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Continuously Variable Transmission

Push belt continuously variable transmission with independent radii control

  • Library:
  • Powertrain Blockset / Transmission / Transmission Systems

Description

The Continuously Variable Transmission block implements a push belt continuously variable transmission (CVT) with independent radii control. Use the block for control system design, powertrain matching, and fuel economy studies. You can configure the block for internal or external control:

  • Internal — Input direction and pulley ratio requests

  • External — Input direction and pulley displacement requests

The table summarizes the pulley kinematic, speed reduction, and dynamic calculations made by the Continuously Variable Transmission block.

CalculationPulley KinematicsReverse and Final Speed ReductionDynamics
Final angular speed ratio
Belt torque applied to the secondary and primary pulleys  
Torque applied to the secondary and primary pulleys  
Angular velocity of secondary and primary pulleys
Belt and pulley geometry  
Belt linear speed  
Wrap angle on secondary and primary pulley  
Primary and secondary pulley radii  

The figure shows the CVT variator with two configurations. In the first configuration, which illustrates speed reduction, the variator is set to decrease the primary pulley radius and increase the secondary pulley radius. In the second configuration, which illustrates overdrive, the variator is set to increase the primary pulley radius and decrease the secondary pulley radius.

Pulley Kinematics

Using the physical dimensions of the system, the block calculates the primary and secondary variator positions that meet the pulley ratio request.

The figure and equations summarize the geometric dependencies.

Cdist=rpmax+rgap+rsec_maxL0=f(rpmax,rsmax,rpmin,rsmin,Cdist)ratiocommand=f(ratiorequest,ratiomax,ratiomin)rpri=f(r0,ratiocommand,Cdist)rsec=f(r0,ratiocommand,Cdist)xpri=f(r0,rpri,θwedge)xsec=f(r0,rsec,θwedge)

The equations use these variables.

ratiorequest

Pulley gear ratio request

ratiocommand

Pulley gear ratio command, based on request and physical limitations

rgap

Gap distance between variator pulleys

Cdist

Distance between variator pulley centers

rpmax

Maximum variator primary pulley radius

rsmax

Maximum variator secondary pulley radius

rpmin

Minimum variator primary pulley radius

rsmin

Minimum variator secondary pulley radius

ro

Initial pulley radii with gear ratio of 1

Lo

Initial belt length, resulting from variator specification

xpri

Variator primary pulley displacement, resulting from controller request

xsec

Variator secondary pulley displacement, resulting from controller request

rpri

Variator primary pulley radius, resulting from controller request

rsec

Variator secondary pulley radius, resulting from controller request

Θwedge

Variator wedge angle

Φ

Angle of belt to pulley contact point

L

Belt length, resulting from variator position

Reverse and Final Speed Reduction

The CVT input shaft connects to a planetary gear set that drives the primary pulley. The shift direction determines the input gear inertia, efficiency, and gear ratio. The shift direction is the filtered commanded direction:

DirshiftDir(s)=1τss+1

For forward motion (Dirshift=1):

Ni=1ηi=ηfwdJi=Jfwd

For reverse motion (Dirshift=1):

Ni=Nrevηi=ηrevJi=Jrev

The gear ratio and efficiency determine the input drive shaft speed and torque applied to the primary pulley:

Tapp_pri=ηiNiTi

The block reduces the secondary pulley speed and applied torque using a fixed gear ratio.

Tapp_sec=ToηoNoωo=ωsecNo

The final gear ratio, without slip, is given by:

Nfinal=ωiωo=NiNorsecrpri

The equations use these variables.

Ni

Input planetary gear ratio

Dir

CVT direction command

Dirshift

Direction used to determine planetary inertia, efficiency, and ratio

τs

Direction shift time constant

ηfwd, ηrev

Forward and reverse gear efficiency, respectively

Jfwd, Jrev

Forward and reverse gear inertia, respectively

Nrev

Reverse gear ratio

Tapp_pri, Tapp_sec

Torque applied to primary and secondary pulleys, respectively

Ti

Input drive shaft torque

ωi, ωo

Input and output drive shaft speed, respectively

ωpri, ωsec

Primary and secondary pulley speed, respectively

Nfinal

Total no-slip gear ratio

Dynamics

The maximum torque that the CVT can transmit depends on the friction between the pulleys and belt. According to Prediction of Friction Drive Limit of Metal V-Belt, the torque friction is defined as:

Tfric(rp,μ)=2μFaxrpcos(ϑwedge)

Without macro slip, the tangential acceleration of the pulley is assumed to be equal to the belt acceleration. Once the torque reaches the static friction limit, the belt begins to slip, and the pulley and belt acceleration are independent. During slip, the torque transmitted by the belt is a function of the kinetic friction factor. During the transition from slip to non-slip conditions, the belt and tangential pulley velocities are equal.

The block implements these equations for four different slip conditions.

ConditionEquations

Belt slips on both secondary and primary pulleys

(Jpri+Ji)ω˙pri=Tapp_pri-TBoP_pri-bpriωpriJsecω˙sec=Tapp_sec-TBoP_sec-bsecωsecmbv˙b=TBoP_prirpri+TBoP_secrsec-bbvbrpriωprivbrsecωsecvb

Belt slips on only the primary pulley

(Jpri+Ji)ω˙pri=Tapp_pri-TBoP_pri-bpriωpri(mb+Jsecr2sec)v˙b=TBoP_prirpri+TBoP_secrsec-(bb+bsecr2sec)vbωsec=vbrsecrpriωprivbTBoP_pri=sgn(rpriωprivb)Tfric(rpri,μkin)|TBoP_sec|<Tfric(rsec,μstatic)

Belt slips on only the secondary pulley

(mb+Jpri+Jir2pri)v˙b=Tapp_prirpri+TBoP_secrsec-(bb+bprir2pri)vbJsecω˙b=Tapp_sec+TBoP_sec-bsecωsecωpri=vbrprirsecωsecvbTBoP_sec=sgn(rsecωsecvb)Tfric(rsec,μkin)|TBoP_pri|<Tfric(rpri,μstatic)

Belt does not slip

(mb+Jsecr2sec+Jpri+Jir2pri)v˙b=Tapp_prirpri+Tapp_secrsec-(bb+bsecr2sec+bprir2pri)vbωpri=vbrpriωsec=vbrsec|TBoP_pri|<Tfric(rpri,μstatic)|TBoP_sec|<Tfric(rsec,μstatic)

Slip direction

PriSlipDir={0rpriωpri=vb1rpriωpri>vb1rpriωpri<vbSecSlipDir={0rsecωsec=vb1rsecωsec>vb1rsecωsec<vb

The equations use these variables.

TBoP_pri, TBoP_sec

Belt torque acting on the primary and secondary pulleys, respectively

Tapp_pri, Tapp_sec

Torque applied to primary and secondary pulleys, respectively

Jpri, Jsec

Primary and secondary pulley rotational inertias, respectively

bpri, bsec

Primary and secondary pulley rotational viscous damping, respectively

Fax

Pulley clamp force

μ

Coefficient of friction

μkin, μstatic

Coefficient of kinetic and static friction

vb, аb

Linear speed and acceleration of the belt, respectively

mb

Total belt mass

rpri, rsec

Radii of the primary and secondary pulleys, respectively

Φwrap

Wrap angle of belt to pulley contact point

Φwrap_pri, Φwrap_sec

Primary and secondary pulley wrap angles, respectively

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal DescriptionVariableEquations

PwrInfo

PwrTrnsfrd — Power transferred between blocks

  • Positive signals indicate flow into block

  • Negative signals indicate flow out of block

PwrEng

Engine power

Peng

ωiTi
PwrDiffrntl

Differential power

Pdiff

ωoTo

PwrNotTrnsfrd — Power crossing the block boundary, but not transferred

  • Positive signals indicate an input

  • Negative signals indicate a loss

PwrBltLoss

Belt slip power loss

Pbltloss

(Jin+Jpri)ω˙priωpri+Jsecω˙secωsec+mbv˙bvb+bpriωpri2+bsecωsec2+bbvb2TapppriωpriTappsecωsec
PwrGearInLoss

Input planetary gear mechanical power loss

Pgrinloss

|ωiTiΤapp_priωpri|
PwrGearOutLoss

Output gear reduction mechanical power loss

Pgroutloss

|ωoToΤapp_secωsec|

PwrDampLoss

Mechanical damping loss

Pdamploss

bpriωpri2bsecωsec2bbvb2

PwrStored — Stored energy rate of change

  • Positive signals indicate an increase

  • Negative signals indicate a decrease

PwrStoredTrans

Rate change in rotational kinetic energy

Pstr

(Jin+Jpri)ω˙priωpri+Jsecω˙secωsec+mbv˙bvb

The equations use these variables.

Tapp_pri, Tapp_sec

Torque applied to primary and secondary pulleys, respectively

Ti, To

Input and output drive shaft torque, respectively

Jpri, Jsec

Primary and secondary pulley rotational inertias, respectively

bpri, bsec

Primary and secondary pulley rotational viscous damping, respectively

ωpri, ωsec

Primary and secondary pulley speed, respectively

ωi, ωo

Input and output drive shaft speed, respectively

vb, аb

Linear speed and acceleration of the belt, respectively

rpri, rsec

Radii of the primary and secondary pulleys, respectively

Ports

Inputs

expand all

Direction request, Dirreq, controlling the direction. The block filters the request to determine the direction, forward or reverse. Dir equals 1 for forward motion. Dir equals -1 for reverse.

Dir={1   when Dirreq01  when Dirreq<0

CVT pulley ratio request, ratiorequest.

Dependencies

To create this port, for the Control mode parameter, select Ideal integrated controller.

Variator primary pulley displacement, xpri, in m.

Dependencies

To create this port, for the Control mode parameter, select External control.

Variator secondary pulley displacement, xsec, in m.

Dependencies

To create this port, for the Control mode parameter, select External control.

External torque applied to the input drive shaft, Ti, in N·m.

External torque applied to the output drive shaft, To, in N·m.

Output

expand all

Bus signal containing these block calculations.

SignalDescriptionVariableUnits

EngTrq

Input shaft torque

Ti

N·m

DiffTrq

Output shaft torque

To

N·m

EngSpd

Input shaft speed

ωi

rad/s

DiffSpd

Output shaft speed

ωo

rad/s

PriRadius

Primary pulley radius

rpri

m

PriPhi

Primary pulley wrap angle

Φpri

rad

SecRadius

Secondary pulley radius

rsec

m

SecPhi

Secondary pulley wrap angle

Φsec

rad

BltLngthDelta

Change in belt length

ΔL

m

BltLngth

Belt length

L

m

BltLngthInit

Initial belt length

Lo

m

BltOnPriTrq

Belt torque acting on the primary pulley

TBoP_pri

N·m

BltOnSecTrq

Belt torque acting on the secondary pulley

TBoP_sec

N·m

BltVel

Linear speed of the belt

vb

m/s

PriAngVel

Primary pulley speed

ωpri

rad/s

SecAngVel

Secondary pulley speed

ωsec

rad/s

PriSlipDir

Primary pulley slip direction indicator

PriSlipDir

N/A

SecSlipDir

Secondary pulley slip direction indicator

SecSlipDir

N/A

TransSpdRatio

Total no-slip gear ratio

Nfinal

N/A

PwrInfo

PwrTrnsfrd

PwrEng

Engine power

Peng

W
PwrDiffrntl

Differential power

Pdiff

W
PwrNotTrnsfrdPwrBltLoss

Belt slip power loss

Pbltloss

W
PwrGearInLoss

Input planetary gear mechanical power loss

Pgrinloss

W
PwrGearOutLoss

Output gear reduction mechanical power loss

Pgroutloss

W
PwrDampLoss

Mechanical damping loss

Pdamploss

W
PwrStoredPwrStoredTrans

Rate change in rotational kinetic energy

Pstr

W

Input drive shaft angular speed, ωi, in rad/sec.

Output drive shaft angular speed, ωo, in rad/sec.

Parameters

expand all

Specify the control method, either internal or external.

Dependencies

This table summarizes the port and input model configurations.

Control ModeCreates Ports
Ideal integrated controller

PllyRatioReq

External control

PriDisp

SecDisp

Kinematics

Maximum variator primary pulley radius, rpmax, in m.

Maximum variator secondary pulley radius, rsmax, in m.

Minimum variator primary pulley radius, rpmin, in m.

Minimum variator secondary pulley radius, rsmin, in m.

The gap between the secondary and primary pulleys, rgap, in m. The figure shows the pulley geometry.

Variator wedge angle, Θwedge, in deg.

Dynamics

Primary pulley inertia, Jpri, in kg·m^2.

Secondary pulley inertia, Jsec, in kg·m^2.

Primary pulley damping coefficient, bpri, in N·m·s/rad.

Secondary pulley damping coefficient, bsec, in N·m·s/rad.

Belt damping coefficient, bb, in kg/s.

Static friction coefficient between the belt and primary pulley, μstatic, dimensionless.

Kinetic friction coefficient between the belt and primary pulley, μkin, dimensionless.

Belt mass, mb, in kg.

Pulley clamp force, Fax, in N.

Reverse and Output Ratio

Forward inertia, Jfwd, in kg·m^2.

Reverse inertia, Jrev, in kg·m^2.

Forward efficiency, ηfwd, dimensionless.

Reverse efficiency, ηrev, dimensionless.

Reverse gear ratio, Nrev, dimensionless.

Shift time constant, τs, in s.

Output gear ratio, No, dimensionless.

Output gear efficiency, ηo, dimensionless.

References

[1] Ambekar, Ashok G. Mechanism and Machine Theory. New Delhi: Prentice-Hall of India, 2007.

[2] Bonsen, B. Efficiency optimization of the push-belt CVT by variator slip control. Ph.D. Thesis. Eindhoven University of Technology, 2006.

[3] CVT How Does It Work. CVT New Zealand 2010 Ltd, 10 Feb. 2011. Web. 25 Apr. 2016. http://www.cvt.co.nz/cvt_how_does_it_work.htm

[4] Klaassen, T. W. G. L. The Empact CVT: Dynamics and Control of an Electromechanically Actuated CVT. Ph.D. Thesis. Eindhoven University of Technology, 2007.

[5] Sakagami, K. Prediction of Friction Drive Limit of Metal V-Belt. Warrendale, PA: SAE International Journal of Engines 8(3):1408-1416, 2015.

Extended Capabilities

C/C++ Code Generation
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