摇动摆杆运动分析

2021-05-01 杂谈机械原理 Comments

提出问题：

在如左图所示的摆杆中，已知:

$\iota_1$ = 0.6 m
$\iota_4$ = 1.2 m
$\omega_1$= 3 rad/s

假如原动杆以$\omega_1$的恒定角速度转动，尝试导出机构3的角位置 $\theta_3$、角速度 $\omega_3$、角加速度$\alpha_3$

建立模型

1、首先要将运动机构简化为数学模型，可以很轻易得出如右图所示模型；
2、求出 $\theta_3$ ：

$\frac{cos\theta_3}{\iota_1}$=$\frac{sin(\theta_3-\theta_1)}{\iota_4}$ $\Downarrow$ 2cos$\theta_3$=sin($\theta_3-\theta_1$) $\Downarrow$ $2cos\theta_3 =sin\theta_3cos\theta_1-sin\theta_1cos\theta_3$ $\Downarrow$ $tan\theta_3 = \frac{2+sin\theta_1}{cos\theta_1}$ $\Downarrow$ $\theta_3 = arctan\frac{2+sin\theta_1}{cos\theta_1}$

3、因为$\theta_3$要满足如此条件： ($\frac{\pi}{3} \leqslant \theta_3 \leqslant \frac{2\pi}{3}$)。且用反三角函数解出来的 $\theta_3$ 有一跃变的过程。故，对 $\theta_3$ 对弧度值小于0的 $\theta_3$ 进行 $\theta_3$ = $\theta_3$ +$\pi$ 处理，可以得到正确的 $\theta_3$ 值。

4、求出 $\theta_3$ 后，则 $\omega_3$ 、$\alpha_3$ 求法就简单：

$\omega_3 = \omega_1 * cos (\theta_3-\theta_1)$ $\alpha_3 = \omega_1 * sin (\theta_3-\theta_1)$

5、对以上数学模型进行处理，分别带入到Excel、Matlab中，可以输出运行结果‘

Excel计算数据

\theta_1 (\cric)	\theta_3 (rad)	\omega_3 (rad/s)	\alpha_3 (rad/s^-2)
0	1.107148718	1.341640786	2.683281573
5	1.125474919	1.523293436	2.584487784
10	1.145395003	1.693764615	2.476118218
15	1.166705008	1.8532794	2.359100563
20	1.189228628	2.002024817	2.234255275
25	1.212812651	2.140153305	2.10231868
30	1.237323155	2.267786838	1.963961012
35	1.262642349	2.385021177	1.819800535
40	1.288665945	2.491929983	1.670414607
45	1.315300953	2.588568628	1.516348395
50	1.342463822	2.674977634	1.358121738
55	1.370078875	2.751185712	1.196234584
60	1.398076962	2.817212405	1.031171308
65	1.426394312	2.873070353	0.863404163
70	1.45497152	2.918767192	0.693396048
75	1.48375266	2.954307121	0.521602754
80	1.512684489	2.979692149	0.348474818
85	1.541715716	2.994923043	0.174459074
90	1.570796327	3	0
95	1.599876938	2.994923043	-0.174459074
100	1.628908165	2.979692149	-0.348474818
105	1.657839993	2.954307121	-0.521602754
110	1.686621134	2.918767192	-0.693396048
115	1.715198341	2.873070353	-0.863404163
120	1.743515692	2.817212405	-1.031171308
125	1.771513779	2.751185712	-1.196234584
130	1.799128831	2.674977634	-1.358121738
135	1.8262917	2.588568628	-1.516348395
140	1.852926708	2.491929983	-1.670414607
145	1.878950305	2.385021177	-1.819800535
150	1.904269499	2.267786838	-1.963961012
155	1.928780002	2.140153305	-2.10231868
160	1.952364025	2.002024817	-2.234255275
165	1.974887646	1.8532794	-2.359100563
170	1.99619765	1.693764615	-2.476118218
175	2.016117734	1.523293436	-2.584487784
180	2.034443936	1.341640786	-2.683281573
185	2.050939135	1.148541591	-2.771435046
190	2.065326477	0.943691779	-2.847708873
195	2.077281554	0.726754602	-2.910640436
200	2.086423264	0.497376043	-2.958482224
205	2.092303368	0.255215338	-2.989124476
210	2.094395102	-1.1485E-15	-3
215	2.092081706	-0.268380304	-2.98797122
220	2.084646809	-0.549719909	-2.949204642
225	2.071270364	-0.843253913	-2.879048947
230	2.051036647	-1.147288675	-2.771953949
235	2.022965068	-1.458669382	-2.621504079
240	1.986079566	-1.772071484	-2.420694664
245	1.939536273	-2.07917744	-2.162642174
250	1.882826066	-2.367960099	-1.841945974
255	1.816048477	-2.622540252	-1.456805623
260	1.740206377	-2.824297227	-1.011605247
265	1.657404473	-2.954765565	-0.518999474
270	1.570796327	-3	-3.67545E-16
275	1.484188181	-2.954765565	0.518999474
280	1.401386276	-2.824297227	1.011605247
285	1.325544177	-2.622540252	1.456805623
290	1.258766587	-2.367960099	1.841945974
295	1.202056381	-2.07917744	2.162642174
300	1.155513088	-1.772071484	2.420694664
305	1.118627586	-1.458669382	2.621504079
310	1.090556007	-1.147288675	2.771953949
315	1.07032229	-0.843253913	2.879048947
320	1.056945844	-0.549719909	2.949204642
325	1.049510948	-0.268380304	2.98797122
330	1.047197551	-5.51317E-16	3
335	1.049289285	0.255215338	2.989124476
340	1.05516939	0.497376043	2.958482224
345	1.064311099	0.726754602	2.910640436
350	1.076266177	0.943691779	2.847708873
355	1.090653519	1.148541591	2.771435046
360	1.107148718	1.341640786	2.683281573

Matlab程序运算、作图

示例程序：

clear
clc
%已知：
l1=0.6;
l4=1.2;
w1=3;
th=[0:5:360]
th1=deg2rad(th)
th3=atan(((l4/l1)+sin(th1))./cos(th1));
th3(th3<0)=th3(th3<0)+pi;
om3=(w1.*cos(th3-th1));
al3=w1.*sin(th3-th1)



figure(1)       %得到机构3的转动角度\theta_3的角度
plot(th,th3,'r')
title('角位移图线');
xlabel('原动杆转角 \theta_1 /\circ')
ylabel('从动杆转角 \theta_2 ')
grid on;

figure(2)       %得到机构3的角速度\omega_3
plot(th,om3,'b')
title('角速度图像');
xlabel('原动杆转动角度 \theta_1 /\circ')
ylabel('从动杆转角速读 \omega_2 /rad/s')
grid on;


figure(3)       %得到机构3的角加速度\alpha_3
plot(th,al3,'black')
title('角加速度图线');
xlabel('原动杆转角度 \theta_1 /\circ')
ylabel('从动杆转角加速度 \alpha_3 rad/s^2')
grid on;

figure(4)       %合并以上三个图像
plot(th,th3,'r',th,om3,'b',th,al3,'black')
title('摆动导杆运动分析');
text(100,1.3,'\theta_3')
text(100,2.7,'\omega_3')
text(200,-2.5,'\alpha_3')
grid on;

disp('原转动角度 - 从转动角度 - 角速度 - 角加速度')
y=[th',th3',om3',al3'];
disp(y)