Mumbai University > Mechanical Engineering > Sem 7 > Machine Design 2

Marks: 10M

Year: Dec 2016

Given

P = 36800

N = 30 x 60 = 1800 rpm

i = 4

$\beta = 15^0$

$\alpha = 20^0$

Material C50

Solution

Design bending stress

$[\sigma_b] = \frac{1.4K_{b1}}{nk_{\sigma}} \sigma_{-1}$

For C50

$\sigma_u = 700 N/mm^2 \\ \sigma_y = 380 N/mm^2$ B4N = 241

$\sigma_{-1} = 0.22 (\sigma_u + \sigma_y) + 500 = 523.76$ For cast steel.

n = 2 (Normalized cast steel)

$k_\sigma = 1.5$ (For addendum modification o)

$K_{b1} = 1$ (For BHN < 350 and > $10^7$ cycle)

$\therefore [\sigma_b] = 287.6 N/mm^2$

Design contact stress

$[\sigma_c] = C_B HB K_{c1}\\ C_B = 25 \text{for HB $\leq$ 350}\\ K_{c1} = 1 \text{for \gt $10^7$ cycles}$

$[\sigma_c] = 6025 kgf/cm^2 = 602.5 N/mm^2$

Virtual no. of teeth of gear

$Z_{v1} = \frac{2}{sin^2 \alpha} = 17.09$

Actual no. of teeth on pinion

$Z_1 = Z_{v1} cos^3 \beta \\ z_1 = 15.407\\ Z_1 = 16 \\ Z_2 = iz_1 = 64 \approx 65$

Corrected i = 4.0625

$Z_{v2} = 72.124$

Form factor

$Y_{v1} = \pi \bigg[ 0.154 - \frac{0.912}{Z_v1}\bigg]\\ Y_{v1} = 0.3161\\ Y_{v1} = 0.444$

Strength of pinion = $Y_{v1} [\sigma_{b1}]=90.91$ Wheel = 127.717

Calculation of module based on beam strength

$m_n \geq 1.15 cos \beta \sqrt[3]{\frac{[M_t]}{Y [\sigma_b] \psi z_1}}$

$[M_t] = 1.5 \times \frac{P \times 60}{2 \pi N}\\ = 292.845 \times 10^3\\ M_n = 4.54\\ M_t = \frac{M_n}{cos \beta} = 4.71 \approx 5\\ M_n = 4.829$

Checking for contact stress induced

$\sigma_c = 0.7 \frac{i+1}{a} \sqrt{\frac{i+1}{ib} E [M_t]}\\ a = \frac{m(z_1 + z_2)}{2} = 202.5\\ b = 12\times 4.829 = 57.948 \approx 58\\ b_c = 643.647 N/mm^2\\ \sigma_c \gt [\sigma_c]\\ M_t = 6 mm\\ M_n = 5.7955 mm\\ b = 70 mm \\ a = 243\\ \sigma_c = 488.238 N/mm^2$

Safe.

Principal dimension

module $M_n = 5.7955$

Transverse $M_t = 6mm$

centre distance a = 243 mm

Bottom clearance c = 1.44 mm

tooth depth h = 13.039 mm

PCD $d_1 = m_t z_1 = 96 \hspace{1.5cm} d_2 = m_t z_2 = 390$

ACD $d_{a1} = 107.59 \hspace{2cm} d_{a2}= 401.587$

DCD $d_{f1} = 81.528 \hspace{2cm} d_{f2} = 375.525$