Research ArticleThe Parameters Optimizing Design of Double SuspensionArm Torsion Bar in the Electric Sight-Seeing Car by RandomVibration Analyzing Method
Shui-Ting Zhou12 Yi-Jui Chiu23 and I-Hsiang Lin45
1State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan University Changsha 410082 China2School of Mechanical and Automotive Engineering Xiamen University of Technology Xiamen Fujian 361024 China3State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University No 28 Xianning West RoadXirsquoan 710049 China4School of Economics and Management Xiamen University of Technology Fujian China5Fujian Key Laboratory of Advanced Design and Manufacture for Bus Fujian China
Correspondence should be addressed to Yi-Jui Chiu chiuyijuixmuteducn
Received 28 March 2017 Accepted 27 June 2017 Published 12 September 2017
Academic Editor Matteo Filippi
Copyright copy 2017 Shui-Ting Zhou et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This study is about the impact of the performance and the sensitivity analysis for parameters of the torsion bar suspension in theelectric sight-seeing car which the authorsrsquo laboratory designed andwhich is used in the authorsrsquo universityThe suspension stiffnesswas calculated by using the virtual work principle the vector algebra and tensor of finite rotation methods and was verified by theADAMS software Based on the random vibration analysis method the paper analyzed the dynamic tire load (DTL) suspensionworking space (SWS) and comfort performance parameters For the purpose of decreasing the displacement of the suspension andlimiting the frequency of impacting the stop block the paper examined the three parameters and optimized the basic parametersof the torsion bar The results show that the method achieves a great effect and contributes an accurate value for the general layoutdesign
1 Introduction
The common problems of vehicles including suspensionstatic deformation front suspension weakness tendency offrequently hitting stop blocks and location angle of frontwheels changing out of its limits are mainly caused byexcessively low biased frequency design value and suspensionstiffness
Since the year 2000 some conference papers focus onsmall but specific studies Wang et al [1] explored kinematicscharacteristics of suspension with double trailing arms forlight off-road vehicles they found that the double trailingarm lengths the angles between double trailing arms hor-izontalvertical plane and bushing stiffness of the doubletrailing arms linking with frame are four major parametersthat affect the caster and front-wheel steering angle Tian et al[2] optimized analysis of the wheeled armored vehicle double
wishbone independent suspension system Guo and Sun [3]simulated model of the double wishbone independent frontsuspension with a single degree of freedom Through theoptimization and analysis of the upper cross arm and the lowcross arm they observed the goal of decreasing the sidewaysdisplacement and the tire wear Cherian et al [4] developeda nonlinear model of a double wishbone suspension andinvestigated the effects of variation of suspension parameteron the transmission and distribution of tire forces actingon the wheel spindle to the steering system and the vehiclechassis Burgess et al [5] described the development of avehicle suspension analysis tool they selected points andset tolerances to study the suspension model system in theinteraction environment Mohamad and Farhang [6] builta 2D suspension tire system model and investigated thedynamic interaction between the suspension and the tire ofan automotive system
HindawiShock and VibrationVolume 2017 Article ID 8153756 9 pageshttpsdoiorg10115520178153756
2 Shock and Vibration
(a) (b)
Figure 1 (a) The electric sight-seeing car (b) the double suspension arm torsion bar of car
Linear or nonlinear 2-DOF quarter-carmodels have beenwidely used to study the conflicting dynamic performances ofa vehicle suspension In recent years Balike et al [7] built theformulation of a comprehensive kinetic-dynamic quarter-carmodel to study the kinematics and dynamic properties of alinkage suspension Zhao et al [8] studied the lower controlarm of front suspension they optimized original structureand observed that the strength and stiffness were increasedsignificantly while the mass was almost unchanged Wanget al [9] developed a model and calculation method forobtaining the joint forces and moments for a multilink sus-pension Tang et al [10] proposed and developed a modelingand calculation method for obtaining the forces applied tothe ball joints and the bushings and moments applied tothe bushings The calculation and comparison demonstratethat the bushing nonlinear and rotational stiffness mustbe modeled for precise calculation of ball joint force andmoments Liu and Zhao [11] took an optimization design ofthe incompatibility of themovement between suspension andsteering linkage and the trend of tiresrsquo nonpure rolling toreduce the tiresrsquo wear and improve the moving compatibilitybetween suspension and steering system Sun et al [12]designed and strengthened analysis of FSAE suspensionKang et al [13] presented a comprehensive analysis ofdifferent independent front suspension linkages that havebeen implemented in various off-road vehicles including acomposite linkage a candle a trailing arm and a doublewishbone suspension applied to a truck Design fabricationand testing of the suspension subsystem by Das et al [14]had double wishbone geometry for the front half of the carand a semitrailing arm geometry for the rear half Wu etal [15] proposed a new design optimization framework forsuspension systems considering the kinematics characteris-tics The results showed the effectiveness of the proposeddesign method Kang et al [16] investigated relative kine-matics properties of four different independent front axialsuspensions
In all researches mentioned above they did not inputthe road situation as random vibration However the realroad situation is random In this paper suspension stiff-ness was calculated with principle of virtual work andthe result was verified from result simulation in ADAMSConsidering the road situation as random vibration thepower spectral function was used to establish correlation
function as one of constraint conditions to optimize cal-culating basic size of torsion bar for increasing suspensionstiffness decreasing frequency of hitting stop block andmeanwhile limiting the change of location angle of frontwheels
2 Establishing a Mathematical Model
Based on the electric sight-seeing car (Kin-Xia No 1 in Fig-ure 1(a)) which the authorsrsquo laboratory designed and whichis used in the authorsrsquo university the authors established amathematical model of double-arm torsion bar suspensionguide mechanism kinematics as shown in Figure 1(b) Theguide mechanism of double-arm suspension is consideredas a spatial four-bar mechanism The suspension guiderob kinematics relations are established with transcendentalequation of input axis turning angle and output turningangle calculated with MVA combined with finite rotationtensor method The variation function of hard spots isobtained through geometrical relationship The arms undersuspension are connected with spindle 1 and the upper arms119862119863 are connected with spindle 2 with the steering knucklelink point of point 119864 in Figure 2 119901 119902 of unit vectors aredefined as forward direction rotating along spindle 1 andspindle 2 separately The 120572 angle is the rotation angle ofupper arms 119862119863 moving around spindle 2 with the rotationcenter of point 119862 The upper arms 119862119863 are denoted as 1198621198631015840after rotation The 120573 angle is the rotation angle of netherarms 119860119861 moving around spindle 1 with the rotation centerof point 119860 The nether arms 119860119861 are denoted as 1198601198611015840 afterrotation1198741015840 is the wheel center and 119864 point is the intersectionof wheel centerline extension and the virtual swizzle link119863119861
Next we solved the hard point of guide mechanism inthe process of suspension movement The first step is tosolve the point coordinate of exterior ball joints of upperarms 119863 point coordinate is solved according to geomet-ric characteristics since the upper exterior point 119863 onlyrotates around spindle 2 in the process of bobbling Thecoordinate variation function is obtained according to vectormethod
[119909119863 119910119863 119911119863 1] = 119867119900119888 times 119867minus11 times 119867119885 times 1198671 times 1198671198880 (1)
Shock and Vibration 3
E
B
C
A
O
D
D
B
0
0
Figure 2 Structure diagram of double suspension arm
The displacement transformation matrixes of the point 119862coordinate system to the point119874 coordinate of full vehicle are
The displacement transformation matrix of rotationalangle is obtained with minus1205731199001015840 as the rotational angle of 119862 pointcoordinate system moving around the 119910-axis of 119874 pointcoordinate system and minus1205721199001015840 as rotational angle around 119911-axis
The point coordinate of exterior ball joints is obtainedThenwe solved the point coordinate of exterior ball joints
of nether arms 119861 point coordinate is obtained in the sameway since the nether outer point 119861 is rotating around spindle1 in the process of bobbling
The displacement equation of wheel centers is obtainedSince the distances between wheel centers and upper or
B2
DC
PE
B
A
Q
M
m
L
0
dpFs
Figure 3 Force model of the suspension
nether arm points of suspensions and central points ofsteering knuckles are invariant according to the structuredimension separately the 1198741015840 coordinate is obtained onthe basis of vector module invariance The mathematicalexpression of the coordinate of 1198741015840 as the center point of thewheel is as follows
where 1198891199001015840119894 is the distance and 119909119894 119910119894 and 119911119894 119894 = (1198741015840 119861 119863 119864)are the coordinates of corresponding points in the full vehiclecoordinate system that is
(1199091199001015840 minus 119909119861)2 + (1199101199001015840 minus 119910119861)2 + (1199111199001015840 minus 119911119861)2 = 11988921199001015840119861(1199091199001015840 minus 119909119863)2 + (1199101199001015840 minus 119910119863)2 + (1199111199001015840 minus 119911119863)2 = 11988921199001015840119863(1199091199001015840 minus 119909119864)2 + (1199101199001015840 minus 119910119864)2 + (1199111199001015840 minus 119911119864)2 = 11988921199001015840119864
(6)
The coordinate of 1198741015840 as the center point of the wheel isobtained
The vertical displacement variation is expressed as
119889119891 = Δ119904 = Δ1199111199001015840 = (11991111990010158401 minus 11991111990010158402) (7)
where 11991111990010158401 11991111990010158402 are the 119911-axis coordinates of the central pointof wheel
This paper also used virtual work principle to solve thestiffness of suspensions The spring stiffness of double-armindependent suspension is assumed with a virtual stiffness of119862119904 acting on wheels vertically with 119875 point as intersection ofupper and nether arm extension lines separately 119897 as distancebetween intersection 119901 to 119861 point of the nether arm 119898 aslength of the nether arm 119901 as horizontal range from theintersection to the ground connection point of the wheel 119861as wheel span and 119862119879 as stiffness of torsion bars These areshown in Figure 3
According to the virtual work principle the rotationalvirtual angle of the force119865119904 acting on the terrain point119876 is 120575Φ
4 Shock and Vibration
Figure 4 The dynamic model of front suspension and steeringsystem by ADAMS
and the rotational virtual angle of the nether arm suspensionaround point 119860 is 1205751015840 then
3 Simulation Model of Front Suspension andSteering System Establish by ADAMS
The entity structure of automobile front suspension is morecomplicated In order to establish a reasonable model thispaper simplified and assumed the suspension as follows (1)All components of the system are rigid bodies (2) all con-nections between components in the system are simplified ashinges (3) internal clearance is negligible (4) ignoring thedeformation of the guide rod and the absorber is simplified toa damping and linear spring (5) the friction force between thekinematic pairs in the system is ignored and the connectionbetween the motion pairs is rigid (6) the body of the cardoes not move relative to the ground and the tire is simplifiedas a rigid body In the modeling the opposite direction ofthe vehicle is the 119909-axis the left side of the car is the 119910-axisand the vertical upward direction is the 119911-axis This researchbuilds double wishbone torsion bar suspension system andsteering system in the ADAMSCar standardmodeThen onthe basis of the system model established by the templatethe coordinate values of the system model are modifiedaccording to the hard point coordinates as shown in Table 1The dynamic model of the double wishbone torsion barsuspension and steering system is shown in Figure 4
Table 1 The hard point coordinates of front suspension
Hard point 119883coordinate
119884coordinate
119885coordinate
Torsion bar 9500 minus2370 minus490FulcrumLower control arm minus537 minus2370 minus490Front fulcrumLower control arm minus50 minus5940 minus1035Outside fulcrumLower absorber minus37 minus4570 minus633Mounting pointInner spherical head minus1045 minus2570 minus66PointOuter spherical head minus119 minus6315 minus328PointUpper absorber minus37 minus3806 2360Mounting pointUpper control arm minus940 minus3534 1100Front fulcrumUpper control arm 65 minus5675 1160Outer fulcrumUpper control arm 1060 minus3534 1100Back fulcrumWheel center 00 minus6900 minus20
Figure 5 shows that the ADAMS emulation result issimilar to the result calculated mathematically in MATLABwith an error range within plusmn10 in the bobbling process ofplusmn60mm The error is mainly caused by the stiffness changeof rubber boot outside the range of plusmn20mm This figureindicates that our mathematical model is feasible withinnormal range of bobbling
4 Analyzing the Influence of Torsion Bar onSuspension Stiffness
The suspension stiffness is nonlinear based on the nonlin-earity of the movement of space guidance mechanism in theprocess of tire bobbling The variation tendency of front-wheel alignment parameters is relatively small in order toensure the suspension bobbling confirming demand of thearrangement design On the condition that the coordinate ofsuspension hard point is invariant 119889 as the diameter torsionbar along with 119871119890 as the torsion bar effective length and 120572 aspretwist angle are analyzed The swing arm length of netherarms is not analyzed as an effect variable in this paper becausechanging the swing arms of nether arms is equal to changingthe suspension hard point which involves the coordination ofhard points of suspension and steering
The influence of parameter on suspension stiffness isshown in Figure 6 and the parameter variation ranges areplusmn10 Taking full-load suspension stiffness as assessmentcriteria the variable value of stiffness is 15 with the torsion
Figure 6 Stiffness curve of the suspension with variable diameter
bar diameters of 25mm 26mm and 27mm respectively andwith the diameter variable value of 385 compared with themedian of 26mm
Figure 7 shows stiffness curve of the suspension withvariable length The effective lengths of torsion bar are9135mm 950mm and 9865mm respectively with thevariable value of 385 compared with median value Thestiffness variable value of full-loan suspension is 256
Figure 8 shows stiffness curve of the suspension withvariable angle The pretwist angle has no influence onsuspension stiffness in fact with the torsion bar pretwistangle of 841∘ 857∘ and 909∘ with the variable value of385 compared with median value It is probably because
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
L = 9135 mm
L = 9865 mmL = 950mm
Figure 7 Stiffness curve of the suspension with variable length
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
Alfa = 841∘
Alfa = 875∘
Alfa = 909∘
Figure 8 Stiffness curve of the suspension with variable angle
the change of pretwist angle actually changes the workingstate of suspension nether arms and adjusts the position ofinitial state of suspension space that is it changes theworkinginterval instead of suspension stiffness curve
It can be concluded from the above analysis that theparameters with greater influences on suspension stiffnessinclude 119889 as diameter of torsion bar and 119871119890 as effective lengthof torsion bar Considering limits of the general arrangementand the changes of nether arms involving redesign of hard
6 Shock and Vibration
h
m1
x1
x2
k2
m2
k1c1
Figure 9 14 physical model of the vehicle
points both 119889 as diameter of torsion bar and 119871119890 as effectivelength of torsion bar are selected as efficient variables foroptimizing in this paper
5 Establishing the OptimizingMathematical Model
The problems in the test run include the obvious parametervariation of front-wheel alignment and great variation ofvehicle height caused by frontier suspension softness andexcessive wheel bobbling separately The suspension stiffnessis optimized and the diameter and effective length of torsionbar are selected appropriately with the optimizing targetof vertical displacement variations of suspension from zeroloads to no load and from no load to full load combined withdynamic load of tires and riding comfort parameters in thispaper
First the authors established the constrained function ofsuspension performance parameters The evaluation indexformula is established and dynamic rate parameters of sus-pension are analyzed from the perspective of riding safetyand comfort The performance in the working conditionof no load (drivers included) half load and full load isanalyzed with the evaluation index of power spectral densityin frequency domain and with the introduction of newobjective function method
The 14 vehicle model is established as demonstrated inFigure 9 with ℎ as road input 1198961 1198962 as suspension stiffnessand tire stiffness separately 1198881 as suspension damping factor1198981 as bodymass1198982 as wheel mass 1199091 as body displacementand 1199092 as wheel displacement
minus 1198961119898111989611198981 minus 11988811198981
11988811198981
]]]]]
119863 = [[[[[
00
minus 11989621198982
]]]]]
(13)
The road input ℎ is a spectral density function
Φℎ (Ω) = Φℎ (Ω0) [ ΩΩ0 ]minus119908 (14)
where Ω0 is the standard travel circular frequency Φℎ(Ω0) isthe scale of road roughness and119908 is the scale ofwavinessTheroad condition is relatively good considering the driving con-dition as schoolyards and tourist attractions The pavementis comparatively flat and the jerk value is comparatively lowwith the reference input of 119861 scale pavement Next the speedof a campus car is comparatively low because of the speedlimit The vehicle speed V here is set as 10ms
The road spectrum travel power is transformed into timedomain power spectrum as
Φℎ (119908) = 1VΦℎ (Ω) (15)
The standard deviation of evaluation criteria is obtainedon the basis of amplitude-frequency characteristic of roadspectrum and vibration parameters
The scales of driving safety of cars are defined as the rootmean square of weighed acceleration of vertical accelerome-ter
119885 direction of seat 126 11119885 direction of foot 126 13119885 direction of hand 126 075119883 and 119884 direction of seat 123 147119883 direction of back seat 125 109119883 and 119884 direction of foot 128 128Pitching motion 123 116Roll motion 123 098
2
4
6
8
10
12
14
16
18
Wei
ghtin
g co
effici
ent
5 10 15 200Excitation frequency (Hz)
Figure 10 Evaluation value of the vibration
Since the frequency range of random vibration is wideand randomvibration bringsmore discomfort than harmonicvibration for this reason an evaluation function is intro-duced in Figure 10 and 119861foot = 120577119894119861119898 where 120577119894 is randomcorrection factor as shown in Table 2
The vibration analysis in the mathematical model of thispaper is mainly focused on 119911 orientation of feet Correspond-ing parameters are selected according to the above icons
Evaluation index of suspension dynamic travel is definedas quadratic mean of difference between wheel displacementand body displacement the difference of displacement vari-ation caused by road surface irregularity on the condition ofinvariant load
SWS = int501205870
ΦSWS (119908)
= int501205870
[1199091 (119908) minus 1199092 (119908)ℎ (119908) ]2Φℎ (119908) 119889119908
(17)
The two parameters listed below are taken as objectivefunction since the suspension bobbling travel is excessivelylong and the body height variation is excessively obvious inthe trial-production process
Table 3 The optimized result
Torsion bar diameter 265mmTorsion bar effective length 895mm
(1) The vertical displacement variation of nether balljoint point from zero load to full load is
Δ119878 = 119878zerozai minus 119878kongzai (18)
(2) The vertical displacement variation of nether balljoint point from empty load to full load is
Δ119878 = 119878manzai minus 119878kongzai (19)
Next the authors determined the constraint
(1) The longitudinal set of torsion bar is decided bythe spatial arrangement of general arrangement Thelengths of torsion bar (119871119890) in the long direction arelimited in the range of 800mm to 1000mm
(2) Considering the height limit of the distance to nethersurface of the frame the diameters of torsion bar arelimited to the range of 0 lt 119889 lt 40
(3) The diameters are selected within integers or nonin-tegers with decimal parts of 05
(4) The torsion bar is made of 45CrNiMoVa alloy steelwith good quality
(5) The allowable stress of the torsion bar [120591119904] is withinthe range of 1000 to 1250Nmm2
(6) The possibility of hitting a stock is comparativelysmall since the SWS variation of suspension dynamictravel variation is comparatively small
The optimized result is obtained with the MATLABnonlinear multitarget optimizing tool box which is shown inTable 3 The torsion bar diameter is set to be 265mm thetorsion bar effective length is set to be 895mm
6 Analysis of Optimized Result
In Table 4 the optimized result is also obtained with theMATLAB nonlinear multitarget optimizing tool box wenoticed the following
(1) Before optimizing the vertical displacement variationof swizzle nether ball joint points from suspensionzero load to suspension empty load is 775mm andthe vertical displacement variation from empty loadto full load is 88mm before optimizing After opti-mizing the vertical displacement variation of swizzlenether ball joint points from suspension zero loadto suspension empty load becomes 70mm and thevertical displacement variation from empty load tofull load becomes 765mm the value of vertical dis-placement variation is 967 and 131 respectivelyless than the original value
8 Shock and Vibration
Table 4 Property parameter before and after optimization
Property parameter before and after optimizationOptimization 119865119885 (N) SWS (m) 119870foot (dimensionless)
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
Figure 1 (a) The electric sight-seeing car (b) the double suspension arm torsion bar of car
Linear or nonlinear 2-DOF quarter-carmodels have beenwidely used to study the conflicting dynamic performances ofa vehicle suspension In recent years Balike et al [7] built theformulation of a comprehensive kinetic-dynamic quarter-carmodel to study the kinematics and dynamic properties of alinkage suspension Zhao et al [8] studied the lower controlarm of front suspension they optimized original structureand observed that the strength and stiffness were increasedsignificantly while the mass was almost unchanged Wanget al [9] developed a model and calculation method forobtaining the joint forces and moments for a multilink sus-pension Tang et al [10] proposed and developed a modelingand calculation method for obtaining the forces applied tothe ball joints and the bushings and moments applied tothe bushings The calculation and comparison demonstratethat the bushing nonlinear and rotational stiffness mustbe modeled for precise calculation of ball joint force andmoments Liu and Zhao [11] took an optimization design ofthe incompatibility of themovement between suspension andsteering linkage and the trend of tiresrsquo nonpure rolling toreduce the tiresrsquo wear and improve the moving compatibilitybetween suspension and steering system Sun et al [12]designed and strengthened analysis of FSAE suspensionKang et al [13] presented a comprehensive analysis ofdifferent independent front suspension linkages that havebeen implemented in various off-road vehicles including acomposite linkage a candle a trailing arm and a doublewishbone suspension applied to a truck Design fabricationand testing of the suspension subsystem by Das et al [14]had double wishbone geometry for the front half of the carand a semitrailing arm geometry for the rear half Wu etal [15] proposed a new design optimization framework forsuspension systems considering the kinematics characteris-tics The results showed the effectiveness of the proposeddesign method Kang et al [16] investigated relative kine-matics properties of four different independent front axialsuspensions
In all researches mentioned above they did not inputthe road situation as random vibration However the realroad situation is random In this paper suspension stiff-ness was calculated with principle of virtual work andthe result was verified from result simulation in ADAMSConsidering the road situation as random vibration thepower spectral function was used to establish correlation
function as one of constraint conditions to optimize cal-culating basic size of torsion bar for increasing suspensionstiffness decreasing frequency of hitting stop block andmeanwhile limiting the change of location angle of frontwheels
2 Establishing a Mathematical Model
Based on the electric sight-seeing car (Kin-Xia No 1 in Fig-ure 1(a)) which the authorsrsquo laboratory designed and whichis used in the authorsrsquo university the authors established amathematical model of double-arm torsion bar suspensionguide mechanism kinematics as shown in Figure 1(b) Theguide mechanism of double-arm suspension is consideredas a spatial four-bar mechanism The suspension guiderob kinematics relations are established with transcendentalequation of input axis turning angle and output turningangle calculated with MVA combined with finite rotationtensor method The variation function of hard spots isobtained through geometrical relationship The arms undersuspension are connected with spindle 1 and the upper arms119862119863 are connected with spindle 2 with the steering knucklelink point of point 119864 in Figure 2 119901 119902 of unit vectors aredefined as forward direction rotating along spindle 1 andspindle 2 separately The 120572 angle is the rotation angle ofupper arms 119862119863 moving around spindle 2 with the rotationcenter of point 119862 The upper arms 119862119863 are denoted as 1198621198631015840after rotation The 120573 angle is the rotation angle of netherarms 119860119861 moving around spindle 1 with the rotation centerof point 119860 The nether arms 119860119861 are denoted as 1198601198611015840 afterrotation1198741015840 is the wheel center and 119864 point is the intersectionof wheel centerline extension and the virtual swizzle link119863119861
Next we solved the hard point of guide mechanism inthe process of suspension movement The first step is tosolve the point coordinate of exterior ball joints of upperarms 119863 point coordinate is solved according to geomet-ric characteristics since the upper exterior point 119863 onlyrotates around spindle 2 in the process of bobbling Thecoordinate variation function is obtained according to vectormethod
[119909119863 119910119863 119911119863 1] = 119867119900119888 times 119867minus11 times 119867119885 times 1198671 times 1198671198880 (1)
Shock and Vibration 3
E
B
C
A
O
D
D
B
0
0
Figure 2 Structure diagram of double suspension arm
The displacement transformation matrixes of the point 119862coordinate system to the point119874 coordinate of full vehicle are
The displacement transformation matrix of rotationalangle is obtained with minus1205731199001015840 as the rotational angle of 119862 pointcoordinate system moving around the 119910-axis of 119874 pointcoordinate system and minus1205721199001015840 as rotational angle around 119911-axis
The point coordinate of exterior ball joints is obtainedThenwe solved the point coordinate of exterior ball joints
of nether arms 119861 point coordinate is obtained in the sameway since the nether outer point 119861 is rotating around spindle1 in the process of bobbling
The displacement equation of wheel centers is obtainedSince the distances between wheel centers and upper or
B2
DC
PE
B
A
Q
M
m
L
0
dpFs
Figure 3 Force model of the suspension
nether arm points of suspensions and central points ofsteering knuckles are invariant according to the structuredimension separately the 1198741015840 coordinate is obtained onthe basis of vector module invariance The mathematicalexpression of the coordinate of 1198741015840 as the center point of thewheel is as follows
where 1198891199001015840119894 is the distance and 119909119894 119910119894 and 119911119894 119894 = (1198741015840 119861 119863 119864)are the coordinates of corresponding points in the full vehiclecoordinate system that is
(1199091199001015840 minus 119909119861)2 + (1199101199001015840 minus 119910119861)2 + (1199111199001015840 minus 119911119861)2 = 11988921199001015840119861(1199091199001015840 minus 119909119863)2 + (1199101199001015840 minus 119910119863)2 + (1199111199001015840 minus 119911119863)2 = 11988921199001015840119863(1199091199001015840 minus 119909119864)2 + (1199101199001015840 minus 119910119864)2 + (1199111199001015840 minus 119911119864)2 = 11988921199001015840119864
(6)
The coordinate of 1198741015840 as the center point of the wheel isobtained
The vertical displacement variation is expressed as
119889119891 = Δ119904 = Δ1199111199001015840 = (11991111990010158401 minus 11991111990010158402) (7)
where 11991111990010158401 11991111990010158402 are the 119911-axis coordinates of the central pointof wheel
This paper also used virtual work principle to solve thestiffness of suspensions The spring stiffness of double-armindependent suspension is assumed with a virtual stiffness of119862119904 acting on wheels vertically with 119875 point as intersection ofupper and nether arm extension lines separately 119897 as distancebetween intersection 119901 to 119861 point of the nether arm 119898 aslength of the nether arm 119901 as horizontal range from theintersection to the ground connection point of the wheel 119861as wheel span and 119862119879 as stiffness of torsion bars These areshown in Figure 3
According to the virtual work principle the rotationalvirtual angle of the force119865119904 acting on the terrain point119876 is 120575Φ
4 Shock and Vibration
Figure 4 The dynamic model of front suspension and steeringsystem by ADAMS
and the rotational virtual angle of the nether arm suspensionaround point 119860 is 1205751015840 then
3 Simulation Model of Front Suspension andSteering System Establish by ADAMS
The entity structure of automobile front suspension is morecomplicated In order to establish a reasonable model thispaper simplified and assumed the suspension as follows (1)All components of the system are rigid bodies (2) all con-nections between components in the system are simplified ashinges (3) internal clearance is negligible (4) ignoring thedeformation of the guide rod and the absorber is simplified toa damping and linear spring (5) the friction force between thekinematic pairs in the system is ignored and the connectionbetween the motion pairs is rigid (6) the body of the cardoes not move relative to the ground and the tire is simplifiedas a rigid body In the modeling the opposite direction ofthe vehicle is the 119909-axis the left side of the car is the 119910-axisand the vertical upward direction is the 119911-axis This researchbuilds double wishbone torsion bar suspension system andsteering system in the ADAMSCar standardmodeThen onthe basis of the system model established by the templatethe coordinate values of the system model are modifiedaccording to the hard point coordinates as shown in Table 1The dynamic model of the double wishbone torsion barsuspension and steering system is shown in Figure 4
Table 1 The hard point coordinates of front suspension
Hard point 119883coordinate
119884coordinate
119885coordinate
Torsion bar 9500 minus2370 minus490FulcrumLower control arm minus537 minus2370 minus490Front fulcrumLower control arm minus50 minus5940 minus1035Outside fulcrumLower absorber minus37 minus4570 minus633Mounting pointInner spherical head minus1045 minus2570 minus66PointOuter spherical head minus119 minus6315 minus328PointUpper absorber minus37 minus3806 2360Mounting pointUpper control arm minus940 minus3534 1100Front fulcrumUpper control arm 65 minus5675 1160Outer fulcrumUpper control arm 1060 minus3534 1100Back fulcrumWheel center 00 minus6900 minus20
Figure 5 shows that the ADAMS emulation result issimilar to the result calculated mathematically in MATLABwith an error range within plusmn10 in the bobbling process ofplusmn60mm The error is mainly caused by the stiffness changeof rubber boot outside the range of plusmn20mm This figureindicates that our mathematical model is feasible withinnormal range of bobbling
4 Analyzing the Influence of Torsion Bar onSuspension Stiffness
The suspension stiffness is nonlinear based on the nonlin-earity of the movement of space guidance mechanism in theprocess of tire bobbling The variation tendency of front-wheel alignment parameters is relatively small in order toensure the suspension bobbling confirming demand of thearrangement design On the condition that the coordinate ofsuspension hard point is invariant 119889 as the diameter torsionbar along with 119871119890 as the torsion bar effective length and 120572 aspretwist angle are analyzed The swing arm length of netherarms is not analyzed as an effect variable in this paper becausechanging the swing arms of nether arms is equal to changingthe suspension hard point which involves the coordination ofhard points of suspension and steering
The influence of parameter on suspension stiffness isshown in Figure 6 and the parameter variation ranges areplusmn10 Taking full-load suspension stiffness as assessmentcriteria the variable value of stiffness is 15 with the torsion
Figure 6 Stiffness curve of the suspension with variable diameter
bar diameters of 25mm 26mm and 27mm respectively andwith the diameter variable value of 385 compared with themedian of 26mm
Figure 7 shows stiffness curve of the suspension withvariable length The effective lengths of torsion bar are9135mm 950mm and 9865mm respectively with thevariable value of 385 compared with median value Thestiffness variable value of full-loan suspension is 256
Figure 8 shows stiffness curve of the suspension withvariable angle The pretwist angle has no influence onsuspension stiffness in fact with the torsion bar pretwistangle of 841∘ 857∘ and 909∘ with the variable value of385 compared with median value It is probably because
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
L = 9135 mm
L = 9865 mmL = 950mm
Figure 7 Stiffness curve of the suspension with variable length
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
Alfa = 841∘
Alfa = 875∘
Alfa = 909∘
Figure 8 Stiffness curve of the suspension with variable angle
the change of pretwist angle actually changes the workingstate of suspension nether arms and adjusts the position ofinitial state of suspension space that is it changes theworkinginterval instead of suspension stiffness curve
It can be concluded from the above analysis that theparameters with greater influences on suspension stiffnessinclude 119889 as diameter of torsion bar and 119871119890 as effective lengthof torsion bar Considering limits of the general arrangementand the changes of nether arms involving redesign of hard
6 Shock and Vibration
h
m1
x1
x2
k2
m2
k1c1
Figure 9 14 physical model of the vehicle
points both 119889 as diameter of torsion bar and 119871119890 as effectivelength of torsion bar are selected as efficient variables foroptimizing in this paper
5 Establishing the OptimizingMathematical Model
The problems in the test run include the obvious parametervariation of front-wheel alignment and great variation ofvehicle height caused by frontier suspension softness andexcessive wheel bobbling separately The suspension stiffnessis optimized and the diameter and effective length of torsionbar are selected appropriately with the optimizing targetof vertical displacement variations of suspension from zeroloads to no load and from no load to full load combined withdynamic load of tires and riding comfort parameters in thispaper
First the authors established the constrained function ofsuspension performance parameters The evaluation indexformula is established and dynamic rate parameters of sus-pension are analyzed from the perspective of riding safetyand comfort The performance in the working conditionof no load (drivers included) half load and full load isanalyzed with the evaluation index of power spectral densityin frequency domain and with the introduction of newobjective function method
The 14 vehicle model is established as demonstrated inFigure 9 with ℎ as road input 1198961 1198962 as suspension stiffnessand tire stiffness separately 1198881 as suspension damping factor1198981 as bodymass1198982 as wheel mass 1199091 as body displacementand 1199092 as wheel displacement
minus 1198961119898111989611198981 minus 11988811198981
11988811198981
]]]]]
119863 = [[[[[
00
minus 11989621198982
]]]]]
(13)
The road input ℎ is a spectral density function
Φℎ (Ω) = Φℎ (Ω0) [ ΩΩ0 ]minus119908 (14)
where Ω0 is the standard travel circular frequency Φℎ(Ω0) isthe scale of road roughness and119908 is the scale ofwavinessTheroad condition is relatively good considering the driving con-dition as schoolyards and tourist attractions The pavementis comparatively flat and the jerk value is comparatively lowwith the reference input of 119861 scale pavement Next the speedof a campus car is comparatively low because of the speedlimit The vehicle speed V here is set as 10ms
The road spectrum travel power is transformed into timedomain power spectrum as
Φℎ (119908) = 1VΦℎ (Ω) (15)
The standard deviation of evaluation criteria is obtainedon the basis of amplitude-frequency characteristic of roadspectrum and vibration parameters
The scales of driving safety of cars are defined as the rootmean square of weighed acceleration of vertical accelerome-ter
119885 direction of seat 126 11119885 direction of foot 126 13119885 direction of hand 126 075119883 and 119884 direction of seat 123 147119883 direction of back seat 125 109119883 and 119884 direction of foot 128 128Pitching motion 123 116Roll motion 123 098
2
4
6
8
10
12
14
16
18
Wei
ghtin
g co
effici
ent
5 10 15 200Excitation frequency (Hz)
Figure 10 Evaluation value of the vibration
Since the frequency range of random vibration is wideand randomvibration bringsmore discomfort than harmonicvibration for this reason an evaluation function is intro-duced in Figure 10 and 119861foot = 120577119894119861119898 where 120577119894 is randomcorrection factor as shown in Table 2
The vibration analysis in the mathematical model of thispaper is mainly focused on 119911 orientation of feet Correspond-ing parameters are selected according to the above icons
Evaluation index of suspension dynamic travel is definedas quadratic mean of difference between wheel displacementand body displacement the difference of displacement vari-ation caused by road surface irregularity on the condition ofinvariant load
SWS = int501205870
ΦSWS (119908)
= int501205870
[1199091 (119908) minus 1199092 (119908)ℎ (119908) ]2Φℎ (119908) 119889119908
(17)
The two parameters listed below are taken as objectivefunction since the suspension bobbling travel is excessivelylong and the body height variation is excessively obvious inthe trial-production process
Table 3 The optimized result
Torsion bar diameter 265mmTorsion bar effective length 895mm
(1) The vertical displacement variation of nether balljoint point from zero load to full load is
Δ119878 = 119878zerozai minus 119878kongzai (18)
(2) The vertical displacement variation of nether balljoint point from empty load to full load is
Δ119878 = 119878manzai minus 119878kongzai (19)
Next the authors determined the constraint
(1) The longitudinal set of torsion bar is decided bythe spatial arrangement of general arrangement Thelengths of torsion bar (119871119890) in the long direction arelimited in the range of 800mm to 1000mm
(2) Considering the height limit of the distance to nethersurface of the frame the diameters of torsion bar arelimited to the range of 0 lt 119889 lt 40
(3) The diameters are selected within integers or nonin-tegers with decimal parts of 05
(4) The torsion bar is made of 45CrNiMoVa alloy steelwith good quality
(5) The allowable stress of the torsion bar [120591119904] is withinthe range of 1000 to 1250Nmm2
(6) The possibility of hitting a stock is comparativelysmall since the SWS variation of suspension dynamictravel variation is comparatively small
The optimized result is obtained with the MATLABnonlinear multitarget optimizing tool box which is shown inTable 3 The torsion bar diameter is set to be 265mm thetorsion bar effective length is set to be 895mm
6 Analysis of Optimized Result
In Table 4 the optimized result is also obtained with theMATLAB nonlinear multitarget optimizing tool box wenoticed the following
(1) Before optimizing the vertical displacement variationof swizzle nether ball joint points from suspensionzero load to suspension empty load is 775mm andthe vertical displacement variation from empty loadto full load is 88mm before optimizing After opti-mizing the vertical displacement variation of swizzlenether ball joint points from suspension zero loadto suspension empty load becomes 70mm and thevertical displacement variation from empty load tofull load becomes 765mm the value of vertical dis-placement variation is 967 and 131 respectivelyless than the original value
8 Shock and Vibration
Table 4 Property parameter before and after optimization
Property parameter before and after optimizationOptimization 119865119885 (N) SWS (m) 119870foot (dimensionless)
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
The displacement transformation matrix of rotationalangle is obtained with minus1205731199001015840 as the rotational angle of 119862 pointcoordinate system moving around the 119910-axis of 119874 pointcoordinate system and minus1205721199001015840 as rotational angle around 119911-axis
The point coordinate of exterior ball joints is obtainedThenwe solved the point coordinate of exterior ball joints
of nether arms 119861 point coordinate is obtained in the sameway since the nether outer point 119861 is rotating around spindle1 in the process of bobbling
The displacement equation of wheel centers is obtainedSince the distances between wheel centers and upper or
B2
DC
PE
B
A
Q
M
m
L
0
dpFs
Figure 3 Force model of the suspension
nether arm points of suspensions and central points ofsteering knuckles are invariant according to the structuredimension separately the 1198741015840 coordinate is obtained onthe basis of vector module invariance The mathematicalexpression of the coordinate of 1198741015840 as the center point of thewheel is as follows
where 1198891199001015840119894 is the distance and 119909119894 119910119894 and 119911119894 119894 = (1198741015840 119861 119863 119864)are the coordinates of corresponding points in the full vehiclecoordinate system that is
(1199091199001015840 minus 119909119861)2 + (1199101199001015840 minus 119910119861)2 + (1199111199001015840 minus 119911119861)2 = 11988921199001015840119861(1199091199001015840 minus 119909119863)2 + (1199101199001015840 minus 119910119863)2 + (1199111199001015840 minus 119911119863)2 = 11988921199001015840119863(1199091199001015840 minus 119909119864)2 + (1199101199001015840 minus 119910119864)2 + (1199111199001015840 minus 119911119864)2 = 11988921199001015840119864
(6)
The coordinate of 1198741015840 as the center point of the wheel isobtained
The vertical displacement variation is expressed as
119889119891 = Δ119904 = Δ1199111199001015840 = (11991111990010158401 minus 11991111990010158402) (7)
where 11991111990010158401 11991111990010158402 are the 119911-axis coordinates of the central pointof wheel
This paper also used virtual work principle to solve thestiffness of suspensions The spring stiffness of double-armindependent suspension is assumed with a virtual stiffness of119862119904 acting on wheels vertically with 119875 point as intersection ofupper and nether arm extension lines separately 119897 as distancebetween intersection 119901 to 119861 point of the nether arm 119898 aslength of the nether arm 119901 as horizontal range from theintersection to the ground connection point of the wheel 119861as wheel span and 119862119879 as stiffness of torsion bars These areshown in Figure 3
According to the virtual work principle the rotationalvirtual angle of the force119865119904 acting on the terrain point119876 is 120575Φ
4 Shock and Vibration
Figure 4 The dynamic model of front suspension and steeringsystem by ADAMS
and the rotational virtual angle of the nether arm suspensionaround point 119860 is 1205751015840 then
3 Simulation Model of Front Suspension andSteering System Establish by ADAMS
The entity structure of automobile front suspension is morecomplicated In order to establish a reasonable model thispaper simplified and assumed the suspension as follows (1)All components of the system are rigid bodies (2) all con-nections between components in the system are simplified ashinges (3) internal clearance is negligible (4) ignoring thedeformation of the guide rod and the absorber is simplified toa damping and linear spring (5) the friction force between thekinematic pairs in the system is ignored and the connectionbetween the motion pairs is rigid (6) the body of the cardoes not move relative to the ground and the tire is simplifiedas a rigid body In the modeling the opposite direction ofthe vehicle is the 119909-axis the left side of the car is the 119910-axisand the vertical upward direction is the 119911-axis This researchbuilds double wishbone torsion bar suspension system andsteering system in the ADAMSCar standardmodeThen onthe basis of the system model established by the templatethe coordinate values of the system model are modifiedaccording to the hard point coordinates as shown in Table 1The dynamic model of the double wishbone torsion barsuspension and steering system is shown in Figure 4
Table 1 The hard point coordinates of front suspension
Hard point 119883coordinate
119884coordinate
119885coordinate
Torsion bar 9500 minus2370 minus490FulcrumLower control arm minus537 minus2370 minus490Front fulcrumLower control arm minus50 minus5940 minus1035Outside fulcrumLower absorber minus37 minus4570 minus633Mounting pointInner spherical head minus1045 minus2570 minus66PointOuter spherical head minus119 minus6315 minus328PointUpper absorber minus37 minus3806 2360Mounting pointUpper control arm minus940 minus3534 1100Front fulcrumUpper control arm 65 minus5675 1160Outer fulcrumUpper control arm 1060 minus3534 1100Back fulcrumWheel center 00 minus6900 minus20
Figure 5 shows that the ADAMS emulation result issimilar to the result calculated mathematically in MATLABwith an error range within plusmn10 in the bobbling process ofplusmn60mm The error is mainly caused by the stiffness changeof rubber boot outside the range of plusmn20mm This figureindicates that our mathematical model is feasible withinnormal range of bobbling
4 Analyzing the Influence of Torsion Bar onSuspension Stiffness
The suspension stiffness is nonlinear based on the nonlin-earity of the movement of space guidance mechanism in theprocess of tire bobbling The variation tendency of front-wheel alignment parameters is relatively small in order toensure the suspension bobbling confirming demand of thearrangement design On the condition that the coordinate ofsuspension hard point is invariant 119889 as the diameter torsionbar along with 119871119890 as the torsion bar effective length and 120572 aspretwist angle are analyzed The swing arm length of netherarms is not analyzed as an effect variable in this paper becausechanging the swing arms of nether arms is equal to changingthe suspension hard point which involves the coordination ofhard points of suspension and steering
The influence of parameter on suspension stiffness isshown in Figure 6 and the parameter variation ranges areplusmn10 Taking full-load suspension stiffness as assessmentcriteria the variable value of stiffness is 15 with the torsion
Figure 6 Stiffness curve of the suspension with variable diameter
bar diameters of 25mm 26mm and 27mm respectively andwith the diameter variable value of 385 compared with themedian of 26mm
Figure 7 shows stiffness curve of the suspension withvariable length The effective lengths of torsion bar are9135mm 950mm and 9865mm respectively with thevariable value of 385 compared with median value Thestiffness variable value of full-loan suspension is 256
Figure 8 shows stiffness curve of the suspension withvariable angle The pretwist angle has no influence onsuspension stiffness in fact with the torsion bar pretwistangle of 841∘ 857∘ and 909∘ with the variable value of385 compared with median value It is probably because
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
L = 9135 mm
L = 9865 mmL = 950mm
Figure 7 Stiffness curve of the suspension with variable length
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
Alfa = 841∘
Alfa = 875∘
Alfa = 909∘
Figure 8 Stiffness curve of the suspension with variable angle
the change of pretwist angle actually changes the workingstate of suspension nether arms and adjusts the position ofinitial state of suspension space that is it changes theworkinginterval instead of suspension stiffness curve
It can be concluded from the above analysis that theparameters with greater influences on suspension stiffnessinclude 119889 as diameter of torsion bar and 119871119890 as effective lengthof torsion bar Considering limits of the general arrangementand the changes of nether arms involving redesign of hard
6 Shock and Vibration
h
m1
x1
x2
k2
m2
k1c1
Figure 9 14 physical model of the vehicle
points both 119889 as diameter of torsion bar and 119871119890 as effectivelength of torsion bar are selected as efficient variables foroptimizing in this paper
5 Establishing the OptimizingMathematical Model
The problems in the test run include the obvious parametervariation of front-wheel alignment and great variation ofvehicle height caused by frontier suspension softness andexcessive wheel bobbling separately The suspension stiffnessis optimized and the diameter and effective length of torsionbar are selected appropriately with the optimizing targetof vertical displacement variations of suspension from zeroloads to no load and from no load to full load combined withdynamic load of tires and riding comfort parameters in thispaper
First the authors established the constrained function ofsuspension performance parameters The evaluation indexformula is established and dynamic rate parameters of sus-pension are analyzed from the perspective of riding safetyand comfort The performance in the working conditionof no load (drivers included) half load and full load isanalyzed with the evaluation index of power spectral densityin frequency domain and with the introduction of newobjective function method
The 14 vehicle model is established as demonstrated inFigure 9 with ℎ as road input 1198961 1198962 as suspension stiffnessand tire stiffness separately 1198881 as suspension damping factor1198981 as bodymass1198982 as wheel mass 1199091 as body displacementand 1199092 as wheel displacement
minus 1198961119898111989611198981 minus 11988811198981
11988811198981
]]]]]
119863 = [[[[[
00
minus 11989621198982
]]]]]
(13)
The road input ℎ is a spectral density function
Φℎ (Ω) = Φℎ (Ω0) [ ΩΩ0 ]minus119908 (14)
where Ω0 is the standard travel circular frequency Φℎ(Ω0) isthe scale of road roughness and119908 is the scale ofwavinessTheroad condition is relatively good considering the driving con-dition as schoolyards and tourist attractions The pavementis comparatively flat and the jerk value is comparatively lowwith the reference input of 119861 scale pavement Next the speedof a campus car is comparatively low because of the speedlimit The vehicle speed V here is set as 10ms
The road spectrum travel power is transformed into timedomain power spectrum as
Φℎ (119908) = 1VΦℎ (Ω) (15)
The standard deviation of evaluation criteria is obtainedon the basis of amplitude-frequency characteristic of roadspectrum and vibration parameters
The scales of driving safety of cars are defined as the rootmean square of weighed acceleration of vertical accelerome-ter
119885 direction of seat 126 11119885 direction of foot 126 13119885 direction of hand 126 075119883 and 119884 direction of seat 123 147119883 direction of back seat 125 109119883 and 119884 direction of foot 128 128Pitching motion 123 116Roll motion 123 098
2
4
6
8
10
12
14
16
18
Wei
ghtin
g co
effici
ent
5 10 15 200Excitation frequency (Hz)
Figure 10 Evaluation value of the vibration
Since the frequency range of random vibration is wideand randomvibration bringsmore discomfort than harmonicvibration for this reason an evaluation function is intro-duced in Figure 10 and 119861foot = 120577119894119861119898 where 120577119894 is randomcorrection factor as shown in Table 2
The vibration analysis in the mathematical model of thispaper is mainly focused on 119911 orientation of feet Correspond-ing parameters are selected according to the above icons
Evaluation index of suspension dynamic travel is definedas quadratic mean of difference between wheel displacementand body displacement the difference of displacement vari-ation caused by road surface irregularity on the condition ofinvariant load
SWS = int501205870
ΦSWS (119908)
= int501205870
[1199091 (119908) minus 1199092 (119908)ℎ (119908) ]2Φℎ (119908) 119889119908
(17)
The two parameters listed below are taken as objectivefunction since the suspension bobbling travel is excessivelylong and the body height variation is excessively obvious inthe trial-production process
Table 3 The optimized result
Torsion bar diameter 265mmTorsion bar effective length 895mm
(1) The vertical displacement variation of nether balljoint point from zero load to full load is
Δ119878 = 119878zerozai minus 119878kongzai (18)
(2) The vertical displacement variation of nether balljoint point from empty load to full load is
Δ119878 = 119878manzai minus 119878kongzai (19)
Next the authors determined the constraint
(1) The longitudinal set of torsion bar is decided bythe spatial arrangement of general arrangement Thelengths of torsion bar (119871119890) in the long direction arelimited in the range of 800mm to 1000mm
(2) Considering the height limit of the distance to nethersurface of the frame the diameters of torsion bar arelimited to the range of 0 lt 119889 lt 40
(3) The diameters are selected within integers or nonin-tegers with decimal parts of 05
(4) The torsion bar is made of 45CrNiMoVa alloy steelwith good quality
(5) The allowable stress of the torsion bar [120591119904] is withinthe range of 1000 to 1250Nmm2
(6) The possibility of hitting a stock is comparativelysmall since the SWS variation of suspension dynamictravel variation is comparatively small
The optimized result is obtained with the MATLABnonlinear multitarget optimizing tool box which is shown inTable 3 The torsion bar diameter is set to be 265mm thetorsion bar effective length is set to be 895mm
6 Analysis of Optimized Result
In Table 4 the optimized result is also obtained with theMATLAB nonlinear multitarget optimizing tool box wenoticed the following
(1) Before optimizing the vertical displacement variationof swizzle nether ball joint points from suspensionzero load to suspension empty load is 775mm andthe vertical displacement variation from empty loadto full load is 88mm before optimizing After opti-mizing the vertical displacement variation of swizzlenether ball joint points from suspension zero loadto suspension empty load becomes 70mm and thevertical displacement variation from empty load tofull load becomes 765mm the value of vertical dis-placement variation is 967 and 131 respectivelyless than the original value
8 Shock and Vibration
Table 4 Property parameter before and after optimization
Property parameter before and after optimizationOptimization 119865119885 (N) SWS (m) 119870foot (dimensionless)
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
3 Simulation Model of Front Suspension andSteering System Establish by ADAMS
The entity structure of automobile front suspension is morecomplicated In order to establish a reasonable model thispaper simplified and assumed the suspension as follows (1)All components of the system are rigid bodies (2) all con-nections between components in the system are simplified ashinges (3) internal clearance is negligible (4) ignoring thedeformation of the guide rod and the absorber is simplified toa damping and linear spring (5) the friction force between thekinematic pairs in the system is ignored and the connectionbetween the motion pairs is rigid (6) the body of the cardoes not move relative to the ground and the tire is simplifiedas a rigid body In the modeling the opposite direction ofthe vehicle is the 119909-axis the left side of the car is the 119910-axisand the vertical upward direction is the 119911-axis This researchbuilds double wishbone torsion bar suspension system andsteering system in the ADAMSCar standardmodeThen onthe basis of the system model established by the templatethe coordinate values of the system model are modifiedaccording to the hard point coordinates as shown in Table 1The dynamic model of the double wishbone torsion barsuspension and steering system is shown in Figure 4
Table 1 The hard point coordinates of front suspension
Hard point 119883coordinate
119884coordinate
119885coordinate
Torsion bar 9500 minus2370 minus490FulcrumLower control arm minus537 minus2370 minus490Front fulcrumLower control arm minus50 minus5940 minus1035Outside fulcrumLower absorber minus37 minus4570 minus633Mounting pointInner spherical head minus1045 minus2570 minus66PointOuter spherical head minus119 minus6315 minus328PointUpper absorber minus37 minus3806 2360Mounting pointUpper control arm minus940 minus3534 1100Front fulcrumUpper control arm 65 minus5675 1160Outer fulcrumUpper control arm 1060 minus3534 1100Back fulcrumWheel center 00 minus6900 minus20
Figure 5 shows that the ADAMS emulation result issimilar to the result calculated mathematically in MATLABwith an error range within plusmn10 in the bobbling process ofplusmn60mm The error is mainly caused by the stiffness changeof rubber boot outside the range of plusmn20mm This figureindicates that our mathematical model is feasible withinnormal range of bobbling
4 Analyzing the Influence of Torsion Bar onSuspension Stiffness
The suspension stiffness is nonlinear based on the nonlin-earity of the movement of space guidance mechanism in theprocess of tire bobbling The variation tendency of front-wheel alignment parameters is relatively small in order toensure the suspension bobbling confirming demand of thearrangement design On the condition that the coordinate ofsuspension hard point is invariant 119889 as the diameter torsionbar along with 119871119890 as the torsion bar effective length and 120572 aspretwist angle are analyzed The swing arm length of netherarms is not analyzed as an effect variable in this paper becausechanging the swing arms of nether arms is equal to changingthe suspension hard point which involves the coordination ofhard points of suspension and steering
The influence of parameter on suspension stiffness isshown in Figure 6 and the parameter variation ranges areplusmn10 Taking full-load suspension stiffness as assessmentcriteria the variable value of stiffness is 15 with the torsion
Figure 6 Stiffness curve of the suspension with variable diameter
bar diameters of 25mm 26mm and 27mm respectively andwith the diameter variable value of 385 compared with themedian of 26mm
Figure 7 shows stiffness curve of the suspension withvariable length The effective lengths of torsion bar are9135mm 950mm and 9865mm respectively with thevariable value of 385 compared with median value Thestiffness variable value of full-loan suspension is 256
Figure 8 shows stiffness curve of the suspension withvariable angle The pretwist angle has no influence onsuspension stiffness in fact with the torsion bar pretwistangle of 841∘ 857∘ and 909∘ with the variable value of385 compared with median value It is probably because
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
L = 9135 mm
L = 9865 mmL = 950mm
Figure 7 Stiffness curve of the suspension with variable length
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
Alfa = 841∘
Alfa = 875∘
Alfa = 909∘
Figure 8 Stiffness curve of the suspension with variable angle
the change of pretwist angle actually changes the workingstate of suspension nether arms and adjusts the position ofinitial state of suspension space that is it changes theworkinginterval instead of suspension stiffness curve
It can be concluded from the above analysis that theparameters with greater influences on suspension stiffnessinclude 119889 as diameter of torsion bar and 119871119890 as effective lengthof torsion bar Considering limits of the general arrangementand the changes of nether arms involving redesign of hard
6 Shock and Vibration
h
m1
x1
x2
k2
m2
k1c1
Figure 9 14 physical model of the vehicle
points both 119889 as diameter of torsion bar and 119871119890 as effectivelength of torsion bar are selected as efficient variables foroptimizing in this paper
5 Establishing the OptimizingMathematical Model
The problems in the test run include the obvious parametervariation of front-wheel alignment and great variation ofvehicle height caused by frontier suspension softness andexcessive wheel bobbling separately The suspension stiffnessis optimized and the diameter and effective length of torsionbar are selected appropriately with the optimizing targetof vertical displacement variations of suspension from zeroloads to no load and from no load to full load combined withdynamic load of tires and riding comfort parameters in thispaper
First the authors established the constrained function ofsuspension performance parameters The evaluation indexformula is established and dynamic rate parameters of sus-pension are analyzed from the perspective of riding safetyand comfort The performance in the working conditionof no load (drivers included) half load and full load isanalyzed with the evaluation index of power spectral densityin frequency domain and with the introduction of newobjective function method
The 14 vehicle model is established as demonstrated inFigure 9 with ℎ as road input 1198961 1198962 as suspension stiffnessand tire stiffness separately 1198881 as suspension damping factor1198981 as bodymass1198982 as wheel mass 1199091 as body displacementand 1199092 as wheel displacement
minus 1198961119898111989611198981 minus 11988811198981
11988811198981
]]]]]
119863 = [[[[[
00
minus 11989621198982
]]]]]
(13)
The road input ℎ is a spectral density function
Φℎ (Ω) = Φℎ (Ω0) [ ΩΩ0 ]minus119908 (14)
where Ω0 is the standard travel circular frequency Φℎ(Ω0) isthe scale of road roughness and119908 is the scale ofwavinessTheroad condition is relatively good considering the driving con-dition as schoolyards and tourist attractions The pavementis comparatively flat and the jerk value is comparatively lowwith the reference input of 119861 scale pavement Next the speedof a campus car is comparatively low because of the speedlimit The vehicle speed V here is set as 10ms
The road spectrum travel power is transformed into timedomain power spectrum as
Φℎ (119908) = 1VΦℎ (Ω) (15)
The standard deviation of evaluation criteria is obtainedon the basis of amplitude-frequency characteristic of roadspectrum and vibration parameters
The scales of driving safety of cars are defined as the rootmean square of weighed acceleration of vertical accelerome-ter
119885 direction of seat 126 11119885 direction of foot 126 13119885 direction of hand 126 075119883 and 119884 direction of seat 123 147119883 direction of back seat 125 109119883 and 119884 direction of foot 128 128Pitching motion 123 116Roll motion 123 098
2
4
6
8
10
12
14
16
18
Wei
ghtin
g co
effici
ent
5 10 15 200Excitation frequency (Hz)
Figure 10 Evaluation value of the vibration
Since the frequency range of random vibration is wideand randomvibration bringsmore discomfort than harmonicvibration for this reason an evaluation function is intro-duced in Figure 10 and 119861foot = 120577119894119861119898 where 120577119894 is randomcorrection factor as shown in Table 2
The vibration analysis in the mathematical model of thispaper is mainly focused on 119911 orientation of feet Correspond-ing parameters are selected according to the above icons
Evaluation index of suspension dynamic travel is definedas quadratic mean of difference between wheel displacementand body displacement the difference of displacement vari-ation caused by road surface irregularity on the condition ofinvariant load
SWS = int501205870
ΦSWS (119908)
= int501205870
[1199091 (119908) minus 1199092 (119908)ℎ (119908) ]2Φℎ (119908) 119889119908
(17)
The two parameters listed below are taken as objectivefunction since the suspension bobbling travel is excessivelylong and the body height variation is excessively obvious inthe trial-production process
Table 3 The optimized result
Torsion bar diameter 265mmTorsion bar effective length 895mm
(1) The vertical displacement variation of nether balljoint point from zero load to full load is
Δ119878 = 119878zerozai minus 119878kongzai (18)
(2) The vertical displacement variation of nether balljoint point from empty load to full load is
Δ119878 = 119878manzai minus 119878kongzai (19)
Next the authors determined the constraint
(1) The longitudinal set of torsion bar is decided bythe spatial arrangement of general arrangement Thelengths of torsion bar (119871119890) in the long direction arelimited in the range of 800mm to 1000mm
(2) Considering the height limit of the distance to nethersurface of the frame the diameters of torsion bar arelimited to the range of 0 lt 119889 lt 40
(3) The diameters are selected within integers or nonin-tegers with decimal parts of 05
(4) The torsion bar is made of 45CrNiMoVa alloy steelwith good quality
(5) The allowable stress of the torsion bar [120591119904] is withinthe range of 1000 to 1250Nmm2
(6) The possibility of hitting a stock is comparativelysmall since the SWS variation of suspension dynamictravel variation is comparatively small
The optimized result is obtained with the MATLABnonlinear multitarget optimizing tool box which is shown inTable 3 The torsion bar diameter is set to be 265mm thetorsion bar effective length is set to be 895mm
6 Analysis of Optimized Result
In Table 4 the optimized result is also obtained with theMATLAB nonlinear multitarget optimizing tool box wenoticed the following
(1) Before optimizing the vertical displacement variationof swizzle nether ball joint points from suspensionzero load to suspension empty load is 775mm andthe vertical displacement variation from empty loadto full load is 88mm before optimizing After opti-mizing the vertical displacement variation of swizzlenether ball joint points from suspension zero loadto suspension empty load becomes 70mm and thevertical displacement variation from empty load tofull load becomes 765mm the value of vertical dis-placement variation is 967 and 131 respectivelyless than the original value
8 Shock and Vibration
Table 4 Property parameter before and after optimization
Property parameter before and after optimizationOptimization 119865119885 (N) SWS (m) 119870foot (dimensionless)
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
Figure 6 Stiffness curve of the suspension with variable diameter
bar diameters of 25mm 26mm and 27mm respectively andwith the diameter variable value of 385 compared with themedian of 26mm
Figure 7 shows stiffness curve of the suspension withvariable length The effective lengths of torsion bar are9135mm 950mm and 9865mm respectively with thevariable value of 385 compared with median value Thestiffness variable value of full-loan suspension is 256
Figure 8 shows stiffness curve of the suspension withvariable angle The pretwist angle has no influence onsuspension stiffness in fact with the torsion bar pretwistangle of 841∘ 857∘ and 909∘ with the variable value of385 compared with median value It is probably because
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
L = 9135 mm
L = 9865 mmL = 950mm
Figure 7 Stiffness curve of the suspension with variable length
10
15
20
25
Stiff
ness
(Nm
m)
0 50 100minus50minus100Displacement (mm)
Alfa = 841∘
Alfa = 875∘
Alfa = 909∘
Figure 8 Stiffness curve of the suspension with variable angle
the change of pretwist angle actually changes the workingstate of suspension nether arms and adjusts the position ofinitial state of suspension space that is it changes theworkinginterval instead of suspension stiffness curve
It can be concluded from the above analysis that theparameters with greater influences on suspension stiffnessinclude 119889 as diameter of torsion bar and 119871119890 as effective lengthof torsion bar Considering limits of the general arrangementand the changes of nether arms involving redesign of hard
6 Shock and Vibration
h
m1
x1
x2
k2
m2
k1c1
Figure 9 14 physical model of the vehicle
points both 119889 as diameter of torsion bar and 119871119890 as effectivelength of torsion bar are selected as efficient variables foroptimizing in this paper
5 Establishing the OptimizingMathematical Model
The problems in the test run include the obvious parametervariation of front-wheel alignment and great variation ofvehicle height caused by frontier suspension softness andexcessive wheel bobbling separately The suspension stiffnessis optimized and the diameter and effective length of torsionbar are selected appropriately with the optimizing targetof vertical displacement variations of suspension from zeroloads to no load and from no load to full load combined withdynamic load of tires and riding comfort parameters in thispaper
First the authors established the constrained function ofsuspension performance parameters The evaluation indexformula is established and dynamic rate parameters of sus-pension are analyzed from the perspective of riding safetyand comfort The performance in the working conditionof no load (drivers included) half load and full load isanalyzed with the evaluation index of power spectral densityin frequency domain and with the introduction of newobjective function method
The 14 vehicle model is established as demonstrated inFigure 9 with ℎ as road input 1198961 1198962 as suspension stiffnessand tire stiffness separately 1198881 as suspension damping factor1198981 as bodymass1198982 as wheel mass 1199091 as body displacementand 1199092 as wheel displacement
minus 1198961119898111989611198981 minus 11988811198981
11988811198981
]]]]]
119863 = [[[[[
00
minus 11989621198982
]]]]]
(13)
The road input ℎ is a spectral density function
Φℎ (Ω) = Φℎ (Ω0) [ ΩΩ0 ]minus119908 (14)
where Ω0 is the standard travel circular frequency Φℎ(Ω0) isthe scale of road roughness and119908 is the scale ofwavinessTheroad condition is relatively good considering the driving con-dition as schoolyards and tourist attractions The pavementis comparatively flat and the jerk value is comparatively lowwith the reference input of 119861 scale pavement Next the speedof a campus car is comparatively low because of the speedlimit The vehicle speed V here is set as 10ms
The road spectrum travel power is transformed into timedomain power spectrum as
Φℎ (119908) = 1VΦℎ (Ω) (15)
The standard deviation of evaluation criteria is obtainedon the basis of amplitude-frequency characteristic of roadspectrum and vibration parameters
The scales of driving safety of cars are defined as the rootmean square of weighed acceleration of vertical accelerome-ter
119885 direction of seat 126 11119885 direction of foot 126 13119885 direction of hand 126 075119883 and 119884 direction of seat 123 147119883 direction of back seat 125 109119883 and 119884 direction of foot 128 128Pitching motion 123 116Roll motion 123 098
2
4
6
8
10
12
14
16
18
Wei
ghtin
g co
effici
ent
5 10 15 200Excitation frequency (Hz)
Figure 10 Evaluation value of the vibration
Since the frequency range of random vibration is wideand randomvibration bringsmore discomfort than harmonicvibration for this reason an evaluation function is intro-duced in Figure 10 and 119861foot = 120577119894119861119898 where 120577119894 is randomcorrection factor as shown in Table 2
The vibration analysis in the mathematical model of thispaper is mainly focused on 119911 orientation of feet Correspond-ing parameters are selected according to the above icons
Evaluation index of suspension dynamic travel is definedas quadratic mean of difference between wheel displacementand body displacement the difference of displacement vari-ation caused by road surface irregularity on the condition ofinvariant load
SWS = int501205870
ΦSWS (119908)
= int501205870
[1199091 (119908) minus 1199092 (119908)ℎ (119908) ]2Φℎ (119908) 119889119908
(17)
The two parameters listed below are taken as objectivefunction since the suspension bobbling travel is excessivelylong and the body height variation is excessively obvious inthe trial-production process
Table 3 The optimized result
Torsion bar diameter 265mmTorsion bar effective length 895mm
(1) The vertical displacement variation of nether balljoint point from zero load to full load is
Δ119878 = 119878zerozai minus 119878kongzai (18)
(2) The vertical displacement variation of nether balljoint point from empty load to full load is
Δ119878 = 119878manzai minus 119878kongzai (19)
Next the authors determined the constraint
(1) The longitudinal set of torsion bar is decided bythe spatial arrangement of general arrangement Thelengths of torsion bar (119871119890) in the long direction arelimited in the range of 800mm to 1000mm
(2) Considering the height limit of the distance to nethersurface of the frame the diameters of torsion bar arelimited to the range of 0 lt 119889 lt 40
(3) The diameters are selected within integers or nonin-tegers with decimal parts of 05
(4) The torsion bar is made of 45CrNiMoVa alloy steelwith good quality
(5) The allowable stress of the torsion bar [120591119904] is withinthe range of 1000 to 1250Nmm2
(6) The possibility of hitting a stock is comparativelysmall since the SWS variation of suspension dynamictravel variation is comparatively small
The optimized result is obtained with the MATLABnonlinear multitarget optimizing tool box which is shown inTable 3 The torsion bar diameter is set to be 265mm thetorsion bar effective length is set to be 895mm
6 Analysis of Optimized Result
In Table 4 the optimized result is also obtained with theMATLAB nonlinear multitarget optimizing tool box wenoticed the following
(1) Before optimizing the vertical displacement variationof swizzle nether ball joint points from suspensionzero load to suspension empty load is 775mm andthe vertical displacement variation from empty loadto full load is 88mm before optimizing After opti-mizing the vertical displacement variation of swizzlenether ball joint points from suspension zero loadto suspension empty load becomes 70mm and thevertical displacement variation from empty load tofull load becomes 765mm the value of vertical dis-placement variation is 967 and 131 respectivelyless than the original value
8 Shock and Vibration
Table 4 Property parameter before and after optimization
Property parameter before and after optimizationOptimization 119865119885 (N) SWS (m) 119870foot (dimensionless)
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
points both 119889 as diameter of torsion bar and 119871119890 as effectivelength of torsion bar are selected as efficient variables foroptimizing in this paper
5 Establishing the OptimizingMathematical Model
The problems in the test run include the obvious parametervariation of front-wheel alignment and great variation ofvehicle height caused by frontier suspension softness andexcessive wheel bobbling separately The suspension stiffnessis optimized and the diameter and effective length of torsionbar are selected appropriately with the optimizing targetof vertical displacement variations of suspension from zeroloads to no load and from no load to full load combined withdynamic load of tires and riding comfort parameters in thispaper
First the authors established the constrained function ofsuspension performance parameters The evaluation indexformula is established and dynamic rate parameters of sus-pension are analyzed from the perspective of riding safetyand comfort The performance in the working conditionof no load (drivers included) half load and full load isanalyzed with the evaluation index of power spectral densityin frequency domain and with the introduction of newobjective function method
The 14 vehicle model is established as demonstrated inFigure 9 with ℎ as road input 1198961 1198962 as suspension stiffnessand tire stiffness separately 1198881 as suspension damping factor1198981 as bodymass1198982 as wheel mass 1199091 as body displacementand 1199092 as wheel displacement
minus 1198961119898111989611198981 minus 11988811198981
11988811198981
]]]]]
119863 = [[[[[
00
minus 11989621198982
]]]]]
(13)
The road input ℎ is a spectral density function
Φℎ (Ω) = Φℎ (Ω0) [ ΩΩ0 ]minus119908 (14)
where Ω0 is the standard travel circular frequency Φℎ(Ω0) isthe scale of road roughness and119908 is the scale ofwavinessTheroad condition is relatively good considering the driving con-dition as schoolyards and tourist attractions The pavementis comparatively flat and the jerk value is comparatively lowwith the reference input of 119861 scale pavement Next the speedof a campus car is comparatively low because of the speedlimit The vehicle speed V here is set as 10ms
The road spectrum travel power is transformed into timedomain power spectrum as
Φℎ (119908) = 1VΦℎ (Ω) (15)
The standard deviation of evaluation criteria is obtainedon the basis of amplitude-frequency characteristic of roadspectrum and vibration parameters
The scales of driving safety of cars are defined as the rootmean square of weighed acceleration of vertical accelerome-ter
119885 direction of seat 126 11119885 direction of foot 126 13119885 direction of hand 126 075119883 and 119884 direction of seat 123 147119883 direction of back seat 125 109119883 and 119884 direction of foot 128 128Pitching motion 123 116Roll motion 123 098
2
4
6
8
10
12
14
16
18
Wei
ghtin
g co
effici
ent
5 10 15 200Excitation frequency (Hz)
Figure 10 Evaluation value of the vibration
Since the frequency range of random vibration is wideand randomvibration bringsmore discomfort than harmonicvibration for this reason an evaluation function is intro-duced in Figure 10 and 119861foot = 120577119894119861119898 where 120577119894 is randomcorrection factor as shown in Table 2
The vibration analysis in the mathematical model of thispaper is mainly focused on 119911 orientation of feet Correspond-ing parameters are selected according to the above icons
Evaluation index of suspension dynamic travel is definedas quadratic mean of difference between wheel displacementand body displacement the difference of displacement vari-ation caused by road surface irregularity on the condition ofinvariant load
SWS = int501205870
ΦSWS (119908)
= int501205870
[1199091 (119908) minus 1199092 (119908)ℎ (119908) ]2Φℎ (119908) 119889119908
(17)
The two parameters listed below are taken as objectivefunction since the suspension bobbling travel is excessivelylong and the body height variation is excessively obvious inthe trial-production process
Table 3 The optimized result
Torsion bar diameter 265mmTorsion bar effective length 895mm
(1) The vertical displacement variation of nether balljoint point from zero load to full load is
Δ119878 = 119878zerozai minus 119878kongzai (18)
(2) The vertical displacement variation of nether balljoint point from empty load to full load is
Δ119878 = 119878manzai minus 119878kongzai (19)
Next the authors determined the constraint
(1) The longitudinal set of torsion bar is decided bythe spatial arrangement of general arrangement Thelengths of torsion bar (119871119890) in the long direction arelimited in the range of 800mm to 1000mm
(2) Considering the height limit of the distance to nethersurface of the frame the diameters of torsion bar arelimited to the range of 0 lt 119889 lt 40
(3) The diameters are selected within integers or nonin-tegers with decimal parts of 05
(4) The torsion bar is made of 45CrNiMoVa alloy steelwith good quality
(5) The allowable stress of the torsion bar [120591119904] is withinthe range of 1000 to 1250Nmm2
(6) The possibility of hitting a stock is comparativelysmall since the SWS variation of suspension dynamictravel variation is comparatively small
The optimized result is obtained with the MATLABnonlinear multitarget optimizing tool box which is shown inTable 3 The torsion bar diameter is set to be 265mm thetorsion bar effective length is set to be 895mm
6 Analysis of Optimized Result
In Table 4 the optimized result is also obtained with theMATLAB nonlinear multitarget optimizing tool box wenoticed the following
(1) Before optimizing the vertical displacement variationof swizzle nether ball joint points from suspensionzero load to suspension empty load is 775mm andthe vertical displacement variation from empty loadto full load is 88mm before optimizing After opti-mizing the vertical displacement variation of swizzlenether ball joint points from suspension zero loadto suspension empty load becomes 70mm and thevertical displacement variation from empty load tofull load becomes 765mm the value of vertical dis-placement variation is 967 and 131 respectivelyless than the original value
8 Shock and Vibration
Table 4 Property parameter before and after optimization
Property parameter before and after optimizationOptimization 119865119885 (N) SWS (m) 119870foot (dimensionless)
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
119885 direction of seat 126 11119885 direction of foot 126 13119885 direction of hand 126 075119883 and 119884 direction of seat 123 147119883 direction of back seat 125 109119883 and 119884 direction of foot 128 128Pitching motion 123 116Roll motion 123 098
2
4
6
8
10
12
14
16
18
Wei
ghtin
g co
effici
ent
5 10 15 200Excitation frequency (Hz)
Figure 10 Evaluation value of the vibration
Since the frequency range of random vibration is wideand randomvibration bringsmore discomfort than harmonicvibration for this reason an evaluation function is intro-duced in Figure 10 and 119861foot = 120577119894119861119898 where 120577119894 is randomcorrection factor as shown in Table 2
The vibration analysis in the mathematical model of thispaper is mainly focused on 119911 orientation of feet Correspond-ing parameters are selected according to the above icons
Evaluation index of suspension dynamic travel is definedas quadratic mean of difference between wheel displacementand body displacement the difference of displacement vari-ation caused by road surface irregularity on the condition ofinvariant load
SWS = int501205870
ΦSWS (119908)
= int501205870
[1199091 (119908) minus 1199092 (119908)ℎ (119908) ]2Φℎ (119908) 119889119908
(17)
The two parameters listed below are taken as objectivefunction since the suspension bobbling travel is excessivelylong and the body height variation is excessively obvious inthe trial-production process
Table 3 The optimized result
Torsion bar diameter 265mmTorsion bar effective length 895mm
(1) The vertical displacement variation of nether balljoint point from zero load to full load is
Δ119878 = 119878zerozai minus 119878kongzai (18)
(2) The vertical displacement variation of nether balljoint point from empty load to full load is
Δ119878 = 119878manzai minus 119878kongzai (19)
Next the authors determined the constraint
(1) The longitudinal set of torsion bar is decided bythe spatial arrangement of general arrangement Thelengths of torsion bar (119871119890) in the long direction arelimited in the range of 800mm to 1000mm
(2) Considering the height limit of the distance to nethersurface of the frame the diameters of torsion bar arelimited to the range of 0 lt 119889 lt 40
(3) The diameters are selected within integers or nonin-tegers with decimal parts of 05
(4) The torsion bar is made of 45CrNiMoVa alloy steelwith good quality
(5) The allowable stress of the torsion bar [120591119904] is withinthe range of 1000 to 1250Nmm2
(6) The possibility of hitting a stock is comparativelysmall since the SWS variation of suspension dynamictravel variation is comparatively small
The optimized result is obtained with the MATLABnonlinear multitarget optimizing tool box which is shown inTable 3 The torsion bar diameter is set to be 265mm thetorsion bar effective length is set to be 895mm
6 Analysis of Optimized Result
In Table 4 the optimized result is also obtained with theMATLAB nonlinear multitarget optimizing tool box wenoticed the following
(1) Before optimizing the vertical displacement variationof swizzle nether ball joint points from suspensionzero load to suspension empty load is 775mm andthe vertical displacement variation from empty loadto full load is 88mm before optimizing After opti-mizing the vertical displacement variation of swizzlenether ball joint points from suspension zero loadto suspension empty load becomes 70mm and thevertical displacement variation from empty load tofull load becomes 765mm the value of vertical dis-placement variation is 967 and 131 respectivelyless than the original value
8 Shock and Vibration
Table 4 Property parameter before and after optimization
Property parameter before and after optimizationOptimization 119865119885 (N) SWS (m) 119870foot (dimensionless)
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
Half load Before 9416 00023 26229After 9536 00023 27706
Full load Before 9795 00026 24907After 1005 00026 26803
(2) The variation of tire dynamic load of three assessmentindices is 135 and the comfort parameter variationis 583 The suspension dynamic travel is basicallyinvariant based on the good condition of campusroads as driving road
(3) The suspension softness is effectively reduced onthe condition that there is almost no effect on thesuspension performance
7 Conclusion
The stiffness of double-arm torsion bar suspension in theelectric sight-seeing car is calculatedwith vector algebra com-bined with virtual work principle and finite tensor methodand is tested with ADAMS software to prove the feasibilityof mathematical calculating method in this paper The effectof torsion bar basic parameters on suspension stiffness isanalyzed through calculation and the parameters needingoptimizing are obtained through sensitivity analysis
The tire dynamic load is proposed according to the actualcondition of driving road The new evaluation function andindex are establishedwith the constraint condition of suspen-sion dynamic travel and comfort parameters and the objectivefunction of suspension jerk value and the optimizing modelof torsion bar basic size of double-arm torsion bar suspensionis established
The torsion bar basic size which is optimized withMATLAB multitarget optimizing tool box decreases verticaldisplacement travel by 967 from suspension zero loadto empty load and vertical displacement travel by 131from suspension empty load to full load The optimizedsize increases suspension stiffness effectively and reduces thephenomenon of suspension blocks being hit restricts thevariation of front-wheel alignment angle and meanwhileeffectively improves the phenomenon of frontier suspensionsoftness on the condition of no effects on suspension perfor-mance
Conflicts of Interest
This manuscript did not lead to any conflicts of interestregarding the publication
Acknowledgments
This work was supported by the Fujian Nature Foundationno 2016J01039 Xiamen City Project no 3502Z20173037 and
National Natural Science Foundation of China under Grantnos 51475399 and 51405410
References
[1] R L Wang J Z Zhao G Y Wang X K Chen and L LildquoModeling and kinematics simulation analyze of conventionalsuspensionwith double trailing arms for light off-road vehiclesrdquoApplied Mechanics and Materials vol 312 pp 673ndash678 2013
[2] G Tian Y Zhang J-H Liu and X-J Shao ldquoDouble wishboneindependent suspension parameter optimization and simula-tionrdquo Applied Mechanics and Materials vol 574 pp 109ndash1132014
[3] Z Z Guo and Y F Sun ldquoOptimization and analysis of doublewishbone independent front suspension based on virtual pro-totyperdquoAppliedMechanics andMaterials vol 490-491 pp 832ndash835 2014
[4] V Cherian I Haque andN Jalili ldquoDevelopment of a non-linearmodel of a double wishbone suspension for the characterizationof force transmission to the steering column and chassisrdquoin Proceedings of the 2004 ASME International MechanicalEngineering Congress and Exposition IMECE pp 775ndash780Anaheim Ca USA November 2004
[5] M J Burgess N P Fleming M Wootton and S J Williams ldquoAtool for rapid vehicle suspension designrdquo SAE Technical Papers2004
[6] N Mohamad and K Farhang ldquoA vibration model of asuspensionmdashTire systemrdquo in Proceedings of the ASME DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference vol 2B pp 1465ndash1476 Salt LakeCity Utah USA 2004
[7] K P Balike S Rakheja and I Stiharu ldquoDevelopment of kineto-dynamic quarter-car model for synthesis of a double wishbonesuspensionrdquo Vehicle System Dynamics vol 49 no 1-2 pp 107ndash128 2011
[8] L Zhao S Zheng J Feng and Q Hong ldquoDynamic structureoptimization design of lower control arm based on ESLrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 22 pp 4871ndash4878 2012
[9] X-L Wang L Dai and W-B Shangguan ldquoCalculation ofjoint forces of a multi-link suspension for strength and fatigueanalysis of bushings and control armsrdquo International Journal ofVehicle Design vol 66 no 3 pp 217ndash234 2014
[10] L TangW-B Shangguan and L Dai ldquoA calculation method ofjoint forces for a suspension considering nonlinear elasticity ofbushingsrdquoProceedings of the Institution ofMechanical EngineersPart K Journal of Multi-body Dynamics vol 226 no 4 pp 281ndash297 2012
Shock and Vibration 9
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
[11] Z Liu and L Zhao ldquoMotion compatibleness analysis andoptimization of double front axle steering mechanism andsuspension systemrdquo China Mechanical Engineering vol 24 no16 pp 2164ndash2167 2013
[12] L Sun Z Deng andQ Zhang ldquoDesign and strength analysis ofFSAE suspensionrdquoOpenMechanical Engineering Journal vol 8article A414 pp 414ndash418 2014
[13] Y Kang S Rakheja and W Zhang ldquoRelative performanceanalyses of independent front axle suspensions for a heavy-duty mining truckrdquo SAE International Journal of CommercialVehicles vol 7 no 2 2014
[14] A Das N Unnikrishnan B Shankar and J D FreemanldquoDesign fabrication and testing of the suspension subsystem ofa single seater off-road buggyrdquo International Journal of AppliedEngineering Research vol 9 no 5 pp 525ndash536 2014
[15] J Wu Z Luo Y Zhang and N Zhang ldquoAn interval uncertainoptimization method for vehicle suspensions using Chebyshevmetamodelsrdquo Applied Mathematical Modelling vol 38 no 15-16 pp 3706ndash3723 2014
[16] Y Kang W Zhang and S Rakheja ldquoRelative kinematic andhandling performance analyses of independent axle suspen-sions for a heavy-duty mining truckrdquo International Journal ofHeavy Vehicle Systems vol 22 no 2 pp 114ndash136 2015
![Page 1: The Parameters Optimizing Design of Double Suspension Arm ...downloads.hindawi.com/journals/sv/2017/8153756.pdf · between suspension and steering system. Sun et al. [12] designed](https://reader034.documents.pub/reader034/viewer/2022052104/603f2239b108b802da6e7210/html5/thumbnails/1.jpg)
![Page 2: The Parameters Optimizing Design of Double Suspension Arm ...downloads.hindawi.com/journals/sv/2017/8153756.pdf · between suspension and steering system. Sun et al. [12] designed](https://reader034.documents.pub/reader034/viewer/2022052104/603f2239b108b802da6e7210/html5/thumbnails/2.jpg)
![Page 3: The Parameters Optimizing Design of Double Suspension Arm ...downloads.hindawi.com/journals/sv/2017/8153756.pdf · between suspension and steering system. Sun et al. [12] designed](https://reader034.documents.pub/reader034/viewer/2022052104/603f2239b108b802da6e7210/html5/thumbnails/3.jpg)
![Page 4: The Parameters Optimizing Design of Double Suspension Arm ...downloads.hindawi.com/journals/sv/2017/8153756.pdf · between suspension and steering system. Sun et al. [12] designed](https://reader034.documents.pub/reader034/viewer/2022052104/603f2239b108b802da6e7210/html5/thumbnails/4.jpg)
![Page 5: The Parameters Optimizing Design of Double Suspension Arm ...downloads.hindawi.com/journals/sv/2017/8153756.pdf · between suspension and steering system. Sun et al. [12] designed](https://reader034.documents.pub/reader034/viewer/2022052104/603f2239b108b802da6e7210/html5/thumbnails/5.jpg)
![Page 6: The Parameters Optimizing Design of Double Suspension Arm ...downloads.hindawi.com/journals/sv/2017/8153756.pdf · between suspension and steering system. Sun et al. [12] designed](https://reader034.documents.pub/reader034/viewer/2022052104/603f2239b108b802da6e7210/html5/thumbnails/6.jpg)
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