石川公式只适用于外啮合吗

shine668899

本人研究行星齿轮动力学，根据石川公式编代码的时候，发现渐开线展开的范围只适用于重合度1-2的齿轮，请问是否适用于重合度大于2的齿轮。内外啮合需要修改哪那些地方

猫头鹰先生

clear;
z1=19;z2=45;%两齿轮的齿数
m=1.75;
b=m*z1*0.6;
ha=m;
c=0.25*m;
d1=m*z1;d2=m*z2;
r1=m*z1/2;r2=m*z2/2;%分度圆半径
hf1=1.25*m;
hf2=1.25*m;
alpha=20*pi/180;%分度圆压力角
invalpha=tan(alpha)-alpha;
db1=d1*cos(alpha);
db2=d2*cos(alpha);
rb1=db1/2;
rb2=db2/2;%基圆半径
rf1=r1-hf1;
rf2=r2-hf2;%齿根圆半径
da1=d1+2*ha;
da2=d2+2*ha;
ra1=da1/2;
ra2=da2/2;%齿顶圆半径
alpha_a1=acos(rb1/ra1);
alpha_a2=acos(rb2/ra2);%齿顶圆压力角
alpha_f1=acos(rb1/rf1);
alpha_f2=acos(rb2/rf2);%齿根圆压力角
s=pi*m/2;%分度圆弧齿厚
e=s;%分度圆齿槽宽
sk1=ra1*(s/r1+2*((tan(alpha)-alpha)-(tan(alpha_a1)-alpha_a1)));
sk2=ra2*(s/r2+2*((tan(alpha)-alpha)-(tan(alpha_a2)-alpha_a2)));%齿顶圆齿厚
PB1=r1*cos(alpha)*(tan(alpha_a1)-tan(alpha));
PB2=r2*cos(alpha)*(tan(alpha_a2)-tan(alpha));
B1B2=PB1+PB2;
Pb=pi*m*cos(alpha);%基圆齿距
Epsilona=B1B2/Pb;
N1B1=sqrt(ra1^2-rb1^2);
N1B2=N1B1-B1B2;
N2B2=sqrt(ra2^2-rb2^2);
N2B1=N2B2-B1B2;
rF2=sqrt(rb2^2+N2B1^2);
rF1=sqrt(rb1^2+N1B2^2);%有效齿根圆半径
alpha_F1=acos(rb1/rF1);
alpha_F2=acos(rb2/rF2);
sf1=2*rF1*sin(pi/2/z1+invalpha-tan(alpha_F1)+alpha_F1);
sf2=2*rF2*sin(pi/2/z2+invalpha-tan(alpha_F2)+alpha_F2);
h1=sqrt(ra1^2-(sk1/2)^2)-sqrt(rf1^2-(sf1/2)^2);
h2=sqrt(ra2^2-(sk2/2)^2)-sqrt(rf2^2-(sf2/2)^2);
hr1=sqrt(rF1^2-(sf1/2)^2)-sqrt(rf1^2-(sf1/2)^2);
hr2=sqrt(rF2^2-(sf2/2)^2)-sqrt(rf2^2-(sf2/2)^2);
hi1=(h1*sf1-hr1*sk1)/(sf1-sk1);
hi2=(h2*sf2-hr2*sk2)/(sf2-sk2);
N2C=N2B1+Pb;
B2C=B1B2-Pb;%双齿啮合区
CD=Pb-B2C;%单齿啮合区
N1C=N1B1-Pb;
Fn=754;%外力
E=2e+008;%弹性模量
v=0.26;%泊松比
n=100;
step=B2C/n;%B2C为双齿啮合区
nz1=3600;%齿轮1转速r/min
Tz=60/z1/nz1;%循环的周期
t1=B2C/Pb*Tz;
t2=CD/Pb*Tz;
step2=t1/n;
step4=t2/n;
%双齿啮合区设一啮合为i点，一啮合点为j点。
for i=1:n
   x(i)=i*step;
    tt(i)=i*step2;
    xx(i)=Pb+i*step;
    N1Bi(i)=N1B2+i*step;%双齿啮合区i啮合点公式中具体参数的计算
    O1Bi(i)=sqrt(N1Bi(i)*N1Bi(i)+rb1^2);
    ai1(i)=acos(rb1/O1Bi(i));
    gamai1(i)=pi/2/z1+invalpha-tan(ai1(i))+ai1(i);
    miui1(i)=ai1(i)-gamai1(i);
    rxi1(i)=O1Bi(i);
    hxi1(i)=rxi1(i)*cos(gamai1(i))-sqrt(rf1^2-(sf1/2)^2);
    N2Bi(i)=N2B2-i*step;
    O2Bi(i)=sqrt(N2Bi(i)^2+rb2^2);
    ai2(i)=acos(rb2/O2Bi(i));
    gamai2(i)=pi/2/z2+invalpha-tan(ai2(i))+ai2(i);
    miui2(i)=ai2(i)-gamai2(i);
    rxi2(i)=O2Bi(i);
    hxi2(i)=rxi2(i)*cos(gamai2(i))-sqrt(rf2^2-(sf2/2)^2);
    N1Bj(i)=N1Bi(i)+Pb;%双齿啮合区j啮合点公式中具体参数的计算
    O1Bj(i)=sqrt(N1Bj(i)^2+rb1^2);
    aj1(i)=acos(rb1/O1Bj(i));
    gamaj1(i)=pi/2/z1+tan(alpha)-alpha-tan(aj1(i))+aj1(i);
    miuj1(i)=aj1(i)-gamaj1(i);
    rxj1(i)=O1Bj(i);
    hxj1(i)=rxj1(i)*cos(gamaj1(i))-sqrt(rf1^2-(sf1/2)^2);
    N2Bj(i)=N2Bi(i)-Pb;
    O2Bj(i)=sqrt(N2Bj(i)^2+rb2^2);
    aj2(i)=acos(rb2/O2Bj(i));
    gamaj2(i)=pi/2/z2+tan(alpha)-alpha-tan(aj2(i))+aj2(i);
    miuj2(i)=aj2(i)-gamaj2(i);
    rxj2(i)=O2Bj(i);
    hxj2(i)=rxj2(i)*cos(gamaj2(i))-sqrt(rf2^2-(sf2/2)^2);
    sigmabri1(i)=12*Fn*cos(miui1(i))^2*(hxi1(i)*hr1*(hxi1(i)-hr1)+hxi1(i)^3/3)/b/E/sf1^3;
    sigmabri2(i)=12*Fn*cos(miui2(i))^2*(hxi2(i)*hr2*(hxi2(i)-hr2)+hxi2(i)^3/3)/b/E/sf2^3;
    sigmabti1(i)=6*Fn*cos(miui1(i))^2*((hi1-hxi1(i))/(hi1-hr1)*(4-(hi1-hxi1(i))/(hi1-hr1))-2*log((hi1-hxi1(i))/(hi1-hr1))-3)*(hi1-hr1)^3/b/E/sf1^3;
    sigmabti2(i)=6*Fn*cos(miui2(i))^2*((hi2-hxi2(i))/(hi2-hr2)*(4-(hi2-hxi2(i))/(hi2-hr2))-2*log((hi2-hxi2(i))/(hi2-hr2))-3)*(hi2-hr2)^3/b/E/sf2^3;
    sigmasi1(i)=2*(1+v)*Fn*cos(miui1(i))^2*(hr1+(hi1-hr1)*log((hi1-hr1)/(hi1-hxi1(i))))/b/E/sf1;
    sigmasi2(i)=2*(1+v)*Fn*cos(miui2(i))^2*(hr2+(hi2-hr2)*log((hi2-hr2)/(hi2-hxi2(i))))/b/E/sf2;
    sigmagi1(i)=24*Fn*hxi1(i)*cos(miui1(i))^2/pi/b/E/sf1^2;
    sigmagi2(i)=24*Fn*hxi2(i)*cos(miui2(i))^2/pi/b/E/sf2^2;
    sigmap=4*Fn*(1-v^2)/pi/b/E;
    sigmabrj1(i)=12*Fn*cos(miuj1(i))^2*(hxj1(i)*hr1*(hxj1(i)-hr1)+hxj1(i)^3/3)/b/E/sf1^3;
    sigmabrj2(i)=12*Fn*cos(miuj2(i))^2*(hxj2(i)*hr2*(hxj2(i)-hr2)+hxj2(i)^3/3)/b/E/sf2^3;
    sigmabtj1(i)=6*Fn*cos(miuj1(i))^2*((hi1-hxj1(i))/(hi1-hr1)*(4-(hi1-hxj1(i))/(hi1-hr1))-2*log((hi1-hxj1(i))/(hi1-hr1))-3)*(hi1-hr1)^3/b/E/sf1^3;
    sigmabtj2(i)=6*Fn*cos(miuj2(i))^2*((hi2-hxj2(i))/(hi2-hr2)*(4-(hi2-hxj2(i))/(hi2-hr2))-2*log((hi2-hxj2(i))/(hi2-hr2))-3)*(hi2-hr2)^3/b/E/sf2^3;
    sigmasj1(i)=2*(1+v)*Fn*cos(miuj1(i))^2*(hr1+(hi1-hr1)*log((hi1-hr1)/(hi1-hxj1(i))))/b/E/sf1;
    sigmasj2(i)=2*(1+v)*Fn*cos(miuj2(i))^2*(hr2+(hi2-hr2)*log((hi2-hr2)/(hi2-hxj2(i))))/b/E/sf2;
    sigmagj1(i)=24*Fn*hxj1(i)*cos(miuj1(i))^2/pi/b/E/sf1^2;
    sigmagj2(i)=24*Fn*hxj2(i)*cos(miuj2(i))^2/pi/b/E/sf2^2;
    ki1(i)=Fn/(sigmabri1(i)+sigmabri2(i)+sigmabti1(i)+sigmabti2(i)+sigmasi1(i)+sigmasi2(i)+sigmagi1(i)+sigmagi2(i)+sigmap);%双齿啮合时i点刚度
    kj(i)=Fn/(sigmabrj1(i)+sigmabrj2(i)+sigmabtj1(i)+sigmabtj2(i)+sigmasj1(i)+sigmasj2(i)+sigmagj1(i)+sigmagj2(i)+sigmap);%双齿啮合时j点刚度
    k(i)=ki1(i)+kj(i);
end
step3=CD/n;%CD为单齿啮合区
for i=1:n
    xxx(i)=B2C+i*step3;
    ttt(i)=t1+i*step4;
    N1Bi(i)=N1C+i*step3;
    O1Bi(i)=sqrt(N1Bi(i)^2+rb1^2);
    ai1(i)=acos(rb1/O1Bi(i));
    gamai1(i)=pi/2/z1+invalpha-tan(ai1(i))+ai1(i);
    miui1(i)=ai1(i)-gamai1(i);
    rxi1(i)=O1Bi(i);
    hxi1(i)=rxi1(i)*cos(gamai1(i))-sqrt(rf1^2-(sf1/2)^2);
    N2Bi(i)=N2C-i*step3;
    O2Bi(i)=sqrt(N2Bi(i)^2+rb2^2);
    ai2(i)=acos(rb2/O2Bi(i));
    gamai2(i)=pi/2/z2+invalpha-tan(ai2(i))+ai2(i);
    miui2(i)=ai2(i)-gamai2(i);
    rxi2(i)=O2Bi(i);
    hxi2(i)=rxi2(i)*cos(gamai2(i))-sqrt(rf2^2-(sf2/2)^2);
    sigmabri1(i)=12*Fn*cos(miui1(i))^2*(hxi1(i)*hr1*(hxi1(i)-hr1)+hxi1(i)^3/3)/b/E/sf1^3;
    sigmabri2(i)=12*Fn*cos(miui2(i))^2*(hxi2(i)*hr2*(hxi2(i)-hr2)+hxi2(i)^3/3)/b/E/sf2^3;
    sigmabti1(i)=6*Fn*cos(miui1(i))^2*((hi1-hxi1(i))/(hi1-hr1)*(4-(hi1-hxi1(i))/(hi1-hr1))-2*log((hi1-hxi1(i))/(hi1-hr1))-3)*(hi1-hr1)^3/b/E/sf1^3;
    sigmabti2(i)=6*Fn*cos(miui2(i))^2*((hi2-hxi2(i))/(hi2-hr2)*(4-(hi2-hxi2(i))/(hi2-hr2))-2*log((hi2-hxi2(i))/(hi2-hr2))-3)*(hi2-hr2)^3/b/E/sf2^3;
    sigmasi1(i)=2*(1+v)*Fn*cos(miui1(i))^2*(hr1+(hi1-hr1)*log((hi1-hr1)/(hi1-hxi1(i))))/b/E/sf1;
    sigmasi2(i)=2*(1+v)*Fn*cos(miui2(i))^2*(hr2+(hi2-hr2)*log((hi2-hr2)/(hi2-hxi2(i))))/b/E/sf2;
    sigmagi1(i)=24*Fn*hxi1(i)*cos(miui1(i))^2/pi/b/E/sf1^2;
    sigmagi2(i)=24*Fn*hxi2(i)*cos(miui2(i))^2/pi/b/E/sf2^2;
    sigmap=4*Fn*(1-v^2)/pi/b/E;
    ki2(i)=Fn/(sigmabri1(i)+sigmabri2(i)+sigmabti1(i)+sigmabti2(i)+sigmasi1(i)+sigmasi2(i)+sigmagi1(i)+sigmagi2(i)+sigmap);
end
o1=polyfit(tt,k,3);                                    
o2=polyfit(ttt,ki2,3);
o3=polyfit(tt,ki1,3);
o4=polyfit(tt,kj,3);
%plot(2,2,1);plot(x,k);hold on;plot(xxx,ki2);xlabel('啮合线位移/(mm)');ylabel('线性啮合刚度k');

%plot(tt,k);
%hold on;
%plot(ttt,ki2);
%hold on;
T=0.0008772;

for i=0:1;
    A=polyfit(tt,k,2)
    for TT=0.000048:0.0000005:0.000512;
        K=-187835724775138*TT.*TT+80690343132.3420*TT+502073826.920299;
        S1=6.15*0.000001*sin(2*pi*60*19*(TT+i*T));
        if K.*S1<0
           P1=0;
        else
           P1=K.*S1;
        end
        plot(TT+i*T,P1,'r');xlabel('啮合时间/(s)');ylabel('激励/（N）');
        hold on;
    end
   
    for A1=0.000512:0.0000005:0.0006004;
        A2=1898981472963.80*A1+1466425788.53;
        S2=6.15*0.000001*sin(2*pi*60*19*(A1+i*T));
        if A2.*S2<0
           P2=0;
        else
           P2=A2.*S2;
        end
        plot(A1+i*T,P2,'r');
        hold on;
    end
   
    for TTT=0.0006004:0.0000005:0.0008344;
        B=polyfit(ttt,ki2,2);
        KI2=-472401734368030*TTT.*TTT+659557708465.797*TTT+100570316.788086;
        S3=6.15*0.000001*sin(2*pi*60*19*(TTT+i*T));
        if KI2.*S3<0
           P3=0;
        else
           P3=KI2.*S3;
        end
        plot(TTT+i*T,P3,'r');
        hold on;
    end
   
    for A3=0.0008344:0.0000005:0.0009252;
        A4=2020991673127.75*A3-1364307306.1;
        S4=6.15*0.000001*sin(2*pi*60*19*(A3+i*T));
        if A4.*S4<0
           P4=0;
        else
           P4=A4.*S4;
        end
        plot(A3+i*T,P4,'r');
    end
end