Mad Revisedword

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1

NAMIBIA UNIVERSITY OF SCIENCE AND

TECHNOLOGY

SCHOOL OF ENGINEERING

DEPARTMENT OF MECHANICAL ENGINEERING

MACHINE DESIGN (MAD710S)

ASSIGNMENT 1

NAME: HAIDULA JOHANNES

STUDENT NUMBER: 21305!

1



2

Date Submitted: 11 March 2016

Lecturer: Mr Nikanor Abiatar "UESTION 1

1 21

61 21

1 1 12 21 12 21 16 61 16 61

21 21 1 1

21

part 1

0 0

0 0

0 ( ) ( ) ( ) 0

0 ( 260 360 ) (160 260 ) (230 380 ) 0

0 160 260

x x x Hx

y y y Hy

z Hx Hy Hy Hx x y y x x y y x

z Hy Hx y x y x

z y

F F F F

F F F F

M R F R F R F R F R F R F

M F F F F F F

M F

= → + + =

= → + + =

= → − + − + − =

= → − − + + + + =

= → +

∑∑∑∑∑ 21 1 1

230 380 3!"6#""83 x y x F F F + + = −

2



3

12 32

12 32

21 12 13 32

12 32

part 2

0 0

0 0

0 0

0 "80 "$0 0

x x x

y y y

z x y x y

z y y

F F F

F F F

M R F R F

M F F

= → + =

= → + =

= → + =

= → − + =

∑∑∑∑

3



4

23 3

23 !3 3

32 23 32 23 36 63 3! !3

23 23 63 !3

3

0 0

0 0

( ) 0

( 3$0 280 ) 320 60 0

x x x

y y y y

z x y y x x y x y

z y x y y

Part

F F F

F F F F

M R F R F R F R F

M F F F F

−

= → + =

= → + + =

= − + + =

= − − + + =

∑

∑

∑

∑

6!

3! 6! "!

!" "! !6 6! 6! 6! !3 3!

"! 6! 3!

!

0 0

0 0

0 ( ) 0

0 330 1130 "0 0

x x

y y y y

z x y x y y x x y

z y y y

Part

F F

F F F F

M R F R F R F R F

M F F F

−

= → =

= → + + =

= → + − + =

= → − + =

∑

∑

∑∑

4



5

6" !"

"

0 0 e%uation $ y y y

Part

F F F

−

= → + =∑

12 21

12 21

"! !"

32 23

32 23

3! !3

y y

x x

y y

y y

x x

y y

F F

F F

F F

F F

F F

F F

= −

= −

= −

= −

= −

= −

21 23

21 23

21 23

2

0 0

0 0

0 "80 "$0 0

x x x

y y y

z y y

Part

F F F

F F F

M F F

−

= → − − =

= → − − =

= → − =

∑

∑

∑

5



6

!3 !6 !"

!" 6! !3

61 21

61 21

!

0 0

0 330 1130 "0 0

co&&ectin' a&& the e%uation

0co22" 63#6361031

0in 22" 63#6361031

63#6361031 1

63#6361031 2

y y y y

z y y y

Hx

Hy

x x

y y

Part

F F F F

M F F F

F

F

F F

F F

−

= → − + − =

= → − − − =

= = −

= = −

+ = →

+ = →

∑

∑

21 21 61 61

23 3

6" !"

23 !3 3

23 23 63 !3

21 23

21 23

21

160 260 230 380 3!"6#""83 3

0 !

0 "

0 6

3$0 280 320 60 0 $

0 8

0

"80

y x y x

x x

y y

y y y

y x y y

x x

y y

F F F F

F F

F F

F F F

F F F F

F F

F F

F

+ + + = − →

+ = →

+ = →

+ + = →

− − + + = →

− − = →

− − = →

23

!3 !6 !"

!" 6! !3

"$0 0 10

0 11

330 1130 "0 0 12

y y

y y y

y y y

F

F F F

F F F

− = →

− + − = →

− − = →

Now we have 12 equations and 12 unknowns, solving them simultaneously

1 1 0 0 0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 0 0 0 0

260 380 160 230 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0

0 0 1 0 0 1 0 0 0 0 0 0

0 0 "80 0 0 "$0 0 0 0 0 0 0

0 0 0 0 1 0 1 0 0 0 0 0

0 0 0 0 0 1 0 1 1 0 0 0

0 0 0 0 280 3$0 0 320 60 0 0 0

0 0 0 0 0 0 0 0 1 1 1 0

0 0 0 0 0 0 0 0 "0 1130 330 0

0 0 0 0 0 0 0 0 0 0 1 1

− −

− −

−

− − − −

− − −

21

61

21

61

23

23

63

63

!3

6!

!"

6"

63#6361

63#6361

3!"6#""83

0

0

0

0

0

0

0

0

0

x

x

y

y

x

y

x

y

y

y

y

y

F

F

F

F

F

F

F

F

F

F

F

F

− =

6



7

21 x F

=652.306 N

61 x F

=!5"".666 N

21 y F

=0

61 y F

=63.63#61 N

23 x F

=!652.306 N

23 y F

=0 N

63 x F =652.306 N

63 y F

=!702.4"3 N

!3 y F

=702.4"34 N

6! y F

=!125.1 N

!" y F

=!"27.5"3 N

6" y F

="27.5"32 N

$he %utting &o'%e is equal to "27.5"3 N

7



"

"#$%&' 2

(a)

(*)

1 1 1 2

0 0 0 1

1

1 1 1 0

1 2

1 2

&oad e%uation uin' in'ura&it *unction

( ) 0

( ) 0

( ) 0

0 and

AY cy BY

AY cy BY

AY cy BY

q x R x F x a R x L M x a

v x R x F x a R x L M x a c

M x R x F x a R x L M x a c x c

c c

− − − −

−= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ +

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ + +

= 0 becaue the reaction +ere inc&uded in the &oadin' e%uation

at 0 and 0 x L v M +

=

= = =

"



#

1 1 1 0

1 2

,rom

0 0

( )

AY cy BY

cy

AY

M

R L F L a R L L M L a c L c

F L a M

R L L

+ + + + += ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ + +

−

= +

0 0 0 1

1

,-.M

0 0 AY cy BY

BY cy AY

v

R L F L a R L L M L a c

R F R

+ + + + −= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ +

= −

2 2 2 1

3

3 3 3 2

3 !

( ) 02 2 2

( ) 06 6 6 2

cy AY BY

cy AY BY

F R R EI x x x a x L M x a c

F R R M EI x x x a x L x a c x c

θ

δ

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ + ⟨ − ⟩ +

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ + ⟨ − ⟩ + +

3 3 3 2

3 !

!

at 0/ 0

0 0 0 0 0 0 ( 0)6 6 6 2

0

cy AY BY

x

F R R M a L a c c

c

δ = =

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ + ⟨ − ⟩ + × +

=

3 3 3 2

3 !

3 3 2

3

at / 0

0 0 ( )6 6 6 2

( ) ( 0) ( )6 6 2

cy AY BY

cy AY

x L

F R R M L L a L L L a c L c

F R M c L a L L a

L L L

δ = =

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ + × +

= − − − + −

#



10

( ) #in'* ( 1/0/ ) #in'* ( 1/0#6/ ) #in'* ( 1/1/ ) #in'* ( 2/0#6/ )

( ) #in'* (0/0/ ) #in'* (0/0#6/ ) #in'* (0/1/ ) #in'* ( 1/0#6/ )

( ) #in'* (1/0/ ) #in'* (1/0#6/ ) #i

AY cy BY

AY cy BY

AY cy BY

q x R x F x R x M x

V x R x F x R x M x

M x R x F x R

= − − − + − − −

= − + − −

= − +

3

3

n'* (1/1/ ) #in'* (0/ 0#6/ )

1

( ) #in'* (2/0/ ) #in'* (2/0#6/ ) #in'* (2/1/ ) #in'* (1/0#6/ )2 2 2

1( ) #in'* (3/0/ ) #in'* (3/0#6/ ) #in'* (3/1/ ) #in'* (2/0#6/ )

6 6 6 2

cy AY BY

cy AY BY

x M x

F R R

x x x x M x C EI

F R R M x x x x x C x

EI

θ

δ

−

= − + − +

= − + − +

10



11

1 1 1 2

0 0 0 1

1

1 1 1 0

1 2

1 2

&oad e%uation uin' in'ura&it *unction

( ) 0

( ) 0

( ) 0

0 and

AY cy BY

AY cy BY

AY cy BY

q x R x F x a R x L M x a

v x R x F x a R x L M x a c

M x R x F x a R x L M x a c x c

c c

− − − −

−

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ +

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ + +

= =

1 1 1 0

1 2

0 0 0

0 becaue the reaction +ere inc&uded in the &oadin' e%uation

at 0 and 0

,rom

0 0

( )

,-.M

0 0

AY cy BY

cy

AY

AY cy BY

x L v M

M

R L F L a R L L M L a c L c

F L a M R

L L

v

R L F L a R L L M L

+

+ + + + +

+ + + +

= = =

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ + +

−= +

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ 1

1

2 2 2 1

3

3 3 3 2

3 !

3 3 3 2

3 !

!

( ) 02 2 2

( ) 06 6 6 2

at 0/ 0

0 0 0 0 0 0 ( 0)6 6 6 2

0

at /

BY cy AY

cy AY BY

cy AY BY

cy AY BY

a c

R F R

F R R EI x x x a x L M x a c

F R R M EI x x x a x L x a c x c

x

F R R M a L a c c

c

x L

θ

δ

δ

−− ⟩ +

= −

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ + ⟨ − ⟩ +

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ + ⟨ − ⟩ + +

= =

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ + ⟨ − ⟩ + × +

=

=

3 3 3 2

3 !

3 3 2

3

0

0 0 ( )6 6 6 2

( ) ( 0) ( )6 6 2

( ) #in'* ( 1/0/ ) #in'* ( 1/0#6/ ) #in'* ( 1/1/ ) #in'* ( 2/0#6/ )

( ) #in'* (0/0/ ) #i

cy AY BY

cy AY

AY cy BY

AY cy

F R R M L L a L L L a c L c

F R M c L a L L a

L L L

q x R x F x R x M x

V x R x F

δ =

= ⟨ − ⟩ − ⟨ − ⟩ + ⟨ − ⟩ − ⟨ − ⟩ + × +

= − − − + −

= − − − + − − −

= − n'* (0/0#6/ ) #in'* (0/1/ ) #in'* ( 1/0#6/ )

( ) #in'* (1/0/ ) #in'* (1/0#6/ ) #in'* (1/1/ ) #in'* (0/0#6/ )

1( ) #in'* (2/0/ ) #in'* (2/0#6/ ) #in'* (2/1/ ) #in'* (1/0#6/

2 2 2

BY

AY cy BY

cy AY BY

x R x M x

M x R x F x R x M x

F R R x x x x M

EI θ

+ − −

= − + −

= − + − 3

3

)

1( ) #in'* (3/0/ ) #in'* (3/0#6/ ) #in'* (3/1/ ) #in'* (2/0#6/ )

6 6 6 2

cy AY BY

x C

F R R M x x x x x C x

EI δ

+

= − + − +

11



12

12

% ( ) -a in'* 1− 0, ,( )⋅ ,c in'* 1− 0#6, ,( )⋅− M in'* 2− 0#6, ,( )⋅− -b in'* 1− 1, ,( )⋅+:=

0 0#2 0#! 0#6 0#8 1

!00−

200−

200

% ( )

( ) -a in'* 0 0, ,( )⋅ ,c in'* 0 0#6, ,( )⋅− M in'* 1− 0#6, ,( )⋅ -b in'* 0 1, ,( )⋅+( )−:=

0 0#2 0#! 0#6 0#8 1

200−

100−

100

200

( )

m ( ) -a in'* 1 0, ,( )⋅ ,c in'* 1 0#6, ,( )⋅− M in'* 0 0#6, ,( )⋅ -b in'* 1 1, ,( )⋅+( )−:=

0 0#2 0#! 0#6 0#8 1

"0

100

1"0

m ( )



13

13

θ ( )1

4⋅

-a

2in'* 2 0, ,( )⋅

,c

2 in'* 2 0#6, ,( )⋅− M in'* 1 0#6, ,( )⋅−

-b

2in'* 2 1, ,( )⋅+ 53+ :=

0 0#2 0#! 0#6 0#8 1

3− 103−×

1− 103−×

1 103−×

3 103−×

θ ( )

ε ( )1

4⋅

-a

6in'* 3 0, ,( )⋅

,c

6 in'* 3 0#6, ,( )⋅−

M

2in'* 2 0#6, ,( )⋅−

-b

6in'* 3 1, ,( )⋅+ 53 ⋅+ :=

0 0#2 0#! 0#6 0#8 11− 10

3−×

8− 10!−×

6− 10!−×

!− 10!−

×

2− 10!−×

0

ε ( )



14

(+) a-imum shea'

1"# N

a-imum moment

112.455 N

a-imum sloe

32#$ 10−× /adians

a-imum dee%tion

!8#$1! 10−× ete's

(d)

2

tranere &oadin'

112#!"" 0#01" !#2"881806 Mpa

0#0000000360$8202

!7 ! 183"6"0$#0$2" a

3 3

Aia& &oadin'

10001!1!$10#60" a

0#033

!

z

x

Mc I

A A

F

A

σ

τ

σ π

=

×=

×= = =

×

= = =×

×

14



15

!

9oriona& &oadin'

(2$0 N 0#"0 #01")2"!6!$0#8 a

0#03

32

T r

J τ

π

× × ×= = =

×

(e) +'oss se%tions shows the %'iti%al oint whe'e ma-imum st'ess is e-e'ien%ed

15



16

1 3

6

ma ma

Sa*et ,actor *or the ha*t made o* ducti&e materia&:

oint A

1#Maimum Shear Stre 9heor (MSS)

0#"

0#" 0#"(!3" 10 )8#"2$1803

2""0!!66#"083

y

sy y

sy y

S

S S

S S N

σ σ

τ τ

− ≥

=

×∴ = = = =

( )

( )

( )

: 2 2

1 1 3 3

2 2

1"

6

:

2#7on Mie 9heor or Ditortion;ner' 9heor (D)

283!1231#8!08 283!1231#8!08 2266$$02#288! 283!1231#8!08

2#2!8881!" 10

!$!223$2#8a

!3" 10#1$288""$1

!$!223$2#8

yS

N

σ σ σ σ σ

σ

= − +

= − ×− +

= ×

=

×∴ = = =

ma

ma

ma

6

ma

oint <

oint < i in pure hear o the tota& hear tre i

!7

3

2"!6!$0#8 a=3"6"0$#0$2" a > 2"8212$#6a

?in' maimum hear theor@

0#" 0#" !3" 10

2"8

torsion bn!in"

y

T r

J A

S N

τ τ τ

τ

τ

τ

= +

×= +

=

× ×= =

6

ma

8#!232$121212$#6

?in' Ditortion;ner' theorem@

0#"$$ 0#"$$ !3" 10#$20!6!106

2"8212$#6

yS N

τ

=

× × ×= = =

16



17

(&)

6

1

1 ma

6

1

Sa*et ,actor *or the ha*t made o* britt&e materia&:

oint A

9he Modi*ied Mohr 9heor

6$0 1023#6!0!68""

283!1231#8!08

oint <

at point <

6$0 102"#!$"$2!$

2"8212$#6

#t

#t

S N

S N

σ

σ τ

σ

×∴ = = =

=

×= = =

17

Mad Revisedword

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