EMA 3702 Mechanics & Materials Science (Mechanics of ...

EMA 3702

Mechanics & Materials Science

(Mechanics of Materials)

Chapter 3 Torsion

Homework Answers

EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion

Homework 3.1

Electric motor applies a torque of 2.8 kN•m on shaft EF at E. Knowing

each shaft section is solid. Based on the additional torques as

depicted, please determine the maximum shearing stress in (a) shaft

EF (radius cEF = 25 mm), (b) shaft FG (radius cFG=23 mm), and c) shaft

GH (radius cGH=21 mm)

For EF, net internal torque TEF = ??

E

E F G H

TF = 1.6 kN•m TG = 0.8 kN•m TH = 0.4 kN•m

PaEF

8

max_1014.1


Homework 3.2

For section FG, net internal torque TFG = ??

For section GH, net internal torque TGH = ??

E

E F G H

TF = 1.6 kN•m TG = 0.8 kN•m TH = 0.4 kN•m

PaFG

7

max_1028.6

PaGH

7

max_1075.2


Homework 3.2

A component with both solid rod EF and

hollow tube with flange GH welded onto a flat

end plate of GFI, and the rod EF and tube GH

share the same longitude axis, as illustrated.

The hollow tube GH is fixed at rigid flange of

AA’BB’ at H using bolts. The solid rod EF has

a diameter dEF = 3.81 cm and is made of steel

with allowable shearing stress of 82.7 MPa.

Tube GH is made of copper alloy with OD

d2=7.62 cm and ID d1=6.35 cm with allowable

shearing stress of 48.3 MPa. Determine the

largest torque T that can be applied at E.

For section EF,

E

F

H

G

T

I

A A’ B B’

MPa7.82max

mNT 941


Homework 3.2 (2)

For section GH,

Therefore, max torque that can be applied

at EF is 941 N•m

E

F

H

G

T

I

A A’ B B’

MPa3.48max

mNT 2172


Homework 3.3 (1)

Two solid shafts GH and IJ are connected by gears K and L as illustrated.

The radius for gear K is RK = 4 in, while radius for gear L is RL= 2.5 in.

Both two shafts of GH and IJ are made of same steel with allowable

shearing stress of 8 ksi. The radius for the shafts are cGH = 0.8 in and cIJ =

0.625 in. Please determine the largest torque that can be applied at H.

For GH, G H

RK = 4.0 in

I J

RL = 2.5 in

K

L

inlbT 6430


Homework 3.3 (2)

For section IJ, due to the two-gear connection,

Therefore, max T can be applied at H is 4906 lb•in

G H

RK = 4.0 in

I J

RL = 2.5 in

K

L

L

K

LR

R

T

T

inlbT 4906


Homework 3.4 (1)

Motor applies torque T = 500 N•m on the HGFE shaft when it is rotating at

a constant speed. Knowing shear modulus G = 27 GPa and the torques

exerted on pulleys B and C are shown. Please calculate the angle of

twist between (a) F and G, and (b) between F and H. Knowing shaft

radius cGF = 2.2 cm; cHG = 2.4 cm; cEF = 2.0 cm;

H G F E

TG = 300 N•m TF= 200 N•m Motor

T = 500 N•m

1.2 m 0.9 m


Homework 3.4 (2)

Between F and G, net internal torque = ??

Between G and H, net internal torque = ??

Between F and H, total angle of twist:

38.10242.0 rad

FG

83.10320.0 rad

GH

22.30562.0 rad

FH


Homework 3.5

Solid aluminum rod (GAl=26 GPa) is bonded to solid copper alloy rod

(GCu=39 GPa). The radius for both rods are 10 mm. Calculate the

angle of twist with respect to base D at E point and at F point,

respectively.

Between DE,

Between EF

Therefore, for DF,

D E

TF = 50 N•m

0.4 m 0.2 m

F copper aluminum

94.00163.0 rad

DE

81.2049.0 rad

EF

74.30653.0 rad

DF


Homework 3.6 (1)

A 6 foot long composite shaft consists of 0.2 inch thick copper shell (GCu

= 5.6 106 psi) bonded to 1.2 inch (GSteel = 11.2 106 psi) diameter iron

core. If the shaft is subject to 5000 lb•in torque. Please calculate the

maximum shearing stress in the steel core and the angle of twist of one

end versus the other.

From statics point of view

Consider geometry

T

Cu shell

Fe core

6 ft

?CuFe

TT

? vsCuFe

FeCuTT ??

inlbTFe

2404


Homework 3.6 (2)

Max shearing stress in steel core

Angle of twist:

psiFe

7088max_

35.4076.0 rad

CuFe


Homework 3.7

Determine maximum shearing stress in a solid shaft of 10 mm diameter

as it transmits 2.4 kW at a frequency of (a) 30 Hz and (b) 60 Hz.

Max shearing stress

When f = 30 Hz

When f = 60 Hz

??max

Pa7

max1049.6

Pa7

max1025.3


Homework 3.8 (1)

Two solid shafts EF and IJ and gears G (RG = 3 inch) and H (RH = 5 inch)

are used to transmit 16 hp (1 hp = 6600 lbin/sec) from motor at E

operating at 1200 rpm (f = 20 Hz) to machine tool at J. Knowing the

maximum allowable shearing stress is 7.5 ksi. Please calculate the

required radius for shaft EF and shaft IJ, respectively.

For EF,

I J

RH = 5.0 in

F E

RG = 3.0 in

H

G

incEF

415.0


Homework 3.8 (2)

For IJ

due to gear H and G connection,

I J

RH = 5.0 in

F E

RG = 3.0 in

H

G

EFIJff ??

incIJ

492.0

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