J. Cent. South Univ. (2015) 22: 180−188 DOI: 10.1007/s11771-015-2508-8

Parameter design and performance analysis of zero inertia continuously variable transmission system

HU Jian-jun(胡建军), JI Yi(吉毅), YAN Jiu-jiang(晏玖江)

State Key Laboratory of Mechanical Transmission (Chongqing University), Chongqing 400044, China

© Central South University Press and Springer-Verlag Berlin Heidelberg 2015

Abstract: In order to solve the problem of weak power performance of vehicle equipped with continuously variable transmission (CVT) system working under transient operating conditions, a new CVT equipped with planetary gear mechanism and flywheel was researched, a design method of transmission parameter optimization was proposed, and the comprehensive matching control strategy was established for the new transmission system. Fuzzy controllers for throttle opening and CVT speed ratio were designed, and power performance and fuel economy of both vehicles respectively equipped with conventional CVT system and new transmission system wrere compared and analyzed by simulation. The results show that power performance and fuel economy of the vehicle equipped with new transmission system are better than that equipped with conventional CVT, thus the rationality of the parameter design method and control algorithm are verified. Key words: continuously variable transmission; inertia; optimization; fuzzy control; simulation

1 Introduction

Reducing both fuel consumption and exhaust emissions has been one of the main directions for the vehicle research nowadays. Vehicle equipped with continuously variable transmission (CVT) system can obtain the best fuel economy by controlling engine to work along its optimal economic curve, but its reserve- power to meet the power performance is very small [1−2]. When the vehicle needs hard acceleration, the CVT speed ratio will be decreased. However, when the change rate of the CVT speed ratio is very high, much power will be consumed to overcome the rotational inertia of engine owing to the high angular acceleration of the engine speed, thus causing the decrease of the vehicle acceleration and the deterioration of driving performance. When the change rate of the CVT speed ratio is too low, the acceleration response will be very slow and the engine speed will deviate from the optimal economic working point for a long time. The research shows that the weak power performance of the vehicle equipped with CVT under transient operating conditions is one of important reasons, for which the vehicle equipped with CVT cannot be generally accepted by consumers [2]. Therefore, it is an urgent problem how to improve power performance of the vehicle equipped with CVT under transient operating conditions.

The control of speed ratio is one of the key technologies for CVT system, and the change rate of the CVT speed ratio has an important impact on the dynamic characteristics of vehicle [3−4]. Many researchers have tried to improve power performance of the vehicle equipped with CVT under transient operating conditions by appropriately controlling the CVT speed ratio [5−7], but the economy and acceleration response speed would be decreased by the method. In order to improve power performance of the vehicle equipped with CVT under transient operating conditions and simultaneously keep the economy and acceleration response speed, the supplement of extra power is needed. The hybrid electric vehicle can meet the requirements above, that is to say, the engine works along its optimal economic curve under steady-state conditions and the extra power can be supplied by electric driving system under transient operating conditions. But it is difficult for hybrid electric vehicle to be widely used because of its own disadvantages such as high weight, high cost and complicated system.

The problem that both fuel economy and power performance of the vehicle equipped with conventional CVT system cannot be simultaneously guaranteed under transient operating conditions can be solved by a zero inertia CVT system. Based on the structure and dynamic analyses of the zero inertia flywheel booster system, the optimization design of the system parameter is carried

Foundation item: Project(2011BA3019) supported by the Chongqing Natural Science Foundation, China Received date: 2013−09−17; Accepted date: 2013−12−27 Corresponding author: HU Jian-jun, PhD; Tel: +86−13996073282; E-mail address: hujianjun@cqu.edu.cn

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out, and the control strategy for the new transmission system is established and analyzed by simulation. The results show that the power performance and fuel economy of vehicle equipped with new transmission system are better than that equipped with conventional CVT. 2 Structure and dynamics analysis of

flywheel booster system

The structure of the flywheel booster system is shown in Fig. 1. The flywheel is connected to the sun gear of planetary gear mechanism, the planet carrier is connected to the secondary shaft (driven shaft) of the CVT and the annulus gear is connected to the primary shaft (driving shaft), respectively. When the power of engine needs to be output fast, the driving shaft speed of CVT can be rapidly increased through absorbing the energy released by the flywheel rather than only absorbing the energy provided by the engine relative to the conventional CVT system, so that the power shortage of vehicle caused by the decrease of actual power output by transmission system can be avoided. In the model, the inertia of engine is overcome by the inertia of flywheel, so that the whole system is manifested as zero inertia. Therefore, the new transmission system is called zero inertia CVT (ZI-CVT) [8].

Fig. 1 Structure of zero inertia CVT system

The transmission system equipped with ZI-CVT

consists of an engine, a torque converter, a drive-neutral- reverse (DNR) set, a CVT with metal pushing belt type, a final drive, a differential, a planetary gear mechanism and a flywheel, of which core structure is shown in Fig. 2.

In Fig. 2, rcvt is the speed ratio of CVT ranging from 0.4 to 2.5 [9]; rd is the constant speed ratio. The equations can be given by

cvt s p d w s/ , /r r (1) where ωp is the driving shaft speed of CVT (rad·s−1); ωs is the driven shaft speed of CVT (rad·s−1); and ωw is the wheel speed (rad·s−1).

Fig. 2 Core structure of zero inertia CVT system

It is defined that ra=ωp/ωa and rc=ωs/ωc. Based on

the motion relationship among the each part of the planetary gear mechanism, the flywheel speed ωf can be obtained by

pf s s p p s s

cvtr

(2)

where s a s c( / 1) /R R r and p a s a/( );R R r ωa is the annulus gear speed (rad·s−1); ωc is the carrier speed (rad·s−1); Ra is the annulus gear radius (m); Rs is the sun gear radius (m). When ωf is equal to zero, rgn=αp−αs.

The power balance equation of the system can be obtained by

a p c s f f 0T T T (3) where Ta is the torque of the annulus gear (N·m); Tc is the torque of the planet carrier (N·m); Tf is the torque of the flywheel (N·m). Based on Eqs. (2) and (3), the torques of the annulus gear, the planet carrier and the flywheel are related by

a p f p f f

c s f s f f

T T J

T T J

(4)

It is known that the planetary gear mechanism splits

the system power. The primary shaft can be provided to the assisted torque by the flywheel through controlling the change rate of CVT speed ratio.

Based on the torque balance of the secondary shaft, the external drive or load torque Tp of the CVT driving shaft and the external drive or load torque Ts of the CVT driven shaft are related by

p 1 ps 2 s

cvt

T JT J

r

(5)

where 1 p p p s cvt f( ) ;J J r J p2 s s

cvt

J Jr

s f ;J

Jf is the rotational inertia of the flywheel

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(kg·m2); Jp and Js are the rotational inertia of the CVT driving shaft, and driven shaft respectively (kg·m2). 3 Parameter optimization design

Based on the analysis of the core components of ZI-CVT system, the energy efficiency of flywheel is determined by the parameters such as Jf, Ra/Rs, ra, and rc. In order to give full play to the function of flywheel, it is necessary to optimize the structure and parameters of the planetary gear mechanism for different vehicle types [10−11]. 3.1 Energy function

The main purpose of system design is to offset the effect of inertia on the driving shaft by using flywheel.

The energy of the primary shaft of CVT is expressed as

22 s

p e p t p e p t 2cvt

1 1( ) ( )

2 2E J J J J J J

r

(6)

where Jt is the rotational inertia of torque converter (kg·m2), and Je is the rotational inertia of input terminal (the sum of the rotational inertia of the engine, torque converter, DNR set, CVT driving disc and annulus gear) (kg·m2).

The energy of flywheel is expressed as

2 2 2 2 2f f f f s gn s

gn cvt

1 1 1 1( )

2 2E J J r

r r (7)

The total energy of the output terminal (including

whole vehicle) is expressed as

2 2 2s s v d s v v w w

1( ) , 2

2E J J r J m R J (8)

where mv is the gross mass of vehicle (kg); Rw is the wheel radius(m); Jw is the rotational inertia of the output terminal of wheel (the sum of the rotational inertia of the planet carrier, CVT driven shaft, final drive, differential and wheel) (kg·m2).

The specific energy ep of the primary shaft and specific energy ef of the flywheel relative to the output terminal, and the total specific energy e are expressed as

p 2fp f f2

s s gn cvtcvt

1 1, ( ) ,

E Ee e

E E r rr

f pe e e (9)

where e p t 2 f

f p2e p ts v d

and .J J J J

J J JJ J r

The relationship between the specific energy of system and CVT speed ratio rcvt can be obtained based on Eq. (9), which is shown in Fig. 3. For the designed

planetary wheel set, the total specific energy e is only determined by rcvt. When rcvt is equal to rzi, e reaches the minimum value. rzi can be given by

fzi gn

f

1r r

(10)

Fig. 3 Relation ship between specific energy and CVT speed

ratio

The objective function is defined as

LB B B L= / + ,W d d where WB is weight coefficient, dB is the difference value of total specific energy when rcvt=rod (the maximum speed ratio）and rcvt=rzi, and dL is the difference between the total specific energy and the specific energy of the primary shaft when rcvt=rud (The minimum speed ratio).

It is indicated that the more energy that flywheel releases when the speed ratio of CVT decreases and the higher the effectiveness of flywheel is; while, the energy efficiency of flywheel is related to the value of rgn, as shown in Fig. 3. When rcvt>rgn, ef gradually decreases with the decrease of CVT speed ratio, and flywheel releases the energy to increase ep. The larger the dB is, the more the e is reduced, that is to say, the more the energy stored in flywheel can be released, the higher the energy utilization efficiency is. While vehicle is starting, rcvt<rgn, the flywheel rotates reversely to absorb energy, and flywheel speed increases with the decrease of rcvt. So, the flywheel speed will be increased when vehicle starts, which is unnecessary. The energy absorbed by flywheel should be reduced at this time, that is to say, the value of dL in Fig. 3 should be reduced to improve energy utilization efficiency. Therefore, in order to make full use of the energy of flywheel, the value of the design objective functions should be the minimum to meet the requirements of the starting performance and accelerating performance of the transmission system. 3.2 Parameter optimization

The expression of objective function ΓLB is expressed as

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2 2

B f fLB f f

od gn gn ud

1 1 1(1+ ) +

W

r r r r

(11)

The constraints of design parameters can be

obtained from Fig. 3:

od zi gn ud f, 0r r r r (12)

In order to achieve optimal matching of flywheel and transmission system, the maximum values allowed for the rotational inertia and the speed of the flywheel are Jf,max=0.4 kg·m2, ωf,max=750 rad/s, respectively [12]. The following expression can be obtained by the constraint conditions such as ωf≤ωf,max, αs≤αs,max:

s

f ,maxs,max

s gn E line s( ( ))max r

(13)

where ωE−line(ωs) is the engine speed of the optimal economic curve (rad/s).

When Jf≤Jf,max, the following expression can be obtained by Eq. (9):

f,max2f s,max gn

e p t

( )J

rJ J J

(14)

According to the constraint condition which is

composed of Eqs. (12) and (14), the objective function ΓLB is optimized, thus rgn and γf can be got correspondingly by the optimization toolbox of Matlab. The partial calculation results of different WB and Jf,max are shown in the Table 1 correspondingly. As it is shown in the table, when the maximum permissible rotational inertia of flywheel Jf,max increases, dB increases, and dL decreases. So, the utilization ratio of the flywheel energy is improved. When the weight coefficient WB decreases, dB decreases, dL decreases and but the flywheel energy utilization ratio is still improved because the latter decreases more. Comprehensively considering the influence of Jf,max and WB, the value of the objective function ΓLB in the group 4 is the minimum in Table 1, which means that the optimization results is the best. Table 1 Optimization results of objective function

No. Jf,max/(kg·m2) WB dB dL ΓLB rgn rzi

1 0.20 0.02 0.032 0.246 0.866 0.901 1.143

2 0.20 0.004 0.021 0.098 0.287 0.766 1.142

3 0.40 0.020 0.056 0.112 0.448 0.698 0.940

4 0.40 0.004 0.040 0.036 0.136 0.494 0.948

The design variables include Jf, Ra/Rs, ra, and rc. The

rgn and γf for different WB and Jf,max can be got correspondingly by the optimization toolbox of Matlab.

The αp can be got based on the expression of γf in Eq. (9). Eventually, the Ra/Rs, ra, and rc can be got based on the expression of rgn, αp and αs in Eq. (2). 4 Study on control strategy for transmission

system 4.1 Determination of control goal

Generally, vehicle acceleration is treated as the evaluation indicator of drivability. Meanwhile, the vehicle acceleration is affected by wheel load torque Trv and the wheel driving torque Td. However, the wheel load torque Trv is uncontrolled to change the pedal position aims at changing the vehicle speed rather than accurately determining the vehicle acceleration; therefore, the wheel driving torque Td is treated as the control variable.

In the actual control process, the target torque Td,d is treated as the input value and the actual torque Td is treated as the feedback quantity, so that the target torque Td,d can be better tracked. Furthermore, the target torque Td,d is obtained by changing the pedal position δ. The relationship of target torque with pedal position δ and actual wheel speed ωw is shown in Fig. 4.

Fig. 4 Relation of wheel target torque with pedal position δ and

actual speed ωw

4.2 Determination of control strategy

For the CVT system, the reserve torque which reflects vehicle drivability is related to the changing rate of CVT speed ratio. When the required driving torque is greater than the reserve torque, the vehicle can only be accelerated by decreasing the speed ratio of CVT. The vehicle acceleration is obtained by

tot w e rv eq w3J T T J r

r r

(15)

where cvt d tot wq eq wq w gn2, ; (1 / )r r r J J J J J r r

r

2w f ;J 2

eq e gn e f(1 /( )) ;J J r r J r is the overall

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transmission ratio of driveline; Jtot is the total equivalent rotational inertia (kg·m2); Jwq is the equivalent rotational inertia of vehicle (kg·m2); Jeq is the equivalent rotational inertia of engine side (kg·m2); Te is the engine output torque (N·m); αe is the torque magnification factor from flywheel to the driving shaft of CVT; αw is the torque magnification factor from flywheel to the wheel; η is the transmission efficiency.

In order to ensure w 0, Eq. (15) can be transformed into

2d rveq cvt e cvt

w

( )r rT

J r T r

(16)

If the change of wheel load torque Trv is small and

slow, the value of Jeq(r) is always positive for conventional CVT system when the accelerator opening is increased, so that the change rate of the CVT speed ratio should be limited. However, the value of Jeq(r) is negative for the ZI-CVT when rcvt>rzi, which means that the change rate of the CVT speed ratio is not limited.

In order to obtain better drivability, both the engine torque Te and the change rate of CVT speed ratio should be synthetically controlled for conventional CVT system. The torque converter is locked up under normal driving condition, that is to say ωp=ωe. Because the fixed reduction ratio is constant, the change rate of CVT speed ratio r can be controlled directly. The expression can be got from Eq. (15):

2wq e e rv eq wq e

d( )

d

rJ r T rT J J r

t (17)

For the conventional CVT system, the output torque

of engine Te is limited by the minimal torque of engine Tα=0 and the engine torque TWOT at full load. The control strategy is expressed as,

d,de WOT e eq p OLmin( ( ),max( , )), 0T T T r J T (18)

e 0 e d,d eq p OLmax( ( ),min( , )), 0T T T r J T (19) where TOL is the engine torque operating under optimal fuel economy curve (N·m).

Otherwise, the control law for the ZI-CVT system should be considered respectively under the situations such as rzi<rcvt<rod and rud<rcvt<rzi. The control law of engine torque is expressed as

2 w cvte OL

d

J rT T

r

(20)

The change rate of CVT speed ratio should meet the

equation Td=Td,d; finally, the control law is described by 3 2 2cvt d cvt d

cvt d rv wq d,d ew eq w eq w

( )r r r r

r J T J T TJ J J

(21)

where Jd= Jwq− Jw. For Te and cvt w,r is equal to zero or

is very small when the transmission system operates under steady state or semi-steady state, so that the change rate of CVT speed ratio can be controlled based on Te=TOL.

The target driving torque Td,d can be tracked well by the proposed control strategy when rzi<rcvt<rod. The control method is the same as the traditional CVT system owing to Jeq>0 when rud<rcvt<rod, which means that the change rate of CVT speed ratio should be limited. 4.3 Controller design

Generally, the fuzzy controller has strong stability and robustness. Consequently, the feedback fuzzy controller is adopted to control the change rate of engine speed and the throttle opening.

The error between target engine speed and actual engine speed Δωe and its change rate e are treated as the input values of CVT speed ratio controller, and the change rate of engine speed e is treated as the output value. The fuzzy control law obtained according to the experience is shown in Table 2. Table 2 Fuzzy control laws

)(e Δωe(ΔTe)

N Z P

)( ee T

N PL PS Z

Z PS Z NS

P Z NS NL

The error between target engine torque and actual

engine torque ΔTe and its change rate eT are treated as the input values of throttle opening controller, and the change rate of throttle opening is treated as the output value. The fuzzy control law obtained according to the experience is shown in Table 2.

The input and output of the two controllers can be defined as follows.

1) The input variable Δωe and ΔTe are divided into three fuzzy subsets namely EN (error negative), EZ (error zero) and EP (error positive), and the input variables )( and ee T are divided into another three fuzzy subsets, namely CN, CZ and CP.

2) The output variables e and are divided into five fuzzy subsets, namely NL (negative large), NS (negative small), Z(zero), PS(positive small) and PL (positive large).

3) The fuzzy domain of both the input and output of controller is [−1, 1].

4) The membership functions of the two input variables Δωe and e are the same, and the membership functions of the two input variables ΔTe and

eT are also the same. The membership functions of the two controllers are shown in Figs. 5 and 6.

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Fig. 5 Membership functions of input variable and output

variable for CVT speed ratio controller

4.4 Simulation and analysis of acceleration and

deceleration process In the simulation, the initial value of acceleration

pedal opening is set as 0.15. When the time is 1.5 s, the value of acceleration pedal opening increases to 1 from 0.15 and then remains unchanged. When the time is 6.5 s, the value of acceleration pedal opening quickly decreases to 0.15 from 1 and then remains unchanged again. This simulation results are shown in Fig. 7. The change rate of speed ratio for the conventional CVT is

Fig. 6 Membership functions of input and output variable of

throttle opening controller

higher than that for the ZI-CVT, and that the deviation between the target and the actual driving torque for the conventional CVT is larger than that for the ZI-CVT. Furthermore, the acceleration of the vehicle equipped with the ZI-CVT is rapidly increased to 0.9 m/s2 after the acceleration pedal is pressed for 1.5 s, and that the acceleration of the vehicle equipped with the ZI-CVT rapidly decreases to less than −0.25 m/s2 after the acceleration pedal is loosened for 6.5 s. However, the acceleration of the vehicle equipped with the conventional

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CVT increases after slightly decreased when the acceleration pedal is pressed. It is caused by the fact that the vehicle is accelerated only after the primary shaft is accelerated by engine. Moreover, the acceleration of the vehicle equipped with the conventional CVT causes an inverse response after the acceleration pedal is released. The reason for this phenomenon is that inertia moment from the primary shaft of the conventional CVT is transmitted to the wheels and consequently the vehicle

generates slightly accelerating responses. The distribution maps of the engine operating point

under two transient operating conditions are shown in Fig. 8 and the engine operating point for the conventional CVT under transient operating conditions is shown in Fig. 8(a). It is noted that the initial operating point is respectively Te=59.6 N·m and ωe=201 rad/s. For the conventional CVT, the engine operating point moves quickly to the external characteristic curve at 1.5 s and

Fig. 7 Simulation results of two types of transmission systems

Fig. 8 Distribution map of engine operating point for two transmissions under transient operating conditions: (a) Conventional CVT;

(b) ZI-CVT

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moves along this curve. While the target path of engine torque affected by the driving shaft of CVT has somewhat offset at 6.5 s, and the engine torque mainly returns the initial operating point according to the equivalent power curve. However, for the ZI-CVT, the initial engine torque is larger than the target torque corresponding to the optimum operating curve owing to the increase of the equivalent rotational inertia of the secondary shaft. When the flywheel assists driveline, the engine operating point starts to work along the optimum operating curve. And at 6.5 s, the engine torque keeps constant to return the initial operating point when the acceleration pedal rapidly returns the initial position. 4.5 Simulation for power performance and economy

The acceleration time for which the vehicle is accelerated with the largest accelerator opening from 48 km/h to 112 km/h is treated as the measurement standard for vehicle power performance [13−14].

Simulation results of the vehicle power performance are shown in Fig. 9 according to the above request. It is known that the acceleration time for the conventional CVT and the ZI-CVT is respectively 12.09 s and 9.2 s, so the overtaking time is reduced by 2.89 s.

The evaluation index of the vehicle fuel economy usually uses the fuel consumption per 100 km at constant

Fig. 9 Simulation results of overtaking acceleration

speed and under certain driving cycle [15−16]. Simulation tests were performed for two types of transmission system under the ECE driving cycle. The results indicate that the fuel consumption per 100 km for the conventional CVT system is 8.6 L and for the ZI-CVT is 8.139 L, and that the fuel consumption is decreased by 5.3%. 5 Conclusions

1) The dynamic characteristics of a new ZI-CVT structure are analyzed, and the parameters of the structure are optimally designed.

2) The fuzzy controller of CVT speed ratio and the fuzzy controller of throttle opening are designed on the basis of the proposed control strategy for two kinds of transmissions. Simulation results indicate that the target torque of the ZI-CVT could be tracked by actual torque well.

3) The power performance and fuel economy of the vehicle equipped with the ZI-CVT and the conventional CVT are respectively analyzed. The results indicate that the acceleration time of the vehicle equipped with the ZI-CVT decreases by 2.89 s and the fuel consumption under the ECE driving cycle decreases by 5.3% in comparison with the vehicle equipped with the conventional CVT, so the power performance under transient operating conditions and fuel economy of the former are obviously improved. Therefore, the rationality of control strategy can be verified. References [1] JAMES D, MICHAEL A. Switch-mode continuously variable

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(Edited by YANG Hua)