Download - Vibration Theory
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3102
1
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snoitrabiV fo yroehT
I 51 , . . . . . . . . . . . . . . . . . . . . 51.1 : )y ,x(f = z . . . . . . . 82 )trebmelA'd( . . . . . . . . . . . . . . . . . . 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.4 : )x(f = y . . . . . . . . 512.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4 : . . . . . . . . . . 815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 021.5 : . . . . . . . . . . . . . . . . . . . . . . 122.5 : )y ,x(f = z . . . . . . . . . . . . . . . . . . . 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 egnargaL . . . . . . . . . . . . . . . . . . . . . . 421.7 . . . . . . . . . . . . . . . . . . . . . . . . 522.7 . . . . 723.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 034.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.01 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.01 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.01 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.01 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.01 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841.11 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.11 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.11 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 054.11 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
II 351 . . . . . . . . . . . . . . . . . . . . . . . . 352 . . . . . . . . . . . . . . . . . . . 451.2 ecnanoseR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 . . . . . . . . . . . . . . . . . . . . . . . . 651.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 . . . . . . . . . . . . . . . . . . . . . . 061.4 . . . . . . . . . . . . . . . . . . . . . . . . 265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.5 . . . . . . . . . . . . . . . . . . . . . . 462.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07
III 371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.4 . . . . . . . . . . . . . . 973.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.5 . . . . . . . . . . . . . . 382.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 . . . . . . . . . . . . . . . . . . . . . . 486 . . . . . . . . . . . . . . . . . . . . . . . 781.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.7 . . . . . . . . . . . . . . . . . . . . 001
3hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
VI 4011 . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.1 1 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.1 1 < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.1 1 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0111.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.3 . . . . . . . . . . . . . . . . . . . 3112.3 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.4 esnopser eslupmI . . . . . . . . . . . . . . . . . . . . . 6112.4 . . . . . . . . . . . . . . . . . . . . . . . . 7113.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911
V 2211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
4hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
I
1 ,
P 3R. . ,
P3 . . / N ) P3 < N( . 1.1, )a( 4 )2y ,1y ,2x ,1x(, )b( 4 L 2L = 2)2y 1y( + 2)2x 1x(
3 , ) ,1y ,1x( .
x
y
1m
2m
2x 1x)a(
1y
2y
x
y
1m
2m
2x 1x)b(
1y
2yL
1.1:
. 2.1,
2 ,1 . N 1=nN}nq{ N 1=nN}nq{ . , 1=nN}nq{
.
5hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
1
2
2.1:
P :
P , . . . ,1 = i im i.
P , . . . ,1 = i ir i.
P , . . . ,1 = i i f i.
ir = ir(t , Nq , . . . ,1q
))1.1( ,
, 1=nN}nq{.
q( ir = ir)t ,n
. 5 . i
irdtd
=N1=n
irnq
+ nqirt)2.1( ,
tir, i. ,
. j)t(nis r +i)t(soc r = r.
r
tj)t(soc r +i)t(nis r =
6 hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
.
q( ir = ir. )t ,mq ,n
= nqir irnq(t , Nq , . . . ,1q
) mqir , 2.1
(irnq
)mq
,0 =
q , n
= mq
m =6 n 0m = n 1irmq
=irmq
)3.1( .
ir mq
irmq
=
mq(ir)
=
mq
(N1=n
irnq
+ nqirt
)
=N1=n
ir2nqmq
+ nqir2tmq
1.1
irmq
=irmq
(t ,nq , . . . ,1q
),
mqir =
irmq
=N1=n
nq+ nq
t=d
td=
d
td
(irmq
),
mq
(irdtd
)=
d
td
(irmq
))4.1( .
7hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
1.1 : )y ,x(f = z
x
y
z
m
)y ,x( f = z
3.1:
m )y ,x(f = z.
.
kz + jy +ix = r
kz + jy +ix = r = v
)y ,x(f = z y ,x , , .
,y = 2q ,x = 1q
,k)2q ,1q(f + j2q +i1q = r
8 hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
= rr
1q+ 1q
r
2q2q
=
(+i
f
1qk
)+ 1q
(+ j
f
2qk
)2q
+ j2q +i1q =
(f
1q+ 1q
f
2q2q).k
2q ,1q r 2q ,1q.
2 )trebmelA'd(
. .
1. i iu. nS ,
N , . . . ,1 = n nS iu
= iuN1=n
irnq
)1.2( ,nS
iu , 1.2.
illuonreB 3071 3471 rebmelA'd, , rebmelA'd " "
. i
,irim = i f
1 : , ,
nq.
9 hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
irim
0 = irim i f
iu P , . . . ,1 = i
P1=i
= iu i fP1=i
)2.2( .yticoliv lautriv elbatpecca 1=iP}iu{ iu irim
2.2 . 1.2 2.2
P1=i
i fN1=n
irnq
= nSP1=i
irimN1=n
irnq
,nS
N1=n
[P1=i
i firnq
]= nS
N1=n
[P1=i
ir irimnq
]nS
N , . . . ,1 = n nS
P1=i
i firnq
=P1=i
ir irimnq
.N , . . . ,1 = n
= nQP1=i
i firnq
,
N
SnQ1=n nq n
.
= nQP1=i
i firnq
=P1=i
ir irimnq
.
3.1
= nQP1=i
i firnq
=P1=i
i f1rnq
,
01 hcalaF roiL dna vegeS nevueRc
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Theory of Vibartions
3
P T
T =Pi=1
miri ri
2.
Qn =Pi=1
miri riqn
=d
dt
(T
qn
) Tqn
.
(ri ri
)qn
=riqn ri + ri ri
qn= 2ri ri
qn.
1.4 ,1.3
qm
(dridt
)=
d
dt
(riqm
)riqm
=riqm
.
d
dt
(riqm
)=
d
dt
(riqm
)=
riqm
.
ddt
(Tqn
)
d
dt
(T
qn
)=
d
dt
(Pi=1
1
2mi(ri ri
)qn
)
=d
dt
(Pi=1
1
2mi2ri ri
qn
)
=Pi=1
miri riqn
+Pi=1
miri ddt
(riqm
)
=Pi=1
miri riqn
+Pi=1
miri riqn
.
cReuven Segev and Lior Falach 11
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snoitrabiV fo yroehT
. nqT
T
nq=
nq
(P1=i
1
2ir irim
)
=P1=i
ir irimnq
.
d
td
(T
nq
)T nq
=P1=i
ir irimnq
.
= nQd
td
(T
nq
)T nq
.N , . . . ,1 = n
egnargaL.
1.3 1.1
k)2q ,1q(f + j2q +i1q = r
r
1q+i =
f
1qk
r
2q+ j =
f
2q.k
,kzF + j yF +ixF = F
r F = 1Q1q
zF + xF =f
1q,
r F = 2Q2q
zF + yF =f
2q.
+ j2q +i1q = r
(f
1q+ 1q
f
2q2q),k
21 hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
= Tm
2
([1q2)
+(2q2)
+
(f
1q+ 1q
f
2q2q]2)
.
T
1qm =
[+ 1q
(f
1q+ 1q
f
2q2q)f
1q
]
d
td
(T
1q
)m =
[+ 1q
(f2
)1q( 2
(1q2)
2 +f2
1q2q+ 1q1q
f2
)2q( 2
(2q2)
+f
1q+ 1q
f
2q2q
)(f
1q+ 1q
f
2q2q()
f2
)1q( q 2
+ 1f2
1q2q2q
])
T
1qm =
(f
1q+ 1q
f
2q2q()
f2
)1q( q 2
+ 1f2
1q2q2q
).
, 1q
m = 1Q
[+ 1q
(f2
)1q( 2
(1q2)
2 +f2
1q2q+ 1q1q
f2
)2q( 2
(2q2)
+f
1q+ 1q
f
2q2q
]).
4
i f
.nocnon,i f + noc,i f = i f
= U = noc f[U
x+i
U
y+ j
U
zk
] )z ,y ,x(U = U. ,
) Pz , Py , Px , . . . ,2z ,2y ,2x ,1z ,1y ,1x( U = U
31hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
i
f i,con = iU = [U
xii+
U
yij +
U
zik
].
N x1, . . . , xP , y1, . . . , yP , z1, . . . , zP Uqn ,
U
qn=
pi=1
[U
xi
xiqn
+U
yi
yiqn
+U
yi
yiqn
].
riqn =xiqn i+
yiqn j +
ziqn k
Qn =Pi=1
f i riqn
=Pi=1
(fi,xi+ fi,y j + fi,zk
)(xiqn
i+yiqn
j +ziqn
k
)
= Pi=1
(U
xii+
U
yij +
U
zik
)(xiqn
i+yiqn
j +ziqn
k
)
= Pi=1
[U
xi
xiqn
+U
yi
yiqn
+U
yi
yiqn
]= U
qn
Qn
Qn = Uqn
.
cReuven Segev and Lior Falach14
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snoitrabiV fo yroehT
1.4 : )x(f = y
1m2m
k k))1x( f ,1x(
))2x( f ,2x(
1.4: )x(f = y
2x = 2q ,1x = 1q .
()2q(f ,2q
)()1q(f ,1q
)
= U1
2y + 12x(k
2+ )1
1
2k(
2)1y 2y( + 2)1x 2x()
=1
2k((1q2)
+()1q(f
)2)+
1
2k((. )2))1q(f )2q(f( + 2)1q 2q
U = 1Q1q
))1q( f ))1q(f )2q(f( +)1q 2q(( k ))1q( f)1q(f + 1q(k =k =
[. ])1q( f ))2q(f )1q(f2( + 2q 1q2
U = 2Q2q
k =[. ])2q( f ))1q(f )2q(f( + 1q 2q
51 hcalaF roiL dna vegeS nevueRc
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snoitrabiV fo yroehT
q = iri(j f +i
)
= T21=i
im2
(iq( 2)
2 f + 1).
T
iqqim =
i(2 f + 1
)d
td
(T
iq
)im =
[iq(2 f + 1
)iq f f2iq +
]im =
[iq(2 f + 1
)2 +
(iq2) f f
]T
iq=
im2
(iq( 2)
f f2)
im =(iq2) f f
d
td
(T
iq
)T iq
im =
[iq(2 f + 1
)+(iq2) f f
].iQ =
) Pr , . . . ,1r(U = U 0 = nqU
U nocnon,nQnq
=d
td
(T
nq
)T nq
d
td
()U T(
nq
))U T(
nq.nocnon,nQ =
naignargaL U T = L d
td
(L
nq
)L nq
)1.4( .N , . . . ,1 = n nocnon,nQ =
gnargaL.: gnargaL naignargaL. gnargaL . gnargaL .
61hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
2.4 :
1m
2m
L
F
k k
x
y
x
2.4:
2.4 1m k. L 2m .
,x .
,ix = 1r
,j soc L i ) nis L +x( = 2r
,ix = 1r
= 2r( soc L +x
).j nis L +i
, i F = 2f ,0 = 1f
1f = xQ1rx
2f +2rx
F = i i F =
1f = Q1r
2f +2r
i F =[j nis L +i soc L
] soc LF =
= T21=i
ir irim2
=2m+ 1m
2+ soc xL2m+ 2x
2m222L
71 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
U = m2gL cos + k2
(2x2)
x
L
x=T
x= (m1 +m2)x+m2L cos
d
dt
(L
x
)=
d
dt
(T
x
)= (m1 +m2)x+m2L
( cos 2 sin
)L
x= U
x= 2kx
(m1 +m2)x+m2L( cos 2 sin
)+ 2kx = F
L
=T
= m2Lx cos +m2L
2
d
dt
(L
)=
d
dt
(T
)= m2L
(x cos x sin
)+m2L
2
L
=T
U
= m2Lx sin m2gL sin
m2Lx cos +m2L2 +m2gL sin = FL cos .
: 4.3
m2
m1
:4.3
cReuven Segev and Lior Falach18
-
snoitrabiV fo yroehT
3.4 1m , , 2m . 4.4.
2m
1m
x
y
g1m 1N
g2m2N
x
y
4.4:
, 2m,xa2m = )(nis 1N = xF.0 = )(soc 1N g2m 2N = yF
1m, xa1m = )(nis 1N = xF. ya1m = g1m)(soc 1N = yF
,
= )(naty,x x
( ,y = )(nat )x x
xa2m = )(nis 1Nxa1m = )(nis 1N
)xa xa()(nat 1m = g1m)(soc 1N
91hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1N , xa ,xa. 2q 1q 1m
5.4.
2m
1m
1q
2q
5.4:
1m 2m
q( = 1r,i2q = 2r ,j)(nis 1q i)2q + )(soc 1
q( = 1r.i2q = 2r ,j)(nis 1q i)2q + )(soc 1
= T1m2
([2q + )(soc 1q
2)+()(nis 1q
]2)+2m2
(2q2)
=2m+ 1m
2
(2q2)
+1m2
([1q2)
)(soc 2q1q2 +],
.)(nis 1qg1m = U
d
td
(L
1q
)L 1q
1m =[)(soc 2q + 1q
],0 = )(nis g1m+
d
td
(L
2q
)L 2q
q)(soc 1m =q)2m+ 1m( + 1
.0 = 2
5
) 1.1( .
q(ir = ir,) Nq , . . . ,1
02 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
= irN1=n
irnq
.nq
= TP1=i
r rim2
=1
2
P1=i
im
([N1=n
irnq
nq
)(
N1=m
irmq
mq
])
=1
2
N1=n
N1=m
P1=i
im
(irnqir mq
).mqnq
]g[
= nmgP1=i
im
(irnqir mq
).
= q[Nq , . . . ,1q
T]Nq , . . . ,1q[ = q T] , ,
= T1
2
{qT}
]g[{q}
=1
2
N1=m,k
qmkg)1.5( .mqk
kmg = mkg , q ])q(g[
.0 > T])q(g[ = ])q(g[
1.5 :
= T1
2x1m
2+ 1
1
2x2m
22
12 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1x
2x
2m 1m
2q 1q
1.5:
= jig 2
nm1=n(nriqnr iq
)= ixnr
i =6 n 0i = n 1= ]g[
[0 1m
2m 0
].
q 1x = 1q1x 2x = 2
= T1
21m(1q2)
+1
22m(2q + 1q
2).
nriq
1r1q
1 =1r2q
0 =2r1q
1 =2r2q
1 =
= 11g2
1=n
nm
(nr1qnr 1q
)2m+ 1m =
= 21g2
1=n
nm
(nr1qnr 2q
)2m =
= 12g2
1=n
nm
(nr2qnr 1q
)2m =
= 22g2
1=n
nm
(nr2qnr 2q
)2m =
= T1
2
[2q 1q
2m 2m+ 1m [ ]2m 2m
[]1q
2q
]
22hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
2.5 : )y ,x(f = z
3.1 y = 2q ,x = 1q
k)2q ,1q(f + j2q +i1q = r
r
1q+i =
f
1q,k
r
2q+ j =
f
2q.k
m = 11gr
1qr 1q
m =
[+ 1
(f
1q
]2)
m = 21gr
1qr 2q
m =f
1qf
2q
m = 22gr
2qr 2q
m =
[+ 1
(f
2q
]2)
6
: vm = p i
ixm = ivm = ip
= ipT
ix=
ix
j
m
2j2x
.ixm = , nP
32 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
Pn =T
qn=
1
2
Nk,m=1
gkmqk
qnqm +
Nk,m=1
gkmqk q
m
qn
.=
1
2
[Nm=1
gnmqm +
Nk=1
gknqk
]
=Nm=1
gnmqm
Pn =T
qn=
Nm=1
gnmqm.
Lagrange 7
Lagrange
d
dt
(T
qn
) (T )
qn=
d
dt
(Nm=1
gnmqm
) 1
2
Nm=1
Nk=1
gkmqn
qmqk
=Nm=1
d (gnm)
dtqm +
Nm=1
gnmqm 1
2
Nm=1
Nk=1
gkmqn
qmqk
=Nm=1
Nk=1
gnmqk
qkqm +Nm=1
gnmqm 1
2
Nm=1
Nk=1
gkmqn
qmqk.
Lagrange
d
dt
(T
qn
) (T )
qn=
Nm=1
Nk=1
gnmqk
qkqm +Nm=1
gnmqm 1
2
Nm=1
Nk=1
gkmqn
qmqk
=Nm=1
Nk=1
[gnmqk
12
gkmqn
]qkqm +
Nm=1
gnmqm = Qn (7.1)
N
nkm(q1, . . . , qN ) =
gnmqk
12
gkmqn
cReuven Segev and Lior Falach 24
-
snoitrabiV fo yroehT
N1=m
N1=k
qmkn+ mqk
N1=m
qmngnQ = m
, 0q 0 = )0q(nQ.
N1=m
N 1=k
nqmk
mqk )0q(mng )q(mng N1=m
q)0q(mng.nQ = m
1.7
. 0q = q
.N , . . . ,1 = n ,0 = nq ,0 = nq
, " . ". N , . . . ,1 = n ,0 = nq ,0 = nq
0 = )0q(nQ ".
N1=m
N1=k
qmkn+ mqk
N1=m
qmng m
[N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
]0=q=q,0q
+N1=l
[
lq
(N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
])0=q=q,0q
lq
+N1=l
[
lq
(N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
])0=q=q,0q
lq
+N1=l
[
lq
(N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
])0=q=q,0q
lq
[ N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
]0=q=q,0q
0 =
52 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
N1=l
[
lq
(N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
])0=q=q,0q
= lq
N1=l
([N1=m
N1=k
mknlq
+ mqkqN1=m
mnglq
mq
])0=q=q,0q
.0 = lq
[
lq
(N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
])0=q=q,0q
=
([N1=m
N1=l
qmln2m
])0=q=q,0q
.0 =
N1=l
[
lq
(N1=m
N1=k
qmkn+ mqk
N1=m
qmngm
])0=q=q,0q
= lqN1=l
,lq 0q| lng
= nQN1=m
N1=k
qmkn+ mqk
N1=m
qmng m
N1=m
)2.7( .mq 0q| mng
0q|g = M. " 0q 0q + = q 0 = ",
0 = q "
= nmMP1=i
im
(irnqir mq
)0=q
.
egnargaL
d
td
(T
nq
)) T(
nq=
N1=m
qmnM.nQ = m
= T1
2
N1=m,n
1 = nqnq 0=q| mng2
N1=m,n
qmnMnqn
= mnMT2
mqnq. 0=q|
0 > T] M[ = ] M[.
62hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
2.7
, , , ,
.
"
1.5
= T1
2
N1=m,k
qmkgmqk
+ )0 = q( T = Tm1=l
T
lqT.O.H+ lq 0q|
1 = )0 = q( T T2
N1=n
N1=m
P1=i
im
(irnqir mq
)mqnq 0q|
= mnMP1=i
im
(irnqir mq
)0=q|
1 T2
N1=m,n
qmnM)3.7( .mqn
"
. "0 = 0q 0 =0| nqU = nQ 0 = )0q(U )
C = )0q(U C U = U U = U(. 0q U
,0 = )0q(UU
nq0 = nQ =0q|
72hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
. 0q
. .
: .
: 0U ,0T
U + T = 0U + 0T
)0U U( = 0T T 0U < U 0 > 0T T . 0 < 0T T 0 T 0 ) T (.
. 1.7 ".
A
B
C
1.7: A " ,B " , C "
+ )0 = q(U = ) Nq , . . . ,1q(UN1=l
U
lq1 + lq 0=q|
2
N1=k,l
U2
kqlqT.O.H+ kqlq 0=q|
82 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
0 = )0 = q(U 0 = nQ = nqU
= ) Nq , . . . ,1q(U1
2
N1=k,l
U2
kqlq)4.7( .T.O.H+ kqlq 0=q|
1 ) Nq , . . . ,1q(U2
N1=k,l
U2
kqlq.kqlq 0=q|
]K[
= mnKU2
mqnq,0=q|
1 U2
N1=m,n
qmnK)5.7( .mqn
= noc,nQN1=m
qmnK.m
2.7 N1=l
qlnM+ l
N1=l
qlnK.nocnon,nQ = l
5.7 3.7
1 = U T = L2
N1=l,k
qlkM1 lqk
2
N1=l,k
qlkK.lqk
L
nq=
T
nq=
nq
12
N1=l,k
qlkMlqk
=
1
2
N 1=l,k
lkMkq
nq+ lq
N1=l,k
qlkMq k
l
nq
=
1
2
[N1=l
qlnM+ l
N1=k
qnkMk
]=
N1=l
qlnMl
92 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
d
td
(L
nq
)=
d
td
(N1=l
qlnMl
)=
N1=l
qlnM.l
L nq
=U
nq=
nq
12
N1=l,k
qlkKlqk
=
1
2
N 1=l,k
lkKkq
nq+ lq
N1=l,k
qlkKq k
l
nq
=
1
2
[N1=l
qlnK+ l
N1=k
qnkKk
]=
N1=l
qlnKl
d
td
(L
nq
)L nq
=N1=l
qlnM+ l
N1=l
qlnK.nocnon,nQ = l
, . M K .
.
3.7
2m 1m2k 1k
2x 1x
)t(y
)t(y.
= T1
2x1m
2+ 1
1
2x2m
22
= U1
21 + 2)y 1x(1k
22)1x 2x(2k
03hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
)y ,2x ,1x(U = U y
3 .
= M
T21x
2T2
2x1xT2y1x
T22x
2T2y2x
mysT22y
= 0 2m0 0 1m
0 mys
,
= K
U212x
U22x1x
U2y1x
U222x
U2y2x
mysU22y
= 0 2k1k 2k 2k + 1k
1k mys
, 0 2m0 0 1m
0 mys
2x1x
y
+0 2k1k 2k 2k + 1k
1k mys
2x1x
y
= 00
0
. 2 2
[, 0 1m
2m
[]1x
2x
]+
[2k 2k + 1k
2k
[]1x
2x
]=
[y1k
0
].
4.7
2m,1m l 3m . 3m )t( F. 54 = .
.
13hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
m1 m2
m3
m3
k1 k2k3
F (t)
l l
l l
m3 m1 ( )m1 q1 = x () m3 .( )
r3 = xi+ k (l cos()i+ l sin()j)
m2
r2 = r3 + 1k (l cos()i l sin()j)= xi+ k (l cos()i+ l sin()j) + 1k (l cos()i l sin()j)= i
(x l sin() + l sin()1
)+ j
(l cos() + l cos()1
) = 1 i m2
r2 = i(x 2l sin()
),
r3 =(x l sin()
)i+ l cos()j.
m3
r3d =(x l sin()
)i l cos()j.
T =m1r1 r1
2+m2r2 r2
2+ 2
m3r3 r32
=m1 (x)
2
2+m2(x 2l sin()
)22
+m3
[(x l sin()
)2+(l cos()
)2] [
m1 +m2 + 2m3 2l sin() (m2 +m3)sym 4l2 sin2()m2 + 2m2l
2
].
cReuven Segev and Lior Falach 32
-
snoitrabiV fo yroehT
= Ux1k
21
2+x2k
22
2+)3y2( 3k
2
2
.)(soc l = 3y ,)(nis l2 x = 2x ,x = 1x
= Ux1k
2
2+2))(nis l2 x( 2k
2+))(soc l2( 3k
2
2
[ )(nis 2kl2 2k + 1k2l4 mys
(nis 2k
soc 3k + )(2)(2
. ] )
3r F = xQx
j)t( F =(i)
0 =
3r F = Q
j)t( F =(j)(soc l +i)(nis l
).)(soc l)t( F =
8
. 6
= T1
2rT+ rM
1
2.cI T
cI .
ji f j i iJ , . . . ,1 = j.
ir i.
jir j i.
i M J , . . . ,1 = j.
33 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
iA i.
i i, ) R = )R( TAA R(.
N P .
,) Nq , . . . ,1q(ir = ir
.) Nq , . . . ,1q(iA = iA
iJ1=j
)1.8( P , . . . ,1 = i ,irim = ji f
iJ1=j
)2.8( .P , . . . ,1 = i ,iH = i M+ ji f jir
i
= iuN1=n
irnq
)3.8( ,nS
i iv
= ivN1=n
inq
)4.8( .nS
1.8 3.8 2.8 4.8 i.
P1=i
iJ 1=j
ji f jir
N 1=n
inq
i M+ nSN1=n
inq
+ nS
iJ1=j
ji f
N1=n
irnq
nS
= N1=n
P1=i
iJ 1=j
ji f jir
i nq
+
iJ1=j
ji firnq
i i M+nq
= nSN1=n
P1=i
iJ 1=j
ji f(inqjir
)+
iJ1=j
ji firnq
i i M+nq
= nSN1=n
P1=i
iJ 1=j
ji f(irnq
+inqjir
)i i M+
nq
.nS
43 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
f ij riqn +
iqn rij
.qn i iqn qn
Pi=1
[miri
Nn=1
riqn
Sn + Hi Nn=1
iqn
Sn
]=
Nn=1
{Pi=1
[miri ri
qnSn + Hi i
qn
]}Sn.
Nn=1
Pi=1
Jij=1
f ij (riqn
+iqn rij
)+M i i
qn
Sn =Nn=1
{Pi=1
[miri ri
qnSn + Hi i
qn
]}Sn,
Sn
Pi=1
Jij=1
f ij (riqn
+iqn rij
)+M i i
qn
= Pi=1
[miri ri
qn+ Hi i
qn
].
T =Pi=1
[miri ri
2+Ti Ii i
2
] 3 . i Ii
d
dt
(
qn
(Pi=1
miri ri2
)) qn
(Pi=1
miri ri2
)=
Pi=1
miri riqn
d
dt
(
qn
(Pi=1
Ti Ii i2
)) qn
(Pi=1
Ti Ii i2
)=
Pi=1
Hi iqn
cReuven Segev and Lior Falach 35
-
snoitrabiV fo yroehT
H, ,I , I
d
td
(
nq
(P1=i
i iI iT2
))=
P1=i
1
2
d
td
(inq
T
+ iiITiI i
inq
),
=P1=i
d
td
(inq
T
iiI
),
=P1=i
d
td
(inq
T
iH
)=
P1=i
(d
td
(inq
T)i + iH
nq
T
iH ),
=P1=i
d
td
(inq
T
iH
)=
P1=i
(inq
T
i + iHnq
T
iH ),
.
nq
(P1=i
i iI iT2
)=
P1=i
1
2
[inq
T
+ iiITiI i
inq
]
=P1=i
[inq
T
iiI
]
=P1=i
[inq
T
iH
]
d
td
(T
nq
)T nq
.N , . . . ,1 = n nQ =
= nQI1=i
iJ 1=j
ji fjirnq
+J1=j
i i Mnq
nq.
4.
1.8
M L 2k ,1k )t( F )t( T 1.8.
.
63 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
k1 k2
L
a
F (t)
T (t)
M
:8.1
k1 k2
L
a
F (t)
T (t)
M
x1 x2
k1 k2
L
a
F (t)
T (t)
M
y
I II
8.1 :8.2
cReuven Segev and Lior Falach 37
-
snoitrabiV fo yroehT
I ,
= Tcv cvM
2+cI
2
2.
j1x = 1r j2x = 2r
iL k + j1x = 21r + 1r = 2r
= 1x 2xL
2x + 1x = c1r + 1r = cv2
.j
= TM
2
(2x + 1x
2
2)+cI2
(1x 2xL
2),
= U1k2
)1x(+ 2
2k2
)2x(. 2
= a1r + 1r = ar[+ 1x
(1x 2xL
)a
]j
= 1x 2xL
k
ar F = 1Q1x
T +1x
j)t( F =(
a 1L
) k)t( T + j
(1 Lk
)=)a L()t( F
L)t( T
L
ar F = 2Q2x
T +2x
j)t( F =a (L
) k)t( T + j
(1
Lk
)=)t( Fa
L+)t( T
L
II
= T2)y( M
2+cI(2)
2
= U1k2
(L y
2
2)+2k2
( + y
L
2
2)
83 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
ar F = yQy
T +y
k)t( T + j j)t( F =(
k0)
)t( F =
ar F = Q
T +
j)t( F =((L
2a ))
k)t( T + j(k)
)t( F =
(L a
2
))t( T +
9
M 0ml ,ml ,mk , m, . m 2)0ml ml( 2mk = mU
= UM1=m
= mUM1=m
mk2
. 2)0ml ml(
= jiKU2
jqiq=0=q|
M1=m
mU2jqiq
=0=q|M1=m
, jimK
= jimKmU2jqiq
=0=q|2
jqiq
(mk2
2)0ml ml()0=q|
=
iq
(ml )0ml ml( mk
jq
)0=q|
mk =
( )0ml ml(
ml2iqjq
+mljq
mliq
)0=q|
mk =mljq
mliq. 0=q|
1.9
93hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
rk
rs
um
kmrs
rk
:9.1
lm = (rs rk) um, ,
um
um =rs rkrs rk
um um = 0 um
lm =(rs rk
) um + (rs rk) um = (rs rk) um. lm
qi= lm
qi rs
qi= rs
qi
lmqi|q=0= lm
qi|q=0=
((rs rk
) um)qi
|q=0=(rsqi rkqi
) um |q=0 .
Kij =Mm=1
kmlmqj
lmqi|q=0, lm
qi=
(rsqi rkqi
) um
cReuven Segev and Lior Falach40
-
Theory of Vibartions
9.1
k1
k2
a,m2
m1
x
u
r2
r1
:9.2
2 9.2
q1 = x, q2 =
l1 = l10 + q1 l1 l1q1
= 1,l1q2
= 0
K1 =
[k1 0
0 0
]
l2 = u2 (r2 r1
) l2
q1, l2q2
k2
u2 |q=0= cos i+ sin jr1 = q
1i
r2 = aq2[ sin ( + q2) i+ cos ( + q2) j]
r1q1|q=0= i , r1
q2|q=0= 0,
r2q1|q=0= 0, r2
q2|q=0= a
[ sini+ cosj
].
l2q1|q=0 = u2
(r2q1 r1q1
)q=0
= cos ,
l2q2|q=0 = u2
(r2q2 r1q2
)q=0
= a (cos sin sin cos ) = a sin( ).
cReuven Segev and Lior Falach 41
-
snoitrabiV fo yroehT
2k = 2K
[ soc ) (nis a 2soc
) (2nis 2a soc ) (nis a
]
= T1m2
(1q2)
+2I2
(2q2)
= M
[0 1m
2I 0
].
01
, nq . .
= TM1=n
1
2q( nm
+ 2)nN
1+M=n
1
2q( nI
2)n
, . \ \ .
)(
U = nFnq
=N1=m
qmnKm
= mq lnK = nF. l =6 m 0l = m 1
lnK , , nq lq.
:
24hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
m . .
.
n m mnK.
1.01 1
m
1k
2k
3k
x
m
1f
2f
3f
1 = 3 = 2 = 1
= K31=i
3k + 2k + 1k = )3k 2k 1k( = if
2.01 2
a
b
ak
bk
34 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
.1 b = 2 ,1 a = 1
.2bkb = 2M ,1aka = 1M
= K21=i
aak = iMbbk + 2
.2
3.01 3
aRbR
cR
aL ,aJ ,aGcL ,cJ ,cG
bL ,bJ ,bG
bm amcm
cR ,bR ,aR . iL ,iJ ,iG c ,b ,a = i. a .
44hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
bRaR
cR
1 = b= c
bRcR= b
bRcR
= abRaR= b
bRaR
bT
cT
cF
cFaT
aF
aF
a
= K
)bRaF bRcF bT( = M
= bT, bJbGbL
= bbJbGbL
= aTaJaGaL
= aaJaGaL
bRaR
= baJaGaL
bRaR
= cTcJcGcL
= ccJcGcL
bRcR= b
cJcGcL
bRcR
. aF ,cF c ,a
= cFcTcR
=cJcGcL
bRc2R
= aFaTaR
=aJaGaL
bRa2R
)bRaF bRcF aT( = K=
bJbGbL
+cJcGcL
b2Rc2R
+aJaGaL
b2Ra2R
.
. ): (.
4.01 4
54 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
R
k1k2 k3
m1m2
x
a
= 1, x = 0 .q2 = x q1 = , 2
1 = (a+R), 2 = R, 3 = 0
K11 =
M = ((R + a)k11 +Rk22) = k1(R + a)2 + k2R2K21 = f = (k22) = k2R
= 0, x = 1
1 = 0, 2 = 1, 3 = 1
K12 =
M = (Rk22) = k2RK22 =
f = (k22 k33) = k3 + k2,
K =
[k1(R + a)
2 + k2R2 k2R
k2R k3 + k2
].
5 10.5
m
k1
k2 k1
k2
x
y
cReuven Segev and Lior Falach 46
-
Theory of Vibartions
2
q1 = x, q2 = y.
M =
[m 0
0 m
].
x
k1 1
1
k2
1
2
k22
k22
k11
k11
.1 = cos,2 = cos
K11 =
Fx = [2k11 cos 2k22 cos ] = 2k1 cos2 + 2k2 cos2 K21 =
Fy = [2k11 sin + 2k22 sin ] = k1 sin(2) k2 sin(2)
y
k1
1
1
k2
1
2
k22
k22
k11
k11
1 = sin,2 = sin
K12 =
Fx = [2k11 cos + 2k22 cos ] = k1 sin(2) k2 sin(2)K22 =
Fy = [2k11 sin 2k22 sin ] = 2k1 sin2 + 2k2 sin2 ,
K =
[2k1 cos
2 + 2k2 cos2 k1 sin(2) k2 sin(2)
k1 sin(2) k2 sin(2) 2k1 sin2 + 2k2 sin2
]
cReuven Segev and Lior Falach47
-
snoitrabiV fo yroehT
11
. , .
. nF nq
= nG, N
qmnK1=m nG nF = nG, m
= nFN1=m
qnmKm
)q( K = F
1K = ) F( = q
= nqN1=m
mFmn
mn nq mq mF.
Q = qK+ qM
M(q)
Q = q +
M(q)
0 = q +
84hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1.11 1
m
2k 1k
x
f x f = 2, 1k1 = 1 x
12k
= 2 + 1 = 1
1k+
1
2k=2k + 1k2k1k
,
= K1
=
2k1k2k + 1k
.
2.11 2
a b
m
1k 2k
x
f x .
94hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
a b
2T1T f
21
,1 = f = 2T + 1T
.0 = b2T a1T
= 1Tb
b +a= 2T ,
a
b +a,
= 11T1k
=b
)b +a( 1k= 2 ,
2T2k
=a
)b +a( 2k.
+ 1 = a)1 2(
b +a=
b
)b +a( 1k+
a1k b2k)b +a( 1k2k
a2
=b2k
2a1k ba2k1k2 + 22)b +a(2k1k
,
= K1
=
)b +a(2k1k2
. 2a1k ba2k1k2 + 2b2k
3.11 3
.
05hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
m
1k
2k
x
f x f2 = 2f = 1f
= 11f1k
=2
1k= 2 ,
2f2k
=2
2k.
= 22 + 12 = )2k + 1k(4
2k1k,
= K1
=
2k1k)2k + 1k(4
.
4.11 4
L3
L6
L2
2m 1m
:
15hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
a bf
L
y(x) =
fbx
6EIL
[L2 b2 x2] x a
fb6EIL
[Lb (x a)3 +
(L2 b2)x x3] x a
a = L3 , b =2L3 m1 f = 1
11 = y
(x =
L
3
)=
4L3
243EI,
21 = y
(x =
L
2
)=
23
1296
L3
EI.
a = L2 , b =L2 m2 f = 1
12 = y
(x =
L
3
)=
23
1296
L3
EI,
22 = y
(x =
L
2
)=
1
48
L3
EI.
cReuven Segev and Lior Falach52
-
snoitrabiV fo yroehT
II
.
1
.)t(f = xk +xc +xm
, 0 = )t(f . 0 = c
.
,0 = xk +xm
,)t(nis B+ )t(soc A = )t(x
)t(soc B + )t(nis A = )t(x)t(nis B2 )t(soc A2 = )t(x
0 = ))t(nis B+ )t(soc A()2m k(
= 0 = m2 kk
m.
cesdar = ][ .
= f
pi2
= ] f[ f1 = T.1
zH = ces
35 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
0x
0v
D
1.1:
,0v = )0(x ,0x = )0(x
+ )t(soc 0x = )t(x0v
.)t(nis
1nat = (0x/0v
)= D ,
+ 02x
0v(
2).
= )(nis0xD= )(soc ,
/0v
D,
])t(nis )(soc + )t(soc )(nis[ D = )t(x
. ) +t( nis D =
D .
2
.
,)t(nis 0F = xk +xm
45 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
mk
)t(nis 0F = )t( F
1.2:
,)t(px + )t(hx = )t(x
)t(hx )t(px .
)t(nis D+ )t(soc C = )t(px
( .)t(nis 0F = )t( nis D)m2 k( + )t(soc C)m2 k
= D0F
,0 = C ,m2 k
= )t(x0F
)t(nis B+ )t(soc A + )t(nis m2 k
B,A . ) (
,0x = )0(hx + )0(px = )0(x
.0v = )0(hx + )0(px = )0(x
55 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1.2 ecnanoseR
0 = 0x0 = 0v
= B ,0 = A
0F= m2 k
0F
. )2 2( m
)t(x
= )t(x0F
)t(nis )2 2( m0F
= )t(nis )2 2( m0Fm
()t(nis )t(nis /
2 2)
)t(x mil latipoH'l
mil
mil = )t(x
0Fm
()t(soc t )t(nis /1
2
)=
0F2m2
))t(soc t )t(nis(
001 08 06 04 02 0001
08
06
04
02
0
02
04
06
08
001
]ces[t
1= rof ecnanoseR
2.2:
) (
.
3
65 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1m3k 2k 1k
2m
2x 1x
1.3: 2
2x2k + 1x)2k + 1k( = )2x 1x(2k 1x1k = 1x1m2x)3k + 2k( 1x2k = 2x3k )1x 2x(2k = 2x2m
[ 0 1m
2m 0
[]1x
2x
]+
[2k 2k + 1k3k + 2k 2k
[]1x
2x
]=
[0
0
])1.3( .
.) +t(nis 2A = 2x ,) +t(nis 1A = )t(1x
) +t(nis 2A2 = 2x ) +t(nis 1A2 = )t(1x[ 1.3
0 1m
2m 0
[]) +t(nis 1A2) +t(nis 2A2
]+
[2k 2k + 1k3k + 2k 2k
[]) +t(nis 1A
) +t(nis 2A
]=
[0
0
][ t
2k 21m 2k + 1k22m 3k + 2k 2k
[]1A
2A
]=
[0
0
].
. ) (.
ted
[2k 21m 2k + 1k
22m 3k + 2k 2k]
=(21m 2k + 1k
( )22m 3k + 2k
0 = 22k )75hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
, .
k = 3k = 2k = 1k m = 2m = 1m
0 = ]k + 2m k2[ ]k 2m k2[ = 2k 2)2m k2(
= 12k
m= 22 ,
k3
m.
]2A ,1A[ = A 2 ,1 A . T
[ k 2m k22m k2 k
[]1A
2A
]=
[0
0
].
= 12 2A = 1A
k 2A ,1A m
mk3 = 22 2A = 1A. , A R A .= 1 2x ,1x
k m
= 2 2x ,1x .
k3 m
1.3
jiA i j . 1+n < n.
= 1
k
m= 2 ,
k3
m.
)2 +t2(nis 21A + )1 +t1(nis 11A = )t(1x
)2 +t2(nis 22A + )1 +t1(nis 12A = )t(2x
12A = 11A 22A = 21A B = 12A = 11A A = 22A = 21A
)2 +t2(nis A + )1 +t1(nis B = )t(1x
)2 +t2(nis A )1 +t1(nis B = )t(2x
85 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
0 = )0(2x = )0(1x 1 = )0(2x = )0(1x
)2(nis A + )1(nis B = 1
)2(nis A )1(nis B = 1)2(soc A2 + )1(soc B1 = 0
)2(soc A2 )1(soc B1 = 0
)1(soc B = 0 )1(nis B = 1
2pi = 1 1 = B.
)2(soc A = 0 )1(nis A = 0
0 = A 2.
nis = )t(2x = )t(1x(+t1
pi
2
).)t1(soc =
.
1 = )0(2x = )0(1x 0 = )0(2x = )0(1x
)2(nis A + )1(nis B = 0
)2(nis A )1(nis B = 0)2(soc A2 + )1(soc B1 = 1
)2(soc A2 )1(soc B1 = 1
0 = B 0 = 2 21 = A,
= )t(1x1
21 = )t(2x )t2(nis
2.)t2(nis
.
95 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
0 = )0(2x = )0(1x 0 = )0(2x 1 = )0(1x
)2(nis A + )1(nis B = 1
)2(nis A )1(nis B = 0)2(soc A2 + )1(soc B1 = 0
)2(soc A2 )1(soc B1 = 0
0 = )1(soc B )1(nis B2 = 1
= 1 21 = B. pi 2
0 = )2(soc A )2(nis A2 = 1
= 2 21 = A pi 2
= )t(1x1
2nis(+t1
pi
2
)+
1
2nis(+t2
pi
2
)1 =
2])t2(soc + )t1(soc[
= )t(2x1
2nis(+t1
pi
2
)1
2nis(+t2
pi
2
)1 =
2. ])t2(soc )t1(soc[
4
0 = x]K[ +x ] M[
)+t(nis A = )t(x )+t(nis iA = )t(ix . )+t(nis A2 = x
0 = A]K[ ) +t(nis +A] M[ ) +t(nis 2
t
[)1.4( 0 = A]] M[ 2 ]K[06hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1] M[ 2 ]K[ 1] M[.
1.4
1. n n nA , . . . ,1A.
2. .
3. .
ted n 2 2 n [ 0 = ]] M[ 2 ]K[
) ]K[ 1] M[ etined evitisoP ( ,
,n2 < < 12
j2 1 n n jiA i j. 0 =6 j1A juj1A = jA ju
(Aj)
1A j1A =
j
= ju
1
j1A/j2A.
.
.
j1A/j,1nAj1A/jnA
= ] U[[nu . . . 1u
]=
1 . . . 1 1
n1A/n2A 21A/22A 11A/12A.
.
.
.
.
.
.
.
.
.
.
.
n1A/nnA 21A/2nA 11A/1nA
.
ju j. ] U[ .
= )t(ixn1=j
= )j +tj(nis jiAn1=j
,)j +tj(nis jiUj1A
jiU n2 j1A j, n2 .
16 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1.4
n
A]K[j
A] M[ j2 =j
juj1A = jA
u] M[ j2 = ju]K[j
Tiu
uj2 = ju]K[ Tiu)2.4( ju] M[ Ti
ju ,iu
u i2 = iu]K[ Tju)3.4( iu] M[ Tj
: D,C D C , D C
. TC TD = T]D C[
[ iu]K[ Tju
T][ju]K[ Tiu = juT]K[ Tiu =
iu] M[ TjuT]
ju] M[ Tiu = juT] M[ Tiu =
] M[ ,]K[ . 2.4 3.4
.iu] M[ Tju i2 ju] M[ Tiuj2 = iu]K[ Tju ju]K[ Tiu
[ R iu] M[ Tju ,iu]K[ Tju iu]K[ Tju
T][,iu]K[ Tju =
iu] M[ TjuT]
.iu] M[ Tju =
26 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
ju]K[ Tiu = iu]K[ Tju ju]K[ Tiu[iu]K[ Tju
T]0 = ju]K[ Tiu ju]K[ Tiu =
= 0(i2 j2
), ju] M[ Tiu
j =6 i 0 =6 i2 j2
.j =6 i 0 = ju] M[ Tiu
" ". ]I[ ] M[ n .
u] M[ j2 = ju]K[j
Tju
uj2 = ju]K[ Tju.0 = ju] M[ Tj
ju] M[ Tiu ) ]I[ = ] M[ (.
u 1 = 1ju j
ju] M[ Tiu
.1 = iu] M[ Tiu
5
, N , , n lx ,mx N , . . . ,1 = l ,m.
.: N , . . . ,1 = j ju .
36 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
: , Na , . . . ,1a
N1=j
uja,0 = j
[iuT] M
[iuT]M
N 1=j
ujaj
N = 1=j
ja[iuT]ia = juM
[iuT]0 = iuM
0 = ia N , . . . ,1 = i
N1=j
ujaN , . . . ,1 = i 0 = ia 0 = j
nR nR y
= yN1=i
uin.n] U[ = i
in y .
in , ] M[ Tju
= y] M[ TjuN1=i
uiniu] M[ Tj
0 = iu] M[ Tju j =6 i
= iny] M[ Tiu
iu] M[ Tiu.
1.5
)t(f
.)t(f = x]K[ +x] M[
46 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
= )t(xN1=i
u)t(in,i
= )t(x n
u)t(in 1=i N , . . . 1 = i ,iu . i
] M[N1=i
u)t(in]K[ + i
N1=i
u)t(in)t(f = i
N1=i
u] M[)t(in+ i
N1=i
u]K[)t(in.)t(f = i
Tju
N1=i
u)t(in+ iu] M[ Tj
N1=i
u)t(in)t(f Tju = iu]K[ Tj
i =6 j 0 = iu] M[ Tju = iu]K[ Tju
u)t(jnu)t(jn + ju] M[ Tj
)t(f Tju = ju]K[ Tj
n , . . . ,1 = j
)t(jn + )t(jnju]K[ Tju
ju] M[ Tju=
)t(f Tju
ju] M[ Tju
ju 0 = ju] M[ j2 ju]K[
ju]K[ Tju
ju] M[ Tju=u] M[ j2Tju
j
ju] M[ Tju, j2 =
+ )t(jn2= )t(jnj
)t(f Tju
ju] M[ Tju,
n + )tj(soc jD+ )tj(nis jC = )t(jnp,)t(j
)t(jpn .
,0v = )0(x ,0x = )0(x
56 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
= 0xN1=i
u)0(ini
] M[ Tju
u = 0x] M[ Tju] M[ Tj
u)0(in
= iN1=i
u)0(inu)0(jn = iu] M[ Tj
ju] M[ Tj
= )0(jn0x] M[ Tju
ju] M[ Tju.
= )0(jn0v] M[ Tju
ju] M[ Tju.
2.5
mmk k k
2x 1x
)t(2F )t(1F
1.5: 2 .
[ 0 m
m 0
[]1x
2x
]+
[k k2k2 k
[]1x
2x
]=
[)t(1F
)t(2F
].
66 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
[m 0
0 m
][x1
x2
]+
[2k kk 2k
][x1
x2
]=
[0
0
]
2 =
3km 1 =
km
u1 =
[1
1
], u2 =
[1
1
].
u1T [M ]u1 = 2m u2T [M ]u2 = 2m.
u1T f(t) = F1(t) + F2(t),
u2T f(t) = F1(t) F2(t).
n1 + 21n1 =
F1(t) + F2(t)
2m,
n2 + 22n2 =
F1(t) F2(t)2m
.
x0 =
[x1
x2
], v0 =
[v1
v2
],
n1(0) =u1T [M ]x0
u1T [M ]u1=x1 + x2
2,
n1(0) =u1T [M ]v0
u1T [M ]u1=v1 + v2
2,
n2(0) =u2T [M ]x0
u2T [M ]u2=x1 x2
2,
n2(0) =u2T [M ]v0
u2T [M ]u2=v1 v2
2.
F2(t) = D2 sin(2t) F1(t) = D1 sin(1t)
nP1 (t) =D1
2m(21 21)sin(1t) +
D22m(21 22)
sin(2t)
nP2 (t) =D1
2m(22 21)sin(1t) D2
2m(22 22)sin(2t)
cReuven Segev and Lior Falach 67
-
Theory of Vibartions
nH1 (t) = A1 sin(1t) +B1 cos(1t),
nH2 (t) = A2 sin(2t) +B2 cos(2t).
n1(0) = nP1 (0) + n
H1 (0) = B1 =
x1 + x22
n2(0) = nP2 (0) + n
H2 (0) = B2 =
x1 x22
n1(0) = nP1 (0) + n
H1 (0) = 1A1 +
1D12m(21 21)
+2D2
2m(21 22)=v1 + v2
2
n2(0) = nP2 (0) + n
H2 (0) = 2A2 +
1D12m(22 21)
2D22m(22 22)
=v1 v2
2
A1 =v1 + v2
21 1D1
2m(21 21)1 2D2
2m(21 22)1,
A2 =v1 v2
22 1D1
2m(22 21)2 2D2
2m(22 22)2.
5.3
L2
L2
L2
L2
k 2k 3k
x1 x2
F0 sin(t)
m 2m
=1
k
[3/8 1/8
1/8 5/24
], M = m
[1 0
0 2
], f(t) =
[0
F0 sin(t)
]
[] [M ] x+ x = [] f(t)
cReuven Segev and Lior Falach 68
-
snoitrabiV fo yroehT
m
k
[8/2 8/3
42/01 8/1
[]1x
2x
]+
[1x
2x
]=
[8/1
42/5
]0Fk
.)t(nis
] M[ ][ .
ted[0 = ]] M[ 2 I
km2 =
1
891 2
42.0 = 1 +
47.1 = 12k
m95.4 = 22 ,
k
m.
= 1u
[1
8.0
]= 2u ,
[1
826.0
].
m87.1 = 2u] M[ T2u m82.2 = 1u] M[ T1u
,)t(nis 0F8.0 = )t(f T1u
.)t(nis 0F826.0 = )t(f T2u
47.1 + 1nk
m= 1n
0F8.0m82.2
,)t(nis
95.4 + 2nk
m= 2n
0F826.0m87.1
.)t(nis
96hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
4.5
.
54
k2
k4k6
)t(nis 0F = )t( F
x
y
m
.y = 2q ,x = 1q
.
0 = y ,1 = x 1 = y ,0 = x
k2
)54(soc k3 )54(soc k3
k4
07 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
0 = y ,1 = x
= 11K
= xFk7 = ))54(2soc k6 k4(
= 12K
= yFk3 = ))54(2soc k6(
1 = y ,0 = x
= 22K
= yF.k5 = ))54(2soc k6 k2(
= M
[0 m
m 0
]= F ,
[)(soc )t( F
)(nis )t( F
].
ted[k3 m2 k7 [ted = ]M2 K
m2 k5 k3
]0 = 2k62 + km221 4 =
38.2 = 12k
m61.9 = 22 ,
k
m.
[ Mj2 K
]0 = ju
= 1u
[1
27.0
]= 2u ,
[1
83.1
]
17 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
m29.2 = 2uMT2u ,m25.1 = 1u] M[ T1u
FT1u
1u] M[ T1u])( nis 27.0 )(soc[ )t( F66.0 =
FT2u
2uMT2u])( nis 83.1 + )(soc[ )t( F43.0 =
+ 1n2)t(1F = )t(nis 0F])( nis 27.0 )(soc[ 66.0 = 1n1
+ 2n2)t(2F = )t(nis 0F])( nis 83.1 + )(soc[ 43.0 = 2n2
2.45 = 0 = )t(1F )t(2F , 2u 7.53 =
1u 0 = )t(2F )t(1F .
27hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
III
.
1
)t ,x(v
L
x
1.1:
T , T ) (. )t ,x(v x t.
x . .
x
x + x x
)x()x + x(
T
T
2.1: "
x = ))x((nis T ))x +x((nis Tv2
2t,
37 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
v
x,)(nis ' )(nat =
( Tv
xv )x +x(
xx = )
v2
2t
x 0 x
mil0x
(Tv)t ,x(xv )t ,x +x( x
x
)T =
v2
2x =
v2
2t
T = 2c
v2
2t2c =
v2
2x,
. A = A EA = T= AEA = 2c
E E
c .
2
L)t ,x + x(N
E,A
x
)t ,x(N
x + x x
)t ,x(f
)t ,x(f
1.2:
47 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
x
x = )t ,x(N)t ,x +x(N+x)t ,x(fu2
2t
. x 0 x
+ )t ,x(fN
x =
u2
2t
)t ,x(EA = )t ,x(N xu =
u2
2t=AE
u2
2x+)t ,x(f
= AE = 2c E A =
u2
2t2c =
u2
2x+)t ,x(f
.
3
L
x + x x
)x( T)x + x( T
R
)t ,x( J ,G
)t ,x(
1.3:
57hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
)t ,x( x t
R
L
G
R2pi = JJ 4
= x .T= JG
LT JG
x
I = x)t ,x( + )x( T )x +x( T2
2tJx =
2
2t
x 0 x
mil0x
[)x( T )x +x( T
x)t ,x( +
]=T
x= )t ,x( +
(x JG
)x
J = )t ,x( +2
2t
G = 2c
2
2t2c =
2
2x+)t ,x(
J.
4
2
2t2c =
2
2x)1.4(
. ,
)t( T)x(X = )t ,x(
)x( X)t( T2c = )x(X)t( T
67 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
)t( T)x(X
)t( T)t( T
2c =)x( X)x(X
.
, t x
)t( T)t( T
2c =)x( X)x(X
. =
. )t( T ,)x(X )0 ,,+( . 0 < ,
T.)t( = )t( T
0 > 2 = teB + teA = )t( T )t( T tmil.
0 = B+tA = )t( T .
0 < 0 < 2 = )t(soc D+ )t(nis C = )t( T.
)t( T)t( T
2c =)x( X)x(X
.2 =
, t
0 = )t( T2 + )t( T
x
+ )x( X(c
2).0 = )x(X
nis A = )x(X(cx)
soc B+(cx))2.4( .
.
77 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1.4
0 = )t ,L( = )t ,0(
)t( T)x(X = )t ,x(
,0 = )t( T)0(X = )t ,0(
.0 = )t( T)L(X = )t ,L(
0 = )L(X = )0(X 2.4
0 = B = )0(X
nis A = )L(X(cL)
soc B+(cL)
0 =
0 = B
c. . . ,2 ,1 = n ,npi = L
n = ncpi
L,
nis nA = )x(nXn(cx)
nis nA =npi(Lx),
)x(nX )( n.
= )t ,x(1=n
)t(nT)x(nX
=1=n
nisn(cx)
. ])tn(soc nD+ )tn(nis nC[
87hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0
1
8.0
6.0
4.0
2.0
0
2.0
4.0
6.0
8.0
1
L/x
n X)x(
1=n2=n3=n4=n
1.4:
2.4
L
x
L
x
nn
)t ,L = x( T
)t ,L = x(N
2.4:
L = x 0 = )t ,L = x(N 0 = )t ,L = x( T,
JG
x 0 =L=x|
x,0 =L=x|
AEu
xu 0 =L=x|
x.0 =L=x| 0 = x
.0 = )t ,0 = x(u ,0 = )t ,0 = x(
97 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
= )t ,0(
x.0 = )t ,L(
0 = B = )0(X
= )L( X
csoc A
(cL)
+
cnis B
(cL)
0 =
soc (Lc)
0 =
cpi = L
2. . . ,2 ,1 = n npi +
c = n)1 n2( pi
L2. . . ,2 ,1 = n
nis A = )x(nXn(cx).
= )t ,x(1=n
)t(nT)x(nX
=1=n
nisn(cx)
. ])tn(soc nD+ )tn(nis nC[
1 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0
1
8.0
6.0
4.0
2.0
0
2.0
4.0
6.0
8.0
1
L/x
n X)x(
1=n2=n3=n4=n
3.4:
08hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
3.4 x
Tk
k
x
L = x .
)t ,L = x(uk = sF = )t ,L = x(N
xuAE = )t ,x(N
AEu
x)t ,L = x(uk = )t ,L = x(
AEu
x.0 = )t ,L(uk + )t ,L(
)t ,L = x( Tk = sM = )L = x( T
x JG = )t ,x( T
JG
x)t ,L = x( Tk = )t ,L = x(
JG
x.0 = )t ,L( Tk + )t ,L(
,
0 = )0(X 0 = )t( T)0(X = )t ,0(JG
xXJG = )t ,L( Tk + )t ,L(
0 = )L(XTk + )L( XJG 0 = )t( T)L(XTk + )t( T)L(
18 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
5.4 4 5.3 3 5.2 2 5.1 1 5.0 05
4
3
2
1
0
1
2
3
4
5
c/
4.4: 3.4
0 = B = )0(X
JG = )L(XTk + )L( XJG
csoc A
(cL)
nis ATk +(cL)
0 =
nat c(Lc)
JG =
Tk)3.4( .
5
1.4
2n
2cX = )x(nX
)x(n
L
)x(mX xd 0
2n
2c
L 0= xd)x(mX)x(nX
L 0xd)x(mX)x(n X
2n
2c
L 0= xd)x(mX)x(nX
[X)x(mX
)x(n
L]0L
0X)x(n X
.xd)x(m
28 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
n,m
2m
2c
L 0= xd)x(mX)x(nX
[X)x(nX
)x(m
L]0L
0X)x(n X
.xd)x(m
n2 m22c
L 0= xd)x(mX)x(nX
[X)x(mX
)x(n
L]0)1.5( 0L])x(m X)x(nX[
, .
1.5
[ 0 = )L(mX ,0 = )0(nX N m,n X)x(mX
)x(n
L]0
,0 =
[ 0 = )L(m X ,0 = )0(nX N m,n X)x(mX
)x(n
L]0
.0 =
( 1.5 n2 m2
2c
L )0.0 = xd)x(mX)x(nX
m =6 n m =6 n L
0, 0 = xd)x(mX)x(nX
n = m L
00 =6 xd)x(nX)x(nX
0 )x(nX)x(nX.
2.5
, L = x
0 = )L(XTk + )L( XJG
38 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
)L(XJGTk = )L( X
= )L(n XTkJG
.)L(nX
[ 0 = x 0 = )0(mX X)x(mX
)x(n
L]0)L(nXJGTk [ )L(mX = 0L])x(m X)x(nX[
][Tk JG
)L(mX
],0 = )L(nX
L 0.0 = xd)x(mX)x(nX
: , .
0b + )t ,0(0a)t ,0(
x,0 =
Lb + )t ,L(La)t ,L(
x.0 =
L
0.m =6 n ,0 = xd)x(mX)x(nX
3.5 x
x
m0I
, "
48hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
x
x
m
)t ,L(N)t ,L( T0I
m = )t ,L(Nu2
2t)t ,L(
xuAE = )t ,x(N
uAEx
m = )t ,L(u2
2t)t ,L(
u22t
2c =u22x
uAEx
2cm = )t ,L(u2
2x)t ,L(
2cmu2
2xEA + )t ,L(
u
x.0 = )t ,L(
JG = )t ,L( Tx
0I = )t ,L(2
2tc0I = )t ,L(
22
2x)t ,L(
c0I 2
2
2xGJ + )t ,L(
x.0 = )t ,L(
.
0 = )0(X 0 = )t( T)0(X = )t ,0(c0I
22
2xGJ + )t ,L(
xc0I = )t ,L(
0 = )L( XGJ + )L( X2c0I 0 = )t( T)L( XGJ + )t( T)L( X2
58 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
X = 2c2X .X(0) = B = 0
I0c2X (L) + JGX (L) = I02X(L) + JGX (L)
= I02A sin(cL)
+
cJGA cos
(cL)
= 0
tan(cL) JGI0c
= 0.
X(0) = 0(
2m 2nc2
) L1Xn(x)Xm(x)dx = Xm(L)X
n(L)X m(L)Xn(L).
I02X(L) + JGX (L) = 0
X n(L) =I0
2n
JGX(L)
(2m 2n
c2
) L0Xn(x)Xm(x)dx =
(2n 2m
) I0JG Xm(L)Xn(L)
(2n 2m
) [ 1c2
L0Xn(x)Xm(x)dx+
I0JG
Xm(L)Xn(L)
]= 0,
n 6= m [1
c2
L0Xn(x)Xm(x)dx+
I0JG
Xm(L)Xn(L)
]= 0
2n
c2
L0Xn(x)Xm(x)dx =
L0X n(x)Xm(x)dx
=[Xm(x)X
n(x)
]L0 L
0X n(x)X
m(x)dx
= Xm(L)Xn(L)
L0X n(x)X
m(x)dx
=I0
2n
JGXn(L)Xm(L)
L0X n(x)X
m(x)dx
cReuven Segev and Lior Falach 86
-
snoitrabiV fo yroehT
n2
[1
2c
L 0+xd)x(mX)x(nX
0IGJ
)L(mX)L(nX
]=
L 0X)x(n X
xd)x(m
L 0X)x(n X
.0 = xd)x(m
6
N , ) (.
= )t ,x(n
)t(nT)x(nX
)t(nT )t(nX .
nX , , )x(X
= )x(Xn
,)x(nXng
ng . ellivuoiL-mrutS.
1.6 .
2
2t2c =
2
2x)t ,x(g +
= )t ,x(
)t ,x(g . )t(nT)x(nX1=n
n
T)x(nXc = )t( n
2n
)t ,x(g + )t(nT)x(n X
78 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
= n X 2n2c nX
n
)x(nX[ + )t( n T
2)t(nTn
].)t ,x(g =
L
)x(mX xd 0
L 0
[n
)x(nX[ + )t( n T
2)t(nTn
]]= xd)x(mX
L 0xd)x(mX)t ,x(g
+ )t( n T2= )t(nTn
L L xd)x(nX)t ,x(g 0
X 02xd)x(n
T + )tn(soc nD+ )tn(nis nC = )t(nTP.)t( n
nD ,nC
)x(h = )0 ,x(
t)x(v = )0 ,x(
= )0 ,x(n
)x(h = )0(nT)x(nX
L
)x(mX xd 0
= )0(nT
L L xd)x(nX)x(h 0
X 02xd)x(n
T + nD =P)0( n
L
L xd)x(nX)x(h 0X 0
2xd)x(n
T + nD =P)0( n
= )0(nT
L L xd)x(nX)x(v 0
X 02xd)x(n
+ nCn =.)0( nPT
88hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
2.6
L
,E ,0A
)t(nis 0Y = )t(y
0A L, E. )t(nis 0Y = )t(y.
.
u2
2t=
0A0AE
u2
2x2c =
u2
2x,
E = 2c,
,)t(y = )t ,0(u
u
x,0 = )t ,L(
,0 = )0 ,x(uu
t.0 = )0 ,x(
,
)t(y + )t ,x(v = )t ,x(u
u2
2t=
v2
2t+)t(y2d
2td=v2
2t)t(nis 0Y2
u2
2x=
v2
2x
v2
2t2c =
v2
2x)t(nis 0Y2 +
.
0 = )t ,0(v )t(y = )t(y + )t ,0(v = )t ,0(uu
x= )t ,L(
v
x0 = )t ,L(
98 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
.
0 = )0 ,x(v 0 = )0(nis 0Y + )0 ,x(v = )0 ,x(uu
t= )0 ,x(
v
tv 0 = )0(soc 0Y + )0 ,x(
t.0Y = )0 ,x(
v2
2t2c =
v2
2x)t(nis 0Y2 +
0 = )t ,0(vv
x0 = )t ,L(
0 = )0 ,x(vv
t0Y = )0 ,x(
nis = )x(nXn(cx)= n ,
cpi
L2. . . ,2 ,1 = n )1 n2(
1=n
)t(nT)x(nX
1=n
)x(nX[ + )t( n T
2)t(nTn
])t(nis 0Y2 =
L )t(mX xd 0
+ )t( n T2 = )t(nTn
)t(nis 0Y2
L L xd)x(nX 0X 0
2xd)x(n
L
0= xd)x(nX
L 0
nisn(cx)= xd
[c n
socn(cxL])
0
=c
n,
L 0= xd)x(n2X
L 0
2nisn(cx)= xd
L 0
[)x cn( soc 1
2
]= xd
L
2.
+ )t( n T2 = )t(nTn
)t(nis 0Y2c2
Ln
09 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
TPn (t) = Gn sin(t) +Hn cos(t)
Gn =2c2Y0
nL (2n 2)
Tn(t) = Gn sin(t) + Cn sin(nt) +Dn cos(nt).
Tn(0) =
L0 v(x, 0)Xn(x)dx L
0 X2n(x)dx
= 0 = Dn
Tn(0) =
L0
vt (x, 0)Xn(x)dx L
0 X2n(x)dx
= 2cY0Ln
= Gn + nCn
Cn = n
[2cY0
Ln+Gn
].
v(x, t) =n=1
[Gn sin(t) + Cn sin(nt)] sin(ncx).
u(x, t) = v(x, t) + Y0 sin(t).
cReuven Segev and Lior Falach91
-
snoitrabiV fo yroehT
7
x
1.7:
. . 2.7.
x
)t ,x(v
x x
y
)x( V z
x x
)x(M
2.7:
)t ,x(v )( x t.
)t ,x( V x t.
)t ,x(M x t.
29 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
v2
2x=M
IE)1.7(
I E .
M
x)2.7( V =
3.7 x.
x + x x
xx
)t ,x( V)t ,x + x( V
)t ,x(M)t ,x + x(M
)t ,x(q
3.7:
x
xA = x)t ,x(q + )t ,x( V )t ,x +x( V = F
v2
2t,
x 0 x mil
mil0x
[)t ,x( V )t ,x +x( V
x
]xA = )t ,x(q +
v2
2t,
V
xA = )t ,x(q +
v2
2t,
39 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
1.7 2.7
V
x =
M2
2x =
2
2x
(IE
M2
2x
) IE =
v4
4x
IEv4
4xA = )t ,x(q +
v2
2t
Av2
2tIE +
v4
4x)t ,x(q =
1.7
Av2
2tIE +
v4
4x0 =
AIE = 2c
v2
2t 2c =
v4
4x
)t( T)x(X = )t ,x(v
T2d
2tdd T2c = X
X4
4xd
)t( T)x(X
T2d
2td1
T1 2c =
X
X4d
4xd
,
T2d
2td1
T1 2c =
X
X4d
4xd =
4 2 = . T2d
2td0 = T2 +
X4d
4xd
2
2c0 = X
49 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
( xeB = )x(X 4
(c
)2)0 = teB
c = 0 = 2) c( 4 i = 4,3 , = 2,1
e1B = )x(Xe2B+ x
xie1B+ xie3B+ x
.)x(soc 4D+ )x(nis 3D+ )x(hsoc 2D+ )x(hnis 1D = )x(X
)t ,x(v x t.
)t ,x(xv = )t ,x( )( x t.
IE = )t ,x(M x t.v22x
)t ,x(
v3 IE = )t ,x( V x t.3x
x
4.7:
59 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
7.4
v(0, t) = 0, v(L, t) = 0, M(0, t) = 0, M(L, t) = 0.
v(x, t) = X(x)T (t)
v(0, t) = 0 X(0) = 0 D2 +D4 = 0M(0, t) = 0 d
2X
dx2(0) = 0 2D2 2D4 = 0
v(L, t) = 0 X(L) = 0 D1 sinh(L) +D2 cosh(L) +D3 sin(L) +D4 cos(L) = 0M(L, t) = 0 d
2X
dx2(L) = 0 2D1 sinh(L) + 2D2 cosh(L) 2D3 sin(L) 2D4 cos(L) = 0
,D2 = D4 = 0
D1 sinh(L) +D3 sin(L) = 0
2D1 sinh(L) 2D3 sin(L) = 0
D3 6= 0 D3 sin(L) = 0 D1 = 0 D1 sinh(L) = 0
L = npi n = 1, 2, . . .
n = c(npiL
)2.
Xn(x) = sin(npixL
).
x
:7.5
cReuven Segev and Lior Falach 96
-
Theory of Vibartions
7.5
v(0, t) = 0, (0, t) = 0, M(L, t) = 0, V (L, t) = 0
v(x, t) = X(x)T (t)
X(x) = D1 sinh(x) +D2 cosh(x) +D3 sin(x) +D4 cos(x).
v(0, t) = 0 X(0) = 0 D2 +D4 = 0(0, t) = 0 dX
dx(0) = 0 D1 + D3 = 0
V (L, t) = 0 d3X
dx3(L) = 0 3D1 cosh(L) + 3D2 sinh(L) 3D3 cos(L) + 3D4 sin(L) = 0
M(L, t) = 0 d2X
dx2(L) = 0 2D1 sinh(L) + 2D2 cosh(L) 2D3 sin(L) 2D4 cos(L) = 0
D1 = D3D2 = D4 D1 [cosh(L) + cos(L)] +D2 [sinh(L) sin(L)] = 0D1 [sinh(L) + sin(L)] +D2 [cosh(L) + cos(L)] = 0.
[cosh(L) + cos(L) sinh(L) sin(L)sinh(L) + sin(L) cosh(L) + cos(L)
]{D1
D2
}= 0
det
[cosh(L) + cos(L) sinh(L) sin(L)sinh(L) + sin(L) cosh(L) + cos(L)
]=
(cosh(L) + cos(L))2 (sinh(L) sin(L)) (sinh(L) + sin(L)) =cosh2(L) + 2 cosh(L) cos(L) + cos2(L) sinh2(L) + sin2(L) =
2 cosh(L) cos(L) + 2 = 0
cosh(L) cos(L) = 1.
1L = 1.875, 2L = 4.694, 3L = 7.855, 4L = 10.99, 5L = 14.137
Xn(x) = sinh(nx) sin(nx) cosh(nL) + cos(nL)sinh(nL) sin(nL) [cosh(nx) cos(nx)] .
cReuven Segev and Lior Falach 97
-
Theory of Vibartions
7.2
(nc
)2Xn(x) =
d4Xndx4
L
0 dx Xm(x) (nc
)2 L0Xm(x)Xn(x)dx =
L0
d4Xndx4
Xmdx
=
L0Xmd
(d3Xndx3
)=
[Xm
d3Xndx3
]L0
L
0
d3Xndx3
d (Xm)
=
[Xm
d3Xndx3
]L0
L
0
d3Xndx3
dXmdx
dx
=
[Xm
d3Xndx3
]L0
L
0
dXmdx
d
(d2Xndx2
)=
[Xm
d3Xndx3
]L0
[dXmdx
d2Xndx2
]L0
+
L0
d2Xndx2
d2Xmdx2
dx
(nc
)2 L0Xm(x)Xn(x)dx =
[Xm
d3Xndx3
dXmdx
d2Xndx2
]L0
+
L0
d2Xndx2
d2Xmdx2
dx
(mc
)2 L0Xm(x)Xn(x)dx =
[Xn
d3Xmdx3
dXndx
d2Xmdx2
]L0
+
L0
d2Xndx2
d2Xmdx2
dx
2n 2mc2
L0Xm(x)Xn(x)dx =
[Xm
d3Xndx3
Xnd3Xmdx3
(dXmdx
d2Xndx2
dXndx
d2Xmdx2
)]L0
n 6= m L
0Xm(x)Xn(x)dx.
cReuven Segev and Lior Falach 98
-
snoitrabiV fo yroehT
3.7
Av2
2tIE +
v4
4x)t ,x(q =
= )t ,x(v1=n
)t(nT)x(nX
)x(nX
A1=n
[)t(nT2d
2td)x(nX
]IE +
1=n
[)t(nT
)x(nX4d
4xd
])t ,x(q =
(nc
2)= )x(nX
nX4d4xd
1=n
[A
)t(nT2d
2td+ )x(nX
n(c
2))t(nTIE
])t ,x(q = )x(nX
)t(nT2d
2td= )t(nTn2 +
L L xd)x(nX)t ,x(q 0
X 02xd)x(n
.
4.7
.
xL
a)t(nis 0F
,A,E
99 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
)x(a)t(nis 0F = )t ,x(q
c = npin(L
2)nis = )x(nX ,
xpin(L
).
L
0nis = )a(nX = xd)x(a)x(nX
apin(L
)L
0= xd)x(n2X
L
2
)t(nT2d
2td= )t(nTn2 +
L L xd)x(nX)t ,x(q 0
X 02xd)x(n
=nis 0F2
(apinL
)L
)t(nis nG = )t(nis
= nG ) Lapin(nis 0F2
L
+ )tn(soc nD+ )tn(nis nB = )t(nTnG
2 n2)t(nis
,0 = )0 ,x(vv
t.0 = )0 ,x(
0 = )0(nT ,0 = )0(nT
= nB ,0 = nDn
nG2 n2
.
5.7
6.7.
001hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
)x( V)x + x( VP
P)x(xv = )x()x + x(xv = )x + x(
7.7: x
P P
6.7:
x 7.7. y
v P )x( V )x +x( Vx
P + )x +x(v
xAx = )x(
v2
2t
P )x( V )x +x( V[v
xv )x +x(
x)x(
]Ax =
v2
2t
= xV M22x
x 0 x
M2
2x P
v2
2xA =
v2
2t
v22x
IEM =
IEv4
4x P
v2
2xA =
v2
2t
v2
2t=IE
A
v4
4x+
P
A
v2
2x
101 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
v(x, t) = X(x)T (t)
d2T
dt2X =
EI
A
d4X
dx4T +
P
A
d2X
dx2T
XT
d2T
dt21
T=EI
A
d4X
dx41
X+
P
A
d2X
dx21
X= 2
T .
T = A cos(t) +B sin(t).
X
EId4X
dx4+ P
d2X
dx2 2AX = 0
X(x) = Cex
EI4 + P2 2AX = 0
2 =P
P 2 + 4EI2A
2EI
21 =P +
P 2 + 4EI2A
2EI= 2 > 0
22 =P
P 2 + 4EI2A
2EI= 2 < 0
= {,, i,i}
X(x) = C1ex + C2e
x + C3eix + C4eix
X(x) = D1 cosh(x) +D2 sinh(x) +D3 cos(x) +D4 sin(x)
cReuven Segev and Lior Falach 102
-
snoitrabiV fo yroehT
0 = )t ,L(M ,0 = )t ,L(v ,0 = )t ,0(M ,0 = )t ,0(v
0 = 3D = 1D 0 = 2D
pin = L
+ PA2IE4 + 2 P
IE2=2pi2n
2L
= n2AIE4
(IE2pi2n2
2LP
2)2I2E4 = 2 P +
pin(L
4)4
pin(L
2)PIE
= n2IE
A
pin(L
4)P A
pin(L
2) 0 < P
.
301hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
VI
.
1
: .
c
x
m
1.1:
1.1
.c = cM xc = cF
2.1
cm
x
k
)t(f
2.1:
401 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
,xm = xk xc )t(f = xF
+xc
m+x
k
m0 = x
n mn2c = 2= n
k m
c = . 2km
+xn2 +x2,0 = xn
, teA = )t(x
+n2 + 220 = n
= 2,1 n2
n24 n2242
n n =n = 1 2
(
1 2
).
1.1 1 >
pxe A = )t(x(n(+
1 2
)t)
pxe B+(n(
1 2
)t)
)tn(pxe =[pxe A
(nt1 2
)pxe B+
(n
t1 2
])
= )(hnise e
2= )(hsoc ,
e + e
2,
)tn(pxe = )t(x[hsoc C
(nt1 2
)hnis D+
(n
t1 2
])
0v = )0(x 0x = )0(x
)tn(pxe = )t(x[hsoc 0x
(nt1 2
)+0xn + 0v
nhnis 1 2
(n
t1 2
]).
.
501 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
01 9 8 7 6 5 4 3 2 1 00
2.0
4.0
6.0
8.0
1
t
(X)t
0=0V,1=0X1=0V ,0=0X
3.1:
2.1 1 <
ni n = 2,1
2 1
)tn(pxe = )t(x[pxe A
(ni
t2 1)
pxe B+(ni
t2 1
]),
C B,A R )t(x
)(nis i + )(soc = )i( pxe
)tn(pxe = )t(x[soc )B+A(
(n
t2 1)
nis )BA(i +(n
t2 1]),
)BA(i = D ,B+A = C
R D,C B A.
n = d
,2 1
601 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
01 9 8 7 6 5 4 3 2 1 01
8.0
6.0
4.0
2.0
0
2.0
4.0
6.0
8.0
1
t
4.1:
. ])td( nis D+ )td( soc C[ )tn(pxe = )t(x
0v = )0(x ,0x = )0(x
)tn(pxe = )t(x[+ )td( soc 0x
0xn + 0vd
)td( nis
].
= hX
+ 02x
(0xn + 0v
d
2)= )(nat ,
0x0xn+0v
d
. ) +td( nis )tn(pxe hX = )t(x
tnemerced cimhtiragoL
, , 2.0 = . 2.0 =
= d
,n89.0 = n22.0 1
701 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
.d n x(t) .
x(t) = Xh [n exp(nt) sin (dt+ ) + d exp(nt) cos (dt+ )] = 0
tan (dt+ ) =dn
tk = t +
kpi
d, k = 1, 2, . . .
x(tk)
x(t) = Xh {n exp(nt) [n sin (dt+ ) + d cos (dt+ )]d exp(nt) [n cos (dt+ ) d sin (dt+ )]}
x(tk) = ndXh exp(ntk) cos (dtk + ) .
x(tk) = 0, x(tk) < 0
d =2pi
d=
2pi
n
1 2 .
i Ai
Ai = X0 exp(nti)
Ai+kAi
=X0 exp(ntj)X0 exp(nti)
= exp (n (ti+k ti))
= exp (n (kd)) = exp(nk 2pi
n
1 2
)= exp
(k 2pi
1 2
)
cReuven Segev and Lior Falach108
-
Theory of Vibartions
ln
ln
(Ai+kAi
)= 2pik
1 2 2pik
12pik
ln
(AiAi+k
)
0 1 2 3 4 5 6 7 8 9 101
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
t
x(t)
Ai
Ai+k
kd
:1.5
= 1 1.3
d 0 1
limd0
x(t) = limd0
exp(nt)[x0 cos (dt) +
v0 + nx0d
sin (dt)
]= exp(nt) lim
d0
[x0 cos (dt) +
v0 + nx0d
sin (dt)
]
limd0
[x0 cos (dt)] = x0
limd0
[v0 + nx0
dsin (dt)
]= (v0 + nx0) lim
d0
[sin (dt)
d
]= (v0 + nx0) lim
d0
[t cos (dt)
1
]= t (v0 + nx0)
cReuven Segev and Lior Falach 109
-
Theory of Vibartions
x(t) = exp(nt) [x0 + (v0 + nx0) t] .
. < 1
2
mx+ cx+ kx = F0 sin(t)
x+ 2nx+ 2nx =
F0m
sin(t) = G0 sin(t)
xP (t) = A sin(t) +B cos(t)
xp(t) = A cos(t) B sin(t)xp(t) = 2A sin(t) 2B cos(t),
[2A 2nB + 2nA] sin(t) + [2B + 2nA+ 2nB] cos(t) = G0 sin(t)
2A 2nB + 2nA = G02B + 2nA+ 2nB = 0
A =G0(2n 2
)(2n 2)2 + (2n)2
=
F0m
(km 2
)(km 2
)2+(cm)2 = F0
(k m2)
(k m2)2 + (c)2,
B = G02n(2n 2)2 + (2n)2
= F0m
cm(
km 2
)2+(cm)2 = F0c(k m2)2 + (c)2 .
cReuven Segev and Lior Falach 110
-
snoitrabiV fo yroehT
= pX= 2B+ 2A
0F2)c( + 2)2m k(
c = )(nat2m k
.) +t(nis pX = )t(px
, ) +td( nis )tn(pxe hX+ ) +t(nis pX = )t(x
,hX . )tn(pxe )t(px = )t(ssx
= pmA0F
2)c( + 2)2m k(=
(k/0F2)k/m( 1
2)+(ck
2)
= n2k
m,
c
mn2 =
c
k=m
k
c
m=
1
n2= n2
2
n
= pmA(k/0F
2)n/( 12)
24 +(n
. 2)
1.2
. )(
= )(S(2 n2
2))n2( +
2
Sd
d,0 = n228 +)2 n2( 4 =111 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
2 8.1 6.1 4.1 2.1 1 8.0 6.0 4.0 2.0 00
1
2
3
4
5
6
/n
mAp
1.0=2.0=3.0=4.0=5.0=6.0=7.0=8.0=9.0=1=
1.2:
,)22 1(n2 = 2
n = ,
22 1
n = r
22 1
. r
= pXk/0F
)22 1( 24 + 2))22 1( 1(=
k/0F
2
= 2 10Fcd
2
0F pXk
1
2.
1 > .2
3
)t(f T ) T + t(f = )t(f t, T , T1 = f fpi2 = .
k2 cd0F = pX.
n2mc k2 = 2 1
cn = 2 1
n2mc = cd = 2 12
211 hcalaF roiL dna vegeS nevueRc
-
Theory of Vibartions
3.1
sin(nt), cos(mt) = 2piT
T0
sin(nt) sin(mt)dt =
0 n 6= mT/2 n = m T
0cos(nt) cos(mt)dt =
0 n 6= mT/2 n = m T
0cos(nt) sin(mt)dt = 0.
f(t)
f(t) = a0 +n=1
[an sin(nt) + bn cos(nt)] ,
a0 =1
T
T0f(t)dt, an =
2
T
T0f(t) sin(nt)dt, bn =
2
T
T0f(t) cos(nt)dt.
mx+ cx+ kx = f(t) = a0 +n=1
[an sin(nt) + bn cos(nt)]
: 3.2
3.1
F0
t
f (t)
:3.1
cReuven Segev and Lior Falach 113
-
Theory of Vibartions
f(t) =F0t 0 t
, = 2pi
a0 =1
0f(t)dt =
F02
0tdt =
F02
an =2
0f(t) sin (nt) dt =
2F02
0t sin (nt) dt = 2F0
21
n
0td cos (nt)
= 2F02
1
n
{[t cos (nt)]0
0
cos (nt) dt
}= 2F0
21
n( cos (n))
= 2F0
1
2pin( cos (2pin)) = F0
pin
bn =2
0f(t) cos (nt) dt =
2F02
0t cos (nt) dt =
2F02
1
n
0td sin (nt)
=2F02
1
n
{[t sin (nt)]0
0
sin (nt) dt
}=
2F02
1
n( sin (n))
=2F0
1
2pin( sin (2pin)) = 0
0 0.5 1 1.5 2 2.5 3 3.5 40.5
0
0.5
1
1.5
2
2.5
3
3.5
t[sec]
f(t)
f(t)n=5n=10n=15n=20
:3.2
cReuven Segev and Lior Falach 114
-
Theory of Vibartions
3.3
:
mx+ cx+ kx = F0 sin(t)
xp(t) =F0 sin(t+ )
(k m2)2 + c22tan() = c
k m2 .
xp(t) =a0k
+n=1
an sin(nt+ n) + bn cos(nt+ n)(k m (n)2
)2+ n2c22
tan(n) = cnk mn22 .
xp(t) =a0k
+n=1
a2n + b2n(k m (n)2
)2+ n2c22
sin(nt+ n)
n = m + tan1( bnan
).
4
(t) =
0 t <
1 < t < +
0 t > +
f(t) . (t) (t) = lim0 (t)
0f(t) (t)dt = lim
0
[ 0f(t)0dt+
+
f(t)
dt+
+
f(t)0dt
]= lim
0
[1
+
f(t)dt
]= lim
0
[1
f(c)( + )
]for some c[ ,+].
= lim0 [f(c)] for some c[ ,+]
= f()
cReuven Segev and Lior Falach115
-
snoitrabiV fo yroehT
)t ,x( f
)t( Fa
1.4:
.
. 1.4
v2
2t2c =
v2
2x+)t ,x(f
)t ,x(f , a = x )x(a)t(0F = )t ,x(f
= )t(nTn + )t(nT1
L L xd)x(nX)t ,x(f 0
X 02xd)x(n
=1
L L xd)x(nX)x(a)t(0F 0
X 02xd)x(n
=)a(nX)t(0F
L
X 02xd)x(n
1.4 esnopser eslupmI
0 = t,
)t(00F = xk +xc +xm
611 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
= x0Fmc )t(0
mk x
m,x
+ )0(x = )t(x
t 0.td)t(x
0 = )0(x ,0 = )0(x, )(x xmil
mil0mil = )(x x
[+ )0(x
0td)t(x
]mil =
0
[0
(0Fmt(0
c )m+x
k
mx
)td]
=0Fm
0 = xk +xc +xm
= )0(x ,0 = )0(x0Fm,
tne = )t(x0Fdm
. )td( nis
)t( 0F = )t(f < t0 = )t(x = t
= )(x ,0 = )(x0Fm,
t
)t(ne = )t(x0Fdm
. )) t( d( nis
2.4
) ( , .
711 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
)1.4( )t(2f + )t(1f = xk +xc +xm
0 = )0(x ,0 = )0(x )t(1x
)2.4( )t(1f = xk +xc +xm
)t(2x
)3.4( )t(2f = xk +xc +xm
)t(2x + )t(1x 1.4 . )t(2x ,)t(1x . . )t(1x )t(2x 2.4 3.4 0x = )0(x 0v = )0(x )t(2x + )t(1x
1.4 0x2 = )0(2x + )0(1x 0v2 = )0(2x + )0(1x. 0v = )0(x ,0x = )0(x )t(0x )t(2x ,)t(1x )t(2x + )t(1x + )t(0x 1.4 . )t(0x
)tn( pxe = )t(0x[+ )td( soc 0x
0xn + 0vd
)td( nis
].
2.4. = t d)(f
0F
= )t(x
. > t )) t( d( nis dm/0F )) t( n( pxe < t 0 = t d)(f
= )t( x
d)(f < t 0dm
. > t )) t( d( nis )) t( n( pxe= )t(1x
)t( x t
-
Theory of Vibartions
f (t)
t
d
f ( )d
:4.2
x0(t) = exp (nt)[x0 cos (dt) +
v0 + nx0d
sin (dt)
]
x(t) =
t=0
f()
mdexp (n (t )) sin (d (t )) d Force responce
+ exp (nt)[x0 cos (dt) +
v0 + nx0d
sin (dt)
] Initial condition
4.3
f(t) =
F0 sin(t), 0 t pi0, t > pi t < pi
x(t) =
t=0
f()
mdexp (n (t )) sin (d (t )) d
=F0md
t=0
sin() exp (n (t )) sin (d (t )) d
cReuven Segev and Lior Falach 119
-
Theory of Vibartions
sin() sin() =1
2[cos( ) cos( + )]
sin() sin (d (t )) = 12
[cos (( + d) dt) cos (( d) + dt)]
x(t) = x1(t) x2(t)
x1(t) =F0
2md
t=0
exp (n (t )) cos (( + d) dt) d,
x2(t) =F0
2md
t=0
exp (n (t )) cos (( d) + dt) d.
t0
exp (a + b) cos(h + g) =a2
a2 + h2
[exp (a + b)
(cos (h + g)
a+h sin(h + g)
a2
)]=t=0
.
x1
a = n, b = nt, h = + d, g = dt
x1(t) =F0
2md
(n)2
(n)2 + ( + d)
2[exp (n (t ))
(cos (( + d) dt)
n+
( + d) sin (( + d) dt)(n)
2
)]=t=0
=F0
2md
(n)2
(n)2 + ( + d)
2[(cos (t)
n+
( + d) sin (t)
(n)2
) exp (nt)
(cos (dt)
n ( + d) sin (dt)
(n)2
)].
x2
a = n, b = nt, h = d, g = dt
cReuven Segev and Lior Falach 120
-
Theory of Vibartions
x2(t) =F0
2md
(n)2
(n)2 + ( d)2[
exp (n (t ))(
cos (( d) + dt)n
+( d) sin (( d) + dt)
(n)2
)]=t=0
=F0
2md
(n)2
(n)2 + ( d)2[(
cos (t)
n+
( d) sin (t)(n)
2
) exp (nt)
(cos (dt)
n+
( d) sin (dt)(n)
2
)].
t > pi
x(t) =
pi
=0
f()
mdexp (n (t )) sin (d (t )) d +
t=pi
f()
mdexp (n (t )) sin (d (t )) d
=
pi
=0
f()
mdexp (n (t )) sin (d (t )) d
x1(t) =F0
2md
(n)2
(n)2 + ( + d)
2[exp (n (t ))
(cos (( + d) dt)
n+
( + d) sin (( + d) dt)(n)
2
)]=pi=0
=F0
2md
(n)2
(n)2 + ( + d)
2[exp
(nt+ pin
)(cos(pi d
(t pi
))n
+( + d) sin
(pi d
(t pi
))(n)
2
)
exp (nt)(
cos (dt)
n ( + d) sin (dt)
(n)2
)]
.x2(t)
cReuven Segev and Lior Falach121
-
snoitrabiV fo yroehT
V
.
1 .
a2
x
1.1:
W .
221hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
a2
2N 1Nx
2N 1NW
2.1:
0 = W 2N+ 1N = yFx +a1N = 2N 0 = )x +a(1N)x a(2N = cM
x a
1N
(+ 1
x +a
x a)
x a W = 1N W =a2
W = 2N,x +a
a2.
x
xW = )2N 1N(a
xm =
=
W
ma=
g
a.
.
321hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
2 .
M
)t(nis 0F m
K
k
M
m
K
k
)t(nis 0Y
X
x
1.2:
[ 0 M
m 0
{]X
x
}+
[k k + Kk k
{]X
x
}=
{)t(nis 0F
0
}
)t(nis 2a = )t(x ,)t(nis 1a = )t(X
0F = )k(2a + )k + K+ 2M(1a0 = )k + 2m(2a + )k(1a
= sX0FK= 2 ,
K
M= 2 ,
k
m= ,
m
M
K k
1a
(+ 1
k
K
2
2
)k 2a
KsX =
2a + 1a(
12
2
)0 =
421 hcalaF roiL dna vegeS nevueRc
-
snoitrabiV fo yroehT
= 1a
(2 1
2
)(SX
2 12
() Kk + 1
2
2
)Kk
= 2a(SX
2 12
() Kk + 1
2
2
)Kk
=
0 = 1a
KSX = 2ak
0F =k.
= Kk
m M
k= m
K M
= 1a
(2 1
2
)(SX
2 12
()2 + 1
2
)
= 2a(SX
2 12
()2 + 1
2
)
( 1
2
2
() + 1
2
2
)=
(
4)(
2)0 = 1 + ) + 2(
(
2)=(
+ 1
2
)+
2
4.
2.0 = 52.1 ,8.0 = .
521hcalaF roiL dna vegeS nevueRc