## 73 Modelling Vehicle Acceleration

7.3.1 Acceleration performance parameters

The acceleration of a car or motorcycle is a key performance indicator, though there is no standard measure used. Typically the time to accelerate from standstill to 60 mph, or 30 or 50kph will be given. The nearest to such a standard for electric vehicles are the 0-30 kph and 0-50kph times, though these times are not given for all vehicles.

Such acceleration figures are found from simulation or testing of real vehicles. For IC engined vehicles this is done at maximum power, or 'wide open throttle' (WOT). Similarly, for electric vehicles performance simulations are carried out at maximum torque.

We have already seen in Chapter 6 that the maximum torque of an electric motor is a fairly simple function of angular speed. In most cases, at low speeds, the maximum torque is a constant, until the motor speed reaches a critical value mc after which the torque falls. In the case of a brushed shunt or permanent magnet DC motor the torque falls linearly with increasing speed. In the case of most other types of motor, the torque falls in such a way that the power remains constant.

The angular velocity of the motor depends on the gear ratio G and the radius of the drive wheel r as in equation (7.6) derived above. So we can say that:

Once this constant torque phase is passed, i.e. m > Mc, or v > rrnc/G, then either the power is constant, as in most brushless type motors, and we have:

m Gv or the torque falls according to the linear equation we met in Section 6.1.2:

T = T0 — km which, when equation (7.6) is substituted for angular speed, gives kG

Now that we have the equations we need, we can combine them in order to find the acceleration of a vehicle. Many of these equations may look quite complex, but nearly all the terms are constants, which can be found or estimated from vehicle or component data.

For a vehicle on level ground, with air density 1.25 kg.m-3, equation (7.9) becomes:

2 G2

ngr2

Substituting equation (7.5) for Fte, and noting that a = dv/dt, we have:

We have already noted that T, the motor torque, is either a constant of a simple function of speed (equations (7.10) and (7.11)). So, equation (7.13) can be reduced to a differential equation, first order, for the velocity v. Thus the value of v can be found for any value

For example, in the initial acceleration phase, when T = Tmax, equation (7.12) becomes:

Provided that all the constants are known, or can reasonably be estimated, this is a very straightforward first-order differential equation, whose solution can be found using many modern calculators, as well as a wide range of personal computer programs. This is also possible for the situation with the larger motors. Two examples will hopefully make this clear.

7.3.2 Modelling the acceleration of an electric scooter

For our first example we will take an electric scooter. No particular model is being taken, but the vehicle is similar to the electric scooters made by Peugeot and EVS, an example of which is shown in Figure 7.3.

• The electric scooter has a mass of 115 kg, with a typical passenger of mass 70 kg, so total mass m = 185 kg.

• The moment of inertia of the motor is not known, so we will adopt the expedient suggested at the end of Section 7.2.5, and increase m by 5% in the linear acceleration term only. A value of 194 kg will thus be used from m in the final term of equation (7.13).

• The drag coefficient Cd is estimated as 0.75, a reasonable value for a small scooter, with a fairly 'sit-up' riding style.

• The frontal area of vehicle and rider = 0.6m2.

• The tyres and wheel bearings give a coefficient of rolling resistance, ¡xrr = 0.007.

• The motor is connected to the rear wheel using a 2:1 ratio belt system, and the wheel diameter is 42 cm. Thus G = 2 and r = 0.21 m.

• The motor is an 18 V Lynch type motor, of the type discussed in Section 6.1.2. Equation (6.8) has been recalculated for 18 V, giving:

As in Section 6.1.2, the maximum current is controlled by the maximum safe current, in this case 250 A, so, as shown in equation (6.9), the maximum torque Tmax is 34 Nm.

Figure 7.3 Electric scooter of the type simulated at various points in this chapter. The photograph was taken in a Berlin car park

The critical motor speed after which the torque falls according to equation (7.14) occurs when:

1.16

• The gear system is very simple, and of low ratio, and so we can assume a good efficiency. A value of ng of 0.98 is estimated. An effect of this will be to reduce the torque, and so this factor will be applied to the torque.

When the torque is constant, equation (7.13) becomes:-

2 2 dv x 0.98 x 34 = 0.007 x 185 x 9.8 + 0.625 x 0.6 x 0.75v2 + 194-

0.21 dt dv

UV O

thus 194— = 304 — 0.281u di dv , so — = 1.57 -0.00145U2 (7.15)

This equation holds until the torque begins to fall when, m = mc = 103rads-1, which corresponds to 103 x 0.21/2 = 10.8 ms-1. After this point the torque is governed by equation (7.14). If we substitute this, and the other constants, into equation (7.12) we obtain:

di dv

There are many practical and simple ways of solving these differential equations. Many modern calculators will solve such equations, remembering that there is a simple initial condition that v = 0 when i = 0. However, the most versatile next step is to derive a simple numerical solution, which can then easily be used in EXCEL® or MATLAB®. The derivative of v is simply the difference between consecutive values of v divided by the time step. Applying this to equation (7.15) gives us:

For a program such as EXCEL® or MATLAB® we need to rearrange this equation to obtain the value of the next velocity from the current velocity. This is done as follows:

This equation holds for velocities up to the critical velocity of 10.8 ms-1, after which we have to use equation (7.16), approximated in exactly the same way as we have just done for equation (7.15), which gives:

The MATLAB script file below shows how to solve these equations using this program. Figure 7.4 is a plot of the solution using a time step di of 0.1 s. Exactly the same

result can be obtained with almost equal ease using EXCEL . It is left as an exercise for the reader to produce an EXCEL® spreadsheet or MATLAB® script file where many of the machine parameters, such as the gear ratio G, are left as easily altered variables, so that the effect of changing them on the vehicle's performance can be noted.

% ScootA - electric scooter acceleration. t=linspace(0,50,501); % 0 to 50 s, in 0.1 s steps vel=zeros(1,501); % 501 readings of velocity d=zeros(1,501);% Array for storing distance traveled dT=0.1;

% Now follow equations 7.17 & 7.18 if vel(n)<10.8 % Torque constant till this point vel(n+1) = vel(n) + dT*(1.57 - (0.00145*(vel(n)*2))); elseif vel(n)>=10.8

vel(n+1)=vel(n)+dT*(7.30-(0.53*vel(n))-(0.00145*(vel(n)A2)));

end;

d(n+1)=d(n) + 0.1*vel(n); % Compute distance traveled. end;

vel=vel.*3.6; % Multiply by 3.6 to convert m/sec to kph plot(t,vel); axis([0 30 0 50]); xlabel('Time/seconds'); ylabel('Velocity/kph');

title('Full power (WOT) acceleration of electric scooter');

The result of this simulation is shown in Figure 7.4, and this shows that the performance is somewhat as might be expected from a fairly low power motor. The acceleration is unspectacular, and the top speed is about 30mph, or 48 kph, on level ground. However, this is reasonably compatible with safe city riding. The acceleration of such vehicles is sometimes given in terms of the standing start 100 m times, and the power of such

Full power (WOT) acceleration of electric scooter

0

0 2 4 6 8 10 12 14 16 18 20 Time/seconds

Figure 7.4 A graph showing the acceleration of a design of electric scooter, being the solution of equations (7.15) and (7.16), as approximated by equations (7.17) and (7.18), with a 0.1 s time step

Full power (WOT) acceleration of electric scooter

Full power (WOT) acceleration of electric scooter

Time/seconds

Figure 7.5 Distance/time graph for an electric scooter, showing the time to cover 10m and 100m from a standing start

Time/seconds

Figure 7.5 Distance/time graph for an electric scooter, showing the time to cover 10m and 100m from a standing start

MATLABW script files is that they can very easily be changed to produce such information. If the plot line in the file above is changed as follows, then Figure 7.5 is obtained.

While we are not claiming that our model exactly represents any particular commercial designs, it is worth noting the following points from the specification of the Peugeot 'SCOOT'ELEC' performance specification:

• 100 m from standing start time, 12 s

It is very clear from Figures 7.4 and 7.5 that the performance of our simulated vehicle is remarkably similar.

### 7.3.3 Modelling the acceleration of a small car

For our second example we will use a vehicle that had an important impact on the recent development of electric cars. The GM EV1 was arguably the first modern electric car from one of the really large motor companies. It incorporated technologies that were quite novel for its time, and is indeed still unsurpassed as a design of battery electric car. Several views of this vehicle are shown in Figure 11.5. Further details of this car are given in Section 11.2, but as far as simulating its performance, the main features are:

• an ultra-low drag coefficient Cd of 0.19;

• a very low coefficient of rolling resistance ¡xrr of 0.0048;

• the use of variable frequency induction motors, operating at very high speed, nearly 12000rpm at maximum speed.

Further data is taken from company information1 about the vehicle:

• vehicle mass = 1400 kg.2 Then add a driver and a passenger, each weighing 70 kg, giving m = 1540 kg;

• the motor's moment of inertia is not known, however, compared to the mass of such a heavy vehicle this will be very low; the wheels are also very light and we will approximate this term by increasing the mass very modestly to 1560 kg in the final term of equation (7.12);

• the gear ratio is 11:1, thus G = 11; the tyre radius is 0.30m;

• for the motor, Tmax = 140 Nm and rnc = 733rads-1 note this means T = Tmax until v = 19.8 ms-1 (= 71.3 kph);

• above 19.8 ms-1 the motor operates at a constant 102 kW, as this is a WOT test, so:

• the efficiency of the single-speed drive coupling between motor and axle is estimated as 95%, so ng = 0.95; the values of the torque T will be reduced by a factor of 0.95; this slightly lower figure is because there is a differential and a higher ratio gear box than in the last example.

These values can now be put into equation (7.12), giving for the first phase while the motor torque is constant:

0 dv

Once the speed has reached 19.8 ms-1 the velocity is given by the differential equation:

2756 2 dv

v dt

At v

1 The two sources are Shnayerson (1996) and the official GM EV1 website at www.gmev.com.

2 It is interesting to note that 594kg, or 42%, of this is the lead acid batteries!

The procedure for finding the acceleration is very similar to the first example; the only extra complication is that when the velocity reaches 35.8 ms-1 it stops rising, as at this point the motor controller limits any further acceleration.

Before any program such as EXCEL® or MATLAB® can be used, the key equations (7.19) and (7.20) must be put into 'finite difference' form. This is done exactly as we did for equations (7.15) and (7.16). The two equations become:

(62.1 2\ vn+1 =vn + 'dt i —--0.046 - 0.000137U2 J (7.22)

The MATLAB script file for these equations is very similar to that for the electric scooter given above, so it is not given here in the main text, but can be found in Appendix 1. The plot of velocity against time is shown in Figure 7.6. Looking at Figure 7.6 we can see that the time taken to reach 96kph, which is 60mph, is just under 9 s. Not only is this a very respectable performance, but it is also exactly the same as given in the official figures for the performance of the real vehicle.

We have thus seen that, although not overly complex, this method of modelling vehicle performance gives results that are validated by real data. We can thus have confidence in this method. However, vehicles are required to do more than just accelerate well

Full power (WOT) acceleration of GM EV1 electric car

Full power (WOT) acceleration of GM EV1 electric car

Figure 7.6 A graph of velocity against time for a GM EV1 at full power. This performance graph, obtained from a simple mathematical model, gives very good agreement with published real performance data

from a standing start, and in the next section we tackle the more complex issue of range modelling.

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