Nomographs for synthesis of epicyclic-type automatic transmissions

Abstract

This paper presents a MATLAB optimization technique for the determination of the gear ratios of epicyclic-type transmission mechanisms for a given set of velocity ratios. First, all of the feasible clutching sequences are enumerated directly without using complicated techniques. Then, for the transmission mechanism with the associated clutching sequence graph, the overall velocity ratios are derived and expressed in terms of the gear ratios of all the mating gears. Next, following the general trend of increased shift stages and a wider range of velocity ratios, the numbers of teeth of all gears are estimated by MATLAB optimization technique in a single run.

Ravigneaux gear mechanisms are used as design examples. The methodology can be applied to any transmission mechanism depending on its kinematic and geometric constraints. New five- and six-velocity automatic transmissions are enumerated from the Ravigneaux gear mechanism. It is a major breakthrough to design a completely satisfactory six-speed automatic transmission from the Ravigneaux gear mechanism since it has only eight links. The new design makes use of the benefits of the Ravigneaux gear train and overcomes the previous art difficulties. This structural design has realized a six-speed automatic transmission, while having minimal number of clutches, and brakes. It is also low cost due to adoption of the conventional available Ravigneaux gear train.

Appendix

(1) MATLAB objective function m.file (ravigneauxs.m)

function f=ravigneauxs(x)

RUD1=(x(1)*x(2))/x(4);

RUD2=((x(1)*x(2))-(x(1)*x(5)))/(x(4)- (x(1)*x(5)));

RUD3=((x(1)*x(2))-x(3))/(x(4)-x(3));

RDD=1;

ROD1=x(3)/(x(3)-x(4));

ROD2=-(x(1)*x(5))/(x(4)-(x(1)*x(5)));

RR=(x(3)/x(4));

DRUD1=3.3000;DRUD2=2.1500;DRUD3=1.6133;DRDD=1.0000;DROD1=0.7333;DROD2=0.5000;

DRR=-2.7500;Theta=110.0000;Z(4)=132;M=2;

f=(RUD1/DRUD1-1)^2+(RUD2/DRUD2-1)^2+(RUD3/DRUD3-1)^2+(RDD/DRDD-1)^2+(ROD1/DROD1-1)^2+(ROD2/DROD2-1)^2+(RR/DRR-1)^2;

(2) MATLAB Nonlinear constraints m.file (ravigneauxs_confun.m)

function [c, ceq]=ravigneauxs_confun(x)

RUD1=(x(1)*x(2))/x(4);

RUD2=((x(1)*x(2))-(x(1)*x(5)))/(x(4)- (x(1)*x(5)));

RUD3=((x(1)*x(2))-x(3))/(x(4)-x(3));

RDD=1;

ROD1=x(3)/(x(3)-x(4));

ROD2=-(x(1)*x(5))/(x(4)-(x(1)*x(5)));

RR=(x(3)/x(4));

M=2;

c =[x(4)-(x(1)*x(2));

x(3)-(x(1)*x(5));

abs((1-1/x(3))^2+((1/x(1)*x(2))-(1/x(1)))^2-(1/(M)^2)*(1-1/x(1))^2)-abs(2*((1/x(1)*x(2))-(1/x(1)))*(1-1/x(3)));

acosd(((1-1/x(3))^2+((1/(x(1)*x(2)))-(1/x(1)))^2-(1/(M)^2)*(1-1/x(1))^2)/(2*((1/(x(1)*x(2)))-(1/x(1)))*(1-1/x(3))))+asind(1/(1-(1/x(2))))+asind(1/(1-(1/x(3))))-110.000;

1.3200-(RUD1/RUD2);1.3200-(RUD2/RUD3);1.3200-(RUD3/RDD);1.3200-(RDD/ROD1);1.3200-(ROD1/ROD2);abs((RUD1/RUD2)-(RUD2/RUD3))-0.3000;abs((RUD2/RUD3)-(RUD3/RDD))-0.3000;abs((RUD3/RDD)-(RDD/ROD1))-0.3000;abs((RDD/ROD1)-(ROD1/ROD2))-0.3000;0.5000-ROD2;abs(RUD2)-abs(RR);abs(RR)-abs(RUD1);];

ceq =[(1/x(4))+(1/x(3))-2;(1/x(5))+(1/x(2))-2;];

(3) MATLAB optimization m.file (ravigneauxs_main_file.m)

function [x,fval,exitflag,output,lambda,grad,hessian]=Ravigneauxs(x0,lb,ub)

x0=[0 0 0 0 0]

lb=[-Inf -Inf -Inf 0 0]

ub=[0 0 0 1 1]

options = optimset;

DRUD1=3.3000;DRUD2=2.1500;DRUD3=1.6133;DRDD=1.0000;DROD1=0.7333;DROD2=0.5000;DRR=-2.7500; Theta=110.0000;Z(4)=132;M=2;

options = optimset(options,’Display’,’iter’);

options = optimset(options,’PlotFcns’,{ @optimplotx @optimplotfunccount @optimplotfval @optimplotconstrviolation @optimplotstepsize @optimplotfirstorderopt });

options = optimset(options,’Algorithm’,’interior-point’);

options = optimset(options,’Diagnostics’,’on’);

[x,fval,exitflag,output,lambda,grad,hessian] = ...

fmincon(@ravigneauxs,x0,[],[],[],[],lb,ub,@ravigneauxs_confun,options);

RUD1=(x(1)*x(2))/x(4);

RUD2=((x(1)*x(2))-(x(1)*x(5)))/(x(4)- (x(1)*x(5)));

RUD3=((x(1)*x(2))-x(3))/(x(4)-x(3));

RDD=1;

ROD1=(x(3)/(x(3)-x(4)));

ROD2=-(x(1)*x(5))/(x(4)-(x(1)*x(5)));

RR=(x(3)/x(4));

THETA=acosd(((1-1/x(3))^2+((1/(x(1)*x(2)))-(1/x(1)))^2-(1/(M)^2)*(1-1/x(1))^2)/(2*((1/(x(1)*x(2)))-(1/x(1)))*(1-1/x(3))))+asind(1/(1-(1/x(2))))+asind(1/(1-(1/x(3))));

y(4)=Z(4);y(6)=(y(4)*x(4));y(1)=-(y(6)/x(3));y(5)=-(y(6)/x(1));y(2)=-(y(5)/x(2));

y(7)=(y(5)/x(5));Z(6)=int16(y(6));Z(1)=int16(y(1));Z(5)=int16(y(5));Z(2)=int16(y(2)); Z(7)=int16(y(7));

y(8)=Z(6)/M;Z(8)=int16(y(8));y(9)=-Z(8)/x(1);Z(9)=int16(y(9));

cx(1)=-Z(6)/Z(5);cx(2)=-Z(5)/Z(2);cx(3)=-Z(6)/Z(1);cx(4)=Z(6)/Z(4);cx(5)=Z(5)/Z(7);

CRUD1=(cx(1)*cx(2))/cx(4);

CRUD2=((cx(1)*cx(2))-(cx(1)*cx(5)))/(cx(4)- (cx(1)*cx(5)));

CRUD3=((cx(1)*cx(2))-cx(3))/(cx(4)-cx(3));

CRDD=1;

CROD1=(cx(3)/(cx(3)-cx(4)));

CROD2=-(cx(1)*cx(5))/(cx(4)-(cx(1)*cx(5)));

CRR=(cx(3)/cx(4));

CM=Z(6)/Z(8);

THETAC=acosd(((1-1/cx(3))^2+((1/(cx(1)*cx(2)))-(1/cx(1)))^2-(1/(M)^2)*(1-1/cx(1))^2)/(2*((1/(cx(1)*cx(2)))-(1/cx(1)))*(1-1/cx(3))))+asind(1/(1-(1/cx(2))))+asind(1/(1-(1/cx(3))));

THETACZ=acosd(((Z(2)+Z(5))^2+(Z(1)+Z(6))^2-(Z(8)+Z(9))^2)/(2*(Z(2)+Z(5))*(Z(1)+Z(6))))+asind(Z(5)/

(Z(2)+Z(5)))+asind(Z(6)/(Z(6)+Z(1)));

e1=CRUD1/CRUD2;e2=CRUD2/CRUD3;e3=CRUD3/CRDD;e4=CRDD/CROD1;e5=CROD1/CROD2;

s1=e1-e2;s2=e2-e3;s3=e3-e4;s4=e4-e5;GR=[M CM]

Desired_Velocity_Ratios_R=[DRUD1 DRUD2 DRUD3 DROD1 DROD2 DRR]

DESIRED_INCLUDED_ANGLE=[Theta]

Optimal_Velocity_Ratios_R=[RUD1 RUD2 RUD3 ROD1 ROD2 RR]

OPTIMAL_INCLUDED_ANGLE=[THETA]

Number_of_teeth=[y(1) y(2) y(4) y(5) y(6) y(7) y(8) y(9)]

Number_of_teeth=[Z(1) Z(2) Z(4) Z(5) Z(6) Z(7) Z(8) Z(9)]

Optimal_Velocity_Ratios_CR=[CRUD1 CRUD2 CRUD3 CRDD CROD1 CROD2 CRR]

SRD=[e1 e2 e3 e4 e5 s1 s2 s3 s4]

OPTIMAL_INCLUDED_ANGLE_C=[THETAC]

INCLUDED_ANGLE_Z=[THETACZ]