University of Florida/Egm3520/s13.team1.r4

Problem R4.1

Mechanics of Materials: 6th Edition textbook. Authors: F.P BEER, E.R. JOHNSTON, J.T. DEWOLF AND D.F. MAZUREK ISBN:9780077565664.

Problem statement: A motor exerts a torque ${\displaystyle \displaystyle T_{f}}$ to shaft FGH attached to a gear with radius ${\displaystyle \displaystyle r_{G}}$ which in turn applies a torque ${\displaystyle \displaystyle T_{d}}$ to a gear with radius ${\displaystyle \displaystyle r_{D}}$ attached to shaft CDE. No slipping occurs between the gears. The Allowable shearing stress in each shaft is ${\displaystyle \displaystyle 10.5ksi}$.

Question statement:

What is required diameter of shaft CDE?

What is required diameter of shaft FGH?

Contents taken from Page 157 in the Mechanics of Materials: 6th Edition textbook. Authors: F.P BEER, E.R. JOHNSTON, J.T. DEWOLF AND D.F. MAZUREK ISBN:9780077565664

Givens:

${\displaystyle \tau =10.5ksi=10,500psi}$

${\displaystyle T_{F}=1200lb*in}$

${\displaystyle r_{D}=8in}$

${\displaystyle r_{G}=3in}$

${\displaystyle \tau ={\frac {Tc}{J}}={\frac {2T}{\pi c^{3}}}\implies c={\sqrt[{3}]{\frac {2T}{\pi \tau }}}}$

(a) Required diameter of shaft CDE

${\displaystyle T_{E}={\frac {r_{D}}{r_{G}}}T_{F}={\frac {8in}{3in}}(1200lb*in)=3200lb*in}$

${\displaystyle c={\sqrt[{3}]{\frac {2T_{F}}{\pi \tau }}}={\sqrt[{3}]{\frac {2(3200lb*in)}{\pi (10,500psi)}}}=0.5789}$

${\displaystyle {\frac {1}{2}}d=c\implies d=2c}$

${\displaystyle d_{E}=2(0.5789)=1.157in}$

(b) Required of diameter shaft FGH

${\displaystyle c={\sqrt[{3}]{\frac {2T_{F}}{\pi \tau }}}={\sqrt[{3}]{\frac {2(1200lb*in)}{\pi (10,500psi)}}}=0.4175in}$

${\displaystyle d_{F}=2c=2(0.4175in)=0.8349in}$

Question 3.25

Mechanics of Materials: 6th Edition textbook. Authors: F.P BEER, E.R. JOHNSTON, J.T. DEWOLF AND D.F. MAZUREK ISBN:9780077565664.

Problem Statement: Given the diagram of gears A and D connected to rods BC and EF. There is a given torque of 5 kip*in for ${\displaystyle T_{C}}$ and ${\displaystyle T_{F}}$ is unknown. The shafts of rods ABC and DEF are solid and their diameters are unknown. Each shaft has an allowable shearing stress of 8500psi

Question Statement:

Using the known values find the required minimum diameter a) of shaft BC and b) shaft EF

Contents taken from Page 158 in the Mechanics of Materials: 6th Edition textbook. Authors: F.P BEER, E.R. JOHNSTON, J.T. DEWOLF AND D.F. MAZUREK ISBN:9780077565664

Givens:

${\displaystyle \tau =8500psi}$

${\displaystyle T_{C}=5kip*in=5000lb*in}$

${\displaystyle \tau ={\frac {Tc}{J}}={\frac {2T}{\pi c^{3}}}\implies c={\sqrt[{3}]{\frac {2T}{\pi \tau }}}}$

(a) Required diameter of shaft BC

${\displaystyle c={\sqrt[{3}]{\frac {2(5000lb*in)}{\pi (8500psi)}}}=0.72079in}$

${\displaystyle d=2c\implies d_{C}=40442in}$

(b) Required diameter of shaft EF

${\displaystyle T_{F}={\frac {4in}{2.5in}}(5000lb*in)=8000lb*in}$

${\displaystyle c={\sqrt[{3}]{\frac {2T_{F}}{\pi \tau }}}={\sqrt[{3}]{\frac {2(8000lb*in)}{\pi (8500psi)}}}=0.84304in}$

${\displaystyle d=2c\implies d_{F}=1.686in}$