平面六杆机构的运动分析 （题号 3 3- - C）
1、 题目说明
下图为一平面六杆机构。一直各构件的尺寸如下表一，又知原动件 1 以角速度为1rad/s 沿逆时针方向回转，要求个从动件的角位移、角速度、及角加速度以及 E点的位移、速度及加速度变化情况。
表一 组号 L 1
L 2
L 2 ’ L 3
L 4
L 5
L 6
α （x G ）L 4 ’ （Y G ）
L 7
3-C 26.5 105.6 65.0 49.0 87.5 48.4 39.0 600 153.5 41.7 题目要求：
三人一组计算出原动件从 0 到 2∏变化时（计算点数 37）所要求的各运动变量的大小，并绘出运动曲线图及 E 点的轨迹曲线。
2 2 、题目分析
1 1）
）
建立封闭图形：
L 2 - -
L 3 = L 4 - -
L 1
L 2 + +
L 2 ’ + +
L 5
- - L 6 + +
L 7 = =
L 4 ’ - -
L 1
2 2 ）
机构运动分析
A．角位移分析 由图形封闭性得：
1 1 7 6 6 5 5 2 2 2 21 1 4 6 6 5 5 2 2 2 21 1 3 3 2 21 1 4 3 3 2 2sin sin sin 2 sin " sincos " cos cos 2 cos " cossin sin sincos cos cos L L L L a L LL L L L a L LL L LL L L L
B．角速度分析 上式对时间求一阶导数，可得速度方程 ：
1 1 1 6 6 6 5 5 5 2"2 2 2 21 1 1 6 6 6 5 5 5 2 2"2 2 2 21 1 1 3 3 3 2 2 21 1 1 3 3 3 2 2 2cos cos cos 2 ) cos( cossin sin sin ) sin( sincos cos cossin sin sin L L L L LL L L a L LL L LL L L 化为矩阵形式为 ：
1 11 11 11 1165326 6 5 5 2 2 2 26 6 5 5 2 2 2 23 3 2 23 3 2 2cossincossincos cos 0 cos cossin sin 0 sin " sin0 0 cos cos0 0 sin sin LLLLL L a L LL L a L LL LL L C．角加速度分析：
角速度矩阵对时间求一阶导数，可得加速度矩阵为：
2 2 3 3 22 2 3 3 32 2 2 2 5 5 6 6 52 2 2 2 5 5 6 6 62 2 3 32 2 3 32 2 2sin sin 0 0cos cos 0 0sin " sin 0 sin sincos " cos 0 cos coscos cos 0 0sin sin 0 0cosL LL LL L a L LL L a L LL LL LL L 21 1 221 1 2 3122 5 5 6 6 1 1522 2 2 2 5 5 6 6 1 16cossin" cos 0 cos cos cossin " sin 0 sin sin sinLLa L L LL L a L L L
D．E 点的运动状态 位移： 5 5 6 65 5 6 6sin sincos cos L L y yL L x xG EG E 速度： 5 5 5 6 6 65 5 5 6 6 6cos cossin sin L L vL L vyxEE 加速度：
5 5 525 5 5 6 6 626 6 65 5 525 5 5 6 6 626 6 6cos sin cos sinsin cos sin cos L L L L aL L L L ayxEE 3 3、 、 流程图
建 创 建 fsolve函数入口文用 调用 MATLAB 系 系数 统函数 fsolve 求解各从动件的角位
4 4、 、 源程序
a. 求从动件的角位移 9*- 第一个 M 文件（function1.m）
function t=function1(th,th1,l1,l2,l20,l3,l4,l40,l5,l6,l7,a)
t=[l2*cos(th(1))-l3*cos(th(2))+l1*cos(th1)-l4;
l2*sin(th(1))-l3*sin(th(2))+l1*sin(th1);
l2*cos(th(1))+l20*cos(a-th(1))+l5*cos(th(3))-l6*cos(th(4))+l1*cos(th1)-l40;
l2*sin(th(1))-l20*sin(a-th(1))+l5*sin(th(3))-l6*sin(th(4))+l1*sin(th1)-l7]; 第二个脚本文件（function2.m）
l1=26.5;l2=105.6;l20=65.0;l3=49.0;l4=87.5;l40=153.5;l5=48.4;l6=39.0;l7=41.7;a=pi/3;
th1=0:pi/18:2*pi;
th2356=zeros(length(th1),4);
options=optimset("display","off");
for m=1:length(th1)
th2356(m,:)=fsolve("function1",[0.656 1.267 2.309 1.934],options,th1(m),l1,l2,l20,l3,l4,l40,l5,l6,l7,a);
end
disp(th2356);
th2=th2356(:,1);
th3=th2356(:,2);
th5=th2356(:,3);
th6=th2356(:,4);
hold on
plot(th1,th2,"r:","LineWidth",2.5)
plot(th1,th3,"b","LineWidth",2.5)
plot(th1,th5,"g-.","LineWidth",2.5)
plot(th1,th6,"k--","LineWidth",2.5)
hold off
利用角速度矩阵 W=A\B， ，w2(m)=W(1),w3(m)=W(2),w5(m)=W(3),w6(m)=W(4)利用角加速度矩阵求出从动件角加速度 将以上各参数分入 别带入 E 点的位移、速度、加速度解析方程式，进行制 求解。绘制 E 点的结
grid on
title("从动件的角位移")
xlabel("{\theta}_{1}(rad)")
ylabel("{\theta}_{2}{\theta}_{3}{\theta}_{5}{\theta}_{6},(rad)")
legend("{\theta}_{2}","{\theta}_{3}","{\theta}_{5}","{\theta}_{6}","Location","NorthWest")
axis([0,2*pi,-2,2*pi]) b．求从动件的角速度 第三个脚本文件（function3.m）
w1=1;l1=26.5;l2=105.6;l20=65.0;l3=49.0;l4=87.5;l40=153.5;l5=48.4;l6=39.0;l7=41.7;a=pi/3;
for m=1:37
A=[-l2*sin(th2(m)),l3*sin(th3(m)),0,0;...
l2*cos(th2(m)),-l3*cos(th3(m)),0,0;...
-l2*sin(th2(m))+l20*sin(a-th2(m)),0,-l5*sin(th5(m)),l6*sin(th6(m));...
l2*cos(th2(m))+l20*cos(a-th2(m)),0,l5*cos(th5(m)),-l6*cos(th6(m))];
B=[l1*sin(th1(m)),-l1*cos(th1(m)),l1*sin(th1(m)),-l1*cos(th1(m))]";
W=A\(w1*B);
w2(m)=W(1);
w3(m)=W(2);
w5(m)=W(3);
w6(m)=W(4); disp(W) end for m=1:37
W(m,1)=w2(m);
W(m,2)=w3(m);
W(m,3)=w5(m);
W(m,4)=w6(m);
end hold on
plot(th1,w2,"r:","LineWidth",2.5)
plot(th1,w3,"b","LineWidth",2.5)
plot(th1,w5,"g-.","LineWidth",2.5)
plot(th1,w6,"k--","LineWidth",2.5)
hold off
grid on
title("从动件的角速度")
xlabel("{\theta}_{1}(rad)")
ylabel("{\omega}_{2}{\omega}_{3}{\omega}_{5}{\omega}_{6}(rad/s)")
legend("{\omega}_{2}","{\omega}_{3}","{\omega}_{5}","{\omega}_{6}","Location","SouthWest")
axis([0,2*pi,-2,5])
c．求从动件的角加速度 第四个脚本文件（function4.m）
w1=1;l1=26.5;l2=105.6;l20=65.0;l3=49.0;l4=87.5;l40=153.5;l5=48.4;l6=39.0;l7=41.7;a=pi/3;
for m=1:37
A=[-l2*sin(th2(m)),l3*sin(th3(m)),0,0;...
l2*cos(th2(m)),-l3*cos(th3(m)),0,0;...
-l2*sin(th2(m))+l20*sin(a-th2(m)),0,-l5*sin(th5(m)),l6*sin(th6(m));...
l2*cos(th2(m))+l20*cos(a-th2(m)),0,l5*cos(th5(m)),-l6*cos(th6(m))];
B1=-[-l2*cos(th2(m))*w2(m),l3*cos(th3(m))*w3(m),0,0;...
-l2*sin(th2(m))*w2(m),l3*sin(th3(m))*w3(m),0,0;...
-l2*cos(th2(m))*w2(m)-l20*cos(a-th2(m))*w2(m),0,-l5*cos(th5(m))*w5(m),l6*cos(th6(m))*w6(m);...
-l2*sin(th2(m))*w2(m)+l20*sin(a-th2(m))*w2(m),0,-l5*sin(th5(m))*w5(m),l6*sin(th6(m))*w6(m)];
B2=[w2(m),w3(m),w5(m),w6(m)]";
B3=[l1*cos(th1(m))*w1;...
l1*sin(th1(m))*w1;...
l1*cos(th1(m))*w1;...
l1*sin(th1(m))*w1];
K=A\(B1*B2)+A\(w1*B3);
k2(m)=K(1);
k3(m)=K(2);
k5(m)=K(3);
k6(m)=K(4); disp(K)
end
for m=1:37
AA(m,1)=k2(m);
AA(m,2)=k3(m);
AA(m,3)=k5(m);
AA(m,4)=k6(m);
end
hold on
plot(th1,k2,"r:","LineWidth",2.5)
plot(th1,k3,"b","LineWidth",2.5)
plot(th1,k5,"g-.","LineWidth",2.5)
plot(th1,k6,"k--","LineWidth",2.5)
hold off
grid on
title("从动件的角加速度")
xlabel("{\theta}_{1}(rad)")
ylabel("{\alpha}_{2}{\alpha}_{3}{\alpha}_{5}{\alpha}_{6}(rad/s^{2})")
legend("{\alpha}_{2}","{\alpha}_{3}","{\alpha}_{5}","{\alpha}_{6}","Location","NorthWest")
axis([0,2*pi,-15,20])
d．求 E 点的轨迹 第五个脚本文件(function5.m) l40=153.5;l7=41.7;l5=48.4;l6=39.0;
for m=1:37
X(m)=l40+l6*cos(th6(m))-l5*cos(th5(m));
Y(m)=l7+l6*sin(th6(m))-l5*sin(th5(m)); disp(X(m));disp(Y(m)); end
plot(X,Y,"b","LineWidth",2.5)
axis([110,190,-30,40])
title("E点的轨迹")
xlabel("Xe(mm)")
ylabel("Ye(mm)")
grid on
e．求 E 点的速度 第六个脚本文件(function6.m) l5=48.4;l6=39.0;
for m=1:37
A=[-l6*sin(th6(m)),l5*sin(th5(m));...
l6*cos(th6(m)),-l5*cos(th5(m))];
B=[w6(m),w5(m)]";
V=A*B;
vx(m)=V(1);
vy(m)=V(2); disp(vx(m));disp(vy(m));
end
V=(vx.^2+vy.^2).^(1/2);disp(V);
hold on
plot(th1,vx,"r","LineWidth",2.5)
plot(th1,vy,"b","LineWidth",2.5)
plot(th1,V,"g","LineWidth",2.5)
hold off
grid on
title("E点的速度")
xlabel("{\theta}_{1}(rad)")
ylabel("Vex Vey V (mm/s)")
legend("Vex","Vey","V","Location","NorthWest")
axis([0,2*pi,-70,75]) f．求 E 点的加速度 第七个脚本文件(function7.m) l5=48.4;l6=39.0;
for m=1:37
A1=[-l6*sin(th6(m)),l5*sin(th5(m));...
l6*cos(th6(m)),-l5*cos(th5(m))];
A2=[k6(m),k5(m)]";
B1=[-l6*cos(th6(m)),l5*cos(th5(m));...
-l6*sin(th6(m)),l5*sin(th5(m))];
B2=[w6(m)^2,w5(m)^2]";
J=A1*A2+B1*B2;
jx(m)=J(1);
jy(m)=J(2); disp(jx(m));disp(jy(m));
end
J=[(jx.^2+jy.^2).^(1/2)]"; disp(J);
hold on
plot(th1,jx,"r","LineWidth",2.5)
plot(th1,jy,"b","LineWidth",2.5)
plot(th1,J,"g","LineWidth",2.5)
hold off
grid on
title("E点的加速度")
xlabel("{\theta}_{1}(rad)")
ylabel("{a}_{ex} {a}_{ey} {a}_{e} (mm/s^{2})")
legend("{a}_{ex} ","{a}_{ey}","{a}_{e}","Location","SouthWest")
axis([0,2*pi,-100,170])
5 5、 、 计算结果和曲线图：
a.各从动件的角位移与θ1 的关系曲线和计算数据：
θ θ1
θ θ2
θ θ3
θ θ5
θ θ6
0 0 0.2535 0.5711 1.2563 -0.4095 10 0.174533 0.1928 0.5315 1.2624 -0.5317 20 0.349066 0.1590 0.5540 1.2268 -0.5623
30 0.523599 0.1435 0.6170 1.1609 -0.5391 40 0.698132 0.1392 0.7031 1.0780 -0.4976 50 0.872665 0.1417 0.8018 0.9851 -0.4595 60 1.047198 0.1487 0.9071 0.8846 -0.4371 70 1.22173 0.1592 1.0156 0.7778 -0.4376 80 1.396263 0.1723 1.1248 0.6660 -0.4646 90 1.570796 0.1879 1.2330 0.5524 -0.5178 100 1.745329 0.2059 1.3388 0.4416 -0.5927 110 1.919862 0.2262 1.4409 0.3393 -0.6817 120 2.094395 0.2490 1.5383 0.2501 -0.7752 130 2.268928 0.2744 1.6297 1.9059 2.9469 140 2.443461 0.3025 1.7141 1.8349 2.8963 150 2.617994 0.3336 1.7903 1.7768 2.8569 160 2.792527 0.3675 1.8574 1.7313 2.8229 170 2.96706 0.4042 1.9144 1.6974 2.7888 180 3.141593 0.4436 1.9606 1.6731 2.7503 190 3.316126 0.4852 1.9954 1.6564 2.7038 200 3.490659 0.5284 2.0183 1.6446 2.6470 210 3.665191 0.5723 2.0291 1.6350 2.5783 220 3.839724 0.6160 2.0275 1.6248 2.4968 230 4.014257 0.6580 2.0133 1.6116 2.4023 240 4.18879 0.6969 1.9862 1.5938 2.2949 250 4.363323 0.7310 1.9457 1.5710 2.1755 260 4.537856 0.7581 1.8910 1.5450 2.0468 270 4.712389 0.7760 1.8211 1.5224 1.9152 280 4.886922 0.7823 1.7348 1.5224 1.8013 290 5.061455 0.7740 1.6304 1.6198 1.7958 300 5.235988 0.7483 1.5067 2.0950 2.2656 310 5.410521 0.7027 1.3628 2.7149 3.3036 320 5.585054 0.6357 1.1996 1.1171 0.5828 330 5.759587 0.5484 1.0219 1.1422 0.3407 340 5.934119 0.4465 0.8416 1.1757 0.0712 350 6.108652 0.3424 0.6811 1.2188 -0.1943 360 6.283185 0.2535 0.5711 1.2563
-0.4095
b.各从动件角速度与θ1 的关系曲线和计算结果：
θ θ1
ω ω2
ω ω3
ω ω5
ω ω6
0 -0. 4344 -0.4344 0.1474 -0.9924 10 -0.2639 -0.0297 -0.0883 -0.4110 20 -0.1327 0.2665 -0.3069 0.0191 30 -0.0513 0.4400 -0.4369 0.2125 40 -0.0023 0.5366 -0.5077 0.2433 50 0.0290 0.5888 -0.5554 0.1821 60 0.0590 0.6152 -0.5948 0.0677 70 0.0680 0.6254 -0.6283 -0.0771 80 0.0826 0.6242 -0.6495 -0.2318 90 0.0962 0.6143 -0.6481 -0.3732 100 0.1096 0.5969 -0.6157 -0.4782 110 0.1234 0.5727 -0.5521 -0.5316 120 0.1379 0.5419 -0.4678 -0.5317 130 0.1533 0.5047 -0.4428 -0.3325 140 0.1694 0.4611 -0.3701 -0.2523 150 0.1860 0.4115 -0.2958 -0.2051 160 0.2026 0.3564 -0.2260 -0.1905
170 0.2184 0.2964 -0.1648 -0.2043 180 0.2325 0.2325 -0.1151 -0.2408 190 0.2436 0.1656 -0.0792 -0.2939 200 0.2506 0.0968 -0.0587 -0.3583 210 0.2520 0.0266 -0.0543 -0.2495 220 0.2468 -0.0449 -0.0649 -0.5041 230 0.2335 -0.1179 -0.0876 -0.5791 240 0.2108 -0.1932 -0.1170 -0.6511 250 0.1773 -0.2719 -0.1432 -0.7142 260 0.1316 -0.3557 -0.1488 -0.7553 270 0.0719 -0.4462 -0.0950 -0.7362 280 -0.0033 -0.5449 0.1470 -0.5051 290 -0.0948 -0.6522 1.2594 0.8061 300 -0.2023 -0.7666 4.0192 4.4993 310 -0.3221 -0.8819 2.5398 3.6258 320 -0.4446 -0.9835 0.1479 -1.2540 330 -0.5501 -1.0414 0.1588 -1.4916 340 -0.6055 -1.0037 0.2257 -1.5689 350 -0.5702 -0.8044 0.2542 -1.4272 360 -0.4344 -0.4344 0.1474 -0.2644
c.各从动件角加速度与θ1 的关系曲线和计算结果 θ θ1
α α2
α α3
α α5
α α6
0 0.9699 2.4050 -1.0835 3.20214 10 0.9042 2.0767 -1.4393 3.1194 20 0.5956 1.3094 -0.9918 1.7454 30 0.3561 0.7360 -0.5373 0.5580 40 0.2196 0.4026 -0.3188 -0.1367 50 0.1470 0.2130 -0.2422 -0.5294 60 0.1087 0.0984 -0.2133 -0.7617 70 0.0889 0.0224 -0.1655 -0.8785 80 0.0797 -0.0335 -0.0674 -0.8716 90 0.0767 -0.0791 0.0918 -0.7252 100 0.0776 -0.1195 0.2804 -0.4620 110 0.0809 -0.1577 0.4378 -0.1486 120 0.0855 -0.1949 0.5123 0.1384 130 0.0903 -0.2316 0.3957 0.5474 140 0.0942 -0.2672 0.4285 0.3671 150 0.0958 -0.3007 0.4171 0.1744 160 0.0937 -0.3307 0.3783 -0.0027 170 0.0866 -0.3559 0.3204 -0.1499 180 0.0733 -0.3755 0.2473 -0.2624 190 0.0531 -0.3894 0.1626 -3415 200 0.0255 -0.3989 0.0713 -0.3922 210 -0.0095 -0.4060 -0.0193 -0.4206 220 -0.0519 -0.4133 -0.0995 -0.4313 230 -0.1018 -0.4237 -0.1562 -0.4250 240 -0.1595 -0.4399 -0.1710 -0.3946 250 -0.2255 -0.4641 -0.1136 -0.3172 260 -0.3006 -0.4977 0.0823 -0.1225 270 -0.3849 -0.5407 0.6315 0.4530 280 -0.4770 -0.5902 2.6068 2.7801 290 -0.5718 -0.6383 12.7820 15.7278 300 -0.6569 -0.6672 5.5340 11.0206 310 -0.7073 -0.6413 -13.6792 -11.4941 320 -0.6779 -0.4949 -0.3316 -1.9177 330 -0.4991 -0.1192 0.3174 -0.9030 340 -0.0935 0.6203 0.3669 0.0843 350 0.5146 1.6871 -0.1450 1.6397 360 0.9699 2.4050 -1.0853 3.2021
d.E 点运动分析结果：
θ θ1
Xe Ye Vex Vey Ve aex aey ae 0 0 174.3046 -19.8534 -8.6248 -37.7102 38.6839 -35.1393 147.0985 151.2374 10 0.174533 172.4238 -24.1896 -12.1997 -12.5182 17.4796 -10.2559 129.7075 130.1123 20 0.349066 170.1713 -24.6581 -13.5386 5.6239 14.6602 -7.3747 78.0565 78.4041 30 0.523599 167.6807 -22.7124 -15.0965 15.5185 21.6501 -10.5266 38.3772 39.7947 40 0.698132 164.8738 -19.5563 -17.1192 19.9629 26.2980 -12.2671 14.7095 19.1533 50 0.872665 161.6988 -15.9271 -19.9629 21.2266 28.6555 -11.8315 0.9899 11.8728 60 1.047198 158.1675 -12.2539 -21.1566 20.6314 29.5509 -9.7997 -7.1073 12.1057 70 1.22173 154.3402 -8.
18.9403 29.4959 -6.7383 -11.8242 13.6094 80 1.396263 150.3092 -5.6779 -23.4732 16.6351 28.7701 -3.0636 -14.2705 14.5957 90 1.570796 146.1857 -2.9976 -23.6635 14.0573 27.5240 0.9102 -15.0173 15.0448 100 1.745329 142.0899 -0.7713 -23.1531 11.4727 25.8397 4.9251 -14.3923 15.2117 110 1.919862 138.1438 1.0195 -21.9551 9.0999 23.7662 8.7551 -12.6258 15.3643 120 2.094395 134.4633 2.4281 -20.1197 7.1198 21.3423 12.1961 -9.9258 15.7247 130 2.268928 131.1529 3.5361 -17.7310 5.6762 18.6174 15.0678 -6.5201 16.4180 140 2.443461 128.2997 4.4466 -14.9019 4.8694 15.6773 17.2225 -2.6775 17.4294 150 2.617994 125.9690 5.2758 -11.7668 4.7488 12.6889 18.5615 1.2869 18.6060 160 2.792527 124.2016 6.1437 -8.4720 5.3055 9.9962 19.0556 5.0252 19.7070 170 2.96706 123.0126 7.1632 -5.1611 6.4692 8.2758 18.7628 8.1864 20.4710
180 3.141593 122.3963 8.4299 -1.9596 8.1112 8.3446 17.8365 10.4589 20.6768 190 3.316126 122.3169 10.0122 1.0419 10.0546 10.1085 16.5157 11.6099 20.1881 200 3.490659 122.7430 11.9450 3.7998 12.0906 12.6737 15.0982 11.5061 18.9827 210 3.665191 123.6296 14.2250 6.3249 13.9956 15.3584 13.8996 10.1071 17.1858 220 3.839724 124.9409 16.8096 8.6815 15.5443 19.8043 13.2116 7.4300 15.1575 230 4.014257 126.6566 19.6174 10.9799 16.5162 19.8328 13.2637 3.4973 13.7170 240 4.18879 128.7785 22.5285 13.3625 16.6907 21.3807 14.1899 -1.7181 14.2935 250 4.363323 131.3349 25.3837 15.9842 15.8348 22.4997 15.9876 -8.3401 18.0323 260 4.537856 134.3801 27.9806 18.9827 13.6834 23.4003 18.4521 -16.6171 24.8315 270 4.712389 137.9877 30.0663 22.4345 9.9174 24.5288 21.0603 -26.9055 34.1678 280 4.886922 142.2348 31.3263 26.2843 4.1537 26.6105 22.7881 -39.5543 45.6491 290 5.061455 147.1694 31.3753 30.2371 -4.0293 30.5044 21.8734 -54.5700 58.7906 300 5.235988 152.7570 29.7585 33.6129 -14.9733 36.7971 15.6656 -70.8534 72.5646 310 5.410521 158.8002 25.9888 35.2113 -28.6334 45.3840 0.9965 -84.7776 84.7934 320 5.585054 164.8468 19.6625 33.3497 -43.9678 55.1849 -23.9979 -88.2107 91.4168 330 5.759587 170.1444 10.7101 26.4306 -58.0233 63.7596 -55.5350 -67.4662 87.3832 340 5.934119 173.7701 -0.1972 14.4321 -65.2381 66.8154 -78.6527 -8.1076 79.0694 350 6.108652 175.0806 -11.2616 0.8040 -58.8565 58.8620 -71.1130 83.4351 109.6288 360 6.283185 174.3046 -19.8534 -8.6248 -37.7102 38.6839 -35.1393 147.0985 151.2374
6 6 、心得体会
通过这次的机械原理大作业的完成，我学习到了很多东西，也有很多感触。
在完成本次六杆机构大作业的过程中，我体会到了 MATLAB 在机械分析分析中的重要性，这也是我第一次将过去学过的编程语言应用在实际解题中，确确实实体会到了计算机解题的优越性。同时，在将程序边好的那一刻，看着自己的劳动成果，心里不由得产生一种成就感。
在这个过程中，我也体会到了学习的重要性，我懂得了我们还有很多不懂得东西，但是我们不能被此局限住，我们要不断的学习，查资料，逐渐完善自己的知识面，而不是等待着老师讲解后再去学习，这样我们才能在以后的实践中有所提高。
能完成本次大作业，还得感谢王老师上课时的耐心讲解，以及同学的帮助。
