Chapter 15, Problem 7P

In straight-bevel gearing, there are some analogs to Eqs. (14-44) and (14-45) pp. 766 and 767, respectively. If we have a pinion core with a hardness of (H_B)₁₁ and we try equal power ratings, the transmitted load W^t can be made equal in all four cases. It is possible to find these relations:

	Core	Case
Pinion	(HB)11	(HB)12
Gear	(HB)21	(HB)22

1. (a)     For carburized case-hardened gear steel with core AGMA bending strength (s_at)_G and pinion core strength (s_at)_P, show that the relationship is

${\left(s_{at}\right)}_G=\left(s_{at}\right)P\frac{J_p}{J_G}{m}_G^{-0.0323}$

This allows (H_B)₂₁ to be related to (H_B)₁₁.

1. (b)     Show that the AGMA contact strength of the gear case (s_ac)_G can be related to the AGMA core bending strength of the pinion core (s_at)_P by

${\left(s_{ac}\right)}_G=\frac{C_p}{{(C_L)}_G{C}_H}\sqrt{\frac{S_H^2{(s_{at})}_P{(K_L)}_P{K}_x{J}_P{K}_T{C}_s{C}_{xc}}{N_{P}I{K}_s}}$

If factors of safety are applied to the transmitted load W_t, then S_H = $\sqrt{S_F}$ and $S_H^2/S_F$ is unity. The result allows (H_B)₂₂ to be related to (H_B)₁₁.

1. (c)     Show that the AGMA contact strength of the gear (s_ac)_G is related to the contact strength of the pinion (s_ac)_P by

${(s_{ac})}_P={(s_{ac})}_G{m}_G^{0.0602}{C}_H$

To determine

The relation between AGMA bending strength of gear with AGMA bending strength of pinion.

Answer to Problem 7P

The relation between AGMA bending strength of gear with AGMA bending strength of pinion is $s_{at}^G=s_{at}^P\times {\left(m_G\right)}^{-0.0323}\times \frac{J^P}{J^G}$.

Explanation of Solution

Write the expression for gear ratio of gear set.

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

Here, the gear ratio is $m_G$.

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

$K_L^P=1.683{\left(N_L^P\right)}^{-0.0323}$ (II)

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

$K_L^G=1.683{\left(\frac{N_L^P}{m_G}\right)}^{-0.0323}$ (III)

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

Write the expression for permissible bending stress for pinion.

$\begin{array}{l}{s}_{wt}^P=\frac{s_{at}^P{K}_L^P}{S_F{K}_T{K}_R}\\ S_F=\frac{s_{at}^P{K}_L^P}{s_{wt}^P{K}_T{K}_R}\end{array}$ (IV)

Here, the permissible bending stress for pinion is $s_{wt}^P$, the allowable bending stress number for pinion is $s_{at}^P$, the reliability factor is $K_R$, the bending safety factor is $S_F$ and the temperature factor is $K_T$.

Write the expression for permissible bending stress for gear.

$\begin{array}{l}{s}_{wt}^G=\frac{s_{at}^G{K}_L^G}{S_F{K}_T{K}_R}\\ S_F=\frac{s_{at}^G{K}_L^G}{s_{wt}^G{K}_T{K}_R}\end{array}$ (V)

Here, the permissible bending stress for gear is $s_{wt}^G$ and the allowable bending stress number for gear is $s_{at}^G$.

Substitute $\frac{s_{at}{K}_L^G}{s_{wt}^G{K}_T{K}_R}$ for $S_F$ in Equation (IV).

$\frac{s_{at}^G{K}_L^G}{s_{wt}^G{K}_T{K}_R}=\frac{s_{at}^P{K}_L^P}{s_{wt}^P{K}_T{K}_R}$ (VI)

Write the expression for bending stress for pinion.

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

Here, the bending stress in pinion is $s_t^P$, the transmitted load is $W_t$, 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.

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

Here, the bending stress in gear is $s_t^G$, the transmitted load is $W_t$ and geometry factor for bending in gear is $J^G$.

For limiting condition bending stress in pinion is equal to allowable bending stress in pinion.

Substitute $s_t^P$ for $s_{wt}^P$ in Equation (VI).

$\frac{s_{at}^G{K}_L^G}{s_{wt}^G{K}_T{K}_R}=\frac{s_{at}^P{K}_L^P}{s_t^P{K}_T{K}_R}$ (IX)

For limiting condition bending stress in gear is equal to allowable bending stress in gear.

Substitute $s_t^G$ for $s_{wt}^G$ in Equation (IX).

$\frac{s_{at}^G{K}_L^G}{s_t^G{K}_T{K}_R}=\frac{s_{at}^P{K}_L^P}{s_t^P{K}_T{K}_R}$ (X)

Substitute $\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^P}$ for $s_t^P$ in Equation (X).

$\begin{array}{l}\frac{s_{at}^G{K}_L^G}{s_t^G{K}_T{K}_R}=\frac{s_{at}^P{K}_L^P}{\left(\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^P}\right)K_T{K}_R}\\ \frac{s_{at}^G{K}_L^G}{s_t^G}=\frac{s_{at}^P{K}_L^P}{\left(\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^P}\right)}\end{array}$ (XI)

Substitute $\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^G}$ for $s_t^P$ in Equation (XI).

$\begin{array}{l}\frac{s_{at}^G{K}_L^G}{\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^G}}=\frac{s_{at}^P{K}_L^P}{\left(\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^P}\right)}\\ s_{at}^G=s_{at}^P\frac{K_L^P}{K_L^G}\frac{J^P}{J^G}\end{array}$ (XII)

Substitute $1.683{\left(N_L^P\right)}^{-0.0323}$ for $K_L^P$ and $1.683{\left(\frac{N_L^P}{m_G}\right)}^{-0.0323}$ for $K_L^G$ in Equation (XII).

$\begin{array}{l}{s}_{at}^G=s_{at}^P\frac{\left\{1.683{\left(N_L^P\right)}^{-0.0323}\right\}}{\left\{1.683{\left(\frac{N_L^P}{m_G}\right)}^{-0.0323}\right\}}\frac{J^P}{J^G}\\ s_{at}^G=s_{at}^P\times {\left(m_G\right)}^{-0.0323}\times \frac{J^P}{J^G}\end{array}$ (XIII)

Conclusion:

Thus, the relation between AGMA bending strength of gear with  AGMA bending strength of pinion is $s_{at}^G=s_{at}^P\times {\left(m_G\right)}^{-0.0323}\times \frac{J^P}{J^G}$.

To determine

The AGMA contact strength of the gear case relation with the AGMA core bending strength of the pinion core.

Answer to Problem 7P

The AGMA contact strength of the gear case is related to the AGMA core bending strength of the pinion core by $s_{ac}=\frac{C_P}{C_L^G}\sqrt{\left\{\left(\frac{S_H{}^2{s}_{at}^P}{S_F}\right)\left(\frac{K_L^P{K}_T}{{\left(C_H^G\right)}^2}\right)\left(\frac{K_x{J}^P}{K_S}\right)\left(\frac{C_S{C}_{xc}}{nI}\right)\right\}}$.

Explanation of Solution

Write the expression for pitch diameter of pinion.

$\begin{array}{l}{d}_P=\frac{n}{P_d}\\ d_P{P}_d=n\end{array}$ (XIV)

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 permissible bending stress for pinion.

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

Here, the permissible bending stress for pinion is $s_{wt}^P$, the allowable bending stress number for pinion is $s_{at}^P$, stress cycle factor for pinion is $K_L^P$, the reliability factor is $K_R$, the bending safety factor is $S_F$ and the temperature factor is $K_T$.

Write the expression for bending stress for pinion.

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

Here, the bending stress in pinion is $s_t^P$, dynamic factor is $K_v$, size factor is $K_S$, load distribution factor is $K_m$, the transmitted load is $W_t$, the face width is $F$, the overload factor is $K_o$, length wise curvature factor is $K_x$ and geometry factor for bending in pinion is $J^P$.

For limiting condition bending stress in pinion is equal to allowable bending stress in pinion.

Substitute $s_t^P$ for $s_{wt}^P$ in Equation (II).

$s_t^P=\frac{s_{at}^P{K}_L^P}{S_F{K}_T{K}_R}$ (XVII)

Substitute $\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^P}$ for $s_t^P$ in Equation (IV).

$\begin{array}{l}\frac{W_t}{F}{P}_d{K}_o{K}_v\frac{K_S{K}_m}{K_x{J}^P}=\frac{s_{at}^P{K}_L^P}{S_F{K}_T{K}_R}\\ W_t=\frac{s_{at}^P{K}_L^{P}F{K}_x{J}^P}{P_d{K}_o{K}_v{K}_S{K}_m{S}_F{K}_T{K}_R}\\ W_t=\left(\frac{s_{at}^P}{S_F}\right)\left(\frac{K_L^P}{K_T{K}_R}\right)\left(\frac{F{K}_x{J}^P}{P_d{K}_o{K}_v{K}_S{K}_m}\right)\end{array}$ (XVIII)

Write the expression for reliability factor for pitting.

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

Here, the reliability factor for pitting is $C_R$ and the reliability factor is $K_R$.

Write the expression for permissible contact stress number for gear.

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

Here, the permissible contact stress for gear is $s_{wc}^G$, the allowable bending stress number is $s_{ac}$, the contact safety factor is $S_H$, the hardness ratio factor for pitting resistance for gear is $C_H^G$, the stress cycle factor for pitting resistance for gear is $C_L^G$.

Write the expression for contact stress for gear.

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

Here, the contact stress in gear is $s_c^G$ size factor for pitting resistance is $C_S$, the crowning factor for pitting is $C_{xc}$, Contact geometry factor is $I$ and the Elastic Coefficient is $C_P$.

For limiting condition contact stress in gear is equal to permissible contact stress in gear.

Substitute $s_c^G$ for $s_{wc}^G$ in Equation (XX).

$s_c^G=\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R}$ (XXII)

Substitute $C_p{\left(\frac{W_t}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}$ for $s_c^G$ in Equation (XXII).

$\begin{array}{l}{C}_p{\left(\frac{W_t}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}=\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R}\\ {\left(\frac{W_t}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}=\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R{C}_p}\\ W_t={\left(\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R{C}_p}\right)}^2\frac{F{d}_{P}I}{K_o{K}_v{C}_S{K}_m{C}_{xc}}\end{array}$ (XXIII)

Substitute ${\left(\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R{C}_p}\right)}^2\frac{F{d}_{P}I}{K_o{K}_v{C}_S{K}_m{C}_{xc}}$ for $W_t$ in Equation (XVIII).

$\begin{array}{l}{\left(\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R{C}_p}\right)}^2\frac{F{d}_{P}I}{K_o{K}_v{C}_S{K}_m{C}_{xc}}=\left\{\begin{array}{l}\left(\frac{s_{at}^P}{S_F}\right)\left(\frac{K_L^P}{K_T{K}_R}\right)\\ \left(\frac{F{K}_x{J}^P}{P_d{K}_o{K}_v{K}_S{K}_m}\right)\end{array}\right\}\\ {\left(\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R{C}_p}\right)}^2=\left(\frac{s_{at}^P}{S_F}\right)\left(\frac{K_L^P}{K_T{K}_R}\right)\left(\frac{K_x{J}^P}{P_d{K}_S}\right)\left(\frac{C_S{C}_{xc}}{d_{P}I}\right)\\ \left(\frac{s_{ac}{C}_L^G{C}_H^G}{S_H{K}_T{C}_R{C}_p}\right)=\sqrt{\left\{\left(\frac{s_{at}^P}{S_F}\right)\left(\frac{K_L^P}{K_T{K}_R}\right)\left(\frac{K_x{J}^P}{P_d{K}_S}\right)\left(\frac{C_S{C}_{xc}}{d_{P}I}\right)\right\}}\\ s_{ac}=\sqrt{\left\{\left(\frac{s_{at}^P}{S_F}\right)\left(\frac{K_L^P}{K_T{K}_R}\right)\left(\frac{K_x{J}^P}{P_d{K}_S}\right)\left(\frac{C_S{C}_{xc}}{d_{P}I}\right)\right\}}\times \frac{S_H{K}_T{C}_R{C}_p}{C_L^G{C}_H^G}\end{array}$ (XXIV)

Substitute $\sqrt{K_R}$ for $C_R$ and $n$ for $P_d{d}_P$ in Equation (XXIV).

$\begin{array}{l}{s}_{ac}=\sqrt{\left\{\left(\frac{s_{at}^P}{S_F}\right)\left(\frac{K_L^P}{K_T{K}_R}\right)\left(\frac{K_x{J}^P}{K_S}\right)\left(\frac{C_S{C}_{xc}}{nI}\right)\right\}}\times \frac{S_H{K}_T\sqrt{K_R}{C}_p}{C_L^G{C}_H^G}\\ s_{ac}=\frac{C_P}{C_L^G}\sqrt{\left\{\left(\frac{S_H{}^2{s}_{at}^P}{S_F}\right)\left(\frac{K_L^P{K}_T}{{\left(C_H^G\right)}^2}\right)\left(\frac{K_x{J}^P}{K_S}\right)\left(\frac{C_S{C}_{xc}}{nI}\right)\right\}}\end{array}$ (XXV)

For equal amount of transmitted load $\frac{S_H{}^2}{S_F}$ as $1$.

Substitute $1$ for $\frac{S_H{}^2}{S_F}$ in in equation (XXV).

$\begin{array}{l}{s}_{ac}=\frac{C_P}{C_L^G}\sqrt{\left\{\left(1\times s_{at}^P\right)\left(\frac{K_L^P{K}_T}{{\left(C_H^G\right)}^2}\right)\left(\frac{K_x{J}^P}{K_S}\right)\left(\frac{C_S{C}_{xc}}{nI}\right)\right\}}\\ s_{ac}=\frac{C_P}{C_L^G{C}_H^G}\sqrt{\left\{\left(1\times s_{at}^P\right)\left(K_L^P{K}_T\right)\left(\frac{K_x{J}^P}{K_S}\right)\left(\frac{C_S{C}_{xc}}{nI}\right)\right\}}\end{array}$ (XXVI)

Conclusion:

Thus, the AGMA contact strength of the gear case is related to the AGMA core bending strength of the pinion core by $s_{ac}=\frac{C_P}{C_L^G}\sqrt{\left\{\left(\frac{S_H{}^2{s}_{at}^P}{S_F}\right)\left(\frac{K_L^P{K}_T}{{\left(C_H^G\right)}^2}\right)\left(\frac{K_x{J}^P}{K_S}\right)\left(\frac{C_S{C}_{xc}}{nI}\right)\right\}}$.

To determine

The AGMA contact strength of gear relation with contact strength of pinion.

Answer to Problem 7P

The AGMA contact strength of gear is related to contact strength of pinion by $s_{ac}^P=s_{ac}^G{C}_H^G{\left(m_G\right)}^{0.0602}$.

Explanation of Solution

Write the expression for gear ratio of gear set.

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

Write the expression for stress cycle factor for pitting resistance for pinion.

$C_L^P=3.4822{\left(N_L^P\right)}^{-0.0602}$ for ${10}^4\le N_L^P\le {10}^{10}$ (XXVIII)

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.

$C_L^G=3.4822{\left(\frac{N_L^P}{m_G}\right)}^{-0.0602}$ for $3\times {10}^6\le N_L^P\le {10}^{10}$ (XXIX)

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

Write the expression for permissible contact stress number for pinion.

$\begin{array}{l}{s}_{wc}^P=\frac{s_{ac}^P{C}_L^P}{S_H{K}_T{C}_R}\\ S_H=\frac{s_{ac}^P{C}_L^P}{s_{wc}^P{K}_T{C}_R}\end{array}$ (XXX)

Here, the permissible contact stress number for pinion is $s_{wc}^P$, the contact safety factor is $S_H$ and the temperature factor is $K_T$.

Write the expression for permissible contact stress number for gear.

$\begin{array}{l}{s}_{wc}^G=\frac{s_{ac}^G{C}_L^G{C}_H^G}{S_H{K}_T{C}_R}\\ S_H=\frac{s_{ac}^G{C}_L^G{C}_H^G}{s_{wc}^G{K}_T{C}_R}\end{array}$ (XXXI)

Substitute $\frac{s_{ac}^G{C}_L^G{C}_H^G}{s_{wc}^G{K}_T{C}_R}$ for $S_H$ in Equation (XXX).

$\frac{s_{ac}^G{C}_L^G{C}_H^G}{s_{wc}^G{K}_T{C}_R}=\frac{s_{ac}^P{C}_L^P}{s_{wc}^P{K}_T{C}_R}$ (XXXII)

Write the expression for contact stress for pinion.

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

Here, the contact stress in pinion is $s_C^P$, the transmitted load is $W_t$, 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.

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

Here, the contact stress in gear is $s_c^G$ and the transmitted load is $W_t$.

For limiting condition contact stress on gear and pinion is equal to permissible contact stress on gear and pinion.

Substitute $s_c^G$ for $s_{wt}^G$ and $s_c^P$ for $s_{wt}^P$ in Equation (XXXII).

$\frac{s_{ac}^G{C}_L^G{C}_H^G}{s_c^G{K}_T{C}_R}=\frac{s_{ac}^P{C}_L^P}{s_c^P{K}_T{C}_R}$ (XXXV)

Substitute $C_p{\left(\frac{W_t}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}$ for $s_c^G$ in Equation (XXXV).

$\frac{s_{ac}^G{C}_L^G{C}_H^G}{\left\{C_p{\left(\frac{W_t}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}\right\}{K}_T{C}_R}=\frac{s_{ac}^P{C}_L^P}{s_c^P{K}_T{C}_R}$ (XXXVI)

Substitute $C_p{\left(\frac{W_t}{F{d}_{P}I}{K}_o{K}_v{C}_S{K}_m{C}_{xc}\right)}^{1/2}$ for $s_c^P$ in Equation (XXXVI).

$\begin{array}{l}\frac{s_{ac}^G{C}_L^G{C}_H^G}{\left[\begin{array}{l}\left\{C_p{\left(\begin{array}{l}\frac{W_t}{F{d}_{P}I}\\ K_o{K}_v{C}_S{K}_m{C}_{xc}\end{array}\right)}^{1/2}\right\}\\ K_T{C}_R\end{array}\right]}=\frac{s_{ac}^P{C}_L^P}{\left[\begin{array}{l}\left\{C_p{\left(\begin{array}{l}\frac{W_t}{F{d}_{P}I}\\ K_o{K}_v{C}_S{K}_m{C}_{xc}\end{array}\right)}^{1/2}\right\}\\ K_T{C}_R\end{array}\right]}\\ s_{ac}^P{C}_L^P=s_{ac}^G{C}_L^G{C}_H^G\end{array}$ (XXXVII)

Substitute $3.4822{\left(N_L^P\right)}^{-0.0602}$ for $C_L^P$ and $3.4822{\left(\frac{N_L^P}{m_G}\right)}^{-0.0602}$ for $C_L^G$ in Equation (XXXVII).

$\begin{array}{l}{s}_{ac}^P\left(3.4822{\left(N_L^P\right)}^{-0.0602}\right)=s_{ac}^G\left(3.4822{\left(\frac{N_L^P}{m_G}\right)}^{-0.0602}\right)C_H^G\\ s_{ac}^P=\frac{s_{ac}^G{C}_H^G}{{\left(m_G\right)}^{-0.0602}}\\ s_{ac}^P=s_{ac}^G{C}_H^G{\left(m_G\right)}^{0.0602}\end{array}$

Conclusion:

Thus, AGMA contact strength of gear is related to contact strength of pinion by $s_{ac}^P=s_{ac}^G{C}_H^G{\left(m_G\right)}^{0.0602}$.