Two gears are mounted on a steel shaft as shown. Helical gears can result in axi

Question

Two gears are mounted on a steel shaft as shown. Helical gears can result in axial forces on shafts (as well as transverse forces and torques). Only the axial forces are depicted, and they have been modified to produce only axial loading of the shaft. (Take the distances to be L1 = 200 mm, L2 = 300 mm, and L3 = 200 mm, and the diameters of the three segments to be dAB = 16 mm,

dBC = 20 mm, and dCD = 18 mm.)

If the point D of the shaft does not displace, determine the displacements of each of the gears, and the displacement of point A.

Explanation / Answer

so displacement of D is zero so

displacement of C/D=FL/EA

E of steel=200Gpa

area=pi/4*d^2=pi/4*(0.018)^2=2.54*10^-4 m^2

displacement of point C =7000*0.2/(200*10^9*2.54*10^-4)=2.75*10^-5 m (left) (let left have sign convension positive)

displacement of point C =2.75*10^-5 m (left) gear at C

force on shaft BC=8+7=15KN

area of BC=pi/4*(0.02)^2=3.14*10^-4

dispacement of B/C =15000*0.3/(200*10^9*3.14*10^-4)=7.165*10^-5 m (expand)

so displacement of B=displacement of B/C+displacement of C=7.165*10^-5+2.75*10^-5=9.915*10^-5 m

so displacement of B=9.915*10^-5 m (left)

displacement of A/B=5000*0.2/(200*10^9*2*10^-4)=2.5*10^-5 (right)

so displacement of pont A =displacement of A/B+displacement of B=-2.5*10^-5+9.915*10^-5=7.415*10^-5m

displacement of point A is 7.415*10^-5m (left)

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