ThreeLink LeafSpring Model for Road Loads(1)

1、Downloaded from SAE International by Brought To You Michigan State Univ, Monday, February 24, 2014 07:27:38 AMLeading Our World In Motion2005-01-0625SAE TECHNICAL PAPER SERIESThree-Link Leaf-Spring Model for Road LoadsP. Jayakumar, J Alanoly, and R JohnsonFord Motor CompanyReprinted From: Loads Simu

2、lation and Analysis in Automotive Engineering (SP-1958)ISBN D-7tA0-lb34-79H780768 0163452005 SAE World Congress Detroit, Michigan April 11-14, 2005400 Commonwealth Drive, Warrendale, PA 15096-0001 U.S.A. Tel: (724) 776-4841 Fax: (724) 776-5760 Web: SAW InternationalThe Engineering Meeting

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8、pring Model for Road LoadsP Jayakumar, J Alanoly and R JohnsonFord Motor CompanyCopyright 2005 SAE InternationalABSTRACTSimulatio n of road loads in truck suspe nsions gen erally requires leaf-spring models This paper presents a simple and accurate leaf-spring model that can be effectively used in a

9、nalytical road load simulations using ADAMS software The model topology is based on the familiar SAE Hthree-link” model. The model parameters are identified from static force-deflection test data. Alternatively, the parameters can be identified from a target design specification or from analytical H

10、testsf, on a detailed finite element model. The new leaf spring model has been validated for static and dynamic performanee using laboratory test data.INTRODUCTIONAnalytical simulation of road loads is an enabling technology for CAE based durability analysis Significant progress has been made in thi

11、s area over the last several years Ten years ago, all road loads were directly measured Today however, semi-analytical loads make up the majority of loads delivered to vehicle programs. Semi-analytical loads are calculated using analytical models that use measurements on prototypes as inputs (Jayaku

12、mar and Alanoly, 1996a, 1996b). Semi-analytical methods have been used on most suspe nsion configurations except Hotchkiss. The reason for not applying these methods to Hotchkiss suspensions is the lack of validated leaf-spring models under large deflection dynamic conditions. This paper presents th

13、e modeling and validation of leaf-springs for road load simulatio n.Vehicle system models built using ADAMS (Mechanical Dynamics, Inc., 1994) currently represent a leaf spring as a ”three-link model1 (Antoun, et al., 1986), or a ”beam model”(Rohweder, 1993). The state-of-the-art is considered to be

14、the beam model, where, each spring leaf is discretized into a series of rigid parts, connected using massless linear beams of uniform cross-sections. Adjacent leaves of the spring are kept from penetrating each other using impact statements This results in a very large, and extremely non linear (vis

15、 a vis intermittent con tacts) model with several hundred degrees-of- freedom .It was found that it is difficult to perform road load simulations using this type of a leaf spring model (Jayakumar and Alanoly, 1997).The SAE Spring Design Manual (Society of Automotive Engineers, Inc., 1982) presents a

16、 three-link discretization of a leaf spring for analyzing its kinematics The manual does not specify how to compute the force-deflection behavior. Antoun, et al. (1986) employed the SAE three- link model with linear torsional springs to model leafsprings for vehicle handling simulations It is expect

17、ed that the linear approximation will not hold in the case of multi-stage springs under large deflections An extension to the three-link model (Sitchin, 1991) uses a nonlinear vertical spring in parallel with the three-link model to account for the non linear force-deflection behavior in the vertica

18、l direction. Although this model is useful in certain applications，the spurious load path introduced through the vertical spring and the likely misrepresentation of Iongitudinal behavior make this model unsuitable for application in road load simulation.Other relevant studies reported in literature

19、include the work by Guo (1991), which presents analytical formulations for leaf spring force-deflection behavior. The results show good correlation with test data. The formulation does not address multi-stage springs, where not all the leaves are always in contact. In addition, ADAMS implementation

20、of this and other analytical formulations is not straightforward Fancher, et al. (1980) presented experimental results for heavy truck leaf springs The focus of the study was on the energy dissipation mechanism in the leaf spring.The leaf-spring model presented here is a three-link model. The geomet

21、ry of this model consists of three rigid links constructed using SAE guidelines (Society of Automotive Engineers, 1982). The leaf-spring compliance is incorporated in the model through two nonlinear torsional springs at the center link joints. A systematic approach is used for computing the paramete

22、rs of the nonlinear springs. The model is validated for static and dynamic behavior using laboratory test data.METHODThe proposed model consists of three rigid links The center link is attached to the end links through torsional springs The geometry of the model is constructed using the physical spr

23、ing dimensions and the SAE guidelines on leaf-spring design. The parameters of the torsional springs are identified from a given design intent vertical force-deflection curve, or from measured force-deflection data. Alternatively, the force-deflection curve may be derived from a detailed nonlinear f

24、inite element model of the spring based on design geometry (Asiani and Alanoly, 1997). The eleganee of the proposed method is that a simple model can be easily constructed to reproduce both the kinematic and compliance properties of the actual leaf-spring.THREE-LINK MODEL GEOMETRYThe basis of the SA

25、E three-link geometry is the assumption that a leaf-spring conforms to the shape of a circular arc under vertical loading (Figure 1) Based on the SAE Spring Design Manual (SAE, 1982), an equivale nt five bar mecha nism (4 movi ng bars, i ncludi ng the shackle) can be con structed that closely approx

26、imates the kinematics of the leaf-spring (Figure 2). To construct the mechanism, only the lengths of the four links are needed The shackle is represented as is, with its physical length. The lengths of the other three- links are determined as described below.DIMENSIONS OF THE THREE丄INK MECHANISMThe

27、dime nsions required to build the three-link mechanism are obtained from the geometry of the leaf- spring.Figure 1: Layout of a Leaf-SpringFigure 1 shows the key geometric information necessary to build the three-link model, whereL = Total spring length, measured along flat main leafm = Front inacti

28、ve lengthn = Rear inactive lengtha = Fixed can tilever len gth, called front length (in cludi ng in active length, m)b = Shackled can tilever len gth, called rear length(in cludi ng in active len gth, n)Based on the SAE guidelines (SAE, 1982), the dime nsions of the equivale nt SAE lin kage mecha ni

29、sm are (Figure 2),Ra = 0.75(a m)% = 0.75(b n)(2)Rc = L 傀 + RJd=(aRa)Figure 2: Layout of the MThree丄ink” ModelTHREE-LINK MODEL COMPLIANCETo simulate the complianee characteristics of the physical leaf-spring, two torsional springs are attached at each end of the center link revolute joints. The param

30、eters of these torsional springs are determined such that the equivale nt three-li nk model matches the vertical static force-deflection behavior of the physical leaf-spring The vertical force-deflection characteristic of a physical spring is specified during the suspension design process, and hence

31、, is available a priori.The three-link model is a five-bar mechanism which has two degrees-of-freedom. Therefore, the position of the leaf-spring can be uniquely determined by specify!ng any two independent coordinates, e.g., the x and y locations of the axle center.The torsional springs at each end

32、 of the center link are represented by the following equations:T4=a1a2a3e4 ef 0T(5)丁6 b? b3 o6 Oq 0q (6)whereT = Torque at the link joint0= Rotation at the link jointa., b. = Torsional spring parameters V /= 1,2, 34, 6 = Node numbers for the center link jointst = Vector tran sposeEquations 5 and 6,

33、where the torques in the link joints are assumed to be third order polynomials of the correspond!ng joint rotations，worked well for the leafspring used in the case study The authors recognize that typical design curves of multi-stage leaf-springs are specified as piecewise linear, rather than smooth

34、. However, polynomials are chosen because measured vertical force-deflection behavior of leaf-springs exhibits smooth nonlinear characteristicsThe polynomial coefficients, a. and V / = 1, 2, 3 are identified by Hcurve-fittingn the force-deflection behavior of the physical leaf-spring. Mathematically

35、, this is posed as an optimization problem where the error between the model and the target curve is to be minimized. The error function in the optimization is defined as:Figure 4 shows the leaf-spri ng test set up. The han ger and shackle brackets were attached to a bedplate The relative locations

36、and orientations of the brackets matched their in-vehicle positions，which were obtained using a coordinate measurement machine (CMM). The leaf-spring and the brackets were placed upside down compared to the vehicle to facilitate fixturing and actuator placement Three servo-hydraulic actuators were u

37、sed to apply forces at the axle, as seen in Figure 4 The Iongitudinal and lateral actuators were installed in such a way that they would be horizontal when a vertical preload, corresp on ding to the comer weight, was applied using the vertical actuator. This meant that in the un loaded condition, th

38、e actuators are in dined, as seen in Figure 4 The actuators also had to be offset at the spring attachment in order to facilitate fixturing. The actuator attachment positions were recorded for proper representation in ADAMS modeling. The masses of all significant fixturing downstream of load cells w

39、ere also measured.(8)error = erxex wherexs = Target values of axle vertical displacements for a series of quasi-static vertical axle force in puts. Target may be the design specification or actual test data.xm = Axle vertical displacements of the three-link ADAMS model for the same axle forcesLEAF-S

40、PRING TESTINGA truck leaf-spring was selected for the modeling, laboratory testing, and correlation study for the current project. Figure 3 shows the rear suspension of this vehicle, including the leaf-springs The leaf-spring has two leaves in the first stage and a single leaf in the sec ond stage.

41、The spri ng has a Berli n eye con struct! on with the shackle under compressio n. The han ger and shackle brackets were instrumented for loads measurements Figure 3: Truck Rear Leaf-Spring SuspensionLoad cells mounted along the axes of the actuators measured the loads while the displacement transduc

42、ers internal to the actuators measured the corresponding actuator displacements. The reaction forces on the han ger and shackle brackets were measured using load cells instrume nted at the brackets. Tri-axial accelerometers were placed at the spring seat in order to measure accelerations during dyna

43、mic tests. The longitudinal actuator was offset from the spring seat in order to provide a wind-up torque in addition to applying a longitudinal load The offset value was selected to produce the maximum wind-up torque recorded on a similar vehicle over the durability surfaceFigure 4: Leaf-Spring Tes

44、t Set UpTable 1: Summary of Leaf-Spring Test SpecificationsTestDirectionFrcqaeBcy (llz)AmplitudeQuasl-stattcVertical0 1500 0 to100 lb stepsLongitudnai*0一 1500-0JSOOtO lb100 lb slopsLateral*0 IOOO IOOO Olb100 lb stepsSinusddalVofttcar0.5.14 in.2 4. 18. 204in.Longitudinal*0.5.11 5 in2. 4. 6.18. 201 5

45、InRandomVerticor0.5 50= 1 in. Spectrum shape = 13 sigma = 2in.Spoctrum ihapo 1 i freq21Longitudnal*0.5 503sigmn = 1.5ln Spoctrum shapo - 1 ii In iklditkMi. there y a vertical pmaxl & / lb to timulile the ccmcr wn曲I ol the vehicleThe leaf-spring was tested in the laboratory under static and dynamic l

46、oad conditions as shown in Table 1 The quasi-static tests were performed in all three directions. Next, the spring was tested dynamically with sinusoidal and random excitations in the vertical and Iongitudinal directions. The sin usoidal in put tests were conducted at several frequencies and amplitu

47、des It should be noted that the ranges of amplitudes and frequencies are deliberately chosen to be representative of rough road durability events. Earlier, Fancher, et al. (1980) had tested leaf-springs with smaller dynamic amplitudes ( 0.4 in) in a smaller frequency range (O1S00 100ft500-：(X)O3500S

48、ttkti* Toar Loftgitudiruil Loading Vdrtieftl CtMner WeightLoncKudlnl Actuator Force (N)Figure 10: Three-Link model predictions in the vertical direction at the hanger bracketin eluded in the optimizati on process though this wouldnl be a desirable process in a product!on environment.THREE-LINK MODEL

49、 SIMULATION OF STATIC LONGITUDINAL TESTThe parameters of a three-link model were identified from vertical static test, as described in the previous section. This model is now used to simulate the static longitudinal test. In the physical test, an unloaded leaf- spring is first loaded in the vertical

50、 directi on to static corner weight of the vehicle The Iongitudinal force is then applied while keeping the vertical force constant An identical loading sequenee is simulated using the model in ADAMSFigure 8: Three-Link model predictions in the longitudinal directi on at the axleStatic Test LonKudln

51、al LoadKg VArtica Comer Weight800-1000-12001400Tbi -Model、-eooc -4000-200002000 4OCO 6000Longitudlra) Actuator Force N)btatic I owt Longitudiniil Loadmo * Virtki Comw Weight200 -r111Figure 9: Three-Link model predict!ons in the Iongitudinal directi on at the shackle bracket20001000TMt Modal-6000OOO 4000 AOOO R06OLongitudinal Adjator Ferea (N)Static Teat Longtnidlnal Loading verttea Comer weight30001111*1s35芟肓A PJP28CMMS8000Figures 8 through 11