Dynamic effect of the non-rigid modified bicycle model

717

Dynamic e€ ect of the non-rigid modiŽ ed bicycle model J P Wideberg Transportation Engineering, School of Engineering, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain

Abstract: In this paper a method to add elastic e€ ects to the classic ‘bicycle model’ for the simulation of the dynamic behaviour of vehicles is presented. The obtained results show that, in the simulations, dynamic e€ ects such as lateral acceleration are more severe when non-rigid models of the vehicle frame are used. This work demonstrates that the modiŽ ed bicycle model is a useful instrument to predict the response of a vehicle. At present, much e€ ort is dedicated to simulation with rigid body dynamic programs. The proposed method o€ ers an easy way to evaluate the dynamic e€ ects in models with  exible frames. The elasticity of the frame has an important impact on the directional response (yaw gain and lateral velocity) of the vehicles. This method is useful for analysis from the behaviour of vehicles with modiŽ cations made after their manufacture. In the design phase the manufacturer has the suitable tools and the experience to avoid designing frames of low rigidity. Nevertheless, in the second-hand market it is very common for trucks and their structure to be modiŽ ed. The change in rigidity of the frame could have considerable e€ ects on directional stability and handling. Keywords: vehicle dynamics, bicycle model, simulation, elastic e€ ect

NOTATION a y b c C f C r DOF E F c F yf F yr FEA I z K L m m f m r MBS R SUV v crit

lateral acceleration (m/s2) distance from the centre of gravity to the front axle (m) distance from the centre of gravity to the rear axle (m) cornering sti€ ness, front tyre ( kN /rad) cornering sti€ ness, rear tyre (kN/rad) degree of freedom Young’s modulus (MPa) centrifugal force (N ) reaction force at the front tyre (N ) reaction force at the rear tyre (N ) Ž nite element analysis moment of inertia (mm4) understeer gradient distance between axles (m) total mass of the vehicle (kg) mass (weight) on the front axle ( kg) mass (weight) on the rear axle (kg) multi-body system radius of curve (m) sport utility vehicle critical speed (m/s)

The MS was received on 9 January 2002 and was accepted after revision for publication on 11 July 2002. D00302 © IMechE 2002

v x v y

velocity in the forward direction (m /s) velocity in the lateral direction (m/s)

a f a r h f

slip angle of the front tyre (rad ) slip angle of the rear tyre (rad ) angle due to deformation of the frame, front axle (rad ) angle due to deformation of the frame, rear axle (rad ) yaw speed (rad/s)

h

r

V

1

INTRODUCTION

A very well-known dynamic model for simulating the cornering of a vehicle is the ‘bicycle model’. This model is discussed in many articles and textbooks (e.g. references [1] and [2]). In this model a vehicle with four wheels and two axles is modelled as a vehicle with two wheels, front and rear, with double sti€ ness (i.e. in the calculations the cornering sti€ ness is multiplied by 2). In 1956, Milliken and Whitcomb [3, 4] and Segel [5, 6 ] of the Cornell Aeronautical Laboratory published the Ž rst major quantitative and theoretical analysis of vehicle handling. The Ž nal paper in the series [4] draws a number of conclusions on automobile stability and control using a two-degree-of-freedom (2DOF ) model (yaw and side slip) with experimentally determined Proc Instn Mech Engrs Vol 216 Part D: J Automobile Engineering

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parameters. Owing to the lack of a roll degree of freedom, Whitcomb and Milliken were able to assume that the car had no width and that the tyres lay on the centreline of the vehicle (a ‘bicycle model’). A set of linearized di€ erential equations was derived using stability derivatives, and the steady state and transient responses were studied. In 1967, Bundorf published a paper relating vehicle design parameters to the characteristic speed and to understeer [7]. This paper utilized the deŽ nitions of understeer and characteristic speed proposed by the SAB publication Vehicle Dynamics Terminology [8]. Methods were proposed to predict understeer quality in vehicle designs and for measuring understeer in existing vehicles. It was noted that the characteristic speed is an attribute associated with a linear vehicle model. Bundorf argued that, under most normal driving conditions, which he characterized as having lateral accelerations below 1/3g, a vehicle can be accurately modelled by a linear model. Bundorf derived an expression for predicting the characteristic speed of a vehicle given the design parameters. The vehicle model used in his derivation was a bicycle model with Ackermann (no slip) steering. Since the early 1980s, a shift in the vehicle modelling process has taken place. The demand for accurate vehicle dynamics models combined with the di culty in deriving the equations of motion for large multi-body systems led to the use of general multi-body simulation codes. A wide range of capabilities is present in modern multibody system (MBS ) codes, including the ability to handle non-inertial reference frames, to incorporate  exible elements in the model, to utilize generalized coordinates and symbolically to generate the equations of motion. Although commercial multi-body simulation codes are available, it is desirable to utilize simpler models that do not require the large quantities of descriptive data associated with the more complicated codes. These models typically use less than 12 degrees of freedom to model the vehicle. They predict the general motion of the vehicle with acceptable accuracy and are useful as design tools, but they lack the accuracy required for optimal design. Also, in order to use MBS codes it is necessary to invest substantial amounts in computers, codes and training of the engineer. This is possible for large corporations, consulting houses and universities, but not for small Ž rms and individual independent engineers. It is very common for older trucks to be modiŽ ed for use other than that originally intended. For instance, a new owner might want to modify the structure by adding a crane. In this process it is common for the frame to be modiŽ ed in such a way that it is made more or less rigid. This is usually done by a Ž rm that has limited time, knowledge and resources to use MBS or Ž nite element analysis ( FEA) codes. For the reasons mentioned above, Proc Instn Mech Engrs Vol 216 Part D: J Automobile Engineering

it would be useful to have a tool available to any engineer that is based on the bicycle model but adds a way to interpret the e€ ect of the sti€ ness of the frame. This model, which is presented in this paper, is very simple to use and can be programmed using FORTRAN, Matlab or even Excel, and is therefore accessible to any engineer. The bicycle model is a very valuable tool to use, and there is a wealth of information about the dynamics of a vehicle that can be derived from this relatively simple model. Although the model is not new, it appears in several recent papers: Vagstedt and Dahlberg [9] use it to derive axle data from measurements, Sobottka and Singh [10] use it in control simulations and Spentzas et al. [11] use it for kinematic analysis of di€ erent types of vehicle. One of the drawbacks of the model is that it is considered as rigid. In the present paper the bicycle model is modiŽ ed to take into consideration the elasticity of the frame, i.e. how the bending of the vehicle about the vertical z axis (according to the SAE system) changes the steering angle of the vehicle. It is important to mention that the elasticity of the frame is of particular importance in trucks with an independent frame ( ladder design) that has a slender design of not too high sti€ ness. On the other hand, a car with an integrated frame and with a shorter distance between the front and rear axles relative to the track width is much sti€ er than the independent frame of a truck. Therefore, the in uence of elasticity of the frame will be much too small to be considered in the present article. It is worth mentioning that sport utility vehicles (SUVs) do have dual frames and that this architecture might be adopted for future passenger cars.

1.1 Steady state cornering Steady state cornering will be derived for the case of driving in a circle with a certain constant radius. Here, the classical bicycle model will be combined with the elasticity of a supported beam subjected to a point load at its centre of gravity (see Fig. 1). The boundary conditions permit free rotation of the beam but do not permit displacements. The contact between the tyre and the road will prevent rotation slightly. The amount of cited rotation is very hard to quantify, and therefore the use of totally free rotation will give more conservative results. For this reason, free rotation can be expected. The Ž xed load on the beam is equivalent to the centrifugal force of the vehicle travelling in a circle of radius R, given by mv2 F =mRV 2= x c R

(1) D00302 © IMechE 2002

DYNAMIC EFFECT OF THE NON-RIGID MODIFIED BICYCLE MODEL

Fig. 1

Bicycle model and supported beam

The equilibrium of forces gives mv2 F +F =F = x yf yr c R

and (2 )

and the equilibrium of moments gives F b­ F c=0 yf yr The solution of the former yields bv2 F =m x yr LR

(3)

(4)

and cv2 F =m x yf LR

(5)

Introducing the slip angles for the rear and the front tyres F a = yf f C f D00302 © IMechE 2002

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(6)

F a = yr r C r mc m= , f L

(7) mb m= r L

( 8)

(where m and m denote the weight on the front and f r rear axle respectively), the following expressions are introduced: m v2 a = f x (9) f C R f m v2 a = r x (10) r C R r If the elastic deformation of the frame is taken into consideration, it is possible, as a Ž rst approximation, to consider the frame as a supported elastic beam subjected to a point load. The formula for this angle is published in the literature, e.g. reference [12]. Adding this to the bicycle model, an extra angle emerges on the front wheel (see Proc Instn Mech Engrs Vol 216 Part D: J Automobile Engineering

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Fig. 1b):

3

A B

(11)

A B

(12)

F c bc[1+(c/L)]mv2 x h = bc 1+ = f 6EI L 6EIR and on the rear wheel: F b bc[1+(b/L)]mv2 x h = bc 1+ = r 6EI L 6EIR

The compatibility will give the following relationship between angles: d=

L +a ­ f R

d=

L mv2 + x ­ R RL

a ­ r

or

C

h­ f

h

(13)

r

bc(b+c) 6EI

+

A

c ­ C r

b C

f

BD

(14)

Another convenient way to write the above expression is

C

d=

L m + ­ R L

=

L +Ka y R

A

bc(b+c) c + ­ C 6EI r

b C

f

BD

a

y (15)

where K is the understeer gradient and a is the lateral y acceleration. A discussion on the physical meaning of the understeer gradient may be found in reference [1]. For vehicles that are oversteered it is very important to evaluate the critical speed (after which the vehicles become uncontrollable) deŽ ned as v

crit

=

S

­

L K

(16)

Finally, the yaw rate will be V v = d L +RKa y 2

(17)

The parameters used in this study are presented in Table 1. These values come from commercial speciŽ cations from a standard European truck with two axles. Of interest is the way in which the elasticity in uences the dynamic response for the truck. For this purpose, the impact on the understeer coe cient, K, and on the critical speed, v , are analysed. Inserting the values for crit the truck in the equations, a critical speed that is approximately 3 per cent lower (see Table 2) is obtained using the elastic theory compared with the ordinary bicycle model. In Table 3 the results of the rotation at the points for both models are speciŽ ed. From the results it can be concluded that the results are very similar. The conclusion is that the elastic beam yields satisfactory results. However, it is important to stress the need to have a realistic value for the moment of inertia for the frame, otherwise the results can be very misleading. Figure 3 shows the e€ ect of the sti€ ness of the frame in a more graphical manner. In this example the vehicle parameters from Table 1 have been used, and the understeer coe cient K is plotted versus the moment of inertia for the frame. Also, the critical speed v has been plotcrit ted in the same Ž gure. In the Ž gure it can be observed that both K and v are approaching values equal to the crit ‘standard bicycle model’ when the moment of inertia I z is high. Lower sti€ ness leads to lower (more negative) K values and subsequently lower critical speed. Table 1

Proc Instn Mech Engrs Vol 216 Part D: J Automobile Engineering

Vehicle parameters

Mass Total mass Geometry Distance from centre of gravity to front axle, b Distance from centre of gravity to rear axle, c Width of frame Total length of vehicle Cornering sti€ ness Vehicle front axle Vehicle rear axle Moment of inertia Bending moment of inertia

Table 2

FINITE ELEMENT MODEL

The frame of a real commercial truck has been modelled using the Ž nite element technique and is depicted in Fig. 2. The model corresponds to a smaller standard truck. This model has mainly been used to calculate the moment of inertia I , of the truck frame. The model is y made using a commercial FEA tool. It is made up of quadrilateral shell elements with four nodes in each element. There are a total of 4572 elements and 4172 nodes with six DOF each. The model will also be used to validate the in uence of the elastic frame, i.e. angles h and h . f r

RESULTS AND COMPARISON

K (kg rad/ N ) v (km/h) crit

1.7 m 2.55 m 0.849 m 7.59 m 80 kN/rad 130 kN/rad 2.6×107 mm4

Result summary, comparison critical speed Standard bicycle model

Elastic bicycle model

Di€ erence (%)

0.00288 138 ­

0.003077 133 ­

­

Table 3

h f h r

7490 kg

6.6 3.6

Result summary, elastic angles

Elastic bicycle model

FEA

Di€ erence (%)

0.01053 0.00921

0.00961 0.00894

8.8 2.8 D00302 © IMechE 2002

DYNAMIC EFFECT OF THE NON-RIGID MODIFIED BICYCLE MODEL

Fig. 2

Fig. 3

4

Finite element model

E€ ect of elasticity on the critical speed

SUMMARY

In this paper it has been shown that the modiŽ ed bicycle model is a useful tool to predict the steering of a vehicle. Nowadays, much e€ ort is made with complicated rigid body simulation programs such as ADAMS (see, for D00302 © IMechE 2002

721

instance, reference [13]), although the method proposed here o€ ers an easy way to evaluate the dynamic e€ ects resulting from handling in the early stages of the design process. It is worth mentioning that the elasticity has an important impact on the steering response of a vehicle, especially in heavy industrial vehicles such as trucks. Proc Instn Mech Engrs Vol 216 Part D: J Automobile Engineering

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This method is aimed principally at the engineer modifying second-hand trucks. Supposedly, the truck manufacturer has the tools and the experience not to design weak frames that would give the e€ ects discussed. Nonetheless, on the second-hand market it is very common for trucks and their structure to be modiŽ ed. Changing the bending sti€ ness of the frame (by removing one of the traverse beams, for instance) could then have considerable e€ ects on directional stability and handling. Also, as pointed out earlier, this method is only useful for two-axle trucks which are to a certain extent weak in the lateral direction. Extending this methodology to vehicles with more than two axles is straightforward if the approach described in this article is followed.

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6

7

8 9

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REFERENCES 1 Gillespie, T. D. Fundamentals of Vehicle Dynamics, 1992 (Society of Automotive Engineers, Warrendale, Pennsylvania). 2 Ellis, J. R. Vehicle Handling Dynamics, 1994 (Mechanical Engineering Publications Limited ). 3 Milliken, W. F. and Whitcomb, D. W. General introduction to a programme of dynamic research. Proc. Auto. Div. Instn Mech. Engrs, 1956, (7), 287–309. 4 Whitcomb, D. W. and Milliken, W. F. Design implications

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of a general theory of automobile stability and control. Proc. Auto. Div. Instn Mech. Engrs, 1956, (7), 367–391. Segel, L. Research in the fundamentals of automobile control and stability. SAE National Summer Meeting, Atlantic City, 5 June 1956. Segel, L. Theoretical prediction and experimental substantiation of the response of the automobile to steering control. Proc. Auto. Div. Instn Mech. Engrs, 1956, (7), 310–330. Bundorf, R. T. The in uence of vehicle design parameters on characteristic speed and understeer. SAE paper 670078, 1967. Vehicle dynamics terminology. SAE J670a, 1965; an updated version is available (SAE J670e). Vagstedt, N.-G. and Dahlberg, E. Heavy Vehicle and Highway Dynamics, SAE Special Publications, November 1997, Vol. 1308, pp. 1–8 (Society of Automotive Engineers, Warrendale, Pennsylvania). Sobottka, C. and Singh, T. Optimal fuzzy logic control for an antilock braking system. In Proceedings of IEEE International Conference on Control Applications, Dearborn, Michigan, 1996, pp. 49–54. Spentzas, K. N., Alkhazali, I. and Dernic, M. Kinematics of four wheel steering vehicles. Forschung im Ingenieurwesen, May 2001, 66(5), 211–216. Roark, R. J. and Young, W. C. Roark’s Formulas for Stress and Strain, 6th edition, 1989 (McGraw-Hill). Antoun, R. J., Hackert, P. B., O’Leary, M. C. and Sitchin, A. Vehicle dynamic handling computer simulation—model development, correlation, and application using ADAMS. SAE technical paper 860574, 1986.

D00302 © IMechE 2002