written 5.8 years ago by |
Determine the velocity and acceleration of the piston when the crank is @ 40° from 10 celsius. Also determine [1] the position of the crank for zero acceleration of piston [2] Angular velocity and angular acceleration of the connecting rod when crank is @ 40° from IDC.
→ Given: r=0.3m=300mm
L=1.5m=1500mm { n=Lr=1500300=5 }
N=180rpm
∴ w = \frac{2 \pi (180)}{60} = 18.85 rad/s
\theta = 40° From IDC.
[1] Vp = \omega .r. [sin \ \theta + \frac{sin 2 \ \theta}{2n}]
= 18.85 (0.3) [sin \ 40 + \frac{sin \ 2 \times 40}{2 \times 5}]
= 4.19 m/s - - - - 1st Ans.
[2] a_p = \omega^2. r [ cos \ \theta + \frac{cos \ 2 \theta}{n}]
= (18.85)^2 (0.3) [cos \ 40 + \frac{cos \ (2 \times 40)}{5}]
= 85.35 \ m/s^2 - - - - 2nd Ans.
[3] Position of the crank for zero acceleration of the piston.
\therefore a_p = \omega^2. r. [ cos \ \theta + \frac{cos \ 2 \theta}{n}]
\therefore 0 = [cos \ \theta + \frac{cos \ 2 \theta}{5}]
\therefore 5 \ cos \ \theta + cos \ 2 \ \theta = 0
5 \ cos \ \theta + 2 \ cos^2 \ \theta – 1 =0
On solving,
cos \ \theta = 0.186 OR cos \ \theta = -2.686 x Invalid.
\therefore \theta \ = cos^{-1} \ (0.186)
\theta = 79.28°
[4] Angular velocity of connecting rod.
\omega_{pc} = \frac{w. \ cos \ \theta}{n} = \frac{18.85 \ cos \ 40}{5} = 2.88 rad/s
[5] Angular acceleration of connecting rod.
\alpha_{pc} = \frac{\omega^2. sin \ \theta}{n} = \frac{(18.85)^2 . sin \ 40}{5}
\alpha_{pc} = 45.68 \ rad/s^2