求解齿轮刚度

z1=25;z2=75 ; %两齿轮的齿数
m=3.2; %模数
b=70; %齿宽
alpha=20*pi/180;%分度圆压力角
Fn=760;%外力负载,单位N
E=2.09e5;%弹性模量
v=0.269;%泊松比

ha=m;%齿顶高，标准齿轮，齿顶高系数是1
c=0.25*m;%顶隙，标准齿轮，顶隙系数是0.25
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;
invalpha=tan(alpha)-alpha;%展角
db1=d1*cos(alpha);
db2=d2*cos(alpha);%基圆直径
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;%实际的啮合线

Pb=pi*m*cos(alpha);%基圆齿距
Epsilona=B1B2/Pb%重合度

N1B1=sqrt(ra1^2-rb1^2);
N1B2=N1B1-B1B2;
N2B2=sqrt(ra2^2-rb2^2);
N2B1=N2B2-B1B2;

rF1=sqrt(rb1^2+N1B2^2);%有效齿根圆半径ACTIVE ROOT DIAMETER
rF2=sqrt(rb2^2+N2B1^2);
alpha_F1=acos(rb1/rF1);
alpha_F2=acos(rb2/rF2);%有效齿根圆压力角

% % % %若rb<=rF时,即z>=2*(1-x)/(1-cos(alpha));
% % % %sf=2*rb*sin((pi+4*x*tan(alph))/2/z+invalpha-invalpha_F)；x为变位系数
% % % %hr=sqrt(rF^2-(sf/2)^2)-sqrt(rf^2-(sf/2)^2);
% % % %若rb>=rF时，即z<=2*(1-x)/(1-cos(alpha));
% % % %sf=2*rb*sin((pi+4*x*tan(alph))/2/z-invalpha)；x为变位系数
% % % %hr=sqrt(rb^2-(sf/2)^2)-sqrt(rf^2-(sf/2)^2);

sf1=2*rF1*sin(pi/2/z1+invalpha-tan(alpha_F1)+alpha_F1);%危险截面长度
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;

n=1000;
step=B2C/n;%B2C为双齿啮合区
nz1=30;%齿轮1转速r/min
Tz=60/z1/nz1;%循环的周期
t1=B2C/Pb*Tz;%双齿啮合区时间
t2=CD/Pb*Tz;%单齿啮合区时间
step2=t1/n;%双齿啮合区,每步时间
step4=t2/n;%单齿啮合区，每步时间

%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;%双啮合区的时间历程
xx(i)=Pb+i*step;%双啮合区的位移

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)^2+rb1^2);%求石川法中的rx1
ai1(i)=acos(rb1/O1Bi(i));%石川法中的am，miu=am-gama
gamai1(i)=pi/2/z1+invalpha-tan(ai1(i))+ai1(i);
miui1(i)=ai1

(i)-gamai1(i);
rxi1(i)=O1Bi(i);%求石川法中的rx
hxi1(i)=rxi1(i)*cos(alpha-miui1(i))-sqrt(rf1^2-(sf1/2)^2);
% 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(alpha-miui2(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+invalpha-tan(aj1(i))+aj1(i);
miuj1(i)=aj1(i)-gamaj1(i);
rxj1(i)=O1Bj(i);
hxj1(i)=rxj1(i)*cos(alpha-miuj1(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(alpha-miuj2(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);%刚度为两个啮合区域的刚度相加
end;

step3=CD/n;%CD为单齿啮合区
for i=1:n
xxx(i)=B2C+i*step3;%单啮合区的位移
ttt(i)=t1+i*step4;%单啮合区的时间历程

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(alpha-miui1(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(alpha-miui2(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;

kt=k'/1e5;
k2t=ki2'/1e5;%刚度缩小了5的数量级


kx=[x',kt];%双啮合区
k2x=[xxx',k2t];%单啮合区
kx1=[kx;zeros(100,2)];
k2x=[zeros(100,2);k2x];
z=kx1+k2x
figure(1)
plot(z(:,1),z(:,2))
title('刚度随位移的变化')
grid on;
xlabel('位移');
ylabel('啮合刚度');
hold on;


kt=[tt',kt];%双啮合区
k2t=[ttt',k2t];%单啮合区
kt1=[kt;zeros(100,2)];
k2t=[zeros(100,2);k2t];
s=kt1+k2t
figure(2)
plot(s(:,1),s(:,2))
title('啮合刚度随时间的变化')
grid on;
xlabel('时间');
ylabel('啮合刚度');
hold on;