STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating VECTOR MECHANICS
V P Q y axis i j k VECTOR – magnitude and direction V Px Py Pz SCALAR – magnitude only Qx Qy Qz Fy F x axis VECTOR REPRESENTATION REPRESENTATION MOMENT OF A FORCE ABOUT A Fx Fz =
×
=
A -VECTOR DEFINED BY TWO POINTS
POINT
z axis B(x2,y2,Fz=2λ )⋅ F F F = Fx ⋅ i + Fy ⋅ j + Fz ⋅ k
y axis
y axis B(x ,y z ) My F 2 2, 2 A(x r 1,y1,z1) Mz O Mx x axis z axis
F = Fcosθ x ⋅ i + Fcosθ y ⋅ j + Fsinθ z ⋅ k
A(x1,y1,z1)
F
=
(x 2 - x1 ) ⋅ i
cosθ x = +
(x 2 - x1 )
2
(y 2 - y1 ) ⋅ j +
x
⋅ i + cosθ y ⋅ j + sinθ z ⋅ k
cos2θ x + cos2θ y + cos2θ z = 1
F=λ ⋅ F
(Fx) 2 + (Fy) 2 + (Fz) 2
λ = cosθ x axis
x
z axis
F =
(y 2 - y1 )
2
+
(z 2 - z1 ) ⋅ k
+
(z 2 - z1 )
2
Fx F
cosθ y =
Fy F
cosθ z =
Fz
Mo
F
F
=
F
=
λ ⋅ F x
⋅
i
+
x
2
y ⋅ j +
y
2
+ +
z z
⋅ 2
k
•
DIRECTION AN!ES AND DIRECTION COSINES
F
r × F i
j
k
r x
r y
r z
Fx
Fy
Fz
•
Mo
VECTOR M"!TIP!ICATION A - VECTOR PRODUCT PRODUCT (CROSS (CROSS PRODUCT)
Mo
B - VECTOR DEFINED BY ONE POINT AND THE ORIIN
y axis z F y x x axis z axis
=
=
=
rFsinθ
=
Fd
B - SCALAR PRODUCT (DOT PRODUCT)
P ⋅ Q = PQcosθ 1# T$% T$% &i'% &i'% a*+i a*+i' ' V is is P ⋅ Q = ( Pxi + Pyj + Pzk ) ⋅ ( Qxi + Qyj + Qzk ) %%'.i*/&a + +$% &a'% *'+ai'i'0 P a'. # P ⋅ Q = PxQx + PyQy + PzQz 2# T$% T$% a0'i a0'i+/. +/.% % V is +$% +$% ./*+ +$% a0'i+/.% P APPLICATIONS APPLICATIONS a'. a'. +$% si'% +$% a'0&% %. 3y P a'. A'0&% %. 3y +5 6%*+s V = PQsinθ 4# T$% .i%*+i' .i%*+i' V is 3+ai'%. 3+ai'%. +$% i0$+-$a'. /&%#
cosθ
=
P⋅Q PQ
STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating 2# P=%*+i' a 6%*+ ' a 0i6%' axis
M OL = λ ⋅ Mo = λ ⋅ (r × F) M OL =
λ x
λ y
λ z
x
y
z
Fx
Fy
Fz
SIT"ATION 1 Fi'. +$% .i%*+i' *si'% +$% 6%*+ +$a+ is %%'.i*/&a + +$% &a'% .%>'%. 3y A 9 4i ? = @ a'. B 9 2i ? 2= @ # Ex%ss y/ a's5% i' Ai ? B= ?C # ANSWER :#24i ? :#24= ? :#74
a# D%+%i'% +$% 6a&/% S x 5$i*$ +$% +$%% 6%*+s a% *&a'a# SIT"ATION T$% &&5i'0 6%*+s A, B a'. C a% /+/a&&y %%'.i*/&a + %a*$ +$%
a# D%+%i'% +$% 6a&/% +$% /''5'
SIT"ATION i6%' +$% *% 6%*+ A 9 i ? POL = Pcosθ 7 j -2k N a'. +$% si+i' P ⋅ Q = PQcosθ = POL Q 6%*+s B 9 2 j ? k a'. SIT"ATION 2 C 9 4i -2 j ?7k # P⋅Q PxQx + PyQy + PzQz A *% is a0'i+/.% : N# = POL = T$% *.i'a+% +$% +ai& is a+ (:, Fi'. +$% .+ ./*+ A a'. B Q Q 7, 4) a'. +$a+ +$% +i is (7#8, :, ( A#B) POL = P ⋅ λ Fi'. +$% +$0'a& *'%'+ 4)# I '% /'i+ i' +$% *.i'a+% B i' sys+% is %/a& + '% %+%, B – MIXED TRIPLE PRODUCT +$% .i%*+i' C 5$a+ is +$% %'+ +$% *% Fi'. +$% a'0&% 3%+5%%' A a'. C a3/+ +$% i0i'G Fi'. +$% *ss ./*+ A a'. B ANSWER 44#4 N- ( A x B ) Fi'. +$% /'i+ 6%*+ +$a+ is SIT"ATION 4 %%'.i*/&a + 3+$ A a'. B T$% 6%*+ P a'. a% +5 Fi'. +$% s*a&a +i&% ./*+ A x a.=a*%'+ si.%s a Sx Sy Sz B#C aa&&%&0a# S ⋅ (P × Q) = Px Py Pz a# D%+%i'% +$% a%a +$% Qx Qy Qz SIT"ATION aa&&%&0a i P 9 -i ? R%% + +$% >0/% s$5' 4= @ 4 a'. 9 2i ? 2= ? MOMENT OF A FORCE ABOUT A 8 AXIS
M
SIT"ATION 7# 3# D%+%i'% +$% 6&/% +$% aa&&%&i%. 5$%' P 9 7i - 4= ? 2 a'. 9 -2i - 8= ? a'. S 9 i ? = @ SIT"ATION 8# i6%' +$% 6%*+s P 9 7i - 2= ? 4 a'. 9 2i ? 7= - 8 a'. S 9 Sxi - = ? 2
7 Y F 9D 1:: 2EN A C 2 2 B ; 7 8 <
a# T$% 6%*+ F is %/a& + 3# T$% %'+ a+ F a3/+ i'+ A is %/a& +
STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating *# T$% %'+ a+ F a3/+ i'+ B is %/a& + RES"!TANT AND E"I!IBRI"M Wi'0 F/&a RES"!TANT
Rx =
∑ Fx :
Ry =
R = Rx + Ry 2
2
∑ Fy
θ
= tn -1
Ry Rx
W$%% Rx 9 x-*'%'+ %s/&+a'+ R Ry 9 y-*'%'+ %s/&+a'+ R J 9 a'0&% +$a+ R a%s 5i+$ +$% x-axis Wi'0 F/&a E"I!IBRI"M
∑ Fx = ! : ∑ Fy = R"d = ∑ Mo R"d = Rx" $i
=
SIT"ATION 2 T$%% *a3&%s a% =i'%. a+ +$% =/'*+i' i'0 C a'. s/+ +$% 4: 0 *y&i'.% a# D%+%i'% +$% +%'si's i' *a3&% AC a'. BC # Ca&*/&a+% +$% %a*+i' i' +$% /&&%y a+ D
SIT"ATION 7 T$% /'i . i' >0/% s$5' 5%i0$s 72:N a'. $as i+s *%'+% 0a6i+y a+ # Fi'. +$% %a*+i' a+ +$% s+$ s/a*% A# Fi'. +$% %a*+i' a+ +$% *'% B# Fi'. +$% +%'si' i' +$% *a3&%#
!
Ry"#i
W$%% R 9 %s/&+a'+ *% D 9 %%'.i*/&a .is+a'*% %%%'*% i'+
SIT"ATION 4 T$% a' 5%i0$+ : N a++%+s + &i+ $is%& a'. +$% s%a+ 5%i0$+ 2:: N /si'0 +$% % a'. /&&%y sys+% s$5'# a# D%+%i'% +$% *% a+ A '%%.%. + . s 3# Fi'. $is %a*+i' ' +$% s%a+
SIT"ATION 8 A 5$%%& 1:* a.i/s *ai%s a &a. 1::: N as s$5'# 1# W$i*$ +$% &&5i'0 0i6%s +$% $iz'+a& *% P a&i%. a+ +$% *%'+% 5$i*$ is '%*%ssay + s+a+ +$% 5$%%& 6% +$% 8* 3&*# A&s >'. +$% %a*+i' a+ +$% 3&*# 2# I +$% *% P ay3% i'*&i'%. a+ a'y a'0&% 5i+$
STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating +$% $iz'+a&, 5$i*$ +$% &&5i'0 0i6%s +$% i'i/ 6a&/% P + s+a+ +$% 5$%%& 6% +$% 3&*# 4# W$i*$ +$% &&5i'0 0i6%s +$% a'0&% +$a+ P a%s 5i+$ +$% $iz'+a&#
SIT"ATION A si0' /'i .%'si+y 5%i0$s 87: N a'. is s/+%. 3y a 3a&&-a'.-s*%+ =i'+ a+ A a'. 3y +5 *a3&%s# SIT"ATION N%0&%*+i'0 +$% 5%i0$+s +$% a%, a# Fi'. +$% +%'si' i' *a3&% CD# 3# D%+%i'% +$% i' %a*+i' a+ A a'. B *# Ca&*/&a+% +$% $i'0% %a*+i's a+ E a'. F
1# D%+%i'% +$% +%'si' a+ CE
2# !ii+i'0 i*+i' @ is +$% axi/ 6a&/% s+a+i* i*+i' +$a+ **/s 5$%' +i' is i%'.i'0# 4# Ki'%+i* i*+i' @ is +$% +a'0%'+ia& *% 3%+5%%' +5 3.i%s a+% +i' 3%0i's# I+ is &%ss +$a' s+a+i* i*+i'# 7# A'0&% Fi*+i' ( L ) @ is +$% a'0&% 3%+5%%' +$% a*+i' &i'% +$% ++a& %a*+i' '% 3.y ' a'+$% a'. +$% 'a& + +$% *' +a'0%'+ 3%+5%%' +$% 3.i%s 5$%' +$% +i' is i%'.i'0# 8# C%*i%'+ S+a+i* i*+i'- is +$% a+i' +$% s+a+i* i*+i' ( &ii+i'0 i*+i' ) + +$% Na& *% ( N) # 9 Fs N# # C%*i%'+ Ki'%+i* i*+i' @ is +$% a+i +$% i'%+i* i*+i' + +$% 'a& *%# # A'0&% R%s% ( ) is +$% a'0&% + 5$i*$ a' i'*&i'%. &a'% ay 3% ais%. 3%% a' 3=%*+ %s+i'0 ' i+ 5i&& 6% ( +$a+ is, +$% axi/ a'0&% +$a+ +$% &a'% ay 3% i'*&i'%. 5i+$/+ *a/si'0 +$% 3.y + s&i.% .5' )# "'.% +$% a*+i' +$% *% 0a6i+y a'. +$% %a*+i' +$% &a'%, By Ti0'%+y #
DRY FRICTION @ %%s + +$% i*+i' *% +$a+ %xis+ 3%+5%%' +5 /'&/3i*a+%. s&i. s/ia*%s#
FRICTION T$% %sis+a'*% + s&i.i'0 3%+5%%' *'+a*+i'0 s/a*%s 5$%' +$% %sis+a'*% is ./% + A- STATIC CASE +$% 'a+/% +$% s/a*%s a'. '+ +$%i s$a% # F ≤ F'x = & s % SIT"ATION N%0&%*+i'0 +$% 5%i0$+s +$% %3%s a# D%+%i'% +$% *%s a*+i'0 ' +$% *y&i'.% a+ A a'. B
W$%%
TERMS Fax @ axi/ s+a+i* i*+i' 1# S+a+i* i*+i' - B%+5%%' +5 s 9 *%*i%'+ s+a+i* i*+i' 3.i%s is +$% +a'0%'+ia& *% 5$i*$ N 9 'a& %a*+i' s%s +$% s&i.i'0 '% 3.y B- IMPENDIN S!IDIN %&a+i6% + +$% +$%#
STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating F = F'x = & s % C- DYNAMIC CASE
A 2:: N 3&* is s%+ ' a 4: i'*&i'%. &a'% 5i+$ a *%*i%'+ i*+i' :#4:#
SIT"ATION 7 (CE BOARD MAY 2::) P F = Fk = & k % T$% /'i 8: 0 &a' is %s+i'0 ' /0$ s/a*%s a+ A W$%% a'. B 5$s% *%Q# s+a+i* F @ i'%+i*.y'ai* i*+i' i*+i' a% as s$5'# A 1:: 0 9 *%*i%'+ i'%+i* $s% s+a+s /''i'0 +5a.s B# i*+i' a# W$a+ is +$% %a*+i' a+ BG N 9 'a& %a*+i' 3# W$a+ is +$% %a*+i' a+ AG *# W$a+ is +$% 6a&/% x 5$%' +$% D- F"RTHER DISC"SSION &a' 5i&& s+a+ + s&i.%G 1# T$% i'i/ 6a&/% P +$a+ 5i&& $&. +$% 3&* i' &a*% i J is z% is '%a%s+ 2# T$% sa&&%s+ ssi3&% 6a&/% P +$a+ 5i&& ./*% i%'.i'0 +i' / +$% &a'% i J *a' 3% a.=/s+%. is '%a%s+ + 4# T$% i'i/ 6a&/% P +$a+ 5i&& ./*% i%'.i'0 +i' / CAB!ES "NDER CONCENTRATED +$% &a'% i J is z% is '%a%s+ !OADS + SIT"ATION 1 T$% 1-0 /'i 3a is si+i' SIT"ATION 4 A# ENERA! DISC"SSION T$% si+i' +$% a*$i'% as s$5' a'. is &a.%. 5i+$ a 3&* B is a.=/s+%. 3y 6i'0 *% P i'*&i'%. 8: 5i+$ +$% +$% 5%.0% A# I +$% *%*i%'+ $iz'+a&# s+a+i* i*+i' is :#48 3%+5%%' a# D%+%i'% +$% &a0%s+ P s a&& s/a*%s *'+a*+, +$a+ +$% 3a 5i&& %ai' i' a# D%+%i'% +$% *% P %/i%. %/i&i3i/ si+i'# a# 121#2: N 3# + ais% +$% 3&* B# 3# D%+%i'% +$% *% P %/i%. 118#4 N + &5% +$% 3&* B# *# # .# 2#1 N CAB!ES "NDER DISTRIB"TED !OADS
SIT"ATION 2
STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating
cos θ = O sin θ =
tnθ =
=
O + 2
o 2
B# PARABO!IC CAB!ES
PROPERTIES OF AN AREA CENTROIDS
tnθ =
=
y=
* o "x
Ax = a1 x1 + a2 x2 """" + an xn = Σax = Qy
o
Ay = a1 y1 + a2 y2 """" + an yn = Σay = Qx
O
* o x
2
+
(* o x) 2
MOMENT OF INERTIA
2
2o
Ix =
∫ y dA 2
Iy =
∫ x dA 2
SIT"ATION 2# D%+%i'% +$% x *.i'a+% +$% *%'+i. +$% &i'%#
TRANSFER FORM"!A SIT"ATION 1 F +$% *a3&% &a.%. as s$5' 2 a# D%+%i'% +$% a'0&%s 1 a'. , y = , y + +d 2 , x = , x o + +d o 2 , +$% *% i' %a*$ s%0%'+ a'. +$% &%'0+$ +$% *a3&%# PROD"CT OF INERTIA
∫
Ixy = xydA
TRANSFER FORM"!A
SIT"ATION 4 !*a+% +$% *%'+i. +$% */6% s$5'
Pxy = Po + +xy SIT"ATION 2 i6%' +$% aa3&i* *a3&% s$5', a# H5 a + +$% i0$+ A is +$% &5%s+ i'+G 3# W$a+ is +$% +%'si' a+ +$% &5%s+ i'+G *# W$a+ is +$% +%'si' a+ AG .# W$a+ is +$% +%'si' a+ BG %# W$a+ is +$% s&% *a3&% 2: %+%s + +$% &%+ BG
SIT"ATION 1 D%+%i'% +$% &*a+i' +$% *%'+i. +$% a%a s$5'#
SIT"ATION 4
STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating T$% >0/% s$5' is a i0$+ +ia'0&% 5i+$ +$% .i%'si's as i'.i*a+%.#
a# W$a+ a% +$% 6a&/%s +$% %'+s i'%+ia a3/+ +$% x a'. y ax%sG 3# W$a+ is +$% 6a&/% +$% ./*+ i'%+ia Pxy 5i+$ %s%*+ + x a'. y ax%sG *# W$a+ a% +$% axi/ a'. i'i/ %'+s i'%+ia +$% s%*+i'G SIT"ATION 7 D%+%i'% +$% ./*+ i'%+ia +$% s$a.%. a%a a3/+ +$% x a'. y ax%s#
SIT"ATION D%+%i'% +$% ./*+ i'%+ia +$% /a+% *i*/&a a%a 5i+$ %s%*+ + +$% x a'. y ax%s#
SIT"ATION i6%' +$% >0/% s$5', */+% +$% &&5i'0
SIT"ATION D%+%i'% +$% s/a*% a%a +$% *$ai' &i' s$5', 5$i*$ is a.% a - .ia%+% 3a, i R 9 1: a'. ! 9 4: #
SIT"ATION 8 SIT"ATION D%+%i'% +$% ./*+ i'%+ia D%+%i'% +$% 6&/% +$% +$% a%a s$5' 3%&5 5i+$ *'*%+% a*$ .a %s%*+ + +$% *%'+i.a& x a'. y ax%s#
a# C%'+i.a& %'+s i'%+ia 3# C%'+i.a& ./*+ i'%+ia *# Maxi/ a'. i'i/ %'+s i'%+ia .# Maxi/ ./*+ i'%+ia %# Mi'i/ a.i/s 0ya+i'
STATICS OF RIGID BODIES (LECTURE 1) Lecturer: Engr. MMRTating
