitrary position, therefore, fails to provide the method for designing in the reinforced main spring of sheet less of root major and minor spring gap at end flat segments with auxiliary spring contact point the most always.

The present invention can be according to the root reinforced few physical dimension of the main spring of sheet variable cross-section, elastic modelling quantity, it is first determined go out each main spring deformation coefficient at end points, and the deformation coefficient that the main spring of N is at end flat segments with auxiliary spring contact point；Then, by the deformation coefficient at each end points and rigidity, the load that the main spring of N sheet is shared is obtained at end points；Subsequently, root thickness according to the main spring of N sheet and the shared load of end points, and the deformation coefficient at end flat segments with auxiliary spring contact point, few sheet variable cross-section main spring major and minor spring gap end flat segments with auxiliary spring contacting points position at reinforced to root is designed.

By design example and ANSYS simulating, verifying, the reinforced few main spring of sheet variable cross-section of root accurate, reliable major and minor spring gap parameter design load at end flat segments with auxiliary spring contact point that the method can get, for the major and minor spring gap design in the flat segments of end of the reinforced few sheet variable-section steel sheet spring of root, provide reliable method for designing, and establish reliable technical foundation for CAD software exploitation.Utilize the method, the design level of few major and minor leaf spring of sheet variable cross-section, product quality and performances can be improved, reduce bearing spring quality and cost, improve vehicle ride performance；Meanwhile, also reduce design and testing expenses, accelerate product development speed.

Accompanying drawing explanation

In order to be more fully understood that the present invention, below in conjunction withAccompanying drawingIt is described further.

Figure 1It it is the root reinforced few main spring of the sheet design cycle in end flat segments Yu auxiliary spring gapFigure；

Figure 2It it is the half structural representation of the reinforced few sheet variable cross-section major-minor spring of rootFigure；

Figure 3It it is the deformation simulation cloud of the reinforced few main spring of sheet variable cross-section of embodiment one rootFigure；

Figure 4It it is the deformation simulation cloud of the reinforced few main spring of sheet variable cross-section of embodiment two rootFigure。

Specific embodiments

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

Embodiment one: the main reed number N=2 of certain few reinforced variable-section steel sheet spring of sheet root, wherein, half length L=575mm of each main spring, width b=60mm, elastic modulus E=200GPa, the thickness h of root flat segments₂=11mm, half l of installing space₃=55mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2p=L-l₃-Δ l=490mm, the root of oblique line section is to distance l of main spring end points₂=L-l₃=520mm；Root thickness h of parabolic segment_2p=10.23mm, the thickness of oblique line section is than γ=h_2p/h₂=0.93；The thickness h of the end flat segments of the 1st main spring₁₁=7mm, the thickness of the parabolic segment of the 1st main spring compares β₁=h₁₁/h_2p=0.69；The thickness h of the end flat segments of the 2nd main spring₁₂=6mm, the thickness of the parabolic segment of the 2nd main spring compares β₂=h₁₂/h_2p=0.59；Half length L of auxiliary spring_AHorizontal range l of=465mm, auxiliary spring contact and main spring end points₀=L-L_A=110mm, auxiliary spring is positioned at main spring end flat segments with the contact point of main spring.The designed auxiliary spring required works the half the most single-ended point load P=1200N of load, is designed the main spring of this few reinforced variable-section steel sheet spring of the sheet root major-minor spring gap between end flat segments and auxiliary spring contact.

The reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of root that present example is provided, its design cycleSuch as figure 1Shown in, specifically comprise the following steps that

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

According to half length L=575mm of few sheet root main spring of reinforced variable cross-section, width b=60mm, elastic modulus E=200GPa, half l of installing space₃=55mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2p=490mm, the root of oblique line section is to distance l of main spring end points₂=520mm, the thickness of oblique line section compares β than γ=0.93, the thickness of the parabolic segment of the 1st main spring₁The thickness of the parabolic segment of the=0.69, the 2nd main spring compares β₂=0.59, the end points deformation coefficient G to the 1st and the 2nd main spring_x-E1、G_x-E2Calculate, be respectively

(2) the 2nd main springs are at the deformation coefficient G of end flat segments Yu auxiliary spring contact point_x-DECalculate:

According to half length L=575mm of few sheet root main spring of reinforced variable cross-section, width b=60mm, elastic modulus E=200GPa, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2p=490mm, the root of oblique line section is to distance l of main spring end points₂=520mm, the thickness of oblique line section compares β than γ=0.93, the thickness of the parabolic segment of the 2nd main spring₂=0.59, auxiliary spring contact and horizontal range l of main spring end points₀=110mm, to the 2nd main spring deformation coefficient G at end flat segments with auxiliary spring contact point_x-DECalculate, i.e.

(3) auxiliary spring works end points power F of the 2nd main spring under load_NCalculate:

I step: according to the thickness h of the root flat segments of few sheet root main spring of reinforced variable cross-section₂G obtained by calculating in=11mm, and step (1)_x-E1=107.53mm⁴/ N and G_x-E2=113.42mm⁴/ N, determines the half stiffness K of the 1st, the 2nd main spring_M1、K_M2It is respectively calculated, i.e.

II step: the half the most single-ended point load P=1200N of the load that works according to auxiliary spring, and K determined by I step_M1=12.38N/mm and K_M2=11.74N/mm, end points power F of the 2nd main spring under load that auxiliary spring is worked₂Calculate, i.e.

(4) the root reinforced few main spring of sheet variable cross-section major-minor spring gap delta design between end flat segments and auxiliary spring contact:

The thickness h of the root flat segments according to main spring₂=11mm, II step calculates obtained F₂G obtained by calculating in=584.08N, and step (2)_x-DE=72.59mm⁴/ N, few sheet variable cross-section main spring major-minor spring gap delta end flat segments and auxiliary spring contact between reinforced to this root is designed, i.e.

Utilize ANSYS finite element emulation software, main spring structure parameter according to this few reinforced variable-section steel sheet spring of sheet root and material characteristic parameter, set up ANSYS phantom, grid division, and at the root applying fixed constraint of phantom, applying concentrfated load P=1200N at free end, the deformation to this few sheet root main spring of reinforced variable-section steel sheet spring carries out ANSYS emulation, obtained deformation simulation cloudFigure,Such as figure 3Shown in, wherein, this main spring deflection δ=32.11mm at distance end position 110mm.

Understand, under same load, the ANSYS simulating, verifying value δ=32.11mm of this leaf spring main spring deflection, match with major-minor spring gap design load δ=31.85mm, relative deviation is only 0.81%；Result shows that the reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of root that this invention is provided is correct, and parameter designing value is accurately and reliably.

Embodiment two: the main reed number N=2 of certain few reinforced variable-section steel sheet spring of sheet root, wherein, half length L=600mm of each main spring, width b=60mm, elastic modulus E=200GPa, root thickness h₂=14.78mm, half l of installing space₃=60mm, oblique line segment length Δ l=30mm, the root of parabolic segment is to distance l of main spring end points_2p=L-l₃-Δ l=510mm, the root of oblique line section is to distance l of main spring end points₂=L-l₃=540mm；Root thickness h of parabolic segment_2p=13.3mm, the thickness of oblique line section is than γ=h_2p/h₂=0.90；The thickness h of the end flat segments of the 1st main spring₁₁=8mm, the thickness of the parabolic segment of the 1st main spring compares β₁=h₁₁/h_2p=0.60；The thickness h of the end flat segments of the 2nd main spring₁₂=6.5mm, the thickness of the 2nd main spring parabolic segment compares β₂=h₁₂/h_2p=0.49；Half length L of auxiliary spring_A=510mm, auxiliary spring contact is to horizontal range l of main spring end points₀=L-L_A=90mm, auxiliary spring contact contacts with main spring certain point in the flat segments of end.Auxiliary spring required by design works the half the most single-ended point load P=3000N of load, is designed the major-minor spring gap between this main spring end flat segments and auxiliary spring contact.

Using the method for designing identical with embodiment one and step, this main spring gap at end flat segments with auxiliary spring contact point to be designed, specific design step is as follows:

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

According to half length L=600mm of few sheet root main spring of reinforced variable cross-section, width b=60mm, elastic modulus E=200GPa, half l of installing space₃=60mm, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2p=510mm, the root of oblique line section is to distance l of main spring end points₂=540mm, the thickness of oblique line section compares β than γ=0.90, the thickness of the parabolic segment of the 1st main spring₁The thickness of the parabolic segment of the=0.60, the 2nd main spring compares β₂=0.49, the end points deformation coefficient G to the 1st and the 2nd main spring_x-E1And G_x-E2Calculate, be respectively

(2) the 2nd main springs are at the deformation coefficient G of end flat segments Yu auxiliary spring contact point_x-DECalculate:

According to half length L=600mm of few sheet root main spring of reinforced variable cross-section, width b=60mm, elastic modulus E=200GPa, the length Δ l=30mm of oblique line section, the root of parabolic segment is to distance l of main spring end points_2p=510mm, the root of oblique line section is to distance l of main spring end points₂=540mm, the thickness of oblique line section compares β than γ=0.90, the thickness of the parabolic segment of the 2nd main spring₂=0.49, auxiliary spring contact and horizontal range l of main spring end points₀=90mm, to the 2nd main spring deformation coefficient G at end flat segments with auxiliary spring contact point_x-DECalculate, i.e.

(3) auxiliary spring works end points power F of the 2nd main spring under load₂Calculate:

I step: according to the thickness h of the root flat segments of few sheet root main spring of reinforced variable cross-section₂G obtained by calculating in=14.78mm, and step (1)_x-E1=137.44mm⁴/ N and G_x-E2=143.40mm⁴/ N, determines the half stiffness K of the 1st, the 2nd main spring_M1、K_M2, it is respectively

II step: the half the most single-ended point load P=3000N of the load that works according to auxiliary spring, and K determined by I step_M1=23.49N/mm and K_M2=22.52N/mm, end points power F of the 2nd main spring under load that auxiliary spring is worked₂Calculate, i.e.

(4) the root reinforced few main spring of sheet variable cross-section major-minor spring gap delta design between end flat segments and auxiliary spring contact:

The thickness h of the root flat segments according to main spring₂=14.78mm, II step calculates obtained F₂G obtained by calculating in=1468.40N, and step (2)_x-DE=99.45mm⁴/ N, few sheet variable cross-section main spring major-minor spring gap delta end flat segments and auxiliary spring contact between reinforced to this root is designed, i.e.

Utilize ANSYS finite element emulation software, main spring structure parameter according to this few reinforced variable-section steel sheet spring of sheet root and material characteristic parameter, set up ANSYS phantom, grid division, and at the root applying fixed constraint of phantom, applying concentrfated load P=3000N at free end, the deformation to this few sheet root main spring of reinforced variable-section steel sheet spring carries out ANSYS emulation, obtained deformation simulation cloudFigure,Such as figure 4Shown in, wherein, this main spring deflection δ=45.65mm at distance end position 90mm.

Understand, under same load, the ANSYS simulating, verifying value δ=45.65mm of this leaf spring main spring deflection, match with major-minor spring gap design load δ=45.23mm, relative deviation is only 0.92%；Result shows that the reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of root that this invention is provided is correct, and parameter designing value is accurately and reliably.

Claims (1)

1. the reinforced few sheet variable cross-section main spring method for designing in end Yu auxiliary spring gap of root, wherein, becomes at the reinforced few sheet of root Be provided with oblique line strengthening segment between root flat segments and the parabolic segment of the main spring in cross section, of its symmetrical structure by root flat segments, Oblique line section, parabolic segment and end flat segments 4 sections are constituted；The end flat segments of each main spring is the non-structure that waits, i.e. the 1st The thickness of the end flat segments of main spring and length, more than other thickness of each and length；The end flat segments of main spring and auxiliary spring end It is designed with certain major-minor spring gap between contact, portion, works the design requirement of load meeting auxiliary spring；The each knot at main spring Structure parameter, material characteristic parameter, auxiliary spring length and the auxiliary spring required by design work load given in the case of, root is strengthened The few sheet variable cross-section main spring major-minor spring gap between end flat segments and auxiliary spring contact of type is designed, and specific design step is such as Under:

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

According to half length L of few sheet root main spring of reinforced variable cross-section, width b, half l of installing space₃, the length of oblique line section Δ l, the root of parabolic segment is to distance l of main spring end points_2p=L-l₃-Δ l, the root of oblique line section is to distance l of main spring end points₂=L- l₃, elastic modulus E, the thickness of oblique line section compares β than γ, the thickness of the parabolic segment of i-th main spring_i, wherein, i=1,2 ..., N, N are main reed number, the end points deformation coefficient G to each main spring_x-EiCalculate, i.e.

$\begin{array}{c}{G}_{x-Ei}=\frac{4(L^3-l_2^3)}{Eb}+\frac{4l_{2p}^3(2-{\mathrm{\β}}_i^3)}{{\mathrm{Eb\γ}}^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}^2{(\mathrm{\γ}-1)}^3}(4l_{2p}^2\mathrm{\γ}-l_{2p}^2-3l_{2p}^2{\mathrm{\γ}}^2+3l_2^2{\mathrm{\γ}}^2-4l_2^2{\mathrm{\γ}}^3+l_2^2{\mathrm{\γ}}^4-2l_{2p}{l}_2\mathrm{\γ}+\\ 2l_{2p}^2{\mathrm{\γ}}^2\ln \mathrm{\γ}+2l_2^2{\mathrm{\γ}}^2\ln \mathrm{\γ}+2l_{2p}{l}_2{\mathrm{\γ}}^3-4l_{2p}{l}_2{\mathrm{\γ}}^2\ln \mathrm{\γ}),i=1,2,\mathrm{...},N;\end{array}$

(2) the main springs of N sheet are at the deformation coefficient G of end flat segments Yu auxiliary spring contact point_x-DECalculate:

According to half length L of few sheet root main spring of reinforced variable cross-section, width b, the length Δ l of oblique line section, the root of parabolic segment Distance l to main spring end points_2p, the root of oblique line section is to distance l of main spring end points₂, elastic modulus E, the thickness ratio of oblique line section γ, the thickness of the parabolic segment of the main spring of N sheet compares β_N, auxiliary spring contact and horizontal range l of main spring end points₀, spring main to N sheet Deformation coefficient G at end flat segments with auxiliary spring contact point_x-DECalculate, i.e.

$\begin{array}{c}{G}_{x-DE}=\frac{4L^3-6l_0-4l_2^3+6l_0{l}_2^2}{Eb}-\frac{6l_0\mathrm{\Δ}l(l_{2p}+l_2\mathrm{\γ})}{{\mathrm{Eb\γ}}^2}+\frac{2{(l_0-l_{2p}{\mathrm{\β}}_N^2)}^2(2l_{2p}{\mathrm{\β}}_N^2+l_0)}{{\mathrm{Eb\γ}}^3{\mathrm{\β}}_N^3}+\\ \frac{8l_{2p}^2(1-{\mathrm{\β}}_N)(l_{2p}-3l_0+l_{2p}{\mathrm{\β}}_N^2+l_{2p}{\mathrm{\β}}_N)}{{\mathrm{Eb\γ}}^3}+\frac{6\mathrm{\Δ}l}{{\mathrm{Eb\γ}}^2{(\mathrm{\γ}-1)}^3}(4l_{2p}^2\mathrm{\γ}-l_{2p}^2-3l_{2p}^2{\mathrm{\γ}}^2+3l_2^2{\mathrm{\γ}}^2-4l_2^2{\mathrm{\γ}}^3+\\ l_2^2{\mathrm{\γ}}^4-2l_{2p}{l}_2\mathrm{\γ}+2l_{2p}^2{\mathrm{\γ}}^2\ln \mathrm{\γ}+2l_2^2{\mathrm{\γ}}^2\ln \mathrm{\γ}+2l_{2p}{l}_2{\mathrm{\γ}}^3-4l_{2p}{l}_2{\mathrm{\γ}}^2\ln \mathrm{\γ});\end{array}$

(3) auxiliary spring works end points power F of the main spring of N sheet under load_NCalculate:

I step: according to the thickness h of the root flat segments of few sheet root main spring of reinforced variable cross-section₂, and in step (1) calculated respectively The end points deformation coefficient G of the main spring of sheet_x-Ei, determine the half stiffness K of each main spring_Mi, i.e.

$K_{Mi}=\frac{h_2^3}{G_{x-Ei}},i=1,2,...,N;$

II step: the half the most single-ended point load P of the load that works according to auxiliary spring, and the half of each main spring determined by I step Stiffness K_Mi, end points power F of the main spring of N sheet under load that auxiliary spring is worked_NCalculate, i.e.

$F_N=\frac{K_{MN}P}{\underset{i=1}{\overset{N}{\Σ}}{K}_{Mi}},i=1,2,...,N,$

In formula, K_MNIt is the half rigidity of the main spring of N sheet.

(4) the root reinforced few main spring of sheet variable cross-section major-minor spring gap delta design between end flat segments and auxiliary spring contact:

The thickness h of the root flat segments according to main spring₂, the main spring of N sheet that in II step, calculated auxiliary spring works under load End points power F_N, and the G obtained by the middle calculating of step (2)_x-DE, few sheet variable cross-section main spring reinforced to root end flat segments with Major-minor spring gap delta between auxiliary spring contact is designed, i.e.

$\mathrm{\δ}=G_{x-DE}\frac{F_N}{h_2^3}.$