JJ309 Fluid Mechanics Unit 5

7/30/2019 JJ309 Fluid Mechanics Unit 5

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SCREW J1

UNIT 5

SCREW

OBJECTIVES

General Objective : To understand the concept of the wor k ing of a screw.

Specific Objectives : At the en d of thi s u n ityou should be able to :

state the terms that are im portan t for the study of screw.

match the pr in c ip leof screw work ing to that of f r iction o n an in c li n e d plane.

use these s u itable concepts to solve pro blem in v olve.

calculate the answer u s i n gthese concepts correctly.





SCREW J3

Fig. 5.1 Fig. 5.2

The angle α,

Tan α = p

π D

(b) A Two-Start Thread

For a two-start thread the distance mov es axi a lly by the screw i n its nu t i n one

revoluti o nis the lead £ , which i s twice the p itch .

Tan α =£

π D

Tan α =2 p

π D



5.2 THE VEE-THREADED SCREW

For a V-thread, the n or mal force between the nutand the screw i s increased s ince the

axi a lcomponent of t his force must equal W. Thus, i f the semi-angle of the thread i sφ,

Fig. 5.3., then nor mal force = W sec φ .

The f r ictio n force i s therefore i n creased i n the ratio sec φ : 1, so that the V-thread i s

equivalen t to a square thread hav ing a coef f ici e n tof f r ictio n ιv sec φ

Fig. 5.3



INPUT

Tur n i n gthe screw i sequivalen t to moving a mass of weight W along the inclined p la ne by aho rizon talforce P.

5.3 RAISING LOAD

When the load i s raised by the force P the motion i sup the plane.(Fig.5.4).

Motion

RR

ξ ξ χ

Pξ + χ

W

P

W

Fig. 5.4

From the triangle of forces:

Tan (ξ + χ) = P

w

P = W tan (ξ + χ)



Torque

The torque T required to rotate the screw agai n s tthe load i s :

T = P (1 D )2

0

T =1 PD

+

2

T =1

WD tan (Λ + χ)2 1

P

D2

Fig. 5.5: Cross-sectio n of Screw

Efficiency

The eff ici e ncy of the screw i sequal to:

Forward eff i c iency :

1 forward = force P req uired withoutf r iction (χ = 0)force P requi red w ithfr ic t io n

1 =w tanΛ

w tan(Λ + χ)

1 =tanΛ

tan(Λ + χ )

or,

1 = wo r k d o n e o n load W i n1 r e volu ti on

work doneby P i n1 revoluti o n

1 =we

P (π D)

1 =w

xe

P π D



But,

w=

1

P tan(Λ + χ )

and tanΛ = £

τ D

1 =tanΛ

tan(Λ + χ )

Maximum Efficiency

1mak =1 −sin χ

1 + sin χ

Example 5.1

The helix angle of a screw tread is 100. If the coefficient of friction is 0.3 and the meandiameter of the square thread i s72.5mm,

calculate: (a) the p itc hof the tread,

(b) the eff i ciency when rai sin g a load of 1 kN, (c)the torque req u ir ed.

Solution 5.1

G iven :

α = 100 μ = 0.3 Dmean = 72.5mm

(a) tan α = p

τ D p = tan 100 (3.14)(72.5)

= 40.1

mm



0

(b) μ = tan α

α = tan-1

0.3

α = 16.70

Ef f ici e ncy for raise load, 1

=

tanΛ

tan(Λ + χ )

1 =tan10

tan(100 + 16.7

0)

1 =

0.17630.5027

1 = 0.35 x 100%

1 =

35%

(c) Torque, T =

T =

T =

1WD tan (Λ + χ)

21

(1000)(72.5) tan (100 + 16.70)21

(1000)(72.5)(0.5027)2

T = 18.22 x 103

kNmm

T = 18.22 Nm



5.4 LOAD BEING LOWERED

(i) When χ > Ι

P i sapplied to res ist the downward mov emen tof the load. Under

this cond iti o nthe load would just about to move downwards. If P

were no t appliedi n t his di rectio n the load would overhaul , that i smove down o n its own weight.

R

P

Motionχ χ

P

Rχ - Ι W

W

Fig. 5.6

From the tr i angle of forces:

Tan (χ - Ι) = P

w

P = W tan (χ - Ι)

Efficiency

Downward eff ici e ncy :

1downward = wo r k doneb y P i n1 r e volu ti o n work doneon load W i n1 revoluti o n

1 = P (τ D)

w£

1 =tan(χ Ι

)

tanχ



SCREW J1

Example 5.2

Calculate the p itc hof a single-start square threaded screw of a jack w h ich wil l justallow the load to fall un i f or mly under i t s ownweight. The me an diameter of the

threads is 8 cm and μ= 0.08. If the p itch i s 15mm, what is the torque

required to lower a load of 3 kN?

Solution 5.2

If the load i s to fall u n if or mly under i t s ownweight, α = Ι

G iven :

μ = 0.08 μ = tan Ι Ι = 4.57°

Tan α = p

τ Dthen, α = Ι

Tan 4.57° = p

(3.14)(8)

p = 20mm

If the p i t c hi s15mm, Tan α = p

τ D

Tan α =15

(3.14)(8)

Tan α = 5.97

Torque, T =

T =

T =

α = 80.49°

1WD tan (α - Ι)

21

(3000)(8)tan (80.49°- 4.57°)2

1(3000)(8)(tan 75° 92’)

2

T = 2.4

Nm



(ii) When − > ξ

If the angle of f r ictio n i sgreater than the angle of the p lane , then i t c an be seenf rom the tr i angle of force, that force P must be applied to help lower the load.

Motion

P

R

ξ −c ξ

R −c ξ W

ξ ξP

W

Fig. 5.7

From the tr i angle of forces:

Tan (−c ξ ) = P

w

P = W ta n ( −c ξ )

Efficiency

1 terbalik =tan(− ξ )

tanξ



Example 5.3

The me an diameter of a squarecthreaded screw jack i s5 cm. The p itch of the thread i s1 cm. The coef f icie n tof fr ic t io nis 0.15. What force

must be applied at the end of a 70 cm long lever, which i s

perpendicular to the long itu d i n a laxi s of the screw to lower a load of 2to nnes?

Solution 5.3

G iven:

Mean diameter, d = 5 cm Pitch, p = 1 cm

Coefficient of friction, μ = 0.15 = tan 4 Then , 4 = 8.32°

Length of lever, L = 70 cm Load W = 2 t = 2000 kg

Force required to lower the load,

P = W tan (4 – α)

= 2000 ta n (8. 32°c3.39°)

= 2000 ta n 4. 53°

= 2000 x 0.0854

= 170.8 kg



Activity 5A

TEST YOUR UNDERSTANDING BEFORE YOU CONTINUE WITH THE NEXT

INPUT…!

5.1 A nuton a s i nglecstart square thread bolt i s locked tight by a torque of 6 Nm. Thethread p itch i s5mm an d themean diameter 6 cm. Calculate :

(a) the axi a lload o n the screw i n k ilogrammes

(b) the torque required to loosen the nut. μ = 0.1.

5.2 The totalmass of the table of a p la n n ing mach in ean d its attached work p iece i s350

kg. The table i s traversed by a s inglecstart square thread of exter nal diamet er 45mm an d p itch 10mm. The pressure of the cuttin g i s600 N an d the speed of

cuttin g i s6 met ers per minute. The coeff i c ien tof fr ic t io nfor the table i s0.1 an d for the

screw thread i s0.08. Fi n dthe power required.

5.3 The me an radiusof a screw of a square thread screw jack i s2.5 cm. The p itch of thethread i s 7 . 5mm. If the coeff i c ien tof fr ic t io nis 0.12, what effort applied at the en d of a

lever 60 cm length is needed to raise a weight of 2000 kg?.

5.4 A screw press is used to compr ess books. The thread i sdouble thread (square head)

witha p itchof 4mm an d a me an radiusof 25mm. The coeff i c ien tof fr ic t ion for

the contact surfaces of the threads is 0.3. Determine the torque required for a pressureof

500 N.



Feedback to Activity 5A

Have you triedthe questions????? If “YES”, check your answers now

5.1 161 kg, 3.47 Nm

5.2 191 W

5.3 14.07 kg

5.4 4450 Nmm



INPUT

5.5 OVERHAULING OF THE SCREW

When the load moves down an d overcomes the thread f r iction by its ownweight, i t i ssai dto overhaul . When i tmove s downagai n s ta res istin g force P we found that

P = W tan ( χ c χ )

When the load overhauls, P = 0

Therefore,

tan ( χ c χ ) = 0

χ c χ = 0

χ = χ

Hence,

when the angle of the inclined plane is equal to the angle of the friction the load wi l loverhaul .

The eff ici e ncy of such a screw when rai sin g a load i s giv e n by:

1 =tanχ

tan(χ + χ )

but χ = χ

tanχ1=

tan 2χ



Example 5.4

A load of 6 kN i s lifted by a jack hav ing a s i nglecstart square thread screw of 45

mm core diameter an d p itch 10mm. The load revolves withthe screw an d

the coeff i c ien tof f r ictio n at the screw thread i s 0.05. Fi n dthe torque required to li f tthe load.

Show that the load wi l loverhaul.

Solution 5.4

G iven :

W = 6000 N D mean = 45

+

1x 10 = 50mm μ = 0.05 p = 10

mm

2

Then ,

ta nχ = μ tan α = p

τ D

ta nχ = 0.05 tan α =10

(3.14)(50)

χ = 2.86° tan α = 0.0637

α = 3.64°

Torque required, T =1

WD tan (χ + χ)2

=1

(6000)(50) tan (3.64° + 2.86°)2

= 3000(50) tan (6.50°)

= 3000(50) (0.114)

= 17.1 Nm

Ef f ici e ncy, ι =

ι =

tanχ

tan(χ + χ )

tan 3.640

tan(3.640 + 2.86

0)

ι =0.0636

0.114

ι = 0.558



ι = 56% > 50% (overhau ling)



Activity 5B

TEST YOUR UNDERSTANDING BEFORE YOU CONTINUE WITH THE NEXT

INPUT…!

5.5 The drum of a windlassi s10 cm in diameter an d the effort i sappliedto the handle 60

cm fr om the axi s .Fi n dthe effort necessary to support a weight of mass 120 kg.

5.6 A wheel andaxle i sused to raisea load of mass 50 kg. The radiusof the wheel is 50

cm and wh ile i t mak es seven revolutio n sthe load r i s e s3.3 m. What is the smallest

force that w ill support the load?.

5.7 A table carrying a machin e tool i s traversed by a threecstart screw of 6mm p itch

an d exter nal diamet er 23mm. The mass of the table i s200 kg an d the coeff i c ient

of fr ic t io n between the table an d its guidesi s0.1. The screw i s d riven by a mo tor at 12

rev/s. Fi n dthe speed of operatio n of the tool, the power required and the coeff i c ien tof fr ic t io n

for the thread i f the eff ici e ncy i s70 percen t.

5.8 A d if feren tial pulley, the two parts of which have respectively twenty four and twenty f ive

teeth, is used to raise a weight of mass 500 kg. Show by sketch how the apparatus i s

used, and deter min e its velocity ratio. Fi n dalso what effort must be exerted i f the eff i c iency

i s60 percen t.



Feedback To Activity 5B

Have you tried the questions????? If “YES”, check your answers now

5.5 10 gN

5.6 71

gN2

5.7 12.96 rnrmm ,60.5 W, 0.11

5.8 50, 162

gN.3



SELF-ASSESSMENT 5

You are approach mmg success. Try all the questions m mth m sself cassessremt sectmo m am d check

your amswers wmth those gm v e mm mthe Feedback om Self cAssessr em t5 gm v e mo m the mext page.

If you face amy problers, dmscussmt w mth your lecturer. Good luck.

1. Fm m dthe torque to ram s ea load of 6 kN u s m m ga screw jack w m t ha doublecstart square

thread, co m ta m m mmg two threads per centimeter and a mean diameter of 60 mm. μ =

0.12. What ms the torque requ m red to lower the load?

2. A lathe saddle of r ass 30 kg m straversed by a s m mglecstart squarecthread screw of 10rr p mtch am d ream dmareter 40 rr . If the vertmcalforce o m the cuttm m gtool m s250

N, f m m dthe torque at the screw requ m red to traverse the saddle. The coeff mcm e m tof

f r mctmom betweem saddle am d lathe bed am d for the screw thread m s0.15.

3. A doublecstart squarecthread screw has a p mtch of 20 rr am d a ream dmareter of

100 rr , ι = 0.03. Calculate mts effmcm e mcy whem ram smm ga load.

4. A load of 2500 N m sto be ramsed by a screw jack w m t hr eam dmareter of 75 rr

am d p mtch of 12 rr . Fm m dthe eff m c memcy of the screw jack, m f the coeff m c mem tof

fr m c tm o m betweem the screw am d mut ms 0.075.



Feedback to Self-Assessment 5

Have you tried the questions????? If “YES”, check your answers now

1. 31.4 Nr , 12 Nr

2. 0.376 Nr

3. 80.5 %

4. 41.5 %

CONGRATULATIONS!!!!…..

May success be with you

always….

JJ309 Fluid Mechanics Unit 5

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