Inside the Gearbox

P$1.$ Load Analysis

The countershaft with the reverse gears attached is shown in Figure $1$ with the dimensions given in table $1.$ Gear A receives power from another gear on the input shaft with the transmitted force $F_A$ applied at a pressure angle ${}_A.$ The power $P$ is transmitted through the shaft and delivered to gear $B$ on the output shaft through a transmitted force $F_B$ at a pressure angle ${}_B.$ The countershaft rotates at a constant speed $n_D.$

a$)$ Determine the forces $F_A$ and $F_B.$

b$)$ Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.

c$)$ Draw shear$-$force and bending$-$moment diagrams for the countershaft. If needed, make one set for the horizontal plane and another set for the vertical plane.

Five$-$speed manual transmission gearbox of a passenger car.

Cutout of the Gearbox

The blue line shows the path of the torque through the gearbox when the reverse gear speed is engaged.

The gears are attached to either the main shaft or the countershaft.

The black markers are the bearings and their supports, commonly anchored to the transmission gearbox housing.

Project Shaft Reactions at O and C

rmproviding these values for your reference. Check your static analysis results against these values and make: static analysis leading to these values.

Note: This is similar to textbook problem $3-83.$ Check the solution of this

problem to see how static analysis is dove.

hey are cerrect. You still need to show

P$3.$ Deflections

a$)$ Find the slope of the deflection curve of the countershaft at each bearing location using CAE. Use beam elements to analyze the shaft deflection. Assume the bearings constitute simple supports and the steel in the countershaft has a Young's modulus of $207$ GPa.

b$)$ Now assuming that the bearings have a maximum slope specification of $$ mrad for good bearing life $($see table $1).$ Will the bearings have a good life when the reverse gear is engaged?

c$)$ If the slope at the bearings at O and C are larger than the maximum allowable slope $($see table $1)$ for good bearing life, you might need to increase the shaft diameter such that you can guarantee good bearing life. Determine the new shaft $d_{\text{n}\text{e}\text{w}}$ such that it meets the good bearing life requirement. Hint: use eqn. $7-18$ in the textbook.