Parameters for the Characterization of Motor Vehicle Acceleration Ability

Abstract For the characterization of vehicle acceleration, in addition to such common parameters as change of speed and acceleration in time, there should be applied one more parameter: change of acceleration increase in time. Change of acceleration increase in time jp progresses in four specific phases in each gear: I – beginning of run where jp is growing rapidly to the maximum value of the gear; II – jp reduction from the maximum value to the stabilized value; III – speeding up with stabilized value jp and IV – shifting when jp value is changing in a wide range to the maximum negative value and then to zero.


I. INTRODUCTION
As it is known, acceleration ability is a parameter for the dynamism of vehicle run up: reaching high speed during a short period of time [1], [2].
The dynamism of vehicle speeding up and, accordingly, acceleration ability are commonly characterized by: run up acceleration a; run up maximum value amax; speeding up time tie; distance sie for reaching certain speed v, for example, 100 km/h; distance for covering a certain distance, for example, 400 m or 1000 m; distance change sie in time tie [1], [3]- [11]. There is one more parameter used for dynamismacceleration factor ηj. This is a ratio between power of inertia resistance overcoming Nj and power of potential traction supplied for driving Nk (1) [1]: (1) The definition and calculations of those parameters is a task of Theory of Ground Vehicles science sector [4], [12], [13] and particularly of its branches -Theory of Automobile [1]- [3], [5]- [7], [14], [15] and Theory of Motorcycle [9].
For instance, acceleration a of vehicles depending on powers applied to vehicles, namely traction power Pk and all i resistance powers Pi influencing vehicles in particular drive conditions, and reduced mass of vehicles mred, can be defined as (2) [1]: Analysis and calculations are easier if the dynamic factor of vehicles D, road-resistance ratio ψ and coefficient of vehicle gyrating masses δ are known (3) [1]: The maximum value of acceleration amax can be found from the same formula (3) if dynamic factor D in it is replaced by its maximum value Dmax for particular conditions of vehicle operation.
If the adhesion of vehicle movers with the ground surface is limited, acceleration values including the maximum one can be found from formula (3), replacing D with meshing dynamic factor Dφ.
When acceleration values are known, speeding up time tie and starting distance sie values can be determined, in particular for speed changes from v1 to v2 (4), (5) [1], [4] Usually, the above mentioned parameters are sufficient for conventional vehicles with normal dynamism. However, nowadays when acceleration of dynamic vehicles is already characterized by 4 to 5 seconds or even less for reaching a speed of 100 km/h (see Table I), these parameters become insufficient [16]- [19].
In such cases, one of the most important parameters alongside the developed acceleration is an increase of its speed da/dt, namely the parameter which characterizes how fast acceleration can be or has been increased. Therefore, it is now necessary to add to the traditional acceleration ability parameters one more parameterspeed of acceleration gain jp, measured in m/s 3 . Analogue parameter oscillation in theory is called jar or jerk by some mechanics. In this article, we will abstain from searching any other names and, therefore, use term jars.

II. SUPPLEMENT TO THE PARAMETERS OF ACCELERATION ABILITY
Analytical and graph-analytical methods will be used in this article for clearing up the acceleration ability. If acceleration of vehicles a is (6): jars are (7): They can also be expressed as follows (8): The analytical calculation of jars is made somewhat difficult by the fact that graphs of acceleration changes a = f(tie) and a = f(vie) are not usually described analytically. However, the theory of vehicles handles the graphic forms of these functions [1], [13]. Such pictures are obtained during experiments (see Fig. 1) or by using theoretical curves of dynamic factor [1]. If acceleration graph a = f(vie) is known, by way of its graphical differentiation it is also possible to find jar graph coordinates da/dtie v as function jp = da/dtie = f(v).

III. BEHAVIOUR OF JARS
Using a sample oscillogram with curve a = f(tie) [1] and graphical differentiation, we can draw a graph of jars in coordinates jpt (see Fig.1 in which a = j).
We can see from the graph that the maxima of curves a and jp differ by time.
Maximum values a are reached at the initial phase of vehicle acceleration. For light vehicles (motorcycles, light motorcars), it is reached in the 1 st gear during the development of maximum dynamic factor Dmax. For heavy vehicles with multi-stage gear boxes, amax can be reached only in the 2 nd or 3 rd gear.
The maximum values of jars for all vehicles are reached at the very beginning of run (first seconds). During the acceleration of vehicles, a can reach negative values only during gear shift, but jp could be obtained each time after reaching amax in each gear.
The graph of jars for each gear used during acceleration can be divided into 4 phases characterising the mode, and in the graph their times are marked as t1, t2, t3 and tpp. For example, run up time in the 1 st gear (including time used for switching gears to the 2 nd gear) consists of a sum made by the times of the 4 mentioned phases (9): These phases can be characterised by the following indications: 1. Beginning of run (in each gear). During this time period t11 there is a fast increase of jars reaching the maximum value in the appropriate gear.

IV. ANALYSIS AND RESULTS
The desired curve of jar behaviour is influenced by two main factors displayed in Fig. 2. To manufacture vehicles with very good acceleration ability and short run up time for reaching high speed, it is necessary to influence the following two factors:  inclination angle ß of the curve at the phase of jar increase jp;  height of jar average value α. The angle of curve at the beginning of vehicle run up, namely in its 1 st phase, should be as wide as possible because the speed of vehicle acceleration increase is characterized by the tangent of this angle, namely in the elementary range (10): Higher speed can be achieved in shorter time when angle ß is wider. Higher values of angle ß can be achieved by the first developing high usable traction force to movers Pk. and decreasing resistance powers ∑Pi. Theoretically, higher traction force can be developed by enlarging engine torque in the whole speed range by choosing rational total transmission ratios and reducing vehicle mass if we take no notice to factors with apparently lower influence such as the size of movers, radius of wheels, and transmission efficiency.
However, it should be emphasized that this is a matter of practically usable traction force (not a theoretical possibility). It can be quite different from theoretical, because it mainly depends on the adhesion between movers and ground surface characterized by adhesion factor φ. The largest values a and positive jp can be reached only in case when a very good adhesion between the ground (road) and movers (for example, wheels) is provided at least temporarily, when the value of adhesion factor φ is approximately 1 or higher, namely φ > 1. Modern vehicles and roads can ensure the fulfilment of this provision.
Adhesion and consequently practical possibilities for using theoretically possible traction force can be enlarged by the appropriate design of movers (for instance, tread pattern), the material of movers maximally appropriated for establishing adhesion and other methods ensuring good adhesion, for example, an automatic device for preventing mover skidding.
Practically, during tests, inclination angle ß of curve jp can be considerably influenced by driver's actions depending on his/her skills and experience and psychophysiological features and character.
The average value of jars can be calculated on the basis of vehicle run up time required for reaching a certain speed. We will calculate it for 3 characteristic cases when vehicles achieve speed v = 100 km/h = 27.78 m/s during: 1. 20 s (vehicles with quite poor acceleration capability); 2. 10 s (modern vehicles with quite good acceleration capability); 3. 5 s (some modern vehicles with very good acceleration capability). Modern vehicles with very good acceleration ability are powerful motorcycles and some motorcars (mostly sport models), for example, Lansarea CL600 with V12 turbo 5.5 l engine -4.6 s, Audi SLS AMG E-CELL -4 s, Porsche 977GT2 Gemballa -3.5 s as apparent from Table I. The results of calculations are given in Table II. If the maximum values of acceleration are usually 1.5 to 2 times above the average, the maximum values of jars can be considerably (up to 200 times) higher than the average values. The difference between average and maximum jp values is individual for each particular vehicle. Such a difference as well as good acceleration ability are mostly determined by:  1 st phase: ensured adhesion, engine power and torque, fast operation of clutch during switching;  2 nd phase: ensured adhesion, engine power and torque, total transmission ratio;  3 rd phase: engine power and torque, total transmission ratio;  4 th phase: in case of automatic transmissionautomatic operation of gear switching, in case of non-automatic transmissionoperation of gear switching mechanism and nature of operator activity during gear switching. Additionally, jars are affected by the design and their technical condition of vehicles, streamlining form, weight and other less important factors, as well as the condition of ground.
The first and second phases are the most important for providing excellent acceleration capability.

V. CONCLUSION
In order to characterize the acceleration abilities of vehicles, in addition to common parameters, namely change of speed and acceleration in time, it is required to apply one more parameter: change of acceleration increase in time.
Change of acceleration increase in time jp = f(t) progresses in 4 specific phases: 1. 1 st phasebeginning of acceleration run where jp is growing rapidly to the maximum value in the respective gear; 2. 2 nd phasejp decrease from the maximum value to the stabilized value; 3. 3 rd phasespeeding up with stabilized value jp 4. 4 th phaseshifting when value jp is changing in a wide range from the maximum negative value to zero. Factors for providing maximum value jp and good acceleration ability:  sufficient power and torque of engine and also good adhesion between the ground and movers (driving elements), φ > 1 (1 st , 2 nd , 3 rd phase),  fast operation of clutches during switching (1 st phase),  operation of gear switching mechanism during switching (1 st , 4 th phase);  appropriate total transmission ratio (2 nd , 3 rd phase);  automatic operation of gear switching in case of automatic transmission and nature of operator activity during gear switching in case of non-automatic transmission (4 th phase). The first and second phases are the most important for providing excellent acceleration ability. The maximum of good acceleration ability can be reached in conditions when the greatest ascent (ß) of real curve a = f(v) is obtained as a result of optimal combination of gain of torque, adhesion factor and total transmission ratio. Therefore, jp can be increased by increasing the value of adhesion factor φ, ascend of engine torque Me and value of Me up to the maximum usable value from the point of view of adhesion, as well as by choosing the most appropriate transmission ratios, particularly in the first gears.