CN104182597B - The check method of vehicle suspension roll angular rigidity - Google Patents

The check method of vehicle suspension roll angular rigidity Download PDF

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CN104182597B
CN104182597B CN201410476073.8A CN201410476073A CN104182597B CN 104182597 B CN104182597 B CN 104182597B CN 201410476073 A CN201410476073 A CN 201410476073A CN 104182597 B CN104182597 B CN 104182597B
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CN104182597A (en
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周长城
郭剑
提艳
安艳
高炳凯
于曰伟
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Shandong University of Technology
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Shandong University of Technology
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Abstract

The present invention relates to the check method of vehicle suspension roll angular rigidity, belongs to vehicle suspension technical field.Previously fail to provide the parsing calculation and check method of roll angular rigidity always.The present invention utilizes the deformation coefficient G of end part of stabilizer rod according to vehicle and suspension parameter, and the structure and material characterisitic parameter of designed stabiliser bar and rubber bushingWAnd rubber bushing radial rigidity KxAnalytical formula, calculation and check is carried out to forward and backward suspension roll angular rigidity and the total roll angular rigidity of vehicle.Using the calculation and check value of the available accurately and reliably vehicle roll angular rigidity of this method, vehicle suspension and the design level and quality of stabilizer bar system can be not only improved, improves the ride performance and security of vehicle;Meanwhile design and testing expenses can be reduced using this method, accelerate product development speed, and reliable technical support is improved for the exploitation of stabiliser bar CAD software.

Description

Method for checking vehicle suspension roll angle rigidity
Technical Field
The invention relates to a vehicle suspension, in particular to a method for checking the roll angle rigidity of the vehicle suspension.
Background
The design of the vehicle suspension and the stabilizer bar must meet the design requirement on the rigidity of the side inclination angle when the vehicle turns. However, due to the restriction of key problems of deformation analysis calculation of the stabilizer bar, deformation analysis calculation of the rubber bushing and mutual coupling, a reliable analysis calculation method cannot be provided for checking and calculating the vehicle roll angle rigidity. At present, most vehicle roll angle rigidity checking at home and abroad utilizes ANSYS simulation software to perform simulation analysis and verification on roll angle rigidity through entity modeling, and although a reliable simulation numerical value can be obtained by the method, the requirement of CAD software development of a suspension stabilizer bar system cannot be met because an accurate analytic calculation formula cannot be provided. With the rapid development of the vehicle industry and the continuous improvement of the vehicle running speed, higher requirements are put forward on the design of a vehicle suspension system and a stabilizer bar, and vehicle manufacturers urgently need the CAD software of the stabilizer bar system. Therefore, an accurate and reliable method for checking the vehicle suspension roll angle rigidity is required to be established, the design level and quality of products are improved, and the running smoothness and safety of vehicles are improved; meanwhile, the design and test cost is reduced, and the product development speed is accelerated.
Disclosure of Invention
In view of the above-mentioned drawbacks in the prior art, the present invention provides a simple and reliable method for checking the roll stiffness of a vehicle suspension, where a checking calculation flowchart is shown in fig. 1, a rolling motion model of the vehicle is shown in fig. 2, and a structure diagram of a stabilizer bar is shown in fig. 3.
In order to solve the technical problem, the method for checking the roll angle rigidity of the vehicle suspension is characterized by comprising the following calculation steps.
(1) Total roll stiffness required for vehicle suspensionThe calculation of (2):
according to vehicle body mass msDistance h between the center of mass of the vehicle body and the roll axissWheel radius r, lateral acceleration ayAnd the maximum roll angle of the vehicle body required by the vehicle designCalculating the total roll stiffness required for the vehicle suspension, namely:
wherein g is the acceleration of gravity;
(2) roll stiffness of front and rear suspension springsAndthe calculation of (2):
according to the front track B of the vehiclefAnd rear track BrLength of front swing arm T1fLength of rear swing arm T1rDistance T between the mounting positions of the front and rear suspension springs and the hinge point of the swing arm2fAnd T2rAnd linear stiffness k of front and rear suspension springssfAnd ksrCalculating the roll angle stiffness of the front and rear suspension springs respectively, namely:
(3) deformation coefficient G of vertical displacement of end parts of front and rear suspension stabilizer barswfAnd GwrThe calculation of (2):
according to the total length l of the front and rear suspension stabilizer barscfAnd lcrMounting distance l of two middle rubber bushings0fAnd l0rArm length l of front and rear stabilizer bars1fAnd l1rRadius of transition arc R of front and rear stabilizer barsfAnd RrCentral angle theta of transition arcfAnd thetarAnd the material elastic model E and Poisson's ratio mu, the deformation coefficient G to the vertical displacement of the front and rear suspension stabilizer bar endswfAnd GwrThe calculation is carried out as follows:
in the formula,
Q6f=32(μ+1)[Rf(cosθf-1)-l1fsinθf]2[2l1fcosθf-lcf+2Rfsinθf];
Q6r=32(u+1)[Rr(cosθr-1)-l1rsinθr]2[2l1rcosθr-lcr+2Rrsinθr];
(4) radial linear rigidity K of rubber bushing of front stabilizer bar and rear stabilizer barxfAnd KxrThe analytic calculation of (2):
according to the inner circle half of the front and rear rubber bushingsDiameter rafAnd rarOuter radius rbfAnd rbrAxial length LfAnd LrAnd modulus of elasticity E of rubber bushingxPoisson ratio μxAnd respectively calculating the radial linear rigidity of the front and rear suspension rubber bushings, namely:
wherein,
bessel correction function I (0, α)frbf),K(0,αfrbf),I(1,αfrbf),K(1,αfrbf);
I(1,αfraf),K(1,αfraf),I(0,αfraf),K(0,αfraf);
Wherein,
bessel correction function I (0, α)rrbr),K(0,αrrbr),I(1,αrrbr),K(1,αrrbr);
I(1,αrrar),K(1,αrrar),I(0,αrrar),K(0,αrrar);
(5) Roll stiffness of front and rear stabilizer bar systemsAndchecking and calculating:
according to the wheel track B of the front and rear axles of the vehiclefAnd BrDiameter d of front and rear suspension stabilizer barsfAnd drLength l ofcfAnd lcrAnd the installation distance length l of the rubber bushing of the front stabilizer bar and the rear stabilizer bar0fAnd l0rAnd (4) calculating the deformation coefficients G of the end points of the front and rear stabilizer bars obtained in the step (3)wfAnd GwrCalculating the radial rigidity K of the front and rear rubber bushings obtained in the step (4)xfAnd KxrRoll stiffness to front and rear stabilizer bar systemsAndrespectively carrying out check calculation, namely:
(6) vehicle total roll stiffnessChecking and calculating:
according to the roll angle stiffness of the front and rear suspension springs obtained in the step (2)Androll angle stiffness of the front and rear stabilizer bar systems obtained in step (5)Andchecking the stiffness of the total roll angle of the vehicle, i.e.
If the calculated value of the checking of the rigidity of the total roll angle of the vehicleGreater than or equal to the design value required for the vehicle calculated in step (1)Namely, it isThe vehicle roll angle rigidity meets the vehicle design requirements; otherwise, ifThe roll angle stiffness of the vehicle does not meet the vehicle design requirements and requires front and rear suspensionThe stabilizer bar system is designed to be adjustable.
Compared with the prior art, the invention has the advantages that:
at present, most of the vehicle suspension side inclination rigidity checking at home and abroad utilizes simulation software to check and analyze the vehicle side inclination rigidity through modeling simulation, but the method can not provide an analytic calculation formula, so that the requirement of CAD software development of a stabilizer bar system can not be met. The invention can utilize the roll angle rigidity of the front and rear suspension springs and the deformation coefficient G of the end part of the stabilizer bar according to vehicle parameters, suspension parameters, structural parameters and material characteristic parameters of the designed stabilizer bar and rubber bushingwAnd radial stiffness K of the rubber bushingxThe analysis calculation formula of (2) checks and calculates the front and rear suspension side inclination rigidity and the vehicle total side inclination rigidity. The method can obtain an accurate and reliable checking calculation value of the vehicle roll angle rigidity, provides a reliable roll angle rigidity checking calculation method for the design of a vehicle suspension and a stabilizer bar, and lays a technical foundation for the development of CAD software of the stabilizer bar. By using the method, the design level and quality of a vehicle suspension and a stabilizer bar system can be improved, and the driving smoothness and safety of a vehicle are improved; meanwhile, the method can reduce the design and test cost and accelerate the product development speed.
Drawings
For a better understanding of the invention, reference is made to the following further description taken in conjunction with the accompanying drawings.
FIG. 1 is a flow chart of a vehicle suspension roll stiffness check calculation;
FIG. 2 is a diagram of a model of a vehicle roll motion;
FIG. 3 is a schematic structural view of a stabilizer bar system;
fig. 4 is a schematic structural view of the rubber bushing.
Detailed Description
The present invention will be described in further detail by way of examples.
The first embodiment is as follows: body mass m of a vehicles4690kg, lateral acceleration ay0.4g, the distance h between the center of mass of the vehicle body and the roll axiss1069mm, design requirement value of vehicle body roll angleFront suspension swing arm length T1f675mm, spring wire stiffness ksf102.45N/mm, distance T between spring center and swing arm hinge point2f430 mm; rear suspension swing arm length T1r650mm, spring wire stiffness Ksr261N/mm, distance T between spring center and swing arm hinge point2r400 mm; the front axle track B of the vehiclef1650mm rear axle track Br1485 mm; the vehicle is provided with a stabilizer bar system only at the front suspension, and the structure diagram is shown in figure 3, wherein, the diameter d of the stabilizer barf20mm, total length lcf800mm, arm length l1f150mm, transition arc radius Rf50mm, transition arc central angle thetaf60 degrees, the installation distance between the two rubber bushings l0f400mm, the elastic modulus E of the stabilizer bar material is 210GPa, and the Poisson ratio mu is 0.3. The structure of the rubber bushing of the front stabilizer bar is as shown in fig. 4, a stabilizer bar 1, an inner circle sleeve 2, a rubber bushing 3, an outer circle sleeve 4, an inner circle sleeve 2 and a rubber bushing 3 are integrated, and are matched with the stabilizer bar 1 through the inner circle sleeve 2 and installed on the stabilizer bar 1, wherein the axial length L of the rubber bushing 1f25mm, thickness hf10mm, wall thickness Δ l of the inner sleeve 2f2.0mm, i.e. the inner circle radius r of the rubber bushingaf12mm, outer radius rbf22mm, modulus of elasticity Ex7.84MPa, Poisson ratio mux0.47. And checking and calculating the roll angle rigidity of the vehicle suspension system.
The method for checking the vehicle suspension roll angle stiffness provided by the embodiment of the invention has a checking calculation flow as shown in figure 1, and comprises the following specific steps:
(1) total roll stiffness required for vehicle suspensionThe calculation of (2):
according to vehicle body mass ms4690kg, lateral acceleration ay0.4g, the distance h between the center of mass of the vehicle body and the roll axiss1069mm, body roll angleNeglecting unsprung mass muIn this case, the total roll stiffness required for the vehicle suspension is calculated, namely:
(2) roll stiffness of front and rear suspension springs for a vehicleAndthe calculation of (2):
according to the front track B of the vehiclef1650mm and rear track Br1485mm, forearm length T1f675mm, rear swing arm length T1r650mm, distance T between the mounting positions of the front and rear suspension springs and the hinge point of the swing arm2f430mm and T2r400mm, and linear stiffness k of front and rear suspension springssf102.45N/mm and ksrThe roll stiffness of the front and rear suspension springs were calculated, 261N/mm, respectively, as:
(3) coefficient of deformation G of front suspension stabilizer bar endwfThe calculation of (2):
stabilizer bar total length l according to front suspensioncf800mm, the installation distance l of two middle rubber bushings0f400mm, arm length l of the front stabilizer bar1f150mm, radius of transition arc R of front stabilizer barf50mm, central angle theta of transition arcfThe deformation coefficient of the front suspension stabilizer bar end is calculated as 60 °, the material elastic model E is 210GPa and the poisson ratio μ is 0.3, namely:
in the formula,
Q6f=32(u+1)[Rf(cosθf-1)-l1fsinθf]2[2l1fcosθf-lcf+2Rfsinθf]=-0.5624m3
(4) radial linear rigidity K of rubber bushing of front suspension stabilizer barxfThe analytic calculation of (2):
according to the inner circle radius r of the rubber bushing of the front suspension stabilizer baraf12mm, outer radius rbf22mm, axial length Lf25mm and modulus of elasticity E of the rubber bushingx7.84MPa, Poisson ratio mux0.47, radial line stiffness K to front suspension rubber bushingxfThe calculation is carried out, namely:
wherein,
bessel's correction function:
I(0,αfrbf)=25.0434,K(0,αfrbf)=0.0041,
I(1,αfrbf)=22.3175,K(1,αfrbf)=0.0045,
I(1,αfraf)=2.1439,K(1,αfraf)=0.0922,
I(0,αfraf)=2.8801,K(0,αfraf)=0.0769;
(5) roll stiffness for a front stabilizer bar systemChecking and calculating:
according to the wheel track B of the front axle of the vehiclef1650mm, diameter d of the stabilizer barf20mm, length lcf800mm, the installation distance between two rubber bushings is l0fCalculating the deformation coefficient G at the end point of the front stabilizer bar obtained in the step (3) when the diameter is 400mmwf=1.5935×10-12m5/N, and calculating the radial stiffness K of the front rubber bushing obtained in the step (4)xfRoll stiffness to front stabilizer bar system of 2106.8N/mmAnd (3) performing checking calculation, namely:
(6) vehicle total roll stiffnessChecking and calculating:
according to the roll angle rigidity of the front and rear suspension springs calculated in the step (2)And
calculating roll angle stiffness of the resulting front stabilizer bar system in step (5)Checking the stiffness of the total roll angle of the vehicle, i.e.
Checking calculation value of total roll angle rigidity of vehicleGreater than the design value required for the vehicle calculated in step (1)Namely, it isThe vehicle roll angle stiffness meets vehicle design requirements.
Example two: knowing the mass m of a vehicle bodys5000kg, lateral acceleration ay0.4g, the distance h between the center of mass of the vehicle body and the roll axiss1150mm vehicle body roll angleSpring wire stiffness k for vehicle front suspensionsf90.761N/mm, spring wire stiffness k of rear suspensionsr176.23N/mm; front track Bf1650mm, rear track Br1600 mm; front suspension swing arm length T1f660mm, distance T between the center of the front suspension spring and the hinge point of the swing arm at the same side2f450 mm; length T of rear swing arm1r650mm, distance T between the mounting center of the rear suspension spring and the hinge point of the cross arm at the same side2r400 mm. The front and rear suspension stabilizer bars and the rubber bushing have the same parameters as in the first embodiment except for the difference in diameter, wherein the diameter d of the front stabilizer barf22mm, diameter d of rear stabilizer barr19 mm; inner radius r of front rubber bushingaf13mm, outer radius rbf23mm, inner circle radius r of rear rubber bushingar11.5 mm. Radius of the outer circle rbr21.5 mm. And checking and calculating the roll angle rigidity of the vehicle suspension system.
By adopting the steps of the first embodiment, the roll angle rigidity of the vehicle suspension system is checked and calculated, namely:
(1) total roll stiffness required for vehicle suspensionThe calculation of (2):
according to vehicle body mass ms5000kg, lateral acceleration ay0.4g, the distance h between the center of mass of the vehicle body and the roll axiss1150mm maximum roll angle required for vehicle body designCalculating the total roll stiffness required for the vehicle suspension, namely:
wherein g is the acceleration of gravity, g is 9.8m/s2
(2) Roll stiffness of front and rear suspension springs for a vehicleAndthe calculation of (2):
according to the front track B of the vehiclef1650mm and rear track Br1600mm, front swing arm length T1f660mm, rear swing arm length T1r650mm, distance T between the mounting positions of the front and rear suspension springs and the hinge point of the swing arm2f450mm and T2r400mm, and linear stiffness k of front and rear suspension springssf90.761N/mm and ksrThe roll stiffness of the front and rear suspension springs were calculated as 176.23N/mm, respectively, i.e.:
(3) deformation coefficient G of front and rear suspension stabilizer bar endswfAnd GwrThe calculation of (2):
since the parameters of the front and rear suspension stabilizer bars are identical to those of the first embodiment except for the diameter, the deformation coefficients of the end portions of the front and rear suspension stabilizer bars are also identical to those of the first embodiment, that is:
(4) radial linear rigidity K of rubber bushing of front and rear suspension stabilizer barsxfAnd KxrThe analytic calculation of (2):
because the diameters of the front and rear suspension stabilizer bars and the inner circle radius and the outer circle radius of the rubber bushing are different, the radial linear rigidity K of the rubber bushingxfAnd KxrAnd are also different. According to the inner radius r of the front rubber bushingaf13mm, outer radius rbf23mm, inner circle radius r of rear rubber bushingar11.5mm, outer radius rbr21.5mm and the length L of the rubber bushingf25mm, modulus of elasticity Ex7.84MPa, Poisson ratio muxThe calculation method of the first embodiment is adopted to calculate the radial linear stiffness K of the rubber bushing of the front and rear suspension stabilizer bars as 0.47xfAnd KxrThe calculations were performed separately, i.e.:
wherein,
(5) roll stiffness of front and rear stabilizer bar systemsAndchecking and calculating:
according to the wheel track B of the front and rear axles of the vehiclef1650mm and BrDiameter d of front and rear stabilizer bars 1600mmf22mm and dr19mm, length lcf=lcr800mm, the installation distance between two rubber bushings is l0f=l0rCalculating the deformation coefficient G of the end points of the front and rear stabilizer bars obtained in the step (3) when the length is 400mmwf=Gwr=1.5935×10-12m5/N, and calculating the radial rigidity K of the front and rear rubber bushings obtained in the step (4)xf2267.0N/mm and KxrRoll stiffness to front and rear stabilizer bar systems of 2026.7N/mmAndrespectively carrying out check calculation, namely:
(6) vehicle total roll stiffnessChecking and calculating:
according to the roll angle rigidity of the front and rear suspension springs calculated in the step (2)Andcalculating the roll angle rigidity of the front and rear stabilizer bar systems obtained in the step (5)Andchecking the stiffness of the total roll angle of the vehicle, i.e.
Checking calculation value of total roll angle rigidity of vehicleGreater than the design value required for the vehicle calculated in step (1)Namely, it isThe vehicle roll angle stiffness meets vehicle design requirements.
Example three: stabilizer bar diameter d of some vehicle except front suspensionfAnd the inner circle radius r of the rubber bushingafAnd the radius r of the outer circlebfExcept for the difference, all other parameters were identical to those of the second embodiment, in which the diameter d of the front stabilizer barfAbout 21mm, the inner circle of the rubber bushing is halfDiameter raf12.5mm, and outer radius rbf22.5 mm. And checking and calculating the roll angle rigidity of the vehicle suspension system.
And checking and calculating the roll angle rigidity of the vehicle suspension system by adopting the step of the embodiment I:
(1) total roll stiffness required for vehicle suspensionThe calculation of (2):
since the vehicle parameters are exactly the same as those of the second embodiment, the total roll stiffness required for the vehicle suspension is also the same as that of the second embodiment, namely:
(2) roll stiffness of front and rear suspension springs for a vehicleAndthe calculation of (2):
since the vehicle parameters and the suspension spring stiffness of the vehicle are exactly the same as those of the second embodiment, the roll angle stiffnesses of the front and rear suspension springs of the vehicle are also respectively the same as those of the second embodiment, that is:
(3) deformation coefficient G of front and rear suspension stabilizer bar endswfAnd GwrThe calculation of (2):
since the parameters of the vehicle front and rear suspension stabilizer bars are exactly the same as those of the second embodiment except for the diameters thereof, the coefficients of deformation of the end portions of the vehicle front and rear suspension stabilizer bars are also exactly the same as those of the second embodiment, that is:
(4) radial linear rigidity K of rubber bushing of front and rear suspension stabilizer barsxfAnd KxrThe analytic calculation of (2):
the diameter of the rear suspension stabilizer bar and the inner circle radius and the outer circle radius of the rubber bushing are completely the same as those of the second embodiment, so that the radial linear rigidity K of the rubber bushing of the rear suspension stabilizer bar isxrAlso all are exactly the same as in example two, namely:
the diameter of the stabilizer bar of the front suspension of the vehicle and the inner circle radius and the outer circle radius of the rubber bushing are different from those of the second embodiment. According to the inner radius r of the front rubber bushingaf12.5mm, outer radius rbf22.5mm, modulus of elasticity Ex7.84MPa, Poisson ratio muxWhen the linear stiffness K of the rubber bushing of the front suspension stabilizer bar is 0.47, the calculation method of the first embodiment is adoptedxfThe calculation is carried out, namely:
wherein,
(5) roll stiffness of front and rear stabilizer bar systemsAndchecking and calculating:
since the vehicle rear suspension stabilizer bar diameter and the rubber bushing are completely the same as those of the second embodiment, the roll angle rigidity of the rear suspension stabilizer bar systemAlso the same as in example two, namely:
according to the wheel track B of the front axle of the vehiclef1650mm, diameter d of the stabilizer barfLength l of 21mmcf800mm, the installation distance between two rubber bushings is l0fCalculating the deformation coefficient G at the end point of the front stabilizer bar obtained in the step (3) when the diameter is 400mmwf=1.5935×10-12m5/N, and calculating the radial stiffness K of the front rubber bushing obtained in the step (4)xfRoll stiffness to front stabilizer bar system of 2186.9N/mmAnd (3) performing checking calculation, namely:
(6) vehicle total roll stiffnessChecking and calculating:
according to the roll angle rigidity of the front and rear suspension springs calculated in the step (2)Andcalculating the roll angle rigidity of the front and rear stabilizer bar systems obtained in the step (5)Andchecking the stiffness of the total roll angle of the vehicle, i.e.
Checking calculation value of total roll angle rigidity of vehicleLess than the design value required for the vehicle calculated in step (1)Namely, it isThe vehicle roll stiffness does not meet vehicle design requirements.
The diameter of the front suspension stabilizer bar is adjusted and designed, namely the diameter of the front suspension stabilizer bar is adjusted and designed to be dfThe roll angle rigidity of the vehicle is increased to 22mmThe design requirement value of the vehicle roll angle rigidity is met; also hasThe mounting distance between the two rubber bushings of the front suspension stabilizer bar can be adjusted and designed, and the distance between the two rubber bushings is adjusted0Is designed to adjust the installation distance0The mounting distance is increased by 10mm when the vehicle is 420mm, so that the roll angle rigidity of the vehicle is increasedAnd the design requirement value of the vehicle roll angle rigidity is met.

Claims (1)

1. The method for checking the roll angle rigidity of the vehicle suspension comprises the following specific steps:
(1) total roll stiffness required for vehicle suspensionThe calculation of (2):
according to vehicle body mass msDistance h between the center of mass of the vehicle body and the roll axissWheel radius r, lateral acceleration ayAnd the maximum roll angle of the vehicle body required by the vehicle designCalculating the total roll stiffness required for the vehicle suspension, namely:
wherein g is the acceleration of gravity;
(2) roll stiffness of front and rear suspension springsAndthe calculation of (2):
according to the front track B of the vehiclefAnd rear track BrLength of front swing arm T1fLength of rear swing arm T1rDistance T between the mounting positions of the front and rear suspension springs and the hinge point of the swing arm2fAnd T2rAnd linear stiffness k of front and rear suspension springssfAnd ksrCalculating the roll angle stiffness of the front and rear suspension springs respectively, namely:
(3) deformation coefficient G of vertical displacement of end parts of front and rear suspension stabilizer barswfAnd GwrThe calculation of (2):
according to the total length l of the front and rear suspension stabilizer barscfAnd lcrMounting distance l of two middle rubber bushings0fAnd l0rArm length l of front and rear stabilizer bars1fAnd l1rRadius of transition arc R of front and rear stabilizer barsfAnd RrCentral angle theta of transition arcfAnd thetarAnd the material elastic model E and Poisson's ratio mu, the deformation coefficient G to the vertical displacement of the front and rear suspension stabilizer bar endswfAnd GwrThe calculation is carried out as follows:
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<mrow> <msub> <mi>G</mi> <mrow> <mi>w</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>Q</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>4</mn> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow> <mn>5</mn> <mi>r</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mrow> <mn>6</mn> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mi>&amp;pi;</mi> <mi>E</mi> </mrow> </mfrac> <mo>;</mo> </mrow>
in the formula,
<mrow> <msub> <mi>Q</mi> <mrow> <mn>3</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>64</mn> <msub> <mi>R</mi> <mi>f</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> <mo>+</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>R</mi> <mi>f</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> </msub> <msub> <mi>R</mi> <mi>f</mi> </msub> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> <mo>&amp;rsqb;</mo> <mo>,</mo> <msub> <mi>Q</mi> <mrow> <mn>4</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <msub> <mi>l</mi> <mrow> <mn>0</mn> <mi>f</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>0</mn> <mi>f</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mn>3</mn> </mfrac> <mo>,</mo> </mrow>
<mrow> <msub> <mi>Q</mi> <mrow> <mn>5</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>64</mn> <msub> <mi>R</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;mu;</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <msup> <msub> <mi>R</mi> <mi>f</mi> </msub> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>3</mn> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> </mrow> <mn>4</mn> </mfrac> <mo>-</mo> <mn>2</mn> <msub> <mi>sin&amp;theta;</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> </msub> <msub> <mi>R</mi> <mi>f</mi> </msub> <msup> <mi>sin</mi> <mn>4</mn> </msup> <mfrac> <msub> <mi>&amp;theta;</mi> <mi>f</mi> </msub> <mn>2</mn> </mfrac> <mo>&amp;rsqb;</mo> <mo>,</mo> </mrow>
Q6f=32(μ+1)[Rf(cosθf-1)-l1fsinθf]2[2l1fcosθf-lcf+2Rfsinθf];
<mrow> <msub> <mi>Q</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>64</mn> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> <mn>3</mn> </msubsup> </mrow> <mn>3</mn> </mfrac> <mo>,</mo> <msub> <mi>Q</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>64</mn> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <msub> <mi>cos&amp;theta;</mi> <mi>r</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mi>r</mi> </msub> <msub> <mi>sin&amp;theta;</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>0</mn> <mi>r</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mn>3</mn> </mfrac> <mo>,</mo> </mrow>
<mrow> <msub> <mi>Q</mi> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>64</mn> <msub> <mi>R</mi> <mi>r</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> <mo>+</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>R</mi> <mi>r</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <msub> <mi>R</mi> <mi>r</mi> </msub> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> <mo>&amp;rsqb;</mo> <mo>,</mo> <msub> <mi>Q</mi> <mrow> <mn>4</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <msub> <mi>l</mi> <mrow> <mn>0</mn> <mi>r</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mn>0</mn> <mi>r</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>l</mi> <mrow> <mi>c</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mn>3</mn> </mfrac> <mo>,</mo> </mrow>
<mrow> <msub> <mi>Q</mi> <mrow> <mn>5</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>64</mn> <msub> <mi>R</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;mu;</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <msubsup> <mi>R</mi> <mi>r</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>3</mn> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> </mrow> <mn>4</mn> </mfrac> <mo>-</mo> <mn>2</mn> <msub> <mi>sin&amp;theta;</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>l</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>2</mn> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <msub> <mi>l</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <msub> <mi>R</mi> <mi>r</mi> </msub> <msup> <mi>sin</mi> <mn>4</mn> </msup> <mfrac> <msub> <mi>&amp;theta;</mi> <mi>r</mi> </msub> <mn>2</mn> </mfrac> <mo>&amp;rsqb;</mo> <mo>,</mo> </mrow>
Q6r=32(u+1)[Rr(cosθr-1)-l1rsinθr]2[2l1rcosθr-lcr+2Rrsinθr];
(4) radial linear rigidity K of rubber bushing of front stabilizer bar and rear stabilizer barxfAnd KxrThe analytic calculation of (2):
according to the inner circle radius r of the front and rear rubber bushingsafAnd rarOuter radius rbfAnd rbrAxial length LfAnd LrAnd modulus of elasticity E of rubber bushingxPoisson ratio μxAnd respectively calculating the radial linear rigidity of the front and rear suspension rubber bushings, namely:
<mrow> <msub> <mi>K</mi> <mrow> <mi>x</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>y</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>x</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>u</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>y</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>;</mo> </mrow>
wherein,
<mrow> <mi>y</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> </msub> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mn>2</mn> <mi>f</mi> </mrow> </msub> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mn>3</mn> <mi>f</mi> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>f</mi> </msub> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mi> </mi> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>+</mo> <mfrac> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> <mo>)</mo> <mo>&amp;lsqb;</mo> <mi>K</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>-</mo> <mi>K</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>(</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>f</mi> </msub> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
<mrow> <msub> <mi>a</mi> <mrow> <mn>2</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> <mo>+</mo> <mn>1</mn> <mo>)</mo> <mo>&amp;lsqb;</mo> <mi>I</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>-</mo> <mi>I</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>(</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>f</mi> </msub> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
<mrow> <msub> <mi>a</mi> <mrow> <mn>3</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> <mo>)</mo> <mo>(</mo> <msub> <mi>b</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>b</mi> <mrow> <mn>2</mn> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow> <mn>3</mn> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>f</mi> </msub> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>;</mo> </mrow>
<mrow> <msub> <mi>b</mi> <mrow> <mn>1</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>b</mi> <mrow> <mn>2</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>b</mi> <mrow> <mn>3</mn> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mi> </mi> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>f</mi> </mrow> </msub> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>&amp;alpha;</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>2</mn> <msqrt> <mn>15</mn> </msqrt> <mo>/</mo> <msub> <mi>L</mi> <mi>f</mi> </msub> <mo>,</mo> </mrow>
bessel correction function I (0, α)frbf),K(0,αfrbf),I(1,αfrbf),K(1,αfrbf);
I(1,αfraf),K(1,αfraf),I(0,αfraf),K(0,αfraf);
Wherein,
<mrow> <mi>y</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>r</mi> </msub> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mi> </mi> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mfrac> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> <mo>)</mo> <mo>&amp;lsqb;</mo> <mi>K</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>-</mo> <mi>K</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>(</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>r</mi> </msub> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
<mrow> <msub> <mi>a</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> <mo>+</mo> <mn>1</mn> <mo>)</mo> <mo>&amp;lsqb;</mo> <mi>I</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>-</mo> <mi>I</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>(</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>r</mi> </msub> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
<mrow> <msub> <mi>a</mi> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;mu;</mi> <mi>x</mi> </msub> <mo>)</mo> <mo>(</mo> <msub> <mi>b</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>b</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <msub> <mi>&amp;pi;E</mi> <mi>x</mi> </msub> <msub> <mi>L</mi> <mi>r</mi> </msub> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>;</mo> </mrow>
<mrow> <msub> <mi>b</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>b</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msub> <mi>b</mi> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>K</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>I</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>&amp;lsqb;</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow> <mi>b</mi> <mi>r</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mi>ln</mi> <mi> </mi> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>,</mo> </mrow>
<mrow> <msub> <mi>&amp;alpha;</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>2</mn> <msqrt> <mn>15</mn> </msqrt> <mo>/</mo> <msub> <mi>L</mi> <mi>r</mi> </msub> <mo>;</mo> </mrow>
bessel correction function I (0, α)rrbr),K(0,αrrbr),I(1,αrrbr),K(1,αrrbr);
I(1,αrrar),K(1,αrrar),I(0,αrrar),K(0,αrrar);
(5) Roll stiffness of front and rear stabilizer bar systemsAndchecking and calculating:
according to the wheel track B of the front and rear axles of the vehiclefAnd BrDiameter d of front and rear suspension stabilizer barsfAnd drLength l ofcfAnd lcrAnd the installation distance length l of the rubber bushing of the front stabilizer bar and the rear stabilizer bar0fAnd l0rAnd (4) calculating the deformation coefficients G of the end points of the front and rear stabilizer bars obtained in the step (3)wfAnd GwrCalculating the radial rigidity K of the front and rear rubber bushings obtained in the step (4)xfAnd KxrRoll stiffness to front and rear stabilizer bar systemsAndrespectively carrying out check calculation, namely:
(6) vehicle total roll stiffnessChecking and calculating:
according to the roll angle stiffness of the front and rear suspension springs obtained in the step (2)Androll angle stiffness of the front and rear stabilizer bar systems obtained in step (5)Andchecking the stiffness of the total roll angle of the vehicle, i.e.
If the calculated value of the checking of the rigidity of the total roll angle of the vehicleGreater than or equal to the design value required for the vehicle calculated in step (1)Namely, it isThe vehicle roll angle rigidity meets the vehicle design requirements; otherwise, ifThe roll angle stiffness of the vehicle does not meet the vehicle design requirements and the front and rear suspension stabilizer bar systems need to be adjusted and designed.
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Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
SU481171A1 (en) * 1971-03-26 1985-08-23 Краснодарский политехнический институт Centrifuge blade with inertial unloading of sediment
CN101284487A (en) * 2008-04-29 2008-10-15 奇瑞汽车股份有限公司 Torsion girder-like rear suspension
CN101966804A (en) * 2010-10-27 2011-02-09 江苏大学 Vehicle suspension control device and method based on automaton technology
CN102402644A (en) * 2011-08-11 2012-04-04 西北工业大学 Dynamical model modeling method of vehicle driven on mountainous road

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
SU481171A1 (en) * 1971-03-26 1985-08-23 Краснодарский политехнический институт Centrifuge blade with inertial unloading of sediment
CN101284487A (en) * 2008-04-29 2008-10-15 奇瑞汽车股份有限公司 Torsion girder-like rear suspension
CN101966804A (en) * 2010-10-27 2011-02-09 江苏大学 Vehicle suspension control device and method based on automaton technology
CN102402644A (en) * 2011-08-11 2012-04-04 西北工业大学 Dynamical model modeling method of vehicle driven on mountainous road

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