机械原理423.docx
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机械原理423
机械原理作业
基于matlab平面机构力的分析
摘要:
此文系对《机械原理》第七版4-23题的解答,应用matlab软件对构件进行力学分析的过程。
一、问题:
在下图所示的正弦机构中,已知lAB=100mm,h1=120mm,h2=80mm,W1=10rad/s(常数),滑块2和构件3的重量分别为G2=40N和G3=100N,质心S2和S3的位置如图所示,加于构件3上的生产阻力Fr=400N,构件1的重力和惯性力略去不计。
试用解析法求机构在Φ1=60°、150°、220°位置时各运动副反力和需加于构件1上的平衡力偶Mb。
二、问题分析:
分别对三个构件进行受力分析如图:
构件3受力图
构件2受力图
构件1受力图
(1)滑块2:
VS2=LABW1①
as2=LABW12②
构件3:
S=LABsinΦ1③
V3=LABW1COSΦ1④
a3=-LABW12sinΦ1⑤
(2)确定惯性力:
F12=m2as2=(G2/g)LABW12⑥
F13=m3a3=(G3/g)LABW12sinΦ1⑦
(3)各构件的平衡方程:
构件3:
∑Fy=0,FR23=Fr-F13
∑Fx=0,FR4’=FR4
∑MS3=0,FR4=FR23LAcosΦ1/h2
构件2:
∑Fx=0,FR12x=F12cosΦ1
∑Fy=0,FR12y=FR32-F12sinΦ1
构件1:
∑Fx=0,FR41x=FR12x
∑Fy=0,FR41y=FR12y
∑MA=0,Mb=FR32LABcosΦ1
总共有八个方程,八个未知数。
归纳出一元八次方程矩阵:
10000000FR23Fr-F13
01-100000FR4’0
-LABCOSΦ1/h20100000FR40
00010000FR12x=F12cosΦ1
-10001000FR12y-F12sinΦ1
000-10100FR41x0
0000-1010FR41y0
-LABCOSΦ10000001Mb0
由AX=B进而可得:
X=A\B。
接下来运用软件工具进行分析。
三、基于MATLAB的编程分析
首先编写函数F用于实现上述运算功能:
functiony=F(x)
%
%inputparameters
%
%x
(1)=lAB
%x
(2)=h1
%x(3)=h2
%x(4)=W1
%x(5)=G2
%x(6)=G3
%x(7)=Fr
%x(8)=theta1
%
%outputparameters
%
%y
(1)=FR23
%y
(2)=FR4'
%y(3)=FR4
%y(4)=FR12x
%y(5)=FR12y
%y(6)=FR41x
%y(7)=FR41y
%y(8)=Mb
%
A=[10000000;
01-100000;
-x
(1)*cos(x(8))/x(3)0100000;
00010000;
-10001000;
000-10100;
0000-1010;
-x
(1)*cos(x(8))0000001];
B=[x(7)-(x(6)/10)*x
(1)*x(4)^2*sin(x(8));0;0;(x(5)/10)*x
(1)*x(4)^2*cos(x(8));-(x(5)/10)*x
(1)*x(4)^2*sin(x(8));0;0;0];
y=A\B;
接下来运行程序计算Φ1=60°的各未知量值:
lAB=0.1;
h1=0.120;
h2=0.08;
W1=10;
G2=40;
G3=100;
Fr=400;
th1=60*pi/180;
x=[lABh1h2W1G2G3Frth1];
y=F(x)
y=
313.3975
195.8734
195.8734
20.0000
278.7564
20.0000
278.7564
15.6699
得到:
Φ1=60°时FR23=FR32=313.3975N;FR4=FR4’=195.8734N;FR12x=20.0000N;FR12y=278.7564N;FR41x=20.0000N;FR41y=278.7564N;Mb=15.6699N*m。
运行程序计算Φ1=150°的各未知量值:
>>th1=150*pi/180;
>>x=[lABh1h2W1G2G3Frth1];
>>y=F(x)
y=
350.0000
-378.8861
-378.8861
-34.6410
330.0000
-34.6410
330.0000
-30.3109
得到:
Φ1=150°时FR23=FR32=350.0000N;FR4=FR4’=-378.8861N;FR12x=-34.6410N;FR12y=330.0000N;FR41x=-34.6410N;FR41y=330.0000N;Mb=-30.3109N*m。
运行程序计算Φ1=220°的各未知量值:
>>th1=220*pi/180;
>>x=[lABh1h2W1G2G3Frth1];
>>y=F(x)
y=
464.2788
-444.5727
-444.5727
-30.6418
489.9903
-30.6418
489.9903
-35.5658
得到:
Φ1=220°时FR23=FR32=464.2788N;FR4=FR4’=-444.5727N;FR12x=-30.6418N;FR12y=489.9903N;FR41x=-30.6418N;FR41y=489.9903N;Mb=-35.5658N*m。
接下来取Φ1=0~360°范围分析其受力:
h1=0.120;
h2=0.08;
W1=10;
G2=40;
G3=100;
Fr=400;
th1=linspace(0,2*pi,36);
x=zeros(length(th1),8);
forn=1:
36
x(n,:
)=[lABh1h2W1G2G3Frth1(n)];
end
p=zeros(8,length(th1));
fork=1:
36
p(:
k)=F(x(k,:
));
end
>>p
p=
Columns1through8
400.0000382.1443364.8625348.7101334.2061321.8169311.9404304.8943
500.0000470.0039426.9963374.1872314.6014250.8119184.7735117.7719
500.0000470.0039426.9963374.1872314.6014250.8119184.7735117.7719
40.000039.357237.449434.338030.122924.939618.954712.3607
400.0000375.0020350.8075328.1941307.8886290.5436276.7166266.8521
40.000039.357237.449434.338030.122924.939618.954712.3607
400.0000375.0020350.8075328.1941307.8886290.5436276.7166266.8521
40.000037.600334.159729.935025.168120.065014.78199.4218
Columns9through16
300.9050300.1007302.5072308.0472316.5427327.7205341.2215356.6116
50.4893-16.8300-84.1427-151.3378-217.9780-283.0943-345.0675-401.6200
50.4893-16.8300-84.1427-151.3378-217.9780-283.0943-345.0675-401.6200
5.3693-1.7946-8.9008-15.7210-22.0359-27.6425-32.3607-36.0388
261.2670260.1410263.5101271.2661283.1597298.8087317.7101339.2563
5.3693-1.7946-8.9008-15.7210-22.0359-27.6425-32.3607-36.0388
261.2670260.1410263.5101271.2661283.1597298.8087317.7101339.2563
4.0391-1.3464-6.7314-12.1070-17.4382-22.6475-27.6054-32.1296
Columns17through24
373.3963391.0361408.9639426.6037443.3884458.7785472.2795483.4573
-449.9252-486.8273-509.1470-514.0376-499.3489-463.9495-407.9684-332.9190
-449.9252-486.8273-509.1470-514.0376-499.3489-463.9495-407.9684-332.9190
-38.5585-39.8390-39.8390-38.5585-36.0388-32.3607-27.6425-22.0359
362.7548387.4505412.5495437.2452460.7437482.2899501.1913516.8403
-38.5585-39.8390-39.8390-38.5585-36.0388-32.3607-27.6425-22.0359
362.7548387.4505412.5495437.2452460.7437482.2899501.1913516.8403
-35.9940-38.9462-40.7318-41.1230-39.9479-37.1160-32.6375-26.6335
Columns25through32
491.9528497.4928499.8993499.0950495.1057488.0596478.1831465.7939
-241.6872-138.3782-28.034983.7439191.2451289.0952372.6779438.4701
-241.6872-138.3782-28.034983.7439191.2451289.0952372.6779438.4701
-15.7210-8.9008-1.79465.369312.360718.954724.939630.1229
528.7339536.4899539.8590538.7330533.1479523.2834509.4564492.1114
-15.7210-8.9008-1.79465.369312.360718.954724.939630.1229
528.7339536.4899539.8590538.7330533.1479523.2834509.4564492.1114
-19.3350-11.0703-2.24286.699515.299623.127629.814235.0776
Columns33through36
451.2899435.1375417.8557400.0000
484.2616509.2386513.9257500.0000
484.2616509.2386513.9257500.0000
34.338037.449439.357240.0000
471.8059449.1925424.9980400.0000
34.338037.449439.357240.0000
471.8059449.1925424.9980400.0000
38.740940.739141.114140.0000
P矩阵的每一行分别是八个未知量在Φ1等于36个分量下的值。
整理表格如下:
表一
Φ1
FR23/FR32
FR4
FR4’
FR12X
FR12Y
FR41X
FR41Y
Mb
rad
N
N
N
N
N
N
N
N*m
0
0.1795
0.3590
0.5386
0.7181
0.8976
1.0771
1.2566
1.4362
1.6157
1.7952
1.9747
2.1542
2.3338
2.5133
2.6928
2.8723
3.0518
3.2314
3.4109
3.5904
3.7699
3.9494
4.1290
4.3085
4.4880
4.6675
4.8470
5.0265
5.2061
5.3856
5.5651
5.7446
5.9241
6.1037
6.2832
400.0000
382.1443
364.8625
348.7101
334.2061
321.8169
311.9404
304.8943
300.9050
300.1007
302.5072
308.0472
316.5427
327.7205
341.2215
356.6116
373.3963
391.0361
408.9639
426.6037
443.3884
458.7785
472.2795
483.4573
491.9528
497.4928
499.8993
499.0950
495.1057
488.0596
478.1831
465.7939
451.2899
435.1375
417.8557
400.0000
500.0000
470.0039
426.9963
374.1872
314.6014
250.8119
184.7735
117.7719
50.4893
-16.8300
-84.1427
-151.3378
-217.9780
-283.0943
-345.0675
-401.6200
-449.9252
-486.8273
-509.1470
-514.0376
-499.3489
-463.9495
-407.9684
-332.9190
-241.6872
-138.3782
-28.0349
83.7439
191.2451
289.0952
372.6779
438.4701
484.2616
509.2386
513.9257
500.0000
500.0000
470.0039
426.9963
374.1872
314.6014
250.8119
184.7735
117.7719
50.4893
-16.8300
-84.1427
-151.3378
-217.9780
-283.0943
-345.0675
-401.6200
-449.9252
-486.8273
-509.1470
-514.0376
-499.3489
-463.9495
-407.9684
-332.9190
-241.6872
-138.3782
-28.0349
83.7439
191.2451
289.0952
372.6779
438.4701
484.2616
509.2386
513.9257
500.0000
40.0000
39.3572
37.4494
34.3380
30.1229
24.9396
18.9547
12.3607
5.3693
-1.7946
-8.9008
-15.7210
-22.0359
-27.6425
-32.3607
-36.0388
-38.5585
-39.8390
-39.8390
-38.5585
-36.0388
-32.3607
-27.6425
-22.0359
-15.7210
-8.9008
-1.7946
5.3693
12.3607
18.9547
24.9396
30.1229
34.3380
37.4494
39.3572
40.0000
400.0000
375.0020
350.8075
328.1941
307.8886
290.5436
276.7166
266.8521
261.2670
260.1410
263.5101
271.2661
283.1597
298.8087
317.7101
339.2563
362.7548
387.4505
412.5495
437.2452
460.7437
482.2899
501.1913
516.8403
528.7339
536.4899
539.8590
538.7330
533.1479
523.2834
509.4564
492.1114
471.8059
449.1925
424.9980
400.0000
40.0000
39.3572
37.4494
34.3380
30.1229
24.9396
18.9547
12.3607
5.3693
-1.7946
-8.9008
-15.7210
-22.0359
-27.6425
-32.3607
-36.0388
-38.5585
-39.8390
-39.8390
-38.5585
-36.0388
-32.3607
-27.6425
-22.0359
-15.7210
-8.9008
-1.7946
5.3693
12.3607
18.9547
24.9396
30.1229
34.3380
37.4494
39.3572
40.0000
400.0000
375.0020
350.8075
328.1941
307.8886
290.5436
276.7166
266.8521
261.2670
260.1410
263.5101
271.2661
283.1597
298.8087
317.7101
339.2563
362.7548
387.4505
412.5495
437.2452
460.7437
482.2899
501.1913
516.8403
528.7339
536.4899
539.8590
538.7330
533.1479
523.2834
509.4564
492.1114
471.8059
449.1925
424.9980
400.0000
40.0000
37.6003
34.1597
29.9350
25.1681
20.0650
14.7819
9.4218
4.0391
-1.3464
-6.7314
-12.1070
-17.4382
-22.6475
-27.6054
-32.1296
-35.9940
-38.9462
-40.7318
-41.1230
-39.9479
-37.1160
-32.6375
-26.6335
-19.3350
-11.0703
-2.2428
6.6995
15.2996
23.1276
29.8142
35.0776
38.7409
40.7391
41.1141
40.0000
按照上述分析结果输出图像:
输出FR23和FR32的图像,程序如下:
plot(th1,p(1,:
),'--')
ylabel('FR23,FR32/N');
xlabel('\theta1/rad');
输出FR4和FR4’的图像,程序如下:
plot(th1,p(3,:
),'--')
xlabel('\theta1/rad');
ylabel('FR4,FR4''/N');
输出FR12x的图像,程序如下:
plot(th1,p(4,:
),'--')
ylabel('FR12x/N');
xlabel('\theta1/rad');
输出FR12y的图像,程序如下:
plot(th1,p(5,:
),'--')
ylabel('FR12y/N');
xlabel('\theta1/rad');
输出FR41X的图像,程序如下:
plot(th1,p(6,:
),'--')
xlabel('\theta1/rad');
ylabel('FR41x/N');
输出FR41Y的图像,程序如下:
plot(th1,p(7,:
),'--')
xlabel('\theta1/rad');
ylabel('FR41y/N');
输出Mb的图像,程序如下:
plot(th1,p(8,:
),'--')
xlabel('\theta1/rad');
ylabel('Mb/N*m');
参考文献
【1】孙恒,陈作模.机械原理【M】.7版.北京:
高等教育出版社,2006
【2】李团结,贾建援,胡雪梅.机械工程中两类非线性方程组的完全解.西安电子科技大学学报(自然科学版)2005年2月,第32卷,第1期
【3】贺超英.MATLAB应用与实验教程.北京:
电子工业出版社,2010
【4】张志涌,徐彦琴等.MATLAB教程——基于6.x版本.2004年北京北京航空航天大学出版社
【5】《数据结构(c语言版)例题详解与课程设计指导》第二版
袁志祥、秦锋主编中国科学技术大学出版社出版
【6】周品,何正风.MATLAB数值分析.北京:
机械工业出版社,2009
【7】薛定宇《科学运算语言matlab5.3·程序设计与应用》·清华大学出版社·2000
【8】石博强,赵金《matlab数学计算与工程分析范例教程》中国铁道出版社·2005
【9】同