48 3 IC-Engine-Based Propulsion Systems
Diesel engines. Clearly, this is not sufficient for most practical applications.
Excluding the very small speed values when the clutch or the torque converter are slipping, the ratio of the maximum to the minimum speed of a standard vehicle has a value of around 30. Accordingly, a gear box must be added to the powertrain. This device must realize a minimum gear ratio of five to meet the standard driving requirements.
Three types of gear boxes are encountered in practical applications:
• manual gear boxes, which have a finite number of fixed gear ratios and are manually operated by the driver;
• automatic transmissions, which combine a fixed number of gear ratios with a gear shift mechanism and a hydrodynamic torque converter or an automated standard clutch; and
• continuously variable transmissions (CVTs), which are able to realize any desired gear ratio within the limits of this device.
Choosing the gear ratios requires the solution of a complex optimization problem [78]. In a first iteration, the gear box efficiency and its dynamic properties may be neglected. The largest gear ratio (the smallest gear in a manual gear box) is often chosen to meet the towing requirements. Using (2.4), this gear ratio is found to be
γ1= mv·g·rw·sin(αmax)
Te,max(ωe) . (3.9)
The maximum engine torqueTe,maxdepends on the engine speedωe. For this reason iterations may become necessary.
The smallest gear ratio (the highest gear in a manual gear box) can be chosen either to reach the top-speed limit or to maximize the fuel economy.
These two approaches can be combined by choosing the second-smallest gear to satisfy the maximum speed requirements and the smallest gear to maximize the fuel economy (“overdrive” configuration). This approach is chosen in the example illustrated in Fig. 3.4. In this case, to determine the ratio of the fourth gear, first the following equation5must be solved forvmax
Pe,max=vmax·Fmax=vmax·
mvgcr(vmax) +1
2ρaAfcdv2max
. (3.10) In general, this equation must be solved numerically. If cr may be assumed to be constant, a third-order polynomial equation results. Using Cartan’s formula, it can be proven that in this case only one real solution of (3.10) exists such that no ambiguities arise.
Once the achievable top vehicle speed vmax and the top engine speed cm,max are known, the gear ratioγ4 is found using the following equation
5 This equation is similar to (2.13). However, it includes the rolling friction losses that become relevant in this context.
3.2 Gear-Box Models 49 γ4=r·cm,max·π
vmax·S . (3.11)
With this information, the vehicle resistance curves can be plotted in the engine map. Figure 3.4 shows the resulting resistance curves for the light- weight vehicle and the three-cylinder engine used as an example in this section.
The fifth gear can be chosen according to several fuel-economy optimiza- tion criteria. One possibility is to choose a value with which the vehicle can run at the most frequently used city speed without violating the smooth-running limits. In the case illustrated in Fig. 3.4 that speed is assumed to be 50 km/h.
120 km/h
80 km/h 50 km/h
m/s
pme
10
4 bar
! =0.33 0.30
182 km/h
3 kW 10 kW
20 kW
cm
4 8 12 16
0
0.25
202 km/h
5 smooth- 4
running limit
e
Fig. 3.4.Engine map 3.2 combined with two vehicle resistance curves (fourth and fifth gears on horizontal road) for the vehicle specified in Sect. 3.1.2. The grey square indicates the MVEG–95 average operating point.
Of course, there are many other criteria that must be observed when choos- ing the gear ratios. In all cases, the gear spread, i.e., the ratio of two neigh- boring gear ratios, must remain within certain boundaries. A geometric law is often chosen6
γk =κ·γk−1, k= 2, . . . , kmax, κ≈ 2
3 . (3.12)
Moreover, when using gear boxes that have discrete values of gear ratios,gear- box gaps cannot be avoided. As illustrated by the shaded areas in Fig. 3.5, such gear boxes cannot exploit the full traction potential of the engine. The condition (3.12) ensures that these regions do not become too large.
An example of a model-based numeric optimization of the gear ratios is shown in Appendix I in Case Study 8.1.
6 Of course, only rational gear ratios can be realized in practice.
50 3 IC-Engine-Based Propulsion Systems
10%
5%
0%
v Ft
gear 1
3 4
5 P curvemax
driving-resistance curves 2
T curve in gear 2e,max
(constant road inclination)
Fig. 3.5. Traction force Ft as a function of vehicle speedv at maximum engine power Pmax. The hyperbola is realizable using a CVT and the five dome-shaped regions using a manual gear box with five different gear ratios. Also shown are the total vehicle driving-resistance curves for three different values of constant road inclinations. Vehicle parameters{Af ·cd, cr, mv} ={0.4 m2,0.008,850 kg}; engine map as illustrated in Fig. 3.2.
3.2.3 Gear-Box Efficiency
The main dynamic effects caused by gear boxes have been discussed in Sect. 2.1.1. However, the result summarized in (2.11) is only valid if the effi- ciencyηgb of the gear box is 100%. Of course, this is not realistic. The losses caused by gear boxes and similar powertrain components depend on many influencing factors: speed, load, temperature, etc., just to name the most im- portant ones. Figure 3.6 displays the structure of the system analyzed in this section and illustrates the variables that will be important in this analysis.
γ, η T1
Te
Tv
ωe
Θe Θv
ωw gb
ICE vehicle
T2
Fig. 3.6. Illustration of the definitions relevant for the modeling of the gear-box efficiency.
3.2 Gear-Box Models 51 An approximation of the losses in gear boxes can be formulated using an affine dependency between the gear box input and output power
T2·ωw=egb·T1·ωe−P0,gb(ωe), T1·ωe>0, (3.13) where P0,gb is the power that the gear box needs to idle at an engine speed ωe. Equation (3.13) is valid when the vehicle is in traction mode. IfT1·ωe<0 a similar equation can be formulated to describe the losses in the gear box that affect the fuel cut-off torque
T1·ωe=egb·T2·ωw−P1,gb(ωe), T1·ωe<0. (3.14) For automotive cog-wheel gear boxes, typical values foregb are between 0.95 and 0.97. Depending on the size of the gear box and on its lubrication system, the idling lossesP0,gb can reach up to 3% of the rated power of the gear box.
The evaluation of the efficiency of CVTs is discussed in Chap. 5, where hybrid-inertial vehicles are treated.
3.2.4 Losses in Friction Clutches and Torque Converters
During those phases in which the vehicle and the engine speed are not matched, the powertrain has to be kinematically decoupled. For that pur- pose either friction clutches or hydrodynamic torque converters are used in practice. Figure 3.7 illustrates the structure of the powertrain and the corre- sponding main system variables.
γ, η T1,e
ωe
Θv
ωw gb
vehicle ωgb
T1,gb
Fig. 3.7.Illustration of the definitions relevant for the modeling of the clutch and torque converter efficiency.
Friction Clutches
Dry- or wet-friction clutches have no torque amplification capability, i.e., their input and output torques are identical
T1,e(t) =T1,gb(t) =T1(t) ∀t . (3.15) Friction clutches produce substantial losses only during the first acceler- ation phase when the vehicle starts at zero velocity. If the engine speed ωe
52 3 IC-Engine-Based Propulsion Systems
is assumed to be constant during this start phase, the clutch dissipates the following amount of mechanical energy
Ec=1
2 ·Θv·ωw,02 , (3.16)
whereωw,0is that wheel velocity at which the clutch input speed ωeand the output speedωgbcoincide for the first time. The inertiaΘvincludes the vehicle inertia and all inertias due to the rotating parts located after the clutch. The amount of energy dissipated does not depend on the clutch torque profile during the clutch-closing process.
u=1
Δω Ta
Tb
Δω0 Δω0
-
- -
T1,max
u=0
Ta
Tb
Fig. 3.8.Approximation of the maximum torque of a friction clutch.
Note that during all phases in which the clutch is slipping, the torqueT1(t) at the gear box input is not limited by the engine but is defined by the clutch characteristics and its actuation system. As illustrated in Fig. 3.8, the clutch torque T1(t) depends on the speed difference ∆ω(t) =ω1,e(t)−ω1,gb(t) and on the actuation inputu(t)
T1(t) =T1,max(∆ω(t))·u(t), 0≤u(t)≤1. (3.17) The maximum clutch torque can be approximated by
T1,max(t) = sign(∆ω(t))·h
Tb−(Tb−Ta)·e−|∆ω(t)|/∆ω0
i
. (3.18) The parameters ∆ω0,Ta andTb must be determined experimentally. In gen- eral, they depend on the temperature and wear of the clutch.
Torque Converters
Most automatic transmissions consist of an automated cog-wheel gear system and a hydraulic torque converter. The latter device produces additional losses
3.2 Gear-Box Models 53 in those operating phases in which it is not locked up. The losses incurred in these phases can be modeled as shown below.
The torque at the input of the converter may be modeled as follows T1,e(t) =ξ(φ(t))·ρh·d5p·ω2e(t). (3.19) The converter input speedωe(t) (the “pump speed”) has a strong influence on the converter input torque, but the speed ratio
φ(t) = ωgb(t)
ωe(t) (3.20)
is important as well. The parameters ρh anddp stand for the density of the converter fluid and for the pump diameter, respectively. The function ξ(φ) must be determined using experiments. Its qualitative form is illustrated in Fig. 3.9.
The converter output torque T1,gb, which in this case may be larger than the input torque, is determined by the pump–turbine interaction
T1,gb=ψ(φ(t))·T1,e(t). (3.21) The functionψ(φ) must be experimentally determined as well. Qualitatively, it will have a form similar to the one shown in Fig. 3.9. Equations (3.19) and (3.21) are valid in steady-state conditions. However, since the fluid dynamic processes inside the torque converter are substantially faster than the typical time constants of the vehicle longitudinal dynamics, the fluid dynamic effects may often be neglected.
ψ(φ)
φ (−)
0 0.5 1.0
ξ(φ) (-)
Fig. 3.9.Qualitative representation of the main parameters of a torque converter.
With these preparations, the efficiency of the torque converter in traction mode is easily found to be
ηtc=ωgb·T1,gb
ωe·T1,e =ψ(φ)·φ . (3.22)
54 3 IC-Engine-Based Propulsion Systems