METHOD FOR DETERMINING THE LOSSES OF MOTOR VEHICLES DURING ACCELERATION

METHOD FOR DETERMINING THE LOSSES OF MOTOR VEHICLES DURING ACCELERATION

By

J.

HEGEDUS

Department of Motor Vehicles, Technical University Budapest (Received February 27, 1974)

Presented by Prof. Dr. Z. LtVAI

Mathematical equations of the method

From the speed-time diagram plotted for accelerating from stationary position to maximum speed, the acceleration can be determined in the whole range and - complemented by further data - so can be the pure accelerating force.

The momentary acceleration of the motor vehicle can be determined from the following differencial equation:

where F [kp]

~_ F-E 2

- (.)2[m/sec]

dt m

+

1:f --.:..

r

(1)

the tractive effort available on the circumference of the driving wheels

the running resistance of the motor vehicle the mass of the motor vehicle

the moment of inertia of all rotating parts reduced to the circumference of the driving wheel.

The moment of inertia of the rotating parts is taken in consideration with a mass factor

(i)

^{2 [ kp sec2 ]}

m' = m

+

^{1:f -; =} m . km m . (2) The value of the mass factor km is shown by Fig. 1 as a function of the quotient of the motor rpm by the relevant vehicle speed.

n [l/min]

v [kmjh]

rpm of the motor built in,

the speed pertaining to the motor rpm at the mm.

gear transmission.

84

$

1,60 I,SO

~ ',40 .. 1,30

..

~ 1,20

""E 1,10

-

1.00

o

J. HEGEDGS

V

/

1/

, /

...

f-"'"

V

20 '0 60 80 100 120

V n

Fig. 1

The tractive effort available on the driving wheels is:

where

~ [kp]

[kp] (3)

the theoretical tractive effort on the circumference of the driving wheels computed from the characteristics of the motor, measured on the test stand:

the total loss on the different part - groups and equip- ment,-reduced to the circumference of the driving wheels,

dynamic loss, difference of static and dynamic charac- teristics reduced to the circumference of the driving wheel.

The theoretical tractive effort Fm can be determined from the motor's external characteristics:

w-here Pm [HP]

nm [l/min]

itotal [ - ]

rg [m]

Fm

=

Pm . itotal 716,2

nrn . r g

[kp]

motor performance measured on test stand motor rpm

(4)

all gear transmissions ratio between the motor and the driving wheel

dynamical radius of the driving wheel.

The movement on level road of the motor vehicle is opposed by the following running resistances:

where Fw [kp]

Ft

^[kp]

air resistance rolling resistance.

[kp] (5)

LOSSES OF MOTOR VEHICLES

The air resistance of the motor vehicle:

where y [kp/m³]

g [m/sec²]

Cw [ - ]

A [m²]

v [m/sec]

F - Y · C .A. w - - tv V ., ²

2g ^[kp]

specific density of the air gravitational acceleration form factor of the car-hody

frontal surface of the motor vehicle vehicle speed.

The rolling resistance of the motor vehicle:

F_{f =} Ge . fe

+

^Gh •

hz ,

^[kp]

where

loads on the front and rear axle, resp.

85

(6)

(7)

Ge and G_h [kp]

le

and fh [-] rolling resistance factor of the front and rear wheels, resp.

The rolling resistance factor of each axle is computed according to the method elahorated hy Dr. KOJI SEKI [5] with the relationship:

where

P [kp/cm²]

P [kp/cm^2]

G

[kp]

Gm [kp]

b [-]

f=[~~

^0.14

(-cior4]

^G_v^{O.44 • 0.28 G}_m^{-O.48 • b}

Pv 0.75 Pv 1.55 .

Pv=L;

Pm G_v= - ; G

Gm tyre pressure

suggested tyre pressure load on the tyre

suggested load on the tyre wheel-disc width factor b = - 0.05 . e

+

^{1.25 [ - ]}

where

e [inch] ",idth of the wheel-disc.

(8)

(9)

(10)

(11)

86 J. HEGEDOS

Determination of losses F'I and Fd Starting from the differential equation

dv F-E

- - - - -

dt

the motor vehicle's equation of motion on level road is the following:

Fm - Fd - ~ - Fw -

Fj - -

dv

.

m . km =

o.

dt

(12)

(13) To determine the dynamic and the mechanic losses

Fd

and ~, resp, the values of the acceleration and of the accelerating force are needed.

For the sake of illustrating the method of determining the acceleration and the accelerating force from the tested motor vehicle's speed-time diagram (Fig. 2):

v

(km/h)

B

Fa ~~---~~~----+---+

(kp) t (secl

Fig. 2

The acceleration pertaining to speed Va is equal to the tangent of the angle ^(X.

A perpendicular davv'1l to the tangent pertaining to speed Va cuts point C on the horizontal axis.

According to triangle POC:

OC F-E

tg ^(X =

-=

= - - - - OP m· km OC = AB and

- x

OP

= - .

^{Z •} m . km , y

where y

[m~:c]

x[:: ^J

z [ : ; ]

~[kp]

LOSSES OF MOTOR VEHICLES 87

the scale of the speed axis the scale of the time axis

the scale of the accelerating force axis the available accelerating force.

Since the distance OP is proportional to m ·k_{m ,} to the mass of the motor vehicle increased by the iner tia moment of the rotating masses, the distance OC will be identical to the accelerating force pertaining to the investigated speed.

Knowing the forces Fm,

Fj,

Fw and Fm the diagram of the tractive effort, running resistance and accelerating force of the tested motor vehicle can be plotted (Fig. 3.). At the sp eed Vrnax the acceleration is zero, thus

(14) that is

(15) The lost force ~ contains not only losses in the drive train, but all other losses related to the operation of the motor vehicle, to be met by the motor, such as losses of force due to the output demand for the eventual servo- system, air conditioning, ,visher, ventilator, etc.

(kpl

... >"

Fm

Fa

-

x

F ... " E

..::

F,

-

^u..

""

"

.::

"

u.

-

x

"

^E

>

u.. E u.. ~

"

>

~

Vo vmax ^{v (m/sQr.1}

Fig. 3

88 J. HEGEDOS

At a speed 1'a where the vehicle can be accelerated, the equation of motion will be different:

that is

(17) At all speeds where the motor vehicle can be accelerated, an additional loss occurs impairing performance.

(kp)

IS

U1 o

...

v (m/sec) Fig. 4

This loss is due to mixture formation and thermodynamical conditions different during acceleration from those in stable circumstances in a measure- ment on the test stand.

If the motor vehicle is driven from the level road to a gradient of theor- etically any inclination, then obviously a gradient may be found where the final velocity of the vehicle equals the Va value. Since here the acceleration stops the dynamical loss

Fd

"will necessarily stop, too.

Hence in case of an acceleration according to Fig. 3, at a speed Va it can be 'VTitten:

Fm(va) - Fd(va) - ^~(va) -1j(va) - F;,,(va) - Fa(va) ⁼ 0 (16) and for a constant speed:

Fm(va) - ^~(va) - F;v(va) - G[J(va) . ^{cos cc}

+

^{sin cc] = O. (18)}

Equalizing the equations we get:

(19)

LOSSES OF MOTOR VEHICLES 89 To determine the Fd value in the speed range Vrnax

>

^Va

>

0 the inter- related speed and a; values are needed.

According to Fig 4, if the tested motor vehicle driven at a speed VI to a gradient of the angle a; where

(20) the accelerating capacity of the motor vehicle is ('Iff. Since however, the dynamic loss

Fd

was an actual one, it has previously to end. This is manifest by the operating point on the new running resistance curve shifting towards greater speeds.

Mter the dynamical loss ended, the conditions will be stabilized and the vehicle will run at the constant speed v.

Supposing that the loss

Fd

is proportional to the acceleration, or to the generating accelerating force

Fa,

it seems to be likely that the deviation of the speed V compared to the speeds VI and V2 - determined by the intersection point of the E = f3 (v) curve with the

Pg" =

f2(V) and Fm

=

fl(V) curves -

"will be inversely proportional to the accelerations pertaining to the latter two speeds.

Hence, it can be written:

a( v_{2 )} Fa( v_{2 )} F_{d (} v_{2 )}

= - - = - - = - - - a( VI)

F'a(

^{VI) F}d( VI)

(21) and the speed V where the vehicle will run evenly:

V~ . a~

+

^VIal

V -

-

^{-~ ~}

,

a1

+

a2

[m/sec] (22)

"where

a₁ and a₂ are the accelerations (mJsec2 ) pertaining to the speeds VI and _{V 2•}

Starting from Eqs 20 and 22, the boundary speed curve of the motor vehicle under test is:

(23) yielding in the speed range Vrnax

>

^V

>

Vrni!1 the curve

(24) from Eq. 19, and

(25) from the relationship

(26) Analysis of the mechanical and dynamical losses of the motor built into the motor vehicle requires a complex and sophisticated computation.

90 J. HEGEDtJS

To be acquainted with, and to generalize these losses, many measure- ment data were needed of a great number of current up-to-date passenger motor cars.

The high number of data justified the use of a digitalal computer OnRA 1204.

Results

Computer analysis granted the losses

Fd

and ~ of the tested motor vehicles as well as their specific values as a function of speed and acceleration.

Since the final speeds of vehicles of different categories are widely different, the use of the specific values of the losses according to the speed was considered as impractical.

In the seconds after startmg, the value of the maximum acceleration of each motor vehicle, irrespective of category, is determined by the rate of the adhesion between wheel and road; their difference is negligible. Therefore it seemed to be more expedient to analyse, and to generalize the specific values of the losses as a function of acceleration.

Since the conclusions dravm from the investigations are best used at the design stage where the acceleration of the vehicle is still unkno'wn, the presentation of the specific values

Fd

and ~ as a function of the actual

acceleration would entrain inaccuracy.

Initially the theoretical tractive effort computed from the motor's moment and the running resistances are only known. These values and the vehicle rating permitted to determine thc theoretically possible acceleration as a function of speed.

The theoretical acceleration is

(27)

Accordingly, the scatter of theoretical acceleration

(28) and

Fq =

fs(a e)

Fm ⁽²⁹⁾

has been generalized by an analytic program and the results illustrated in Fig. 5 as dynamical efficiencies as a function of the theoretical acceleration (30)

LOSSES OF MOTOR VEHICLES 91 mechanical and

(31) dynamical efficiencies.

Application of the results

Application of the method of preliminary planning will be illustrated on a Peugeot 504 passenger car -with a carburettor Otto engine built in.

According to the investigated vehicle specification, the theoretical acceleration may be determined as a function of speed

[ m/sec²] (27)

As a matter of course this acceleration cannot be achieved under normal circumstances its values are, however, needed in following computations.

At the zero acceleration pertaining to each speed in case of stationary operation, the actual tractive. force value is given by the relationship:

~ = Fm . Y)m [kp] (32)

where the force Fm is multiplied by Y)m pertaining to the theoretical accelera- tion value generated by the theoretical tractive force Fm (Fig 5.)

In case of acceleration, the dynamical loss consumes the tractive force excess F;, - E.

'1m

0,95

Ild

(-)

0,90

0,85 4

...

1/

o

1

I><

...

1 1 1

t

1

1 1

1

1

I I :.'~

I I

1 1

1)d 1

"'k

1

i I "i....

t-... llml 1 ... , :-...1 1 I '

1 T"'-...I 1

1

1 1 I

1

I

2 3

a.m/sec²

J~I

I ['....

I

I

1

j

Fig. 5. Mechanical and dynamical efficiencies in accelerating:operation of Otto engines built into vehicles

3 Periodica Polytechnic. T.E. 2[2

92 _{J. HEGEDOS}

Fm Fv Fgy Fa E 700 Ft (kpl

600

/

^\

1\

500 I~ Fgy

Io-i 400

1'1\

300

200

l'

N".

~ ^~

^I' I'

~ ^!;:

^t:'-I'

tOO

Fa I '

~~

E_J ...

^;;;;f;;: ...

~

~~

f'r-

1-1-!-"" ^I-'"

I -

1-1-

F:t::: 1-1-

J.

0 10 20 30 40 vmax SO v(mfsec)

Fig. 6. Tractive force, accelerating force and running resistance diagrams of a passenger car type Peugeot 504

The actual pure accelerating force is, similarly to the aforesaid:

~ =

(F" -

E) . Y)d' [kp] (33)

The value Y)d pertaining to the vehicle speed in question will be found in this case also on the base of the theoretical acceleration (Fig. 5).

The relevant values are shown in Fig. 6.

The intersection point of curves E and Fv indicates the final speed v possible for the vehicle on level road. If the running resistance changes, the new intersection point 'will give the changed final speed.

The intersection points of the curves F gy pertaining to the different gears present the ideal spots for gear shifting.

The distance between the curves E and

F'gy

shows the neat accelerating force

Fa.

LOSSES OF MOTOR VEHICLES 93

-;:;

'"

~ 0 ....

0

'"

1\

~

^{0 N}

\~~~ _:\~~~~

⁵²

u u

i\ !\:V'J'f~~~

>]

..

0 ~ 1\ 1 g 1\1

~ ~ ~~;g,.:

<l'I

0

~ "- ^{k-' 0}

1\ ..-~

--- '"

^{..-> :0::: 1}

~

-\

_1~

_k - t

"" ^{X KO}

r

" --

^{\ 1.1-v \}

^:

^{. / ~ . / "-V 52}

..J \-

,

/ 1 v _/ '<

J IY' "l ^{' (}

.. hi

1 ⁰ ₀

\J 1 ^N

I

\

_N

:

\ " 1

0 0

I

, ₁₇

I

^I

I I

⁰

I ⁰

~ It

I ^\

^I\i 0 - " , ^a.

0 _ ..n u. "

3*

, . - -

...--<..

I~

500 Fa (kp)

v (m/sec) 50 - - - -

/ /

~O

A-

i::.

~

~---

-7- -,

_{\ 3 0 - - ~}

k\ \ \ \ =----

^~

/\\ -, n

_~

~.'S...:"~

lA .\ ^Jl-:::

I---~

A ' \ \ \ \ \

^{. /}

~

A '" '\\\ 1\ \--\\

~

"" "" ^'\ \ \ \ \-i1 ^v /'l:t

.-t. l-,~ ~ -~-, ^{\ \} .x

r71I

---

1---- ~ "'-

I'-~"" ^I'\\W\ ~~1

~ ~ i'--...~~ ~~

---

^~~

--- ^{~ ~~} _~ ^nl

~OO 300 200 100 ^- 250 ^. o ^-

Fig. 8. Speed - distanee diagram of a passenger cur type Peugeot 504

.-

---

..

~

~

~ ...,

~ c::.

fJJ

5

(m)

1000

750

500

250

LOSSES OF MOTOR VEHICLES

I I I I

f . I

mantl acturers ratmg

V

computed / own measurement

10 20 30 40 t(sec)

Fig. 9. Distance-time diagram of a passenger car type Peugeot 504

95

The actual situation in the period immediately after the start is some- what hard to follow. The method gives a trustworthy information only after the practically slipfree connection of the clutch.

Figs 7 and 8 illustrate the neat accelerating force values of the tested vehicle as a function of speed, the speed-time and the speed-distance dia- grams have been plotted by means of the aheady described graphical differen- tiation.

The time and distance value pairs pertaining to identical speed values and contained in the speed-time and speed-distance diagrams have been applied to plot the distance-time diagram of the motor vehicle (Fig. 9) where also the values taken and measured on road, using a passenger car of identical type made available by the manufacturers, have been indicated.

96 J. HEGEDOS

Summary

The described method makes easier and more accurate the work of predicting the run- ning qualities of motor vehicles in the design stage or to select the one most appropriate for given running qualities without preliminary operation.

The running qualities of a motor vehicle are determined jointly by the motor, drive, wheels, body, etc. as subsystems of a system.

The said units are rather various in construction but varying their assembly offers further possibilities for the designing engineers.

Several kinds of motors are built into the same body or the same motor is built into different bodies. The running qualities of the described varieties are of course different and so will be the losses.

The pure accelerating force needed for the linear acceleration of a given motor vehicle running on a speedway is reflected by the vehicle's speed-·time diagram.

Since the external characteristics of the motor built into the vehicle and measured on test stand are known and the running resistances can be calculated, the difference of the accel- erating force determined from the speed-time diagram and of the loss-free tractive force theoretically available on the driving wheels, reduced by the value of the running resistance, will yield the sum of the losses occurring during acceleration.

References

1. Ltv . .u, Z.: Gepjarmiimechanika: Mernoktovabbkepzo Intezet, Budapest, 1962.

2. CSUDAKOV, K. A.: A gepkocsi elmelet, Akademiai Kiad6, Budapest, 1955.

3. JUREK, A.: Gepjarmiiszerkezetek I-H. BME Jegyzp.t Budapest, 1961.

4.ILOSVAI, L.-DEZSO, K.: Motor es hajt6mu parameterek megvaIasztasa B.rrIE Segedlet, Budapest. 1973

5. SEKI, K.: Automobile Engineer 59, 88 (1969) 6. WALKER, L.: Automobile Engineer 48,215 (1958) 7. KOFF!rUN, J. L.: Automobile Engineer 45, 576 (1955).

8. ORDORIKA, M. A.: SAE Journal, 74, 101 (1966) 9. SETZ, H.: SAE Transactions 69, 361 (1961)

J6zsef HEGEDUS, 1450, Budapest, P.O.B. 93. Hungary