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The quantification of accident risk for road vehicles in cross winds
dOt~RI¢. OF
ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 52 (1994) 93-107
The quantification of accident risk for road vehicles in cross winds C.J. B a k e r Department of Civil Engineering, Nottingham University, University Park, Nottingham N67 2RD, UK
Abstract This paper presents a method for quantifying accident risk for vehicles in high winds, a physical problem for which there are a large number of independent variables. In particular the parameters that specify driver behaviour are almost impossible to quantify with any precision. Also there are several interrelated failure modes for the driver/vehicle system. The method that is described is essentially based on an extensive series of computations for different pairs of these driver parameters, that enables the areas of safety within the driver parameter plane to be determined. The variation of this area of safety with both vehicle speed and wind speed is investigated, using both time domain and frequency domain models. Suggestions are made as to how this method might be used to determine accident risk for other problems within the field of wind engineering.
1. Introduction In recent years the author has developed a model to investigate the behaviour of ground vehicles in cross winds [1-6]. The model, entitled B L O W O V E R , can operate in two distinct modes - without or with a model of driver behaviour. The former mode enables accident wind speeds to be determined by looking at vehicle behaviour within a time T of vehicle entry into a sharp edged step gust of wind velocity. If one of the vehicle wheel reactions falls to zero, the lateral deviation Y exceeds 0.5 m, or the rotational deviation ¢i exceeds 0.5 rads an accident is said to occur. It is thus possible to build up a graph of accident wind speed ua, for different vehicle speeds, v, and different wind directions ft. By combining this data with site meteorological conditions it is possible to find the percentage of time for which this wind speed will be exceeded, and some quantitative measure of accident risk is obtained. However, this is still subject to the basic assumption that driver behaviour is not considered. N o w the second mode of operation of B L O W O V E R allows for driver behaviour through a two parameter driver model, and, for fixed values of these parameters, produces time 0167-6105/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0 1 6 7 - 6 1 0 5 ( 9 3 ) E 0 0 5 9 - 8
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('.J. Baker,'J.
WhTd Eng. bid. Aeroelvn. 52 ( 1 9 9 4 j 93
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histories of Y and ~b in arbitrary wind fields. In this mode of operation, which is probably a better model of reality than the first mode, it is by no means so straightforward to calculate accident risk because of the inherent difficulty in specifying the two (driver dependent) parameters in the driver model. This paper, thus, sets out a method of assessing and quantifying accident risk for this mode of usage of BLOWOVER. It also applies this method to the recently developed variant of the model [7] which, instead of analysing course deviations in the time domain, investigates the behaviour in the frequency domain. In what follows, the B L O W O V E R model is outlined in Section 2, and the different types of solution procedure set out (with and without driver behaviour, and in both the time and frequency domains). The basic method of analysis and risk assessment is then set out in Section 3. Section 4 then looks at the time domain behaviour of lorries in cross winds and the results are presented in the manner suggested in Section 3. Similarly, Section 5 describes the frequency domain behaviour of such vehicles in cross winds and presents the results as in Section 3. The method that is used is quite capable of being generalised and used in other areas of wind engineering, and this is discussed, along with some other points of interest, in Section 6. Finally, conclusions are drawn in Section 7.
2. The BLOWOVER model 2.1. Outline o f the model
The basic model considers the mechanical and aerodynamic behaviour of a single mass vehicle system under the action of its own weight, tyre forces and aerodynamic forces and moments. The six equations of motion are solved, together with a compatibility condition for tyre deflections. These result in the following relatively simple equations: CV1 =
(Ka + K2 + g3) + ?(K4 + K s ) + 6(K5),
CV2 = ( - - K I + K2 + K3) + ?( Cv3 = (K1 - K2 +
K6)
+ ?(
--
Cv4 = ( - K~ - K2 + g6) + 7(
--
K , - Ks) + 6( - Ks),
K4 + Ks) + 6(K5), --
K4 - Ks) + 6( - Ks),
(1) (2) (3) (4)
dgy/i = o~1 --]- ~217y + ~3 ~,
(5)
d 2 ~ / d t "2 -- ct4 + o~5fy + o~6t~,
(6)
d Y / d t = Vy - v sinq~ - vfr/(r + s),
(7)
d ~ / d t = d~a/dt - 6r/(r + s),
(8)
where i-= tV/q, gy = vr/V, C w = V i / p A V 1 and ? - - v r / v . Here, Y, is the lateral displacement and • is the rotational displacement, vy and ~b are, respectively, the wind
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-107
95
induced lateral velocity and rotational displacement, 6 is the steering angle, Vi is the reaction of wheel i (i = 1 : front windward wheel, i = 2 : front leeward wheel, i = 3 : rear windward wheel, i = 4: rear leeward wheel), v is the vehicle velocity, V is the wind velocity relative to the vehicle, r is the distance from the vehicle centre of gravity to the rear axle, s is the distance from that point to the front axle, q is the vehicle centre of gravity height, A is a reference area, and p is the density of air. ~1 to ~6 and K1 to K6 are lengthy but simple functions of the vehicle geometry and aerodynamic characteristics [5]. Driver behaviour is modelled using the equation: 6 = 21(t -- e ) Y + )~2(t - e)dY/dt,
(9)
where e is the driver reaction time (taken in what follows as 0.2 s) and 21 and ~'2 are driver dependent parameters (note 6 = 0 for t < e). Eqs. (1) to (9) form the basic equation set used in the B L O W O V E R model. Through a time marching solution the values of Y, q' and Cvi can be found as a function of time and thus, through the use of suitable accident criteria, accident wind speeds can be determined. An alternative approach to the solution of these equations has been outlined in Ref. [7]. Through a manipulation of the equations one obtains the following differential equation that relates small variations in the lateral displacement yX to small variations in vehicle side force S 1. (a + ib)YXe -'~v/q = (c + id) (qZ/Mv2)Sl.
(10)
Here M is the vehicle mass, co is a frequency and a, b, c and d are again lengthy functions of the vehicle geometry, aerodynamic characteristics and 09 and are given in Ref. I-7]. The assumptions leading to Eq. (10) implicitly assume that vehicle overturning (i.e. Cvi ~ 0) is not a possibility, and thus apply more to cars and small vans than to the lorries that will be considered later. Eqs. (1) to (9), however, are completely general, within the limitations of the basic assumptions.
2.2. Methods of solution 2.2.1. Solutions without driver simulation If driver behaviour is not considered, 6 in Eqs. (1) to (9) is taken as zero. The equations then become relatively simple and equations can be obtained for lateral and rotational course deviation, and wheel reactions, that can be solved fairly simply as a function of time. Calculations of this sort are reported in Ref. [3]. An accident is then said to occur if, within T seconds of a vehicle entering a sharp edged gust normal to the direction of travel: (a) one wheel reaction falls to zero (overturning accident), (b) the lateral deviation exceeds 0.5 m (sideslip accident), (c) the rotational deviation exceeds 0.1 rads (rotational accident). The wind speed at which these accident criteria are exceeded (the accident wind speed Ua) can thus be found, as a function of vehicle speed and wind direction.
96
C.J. Baker/J. Wind Eng. lnd. Aeroctvn. 52 (1994) 93 107
Through the use of meteorological information the percentage of the total time for which this wind speed is exceeded can thus be found, and some quantification made of accident risk. The disadvantage of this method is that the accident wind speeds thus calculated are fairly sensitive to the time period T. This is shown in Fig. 1 for the articulated lorry with the geometric and aerodynamic characteristics outlined in Ref. [3] and given in Table 1. In that paper it was argued that a time period of 0.5 s was appropriate. However, at the higher vehicle speeds Ua can be seen to be fairly sensitive to the value of T. Clearly it would be more rational to use some model of driver behaviour that would eliminate the need for this arbitrary time period T. This is considered in the following subsections.
2.2.2. Time domain solutions with driver simulation Applying B L O W O V E R to obtain time domain solutions with driver simulation is essentially a time marching solution of Eqs. (1) to (9). A block diagram of the solution method is given in Fig. 2a. For a particular wind field the aerodynamic forces and moments are calculated from measured wind tunnel values and appropriate aerodynamic weighting functions [5]. The lateral and rotational time histories are then calculated from Eqs. (1) to (9). The usual mode of application is to consider vehicle entry into a sharp edged step gust. In this case the application of aerodynamic weighting functions has very little effect, except on the initial transients after gust entry. The same accident criteria can then be applied as in Section 2.2.1., within, say, 100 m of vehicle entry into the gust.
2.2.3. Frequency domain solutions with driver simulation The application of B L O W O V E R in the frequency domain is described in detail in Ref. [7]. A block diagram of the procedure is given in Fig. 2b. Essentially the spectrum of the oncoming wind Su(n) is multiplied by: (a) aerodynamic admittance functions X2(n) [5] to give force and moment spectra, (b) vehicle transfer functions TZ(n) obtained from Eq. (10) to give the spectrum of the lateral displacement St(n). These spectra are then integrated to obtain the standard deviation of lateral displacement (at). As this procedure is based on the use of Eq. (10) and, thus, ignores the possibility of vehicle overturning, it is thus most likely to be of use for studying the sensitivity of cars and vans to cross wind conditions rather than lorries. From these values of at, "maximum" displacements Y can be obtained from:
Y = Y + gar/Y,
(11)
where - indicates mean values, g is a gust factor that can either be arbitrarily defined (say, as 3.0) or calculated from a rigorous extreme value analysis. The same accident criteria can then be applied to these values of Y as in 2.2.1. Also a further accident criterion can be defined based on the frequency content of these lateral oscillations, since rapidly varying course deviations are to be avoided. The criterion that is somewhat arbitrarily chosen is that if the maximum of the lateral deviation power spectrum occurs at frequencies greater
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-107
97
than 0.2 Hz an accident is said to occur. This will be discussed further in what follows. 3. Outline of risk assessment method
In the use of either of the methods outlined in 2.2.2. or 2.2.3., the driver dependent parameters, which are not well specified, effectively determine the nature of the
35
IO 0 Q Q. m "f0
@ "0
30 m----
T = 0.1s
8
T = 0.5s
25
20
--
T = 1.0s
•
T = 1.5s
--
T = lOs
"6 0
1
5 0
~ 10
20
30
v(m/s)
Fig. 1. Variation of accident wind speed with vehicle speed and different accident times (driver behaviour not modelled).
Table 1 Vehicle geometric and aerodynamic characteristics Mass Rotational inertia C G height C G to front axle C G to rear axle Semi-axle length Frontal area Longitudinal friction coefficient Lateral friction coefficient
12000 kg 250000 kg/m 2 1.5 m 2.4 m 5.8 m 1.1 m 11.5 m - 0.075 2.5
Side force coefficient Lift force coefficient Drag force coefficient Rolling m o m e n t coefficient Pitching m o m e n t coetficient Yawing m o m e n t coefficient
5.2 sin ~O 1.1(1 - cos 4~k) - 0.5(1 + sin 3~0) 4.4 sin ~b - 2(1 - cos 2#) 6 sin 2 ~O
is the yaw angle, the angle between the direction of travel and the wind direction relative to the vehicle.
98
C.J. Baker/J. Wind Eng. Ind. Aeroc/vn. 52 /1994) 93 107
(a) Aerodynamic forces and moments
Course and rotational deviation calculations
Vehicle geometry
Accident criteria
Site wind characteristics (b)
Wind spectrum
I
Aerodynamic admittance
I
Force spectrum
I
Vehicle transfer function
I
Lateral displacement spectrum
J
Extreme value of displacement
, Maximum spectral frequency ]
Fig. 2. Outline of BLOWOVER calculation methods: (a) time domain, and (b) frequency domain.
solution. To cope with these ill defined parameters, the following procedures are adopted in the paper. B L O W O V E R is run, for a particular vehicle speed and set of wind characteristics, for a grid of values of 21 and 22 in either the time or the frequency domain, and for each set of values of these parameters, whether or not an accident is predicted is noted, together with the accident type if applicable. Typically a 50 x 50 grid of points is used in the ,~t-22 plane, with a run time of around 4 h on a 386 AT
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-I07
99
{a) F1 ~2
S
F2
Ib) A
Fig. 3. Regions of safety (a) and area of safety (b) in parameter space (S = region of safety, F1 and F2 = areas of different failure types).
microcomputer. Thus a picture of the region of "safety" and the regions of different accident types can be built up in the 21-22 plane as shown in Fig. 3a. The "area of safety" A is a quantitative measure of the safety of the vehicle driver system. By varying, say, mean windspeed (u) and keeping the vehicle speed constant, the variation of A with u can be found, and a value of u, determined for which A falls to zero (Fig. 3b). This procedure will be illustrated in what follows.
4. Time domain calculations for lorries in cross winds
In this section, the method of Section 3 will be applied to time domain calculations (Fig. 2a) of a high sided lorry entering a sharp edged wind gust at 90 ° to its direction of travel. The assumed geometric and aerodynamic characteristics of the lorry are given in Table 1. A typical result for lateral and rotational course deviations is shown in Fig. 4, for a vehicle speed v of 15 m/s and a wind speed u of 15 m/s and driver parameters 21 and 22 equal to 0.3 m - 1 and 0.01 s m - 1, respectively. It can be seen that this corresponds to a "safe" situation with the displacements tending to acceptable, small equilibrium values.
100
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93 107 (a)
0.1
0.0
-0.1
-0.2
!
0
10
i
|
20
30
x
(b)
i
40
!
50
i
6O
(m)
0.12 0.10 -
E
O.08 -
0.06 0.04 II 0
0.02 -
0
0.00 4 0
i
i
10
20
I
30
x
i
40
I
50
i
6O
(m)
Fig. 4. Typical time displacement from B L O W O V E R model (for conditions see text): (a) lateral displacement, and (b) rotational displacement.
Figs. 5a and 5b show computer generated 2t-22 plots for v = 15 m/s and 25 m/s with various values of u. The area of the region of safety A shrinks to zero as the wind speed increases as would be expected. The complexity of the regions of different accident types can be seen particularly at the higher value of v. At low values of 21 there is a region of high course deviations (i.e. driver reaction is not "strong" enough) whilst at high values of 21 and 2z overturning occurs i.e. driver reaction is too "strong". Fig. 6 is a plot of the variation of A with u for different values of v. It can be seen, that A varies little with u until a point where a sudden fall occurs, and then A drops rapidly to zero at the accident wind speed ua. This implies that there will be relatively little increase in difficulty of driving until wind speeds quite close to the critical. The variation of this critical wind speed with vehicle speed v is shown in Fig. 7.
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93- 107
101
la)
Fig. 5. Areas of safety and failure in the 21-22 plane for time domain calculations: (a) v = 15 m/s, u = 10 (i), 15 (ii), 16 (iii), 16.5 (iv), 17 (v) m/s; (b) v = 25 m/s, u = 10 (i), 12.5 (ii), 13 (iii) m/s, (x-axis is 0 < 2~ < 0.5 and y-axis is 0 < 22 < 0.1, white regions are regions of safety, blue regions are regions of course deviation or rotation accidents and red regions are regions of overturning accidents).
F r o m results such as these the m e t h o d d e s c r i b e d in this p a p e r c a n be seen to b e a useful a n d i n s t r u c t i v e w a y in p r e s e n t i n g risk a s s e s s m e n t c a l c u l a t i o n s in the t i m e d o m a i n . Its use in the f r e q u e n c y d o m a i n is i l l u s t r a t e d in the n e x t section.
5. Frequency domain calculations for lorries in cross winds N o t w i t h s t a n d i n g w h a t has a l r e a d y b e e n said a b o u t the f r e q u e n c y d o m a i n m e t h o d b e i n g m o r e s u i t e d to cars a n d s m a l l vans, it is a p p l i e d here to the a r t i c u l a t e d l o r r y case
102
C.J. Baker/J. Wind Eng. Ind. Aeroc,lvn. 52 (1994) 93 107
(b)
Fig. 5. (continued).
outlined in Table 1 for two reasons - firstly for comparison with the results of the last section, and secondly because there is insufficient data available for the high cross wind aerodynamic characteristics of vans and small cars to be adequately specified. Again calculations were carried out in the driver parameter 21-22 plane. The accident criteria were adopted as set out in Section 2.2.3. with g, the gust factor, taken as 3.0. Ideally this value could be determined for any particular site by an extreme value analysis as set out in Ref. [5] with an observation period related to the site length, the vehicle speed and the vehicle occupancy of that site. However, such a procedure was felt to be unnecessarily complicated for the present work, and a simple, constant value of g was adopted. Also as noted in Section 2.2.3. if the m a x i m u m of the lateral displacement spectrum was at a frequency greater than 0.2 Hz
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93 107
103
v = 5m/s &
v = 15m/s v = 25m/s
c
E
M
0 m
.01
0
.001
i
!
i
10
12
Accident
14
wind
1
16
speed
2O
(m/., m/s)
Fig. 6. Variations of area of safety with accident wind speed.
tE
-o
15
o o
•"o
10-
e-
3: =
5
o "O
== 0
i
i
i
10
20
30
Vehicle
speed
(m/s)
Fig. 7. Variation of accident wind speeds with vehicle speed.
1.21.0 'a
;;:.;;;;;:.
-,~,~
(' Wind spectrum
0.8
0.6
s
Aerodynamic admittance
=
Displacement spectrum
s
Normalised transfer functk
0.4 0.2 0.0
......
.001
.01
.1
1
l
10
n(hz)
Fig.' 8. Typical values of wind spectrtrm;, aerodynamic admittance, vehicle transfer function (normalised on maximum value) and lateral displacement spectrum (for conditions see text).
104
C.J. Baker/J. Wind Eng. Ind. Aerodvn. 52 (1994) 93 107
Ill
IN
m
D
Fig. 9. Areas of safety and failure in the )-1-22 plane for frequency domain calculations (v = 10 m/s, u = 5 (i), 10 (ii), 15 (iii),20 (vi) and 23 (v) m/s (x-axis is 0 < 2t < 0.5 and y-axis is 0 < 22 < 0.1, white regions are regions of safety, blue regions are regions of excessive displacement, red regions are regions of high frequency oscillations and green regions are regions of both excessive lateral displacement and high frequency oscillations).
an accident was said to occur. Such a criterion attempts to model the fact that if there is too m u c h displacement at high frequencies, this will be difficult for the driver of the vehicle to manage. Fig. 8 shows typical values for the wind spectrum, a e r o d y n a m i c admittance, vehicle transfer function a n d lateral displacement spectrum for u = v = 10 m/s, 21 --- 0.3 m - 1 a n d 22 = 0.01 s m - 1. Fig. 9 shows c o m p u t e r generated plots of the driver p a r a m e t e r plane for a vehicle speed of 10 m/s. It can be seen that the area of safety is rather different in shape to that shown in Fig. 5, in that there is n o u p p e r limit for a "safe"
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-107
t~
I
0
t...d
e~
e~
.=. O
105
106
('.J. Baker~J, Wind Eng. Ind. Aerodvn. 52 (1994) 93
107
value of 22 . This is because for high values of this parameter the vehicle would overturn, an aspect of the physical situation that is not modelled by this variant of BLOWOVER. It can, however, be seen that, as expected the area of safety shrinks as wind speed increases, as the region of high course deviations at low )-1 increases in area to meet the area of high frequency content at high values of 21. Because of this lack of an upper bound to the area of safety, no graph of the variation of area of safety with wind speed has been drawn.
6. Application of this method in other areas of wind engineering The method described in previous sections can be applied in a straightforward manner in other areas of wind engineering where the physical situation is such that there are parameters involved that are difficult to specify with precision and where there are a number of system failure criteria. The basic method is simply to carry out a large number of calculations for individual values of the uncontrolled parameters, and build up a map of regions of safety in parameter space. The area, and the connectivity (i.e., the existence or otherwise of safe islands as in Fig. 3) of these regions is thus a measure of safety. The variation in these quantities can then be investigated as, say, mean wind speed varies. Whilst considerable computational effort may be needed for such calculations they are usually programmable on microcomputers that can be dedicated to the task for the many hours of computer time required. Table 2 gives details of some possible applications, drawn mainly from the authors own research work, and there are doubtless others that could be added. Also it should be noted that the basis of this method has been suggested by Thompson and Soliman [8] who investigated the capsize behaviour of ships.
7. Concluding remarks A method has been presented which allows some appreciation to be obtained of accident risk in situations where there are physical parameters that are difficult to specify, which is very often the case in the field of wind engineering. This method has been applied to the case of road vehicles in cross winds where the variation of the "area of safety" in the driver parameter plane has been investigated using both frequency and time domain calculations, and a number of other possible uses of the method have been suggested.
References [1] C.J. Baker, A simplified analysis of various types of wind induced road vehicle accidents, J. Wind Eng. Ind. Aerodyn., 22 (1986) 69-85. [2] C.J. Baker, Measures to control vehicle movement at exposed sites during windy periods, J. Wind Eng. Ind. Aerodyn., 25 (1986) 151 167.
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93- 107
107
[3] C.J. Baker, High sided articulated lorries in strong cross winds, J. Wind Eng. Ind. Aerodyn., 31 (1988) 67-85. [4] C.J. Baker, Ground vehicles in high cross winds, Part 1, Steady aerodynamic forces, J. Fluids Struct., 5 (1991) 69-90. I5] C.J. Baker, Ground vehicles in high cross winds, Part 2, Unsteady aerodynamic forces, J. Fluids. Struct., 5 (1991) 91-111. I-6] C.J. Baker, Ground vehicles in high cross winds, Part 3, Interaction of aerodynamic forces and the vehicle system, J. Fluids Struct., 5 (1991) 221-241. [-7] C.J. Baker, The behaviour of road vehicles in unsteady cross winds, Bluff Body Aerodynamics and its Applications 2, Melbourne, Australia, 1992. [8] J. Thompson and M. Soliman, Fractal control boundaries of driven oscillators and their relevance to safe engineering design, Proc. Royal Soc. London A, 428 (1990) 1 13. 1'9] BSI, BS8100 Lattice towers and masts code of practice for loading, British Standard, 1986. [10] C.D. Williams, M.J. Soligo and J. Cote, A discussion of the components for a comprehensive pedestrian level comfort criterion, Proc. 8th Int. Conf. on Wind Engineering, London, Canada, 1991. 1'11] X. Yang, C.J. Baker, R. Hoxey, P. Kettlewell and A. Meehan, The aerodynamics and ventilation of poultry transport vehicles, Wind Eng. Soc. Conf., Cambridge, 1992. [12] H.J. Roodbaraky, C.J. Baker, A.R. Dawson and C.J. Wright, Experimental observations of the aerodynamic characteristics of urban trees, Wind Eng. Soc. Conf., Cambridge, 1992.
ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 52 (1994) 93-107
The quantification of accident risk for road vehicles in cross winds C.J. B a k e r Department of Civil Engineering, Nottingham University, University Park, Nottingham N67 2RD, UK
Abstract This paper presents a method for quantifying accident risk for vehicles in high winds, a physical problem for which there are a large number of independent variables. In particular the parameters that specify driver behaviour are almost impossible to quantify with any precision. Also there are several interrelated failure modes for the driver/vehicle system. The method that is described is essentially based on an extensive series of computations for different pairs of these driver parameters, that enables the areas of safety within the driver parameter plane to be determined. The variation of this area of safety with both vehicle speed and wind speed is investigated, using both time domain and frequency domain models. Suggestions are made as to how this method might be used to determine accident risk for other problems within the field of wind engineering.
1. Introduction In recent years the author has developed a model to investigate the behaviour of ground vehicles in cross winds [1-6]. The model, entitled B L O W O V E R , can operate in two distinct modes - without or with a model of driver behaviour. The former mode enables accident wind speeds to be determined by looking at vehicle behaviour within a time T of vehicle entry into a sharp edged step gust of wind velocity. If one of the vehicle wheel reactions falls to zero, the lateral deviation Y exceeds 0.5 m, or the rotational deviation ¢i exceeds 0.5 rads an accident is said to occur. It is thus possible to build up a graph of accident wind speed ua, for different vehicle speeds, v, and different wind directions ft. By combining this data with site meteorological conditions it is possible to find the percentage of time for which this wind speed will be exceeded, and some quantitative measure of accident risk is obtained. However, this is still subject to the basic assumption that driver behaviour is not considered. N o w the second mode of operation of B L O W O V E R allows for driver behaviour through a two parameter driver model, and, for fixed values of these parameters, produces time 0167-6105/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0 1 6 7 - 6 1 0 5 ( 9 3 ) E 0 0 5 9 - 8
94
('.J. Baker,'J.
WhTd Eng. bid. Aeroelvn. 52 ( 1 9 9 4 j 93
107
histories of Y and ~b in arbitrary wind fields. In this mode of operation, which is probably a better model of reality than the first mode, it is by no means so straightforward to calculate accident risk because of the inherent difficulty in specifying the two (driver dependent) parameters in the driver model. This paper, thus, sets out a method of assessing and quantifying accident risk for this mode of usage of BLOWOVER. It also applies this method to the recently developed variant of the model [7] which, instead of analysing course deviations in the time domain, investigates the behaviour in the frequency domain. In what follows, the B L O W O V E R model is outlined in Section 2, and the different types of solution procedure set out (with and without driver behaviour, and in both the time and frequency domains). The basic method of analysis and risk assessment is then set out in Section 3. Section 4 then looks at the time domain behaviour of lorries in cross winds and the results are presented in the manner suggested in Section 3. Similarly, Section 5 describes the frequency domain behaviour of such vehicles in cross winds and presents the results as in Section 3. The method that is used is quite capable of being generalised and used in other areas of wind engineering, and this is discussed, along with some other points of interest, in Section 6. Finally, conclusions are drawn in Section 7.
2. The BLOWOVER model 2.1. Outline o f the model
The basic model considers the mechanical and aerodynamic behaviour of a single mass vehicle system under the action of its own weight, tyre forces and aerodynamic forces and moments. The six equations of motion are solved, together with a compatibility condition for tyre deflections. These result in the following relatively simple equations: CV1 =
(Ka + K2 + g3) + ?(K4 + K s ) + 6(K5),
CV2 = ( - - K I + K2 + K3) + ?( Cv3 = (K1 - K2 +
K6)
+ ?(
--
Cv4 = ( - K~ - K2 + g6) + 7(
--
K , - Ks) + 6( - Ks),
K4 + Ks) + 6(K5), --
K4 - Ks) + 6( - Ks),
(1) (2) (3) (4)
dgy/i = o~1 --]- ~217y + ~3 ~,
(5)
d 2 ~ / d t "2 -- ct4 + o~5fy + o~6t~,
(6)
d Y / d t = Vy - v sinq~ - vfr/(r + s),
(7)
d ~ / d t = d~a/dt - 6r/(r + s),
(8)
where i-= tV/q, gy = vr/V, C w = V i / p A V 1 and ? - - v r / v . Here, Y, is the lateral displacement and • is the rotational displacement, vy and ~b are, respectively, the wind
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-107
95
induced lateral velocity and rotational displacement, 6 is the steering angle, Vi is the reaction of wheel i (i = 1 : front windward wheel, i = 2 : front leeward wheel, i = 3 : rear windward wheel, i = 4: rear leeward wheel), v is the vehicle velocity, V is the wind velocity relative to the vehicle, r is the distance from the vehicle centre of gravity to the rear axle, s is the distance from that point to the front axle, q is the vehicle centre of gravity height, A is a reference area, and p is the density of air. ~1 to ~6 and K1 to K6 are lengthy but simple functions of the vehicle geometry and aerodynamic characteristics [5]. Driver behaviour is modelled using the equation: 6 = 21(t -- e ) Y + )~2(t - e)dY/dt,
(9)
where e is the driver reaction time (taken in what follows as 0.2 s) and 21 and ~'2 are driver dependent parameters (note 6 = 0 for t < e). Eqs. (1) to (9) form the basic equation set used in the B L O W O V E R model. Through a time marching solution the values of Y, q' and Cvi can be found as a function of time and thus, through the use of suitable accident criteria, accident wind speeds can be determined. An alternative approach to the solution of these equations has been outlined in Ref. [7]. Through a manipulation of the equations one obtains the following differential equation that relates small variations in the lateral displacement yX to small variations in vehicle side force S 1. (a + ib)YXe -'~v/q = (c + id) (qZ/Mv2)Sl.
(10)
Here M is the vehicle mass, co is a frequency and a, b, c and d are again lengthy functions of the vehicle geometry, aerodynamic characteristics and 09 and are given in Ref. I-7]. The assumptions leading to Eq. (10) implicitly assume that vehicle overturning (i.e. Cvi ~ 0) is not a possibility, and thus apply more to cars and small vans than to the lorries that will be considered later. Eqs. (1) to (9), however, are completely general, within the limitations of the basic assumptions.
2.2. Methods of solution 2.2.1. Solutions without driver simulation If driver behaviour is not considered, 6 in Eqs. (1) to (9) is taken as zero. The equations then become relatively simple and equations can be obtained for lateral and rotational course deviation, and wheel reactions, that can be solved fairly simply as a function of time. Calculations of this sort are reported in Ref. [3]. An accident is then said to occur if, within T seconds of a vehicle entering a sharp edged gust normal to the direction of travel: (a) one wheel reaction falls to zero (overturning accident), (b) the lateral deviation exceeds 0.5 m (sideslip accident), (c) the rotational deviation exceeds 0.1 rads (rotational accident). The wind speed at which these accident criteria are exceeded (the accident wind speed Ua) can thus be found, as a function of vehicle speed and wind direction.
96
C.J. Baker/J. Wind Eng. lnd. Aeroctvn. 52 (1994) 93 107
Through the use of meteorological information the percentage of the total time for which this wind speed is exceeded can thus be found, and some quantification made of accident risk. The disadvantage of this method is that the accident wind speeds thus calculated are fairly sensitive to the time period T. This is shown in Fig. 1 for the articulated lorry with the geometric and aerodynamic characteristics outlined in Ref. [3] and given in Table 1. In that paper it was argued that a time period of 0.5 s was appropriate. However, at the higher vehicle speeds Ua can be seen to be fairly sensitive to the value of T. Clearly it would be more rational to use some model of driver behaviour that would eliminate the need for this arbitrary time period T. This is considered in the following subsections.
2.2.2. Time domain solutions with driver simulation Applying B L O W O V E R to obtain time domain solutions with driver simulation is essentially a time marching solution of Eqs. (1) to (9). A block diagram of the solution method is given in Fig. 2a. For a particular wind field the aerodynamic forces and moments are calculated from measured wind tunnel values and appropriate aerodynamic weighting functions [5]. The lateral and rotational time histories are then calculated from Eqs. (1) to (9). The usual mode of application is to consider vehicle entry into a sharp edged step gust. In this case the application of aerodynamic weighting functions has very little effect, except on the initial transients after gust entry. The same accident criteria can then be applied as in Section 2.2.1., within, say, 100 m of vehicle entry into the gust.
2.2.3. Frequency domain solutions with driver simulation The application of B L O W O V E R in the frequency domain is described in detail in Ref. [7]. A block diagram of the procedure is given in Fig. 2b. Essentially the spectrum of the oncoming wind Su(n) is multiplied by: (a) aerodynamic admittance functions X2(n) [5] to give force and moment spectra, (b) vehicle transfer functions TZ(n) obtained from Eq. (10) to give the spectrum of the lateral displacement St(n). These spectra are then integrated to obtain the standard deviation of lateral displacement (at). As this procedure is based on the use of Eq. (10) and, thus, ignores the possibility of vehicle overturning, it is thus most likely to be of use for studying the sensitivity of cars and vans to cross wind conditions rather than lorries. From these values of at, "maximum" displacements Y can be obtained from:
Y = Y + gar/Y,
(11)
where - indicates mean values, g is a gust factor that can either be arbitrarily defined (say, as 3.0) or calculated from a rigorous extreme value analysis. The same accident criteria can then be applied to these values of Y as in 2.2.1. Also a further accident criterion can be defined based on the frequency content of these lateral oscillations, since rapidly varying course deviations are to be avoided. The criterion that is somewhat arbitrarily chosen is that if the maximum of the lateral deviation power spectrum occurs at frequencies greater
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-107
97
than 0.2 Hz an accident is said to occur. This will be discussed further in what follows. 3. Outline of risk assessment method
In the use of either of the methods outlined in 2.2.2. or 2.2.3., the driver dependent parameters, which are not well specified, effectively determine the nature of the
35
IO 0 Q Q. m "f0
@ "0
30 m----
T = 0.1s
8
T = 0.5s
25
20
--
T = 1.0s
•
T = 1.5s
--
T = lOs
"6 0
1
5 0
~ 10
20
30
v(m/s)
Fig. 1. Variation of accident wind speed with vehicle speed and different accident times (driver behaviour not modelled).
Table 1 Vehicle geometric and aerodynamic characteristics Mass Rotational inertia C G height C G to front axle C G to rear axle Semi-axle length Frontal area Longitudinal friction coefficient Lateral friction coefficient
12000 kg 250000 kg/m 2 1.5 m 2.4 m 5.8 m 1.1 m 11.5 m - 0.075 2.5
Side force coefficient Lift force coefficient Drag force coefficient Rolling m o m e n t coefficient Pitching m o m e n t coetficient Yawing m o m e n t coefficient
5.2 sin ~O 1.1(1 - cos 4~k) - 0.5(1 + sin 3~0) 4.4 sin ~b - 2(1 - cos 2#) 6 sin 2 ~O
is the yaw angle, the angle between the direction of travel and the wind direction relative to the vehicle.
98
C.J. Baker/J. Wind Eng. Ind. Aeroc/vn. 52 /1994) 93 107
(a) Aerodynamic forces and moments
Course and rotational deviation calculations
Vehicle geometry
Accident criteria
Site wind characteristics (b)
Wind spectrum
I
Aerodynamic admittance
I
Force spectrum
I
Vehicle transfer function
I
Lateral displacement spectrum
J
Extreme value of displacement
, Maximum spectral frequency ]
Fig. 2. Outline of BLOWOVER calculation methods: (a) time domain, and (b) frequency domain.
solution. To cope with these ill defined parameters, the following procedures are adopted in the paper. B L O W O V E R is run, for a particular vehicle speed and set of wind characteristics, for a grid of values of 21 and 22 in either the time or the frequency domain, and for each set of values of these parameters, whether or not an accident is predicted is noted, together with the accident type if applicable. Typically a 50 x 50 grid of points is used in the ,~t-22 plane, with a run time of around 4 h on a 386 AT
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-I07
99
{a) F1 ~2
S
F2
Ib) A
Fig. 3. Regions of safety (a) and area of safety (b) in parameter space (S = region of safety, F1 and F2 = areas of different failure types).
microcomputer. Thus a picture of the region of "safety" and the regions of different accident types can be built up in the 21-22 plane as shown in Fig. 3a. The "area of safety" A is a quantitative measure of the safety of the vehicle driver system. By varying, say, mean windspeed (u) and keeping the vehicle speed constant, the variation of A with u can be found, and a value of u, determined for which A falls to zero (Fig. 3b). This procedure will be illustrated in what follows.
4. Time domain calculations for lorries in cross winds
In this section, the method of Section 3 will be applied to time domain calculations (Fig. 2a) of a high sided lorry entering a sharp edged wind gust at 90 ° to its direction of travel. The assumed geometric and aerodynamic characteristics of the lorry are given in Table 1. A typical result for lateral and rotational course deviations is shown in Fig. 4, for a vehicle speed v of 15 m/s and a wind speed u of 15 m/s and driver parameters 21 and 22 equal to 0.3 m - 1 and 0.01 s m - 1, respectively. It can be seen that this corresponds to a "safe" situation with the displacements tending to acceptable, small equilibrium values.
100
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93 107 (a)
0.1
0.0
-0.1
-0.2
!
0
10
i
|
20
30
x
(b)
i
40
!
50
i
6O
(m)
0.12 0.10 -
E
O.08 -
0.06 0.04 II 0
0.02 -
0
0.00 4 0
i
i
10
20
I
30
x
i
40
I
50
i
6O
(m)
Fig. 4. Typical time displacement from B L O W O V E R model (for conditions see text): (a) lateral displacement, and (b) rotational displacement.
Figs. 5a and 5b show computer generated 2t-22 plots for v = 15 m/s and 25 m/s with various values of u. The area of the region of safety A shrinks to zero as the wind speed increases as would be expected. The complexity of the regions of different accident types can be seen particularly at the higher value of v. At low values of 21 there is a region of high course deviations (i.e. driver reaction is not "strong" enough) whilst at high values of 21 and 2z overturning occurs i.e. driver reaction is too "strong". Fig. 6 is a plot of the variation of A with u for different values of v. It can be seen, that A varies little with u until a point where a sudden fall occurs, and then A drops rapidly to zero at the accident wind speed ua. This implies that there will be relatively little increase in difficulty of driving until wind speeds quite close to the critical. The variation of this critical wind speed with vehicle speed v is shown in Fig. 7.
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93- 107
101
la)
Fig. 5. Areas of safety and failure in the 21-22 plane for time domain calculations: (a) v = 15 m/s, u = 10 (i), 15 (ii), 16 (iii), 16.5 (iv), 17 (v) m/s; (b) v = 25 m/s, u = 10 (i), 12.5 (ii), 13 (iii) m/s, (x-axis is 0 < 2~ < 0.5 and y-axis is 0 < 22 < 0.1, white regions are regions of safety, blue regions are regions of course deviation or rotation accidents and red regions are regions of overturning accidents).
F r o m results such as these the m e t h o d d e s c r i b e d in this p a p e r c a n be seen to b e a useful a n d i n s t r u c t i v e w a y in p r e s e n t i n g risk a s s e s s m e n t c a l c u l a t i o n s in the t i m e d o m a i n . Its use in the f r e q u e n c y d o m a i n is i l l u s t r a t e d in the n e x t section.
5. Frequency domain calculations for lorries in cross winds N o t w i t h s t a n d i n g w h a t has a l r e a d y b e e n said a b o u t the f r e q u e n c y d o m a i n m e t h o d b e i n g m o r e s u i t e d to cars a n d s m a l l vans, it is a p p l i e d here to the a r t i c u l a t e d l o r r y case
102
C.J. Baker/J. Wind Eng. Ind. Aeroc,lvn. 52 (1994) 93 107
(b)
Fig. 5. (continued).
outlined in Table 1 for two reasons - firstly for comparison with the results of the last section, and secondly because there is insufficient data available for the high cross wind aerodynamic characteristics of vans and small cars to be adequately specified. Again calculations were carried out in the driver parameter 21-22 plane. The accident criteria were adopted as set out in Section 2.2.3. with g, the gust factor, taken as 3.0. Ideally this value could be determined for any particular site by an extreme value analysis as set out in Ref. [5] with an observation period related to the site length, the vehicle speed and the vehicle occupancy of that site. However, such a procedure was felt to be unnecessarily complicated for the present work, and a simple, constant value of g was adopted. Also as noted in Section 2.2.3. if the m a x i m u m of the lateral displacement spectrum was at a frequency greater than 0.2 Hz
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93 107
103
v = 5m/s &
v = 15m/s v = 25m/s
c
E
M
0 m
.01
0
.001
i
!
i
10
12
Accident
14
wind
1
16
speed
2O
(m/., m/s)
Fig. 6. Variations of area of safety with accident wind speed.
tE
-o
15
o o
•"o
10-
e-
3: =
5
o "O
== 0
i
i
i
10
20
30
Vehicle
speed
(m/s)
Fig. 7. Variation of accident wind speeds with vehicle speed.
1.21.0 'a
;;:.;;;;;:.
-,~,~
(' Wind spectrum
0.8
0.6
s
Aerodynamic admittance
=
Displacement spectrum
s
Normalised transfer functk
0.4 0.2 0.0
......
.001
.01
.1
1
l
10
n(hz)
Fig.' 8. Typical values of wind spectrtrm;, aerodynamic admittance, vehicle transfer function (normalised on maximum value) and lateral displacement spectrum (for conditions see text).
104
C.J. Baker/J. Wind Eng. Ind. Aerodvn. 52 (1994) 93 107
Ill
IN
m
D
Fig. 9. Areas of safety and failure in the )-1-22 plane for frequency domain calculations (v = 10 m/s, u = 5 (i), 10 (ii), 15 (iii),20 (vi) and 23 (v) m/s (x-axis is 0 < 2t < 0.5 and y-axis is 0 < 22 < 0.1, white regions are regions of safety, blue regions are regions of excessive displacement, red regions are regions of high frequency oscillations and green regions are regions of both excessive lateral displacement and high frequency oscillations).
an accident was said to occur. Such a criterion attempts to model the fact that if there is too m u c h displacement at high frequencies, this will be difficult for the driver of the vehicle to manage. Fig. 8 shows typical values for the wind spectrum, a e r o d y n a m i c admittance, vehicle transfer function a n d lateral displacement spectrum for u = v = 10 m/s, 21 --- 0.3 m - 1 a n d 22 = 0.01 s m - 1. Fig. 9 shows c o m p u t e r generated plots of the driver p a r a m e t e r plane for a vehicle speed of 10 m/s. It can be seen that the area of safety is rather different in shape to that shown in Fig. 5, in that there is n o u p p e r limit for a "safe"
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93-107
t~
I
0
t...d
e~
e~
.=. O
105
106
('.J. Baker~J, Wind Eng. Ind. Aerodvn. 52 (1994) 93
107
value of 22 . This is because for high values of this parameter the vehicle would overturn, an aspect of the physical situation that is not modelled by this variant of BLOWOVER. It can, however, be seen that, as expected the area of safety shrinks as wind speed increases, as the region of high course deviations at low )-1 increases in area to meet the area of high frequency content at high values of 21. Because of this lack of an upper bound to the area of safety, no graph of the variation of area of safety with wind speed has been drawn.
6. Application of this method in other areas of wind engineering The method described in previous sections can be applied in a straightforward manner in other areas of wind engineering where the physical situation is such that there are parameters involved that are difficult to specify with precision and where there are a number of system failure criteria. The basic method is simply to carry out a large number of calculations for individual values of the uncontrolled parameters, and build up a map of regions of safety in parameter space. The area, and the connectivity (i.e., the existence or otherwise of safe islands as in Fig. 3) of these regions is thus a measure of safety. The variation in these quantities can then be investigated as, say, mean wind speed varies. Whilst considerable computational effort may be needed for such calculations they are usually programmable on microcomputers that can be dedicated to the task for the many hours of computer time required. Table 2 gives details of some possible applications, drawn mainly from the authors own research work, and there are doubtless others that could be added. Also it should be noted that the basis of this method has been suggested by Thompson and Soliman [8] who investigated the capsize behaviour of ships.
7. Concluding remarks A method has been presented which allows some appreciation to be obtained of accident risk in situations where there are physical parameters that are difficult to specify, which is very often the case in the field of wind engineering. This method has been applied to the case of road vehicles in cross winds where the variation of the "area of safety" in the driver parameter plane has been investigated using both frequency and time domain calculations, and a number of other possible uses of the method have been suggested.
References [1] C.J. Baker, A simplified analysis of various types of wind induced road vehicle accidents, J. Wind Eng. Ind. Aerodyn., 22 (1986) 69-85. [2] C.J. Baker, Measures to control vehicle movement at exposed sites during windy periods, J. Wind Eng. Ind. Aerodyn., 25 (1986) 151 167.
C.J. Baker/J. Wind Eng. Ind. Aerodyn. 52 (1994) 93- 107
107
[3] C.J. Baker, High sided articulated lorries in strong cross winds, J. Wind Eng. Ind. Aerodyn., 31 (1988) 67-85. [4] C.J. Baker, Ground vehicles in high cross winds, Part 1, Steady aerodynamic forces, J. Fluids Struct., 5 (1991) 69-90. I5] C.J. Baker, Ground vehicles in high cross winds, Part 2, Unsteady aerodynamic forces, J. Fluids. Struct., 5 (1991) 91-111. I-6] C.J. Baker, Ground vehicles in high cross winds, Part 3, Interaction of aerodynamic forces and the vehicle system, J. Fluids Struct., 5 (1991) 221-241. [-7] C.J. Baker, The behaviour of road vehicles in unsteady cross winds, Bluff Body Aerodynamics and its Applications 2, Melbourne, Australia, 1992. [8] J. Thompson and M. Soliman, Fractal control boundaries of driven oscillators and their relevance to safe engineering design, Proc. Royal Soc. London A, 428 (1990) 1 13. 1'9] BSI, BS8100 Lattice towers and masts code of practice for loading, British Standard, 1986. [10] C.D. Williams, M.J. Soligo and J. Cote, A discussion of the components for a comprehensive pedestrian level comfort criterion, Proc. 8th Int. Conf. on Wind Engineering, London, Canada, 1991. 1'11] X. Yang, C.J. Baker, R. Hoxey, P. Kettlewell and A. Meehan, The aerodynamics and ventilation of poultry transport vehicles, Wind Eng. Soc. Conf., Cambridge, 1992. [12] H.J. Roodbaraky, C.J. Baker, A.R. Dawson and C.J. Wright, Experimental observations of the aerodynamic characteristics of urban trees, Wind Eng. Soc. Conf., Cambridge, 1992.