Chapter 15, Problem 10P

To determine

Whether the rated power is valid.

Answer to Problem 10P

The rated power is valid.

Explanation of Solution

Write the expression for pitch diameter of gear.

$d_G=\frac{N}{P_d}$ (I)

Here, the pitch diameter of gear is $d_G$, the number of teeth on gear is $N$ and the diametral pitch is $P_d$.

Write the expression for pitch diameter of pinion.

$d_P=\frac{n}{P_d}$ (II)

Here, the pitch diameter of pinion is $d_P$, the number of teeth on pinion is $n$ and the diametral pitch is $P_d$.

Write the expression for gear ratio of gear set.

$m_G=\frac{N}{n}$ (III)

Here, the gear ratio is $m_G$.

Write the expression for pitch line velocity.

$v_t=\pi d_P{n}_P$ (IV)

Here, the pitch line velocity is $v_t$ and the rotation of pinion per unit time is $n_P$.

Write the expression for dynamic factor.

$K_v={\left(\frac{A+\sqrt{v_t}}{A}\right)}^B$ (V)

Here, dynamic factor is $K_v$ and constants are $A$ and $B$.

Write the expression for constant $B$.

$B=0.25{\left(12-Q_v\right)}^{2/3}$ (VI)

Here, the transmission accuracy number is $Q_v$.

Write the expression for constant $A$.

$A=50+56\left(1-B\right)$ (VII)

Write the expression of size factor for $0.5\le P_d\le 16\text{teeth}/\text{in}$.

$K_S=0.4867+0.2132/P_d$ (VIII)

Here, size factor is $K_S$.

Write the expression for size factor of pitting resistance for $0.5\le F\le 4.5\text{in}$.

$C_S=0.125F+0.4375$ (IX)

Here, size factor for pitting resistance is $C_S$ and the face width is $F$.

Write the expression for load distribution factor.

$K_m=K_{mb}+0.0036F^2$ (X)

Here, load distribution factor is $K_m$, the face width is $F$ and the constant is $K_{mb}$.

Write the general expression for stress cycle factor of pinion for $3\times {10}^6\le N_L^P\le {10}^{10}$.

$K_L^P=1.3558{\left(N_L^P\right)}^{-0.0178}$ (XI)

Here, the stress cycle factor for pinion is $K_L^P$ and the life goal in revolution for pinion is $N_L^P$.

Write the general expression for stress cycle factor of gear for $3\times {10}^6\le N_L^P\le {10}^{10}$.

$K_L^G=1.3558{\left(\frac{N_L^P}{m_G}\right)}^{-0.0178}$ (XII)

Here, the stress cycle factor for gear is $K_L^G$.

Write the expression for reliability factor for bending strength.

$K_R=0.5-0.25\log \left(1-R\right)$ for $0.99\le R\le 0.999$ (XIII)

Here, the reliability factor is $K_R$ and the reliability is $R$.

Write the expression for reliability factor for pitting.

$C_R=\sqrt{K_R}$ (XIV)

Write the expression for stress cycle factor for pitting resistance for pinion for ${10}^4\le N_L^P\le {10}^{10}$.

$C_L^P=3.4822{\left(N_L^P\right)}^{-0.0602}$ (XV)

Here, the stress cycle factor for pitting resistance for pinion is $C_L^P$ and the life goal in revolution for pinion is $N_L^P$.

Write the critical expression for stress cycle factor for pitting resistance for gear for $3\times {10}^6\le N_L^P\le {10}^{10}$.

$C_L^G=3.4822{\left(\frac{N_L^P}{m}\right)}^{-0.0602}$ (XVI)

Here, the stress cycle factor for pitting resistance for gear is $C_L^G$.

Write the expression for hardness ratio factor for pitting resistance for gear.

$C_H^G=1+B_1\left(m_G-1\right)$ (XVII)

Here, hardness ratio factor for pitting resistance for gear is $C_H^G$ and the constant is $B_1$.

Write the expression for constant $B_1$.

$B_1=0.00898\left(H_B^P/H_B^G\right)-0.00829$ (XVIII)

Here, the Brinell harness number for pinion is $H_B^P$ and the Brinell harness number for gear is $H_B^G$.

Write the expression for allowable bending stress number.

$S_{at}=44H_B+2100$ (XIX)

Here, the allowable bending stress number is $S_{at}$ and the Brinell hardness is $H_B$.

Write the expression for permissible bending stress for pinion.

$S_{wt}^P=\frac{S_{at}{K}_L^P}{S_F{K}_T{K}_R}$ (XX)

Here, the permissible bending stress for pinion is $S_{wt}^P$, the bending safety factor is $S_F$ and the temperature factor is $K_T$.

Write the expression for permissible bending stress for gear.

$S_{wt}^G=\frac{S_{at}{K}_L^G}{S_F{K}_T{K}_R}$ (XXI)

Here, the permissible bending stress for gear is $S_{wt}^G$.

Write the expression for bending stress for pinion.

$s_t^P=\frac{{\left(W_t^P\right)}_B}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^P}$ (XXII)

Here, the bending stress in pinion is $S_t^P$, the transmitted load by pinion in bending is ${\left(W_t^P\right)}_B$, the overload factor is $K_o$, length wise curvature factor is $K_x$ and geometry factor for bending in pinion is $J^P$.

Write the expression for bending stress for gear.

$s_t^G=\frac{{\left(W_t^G\right)}_B}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^G}$ (XXIII)

Here, the bending stress in gear is $S_t^G$, the transmitted load on gear in bending is ${\left(W_t^G\right)}_B$ and geometry factor for bending in gear is $J^G$.

Write the expression for transmitted load by pinion in bending.

${\left(W_t^P\right)}_B=33000\frac{{\left(H^P\right)}_B}{v_t}$ (XXIV)

Here, the transmitted load by pinion in bending is ${\left(W_t^P\right)}_B$ and the power in $\text{hp}$ for pinion in bending is ${\left(H^P\right)}_B$.

Write the expression for transmitted load on gear in bending.

${\left(W_t^G\right)}_B=33000\frac{{\left(H^G\right)}_B}{v_t}$ (XXV)

Here, the transmitted load on gear in bending is ${\left(W_t^G\right)}_B$ and the power in $\text{hp}$ for gear in bending is ${\left(H^G\right)}_B$.

Write the expression for allowable contact stress number.

$S_{ac}=341H_B+23620$ (XXVI)

Here, the allowable bending stress number is $S_{ac}$ and the Brinell hardness is $H_B$.

Write the expression for permissible contact stress number for pinion.

$S_{wc}^P=\frac{S_{ac}{C}_L^P{C}_H^P}{S_H{K}_T{C}_R}$ (XXVII)

Here, the permissible contact stress number for pinion is $S_{wc}^P$, the contact safety factor is $S_H$, the hardness ratio factor for pitting resistance for pinion is $C_H^P$ and the temperature factor is $K_T$.

Write the expression for permissible contact stress number for gear.

$S_{wc}^G=\frac{S_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R}$ (XXVIII)

Here, the permissible contact stress for gear is $S_{wc}^G$.

Write the expression for contact stress for pinion.

$s_c^P=C_p{\left(\frac{{\left(W_t^P\right)}_C}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}$ (XXIX)

Here, the contact stress in pinion is $S_C^P$, the transmitted load by pinion for wear is ${\left(W_t^P\right)}_C$, the overload factor is $K_o$, crowning factor for pitting resistance is $C_{xc}$ and geometry factor for pitting resistance is $I$.

Write the expression for contact stress for gear.

$s_c^G=C_p{\left(\frac{{\left(W_t^G\right)}_C}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}$ (XXX)

Here, the contact stress in gear is $S_c^G$ and the transmitted load on gear for wear is ${\left(W_t^G\right)}_C$.

Write the expression for transmitted load by pinion for wear.

${\left(W_t^P\right)}_C=33000\frac{{\left(H^P\right)}_C}{v_t}$ (XXXI)

Here, the transmitted load by pinion for wear is ${\left(W_t^P\right)}_C$ and the power in $\text{hp}$ of pinion for wear is ${\left(H^P\right)}_C$.

Write the expression for transmitted load on gear for wear.

${\left(W_t^G\right)}_C=33000\frac{{\left(H^G\right)}_C}{v_t}$ (XXXII)

Here, the transmitted load on gear for wear is ${\left(W_t^G\right)}_C$ and the power in $\text{hp}$ of gear for wear is ${\left(H^G\right)}_C$.

Write the expression for crowning factor for pitting of uncrowned tooth.

$C_{xc}=2.0$ (XXXIII)

Here, the crowning factor for pitting is $C_{xc}$.

Conclusion:

Substitute $40\text{teeth}$ for $N$ and $10\text{teeth}/\text{in}$ for $P_d$ in Equation (I).

$\begin{array}{c}{d}_G=\frac{40\text{teeth}}{10\text{teeth}/\text{in}}\\ =4\text{in}\end{array}$

Substitute $20\text{teeth}$ for $n$ and $10\text{teeth}/\text{in}$ for $P_d$ in Equation (II).

$\begin{array}{c}{d}_P=\frac{20\text{teeth}}{10\text{teeth}/\text{in}}\\ =2\text{in}\end{array}$

Substitute $40\text{teeth}$ for $N$ and $20\text{teeth}$ for $n$ in Equation (III).

$\begin{array}{c}m=\frac{40\text{teeth}}{20\text{teeth}}\\ =2\end{array}$

Substitute $2\text{in}$ for $d_P$ and $1200\text{rotation}/\text{min}$ for $n_P$ in Equation (IV).

$\begin{array}{c}{v}_t=\pi \left(2\text{in}\right)\left(1200\text{rotation}/\text{min}\right)\\ =\left(7539.82\text{in}/\text{min}\right)\left(\frac{1\text{ft}}{12\text{in}}\right)\\ =628.3\text{ft}/\text{min}\end{array}$

Substitute $5$ for $Q_v$ in Equation (VI).

$\begin{array}{c}B=0.25{\left(12-5\right)}^{2/3}\\ =0.25{\left(7\right)}^{2/3}\\ =0.9148\end{array}$

Substitute $0.9148$ for $B$ in Equation (VII)

$\begin{array}{c}A=50+56\left(1-0.9148\right)\\ =54.77\end{array}$

Substitute $54.77$ for $A$, $0.9148$ for $B$ and $628.3\text{ft}/\text{min}$ for $v_t$ in Equation (V).

$\begin{array}{c}{K}_v={\left(\frac{54.77+\sqrt{628.3\text{ft}/\text{min}}}{54.77}\right)}^{0.9148}\\ ={\left(1.4576\right)}^{0.9148}\\ =1.412\end{array}$

Substitute $10\text{teeth}/\text{in}$ for $P_d$ in Equation (VIII).

$\begin{array}{c}{K}_S=0.4867+0.2132/10\text{teeth}/\text{in}\\ =0.508\end{array}$

Substitute $0.71\text{in}$ for $F$ in Equation (IX).

$\begin{array}{c}{C}_S=0.125\left(0.71\text{in}\right)+0.4375\\ =0.52625\end{array}$

Substitute $1.25$ for $K_{mb}$ for neither member straddle-mounted and $0.71\text{in}$ for $F$ in Equation (X).

$\begin{array}{c}{K}_m=1.25+0.0036{\left(0.71\right)}^2\\ =1.252\end{array}$

Substitute $3\times {10}^6\text{rotation}$ for $N_L^P$ in Equation (XI).

$\begin{array}{c}{K}_L^P=1.3558{\left(3\times {10}^6\text{rotation}\right)}^{-0.0178}\\ =1.04\end{array}$

Substitute $2$ for $m_G$ and $3\times {10}^6\text{rotation}$ for $N_L^P$ in Equation (XII).

$\begin{array}{c}{K}_L^G=1.3558{\left(\frac{3\times {10}^6}{2}\right)}^{-0.0178}\\ =1.3558\times 0.7764\\ =1.053\end{array}$

Substitute $0.99$ for $R$ in Equation (XIII).

$\begin{array}{c}{K}_R=0.5-0.25\log \left(1-0.99\right)\\ =1\end{array}$

Substitute $1$ for $K_R$ in Equation (XIV).

$\begin{array}{c}{C}_R=\sqrt{1}\\ =1\end{array}$

Substitute $3\times {10}^6\text{rotation}$ for $N_L^P$ in Equation (XV).

$\begin{array}{c}{C}_L^P=3.4822{\left(3\times {10}^6\text{rotation}\right)}^{-0.0602}\\ =1.419\end{array}$

Substitute $2$ for $m_G$ and $3\times {10}^6\text{rotation}$ for $N_L^P$ in Equation (XVI).

$\begin{array}{c}{C}_L^G=3.4822{\left(\frac{3\times {10}^6\text{rotation}}{2}\right)}^{-0.0602}\\ =1.479\end{array}$

Substitute $300$ for $H_B^P$ and $300$ for $H_B^G$ in Equation (XVIII).

$\begin{array}{c}{B}_1=0.00898\left(\frac{300}{300}\right)-0.00829\\ =0.00898-0.00829\\ =0.00069\end{array}$

Substitute $0.00069$ for $B_1$ and $2$ for $m_G$ in Equation (XVII).

$\begin{array}{c}{C}_H^G=1+\left(0.00069\right)\left(2-1\right)\\ =1+0.00069\\ \approx 1\end{array}$

Substitute $300$ for $H_B$ in Equation (XXIX).

$\begin{array}{c}{S}_{at}=44\left(300\right)+2100\\ =15300\text{psi}\end{array}$

Substitute $15300\text{psi}$ for $S_{at}$, $1.04$ for $K_L^P$, $1$ for $S_F$, $1$ for $K_T$ and $1$ for $K_R$ in Equation (XX).

$\begin{array}{c}{S}_{wt}^P=\frac{\left(15300\text{psi}\right)\times 1.04}{1\times 1\times 1}\\ =15912\text{psi}\end{array}$

Refer to Figure 15-7 “Bending factor $J\left(Y_J\right)$ for coniflex straight-bevel gears with a $20°$ normal pressure angle and $90°$ shaft angle” to obtain value of $J^P$ as $0.241$.

Substitute $15912\text{psi}$ for $S_t^P$, $0.71\text{in}$ for $F$, $10\text{teeth}/\text{in}$ for $P_d$, 1 for $K_o$, $1.412$ for $K_v$, $0.508$ for $K_S$, $0.241$ for $J^P$ and $1.252$ for $K_m$ in Equation (XXII).

$\begin{array}{l}15912\text{psi}=\frac{{\left(W_t^P\right)}_B}{0.71\text{in}}\times 10\text{teeth}/\text{in}\times 1\times 1.412\times \frac{0.508\times 1.252}{1\times 0.241}\\ {\left(W_t^P\right)}_B=\frac{15912\times 0.71\times 0.241}{10\times 1.412\times 0.508\times 1.252}\text{lbf}\\ {\left(W_t^P\right)}_B=303.18\text{lbf}\end{array}$

Substitute $303.18\text{lbf}$ for ${\left(W_t^P\right)}_B$ and $628.3\text{ft}/\text{min}$ for $v_t$ in Equation (XXIV).

$\begin{array}{l}303.18\text{lbf}=33000\frac{{\left(H^P\right)}_B}{628.3\text{ft}/\text{min}}\\ {\left(H^P\right)}_B=\frac{303.18\times 628.3}{33000}\text{hp}\\ {\left(H^P\right)}_B=5.77\text{hp}\end{array}$

Substitute $15300\text{psi}$ for $S_{at}$, $1.053$ for $K_L^G$, $1$ for $S_F$, $1$ for $K_T$ and $1$ for $K_R$ in Equation (XXI).

$\begin{array}{c}{S}_{wt}^G=\frac{\left(15300\text{psi}\right)\times 1.053}{1\times 1\times 1}\\ =16110.9\text{psi}\end{array}$

Refer to Figure 15-7 “Bending factor $J\left(Y_J\right)$ for coniflex straight-bevel gears with a $20°$ normal pressure angle and $90°$ shaft angle” to obtain value of $J^G$ as $0.201$.

Substitute $16110.9\text{psi}$ for $S_t^G$, $0.71\text{in}$ for $F$, $10\text{teeth}/\text{in}$ for $P_d$, $1$ for $K_o$, $1.412$ for $K_v$, $0.508$ for $K_S$, $0.201$ for $J^G$ and $1.252$ for $K_m$ in Equation (XXIII).

$\begin{array}{l}16110.9\text{psi}=\frac{{\left(W_t^G\right)}_B}{0.71\text{in}}\times 10\text{teeth}/\text{in}\times 1\times 1.412\times \frac{0.508\times 1.252}{1\times 0.201}\\ {\left(W_t^G\right)}_B=\frac{16110.9\times 0.71\times 0.201}{10\times 1.412\times 0.508\times 1.252}\text{lbf}\\ {\left(W_t^G\right)}_B=256.01\text{lbf}\end{array}$

Substitute $256.01\text{lbf}$ for ${\left(W_t^G\right)}_B$ and $628.3\text{ft}/\text{min}$ for $v_t$ in Equation (XXV).

$\begin{array}{l}256.01\text{lbf}=33000\frac{{\left(H^G\right)}_B}{628.3\text{ft}/\text{min}}\\ {\left(H^G\right)}_B=\frac{628.3\times 256.01}{33000}\\ {\left(H^G\right)}_B=4.87\text{hp}\end{array}$

Substitute $300$ for $H_B$ in Equation (XXVI).

$\begin{array}{c}{S}_{ac}=341\left(300\right)+23620\\ =125920\text{psi}\end{array}$

Substitute $125920\text{psi}$ for $S_{ac}$, $1.419$ for $C_L^P$, $1$ for $C_H^P$, $1$ for $S_H$, $1$ for $K_T$ and $1$ for $C_R$ in Equation (XXVII).

$\begin{array}{c}{S}_{wc}^P=\frac{\left(125920\text{psi}\right)\times 1.419\times 1}{1\times 1\times 1}\\ =178680.48\text{psi}\end{array}$

Refer to Table 14-8 “Elastic Coefficient $C_P\left(Z_E\right)$, $\sqrt{\text{psi}}\left(\sqrt{\text{MPa}}\right)$” to obtain value of $C_P$ as $2290\sqrt{\text{psi}}$ for steel material.

Refer to Figure 15-6 “Contact geometry factor $I\left(Z_I\right)$ for coniflex straight-bevel gears with a $20°$ normal pressure angle and a $90°$ shaft angle” to obtain value of $I$ as $0.078$ for pinion teeth of 20 and gear teeth of 60.

Substitute $178680.48\text{psi}$ for $S_c^P$, $0.71\text{in}$ for $F$, $2\text{in}$ for $d_P$, $0.078$ for $I$, $2290\sqrt{\text{psi}}$ for $C_p$, $1$ for $K_o$, $1.412$ for $K_v$, $0.52625$ for $C_S$, $2$ for $C_{xc}$ and $1.252$ for $K_m$ in Equation (XXIX).

$\begin{array}{l}178680.48\text{psi}=\left[2290\sqrt{\text{psi}}{\left(\begin{array}{l}\frac{{\left(W_t^P\right)}_C}{0.71\text{in}\times 2\text{in}\times 0.078}\times 1\times 1.412\\ \times 0.52625\times 1.252\times 2\end{array}\right)}^{1/2}\right]\\ {\left(W_t^P\right)}_C={\left(\frac{178680.48}{2290}\right)}^2\left[\frac{0.71\times 2\times 0.078}{1\times 1.412\times 1.252\times 0.52625\times 2}\right]\text{lbf}\\ {\left(W_t^P\right)}_C=362.41\text{lbf}\end{array}$

Substitute $362.41\text{lbf}$ for ${\left(W_t^P\right)}_C$ and $628.3\text{ft}/\text{min}$ for $v_t$ in Equation (XXXI).

$\begin{array}{l}362.41\text{lbf}=33000\frac{{\left(H^P\right)}_C}{628.3\text{ft}/\text{min}}\\ {\left(H^P\right)}_C=\frac{362.41\times 628.3}{33000}\text{hp}\\ {\left(H^P\right)}_C=6.9\text{hp}\end{array}$

Substitute $125920\text{psi}$ for $S_{ac}$, $1.479$ for $C_L^G$, $1$ for $C_H^G$, $1$ for $S_H$, $1$ for $K_T$ and $1$ for $C_R$ in Equation (XXVIII).

$\begin{array}{c}{S}_{wc}^G=\frac{\left(125920\text{psi}\right)\times 1.479\times 1}{1\times 1\times 1}\\ =186235.68\text{psi}\end{array}$

Substitute $186235.68\text{psi}$ for $S_c^G$, $0.71\text{in}$ for $F$, $2\text{in}$ for $d_P$, $0.078$ for $I$, $2290\sqrt{\text{psi}}$ for $C_p$, $1$ for $K_o$, $1.412$ for $K_v$, $0.52625$ for $C_S$, $2$ for $C_{xc}$ and $1.252$ for $K_m$ in Equation (XXX).

$\begin{array}{l}186235.68\text{psi}=\left[2290\sqrt{\text{psi}}{\left(\begin{array}{l}\frac{{\left(W_t^G\right)}_C}{0.71\text{in}\times 2\text{in}\times 0.078}\times 1\times 1.412\\ \times 0.52625\times 1.252\times 2\end{array}\right)}^{1/2}\right]\\ {\left(W_t^G\right)}_C={\left(\frac{186235.68\text{psi}}{2290}\right)}^2\left[\frac{0.71\times 2\times 0.078}{1\times 1.412\times 1.252\times 0.52625\times 2}\right]\text{lbf}\\ {\left(W_t^G\right)}_C=393.7\text{lbf}\end{array}$

Substitute $393.7\text{lbf}$ for ${\left(W_t^G\right)}_C$ and $628.3\text{ft}/\text{min}$ for $v_t$ in Equation (XXXII).

$\begin{array}{l}393.7\text{lbf}=33000\frac{{\left(H^G\right)}_C}{628.3\text{ft}/\text{min}}\\ {\left(H^G\right)}_C=\frac{628.3\times 393.7}{33000}\text{hp}\\ {\left(H^G\right)}_C=7.49\text{hp}\end{array}$

The power rating for a gearset is defined as minimum of power rating in wear for pinion and gear and in bending for pinion and gear so that the failure by bending and wear of teeth on both gear and pinion can be saved.

The calculated power rating for gearset is $4.87\text{hp}$ which is smaller than rated power.

Thus, the rated power is valid.