Question

Following in each of the following three mechanisms, the crank is the input link, 1) draw vectors with labeled angles 2) write the lop closure equation/vector equation in exponential form. 3) solve the unknown variables

Answer 1

Explanation:

1.$\theta_{4}=90^{\circ}, \theta_{1}=\theta^{\circ}$ (diagram)

2.

vector loop equation is

$\overline{R_{1}}+\overline{R_{2}}+\overline{R_{3}}=\overline{R_{4}}$

the exponential form,

$r_{1} e^{j \exp (0)}+r_{2} e^{j \theta_{2}}+r_{3} e^{j \theta_{3}}=r_{4} e^{j(90)}$

$e=\cos \theta+j \sin \theta$

3.

Real part,$r_{1}+\ r_{2} \cos \theta_{2}+\ r_{3} \cos \theta_{3}=0$   (i)

Imaginary part,$0+r_{2} \sin \theta_{2}+r_{3} \sin \theta_{3}=r_{4}$     (ii)

From above equation,we get unknown variables,

Now,$\theta_{2}, \theta_{3} \& r_{4}$

If $\theta_{2}$ is given,

From (i), $\theta_{3}=\cos ^{-1}\left(\frac{-r_{1}-r_{2} \cos \theta_{2}}{r_{3}}\right)$

and from (ii), we get

$x_{4}=r_{2} \sin \theta_{2}+r_{3} \sin \left[\cos ^{-1}\left(\frac{-\gamma_{1}-r_{2} \cos \theta_{2}}{r_{3}}\right)\right]$