Scholarly article on topic 'Prediction of impulsive vehicle tyre-suspension response to abusive drive-over-kerb manoeuvres'

Prediction of impulsive vehicle tyre-suspension response to abusive drive-over-kerb manoeuvres Academic research paper on "Mechanical engineering"

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Academic research paper on topic "Prediction of impulsive vehicle tyre-suspension response to abusive drive-over-kerb manoeuvres"

Original Article

Prediction of impulsive vehicle tyre-suspension response to abusive drive-over-kerb manoeuvres

Proc IMechE Part K: J Multi-body Dynamics 227(2) 133-149 © IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1464419312469756 pik.sagepub.com

2 2 2 H von Chappuis , G Mavros , PD King and H Rahnejat

Abstract

This article presents a minimal parameter vehicle simulation model to predict the vertical suspension loads expected during abusive driving manoeuvres, such as a kerb strike event. Impulsive suspension loads are applied to tyre and suspension elements under such conditions. In particular, the aim is to specifically study the reactions of jounce bumper-rebound stop and tyre characteristics. For this purpose, a vehicle in-plane pitch dynamics model with 7 degrees of freedom suffices. Non-linear and hysteretic characteristics of the bump-stop elements are included through a new parametric map concept, based on displacement and velocity-dependent hysteresis. Furthermore, a static tyre model is described, tailored to predict the radial stiffness against penetration of an edge with a flat-type rigid body geometry. The tyre model is derived on the basis of classical membrane theory and represented in terms of only a few input parameters. Model validation is supported through experiments at both component and system levels.

Keywords

Minimal parameter vehicle model, map concept for non-linear compliance, abusive manoeuvres, kerb strike Date received: 3 July 2012; accepted: 17 October 2012

Introduction

With increasing computational power, complex simulation models now allow design verification and optimisation in the final vehicle development phase, for instance, for determining a vehicle's handling response.1,2 However, it is desirable to use simulation studies at an earlier stage of development, such as at the initial conceptual phase. Unfortunately, most physical data, other than main geometrical details and inertial properties are unknown at the outset. Therefore, for concept studies, for example, in the early stages of chassis development and prediction of component loads, simple but effective models are preferred. This is the approach outlined in this article, highlighting a minimal parameter vehicle model. In the automotive industry, the application of such models is regarded as advantageous, because of their parsimony, fast simulation run times and ease of interpretation. The purpose of this approach is to estimate an envelope for the expected vertical suspension load levels under various load case scenarios.

Due to the abundance of computational tools such studies are carried out, typically using rather elaborate multi-body models. For example, handling responses are studied using multi-degrees of freedom

(DOFs) models,1 5 whilst durability analysis is performed taking into account the elasto-kinetic behaviour of suspension elements,6,7 particularly in response to abusive driving conditions such as kerb strike events.8,9 All studies are characterised by high-fidelity dynamic models including a detailed description of the suspension, with the associated difficulties in parameterisation, especially at an early stage of development. These models show that simulation of the three-dimensional motion of complex suspension systems, including their compliance has become commonplace. At the same time, although in pure dynamics terms such models are rather elaborate, they generally lack special features such as appropriate tyre models or representative non-linear bump-stop models for dealing with specific situations such as a kerb strike.

'Ford Werke GmbH, SpessartstraBe, Cologne-Merkenich, Germany 2Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, UK

Corresponding author:

H Rahnejat, Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, LEII 3TU, UK. Email: H.Rahnejat@lboro.ac.uk

Suspensions and tyres should possess many attributes for ride comfort, durability and vehicle handling.10 These characteristics can be measured at the later stages of a vehicle's development, when system components are in situ in physical prototypes. Then, detailed multi-body models can be used for design verification against specific load case scenarios, such as vehicle cornering and braking11 or riding over speed bumps12 and other discrete road induced events.

Modelling of non-linear components such as bump-stops has been the subject of research, aiming to develop rheological bumper models.13 The work presented herein focuses on maintaining simplicity and computational efficiency, whilst at the same time include realistic responses of specific components such as the tyres and bump-stops under specific test cases. The result is a tailored simulation method capable of delivering accurate and dependable results at an early stage of development, with minimum computational effort.

The focus of the current contribution is model representation of jounce bumper characteristics13,14 as well as tyre behaviour15 17 under severe impulsive conditions, particularly in a drive-over-kerb (DOK)

Figure 1. Considered key components for vertical load simulation.

manoeuvre (Figure 1). Ford has specified the worst vertical load case manoeuvre to be the DOK. A two-dimensional (planar) vehicle model18,19 is extended to take into account the fore-aft wheel travel. Large scale wheel motions lead to impulsive actions, which can induce significant jounce bumper impact and gross tyre deflections.

The broad aim of this study is to develop an appropriate combination of vehicle, tyre and bump-stop models, capable of accurately predicting the suspension mount loads as a result of driving directly over a kerb of specified height and at a specified impact speed. This aim has been pursued from a perspective of minimum parameterisation effort, so that the resulting simulation tool can be used efficiently at an early design stage in order to carry out 'what if type studies. In order to achieve this aim, work has been carried out in three areas, namely the development of a simple planar vehicle model, an in-plane tyre model specifically designed to deal with kerb-strike situations and a jounce bumper model based on a novel force mapping procedure from simple measurements. The jounce bumper and tyre models are validated independently at component level, whilst a number of kerb-strike simulations are validated against experimental results obtained using a real vehicle driven against a kerbtype obstacle.

Vehicle model

The vehicle is represented by a planar model as illustrated in Figure 2, possessing 7DOFs, as shown in Figure 3. A planar model suffices since in this study, the direction of travel is assumed perfectly perpendicular with respect to the kerb and the vehicle is assumed to be symmetric with respect to its centre plane.

The vehicle model incorporates in-plane pitch and bounce motions,18,19 extended by three additional

Figure 2. The minimum parameter vehicle model.

longitudinal DOF, as shown in Figure 3, in a similar manner to the approach of Sharp20 and Azman et al.21 The longitudinal DOF refers to the fore-aft motion of the vehicle body, as well as those of the front and rear suspensions relative to the vehicle body. Finally, each one of the suspensions possesses a vertical DOF to account for its jounce/rebound motion. The same study could not be performed using a simpler quarter car model, because the asymmetric loading caused by the front end kerb collision generates a moment about the centre of mass of the vehicle which would induce vehicle pitch, coupled with its longitudinal motion. Therefore, the perceived suspension compliance during the collision dictates the magnitude of the peak load, which not only depends on the vertical translational response, but also on the pitch response of the vehicle.

The inertias of the suspension links are lumped with the unsprung wheel mass. The effects of suspension geometry, kinematics and stiffness of individual components are merged into combined vertical and longitudinal stiffness elements. For simplicity, the vertical and longitudinal suspension compliances are represented by idealised spring and damper elements. The model comprises the following:

(a) sprung mass and rotational inertia about the pitch axis;

(b) unsprung compound mass of road wheel, hub, brake and lumped suspension;

(c) constant suspension spring stiffness;

(d) constant shock absorber damping coefficient;

(e) jounce bumper characteristics represented by the map concept described in section 'The jounce bumper force response map';

(f) an undamped rebound stop with non-linear stiffness;

(g) suspension travel confined to jounce and rebound;

(h) longitudinal stiffness and damping of wheel centre with respect to the vehicle body; and

(i) a radial force response tyre model.

A full model parameter list is given in Table 1, where the stiffness and damping values define absolute component properties.

The vehicle's equations of motion are derived using the Newton-Euler approach, so no kinematic constraint formulation is required.

The equations of motion in matrix form become

[M] xj\ +[D] +[K]

Xj } = Pk

TFxj T,myk

where [M] is the inertia matrix and [K] and [D] the effective stiffness and damping matrices. The applied forces constitute the right-hand side of the equation. The matrix [D] is of equivalent structure to [K]. The reaction loads on the right-hand side of equation (1) need to be determined for the solution of equations of motion. For the DOK manoeuvre, these loads are due to gravity and the contact forces developed at the tyre contact and the jounce/rebound bumper.

Contrary to the typical in-plane dynamics' models, the system properties of the model shown in Figure 4 vary depending on the extent of wheel displacement. The free wheel travel, causing jounce bumper and rebound stop engagement, denoted by UJB1 and URS1, can be established in combination with the geometry parameters A, B and the spring compression for the vehicle under the static laden condition.

The full extent of wheel displacement is divided into the following three cases:

Case 1: Free wheel travel, when only the suspension

spring/damper is active.

Case 2: Jounce bumper is in contact.

Case 3: Rebound stop makes a contact.

Figure 5 shows the vehicle front suspension in vertical free travel (left) and when the bump stop is engaged (right).

Figure 3. DOFs and external loads for simple vehicle model. DOFs: degrees of freedom.

Table 1. Simulation model input parameters. DOK vehicle model parameters

Geometry parameters

Wheel base and CoG (m)

a = 1.379 Longitudinal dist CoG

to front WC

b = 1.471 Longitudinal dist CoG

to rear WC

h = 6.89E — 1 CoG height to ground

Suspension front/rear (m)

A = I.297E — I/2.262E — 1 CoG height to jounce

bumper tip

B = 3.8E — I/4.50E — 01 CoG height to uncompressed

lower spring end

Tyre (m)

Ra = 3.I0E — 1 Belt outer radius

ri = 2.39E — 1 Rim radius

bt = I.92E — 1 Belt width

fc = 3.0E — 2 Sidewall concavity

p = 2.0E5 N/m2 Pneumatic pressure

Kinetic parameters

Masses (kg)

mI = 5I.5 Front wheel unsprung mass

m2 = 40 Rear wheel unsprung mass

m3 = I05I (Body + load)/two sprung mass

J = 3938 kg/m2 Rotational inertia

Damping (Ns/m)

DIz = 2.57E03 Front vertical damping

D2z = I.7IE03 Rear vertical damping

DIx = 2.23E03 Front longitudinal damping

D2x = 2.52E03 Rear longitudinal damping

Stiffness (N/m)

CIz = 3.2E04 Front road spring stiffness

C2z = 2.6E04 Rear road spring stiffness

CI x = 2.8E06 Front longitudinal WC stiffness

C2x = 4.8E06 Rear long WC stiffness

CBOIz = I.4E06 Body front top mount

vertical stiffness

CBO2z = I.2E06 Body rear top mount

vertical stiffness

Crim = 3.2E06 Rim radial edge stiffness

CoG: centre of gravity; WC: wheel centre.

With the wheel and vehicle body DOFs z1, z3 and <, as shown in Figure 5, the wheel is in free travel mode, if

UJB1 + z3 + a< — z1 > 0 and

URS1 — z3 — a<p + z1 > 0 => case 1, free travel.

The bumpers are engaged for the following conditions

UJB1 + z3 + a< — z1 < 0 => case 2, jounce bumper contact.

URS1 — z3 — a< + z1 < 0 => case 3, rebound stop engaged.

For the free travel state (case 1), the matrices [K] and [D] do not include stiffness and damping characteristics associated with the jounce and rebound bumpers. These should be included for cases 2 and 3. In order to ensure a linear system so that the stiffness and damping matrices remain unchanged throughout the simulation, the effect of jounce bumper or rebound stop engagement is handled by modifying the forces at the right-hand side of equation (1). Hence, for case 1 (free wheel travel)

Fz1 — m\g

Fz2 — m2 g —m3g

Fe(1) = \ Fx1 (2)

Fx2 0 0

For cases 2 (jounce bumper engagement) and 3 (rebound stop contact), at the front suspension

Fz1 — m1 g — FjB1 Fz2 — m2g —m3g + Fjm

Fe (2) = • FX1 ,

Fjb1 • a Fz1 — m1 g + FRS1 Fz2 — m2g —m3g + Frs1

Fe(3) = j Fx1 (3)

Frs1 • a

The equations of motion (1) are formulated in FORTRAN and are integrated in time domain using a fourth order Runge-Kutta algorithm with the right-hand side switching between cases 1 and 3, as described above. The tyre forces depend on the relative position between of the tyre with respect to the kerb and their detailed derivation is provided in section 'The tyre model'. The jounce bumper force depends both on the vertical position of the suspension and its rate of change and it is based on a map concept, detailed below.

The jounce bumper force response map

The jounce bumper is regarded as a 1-DOF vertical compression-only element. It attenuates the effect of

Figure 4. Free vertical wheel travel JTand RTon a front suspension. JT: jounce travel; RT: rebound travel.

Figure 5. Free vertical wheel travel UJBI and jounce bumper engagement JBIz on a front suspension.

impact load transmission to the vehicle body postapplication of an impulsive event, whether this is as the result of a kerb strike or riding through a pothole. The bumper is made of polyurethane foam, commonly used in vehicle suspensions. Its force response characteristics are a combination of both a polytropic deformation of enclosed gas chambers and compression of the gas volume surrounding the polymeric structure itself. The material properties, such as density, depend on the specific boundary conditions applied during the polymerisation process. It is, therefore, difficult to investigate the bumper force response through other means than physical testing.

Simple conventional simulation models consist of a non-linear spring element combined with viscous damping.5,8 The spring characteristics are derived from quasi-static force response measurements taken from the bumper element. Typically, the damping coefficient is tuned so that the combined elastic/viscous model correlates well with drop tests, or similar transient responses. In general, it is rather difficult to achieve good correlation in terms of measured/simulated hysteresis loops by simply tuning a linear damping coefficient. Therefore, although the described procedure is rather straightforward, hysteresis in general is not successfully addressed by means of a simple viscous damping coefficient. Consequently, simulations have also been performed, using rheological models,13,14 combining Maxwell or Kelvin viscoelastic models with conventional spring elements. This addresses the hysteretic behaviour of polymeric

elements more effectively, but introduces problems related to model complexity and identification of model parameters. In this article, an alternative map concept is proposed. The concept of a jounce bumper force map is used to feedback the bumper force FJB according to the bumper states: fz, zgT into the equations of motion. The map concept takes the non-linearity and hysteresis of the polymeric elements into account and avoids the burden associated with setting up and calibrating a rheological model.

The map concept is based on the idea of constructing a force response map from experimental measurements, with deformation and its rate as the independent variables. The requirements set for the map concept include experimental simplicity, coverage of the whole operating area of the jounce bumper (no extrapolation) and good correlation with transient drop tests.

During a DOK manoeuvre, the jounce bumper may deform to its full extent. In general, bumper excitation during kerb strike may vary according to impact duration, amplitude and the shape of the force trace. Here, a harmonic excitation is applied to generate the map data in conjunction with the rig shown in Figure 6. The validity of this method is confirmed through comparison of the simulation predictions with experimental drop tests, which are considered fairly representative of the excitation time history during the kerb strike.

Figure 6 shows the resulting kinematics due to bumper actuation as well as the physical rig used to

Figure 6. Concept of bumper test rig kinematics and physical test set up.

Figure 7. Force response of a jounce bumper element to sinusoidal actuation.

generate the map data. The rig is based on an in-plane crank driven by an electric dynamometer with constant revolution speed, !. A connecting rod is linked to a vertical grounded beam which causes a quasi-harmonic motion of the bumper. Any small deviations from harmonic response due to higher order crank kinematics are neglected. Only bumper compression is applied. Both wall support and actuation are adjustable in height in order to allow a variation in the actuation amplitude. The bumper force response is recorded by a load cell.

The quotient Vmax/Smax is used to specify the circular actuation frequency relevant for the DOK event. With Vmax being limited by the damper blocking speed and Smax specified by the bumper deflection range, f = m/2n determines the DOK relevant frequency to be around 10 Hz. Thus, the recorded data

contain the velocity-related damping hysteretic characteristics.

Figure 7 shows the bumper force obtained for various harmonic excitation amplitudes S1-S7, as a function of the phase (crank) angle, at a frequency of 10Hz.

Map data pre-processing

The raw data obtained through experiment are of the form shown in Figure 7. This data require additional processing so that an efficient map may be constructed from which the force can be interpolated as a function of bumper deformation and its time rate of change. The phase angle as shown in Figure 7 is divided into equal, narrow interval bands of 10°. For each interval and excitation amplitude (S1-S7), a cubic spline

Figure 8. (a) Spline interpolation band and (b) the organisation of array 'A' for storage of map data cubic spline coefficients.

Harmonic actuation Phase diagram

Figure 9. Generalised velocity in phase diagram.

interpolation is used to condense the recorded data into a third-order polynomial as

F = Ao + Ai • z + A2 • z2 + A3 • z3 (4)

where z represents the crank angle (independent variable) within a narrow band, as illustrated in Figure 8(a). The figure also shows the organisation of data into an easily accessible array, containing the coefficients of the interpolation polynomial in equation (4).

In the three-dimensional array shown in Figure 8(b), each element A3D (i', j, k) indicates a specific polynomial coefficient. The index i refers to the downward progression in the array and is associated with the angle interval, for example, from 70° to 80°. The index j denotes the array's horizontal progression to the right and is associated with the four coefficients for a specific polynomial, corresponding to a given angle band and excitation amplitude. Finally, the index k corresponds to the

backward progression in the array and is associated with different excitation amplitudes; S1-S7.

Map-solver interaction

The stored map feeds back the force response, corresponding to an instantaneous state of vertical suspension deflection and its time rate of change (S and V) at each integration time step within the overall dynamic simulation. These can be defined in terms of the circular frequency, the phase angle 0 = at, and the generalised velocity vg = ■()/«, as

s(t) = So • sin(0)

= So • cos(0) (5)

The phase plane diagram (displacement versus generalised velocity) reveals a concentric circle (Figure 9).

Plotting the force against phase angle for a number of different excitation amplitudes leads to a series of

Figure 10. Force response traces (z) and their vertical projections (x, y) on the phase plane (also showing circumferential and radial interpolations).

Figure 11. Comparison between simulated map concept and those measured through drop test.

concentric circles and shown in Figure 10. Also note that

the three-dimensional plot

tan é = ■

■ (V/!)2

For any pair of states (S, V), values of tan ' and R can be obtained, using equation (6). Then, the angular frequency is used for the generation of the map.

The previously defined three-dimensional array (i.e. A3D) provides interpolation data, allowing the

calculation of the force corresponding to any phase angle, but at discrete amplitudes S1-S7 (or radii R). In the event that a pair of states (S, V) falls between the concentric circles, a second quadratic interpolation is performed, as shown graphically in Figure 10.

Experimental validation of the map concept

To ascertain the validity of the map concept, simulation results are compared with a drop test applied to a jounce bumper. Figure 11 shows good agreement between the simulation results, using the harmonically

Figure 12. Tyre response force feedback to external load vector Fe.

generated map concept and those obtained through measurement.

The tyre model

In a kerb strike, the tyre loads are derived from the radial deformation of the tyre's belt. The kerb is assumed to be a simplified vertical rigid blade, promoting an edge-type line contact. Therefore, with this assumption, the complexity of a combined flat/edge type contact is removed. Later, it is shown that this assumption does not compromise the accuracy of tyre force predictions. As for the case of jounce bumper and rebound stop reactions, the tyre forces are included on the right-hand side (i.e. Fe) in equations of motion (Figure 12).

The tyre model is simplified in order to address its radial force response, neglecting its structural stiffness and inertia. The tyre structure is simplified by including two primary elements, namely the belt and the sidewall, as shown in Figure 13. The model is defined by the pneumatic pressure, p, and four dimensions of the cross-section (Ra, ri, bt and fc). The belt and side-wall are considered as ideal isotropic membranes; free from any resistance to bending. All tyre model derivations are based on these stated simplifications and assumptions.

In general, a tyre requires many additional parameters to be fully describe its characteristics, such as the parameters of a hyperelastic rubber model; damping ratios and others. Not all these properties are important for all types of study. For example, when it comes to a detailed study of the tyre contact and associated friction, rubber properties are the most important. When one is interested in sound radiation and NVH assessment, bending of the belt is important, which points to both rubber stiffness and the role of the reinforcing belt layers. In the case of a kerb impact, there are large deformations, whose shape and extent depends strongly on the inflation pressure for any

Figure 13. Tyre cross-section showing the geometrical parameters required for the model.

given tyre. In turn, the inflation pressure interacts with the tyre carcass via its behaviour as a membrane. Based on these observations, it is postulated that treating the tyre carcass as a simple membrane would not only be beneficial from a parameter identification point of view, but also would not compromise the accuracy of predictions, as is demonstrated later, when a comparison is made with finite element method (FEM) results.

Pneumatic pressure-induced internal stress of an undeformed tyre

In the absence of any radial deformation, the belt contour is assumed to be an ideal right circular cylinder. The internal radial stress at the circumferential interface between the tyre sidewall and belt, as well as the tangential belt force Fg, are derived through application of shell/membrane theory. Timoshenko22 formulated a general in-plane stress relationship for a doubly curved thin shell element applied to a toroidal structure as (Figure 14)

^ + ^ = P (7)

R1 ^ R2 t ( )

where t is the membrane wall thickness and p the pneumatic pressure.

As t is considered small in comparison to the other tyre dimensions, bending effects are neglected in the derivation, so only the membrane stresses are considered here.

Considering equilibrium in the y-direction leads to the radial stress a1, thus

2n • R • t • CT1sin a= p • n • (R2 — Rm2) (8)

Using the geometric relation

R — Rm

sin a =--— (9)

Figure 14. Curvature radii RI, R2 and membrane stress <7|, <r2 in a torus.

the radial stress in the sidewall is obtained as —1 / —m\

=p ■ —1 (1 + —m)

The circumferential belt force, Fg, is derived by considering the equilibrium condition in the z-direction, as shown in Figure 15. A cylindrical shell element, representing the belt rests in equilibrium under the action of sidewall stress, a\, circumferential belt force, Fg, and the pneumatic pressure, p. Hence

Fg = p • Ra • bt — 2 • CTj cos a • Ra • t / cos < • d<

Equation (10) is used with the following geometrical relations substituted in equation (11)

—a + ri

Rm = —-- hc = —a — ri

, hc2 + 4fc2 —1 - fc

—1=-—r1— cos a = -

After integration, the circumferential belt force is obtained as

Fg = p [—a ■ bt -(—a + —m)- (—1 - fc)]

Now using equation (10), the sidewall line loads Qsg and Qsr at the belt and rim attachments become

—1 —m

Qsg = CT1(—=—a)-1 = p ■ -y ■ I 1 + —a

Figure 15. Belt shell structure subjected to sidewall stress <7| and belt force Fg.

Qsr = CT1(—=ri) ■ t = p-

The validity of equations (13) to (15) is confirmed by Figure 16, which shows a comparison of the analytical method with FEM results on basis of a 235/40-R18 tyre. Deviations rarely exceed by 10%.

Tyre force response for an edge type radial contact

For modelling purposes, the DOK manoeuvre is simplified to a drive-over-blade situation, whereby the tyre rolls over a blade-type obstacle. The associated

Figure 16. Analytical results for belt force Fg and sidewall line loads Qsg and Qsr and their percentage deviations from the FEM results.

FEM: finite element method.

tyre force is derived for an idealised edge type line contact of the belt, as shown in Figure 17.

As indicated in the figure, with an increasing radial deflection, f, the projected belt area decreases, as does the associated belt force, Fg. This effect is incorporated by correcting the belt force, Fg as

Fgcorr =Fg

2Ra - f 2Ra

The horizontal belt force component can be obtained by this relationship, whereas the vertical force component remains unknown in the absence of the belt tangential angle. With the assumption of equal belt force magnitudes at the upper and lower belt cross-sections, the deformed belt appears to resemble a rope subjected to a constant line load distribution per unit length, q, as indicated in Figure 18.

The rope indicated by the dashed line in the figure is shown for its half-span width (effective length Le). It is subjected to a horizontal force, H. On the right-hand side of Figure 18, both the upper and lower tyre belt cross-sections are subjected to horizontal belt forces, Fgcorr. Assuming an edge-type contact, the belt is considered to behave similarly to a simply supported rope with a horizontal force H = Fgcorr. Thus, a vertical component V can be estimated for the belt, based on this rope analogy. Taking into account, the equations for a rope as derived for constant line load, q23

TT q ' Le Ai a A /max

H = —-- and tan p = 4 • ——

"/max Le

with /max = Ra - ri

The horizontal belt force at the tyre-blade contact interface can now be derived for the maximum deflection /max (Figure 15) as

q • Le2

8—max

= p • [Ra • bt - (Ra -

— Fgcorr

-Rm)-(R1 - /c)]

2Ra - fm

Figure 17. Simplified edge-contact geometry.

Rearranging this equation and substituting: q = p • bt provides an estimate of the effective length, Le, for the extreme case of belt-to-rim contact

Le = 2-

2 • fm

(Ra + Rm)-(R1 - fc)

2Ra - fm • 2Ra

with -max = Ra - ri

The vertical half belt force component V can then be defined as being due to the maximum deflection as

V=H- tan i

Strictly speaking, the effective length, Le, would be a function of the radial belt deflection f. However, the model's agreement to physical measurements can be improved by making the assumption that Le would remain constant over the radial belt deflection range. Thus, if Le were to be derived with the condition of maximum deflection fmax, in conjunction with the belt force Fg of an undeformed tyre, the following relationship would be obtained

Le = 2 • 2 • fm

(Ra + Rm)-(R1 - fc)

For arbitrary values of edge deflection, f, the radial edge force Fc can be found after tan p has been linearised within the radial deflection range fmax of the rope (belt), (Figure 16), using equations (17), (18), (20) and (21). Hence

Fc = 2 • V

= 2 • Fgc. = 2p • bt •f

• tan p

- f) l( Ra-(Ra + Rm) \ ( J)y\ (R1 - fc)/bt J

Ra ^2 •(Ra - ri)

V uni*.mm Fr A Ra Fgcorr +

H f fmas rgcorr i

Figure 18.

Belt loading analogy to a rope.

It is noted that besides the pneumatic pressure p, all other parameters used in equation (22) can be directly derived by equation (11) from four key dimensions taken from the tyre cross-section shown in Figure 12.

Validation of the tyre model

The validity of the analytical tyre model is assessed by comparing the predicted radial force with that measured, as well as the forces obtained through an

equivalent finite element analysis (FEA). Both physical tests and the FEA study were conducted for three nominal pneumatic pressures (1,2 and 3 bar) and three tyre deflection levels (20, 40 and 60 mm). A comparison between FEA predictions, measured and analytical results is provided in Figure 19. Despite the simplicity of the modelling approach and although the belt bending stiffness was neglected in the model, the radial forces calculated analytically agree well with those measured and predicted by FEA. The agreement with

Figure 19. FEA and analytically calculated and measured edge contact forces with their percentage deviations compared with measurements (a) FEA (green bars) or (b) analytical (blue bars). FEA: finite element analysis.

Figure 20. Representative DOK test set up. DOK: driver-over-kerb.

measurements improves with increasing pressure, which indicates that the influence of the structural force components diminishes in relation to the pneumatic induced forces. This is in perfect agreement with the initial postulate that the interaction between pressure and a membrane structure is primarily responsible for force generation in the case of a DOK event.

DOK manoeuvre and system level model validation

The jounce bumper characteristic map and the analytical tyre model are integrated with the equations of motion (1). The aim is to ascertain the extent of validity of the developed analytical methodology against real representative physical vehicle tests, particularly

Figure 21. Top mount force and vertical wheel centre deflection over time for DOK test. DOK: driver-over-kerb.

abusive driving manoeuvres. For this purpose, a controlled DOK test was performed with a CD340 vehicle (Ford Galaxy). The test was carried out at two different speeds, 25 and 40km/h. The vehicle was driven at a free rolling mode over a beam of rectangular cross-section of height 135mm. The crossing was performed in a direction perpendicular to the beam longitudinal axis with the intention of inducing simultaneous impact of both wheels with the bar. The experimental test set up is shown in Figure 20. All the vehicle parameters are provided in Table 1.

The vehicle was equipped with 235/40-R18 tyres, operating at a pneumatic pressure of 2 bar. The vehicle was loaded according to the Ford specification standards for passenger cars.

Figure 21 shows a comparison between the measured (dark line) and analytical simulated predictions (grey line) for the top mount force time-history during

the traverse of the vehicle over the kerb at the two tested speeds.

The results show very good agreement in terms of monitored force levels. Any discrepancies in the actual shape of the experimental and simulation time-histories can be attributed to the model simplifying assumptions. An additional benefit from the simulation study is that it allows the individual reactive contributions due to the spring, damper and the jounce bumper to be quantified, as illustrated in Figure 22. Figure 22 also reveals the timing of the bumper engagement and thus the switch of the loading conditions between cases 1 and 2. The bumper force appears to be the dominant contributor during the severe kerb-strike event, as would be expected. Individual contributions of resistive elements would be very difficult to obtain through mere measurements, if not impossible.

Figure 22. Predicted contributions of various system compliances (spring, damper and jounce bumper) to the reaction at the top suspension mount during the kerb strike.

Conclusion

A minimal parameter vehicle model is presented, specifically tailored to simulate vertical suspension loads during abusive DOK manoeuvres. Good correlation with experimental data is demonstrated both at component and system level tests. At component level, it is interesting to note that the tyre impact forces can be satisfactorily predicted by membrane mechanics only, in conjunction with inflation pressure.

It is worth noting that the suggested analytical tyre model fulfils the initial goals of computational efficiency and simplicity in parameterisation, whilst it is frequently more accurate than the complex FE model. This is evident in all cases of moderate-to-high inflation pressures and large tyre deformation, where as discussed before, the combination of inflation pressure and membrane mechanics yields more representative results than the FE model. It is clear that a more detailed and better parameterised FE model would have the potential of improving correlation with experimental edge-contact results, to the extent of being more accurate than the current analytical model. However, the current average performance of the already complex FE model illustrates the difficulty of making the most out of computationally demanding, high-fidelity models. At the same time, it shows that proper consideration of the underlying physics can often lead to very simple and accurate description of the necessary processes, whilst avoiding excessive computational burden.

Using the outlined analytical model, parametric studies can be carried out at an early design stage, when only a few design parameters are known. Alternatively, the model can be used at later stages of vehicle development in order to predict the effect of any design changes. In the latter case, calibration of the model based on existing vehicle data would be advisable.

It should be noted that the vehicle planar model approach limits the scope of investigation to driving manoeuvres which are free of roll and yaw modes. In particular, the tyre model is also limited to a simplified blade type contact, representing a kerb strike-induced edge contact. A lateral impact cannot be handled by the proposed tyre model. However, from a durability point of view, lateral kerb strike is usually studied in the context of the worst case rim-pavement impact, where the contribution of the tyre diminishes.

Funding

Funding for this work is provided by Ford Motor Company.

Acknowledgements

The authors would like to express their gratitude to Ford Motor Company for its financial and technical support of this research project.

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Appendix

Notation

a front wheel centre distance to vehicle

A front/rear suspension hardpoint geo-

metry parameter A0,...,3 spline coefficients

b rear wheel centre distance to vehicle

bt tyre belt width

B front/rear suspension hardpoint geo-

metry parameter [D] damping matrix

f frequency (section 'The jounce bumper

force response map') tyre radial deflection (section 'The tyre model')

fc tyre sidewall warpage deformation

F bumper force polynomial

Fc tyre radial edge contact force

Fe external force vector

Fg circumferential tyre belt force

Fgcorr corrected tyre belt force

FJBi,2 front- (1), rear- (2) jounce bumper force

response

Fr rope tangential force

FRSi,2 front- (1), rear- (2) rebound stop force

response

FX1,2,3 longitudinal front wheel- (1), rear

wheel- (2), body (3) force FZi,2,3 vertical front wheel- (1), rear wheel- (2),

body (3) force g gravity constant

h vehicle CoG height over ground

H horizontal-/cirumferential rope-/belt

force component JB1,2z front-, rear jounce bumper vertical

deflection

my m1,2

S1-S7 t

UJB1,2 V

x1,2,3

z1,2,3

stiffness matrix wheel base

hinge bridge span width/effective length pitch moment

vehicle unsprung masses front, rear

vehicle sprung mass (body)

mass matrix

body pitch moment

pneumatic tyre pressure

rope line load

tyre sidewall line load attached to belt

tyre sidewall line load attached to rim

tyre bead radius

instant torus radius

outer tyre belt radius

mean torus radius

curvature radii of double curved shell element

instant jounce bumper deflection bumper deflection range harmonic excitation amplitudes to generate map data shell wall thickness

front- (1), rear- (2) wheel free travel to jounce bumper engagement instant jounce bumper deflection (section 'The jounce bumper force response map')

vertical-/radial rope-/belt force component (section 'The tyre model') generalised velocity damper hydraulic blocking speed longitudinal front wheel- (1), rear wheel- (2), body (3) travel vertical front wheel- (1), rear wheel- (2), body (3) travel

radial sidewall force angle relative to belt

rope-/belt tangential angle

membrane stress in double curved shell

element

body pitch angle (section 'Vehicle model')

force response map phase angle (section 'The jounce bumper force response map')

tyre circumferential angle (section 'The tyre model')

circular frequency of jounce bumper harmonic actuation