CN105889378B - The design method of the few piece reinforcement end auxiliary spring root thickness of ends contact formula - Google Patents

The design method of the few piece reinforcement end auxiliary spring root thickness of ends contact formula Download PDF

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CN105889378B
CN105889378B CN201610321711.8A CN201610321711A CN105889378B CN 105889378 B CN105889378 B CN 105889378B CN 201610321711 A CN201610321711 A CN 201610321711A CN 105889378 B CN105889378 B CN 105889378B
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mrow
msubsup
msub
gamma
spring
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CN105889378A (en
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周长城
于曰伟
赵雷雷
汪晓
袁光明
邵明磊
邵杰
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Shandong University of Technology
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16F—SPRINGS; SHOCK-ABSORBERS; MEANS FOR DAMPING VIBRATION
    • F16F3/00—Spring units consisting of several springs, e.g. for obtaining a desired spring characteristic
    • F16F3/02—Spring units consisting of several springs, e.g. for obtaining a desired spring characteristic with springs made of steel or of other material having low internal friction
    • F16F3/023—Spring units consisting of several springs, e.g. for obtaining a desired spring characteristic with springs made of steel or of other material having low internal friction composed only of leaf springs
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16F—SPRINGS; SHOCK-ABSORBERS; MEANS FOR DAMPING VIBRATION
    • F16F1/00—Springs
    • F16F1/02—Springs made of steel or other material having low internal friction; Wound, torsion, leaf, cup, ring or the like springs, the material of the spring not being relevant
    • F16F1/18—Leaf springs
    • F16F1/185—Leaf springs characterised by shape or design of individual leaves
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/36—Circuit design at the analogue level
    • G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16F—SPRINGS; SHOCK-ABSORBERS; MEANS FOR DAMPING VIBRATION
    • F16F2230/00—Purpose; Design features

Abstract

The design method of the few piece reinforcement end auxiliary spring root thickness of ends contact formula of the present invention, belongs to suspension leaf spring technical field.The present invention can be designed according to the structural parameters of each main spring, the length of auxiliary spring and piece number, modulus of elasticity and major-minor spring complex stiffness design requirement value to the auxiliary spring root flat segments thickness of the few piece reinforcement end major-minor spring of end contact.Pass through example and ANSYS simulating, verifyings, the design method of the few piece reinforcement end auxiliary spring root thickness of the ends contact formula is correct, available accurately and reliably auxiliary spring root thickness design load, is that reliable technical foundation has been established in the few piece reinforcement end major-minor spring CAD software exploitation of ends contact formula.Product design level and performance and vehicle ride performance are improved using this method；Meanwhile, product development speed is accelerated in reduction product design and testing expenses.

Description

The design method of the few piece reinforcement end auxiliary spring root thickness of ends contact formula
Technical field
The present invention relates to the few piece reinforcement end auxiliary spring root thickness of vehicle suspension leaf spring, particularly ends contact formula Design method.
Background technology
Few piece variable-section steel sheet spring is small because of the small, noise that has the advantages that to rub between lightweight, piece, is widely used in car In Leaf Spring Suspension System.In order to meet processing technology, stress intensity, rigidity and the design requirement of hanger thickness, in reality During the engineer applied of border, few piece variable-section steel sheet spring is generally designed as the few piece reinforcement end deformation of ends contact formula and cut Face major-minor spring form.In the case of auxiliary spring length and piece number and parabolic segment thickness ratio are given, the root flat segments thickness of auxiliary spring Influence the complex stiffness and stress intensity and service life and vehicle ride performance of major-minor spring.However, lacking due to the form Complicated, the non-grade structure of end flat segments of each main spring of piece variable cross-section major-minor spring, the length of major-minor spring is unequal, and works as load Lotus is worked more than auxiliary spring after the contact of load major-minor spring, and the internal force of each major-minor spring and deformation have coupling, to its analysis meter It is extremely difficult, do not provided the few piece reinforcement end variable cross-section major-minor spring of reliable ends contact formula always both at home and abroad at present Auxiliary spring root thickness design method.Mostly be previously ignore each main spring end it is non-wait structure, and major-minor spring regarded as it is isometric, Main spring rigidity directly is subtracted using the complex stiffness design requirement value of major-minor spring, the rigidity and root thickness to auxiliary spring carry out approximate Design, so the few piece reinforcement end variable cross-section major-minor spring careful design of ends contact formula and CAD software exploitation can not be met It is required that.Therefore, it is necessary to set up a kind of auxiliary spring root thickness of the few piece reinforcement end major-minor spring of accurate, reliable ends contact formula Design method, meets the requirement of the few piece reinforcement end variable cross-section major-minor spring careful design of ends contact formula and CAD software exploitation, On the premise of product cost is not increased, design level, quality and the performance and vehicle row of few piece variable-section steel sheet spring are improved Sail ride comfort；Meanwhile, product development speed is accelerated in reduction product design and testing expenses.
The content of the invention
For defect present in above-mentioned prior art, the technical problems to be solved by the invention be to provide it is a kind of easy, The design method of the reliable few piece reinforcement end auxiliary spring root thickness of ends contact formula, design flow diagram, as shown in Figure 1.End Contact few piece reinforcement end variable cross-section major-minor spring in portion's is symmetrical structure, and the half symmetrical structure of major-minor spring can see cantilever as Beam, i.e. symmetrical center line are root fixing end, and the end stress point of main spring and the contact of auxiliary spring are respectively as main spring end points and pair Spring end points, the major-minor spring of half symmetrical structure and the schematic diagram in major-minor spring gap, as shown in Fig. 2 wherein, including：Main spring 1, root Portion's pad 2, auxiliary spring 3, end pad 4；The half symmetrical structure of main spring 1 and auxiliary spring 3 be by root flat segments, parabolic segment, tiltedly Line segment, four sections of compositions of end flat segments, oblique line section play booster action to the end of tapered spring；Put down each root of main spring 1 Root shim 2 is equipped between straight section, between the root flat segments of auxiliary spring 3 and between main spring 1 and the root flat segments of auxiliary spring 3； The end flat segments of main each of spring 1 are provided with end pad 4, and the material of end pad 4 is carbon fibre composite, for reducing bullet The frictional noise that spring is produced when working.The width of main spring 1 and auxiliary spring 3 be b, clipping room away from half length be l3, oblique line section Length is Δ l, and modulus of elasticity is E.The half length of main spring 1 is LM, the thickness of the root flat segments of each main spring is h2M, parabolic The root of line segment to main spring end points distance be l2M=LM-l3；Main reed number is m, and the end thickness of each parabolic segment is h1Mpi, the thickness ratio β of parabolic segmenti=h1Mpi/h2M, the end of parabolic segment is to main spring end points apart from l1Mpi=l2Mβi 2；Respectively The thickness and length of the end flat segments of the non-grade main spring of structure, i.e., the 1st of end flat segments of the main spring of piece, more than other each main spring End flat segments thickness and length, wherein, the thickness and length of the end flat segments of each main spring are respectively h1MiAnd l1Mi =l1Mpi-Δl；The thickness ratio γ of oblique line sectionMi=h1Mi/h1Mpi, i=1,2 ..., m.The half length of auxiliary spring 3 is LA, auxiliary spring touch The horizontal range of point and main spring end points is l0=LM-LA, the thickness of the root flat segments of each auxiliary spring is h2AFor parameter to be designed, The root of parabolic segment to auxiliary spring end points distance be l2A=LA-l3, auxiliary spring piece number is n, wherein, the parabolic segment of each auxiliary spring Thickness ratio βAj, the end thickness of parabolic segment is h1Apj=βAjh2A, the end of parabolic segment is to auxiliary spring end points apart from l1Apj =l2AβAj 2, oblique line section thickness ratio bejγAj, the thickness and length of end flat segments are respectively h1Aj=γAjh1ApjAnd l1Aj= l1Apj-Δl.Major-minor spring gap delta is provided between auxiliary spring ends points and the main spring end flat segments of m pieces；When load is more than auxiliary spring During the load that works, auxiliary spring contact is in contact with certain point in the flat segments of main spring end, and major-minor spring cooperation meets complex stiffness Design requirement.The length and piece number, modulus of elasticity of structural parameters, auxiliary spring in each main spring, and the design of major-minor spring complex stiffness In the case of required value is given, each auxiliary spring root flat segments thickness of the few piece reinforcement end major-minor spring of end contact is carried out Design.
In order to solve the above technical problems, the few piece reinforcement end auxiliary spring root thickness of ends contact formula provided by the present invention Design method, it is characterised in that use following design procedure：
(1) the end points deformation coefficient G of each main spring of reinforcement end variable cross-section under end points stressing conditionsx-EiCalculate：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elasticity Modulus E；The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, the The thickness ratio β of the parabolic segment of the main spring of i piecesi, the thickness ratio γ of oblique line sectionMi, oblique line section root to main spring end points distance l1Mpi, oblique line section end to main spring end points apart from l1Mi, i=1,2 ..., m, to each main spring under end points stressing conditions End points deformation coefficient Gx-EiCalculated, i.e.,
(2) the main spring of m piece reinforcement end variable cross-sections under end points stressing conditions is in end flat segments and auxiliary spring contact point The deformation coefficient G at placex-DECalculate：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elasticity Modulus E；The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, the The thickness ratio β of the parabolic segment of the main spring of m piecesm, oblique line section root to main spring end points apart from l1Mpm, oblique line section end to lead Spring end points apart from l1Mm, the thickness ratio γ of oblique line sectionMm；Auxiliary spring contact and the horizontal range l of main spring end points0, to end points stress feelings Deformation coefficient G of the main spring of m pieces at end flat segments and auxiliary spring contact point under conditionx-DECalculated, i.e.,
(3) the end points deformation coefficient of the main spring of m piece reinforcement end variable cross-sections under major-minor spring contact point stressing conditions Gx-EzmCalculate：According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elasticity Modulus E；The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, the The thickness ratio β of the parabolic segment of the main spring of m piecesm, oblique line section root to main spring end points apart from l1Mpm, oblique line section end to lead Spring end points apart from l1Mm, the thickness ratio γ of oblique line sectionMm；Auxiliary spring contact and the horizontal range l of main spring end points0, major-minor spring is contacted Deformation coefficient G of the main spring of m pieces at endpoint location at point under stressing conditionsx-EzmCalculated, i.e.,
(4) the m main springs of piece reinforcement end variable cross-section under major-minor spring contact point stressing conditions are in end flat segments and pair Deformation coefficient G at spring contact pointx-DEzCalculating：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elasticity Modulus E；The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, the The thickness ratio β of the parabolic segment of the main spring of m piecesm, oblique line section root to main spring end points apart from l1Mpm, oblique line section end to lead Spring end points apart from l1Mm, the thickness ratio γ of oblique line sectionMm；Auxiliary spring contact and the horizontal range l of main spring end points0, major-minor spring is contacted Deformation coefficient G of the main spring of m pieces at end flat segments and auxiliary spring contact point under point stressing conditionsx-DEzCalculated, i.e.,
(5) the n pieces under end points stressing conditions are superimposed total end points deformation coefficient G of auxiliary springx-EATCalculate：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elasticity Modulus E；The half length L of auxiliary springA, the root of auxiliary spring parabolic segment is to auxiliary spring end points apart from l2A；Auxiliary spring piece number n, wherein, respectively The thickness ratio β of the parabolic segment of piece auxiliary springA, the thickness ratio γ of oblique line sectionA, oblique line section root to auxiliary spring end points apart from l1Ap, The end of oblique line section is to auxiliary spring end points apart from l1A, total end points deformation coefficient G of auxiliary spring is superimposed to n piecesx-EATCalculated, i.e.,
(6) the auxiliary spring root thickness h of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula2ADesign：
I steps：Equivalent one-chip auxiliary spring root thickness heADesign
According to major-minor spring complex stiffness design requirement value KMAT, main reed number m, the thickness of the root flat segments of each main spring h2M, step (1) is middle to calculate obtained Gx-Ei, step (2) is middle to calculate obtained Gx-DE, step (3) is middle to calculate resulting Gx-Ezm, Obtained G is calculated in step (4)x-DEz, and obtained G is calculated in step (5)x-EAT, to the few piece reinforcement end of end contact The equivalent one-chip auxiliary spring root thickness h of variable cross-section major-minor springeAIt is designed, i.e.,
II steps：Each root thickness h of the few piece reinforcement end variable cross-section auxiliary spring of ends contact formula2ADesign
According to auxiliary spring piece number n, and resulting h is calculated in I stepseA, contact few piece reinforcement end in end is become and cut The thickness h of each auxiliary spring root flat segments of face major-minor spring2AIt is designed, i.e.,
The present invention has the advantage that than prior art
Because the ends contact formula lacks the complicated of piece reinforcement end variable cross-section major-minor spring, the end of each main spring is put down Straight section is non-to wait structure, and the length of major-minor spring is unequal, and after load works the contact of load major-minor spring more than auxiliary spring, each master The internal force of auxiliary spring and deformation have coupling, and it is analyzed and calculates extremely difficult, reliable end had not been provided always both at home and abroad at present The auxiliary spring root thickness method for accurately designing of the few piece reinforcement end variable cross-section major-minor spring of portion's contact.Mostly it had been previously to ignore each The end flat segments of the main spring of piece are non-to wait structure, and major-minor spring is regarded as isometric, is directly wanted using the complex stiffness design of major-minor spring Evaluation subtracts main spring rigidity, and the rigidity and root thickness to auxiliary spring carry out Approximate Design, so it is few to meet ends contact formula Piece reinforcement end variable cross-section major-minor spring careful design and the requirement of CAD software exploitation.The present invention can be according to the knot of each main spring In the case of structure parameter, the length of auxiliary spring and piece number, modulus of elasticity and major-minor spring complex stiffness design requirement value are given, end is connect Each auxiliary spring root flat segments thickness of the few piece reinforcement end major-minor spring of touch is designed.By designing example and ANSYS Knowable to simulating, verifying, the auxiliary spring root of the few piece reinforcement end major-minor spring of accurate, reliable ends contact formula that this method can obtain Thickness design load, the auxiliary spring root thickness design for the few piece reinforcement end variable cross-section major-minor spring of ends contact formula is provided reliably Design method, and auxiliary spring CAD software exploitation for the few piece variable cross-section reinforcement end major-minor spring of ends contact formula establishes Reliable technical foundation.Design level, the product quality and performances of vehicle suspension variable cross-section major-minor spring can be improved using this method, Bearing spring quality and cost are reduced, the conevying efficiency and ride performance of vehicle is improved；Meanwhile, also reduce product design and examination Expense is tested, accelerates product development speed.
Brief description of the drawings
For a better understanding of the present invention, it is described further below in conjunction with the accompanying drawings.
Fig. 1 is the design flow diagram of the auxiliary spring root thickness of the few piece reinforcement end major-minor spring of ends contact formula；
Fig. 2 is the half symmetrical structure schematic diagram of the few piece reinforcement end major-minor spring of ends contact formula；
Fig. 3 is the ANSYS deformation simulation clouds of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula of embodiment one Figure；
Fig. 4 is the ANSYS deformation simulation clouds of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula of embodiment two Figure.
Specific embodiment
The present invention is described in further detail below by embodiment.
Embodiment one：The width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of certain ends contact formula, clipping room away from Half l3=55mm, the length Δ l=30mm of oblique line section, elastic modulus E=200GPa；The half length L of main springM= 575mm, the thickness h of the root flat segments of each main spring2M=11mm, the root of parabolic segment is to main spring end points apart from l2M= LM-l3=520mm；Main reed number m=2, wherein, the end thickness h of the parabolic segment of the 1st main spring1Mp1=6mm, parabolic segment Thickness ratio β1=h1Mp1/h2M=0.55, the end of parabolic segment is to main spring end points apart from l1Mp1=l2Mβ1 2=154.71mm, The thickness h of end flat segments1M1=7mm, the thickness ratio γ of oblique line sectionM1=h1M1/h1Mp1=1.17, the length of end flat segments l1M1=l1Mp1- Δ l=124.71mm；The end thickness h of the parabolic segment of 2nd main spring1Mp2=5mm, the thickness of parabolic segment Compare β2=h1Mp2/h2M=0.45, the end of parabolic segment is to main spring end points apart from l1Mp2=l2Mβ2 2Put down=107.44mm, end The thickness h of straight section1M2=6mm, the thickness ratio γ of oblique line sectionM2=h1M2/h1Mp2=1.20, the length l of end flat segments1M2= l1Mp2- Δ l=77.44mm.The half length L of auxiliary springA=525mm, the distance of the root of auxiliary spring parabolic segment to auxiliary spring end points l2A=LA-l3=470mm, auxiliary spring piece number n=1, wherein, the thickness ratio β of the parabolic segment of the piece auxiliary springA=0.50, parabolic segment End to auxiliary spring end points apart from l1Ap=l2AβA 2=117.50mm, the thickness ratio γ of oblique line sectionA=1.14, end flat segments Length l1A=l1Ap- Δ l=87.50mm；Auxiliary spring contact and the horizontal range l of main spring end points0=LM-LA=50mm.Major-minor spring Complex stiffness design requirement value KMAT=98.56N/mm, to the pair of the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula Spring root thickness is calculated.
The design method for the few piece reinforcement end auxiliary spring root thickness of ends contact formula that present example is provided, it sets Flow is counted as shown in figure 1, specific design step is as follows：
(1) the end points deformation coefficient G of each main spring of reinforcement end variable cross-section under end points stressing conditionsx-EiCalculate：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section =30mm, elastic modulus E=200GPa；The half length L of main springM=575mm, the root of main spring parabolic segment to main spring end points Apart from l2M=520mm, main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 1st main spring1=0.55, oblique line section Thickness ratio γM1=1.17, oblique line section root to main spring end points apart from l1Mp1=154.71mm, oblique line section end to lead Spring end points apart from l1M1=124.71mm；The thickness ratio β of the parabolic segment of 2nd main spring2=0.45, the thickness ratio of oblique line section γM2=1.20, oblique line section root to main spring end points apart from l1Mp2=107.44mm, the end of oblique line section is to main spring end points Apart from l1M2=77.44mm；To the end points deformation coefficient G of the 1st and the 2nd main spring under end points stressing conditionsx-E1And Gx-E2Enter Row is calculated respectively, i.e.,
(2) the main spring of m piece reinforcement end variable cross-sections under end points stressing conditions is in end flat segments and auxiliary spring contact point The deformation coefficient G at placex-DECalculate：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section =30mm, elastic modulus E=200GPa；The half length L of main springM=575mm, the root of main spring parabolic segment to main spring end points Apart from l2M=520mm；Main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 2nd main spring2=0.45, oblique line section Root to main spring end points apart from l1Mp2=107.44mm, oblique line section end to main spring end points apart from l1M2= 77.44mm, the thickness ratio γ of oblique line sectionM2=1.20；Auxiliary spring contact and the horizontal range l of main spring end points0=50mm, to end points by Deformation coefficient G of the 2nd main spring at end flat segments and auxiliary spring contact point in the case of powerx-DECalculated, i.e.,
(3) the end points deformation coefficient of the main spring of m piece reinforcement end variable cross-sections under major-minor spring contact point stressing conditions Gx-EzmCalculate：According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ of oblique line section L=30mm, elastic modulus E=200GPa；The half length L of main springM=575mm, the root to main spring end of main spring parabolic segment Point apart from l2M=520mm；Main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 2nd main spring2=0.45, oblique line The root of section is to main spring end points apart from l1Mp2=107.44mm, oblique line section end to main spring end points apart from l1M2= 77.44mm, the thickness ratio γ of oblique line sectionM2=1.20；Auxiliary spring contact and the horizontal range l of main spring end points0=50mm, to major-minor spring The end points deformation coefficient G of the 2nd main spring under contact point stressing conditionsx-Ez2Calculated, i.e.,
(4) the m main springs of piece reinforcement end variable cross-section under major-minor spring contact point stressing conditions are in end flat segments and pair Deformation coefficient G at spring contact pointx-DEzCalculating：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section =30mm, elastic modulus E=200GPa；The half length L of main springM=575mm, the root of main spring parabolic segment to main spring end points Apart from l2M=520mm；Main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 2nd main spring2=0.45, oblique line section Root to main spring end points apart from l1Mp2=107.44mm, oblique line section end to main spring end points apart from l1M2= 77.44mm, the thickness ratio γ of oblique line sectionM2=1.20；Auxiliary spring contact and the horizontal range l of main spring end points0=50mm, to major-minor spring Deformation coefficient G of the 2nd main spring at end flat segments and auxiliary spring contact point under contact point stressing conditionsx-DEzCalculated, I.e.
(5) the n pieces under end points stressing conditions are superimposed total end points deformation coefficient G of auxiliary springx-EATCalculate：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section =30mm, elastic modulus E=200GPa；The half length L of auxiliary springA=525mm, the root of auxiliary spring parabolic segment to auxiliary spring end points Apart from l2A=470mm, auxiliary spring piece number n=1, the thickness ratio β of the parabolic segment of the piece auxiliary springA=0.50, the thickness of oblique line section Compare γA=1.14, oblique line section root to auxiliary spring end points apart from l1Ap=117.50mm, the end of oblique line section is to auxiliary spring end points Apart from l1A=87.50mm, total end points deformation coefficient G of auxiliary spring is superimposed to n piecesx-EATCalculated, i.e.,
(6) each auxiliary spring root thickness h of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula2ADesign：
I steps：Equivalent one-chip auxiliary spring root thickness heADesign
According to major-minor spring complex stiffness design requirement value KMAT=98.56N/mm, main reed number m=2, the root of each main spring The thickness h of portion's flat segments2MObtained G is calculated in=11mm, step (1)x-E1=107.53mm4/ N and Gx-E2=113.42mm4/ Obtained G is calculated in N, step (2)x-DE=94.37mm4Obtained G is calculated in/N, step (3)x-Ez2=94.37mm4/ N, step (4) G obtained by being calculated inx-DEz=79.78mm4G obtained by being calculated in/N, and step (5)x-EAT=77.51mm4/ N is right The equivalent one-chip auxiliary spring root thickness h of the few piece reinforcement end variable cross-section major-minor spring of ends contact formulaeAIt is designed, i.e.,
II steps：Each root thickness h of the few piece reinforcement end variable cross-section auxiliary spring of ends contact formula2ADesign
According to auxiliary spring piece number n=1, and resulting h is calculated in I stepseA=14mm, to the few piece end of the ends contact formula The auxiliary spring root thickness h of reinforced variable cross-section major-minor spring2AIt is designed, i.e.,
Using ANSYS finite element emulation softwares, according to each main spring of few piece reinforcement end variable cross-section major-minor spring and The structural parameters and modulus of elasticity of auxiliary spring, and the auxiliary spring root thickness h that design is obtained2A=14mm, sets up half symmetrical structure master The ANSYS simulation models of auxiliary spring, grid division sets auxiliary spring end points to be contacted with main spring, and applies solid in the root of simulation model Conclude a contract or treaty beam, concentrfated load F=1840N is applied in main spring end points, to the major-minor of few piece reinforcement end variable-section steel sheet spring The deformation of spring carries out ANSYS emulation, the ANSYS deformation simulation cloud atlas of resulting major-minor spring, as shown in Figure 3；Wherein, major-minor spring Maximum deformation quantity f at endpoint locationDSmax=37.15mm, it is known that, the simulating, verifying value K of the major-minor spring complex stiffnessMAT= 2F/fDSmax=99.06N/mm.
Understand, major-minor spring complex stiffness simulating, verifying value KMAT=99.06N/mm, with design requirement value KMAT= 98.56N/mm matches, and relative deviation is only 0.51%；As a result show that the few piece end of ends contact formula that the invention is provided adds The design method of strong type auxiliary spring root thickness is correct, and the design load of auxiliary spring root thickness is reliable.
Embodiment two：The width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of certain ends contact formula, clipping room away from Half l3=60mm, the length Δ l=30mm of oblique line section, elastic modulus E=200GPa.The half length L of main springM= 600mm, the thickness h of the root flat segments of each main spring2M=12mm, the root of parabolic segment is to main spring end points apart from l2M= LM-l3=540mm；Main reed number m=2, wherein, the end thickness h of the parabolic segment of the 1st main spring1Mp1=7mm, parabolic segment Thickness ratio β1=h1Mp1/h2M=0.58, the end of parabolic segment is to main spring end points apart from l1Mp1=l2Mβ1 2=183.75mm, The thickness h of end flat segments1M1=8mm, the thickness ratio γ of oblique line sectionM1=h1M1/h1Mp1=1.14, the length of end flat segments l1M1=l1Mp1- Δ l=153.75mm；The end thickness h of the parabolic segment of 2nd main spring1Mp2=6mm, the thickness of parabolic segment Compare β2=h1Mp2/h2M=0.50, the end of parabolic segment is to main spring end points apart from l1Mp2=l2Mβ2 2=135mm, end is straight The thickness h of section1M2=7mm, the thickness ratio γ of oblique line sectionM2=h1M2/h1Mp2=1.17, the length l of end flat segments1M2=l1Mp2- Δ l=105mm.The half length L of auxiliary springA=540mm, the root of auxiliary spring parabolic segment is to auxiliary spring end points apart from l2A=LA-l3 =480mm, auxiliary spring piece number n=1, the thickness ratio β of the parabolic segment of the piece auxiliary springA=0.54, the end of parabolic segment to auxiliary spring End points apart from l1Ap=l2AβA 2=139.17mm, the thickness ratio γ of the oblique line section of auxiliary springA=1.14, the length of end flat segments l1A=l1Ap- Δ l=109.17mm；Auxiliary spring contact and the horizontal range l of main spring end points0=LM-LA=60mm, major-minor spring is compound firm Spend design requirement value KMAT=94.74N/mm.According to the structural parameters of each main spring, the length of auxiliary spring and piece number, modulus of elasticity and Major-minor spring complex stiffness design requirement value is thick to the auxiliary spring root of the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula Degree is designed.
Using with the identical design method of embodiment one and step, to the few piece reinforcement end variable cross-section of the ends contact formula The auxiliary spring root thickness of major-minor spring is designed, and specific design step is as follows：
(1) the end points deformation coefficient G of each main spring of reinforcement end variable cross-section under end points stressing conditionsx-EiCalculate：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；The half length L of main springM=600mm, the root of parabolic segment to main spring end points Apart from l2M=540mm, main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 1st main spring1=0.58, oblique line section Thickness ratio γM1=1.14, oblique line section root to main spring end points apart from l1Mp1=183.75mm, oblique line section end to lead Spring end points apart from l1M1=153.75mm；The thickness ratio β of the parabolic segment of 2nd main spring2=0.50, the thickness ratio of oblique line section γM2=1.17, oblique line section root to main spring end points apart from l1Mp2=135mm, oblique line section end to main spring end points away from From l1M2=105mm, to the 1st main spring and the end points deformation coefficient G of the 2nd main spring under end points stressing conditionsx-E1And Gx-E2Point Do not calculated, i.e.,
(2) the main spring of m piece reinforcement end variable cross-sections under end points stressing conditions is in end flat segments and auxiliary spring contact point The deformation coefficient G at placex-DECalculate：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；The half length L of main springM=600mm, the root of parabolic segment to main spring end points Apart from l2M=540mm, main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 2nd main spring2=0.50, oblique line section Root to main spring end points apart from l1Mp2=135mm, oblique line section end to main spring end points apart from l1M2=105mm, oblique line The thickness ratio γ of sectionM2=1.17；Auxiliary spring contact and the horizontal range l of main spring end points0=60mm, under end points stressing conditions Deformation coefficient G of 2 main springs at end flat segments and auxiliary spring contact pointx-DECalculated, i.e.,
(3) the end points deformation coefficient of the main spring of m piece reinforcement end variable cross-sections under major-minor spring contact point stressing conditions Gx-EzmCalculate：According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；The half length L of main springM=600mm, the root of parabolic segment to main spring end points Apart from l2M=540mm, main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 2nd main spring2=0.50, oblique line section Root to main spring end points apart from l1Mp2=135mm, oblique line section end to main spring end points apart from l1M2=105mm, oblique line The thickness ratio γ of sectionM2=1.17；Auxiliary spring contact and the horizontal range l of main spring end points0=60mm, to major-minor spring contact point stress feelings The end points deformation coefficient G of the 2nd main spring under conditionx-Ez2Calculated, i.e.,
(4) the m main springs of piece reinforcement end variable cross-section under major-minor spring contact point stressing conditions are in end flat segments and pair Deformation coefficient G at spring contact pointx-DEzCalculating：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；The half length L of main springM=600mm, the root of parabolic segment to main spring end points Apart from l2M=540mm, main reed number m=2, wherein, the thickness ratio β of the parabolic segment of the 2nd main spring2=0.50, oblique line section Root to main spring end points apart from l1Mp2=135mm, oblique line section end to main spring end points apart from l1M2=105mm, oblique line The thickness ratio γ of sectionM2=1.17；Auxiliary spring contact and the horizontal range l of main spring end points0=60mm, to stress at major-minor spring contact point In the case of deformation coefficient G of the 2nd main spring at end flat segments and auxiliary spring contact pointx-DEzCalculated, i.e.,
(5) the n pieces under end points stressing conditions are superimposed total end points deformation coefficient G of auxiliary springx-EATCalculate：
According to the width b=60mm of the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula, the length of oblique line section Δ l=30mm, elastic modulus E=200GPa；The half length L of auxiliary springA=540mm, the root of auxiliary spring parabolic segment to auxiliary spring End points apart from l2A=480mm, auxiliary spring piece number n=1, the thickness ratio β of the parabolic segment of the piece auxiliary springA=0.54, oblique line section Thickness ratio γA=1.14, oblique line section root to auxiliary spring end points apart from l1Ap=139.17mm, the end of oblique line section is to auxiliary spring End points apart from l1A=109.17mm, total end points deformation coefficient G of auxiliary spring is superimposed to n piecesx-EATCalculated, i.e.,
(6) each auxiliary spring root thickness h of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula2ADesign：
I steps：Equivalent one-chip auxiliary spring root thickness heADesign
According to major-minor spring complex stiffness design requirement value KMAT=94.74N/mm, main reed number m=2, the root of each main spring The thickness h of portion's flat segments2MObtained G is calculated in=12mm, step (1)x-E1=111.50mm4/ N and Gx-E2=116.10mm4/ Obtained G is calculated in N, step (2)x-DE=93.70mm4Obtained G is calculated in/N, step (3)x-Ez2=93.70mm4/ N, step (4) obtained G is calculated inx-DEz=77.25mm4G obtained by being calculated in/N, and step (5)x-EAT=82.17mm4/ N, opposite end The equivalent one-chip auxiliary spring root thickness h of the few piece reinforcement end variable cross-section major-minor spring of portion's contacteAIt is designed, i.e.,
II steps：Each root thickness h of the few piece reinforcement end variable cross-section auxiliary spring of ends contact formula2ADesign
According to auxiliary spring piece number n=1, and resulting h is calculated in I stepseA=13mm, to the few piece end of the ends contact formula The auxiliary spring root thickness h of reinforced variable cross-section major-minor spring2AIt is designed, i.e.,
Using ANSYS finite element emulation softwares, according to the few piece reinforcement end variable cross-section major-minor spring of the ends contact formula The structural parameters and modulus of elasticity of each main spring and auxiliary spring, and the auxiliary spring root thickness h that design is obtained2A=13mm, sets up half The ANSYS simulation models of symmetrical structure major-minor spring, grid division sets auxiliary spring end points to be contacted with main spring, and in simulation model Root applies fixed constraint, and concentrfated load F=1850N is applied in main spring end points, to the few piece reinforcement end of the ends contact formula The deformation of variable cross-section major-minor spring carries out ANSYS emulation, the ANSYS deformation simulation cloud atlas of resulting major-minor spring, as shown in Figure 4； Wherein, maximum deformation quantity f of the major-minor spring at endpoint locationDSmax=39.23mm, it is known that, the emulation of the major-minor spring complex stiffness Validation value KMAT=2F/fDSmax=94.32N/mm.
Understand, major-minor spring complex stiffness simulating, verifying value KMAT=94.32N/mm, with design requirement value KMAT= 94.74N/mm matches, and relative deviation is only 0.44%；As a result show that the few piece end of ends contact formula that the invention is provided adds The design method of strong type auxiliary spring root thickness is correct, and the design load of auxiliary spring root thickness is accurate, reliable.

Claims (1)

1. the design method of the few piece reinforcement end auxiliary spring root thickness of ends contact formula, wherein, the few piece end of ends contact formula The half symmetrical structure of reinforced major-minor spring is made up of 4 sections of root flat segments, parabolic segment, oblique line section and end flat segments, tiltedly Line segment plays booster action to the main spring end of variable cross-section；Put down the non-end for waiting the main spring of structure, i.e., the 1st of the end flat segments of each main spring The thickness and length of straight section, it is complicated to meet the 1st main spring more than the thickness and length of the end flat segments of other each main spring The requirement of stress；Certain major-minor spring gap is provided between auxiliary spring contact and main spring end flat segments, is worked with meeting auxiliary spring The design requirement of load；Auxiliary spring length is less than main spring length, when load works load more than auxiliary spring, auxiliary spring contact and main spring Certain point is in contact in the flat segments of end, to meet the design requirement of major-minor spring complex stiffness；Structural parameters in each main spring, It is few to end contact in the case of the length and piece number of auxiliary spring, modulus of elasticity and major-minor spring complex stiffness design requirement value are given Each auxiliary spring root flat segments thickness of piece reinforcement end major-minor spring is designed, and specific design step is as follows：
(1) the end points deformation coefficient G of each main spring of reinforcement end variable cross-section under end points stressing conditionsx-EiCalculate：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, modulus of elasticity E；The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, i-th The thickness ratio β of the parabolic segment of main springi, the thickness ratio γ of oblique line sectionMi, oblique line section root to main spring end points apart from l1Mpi, The end of oblique line section is to main spring end points apart from l1Mi, i=1,2 ..., m, to the end points of each main spring under end points stressing conditions Deformation coefficient Gx-EiCalculated, i.e.,
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(2) the m main springs of piece reinforcement end variable cross-section under end points stressing conditions are at end flat segments and auxiliary spring contact point Deformation coefficient Gx-DECalculate：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, modulus of elasticity E；The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, m pieces The thickness ratio β of the parabolic segment of main springm, oblique line section root to main spring end points apart from l1Mpm, the end to main spring end of oblique line section Point apart from l1Mm, the thickness ratio γ of oblique line sectionMm；Auxiliary spring contact and the horizontal range l of main spring end points0, under end points stressing conditions Deformation coefficient G of the main spring of m pieces at end flat segments and auxiliary spring contact pointx-DECalculated, i.e.,
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>D</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>L</mi> <mi>M</mi> <mn>3</mn> </msubsup> <mo>-</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msubsup> <mi>L</mi> <mi>M</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>3</mn> </msubsup> <mo>+</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>E</mi> <mi>b</mi> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>8</mn> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> <mo>-</mo> <mn>3</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>+</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mi>E</mi> <mi>b</mi> </mrow> </mfrac> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <msup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>4</mn> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>24</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;Delta;l&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> </mrow> </mfrac> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
(3) the end points deformation coefficient G of the main spring of m piece reinforcement end variable cross-sections under major-minor spring contact point stressing conditionsx-EzmMeter Calculate：According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, elastic modulus E； The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, m piece masters The thickness ratio β of the parabolic segment of springm, oblique line section root to main spring end points apart from l1Mpm, the end of oblique line section is to main spring end points Apart from l1Mm, the thickness ratio γ of oblique line sectionMm；Auxiliary spring contact and the horizontal range l of main spring end points0, at major-minor spring contact point by Deformation coefficient G of the main spring of m pieces at endpoint location in the case of powerx-EzmCalculated, i.e.,
<mrow> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>z</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>L</mi> <mi>M</mi> <mn>3</mn> </msubsup> <mo>-</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msubsup> <mi>L</mi> <mi>M</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>3</mn> </msubsup> <mo>+</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>E</mi> <mi>b</mi> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>8</mn> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> <mo>-</mo> <mn>3</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>+</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mi>E</mi> <mi>b</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>+</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>4</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> </mrow> </mfrac> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
(4) the main spring of m piece reinforcement end variable cross-sections under major-minor spring contact point stressing conditions connects in end flat segments and auxiliary spring Deformation coefficient G at contactx-DEzCalculating：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, modulus of elasticity E；The half length L of main springM, the root of main spring parabolic segment is to main spring end points apart from l2M, main reed number m, wherein, m pieces The thickness ratio β of the parabolic segment of main springm, oblique line section root to main spring end points apart from l1Mpm, the end to main spring end of oblique line section Point apart from l1Mm, the thickness ratio γ of oblique line sectionMm；Auxiliary spring contact and the horizontal range l of main spring end points0, to major-minor spring contact point by Deformation coefficient G of the main spring of m pieces at end flat segments and auxiliary spring contact point in the case of powerx-DEzCalculated, i.e.,
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>D</mi> <mi>E</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>12</mn> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>6</mn> <msubsup> <mi>l</mi> <mn>0</mn> <mn>2</mn> </msubsup> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>12</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>-</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>6</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>l</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>12</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mi>E</mi> <mi>b</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <msup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> <mrow> <msubsup> <mi>Eb&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mi>M</mi> </msub> <mo>-</mo> <msub> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msubsup> <mi>L</mi> <mi>M</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>3</mn> <msub> <mi>L</mi> <mi>M</mi> </msub> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mi>M</mi> </msub> <msub> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>3</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mi>E</mi> <mi>b</mi> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mn>0</mn> <mn>2</mn> </msubsup> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>l</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mn>0</mn> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>l</mi> <mn>0</mn> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>4</mn> </msubsup> <mo>-</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>+</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>4</mn> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>-</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>6</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>0</mn> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>4</mn> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>M</mi> <mi>p</mi> <mi>m</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>3</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Eb&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>m</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>M</mi> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
(5) the n pieces under end points stressing conditions are superimposed total end points deformation coefficient G of auxiliary springx-EATCalculate：
According to the width b of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula, the length Δ l of oblique line section, modulus of elasticity E；The half length L of auxiliary springA, the root of auxiliary spring parabolic segment is to auxiliary spring end points apart from l2A；Auxiliary spring piece number n, wherein, each pair The thickness ratio β of the parabolic segment of springA, the thickness ratio γ of oblique line sectionA, oblique line section root to auxiliary spring end points apart from l1Ap, oblique line The end of section is to auxiliary spring end points apart from l1A, total end points deformation coefficient G of auxiliary spring is superimposed to n piecesx-EATCalculated, i.e.,
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>A</mi> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <msubsup> <mi>L</mi> <mi>A</mi> <mn>3</mn> </msubsup> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>A</mi> </mrow> <mn>3</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mi>E</mi> <mi>b</mi> <mi>n</mi> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>8</mn> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>A</mi> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>2</mn> <mi>A</mi> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mi>E</mi> <mi>b</mi> <mi>n</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> <mn>3</mn> </msubsup> </mrow> <mrow> <msubsup> <mi>Ebn&amp;gamma;</mi> <mi>A</mi> <mn>3</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>A</mi> <mn>3</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;gamma;</mi> <mi>A</mi> </msub> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>3</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Ebn&amp;gamma;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>A</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mi>A</mi> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>6</mn> <mi>&amp;Delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mo>-</mo> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>4</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> </msub> <msub> <mi>&amp;gamma;</mi> <mi>A</mi> </msub> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mi>A</mi> </msub> <mo>+</mo> <mn>2</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mi>A</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>3</mn> </msubsup> <mo>-</mo> <mn>4</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>A</mi> <mi>p</mi> </mrow> </msub> <msubsup> <mi>&amp;gamma;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <msub> <mi>ln&amp;gamma;</mi> <mi>A</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>Ebn&amp;gamma;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <msubsup> <mi>&amp;beta;</mi> <mi>A</mi> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;gamma;</mi> <mi>A</mi> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> </mfrac> <mo>;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
(6) the auxiliary spring root thickness h of the few piece reinforcement end variable cross-section major-minor spring of ends contact formula2ADesign：
I steps：Equivalent one-chip auxiliary spring root thickness heADesign
According to major-minor spring complex stiffness design requirement value KMAT, main reed number m, the thickness h of the root flat segments of each main spring2M, step Suddenly obtained G is calculated in (1)x-Ei, step (2) is middle to calculate obtained Gx-DE, step (3) is middle to calculate resulting Gx-Ezm, step (4) obtained G is calculated inx-DEz, and obtained G is calculated in step (5)x-EAT, contact few piece reinforcement end in end is become and cut The equivalent one-chip auxiliary spring root thickness h of face major-minor springeAIt is designed, i.e.,
<mrow> <msub> <mi>h</mi> <mrow> <mi>e</mi> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mroot> <mfrac> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>M</mi> <mi>A</mi> <mi>T</mi> </mrow> </msub> <mo>-</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>h</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>3</mn> </msubsup> </mrow> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>)</mo> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>A</mi> <mi>T</mi> </mrow> </msub> <msubsup> <mi>h</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>3</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>A</mi> <mi>T</mi> </mrow> </msub> <msubsup> <mi>h</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>6</mn> </msubsup> </mrow> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>M</mi> <mi>A</mi> <mi>T</mi> </mrow> </msub> <mo>-</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>h</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>3</mn> </msubsup> </mrow> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mo>)</mo> <mo>(</mo> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>z</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>D</mi> <mi>E</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>E</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>D</mi> <mi>E</mi> <mi>z</mi> </mrow> </msub> <mo>)</mo> <mo>+</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow> <mi>x</mi> <mo>-</mo> <mi>D</mi> <mi>E</mi> <mi>z</mi> </mrow> </msub> <msubsup> <mi>h</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mn>3</mn> </msubsup> </mrow> </mfrac> <mn>3</mn> </mroot> <mo>;</mo> </mrow>
II steps：Each root thickness h of the few piece reinforcement end variable cross-section auxiliary spring of ends contact formula2ADesign
According to auxiliary spring piece number n, and resulting h is calculated in I stepseA, to the few piece reinforcement end variable cross-section master of end contact The thickness h of each auxiliary spring root flat segments of auxiliary spring2AIt is designed, i.e.,
<mrow> <msub> <mi>h</mi> <mrow> <mn>2</mn> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mfrac> <msub> <mi>h</mi> <mrow> <mi>e</mi> <mi>A</mi> </mrow> </msub> <mroot> <mi>n</mi> <mn>3</mn> </mroot> </mfrac> <mo>.</mo> </mrow> 3
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Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102734364A (en) * 2012-07-17 2012-10-17 山东理工大学 Analytical design method of camber and surface shape of automobile plate spring
CN104239618A (en) * 2014-09-03 2014-12-24 山东理工大学 Dismounting design method of automobile equal-strength superposition steel plate spring
CN104842734A (en) * 2014-02-14 2015-08-19 北汽福田汽车股份有限公司 Composite material-made plate spring and automobile
CN105526290A (en) * 2016-03-13 2016-04-27 周长城 Method for designing gaps of end straight sections of diagonal few-leaf main springs and auxiliary springs
CN105550487A (en) * 2016-03-13 2016-05-04 周长城 Method for designing few-leaf oblique line type variable-section main springs in gaps between oblique line segments and auxiliary spring

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP4914747B2 (en) * 2007-03-26 2012-04-11 バンドー化学株式会社 Method for manufacturing sliding material for belt tensioner and belt tensioner

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102734364A (en) * 2012-07-17 2012-10-17 山东理工大学 Analytical design method of camber and surface shape of automobile plate spring
CN104842734A (en) * 2014-02-14 2015-08-19 北汽福田汽车股份有限公司 Composite material-made plate spring and automobile
CN104239618A (en) * 2014-09-03 2014-12-24 山东理工大学 Dismounting design method of automobile equal-strength superposition steel plate spring
CN105526290A (en) * 2016-03-13 2016-04-27 周长城 Method for designing gaps of end straight sections of diagonal few-leaf main springs and auxiliary springs
CN105550487A (en) * 2016-03-13 2016-05-04 周长城 Method for designing few-leaf oblique line type variable-section main springs in gaps between oblique line segments and auxiliary spring

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
簧上质量对油气弹簧阀系设计参数的影响;周长城，焦学健;《中国机械工程》;20080430;第19卷(第8期);第998-1002页 *

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