Find the angular velocity of gear 5 using first order kinema

Find the angular velocity of gear 5 using first order kinematic coefficients. The centers of gears 4 and 5 are fixed. Given: R_2 = 4\", R_ab = 12\", p_4 = 2\", p_5 = 6\", theta_2 = 60 degree, omega_2 = 10 rad/s ccw

Solution

solution;

1) here it is four bar chain driving a internal gear 4 ,hence loop closure eqaution is given by

R1\'+R2\'+R3\'+R4\'=0

where here real part and imaginary part is given by

Rcosa1+R2cosa2+R3cosa3+R4cosa4=0

R1sina1+R2sina2+R3sina3+R4sina4=0

hence by first order coefficient method differenting eqaution with respect to input angle a2

here

da1/da2=h1,da4/da2=h4 and dR1/da1=f1

as first link is stationary and has constant angle hence h1 and f1 are zero,then eqaution becomes

-2sina4h4=3.464

2cosa4h4=-2

hence taking ratio we get

a4=60

3) here differentiting loop closure equation wrt time we get velocity as follows

R2w2(cosa2+sina2i)+R3w3(cosa3+sina3i)+R4w4(cosa4+sina4i)=0

on putting value and separating real and imaginary part we get

12w3cos45+2w4cos60+20=0

-12w3sin45+2w4sin60+34.64=0

hence w3=4.3833 rad/s

w4=19.99 rad/s

5) for internal gear ,speed ratio is

w4/w5=R5/R4=6/2=3

w5=w4/3=19.99/3=6.66 rad/s