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svxd zwipkna dpiga
2006 x`exat
1 dl`y
ep` .u
∈ U, v1 , v2 ∈ V, w ∈ W
mixehwede
W -e V ,U
miixehwed miagxnd mipezp
zix`ipild dwzrda mipc
2
1
T = ( w ⊗ v ) ◦ ( v ⊗ u ).
(1)
?ef dwzrd dtnn agxn dfi`l agxn dfi`n .1
zervn`a mipezpd mixehwed z` e`ha ,mipeyd miagxna miqiqa exicbd .2
.el` miqiqal ziqgi
gikedl yi .
T
T
dwzrdd ly dvixhnd z` e`vne ,qiqad ixehwe
dwzrdd xear (znieqn dcina xzei heyt) xg` iehia eazk .3
.lirl dwzrdd zxcbdn zexiyi iehiad z`
2 dl`y
dzxfbpy sebd ly dpezp divxebitpew `id
:`ad i`pzd z` miiwnd
B
κ xy`k κ ( B) lr b sebd gek dcy oezp
b0 sebd gek dcy didi dn . F `id
B ly P wlg lk xear
Z
Z
b0 ( X ) dV =
P
-iyne
∂B
sebd zty lr
b(κ ( X )) dv.
κ ( P)
X
dcewpn mi`veid
U, V
zeawra .mixvei mdy ziliawnd ly ghyd z`
?ziliawnd zenc ly
∂B
lr xcbend
t0
.1
sebd lr xcbend
a
miphw mixehwe ipy mipezp .2
A
ici lr onqp .dtyl miw
ghyd didi dn ,lirl
dtyd gek dcy didi dn .∂κ ( B ) lr
,∂B ly
Q
divxebitpewd
dtyd gek dcy oezp .3
wlg lk xear :`ad i`pzd z` miiwnd
Z
Z
t0 dA =
Q
t
κ
t da.
κ ( Q)
3 dl`y
un`nd xeqph ici lr oezp dgepna `vnpd lfep jeza ihhqexcdd ugld
Tij = ρgx3 δij .
.lfepd zetitv `id
ρ
xy`k
?lfepa lretd sebd gek edn .1
izla elcebe lfepd ztyl avip lfepd ly edylk wlg lr dtyd gek ik egiked .2
b dtyl avipa ielz
.n
`vnpy edylk seb lr lirtn mxefdy gekd ik raewd qcnikx` weg z` egiked .3
.dgec sebdy mxefd lwynl deey ekeza
4 dl`y
dwzrdd zxcben
b
n
oezp dcigi xehwe xear
b).
T = 5( I − 2b
n⊗n
?ef dwzrd ly zixletd dcxtdd idn
5 dl`y
ihpnene
m seb ihpnen seb lr milret dtyd zegeke sebd zegekl sqepa ik migipn ep`
`ed sebd lr lretd mihpnend lk jqy jk µ dty
Z
M=
Z
( x × b + m) dv +
κ ( B)
wlga
( x × t + µ) da.
(2)
∂κ ( B)
µ P ( x ) dtyd ihpnen zelz xear iyew zgpdl zibelp`d dgpdd didz dn
? P sebd
(Couple-Stress) mihpnen-un`n xeqph meiw z` egiked iyew htynl dibelp`a
.sebd iwlg lr dtyd hpnen z` lawl ozip epnny τ
.1
.2
?τ xear lwyn ieeiy ly zil`ivpxticd d`eeynd didz dn .3
!!dglvda
aby oae`x
2
1 dl`y oexzt
sqed)
f (r )vw = f 0 − mω02 r-l
gekd iablα
√
'` sirq
2 · β=
oezpd owez dpigaa ,al miy
.(m-d
∑ fr = mar = m(r̈ − rθ̇
2
)
ozep
f 0 − mω02 r = mr̈ − mω02 r.
,r̈
r (t) =
= f 0 /m
o`kn
f0 2
t + Ct + D
2m
dlgzdd i`pz zavdae
'a sirq
N = f θ = maθ = m(r θ̈ + 2ṙ θ̇ ) = m · 2
f0 t
ω0
m
(dpigad oeilb lr miyxz d`x) 2 dl`y
ω̄ = ω̄1 + ω̄2 = ω1 k̂ + ω2 î,
R̄ = R̄OA = Lî +
ˆb
ω̇ = ω̄1 × ω̄2 = ω1 ω2 jk.
L
L
sin 37ĵ − cos 37k̂,
2
2
c
R̄˙ XYZ = V AB,
c = sin 37̂b − cos 37k̂.
AB
c
R̄¨ XYZ = R̈ XYZ AB.
c + ω̄ × (ω̄ × R̄) + ω̄˙ × R̄ + 2ω̄ × (V AB
c ).
r̄¨ = R̈ XYZ AB
mixai`d xzi lk .cg` ixlwq mlrp `ede ziqgid dve`zd `ed
R̈ XYZ
ef d`eeyna
.mireci
∑ f¯ = mr̄¨,
N̄ + mg(−k̂) = mr̄¨
c -l avip `ede wiwlgd lr lirtn hendy gekd xehwe `ed
miltek m` . AB
zixlwq d`eeyn milawne ltep
el yiy)
N̄
N̄
N̄
xy`k
c -a zixlwq dlrnl oinin d`eeynd z`
f` , AB
xehwed z` milawne d`eeyna
r̄¨-l
dxfg eze` miaivn . R̈ XYZ xear zg`
.( x oeeika aikx mb
3
3 dl`y
ω̄ = ω̄0 + ω̄1 = ω0 k̂ + ω1 î,
ω̇ = ω̄1 × ω̄0 = −ω1 ω0 ĵ.
`ed divxpi`d xefph

[I] = 
1
2
4 mr

0
1
2
4 mr
0
0
0
0
0 
1
2
2 mr
,o`kne
1
∑ M̄ = H̄˙ = I (ω̄˙ ) + ω̄ × ( I (ω̄ )) = − 2 mr2 ω0 ω1 ĵ.
ok m` `id zihpeelxd mihpnend z`eeyne ,oaenk zqt`zn dqnd fkxn zve`z
:y xiv aiaq
− Ax
3L
L
1
− Bx = − mr2 ω0 ω1 .
2
2
2
,lawzn
A x = − Bx =
1
mR2 ω0 ω1
2L
(.dwihhqdn
4
Bz = mg
xy`n ueg miqt`zn xzid)