140 5 Non-electric Hybrid Propulsion Systems
Table 5.1.Numerical values for the flywheel parameters.
µ 1.5·10−3
k 4
dw/d 0.08 ρa 1.3 kg/m3 ηa 1.72·10−5Pa s ρ 8000 kg/m3
or downstream shaft, ω2(t). The model output variable is the torque at the output shaft,T2(t).
FW T 2
ω2
Fig. 5.9. Flywheel accumulators: physical causality for dynamic modeling.
The equations developed in the previous section are suitable also for dy- namic modeling. The only dynamic term is related to the derivative of the flywheel speed. It can be easily estimated sinceω2(t) is an input variable.
5.3 Continuously Variable Transmissions 141 Continuously variable transmissions are key components in hybrid-inertial vehicles, where the rotational speed of the flywheel accumulator must be de- coupled from that of the drive train. In hybrid-electric vehicles, though not strictly necessary, the use of CVTs is gaining a notable amount of attention as an additional control device [232, 55].
The core of a modern CVT (see Fig. 5.10) consists of a transmitter element and two V-shaped pulleys. The element that transmits power between the pulleys can be a metal belt [81] or a chain [190]. The pulley linked to the prime mover is referred to as primary pulley, while the one connected to the drive train is the secondary pulley. Each pulley consists of a fixed and an axially slidable sheave. Each of the two moving pulley halves is connected to a hydraulic actuation system, consisting of a hydraulic cylinder and piston. In the simplest hydraulic configuration the secondary pressure is the hydraulic supply pressure. The primary pressure is determined by one or more hydraulic valves, actuated by a solenoidal (electromagnetic) valve, which connect the two circuits with the pump return line [264].
In the i2-CVT used in the ETH-III propulsion system [65], the range of the CVT alone (typically 1:5) is substantially increased by combining the CVT core (a chain converter) with a two-ratio gear box. In the “slow” gear arrange- ment, the overall transmission ratio is given by the product of the CVT ratio and the gear ratio. In the “fast” gear arrangement, the power flow through the CVT is inverted, thus the overall transmission ratio is proportional to the reciprocal of the CVT ratio. This allows for reaching transmission ratios higher than 1:20.
5.3.1 Quasistatic Modeling of CVTs
The causality representation of CVTs in quasistatic simulations is sketched in Fig. 5.11. The input variables are the torqueT2(t) and the speedω2(t) at the downstream shaft. The output variables are the torque T1(t) and the speed ω1(t) required at the input shaft.
A simple quasistatic model of a CVT can be derived neglecting slip at the transmitter but considering torque losses [101, 232, 155]. The definition of the transmission (speed) ratioν implies that
ω1(t) =ν(t)·ω2(t). (5.8) Newton’s second law applied to the two pulleys yields
Θ1· d
dtω1(t) =T1(t)−Tt1(t), (5.9) Θ2· d
dtω2(t) =Tt2(t)−T2(t), (5.10) whereTt1(t) and Tt2(t) represent the torque transmitted by the chain or the metal belt to the pulleys. The torque losses may be taken into consideration by applying them to the primary pulley, as
142 5 Non-electric Hybrid Propulsion Systems
primary pulley
secondary pulley
pump
reservoir return
line hydraulic valve
electrovalve uCVT
Fig. 5.10.Scheme of a CVT system.
ω2 T2
CVT ω 1
T1
Fig. 5.11. CVTs: causality representation for quasistatic modeling.
Tt2(t) =ν(t)·(Tt1(t)−Tl(t)) . (5.11) Combining (5.8)-(5.11) and recalling that ν is a variable quantity, the final expression for the input torque obtained is
T1(t) =Tl(t) +T2(t)
ν(t) +ΘCV T
ν(t) · d
dtω2(t) + Θ1· d
dtν(t)·ω2(t), (5.12) where the secondary reduced inertia is defined as ΘCV T = Θ2+ Θ1·ν2(t).
The variation of the gear ratio ν(t) is determined by the CVT controller,
˙
ν(t) =uCV T(t), thus it is an input variable for the model.
The evaluation of the torque losses requires the evaluation of the losses in the hydraulic part of the CVT, which are dominated by the pump losses, and the friction losses at the mechanical contacts between various CVT compo- nents. Analytical expressions forTl(t) are derived by fitting experimental data, but due to the complexity of the processes involved, at least a second-order dependency of Tl(t) on ν(t),ω1(t) and on the input power P1(t) is required [265, 65].
5.3 Continuously Variable Transmissions 143 An alternative approach is based on the definition of the CVT efficiency ηCV T(ω2, T2, ν), according to which the torque required at the input shaft is evaluated as
T1(t) = T2(t)
ν(t)·ηCV T(ω2(t), T2(t), ν(t)), T2(t)>0, (5.13) T1(t) = T2(t)
ν(t) ·ηCV T(ω2(t), T2(t), ν(t)), T2(t)<0. (5.14) The typical dependency of ηCV T on output speed and torque as well as on the transmission ratio is depicted in Fig. 5.12. The efficiency of the CVT increases with torque at constant speed andν, exhibiting a maximum that is more pronounced at higher speeds. Higher transmission ratios also favorably affect the efficiency. Values around 90% can be reached for high-load, low- speed conditions. Values lower than 70% are typical during low-load operation [265, 190].
ηCVT
T2
ν
100%
0% ω2
ν
ηCVT 100%
Fig. 5.12.Qualitative dependency of CVT overall efficiency on torque, speed, and transmission ratio.
As an example, Fig. 5.13 shows the result of a detailed model of a push-belt CVT [189] of the type discussed in the next section.
The efficiency of CVTs can be approximated using Eqs. (3.13) and (3.14) as well. However, in this case the factor egb is smaller than in conventional gear boxes and depends on the speed and gear ratio of the CVT. At a gear ratio ν = 1, typical values are egb = 0.96 and P0,gb = 0.02·Pmax at the rated speed and egb = 0.94 and P0,gb = 0.04·Pmax at 50% of the rated speed. At the maximum or minimum gear ratio, the idling losses increase, i.e., P0,gb = 0.04·Pmax at the rated speed and P0,gb= 0.06·Pmax at 50% of the rated speed.
144 5 Non-electric Hybrid Propulsion Systems
Fig. 5.13. Efficiency of a push-belt CVT as a function of input speed ωe, input torqueT1, and gear ratioν[189].
5.3.2 Dynamic Modeling of CVTs
The physical causality representation of a CVT is sketched in Fig. 5.14. The model input variables are the rotational speed at the output or downstream shaft,ω2(t), and the torque at the input or upstream shaft,T1(t). The model output variables are the torque at the output shaft, T2(t), and the speed at the input shaft,ω1(t).
ω2 T2 CVT ω 1
T1
Fig. 5.14.CVTs: physical causality for dynamic modeling.
Dynamic CVT models calculate the rate of change of ν(t), and conse- quently the output variables, as a result of a fundamental hydraulic and me- chanical modeling of the system. The hydraulic submodel calculates the forces acting on the primary and secondary pulley halves,F1(t) andF2(t), which ba- sically depend on pressure in the corresponding circuits and pulley speed (via a centrifugal term). The pressure is modeled according to continuity equations in the various branches of the hydraulic circuit, whose geometrical characteris- tics depend on the status of the hydraulic valves and ultimately on the electric control signaluCV T(t) [265].