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R&D Review of Toyota CRDL Vol. 37 No. 1

First Order Analysis for Automotive Suspension Design

Hideki Sugiura, Yoshiteru Mizutani, Hidekazu Nishigaki

ResearchReport

Abstract

The elastic properties of a torsion beam suspension determine the kinematic performance of thesuspension. In this paper, we introduce a design program for the torsion beam suspension basedon the FOA concept, and confirm its usefulness using simple examples and experimental evalua-tions.

Keywords Suspension system, Torsion beam, Design, FOA

Special Issue First Order Analysis

1. Introduction

Torsion beam suspension, which is often used asthe rear suspension in FF-type vehicles, offers theadvantages of having a small parts count and asimple configuration (Fig. 1). This suspensionconsists of two trailing arms that are mounted to thebody with bushings and a single torsion beam thatconnects these arms together. The torsion beammust be stiff enough to support lateral forces duringcornering and, at the same time, be flexible enoughto allow the right- and left-hand wheels to displacedifferently when driving over a bump. When a

torsion beam suspension is being designed,therefore, we must consider the relative balancebetween the stiffness and the flexibility of theconstituent members, in spite of the simpleconfiguration. In the suspension planning stage, it isnecessary to apply performance predictioncalculation for alignment changes and compliancesteer, with this being even more important in thecase of a torsion beam suspension, which isinfluenced by the stiffness of each of its constituentmembers.

To calculate the mechanical properties of a torsionbeam suspension, finite element analysis andmechanical analysis1) have been used to date. Formany design engineers, however, the planning-stageconstruction of a finite element model or acomplicated mechanical analysis model thatincorporates the elastic properties of its constituentmembers, and then performing calculations usingthat model, requires a tremendous amount of timeand technical skill and is often difficult toaccomplish.

Thus, this research proposes First Order Analysis2)

as a CAE tool for design engineers that can be usedwith no special knowledge or skills in modeling oranalysis. We also describe a design tool that wasdeveloped to calculate the properties of a torsionFig. 1 Torsion beam suspension.

R&D Review of Toyota CRDL Vol. 37 No. 1

beam suspension, and confirm the effectiveness ofthe tool using examples and experimentalverifications.

2. FOA for torsion beam suspension

2. 1 Hierarchical data structure of the objectof design

In conventional types of CAE, the configuration ofa model is such that the positional information of itsmembers and the information for the membersthemselves (including cross-sectional and materialproperties) are combined. In FOA, the object of thedesign is configured with a hierarchical datastructure like that shown in Fig. 2, thereby achievingboth efficient and simple operability. Morespecifically, the uppermost layer of the hierarchyhandles only information related to the entireconfiguration, including the layout. If the designengineer wishes to study one of the constituentmembers in more detail, he or she can move to alower layer to study that specific member only.Finally, the design engineer can manipulate theobject of design specialized in a specific member toincorporate that object into the top layer, i.e., theinformation for the entire suspension. In this way,the design engineer can analyze the effect of eachspecific member on the entire suspension.

Such a hierarchical data structure can easily be

achieved by using spreadsheets that are created on apersonal computer using the Microsoft Excelspreadsheet software with which the design engineerwill already be familiar.

2. 2 Configuration of the design toolAn example FOA configuration is shown in Fig. 3.

Figure 3 (a) shows the sheet for the top layer.Simply by moving the appropriate scroll bars on thesheet, the designer can enter a design plan for theentire configuration, such as the layout of eachmember. If, for example, the designer needs toconsider the torsion beam itself, he or she merelyhas to click the displayed torsion beam to replace thedisplayed sheet with that shown in Fig. 3 (b). Usingthis sheet to enter information specific to the torsionbeam, including the cross-sectional shapes, materialproperties, amounts of curvature, and whether to addreinforcements, it is possible to study the entireshape of the torsion beam and to incorporate thatinformation into the design plan for the entiresuspension while considering the dynamic propertiesspecific to open cross-sectional shapes, usingdegree-of-freedom reduction calculation, describedlater. If a cross-sectional shape is to be changedhere, the cross-sectional shape on the sheet may beclicked to display the edit sheet specifically for opencross-sections, shown in Fig. 3 (c). This sheet notonly allows the designer to perform cross-sectionalshape editing with the mouse, in much the same wayas with general CAD programs, but also gives himor her access to the cross-section propertycalculation functions, described later, such as secondmoment of inertia, polar moment of inertia, centroid,and shear center. This makes it possible to study across-sectional shape while referencing the propertyvalues displayed after each design change. Foranother component, the trailing arm, the sameprocedure can be used to study its design. For thebushings and the coil springs, their positions can bedetermined in the top layer, after which only thebushing stiffness and the spring constant need to bedetermined.

2. 3 Characteristic analysis of objects of designAfter a design has been set using the procedure

described above, returning to the top layer of thisdesign tool and then clicking the calculate buttoncauses the suspension properties such as compliance

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Fig. 2 Hierarchal data structure of torsion beamsuspension.

and alignment changes to be calculated anddisplayed on the sheet shown in Fig. 3 (d). Here,after a suspension is represented by multiple beamelements and stiffness matrixes and appropriateboundary conditions are supplied, the complianceproperties are calculated using linear finite elementanalysis after which alignment changes arecalculated using non-linear finite element analysis.In both calculations, a Toyota-CRDL developed

solver that runs on a personal computer is used. Theamount of time needed for calculation is at most tensof seconds.

The above has explained the procedure for usingthis design tool based on the concept of FOA. Fortorsion beams with open cross-sectional shapes,consideration has been given to assisting designengineers by providing dedicated calculationfunctions, as mentioned earlier. The calculation

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R&D Review of Toyota CRDL Vol. 37 No. 1

Fig. 3 Design program for torsion beam suspension.

method is explained briefly below.2. 4 Cross-sectional property calculationIn general, torsion beams have an open cross-

sectional shape to reduce their torsional stiffness.Among the properties of an open cross-section, thesecond moment of inertia and the polar moment ofinertia, which indicate the stiffness balance of thetorsion beam, are important indexes. Also importantis the shear center that has a great effect onalignment changes. The following describes thecalculations that can be performed using thisprogram.

As shown in Fig. 4, a cross-section of a torsionbeam is represented by a group of nmax line elementson the y-z coordinate system. The thickness of then-th line element is assumed to be tn, each lineelement is subdivided into kmax elements, thecentroid of each of the resulting small-line elementsis assumed to be (yck, zck), and the length is assumedto be ∆sk. According to Oden3), the second momentof inertia in this case is given by

· · · · · · · · · · · · · · · · (1)

· · · · · · · · · · · · · · · · (2)

· · · · · · · · · · · · · · · · (3)

where R denotes the radius of curvature of thetorsion beam in the longitudinal direction. On the

Iyz =y ckzck

1 – y ck / R tn∆sk∑

k = 1

kmax

∑n = 1

nmax

Iz =y ck

2

1 – y ck / R tn∆sk∑

k = 1

kmax

∑n = 1

nmax

Iy =zck

2

1 – y ck / R tn∆sk∑

k = 1

kmax

∑n = 1

nmax

other hand, the polar moment of inertia is given by

· · · · · · · · · · · · · · · · · · · · · (4)

for an open section, and by

· · · · · · · · · (5)

for a closed sectionwhere rk denotes the direction vector of the small-line element (sk

n, sk + 1n), ∆sk the vector connecting

the ends of the small-line element (skn, sk + 1

n), and ithe unit vector in the beam longitudinal direction.

Assuming that the centroid is at the origin, theposition of the shear center is where · · · · · · · · · · · (6)

where

· · · · · · · · · · · · · · · · · · (7)

· · · · · · · · · · · · · · · · · · (8)Qy and Qz are the static moments of inertia.

2. 5 Finite element models and degree-of-reduction calculation

To incorporate a torsion beam design plan into adesign plan for an entire suspension, we use degree-of-freedom reduction calculation. First, we create athree-dimensional representation of the torsion beamby longitudinally enlarging the cross-sectional shapedrawn using the editing program mentioned earlier.We displace this shape with a finite element modelwithin the program, and then automatically convertthe stiffness matrix into a stiffness matrix with 12degrees of freedom at both ends of the torsion beam,using degree-of-freedom reduction calculation. Theshape of the finite element model thus created can beconfirmed on a three-dimensional display like thatshown in Fig. 5. In this model, the shell elementsconstituting the torsion beam are connected to twodegenerate points, using sufficiently stiff beamelements, to enable the beam to be incorporatedstiffly with the trailing arms and also enable reliable

Iωz = Qz rk × ∆sk⋅ i∑k = 1

kmax

∑n = 1

nmax

= zck '

1 – yck ' / R∑

k' = 1

k

tn∆sk' rk' × ∆sk'⋅ i∑n' = 1

n

∑k = 1

kmax

∑n = 1

nmax

Iωy = Qy rk × ∆sk⋅ i∑k = 1

kmax

∑n = 1

nmax

= zck '

1 – yck ' / R∑

k' = 1

k

tn∆sk' rk' × ∆sk'⋅ i∑n' = 1

n

∑k = 1

kmax

∑n = 1

nmax

ey = –2IzIωy – IyzIωz

IyIz – Iyz2

, ez = 2IyIωz – IyzIωy

IyIz – Iyz2

J = rk∑k = 1

kmax

× ∆sk ⋅ i∑n = 1

nmax 2

∆sk

tn ∑

k = 1

kmax

∑n = 1

nmax

J = 13

∆sktn 3∑

k = 1

kmax

∑n = 1

nmax

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R&D Review of Toyota CRDL Vol. 37 No. 1

Fig. 4 Cross-sectional model using line segments.

load transmission.We use a finite element model for the torsion beam

in order to incorporate the degrees of designfreedom specific to this suspension type, such as (1)the cross-sectional property that states that thecentroid is displaced from the shear center, (2) theamount of curvature in the torsion beam, and (3) theaddition of reinforcements. In addition, by reducingthe degrees of freedom of the finite element modeland incorporating them, consideration is given toreducing the calculation time and improving thepracticability.

To reduce a finite element model into a stiffnessmatrix with 12 degrees of freedom, we used theapproach proposed by Guyan.4) Let the stiffnessmatrix for a finite element model be K, then, usingthe symbol m to indicate a node that will remain as areduction point, and the symbol s, which indicatesthe node to be removed, K can be represented by thefollowing submatrix:

· · · · · · · · · · · · · · · · · · · · · · · (9)

Using the following transformation matrix,

· · · · · · · · · · · · · · · · · · · · · · · · · · (10)

the stiffness matrix in which the degrees of freedomrelated to the reduction points will be expressed by

K R = T TKT · · · · · · · · · · · · · · · · · · · · · · · · · · (11)Finally, this matrix is automatically incorporatedbetween the trailing arms.

3. Verification

3. 1 OperabilityTable 1 shows the results of comparing this design

tool with conventional CAE from the viewpoint ofoperability. When using this design tool, no morethan two to three hours is needed for the calculation.For a parameter study associated with a simpledesign change, an answer can be obtained in just afew minutes. Design engineers can, therefore, usethe tool for their daily design study.

The fact that this tool can be run on a personalcomputer using familiar spreadsheet software meansthat there is no need for any special knowledge orskills in handling general-purpose CAE software onan engineering work station (EWS). Similarly, thereis no need to bring in other departments. We believethat this offers the possibility of allowing "just alittle bit of inspiration" to be examined mechanicallyand that the tool can be used midway through adesign review. The individual sheets of this designtool can be easily disconnected and connected, sothat they could easily be applied to other suspension

T = Imm

–Kss-1Ksm

K = Kmm Kms

Ksm Kss

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R&D Review of Toyota CRDL Vol. 37 No. 1

Table 1 Comparison between FOA and current CAE.

Fig. 5 Finite element model of torsion beam on 3Dconfirmation program.

Table 2 Properties of suspension compliance.

types.3. 2 Calculation precisionWe will verify the calculation precision for the

compliance properties and alignment changescalculated with this design tool. Table 2 lists thenumerical and experimental results obtained forvarious compliance properties, while Fig. 6 showsthe numerical and experimental results for alignmentchanges if the right- and left-hand wheels aredisplaced in reverse phases.

From Table 2 and Fig. 6, we can see that thenumerical results and experimental results matchwell and that the design tool can be usedsatisfactorily in the planning stage, in which theacceptability of the design plans are decided.

4. Conclusions

In this report, we described a design tool that wasdeveloped for torsion beam suspensions, and whichis an application of First Order Analysis tosuspension mechanical design. We then verified theeffectiveness of the tool using examples. We candraw the following conclusions:

(1) This design tool enables design engineers toquickly perform basic design and propertyevaluation in the suspension planning stage, simplyand easily using a personal computer.

(2) Experiments have shown that the calculatedsuspension properties are of a degree of precisionsufficient to allow us to judge the acceptability ofdesign plans in the planning stage.

References

1) Sugiura, H., Kojima, Y., Nishigaki, H. and Arima, M.: "Trailing Twist Axle Suspension Design UsingADAMS", FISITA F2000 G306 (2000)

2) Nishigaki, H., Nishiwaki, S., Amago, T. and Kikuchi, N. : "First Order Analysis for AutomotiveBody Structure Design", ASME DETC2000/DAC-14533, (2000)

3) Oden, J. T. : "Mechanics of Elastic Structures",(1967), McGRAW-HILL

4) Guyan, R. J. : "Reduction of Stiffness and MassMatrics", AIAA Journal, 3-2(1965)

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R&D Review of Toyota CRDL Vol. 37 No. 1

Fig. 6 Alignment behavior with respect to rollingmotion.

Hideki SugiuraYear of birth : 1967Division : Research-Domain 14Research fields : Development of CAE

for design engineer, Suspensiondesign

Academic society : Jpn. Soc. Mech. Eng.,Soc. Automot. Eng. Jpn.

Yoshiteru MizutaniYear of birth : 1955Division : Research-Domain 14Research fields : Test and research on the

structural strength of the vehicle,Development of the loadmeasurement transducer

Hidekazu NishigakiYear of birth : 1961Division : Research-Domain 14Research fields : Design method of the

body structure in conceptual designstage

Academic degree : Dr. Eng.Academic society : Jpn. Soc. Mech. Eng.,

Soc. Automot. Eng. Jpn.