CN105956259A - Checking calculation method of composite stiffness of end-contact few-leaf diagonal variable cross-section main and auxiliary spring - Google Patents

Abstract

The present invention relates to a checking calculation method of the composite stiffness of an end-contact few-leaf diagonal variable cross-section main and auxiliary spring, and belongs to the technical field of suspension leaf springs. According to the method, checking calculation can be performed on the composite stiffness of the end-contact few-leaf diagonal variable cross-section main and auxiliary spring according to structure parameters and elastic moduli of main springs and auxiliary springs of the end-contact few-leaf diagonal variable cross-section main and auxiliary spring. Instances and simulation verification show that the checking calculation method of the composite stiffness of the end-contact few-leaf diagonal variable cross-section main and auxiliary spring provided by the present invention is correct and that an accurate and reliable main and auxiliary composite stiffness checking calculation value can be obtained; a reliable checking calculation method is provided for the composite stiffness of the end-contact few-leaf diagonal variable cross-section main and auxiliary spring; and by use of the method, the design level, product quality and performance of the end-contact few-leaf diagonal variable cross-section main and auxiliary spring are improved, it is ensured that the composite stiffness of the main and auxiliary spring meets suspension system design requirements and vehicle driving smoothness is improved. Moreover, design and experiment costs can be reduced, and the product development speed can be increased.

Description

The Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of ends contact formula

Technical field

The present invention relates to the few sheet bias type variable cross-section major-minor spring of vehicle suspension leaf spring, particularly ends contact formula be combined The Method for Checking of rigidity.

Background technology

For few sheet variable-section steel sheet spring, in order to meet the requirement of variation rigidity, generally it is designed to major-minor spring, and leads to Crossing major-minor spring gap, it is ensured that after more than certain load, major-minor spring contacts and cooperatively works, and meets vehicle suspension not With design requirement to leaf spring rigidity under load condition.Owing to the 1st its stress of few main spring of sheet variable cross-section is complicated, not only Bearing vertical load, simultaneously also subject to torsional load and longitudinal loading, therefore, the end of the 1st main spring designed by reality is put down The thickness of straight section, generally the thickest than other each main spring, i.e. in actual design with produce, mostly employing end is non- Few sheet variable-section steel sheet spring Deng structure.Sheet variable-section steel sheet spring mainly has two types less, and one is parabolic type, Another is bias type, and wherein, Parabolic stress is iso-stress, more reasonable than bias type of its stress loading.So And, owing to the processing technique of parabolic type variable-section steel sheet spring is complicated, the process equipment of needs is expensive, and bias type variable cross-section The processing technique of steel plate is simple, it is only necessary to simple equipment just can be processed, therefore, under conditions of meeting rigidity and intensity, and can Use the variable-section steel sheet spring of bias type.For few sheet bias type variable cross-section major-minor spring, in order to meet different composite rigidity Design requirement, generally uses different auxiliary spring length, i.e. auxiliary spring also differs with the contact position of main spring, therefore, according to auxiliary spring The major-minor spring that end flat segments can be divided into contact with the contact position of main spring and to contact in oblique line section, i.e. ends contact formula and non-end Portion's contact, wherein, in the case of identical auxiliary spring root thickness, being combined of the few sheet bias type variable cross-section major-minor spring of ends contact formula Rigidity, big than non-ends contact formula.The size of complex stiffness has material impact to vehicle ride performance, therefore, and must The complex stiffness of designed ends contact formula sheet bias type variable cross-section major-minor spring less must be checked, to guarantee to meet vehicle The design requirement of suspension rate.Understand according to institute's inspection information, sheet bias type variable cross-section major-minor spring few for ends contact formula, at present All giving reliable complex stiffness Method for Checking, main cause is by the few sheet bias type variable cross-section major-minor spring of end contact The restriction of the analytical calculation of the end points power of each main spring and auxiliary spring.Because the few sheet bias type variable cross-section major-minor spring of ends contact formula The non-structure that waits of main spring end flat segments, and after load works the contact of load major-minor spring more than auxiliary spring, main spring and the change of auxiliary spring Shape and internal force all have coupling, and therefore, the analytical calculation of each main spring and the end points power of auxiliary spring and deformation is extremely complex, current state Inside and outside do not provide the parsing Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of accurate ends contact formula always.Cause This, it is necessary to set up the Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of a kind of ends contact formula accurate, reliable, Meet Vehicle Industry fast development and the requirement to suspension Precise Design for Laminated Spring, improve setting of few sheet variable-section steel sheet spring Meter level, product quality and performances, it is ensured that the complex stiffness of major-minor spring meets vehicle suspension design requirement, improve vehicle and travel flat Pliable；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 be to provide a kind of easy, The Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of ends contact formula reliably.The few sheet bias type of ends contact formula The half symmetrical structure of variable cross-section major-minor spring includes main spring, root shim, auxiliary spring, end pad；Each of main spring and auxiliary spring is tiltedly Line style variable-section steel sheet spring is to be made up of root flat segments, oblique line section, end flat segments three sections；Each root of main spring is straight Between Duan, between each root flat segments of auxiliary spring and between the root flat segments of main spring and auxiliary spring, it is provided with root shim； Being provided with end pad between each end flat segments of main spring, the material of end pad is carbon fibre composite, to prevent Frictional noise is produced during work.Wherein, the sheet number of main spring is m, and the root thickness of each main spring is h_2M, width is b, elastic modelling quantity For E, a length of L of half_M, half l of installing space₃, the distance of the root of main spring oblique line section to main spring end points is l_2M；Each master The end flat segments of spring is the non-thickness waiting structure, i.e. the end flat segments of the 1st main spring and length, more than other each master's The thickness of end flat segments and length；The thickness of the end flat segments of each main spring is h_1i, the thickness of the oblique line section of each main spring Ratio is β_i=h_1i/h_2M, length l of the end flat segments of each main spring_1i, i=1,2 ..., m.The root flat segments of each auxiliary spring Thickness be h_2A, width is b, and elastic modelling quantity is E, a length of L of half_A, half l of installing space₃, the end of each auxiliary spring is put down Thickness and the length of straight section are respectively h_A1jAnd l_A1j, the thickness of oblique line section is than for β_Aj=h_A1j/h_2A, j=1,2 ..., n, n are secondary Reed number.Auxiliary spring length is less than main spring length, and auxiliary spring contact is l with the horizontal range of main spring end points₀；Auxiliary spring contact and main spring end The spacing of portion's flat segments is provided with certain major-minor spring gap delta, when load works load more than auxiliary spring, and auxiliary spring contact and master In the flat segments of spring end, certain point contacts, to meet the vehicle suspension requirement to major-minor spring complex stiffness.Knot at each main spring In the case of structure parameter, the structural parameters of each auxiliary spring and elastic modelling quantity are given, sheet bias type major-minor spring few to end contact Complex stiffness checks.

For solving above-mentioned technical problem, the few sheet bias type variable cross-section major-minor spring of ends contact formula provided by the present invention is combined The Method for Checking of rigidity, it is characterised in that the following step that checks of employing:

(1) the end points deformation coefficient G of each main spring of bias type variable cross-section under end points stressing conditions_x-DiCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, Main reed number m, distance l of the root of main spring oblique line section to main spring end points_2M=L_M-l₃, the thickness ratio of the oblique line section of i-th main spring β_i, wherein, i=1,2 ..., m, the end points deformation coefficient G to each main spring under end points stressing conditions_x-DiCalculate, i.e.

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

(2) the deformation coefficient G at end flat segments with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions_x-CD Calculate:

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, The root of oblique line section is to distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of the main spring of m sheet compares β_m, auxiliary spring contact and master Horizontal range l of spring end points₀, to the change at end flat segments with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions Shape coefficient G_x-CDCalculate, i.e.

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

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

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, The root of oblique line section is to distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of the main spring of m sheet compares β_m, auxiliary spring contact and master Horizontal range l of spring end points₀, end points deformation coefficient G to the main spring of m sheet under major-minor spring contact point stressing conditions_x-DzmCarry out Calculate, i.e.

$\begin{array}{c}{G}_{x-D_{z}m}=\frac{2l_3(6L_M^2-6L_M{l}_3-6L_M{l}_0+2l_3^2+3l_0{l}_3)}{Eb}+\frac{2{(l_0-L_M{\mathrm{\β}}_m^2+l_3{\mathrm{\β}}_m^2)}^2(l_0+2L_M{\mathrm{\β}}_m^2-2l_3{\mathrm{\β}}_m^2)}{{\mathrm{Eb\β}}_m^3}+\\ \frac{6{(L_M-l_3)}^3{({\mathrm{\β}}_m+1)}^2(3{\mathrm{\β}}_m-2{\mathrm{ln\β}}_m-2{\mathrm{\β}}_m{\mathrm{ln\β}}_m-3)}{Eb}+\frac{6l_0{(L_M-l_3)}^3({\mathrm{\β}}_m-1){({\mathrm{\β}}_m+1)}^2}{{\mathrm{Eb\β}}_m};\end{array}$

(4) deformation at end flat segments with auxiliary spring contact point of the main spring of m sheet under major-minor spring contact point stressing conditions Coefficient G_x-CDzCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, The root of main spring oblique line section is to distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of the main spring of m sheet compares β_m, auxiliary spring contact Horizontal range l with main spring end points₀, to the main spring of m sheet under major-minor spring contact point stressing conditions at end flat segments and auxiliary spring Deformation coefficient G at contact point_x-CDzCalculate, i.e.

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

$\frac{2{\mathrm{\β}}_m^2(6l_{2M}^2{l}_0+9l_{2M}^3-18l_{2M}^3{\mathrm{ln\β}}_m)}{Eb};$

(5) total end points deformation coefficient G of n sheet superposition auxiliary spring_x-DATCalculate:

Half length L according to few sheet bias type variable cross-section auxiliary spring_A, width b, elastic modulus E, half l of installing space₃, Auxiliary spring sheet number n, distance l of the root of auxiliary spring oblique line section to auxiliary spring end points_2A=L_A-l₃, the thickness ratio of the oblique line section of jth sheet auxiliary spring β_Aj, wherein, j=1,2 ..., n, the total end points deformation coefficient G to the n sheet superposition auxiliary spring under end points stressing conditions_x-DATCount Calculate, i.e.

$G_{x-DAT}=\frac{1}{\underset{j=1}{\overset{n}{\mathrm{\Σ}}}\frac{1}{G_{x-DAj}}},j=1,2,...,n;$

In formula, G_x-DAjEnd points deformation coefficient for jth sheet auxiliary spring, it may be assumed that

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

As auxiliary spring sheet number n=1, then total end points deformation coefficient G of n sheet superposition auxiliary spring_x-DAT, equal to the end points of this sheet auxiliary spring Deformation coefficient G_x-DA1, i.e.

G_x-DAT=G_x-DA1；

(6) the complex stiffness checking computations of the few sheet bias type variable cross-section major-minor spring of ends contact formula:

According to main reed number m, the thickness h of the root flat segments of each main spring_2M, the thickness of the root flat segments of each auxiliary spring h_2A, calculated G in step (1)_x-Di, calculated G in step (2)_x-CD, calculated G in step (3)_x-Dzm, step Suddenly calculated G in (4)_x-CDz, and calculated G in step (5)_x-DAT, sheet bias type variable cross-section few to end contact The complex stiffness K of major-minor spring_MATCheck, i.e.

$K_{MAT}=\underset{i=1}{\overset{m-1}{\mathrm{\Σ}}}\frac{2h_{2M}^3}{G_{x-Di}}+\frac{2h_{2M}^3(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}{G_{x-Dm}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)-G_{x-D_{z}m}{G}_{x-CD}{h}_{2A}^3}.$

The present invention has the advantage that than prior art

Because the non-structure that waits of the main spring end flat segments of the few sheet bias type variable cross-section major-minor spring of ends contact formula, and auxiliary spring length Less than main spring length, after load works load more than auxiliary spring, auxiliary spring contact contacts with somewhere in main spring oblique line section, main spring Deformation and internal force with auxiliary spring all have coupling, and the analytical calculation of each main spring and the end points power of auxiliary spring and deformation is extremely complex, Do not provide the parsing checking computations of the few sheet bias type variable cross-section major-minor spring complex stiffness of accurate ends contact formula the most always Method.The present invention can according to each main spring of the few sheet bias type variable cross-section major-minor spring of ends contact formula and the structural parameters of auxiliary spring and Elastic modelling quantity, the complex stiffness of sheet bias type variable cross-section major-minor spring few to end contact checks.By example and emulation Checking understands, and the Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of the ends contact formula that this invention is provided just is True, in the case of each main spring and the structural parameters of auxiliary spring and elastic modelling quantity are given, available major-minor spring accurately and reliably is multiple Closing rigidity checking value, the complex stiffness checking computations for the few sheet bias type variable cross-section major-minor spring of ends contact formula provide checking computations reliably Method.Utilize the method can improve design level, product quality and the property of the few sheet bias type variable cross-section major-minor spring of ends contact formula Can, it is ensured that major-minor spring complex stiffness meets suspension system designs requirement, improve vehicle ride performance；Meanwhile, also can reduce and set Meter and testing expenses, accelerate product development speed.

Accompanying drawing explanation

In order to be more fully understood that the present invention, it is described further below in conjunction with the accompanying drawings.

Fig. 1 is the flow chart of the complex stiffness checking computations of the few sheet bias type variable cross-section major-minor spring of ends contact formula；

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

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

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

Detailed description of the invention

As it is shown in figure 1, the Method for Checking step of the present invention is as follows: sheet bias type variable cross-section major-minor few for ends contact formula Spring, first the end points deformation coefficient G to each main spring of bias type variable cross-section under its end points stressing conditions_x-DiCalculate, secondly, then To the deformation coefficient G at end flat segments with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions_x-CDCalculate, again, End points deformation coefficient G to the main spring of m sheet under major-minor spring contact point stressing conditions_x-DzmCalculate, again, to major-minor spring contact point The main spring of m sheet under stressing conditions is at end flat segments and auxiliary spring contact point deformation coefficient G_x-CDzCalculate, again, to n sheet superposition Total end points deformation coefficient G of auxiliary spring_x-DATCalculating, finally, sheet bias type variable cross-section major-minor spring few to end contact is combined just Degree K_MATChecking computations.Above-mentioned contact point mean that, under state as shown in Figure 2, the end of auxiliary spring connects with the lower surface of main spring The contact point formed when touching, in actual contact process, the arris of auxiliary spring end contacts with the surface of main spring, testing in the present invention Calculate in procedure, be regarded as point cantact and carry out rigidity checking.As in figure 2 it is shown, the few sheet bias type variable cross-section of ends contact formula The half symmetrical structure schematic diagram of major-minor spring, it includes main spring 1, root shim 2, auxiliary spring 3, end pad 4；Main spring 1 and auxiliary spring 3 Each bias type variable-section steel sheet spring be to be made up of root flat segments, oblique line section, end flat segments three sections；Main spring 1 each Between sheet root flat segments, between each root flat segments of auxiliary spring 3 and between the root flat segments of main spring 1 and auxiliary spring 3, if It is equipped with root shim 2；Being provided with end pad 4 between each end flat segments of main spring 1, the material of end pad is carbon fiber Composite, produces frictional noise during to prevent work.Wherein, the sheet number of main spring 1 is m, and the root thickness of each main spring is h_2M, Width is b, and elastic modelling quantity is E, a length of L of half_M, half l of installing space₃, the root of main spring oblique line section is to main spring end points Distance is l_2M；The end flat segments of each main spring is the non-thickness waiting structure, i.e. the end flat segments of the 1st main spring and length, The thickness of end flat segments and length more than other each master；The thickness of the end flat segments of each main spring is h_1i, each master The thickness of the oblique line section of spring is than for β_i=h_1i/h_2M, length l of the end flat segments of each main spring_1i, i=1,2 ..., m.Each The thickness of the root flat segments of auxiliary spring is h_2A, width is b, and elastic modelling quantity is E, a length of L of half_A, half l of installing space₃, Thickness and the length of the end flat segments of each auxiliary spring are respectively h_A1jAnd l_A1j, the thickness of oblique line section is than for β_Aj=h_A1j/h_2A, j =1,2 ..., n, n are auxiliary spring sheet number.Auxiliary spring length is less than main spring length, and auxiliary spring contact is l with the horizontal range of main spring end points₀； Auxiliary spring contact is provided with certain major-minor spring gap delta with the spacing of main spring end flat segments.

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

Embodiment one: the main reed number m=2, auxiliary spring sheet number n=of the few sheet bias type variable cross-section major-minor spring of certain ends contact formula 1；Wherein, half length L of each main spring_M=575mm, width b=60mm, elastic modulus E=200GPa, the one of installing space Half l₃=55mm, the root of oblique line section is to distance l of main spring end points_2M=L_M-l₃=520mm；The root flat segments of each main spring Thickness h_2M=11mm, the thickness of the end flat segments of the 1st main spring is h₁₁=7mm, the thickness ratio of the oblique line section of the 1st main spring For β₁=h₁₁/h_2M=0.64；The thickness of the end flat segments of the 2nd main spring is h₁₂=6mm, the thickness of the oblique line section of the 2nd main spring Degree ratio respectively β₂=h₁₂/h_2M=0.55.Half length L of this sheet auxiliary spring_AThe water of=525mm, auxiliary spring contact and main spring end points Flat distance l₀=L_M-L_A=50mm；Root thickness h of auxiliary spring_2A=14mm, the thickness h of the end flat segments of auxiliary spring_A11=8mm, secondary The thickness of the oblique line section of spring compares β_A1=h_A11/h_2A=0.57；Auxiliary spring and certain point cantact in the flat segments of main spring end, and connect.To this The complex stiffness of the few sheet bias type variable cross-section major-minor spring of ends contact formula checks.

The Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of the ends contact formula that present example is provided, Its checking computations flow process is as it is shown in figure 1, concrete checking computations step is as follows:

(1) the end points deformation coefficient G of each main spring of bias type variable cross-section under end points stressing conditions_x-DiCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=575mm, width b=60mm, elastic modulus E= 200GPa, the root of oblique line section is to distance l of main spring end points_2M=520mm；The thickness of the oblique line section of the 1st main spring compares β₁= The thickness of the oblique line section of the 0.64, the 2nd main spring compares β₂=0.55, to the 1st main spring under end points stressing conditions and the 2nd main spring End points deformation coefficient G_x-D1And G_x-D2It is respectively calculated, i.e.

$G_{x-D1}=\frac{4}{Eb}(L_M^3-l_{2M}^3)+\frac{6l_{2M}^3{({\mathrm{\β}}_1+1)}^2\[3({\mathrm{\β}}_1-1)-2{\mathrm{ln\β}}_1(1+{\mathrm{\β}}_1)\]}{Eb}+\frac{4{\mathrm{\β}}_1^3{l}_{2M}^3}{Eb}=101.68{\mathrm{mm}}^4/N,$

$G_{x-D2}=\frac{4}{Eb}(L_M^3-l_{2M}^3)+\frac{6l_{2M}^3{({\mathrm{\β}}_2+1)}^2\[3({\mathrm{\β}}_2-1)-2{\mathrm{ln\β}}_2(1+{\mathrm{\β}}_2)\]}{Eb}+\frac{4{\mathrm{\β}}_2^3{l}_{2M}^3}{Eb}=109.72{\mathrm{mm}}^4/N;$

(2) the deformation coefficient G at end flat segments with auxiliary spring contact point of the 2nd main spring under end points stressing conditions_x-CD Calculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=575mm, width b=60mm, elastic modulus E= 200GPa, the root of the oblique line section of main spring is to distance l of main spring end points_2M=520mm；The thickness ratio of the oblique line section of the 2nd main spring β₂=0.55, auxiliary spring contact and horizontal range l of main spring end points₀=50mm, to the 2nd main spring under end points stressing conditions at end Deformation coefficient G at portion's flat segments and auxiliary spring contact point_x-CDCalculate, i.e.

$\begin{array}{c}{G}_{x-CD}=\frac{4L_M^3+22l_{2M}^3({\mathrm{\β}}_2^3-1)+6l_{2M}^3\[3{\mathrm{\β}}_2({\mathrm{\β}}_2-1)-2(1+{\mathrm{\β}}_2^3){\mathrm{ln\β}}_2-6{\mathrm{\β}}_2(1+{\mathrm{\β}}_2){\mathrm{ln\β}}_2\]}{Eb}+\\ \frac{2\[l_0^3+3{\mathrm{\β}}_2^2(l_{2M}^2{\mathrm{\β}}_2^2-L_M^2{\mathrm{\β}}_2-l_{2M}^2)l_0\]}{{\mathrm{Eb\β}}_2^3}=91.20{\mathrm{mm}}^4/N;\end{array}$

(3) the end points deformation coefficient G of the 2nd main spring under major-minor spring contact point stressing conditions_x-Dz2Calculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=575mm, width b=60mm, elastic modulus E= 200GPa, half l of installing space₃=55mm, distance l of the root of main spring oblique line section to main spring end points_2M=520mm；2nd The thickness of the oblique line section of main spring compares β₂=0.55, auxiliary spring contact and horizontal range l of main spring end points₀=50mm, contacts major-minor spring The end points deformation coefficient G of the 2nd main spring under some stressing conditions_x-Dz2Calculate, i.e.

$\begin{array}{c}{G}_{x-D_{z}2}=\frac{2l_3(6L_M^2-6L_M{l}_3-6L_M{l}_0+2l_3^2+3l_0{l}_3)}{Eb}+\frac{2{(l_0-L_M{\mathrm{\β}}_2^2+l_3{\mathrm{\β}}_2^2)}^2(l_0+2L_M{\mathrm{\β}}_2^2-2l_3{\mathrm{\β}}_2^2)}{{\mathrm{Eb\β}}_2^3}+\\ \frac{6{(L_M-l_3)}^3{({\mathrm{\β}}_2+1)}^2(3{\mathrm{\β}}_2-2{\mathrm{ln\β}}_2-2{\mathrm{\β}}_2{\mathrm{ln\β}}_2-3)}{Eb}+\frac{6l_0{(L_M-l_3)}^3({\mathrm{\β}}_2-1){({\mathrm{\β}}_2+1)}^2}{{\mathrm{Eb\β}}_2}\\ =91.20{\mathrm{mm}}^4/N;\end{array}$

(4) deformation at end flat segments with auxiliary spring contact point of the 2nd main spring under major-minor spring contact point stressing conditions Coefficient G_x-CDzCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=575mm, width b=60mm, elastic modulus E= 200GPa, the root of oblique line section is to distance l of main spring end points_2M=520mm, the thickness of the oblique line section of the 2nd main spring compares β₂= 0.55, auxiliary spring contact and horizontal range l of main spring end points₀=50mm, to the 2nd master under stressing conditions at major-minor spring contact point Spring deformation coefficient G at end flat segments with auxiliary spring contact point_x-CDzCalculate, i.e.

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_z}=\frac{2{\mathrm{\β}}_2^3{l}_{2M}^3(11-6{\mathrm{ln\β}}_2)}{Eb}-\frac{2(6L_M^2{l}_0-6L_M{l}_0^2+9l_0^2{l}_{2M}+6l_{2M}^3{\mathrm{ln\β}}_2-2L_M^3+11l_{2M}^3)}{Eb}-\\ \frac{2{\mathrm{\β}}_2(3l_0^2{l}_{2M}+9l_{2M}^3+18l_{2M}^3{\mathrm{ln\β}}_2)}{Eb}-\frac{4l_0^3+2{\mathrm{\β}}_2^2(6l_{2M}^2{l}_0-9l_0^2{l}_{2M})-6l_0^2{l}_{2M}{\mathrm{\β}}_2}{{\mathrm{Eb\β}}_2^3}+\\ \frac{2{\mathrm{\β}}_2^2(6l_{2M}^2{l}_0+9l_{2M}^3-18l_{2M}^3{\mathrm{ln\β}}_2)}{Eb}=77.06{\mathrm{mm}}^4/N;\end{array}$

(5) total end points deformation coefficient G of n sheet superposition auxiliary spring_x-DATCalculate:

Half length L according to few sheet bias type variable cross-section auxiliary spring_A=525mm, width b=60mm, elastic modulus E= 200GPa, distance l of the root of the oblique line section of auxiliary spring to auxiliary spring end points_2A=470mm, the thickness of the oblique line section of auxiliary spring compares β_A1= 0.57, auxiliary spring sheet number n=1, then total end points deformation coefficient G of this sheet auxiliary spring_x-DAT, equal to the end points deformation coefficient of this sheet auxiliary spring G_x-DA1, i.e.

$G_{x-DAT}=\frac{1}{\underset{j=1}{\overset{n}{\mathrm{\Σ}}}\frac{1}{G_{x-DAj}}}=G_{x-DA1}=80.73{\mathrm{mm}}^4/N;$

In formula, G_x-DA1For the end points deformation coefficient of this sheet auxiliary spring, i.e.

$G_{x-DA1}=\frac{4}{Eb}(L_A^3-l_{2A}^3)+\frac{6l_{2A}^3{({\mathrm{\β}}_{A1}+1)}^2\[3({\mathrm{\β}}_{A1}-1)-2{\mathrm{ln\β}}_{A1}(1+{\mathrm{\β}}_{A1})\]}{Eb}+\frac{4{\mathrm{\β}}_{A1}^3{l}_{2A}^3}{Eb}=80.73{\mathrm{mm}}^4/N;$

(6) the complex stiffness checking computations of the few sheet bias type variable cross-section major-minor spring of ends contact formula:

According to main reed number m=2, the thickness h of the root flat segments of each main spring_2M=11mm, the root flat segments of auxiliary spring Thickness h_2A=14mm, calculated G in step (1)_x-D1=101.68mm⁴/ N and G_x-D2=109.72mm⁴/ N, step (2) In calculated G_x-CD=91.20mm⁴/ N, calculated G in step (3)_x-Dz2=91.20mm⁴/ N, calculates in step (4) The G obtained_x-CDz=77.06mm⁴Calculated G in/N, and step (5)_x-DAT=80.73mm⁴/ N is few to this ends contact formula The complex stiffness K of sheet bias type variable cross-section major-minor spring_MATCalculate, i.e.

$K_{MAT}=\underset{i=1}{\overset{m-1}{\mathrm{\Σ}}}\frac{2h_{2M}^3}{G_{x-Di}}+\frac{2h_{2M}^3(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}{G_{x-D2}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)-G_{x-D_{z}2}{G}_{x-CD}{h}_{2A}^3}=95.95N/mm.$

After major-minor spring concurs, in the case of main spring end points imposed load P=1840N, utilize complex stiffness meter Calculation value K_MAT=95.95N/mm, the maximum distortion of sheet bias type variable cross-section major-minor spring half few to this ends contact formula is tested Calculate, i.e.

$f_{Dmax}=\frac{2P}{K_{MAT}}=38.35mm.$

Utilize ANSYS finite element emulation software, join according to the major-minor spring structure of this few sheet bias type variable-section steel sheet spring Number and material characteristic parameter, set up the ANSYS phantom of half symmetrical structure major-minor spring, grid division, and at phantom Root apply fixed constraint, at the end points of main spring half apply concentrfated load P=1840N, to this few sheet bias type become cut The deformation of the major-minor spring of face leaf spring carries out ANSYS emulation, the deformation simulation cloud atlas of obtained major-minor spring, as it is shown on figure 3, Wherein, major-minor spring maximum deformation quantity f at endpoint location_DSmax=38.25mm.

Understand, in the case of same load, ANSYS simulating, verifying value f of this major-minor spring maximum distortion_DSmax=38.25mm, With the maximum distortion f under Rigidity Calculation value_DmaxThe relative deviation of=38.35mm is respectively 0.26%, and result shows this invention institute The Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of the ends contact formula provided is correct, and complex stiffness checks Value is accurately and reliably.

Embodiment two: the main reed number m=2, auxiliary spring sheet number n of the few sheet bias type variable-section steel sheet spring of certain ends contact formula =1, wherein, half length L of each main spring_M=600mm, width b=60mm, elastic modulus E=200GPa, installing space Half l₃=60mm, the root of oblique line section is to distance l of main spring end points_2M=L_M-l₃=540mm；The root flat segments of each main spring Thickness h_2M=12mm, the thickness h of the end flat segments of the 1st main spring₁₁=8mm, the thickness of the oblique line section of the 1st main spring compares β₁ =h₁₁/h_2M=0.67；The end flat segments thickness h of the 2nd main spring₁₂=7mm, the thickness of the oblique line section of the 2nd main spring compares β₂= h₁₂/h_2M=0.58.Half length L of this sheet auxiliary spring_AHorizontal range l of=540mm, auxiliary spring contact and main spring end points₀=L_M-L_A =60mm；The thickness h of the root flat segments of auxiliary spring_2A=13mm, the thickness h of the end flat segments of auxiliary spring_A11=8mm, auxiliary spring oblique The thickness of line segment compares β_A1=h_A11/h_2A=0.62.The complex stiffness of sheet bias type variable cross-section major-minor spring few to this ends contact formula Check.

Use the method for designing identical with embodiment one and step, sheet bias type variable cross-section steel plates few to this ends contact formula The complex stiffness of the major-minor spring of spring checks, and concrete checking computations step is as follows:

(1) the end points deformation coefficient G of each main spring of bias type variable cross-section under end points stressing conditions_x-DiCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=600mm, width b=60mm, elastic modulus E= 200GPa, distance l of the root of main spring oblique line section to main spring end points_2M=540mm, the thickness of the oblique line section of the 1st main spring compares β₁ The thickness of the oblique line section of the=0.67, the 2nd main spring compares β₂=0.58, to the 1st main spring under end points stressing conditions and the 2nd master The end points deformation coefficient G of spring_x-D1And G_x-D2It is respectively calculated, i.e.

$G_{x-D1}=\frac{4}{Eb}(L_M^3-l_{2M}^3)+\frac{6l_{2M}^3{({\mathrm{\β}}_1+1)}^2\[3({\mathrm{\β}}_1-1)-2{\mathrm{ln\β}}_1(1+{\mathrm{\β}}_1)\]}{Eb}+\frac{4{\mathrm{\β}}_1^3{l}_{2M}^3}{Eb}=111.62{\mathrm{mm}}^4/N,$

$G_{x-D2}=\frac{4}{Eb}(L_M^3-l_{2M}^3)+\frac{6l_{2M}^3{({\mathrm{\β}}_2+1)}^2\[3({\mathrm{\β}}_2-1)-2{\mathrm{ln\β}}_2(1+{\mathrm{\β}}_2)\]}{Eb}+\frac{4{\mathrm{\β}}_2^3{l}_{2M}^3}{Eb}=120.43{\mathrm{mm}}^4/N;$

(2) the deformation coefficient G at end flat segments with auxiliary spring contact point of the 2nd main spring under end points stressing conditions_x-CD Calculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=600mm, width b=60mm, elastic modulus E= 200GPa, distance l of the root of main spring oblique line section to main spring end points_2M=540mm, the thickness of the oblique line section of the 2nd main spring compares β₂ =0.58, auxiliary spring contact and horizontal range l of main spring end points₀=60mm, to the 2nd main spring under end points stressing conditions in end Deformation coefficient G at flat segments and auxiliary spring contact point_x-CDCalculate, i.e.

$\begin{array}{c}{G}_{x-CD}=\frac{4L_M^3+22l_{2M}^3({\mathrm{\β}}_2^3-1)+6l_{2M}^3\[3{\mathrm{\β}}_2({\mathrm{\β}}_2-1)-2(1+{\mathrm{\β}}_2^3){\mathrm{ln\β}}_2-6{\mathrm{\β}}_2(1+{\mathrm{\β}}_2){\mathrm{ln\β}}_2\]}{Eb}+\\ \frac{2\[l_0^3+3{\mathrm{\β}}_2^2(l_{2M}^2{\mathrm{\β}}_2^2-L_M^2{\mathrm{\β}}_2-l_{2M}^2)l_0\]}{{\mathrm{Eb\β}}_2^3}=97.67{\mathrm{mm}}^4/N;\end{array}$

(3) the end points deformation coefficient G of the 2nd main spring under major-minor spring contact point stressing conditions_x-Dz2Calculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=600mm, width b=60mm, elastic modulus E= 200GPa, half l of installing space₃=60mm, the thickness of the oblique line section of the 2nd main spring compares β₂=0.58, auxiliary spring contact and main spring Horizontal range l of end points₀=60mm, the end points deformation coefficient G to the 2nd main spring under major-minor spring contact point stressing conditions_x-Dz2 Calculate, i.e.

$\begin{array}{c}{G}_{x-D_{z}2}=\frac{2l_3(6L_M^2-6L_M{l}_3-6L_M{l}_0+2l_3^2+3l_0{l}_3)}{Eb}+\frac{2{(l_0-L_M{\mathrm{\β}}_2^2+l_3{\mathrm{\β}}_2^2)}^2(l_0+2L_M{\mathrm{\β}}_2^2-2l_3{\mathrm{\β}}_2^2)}{{\mathrm{Eb\β}}_2^3}+\\ \frac{6{(L_M-l_3)}^3{({\mathrm{\β}}_2+1)}^2(3{\mathrm{\β}}_2-2{\mathrm{ln\β}}_2-2{\mathrm{\β}}_2{\mathrm{ln\β}}_2-3)}{Eb}+\frac{6l_0{(L_M-l_3)}^3({\mathrm{\β}}_2-1){({\mathrm{\β}}_2+1)}^2}{{\mathrm{Eb\β}}_2}\\ =97.67{\mathrm{mm}}^4/N;\end{array}$

(4) the 2nd main spring under major-minor spring contact point stressing conditions is in the deformation system of end flat segments Yu auxiliary spring contact point Number G_x-CDzCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M=600mm, width b=60mm, elastic modulus E= 200GPa, distance l of the root of main spring oblique line section to main spring end points_2M=540mm, the thickness of the oblique line section of the 2nd main spring compares β₂ =0.58, auxiliary spring contact and horizontal range l of main spring end points₀=60mm, to the 2nd master under major-minor spring contact point stressing conditions Spring deformation coefficient G at end flat segments with auxiliary spring contact point_x-CDzCalculate, i.e.

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_z}=\frac{2{\mathrm{\β}}_2^3{l}_{2M}^3(11-6{\mathrm{ln\β}}_2)}{Eb}-\frac{2(6L_M^2{l}_0-6L_M{l}_0^2+9l_0^2{l}_{2M}+6l_{2M}^3{\mathrm{ln\β}}_2-2L_M^3+11l_{2M}^3)}{Eb}-\\ \frac{2{\mathrm{\β}}_2(3l_0^2{l}_{2M}+9l_{2M}^3+18l_{2M}^3{\mathrm{ln\β}}_2)}{Eb}-\frac{4l_0^3+2{\mathrm{\β}}_2^2(6l_{2M}^2{l}_0-9l_0^2{l}_{2M})-6l_0^2{l}_{2M}{\mathrm{\β}}_2}{{\mathrm{Eb\β}}_2^3}+\\ \frac{2{\mathrm{\β}}_2^2(6l_{2M}^2{l}_0+9l_{2M}^3-18l_{2M}^3{\mathrm{ln\β}}_2)}{Eb}=80.78{\mathrm{mm}}^4/N;\end{array}$

(5) total end points deformation coefficient G of n sheet superposition auxiliary spring_x-DATCalculate:

Half length L according to few sheet bias type variable cross-section auxiliary spring_A=540mm, width b=60mm elastic modulus E= 200GPa, distance l of the root of auxiliary spring oblique line section to auxiliary spring end points_2A=480mm, the thickness of the oblique line section of auxiliary spring compares β_A1= 0.62, this auxiliary spring sheet number n=1, then total end points deformation coefficient G of this sheet auxiliary spring_x-DAT, equal to the end points deformation coefficient of this sheet auxiliary spring G_x-DA1, i.e.

$G_{x-DAT}=\frac{1}{\underset{j=1}{\overset{n}{\mathrm{\Σ}}}\frac{1}{G_{x-DAj}}}=G_{x-DA1}=83.74{\mathrm{mm}}^4/N;$

Wherein, G_x-DA1End points deformation coefficient for this sheet auxiliary spring

$G_{x-DA1}=\frac{4}{Eb}(L_A^3-l_{2A}^3)+\frac{6l_{2A}^3{({\mathrm{\β}}_{A1}+1)}^2\[3({\mathrm{\β}}_{A1}-1)-2{\mathrm{ln\β}}_{A1}(1+{\mathrm{\β}}_{A1})\]}{Eb}+\frac{4{\mathrm{\β}}_{A1}^3{l}_{2A}^3}{Eb}=83.74{\mathrm{mm}}^4/N;$

(6) the complex stiffness checking computations of the few sheet bias type variable cross-section major-minor spring of ends contact formula:

The thickness h of the root flat segments according to each main spring of bias type variable cross-section_2M=12mm, the root flat segments of auxiliary spring Thickness h_2A=13mm, calculated G in step (1)_x-D1=111.62mm⁴/ N and G_x-D2=120.43mm⁴/ N, in step (2) Calculated G_x-CD=97.67mm⁴/ N, calculated G in step (3)_x-Dz2=97.67mm⁴/ N, step calculates in (4) The G arrived_x-CDz=80.78mm⁴Calculated G in/N, and step (5)_x-DAT=83.74mm⁴/ N, sheet few to this ends contact formula The complex stiffness K of bias type variable cross-section major-minor spring_MATCheck, i.e.

$K_{MAT}=\underset{i=1}{\overset{2-1}{\mathrm{\Σ}}}\frac{2h_{2M}^3}{G_{x-Di}}+\frac{2h_{2M}^3(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}{G_{x-D2}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)-G_{x-D_{z}2}{G}_{x-CD}{h}_{2A}^3}=93.38N/mm.$

After major-minor spring concurs, in the case of main spring end points imposed load P=1840N, utilize complex stiffness meter Calculation value K_MAT=93.38N/mm, checks, i.e. the maximum distortion of this few sheet bias type variable cross-section major-minor spring half

$f_{Dmax}=\frac{2P}{K_{MAT}}=39.41mm.$

Utilize ANSYS finite element emulation software, join according to the major-minor spring structure of this few sheet bias type variable-section steel sheet spring Number and material characteristic parameter, set up the ANSYS phantom of half symmetrical structure major-minor spring, grid division, and at phantom Root apply fixed constraint, at main spring end points imposed load P=1840N, to this few sheet bias type variable-section steel sheet spring Major-minor spring deforms and carries out ANSYS emulation, the deformation simulation cloud atlas of obtained major-minor spring, as shown in Figure 4；Wherein, major-minor spring exists Maximum deformation quantity f at endpoint location_DSmax=39.23mm.

Understand, in the case of same load, simulating, verifying value f of this major-minor spring maximum distortion_DSmax=39.23mm, with just Maximum distortion f under degree value of calculation_DmaxThe relative deviation of=39.41mm is respectively 0.46%；Result shows that this invention is provided The Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of ends contact formula be correct, major-minor spring complex stiffness Checking computations value is accurately and reliably.

Claims (1)

1. the Method for Checking of the few sheet bias type variable cross-section major-minor spring complex stiffness of ends contact formula, wherein, few sheet bias type change cuts The half symmetrical structure of face leaf spring is to be made up of root flat segments, oblique line section and end flat segments three sections, each main spring The non-thickness waiting structure, i.e. the end flat segments of the 1st main spring of end flat segments and length, more than the end of other each main spring The thickness of flat segments and length；Auxiliary spring length is less than main spring length, and after load works load more than auxiliary spring, auxiliary spring contact Contact with certain point in the flat segments of main spring end；Each main spring and the structural parameters of auxiliary spring, elastic modelling quantity at major-minor spring give In the case of, the complex stiffness of sheet bias type variable cross-section major-minor spring few to end contact checks, and concrete checking computations step is as follows:

(1) the end points deformation coefficient G of each main spring of bias type variable cross-section under end points stressing conditions_x-DiCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, main spring Sheet number m, distance l of the root of main spring oblique line section to main spring end points_2M=L_M-l₃, the thickness of the oblique line section of i-th main spring compares β_i, its In, i=1,2 ..., m, the end points deformation coefficient G to each main spring under end points stressing conditions_x-DiCalculate, i.e.

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

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

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, oblique line The root of section is to distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of the main spring of m sheet compares β_m, auxiliary spring contact and main spring end Horizontal range l of point₀, to the deformation system at end flat segments with auxiliary spring contact point of the main spring of m sheet under end points stressing conditions Number G_x-CDCalculate, i.e.

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

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

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, oblique line The root of section is to distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of the main spring of m sheet compares β_m, auxiliary spring contact and main spring end Horizontal range l of point₀, end points deformation coefficient G to the main spring of m sheet under major-minor spring contact point stressing conditions_x-DzmCalculate, I.e.

$\begin{array}{c}{G}_{x-D_{z}m}=\frac{2l_3\left(6L_M^2-6L_M{l}_3-6L_M{l}_0+2l_3^2+3l_0{l}_3\right)}{Eb}+\frac{2{\left(l_0-L_M{\mathrm{\β}}_m^2+l_3{\mathrm{\β}}_m^2\right)}^2\left(l_0+2L_M{\mathrm{\β}}_m^2-2l_3{\mathrm{\β}}_m^2\right)}{{\mathrm{Eb\β}}_m^3}+\\ \frac{6{\left(L_M-l_3\right)}^3{\left({\mathrm{\β}}_m+1\right)}^2\left(3{\mathrm{\β}}_m-2{\mathrm{ln\β}}_m-2{\mathrm{\β}}_m{\mathrm{ln\β}}_m-3\right)}{Eb}+\frac{6l_0{\left(L_M-l_3\right)}^3\left({\mathrm{\β}}_m-1\right){\left({\mathrm{\β}}_m+1\right)}^2}{{\mathrm{Eb\β}}_m};\end{array}$

(4) deformation coefficient at end flat segments with auxiliary spring contact point of the main spring of m sheet under major-minor spring contact point stressing conditions G_x-CDzCalculate:

Half length L according to few sheet main spring of bias type variable cross-section_M, width b, elastic modulus E, half l of installing space₃, main spring The root of oblique line section is to distance l of main spring end points_2M=L_M-l₃, the thickness of the oblique line section of the main spring of m sheet compares β_m, auxiliary spring contact and master Horizontal range l of spring end points₀, the main spring of m sheet under major-minor spring contact point stressing conditions is contacted with auxiliary spring in end flat segments Deformation coefficient G at Dian_x-CDzCalculate, i.e.

$\begin{array}{c}{G}_{x-{\mathrm{CD}}_z}=\frac{2{\mathrm{\β}}_m^3{l}_{2M}^3\left(11-6{\mathrm{ln\β}}_m\right)}{Eb}-\frac{2\left(6L_M^2{l}_0-6L_M{l}_0^2+9l_0^2{l}_{2M}+6l_{2M}^3{\mathrm{ln\β}}_m-2L_M^3+11l_{2M}^3\right)}{Eb}-\\ \frac{2{\mathrm{\β}}_m\left(3l_0^2{l}_{2M}+9l_{2M}^3+18l_{2M}^3{\mathrm{ln\β}}_m\right)}{Eb}-\frac{4l_0^3+2{\mathrm{\β}}_m^2\left(6l_{2M}^2{l}_0-9l_0^2{l}_{2M}\right)-6l_0^2{l}_{2M}{\mathrm{\β}}_m}{{\mathrm{Eb\β}}_m^3}+\\ \frac{2{\mathrm{\β}}_m^2\left(6l_{2M}^2{l}_0+9l_{2M}^3-18l_{2M}^3{\mathrm{ln\β}}_m\right)}{Eb};\end{array}$

(5) total end points deformation coefficient G of n sheet superposition auxiliary spring_x-DATCalculate:

Half length L according to few sheet bias type variable cross-section auxiliary spring_A, width b, elastic modulus E, half l of installing space₃, auxiliary spring Sheet number n, distance l of the root of auxiliary spring oblique line section to auxiliary spring end points_2A=L_A-l₃, the thickness of the oblique line section of jth sheet auxiliary spring compares β_Aj, Wherein, j=1,2 ..., n, the total end points deformation coefficient G to the n sheet superposition auxiliary spring under end points stressing conditions_x-DATCalculate, i.e.

$G_{x-DAT}=\frac{1}{\underset{j=1}{\overset{n}{\Σ}}\frac{1}{G_{x-DAj}}},j=1,2,...,n;$

In formula, G_x-DAjEnd points deformation coefficient for jth sheet auxiliary spring, it may be assumed that

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

As auxiliary spring sheet number n=1, then total end points deformation coefficient G of n sheet superposition auxiliary spring_x-DAT, deform equal to the end points of this sheet auxiliary spring Coefficient G_x-DA1, i.e.

G_x-DAT=G_x-DA1；

(6) the complex stiffness checking computations of the few sheet bias type variable cross-section major-minor spring of ends contact formula:

According to main reed number m, the thickness h of the root flat segments of each main spring_2M, the thickness h of the root flat segments of each auxiliary spring_2A, Calculated G in step (1)_x-Di, calculated G in step (2)_x-CD, calculated G in step (3)_x-Dzm, step (4) calculated G in_x-CDz, and calculated G in step (5)_x-DAT, sheet bias type variable cross-section master few to end contact The complex stiffness K of auxiliary spring_MATCheck, i.e.

$K_{MAT}=\underset{i=1}{\overset{m-1}{\Σ}}\frac{2h_{2M}^3}{G_{x-Di}}+\frac{2h_{2M}^3(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)}{G_{x-Dm}(G_{x-DAT}{h}_{2M}^3+G_{x-{\mathrm{CD}}_z}{h}_{2A}^3)-G_{x-D_{z}m}{G}_{x-CD}{h}_{2A}^3}.$