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0 = l>
P=2
(3.1.104b)
The amplitudes of the nodal loads can be expressed as F
ii.*) = F> F
<:.,> = 0 -
F
(Lj)=0>
7*4
j = l,2,-,6
Introducing Eqs. (3.1.105) and (3.1.104) into Eq. (3.1.22), we have
(3.1.105a) (3.1.105b)
Bi-periodic Structures
/(i,m)=--7=siniy'F' /(2, m ) =0,
» = U,-,12
113
(3.1.106a)
m = l,2,-,12
(3.1.106b)
Substituting the relevant formulas, i.e., Eqs. (3.1.104), (3.1.106), (3.1.26), (3.1.94b), (3.1.97b) and (3.1.101)~(3.1.103), into Eq. (3.1.92a) with j = A, the amplitude response function for the loaded node can be found in explicit form as v(14)=//(Q)^
(3.1.107a)
where H(D.)
4
(^(1)-Q)(^(7)-Q) 2
3 (^,(1)-Q)[(^(7)-Q) -/: 2 (7)] + (/: i (7)-Q)[(/:,(l)-Q) 2 -K2(l)] tf,(3)-Q
(3.1.107b)
3[(/:,(3)-fi) 2 -/: 2 (3)]
and Kt(m) , K2(m) and Q have the same meanings as those in Eq. (3.1.77). The frequency response curve, v(14) versus Q , is plotted in Fig. 3.1.6. From Fig. 3.1.6, one can easily find that there are five resonance frequencies. When Q is close to these points, H(Q) will approach infinity. The equations of the resonance frequencies can be found by setting the two denominators on the right side of Eq. (3.1.107b) to be zero, that is (*, (1) - «)[(*, (7) - Q) 2 - K2 (7)] + (*, (7) - Cl)[(K{ (1) - Q) 2 - K2 (1)] = 0 (3.1.108a) (/sT,(3)-Q)2 -K2(3) = 0
(3.1.108b)
Recalling Eqs. (3.1.76), (3.1.78) and n=3, and inserting r=\ and m=3 into Eqs. (3.1.76) and (3.1.78), respectively, we can have Eqs. (3.1.108a) and (3.1.108b). The
roots
for Q in Eq. (3.1.108b) are —(11 + 6-72) and
the
roots
in
114
Exact A nalvsis of Bi-periodic Structures
10
H
0 o^r"
.
1
2
1.5
2.5^-
-5
/ -10
n Resonance frequencies: Q =0.3079247,0.3619367, 1.338599, 2.385953, 2.388870.
50 25 H
0
•J
0 •5
0.30 1
J
0.3S
^WO"
"
OA
-25 •50
Q 10
H
0 1. 38
J 1.3385
:
1000 500
H /
J
o
1.: 39
2. 18
-5
-500
-10
-1000
2.385
l
Frequency response curve,
2. »
( ,
ft
Figure 3.1.6
;
£1
H = v(14) — , fl = ol — r
A
Bi-periodic Structures
115
Eq. (3.1.108a) are 0.3619367, 1.338599, and 2.388870. These five resonance frequencies correspond to the symmetric modes of the considered truss. However, there are nine natural frequencies for the truss with «=3. From Eq. (3.1.107b), it can be shown that, when H(Q) approaches a finite value at one natural frequency, the harmonic force is acting at a nodal point of the corresponding mode. In the present case, the loaded node is located at the nodal point of the antisymmetric modes. F When Q approaches zero, v(14) approaches H(0) — , where K H(0)
=
4
* . ( ! ) * , (7)
3K^I)[K?(1)-K2(7)]+K,(7)[K?(1)-K2(\)]
,
*i(3)
=
3{K;Q)-K2(3)]
353 210
(3.1.109) This result is in agreement with the exact static displacement shown in Eq. (3.1.53a). It is well known that when Q approaches infinity, v(, 4) approaches zero. 3.2
Continuous Beam with Equidistant Roller and Spring Supports [10]
The static and dynamic analyses of the continuous beam with equidistant roller supports were investigated by using the U-transformation method [6], [5]. The static problem of the continuous beam having equidistant roller and spring supports was studied [10] where the structure is regarded as bi-periodic and U-transformation is applied twice. Consider a beam with uniform flexural rigidity EI running over N +1 number of roller supports and n elastic supports as shown in Fig. 3.2.1(a), where K denotes the stiffness of the elastic supports and / denotes the span length between any two adjacent roller supports. The distance between any two adjacent elastic supports is pi. It is assumed that each elastic support is located at midspan and N = pn , and a symmetric plane of the continuous beam exists, i.e., p must be an odd number. To form an equivalent system with cyclic bi-periodicity for the beam considered, it is necessary to extend the original beam by its symmetrical image and apply the antisymmetric loading on the corresponding extended part as shown in Fig. 3.2.1(b). Such an equivalent system can be regarded as a cyclic bi-periodic, because the slopes and moments at both extreme ends are the same, namely the first span can be imaginarily next to the last one (2iV-th see Fig. 3.2.1(b)). The simply supported boundary conditions at both extreme ends of the original beam can be satisfied automatically in its equivalent system.
116
Exact Analysis of Bi-periodic Structures
111 111 r«=—=»T*—&i
EI jo
w^i GT%
j»+q ; i)p
jo+(n-l)p
777"
777"
N
:K JTT
1
n
J p+\
N = np, j 0 = (a)
, p is odd number
Original System
symmetric line J»
J»+(Jhl)P
j°+(n-l)p
777"
777"
777"
J
n
N
i
N+l
"£" 1
jo+(2n-l)p
jo+np 777"
777"
777"
n+l
2n-j+l
2n
2N
a =a (b) Figure 3.2.1
3.2.1
Equivalent System
Continuous Beam with Bi-Periodic Supports
Governing Equation and Static Solution
In each span a local coordinate system oxy with the origin o at its midspan, and x axis along the beam, is established. The equilibrium equations for the equivalent system can be expressed as
Bi-periodic Structures
EId
EI
™jX) =FJ(x)-Kwj(0)S(x), ax ^X)=Fj(x), ax
j = j0J0+p,...,j0+(2n-l)p
j*jQ,j0+p,...J0+(2n-V)p,
;mdj = l,2,...,2N
117
(3.2.1a)
(3.2.1b)
where 7 0 = (p +1)/2 and 7 0 denotes the ordinal number of the span with the first elastic support; Wj(x) and Fj(x) denote deflection and loading functions for the 7-th span; S(x) denotes the Dirac delta function; 2N is the total number of spans of the equivalent system. The loading functions must satisfy the antisymmetric condition, i.e., F2N.J+t(x) = -Fj(-x),
j = l,2,...,N
(3.2.2)
in which Fj(x) (j = 1,2,..., N) is the real loading acting on the original beam. The constraint and continuity across the roller supports require the following conditions to be satisfied: w 7 ( - ~ ) = 0;
*;({) = < • . ; ( " ) ;
wj(^) = 0
y = 1,2
w
"j({) = ">"«(-{)
2JV
J = h2,...,2N
(3.2.3a)
(3.2.3b)
where a prime denotes differentiation with respect to x and w2JV+1 s w, due to the cyclic periodicity. Introducing the U-transformation w
Wj(x)
=7 = y e J2N ~x
qm(x)
= -=J^e-iU-^wj(x), \2N
\ ( 4
j = 12,...,2N
y=)
with y/ = JT/N into Eqs. (3.2.1) and (3.2.3) results in
j = l,2,...,2N
(3.2.4a)
(3.2.4b)
118
Exact Analysis of Bi-periodic Structures
EId
q x)
^ ax
=f°m(x)
+
f'm{x),
m = l,2,...,2N
(3.2.5)
=0
(3.2.6a)
and ? m (^) = 0; qm(~)
l'm({) = e"™rq'J-{); q"m({) = e^q"m(-L)
(3.2.6b)
where
1
2Ar
/ » = -^=£e-^-»""'F. (x)
(3.2.8)
If the loading condition is given, the generalized load f'm(x) can be found from Eq. (3.2.8). The formal solution for qm(x) of Eq. (3.2.5) subject to the boundary condition (3.2.6) can be expressed as om(x) = q°Jx) + q:(x)
(3.2.9)
where
q nix)=
°
~in2N%e
"y°Hk
,)p
~l]mv,w^»mc»°+c°»x+c»>x2+c^1+^M3> (3.2.10)
/ 3 .7-cosmu/\ -o=-^7^ -)! 384 2 + coswi^
c
. I2 , sinmi/ . »> =- T7h, —) 64 2 + cos/ny
c
l
,„„«,* (3.2.11a)
Bi-periodic Structures
/ ,5 + cosmw. cm2=~—(r-); 32 2 + coswi^
i , siamw
c
. —) 16 2 + cos/w^
119
,.,<,,,,* (3.2.11b)
m3=77(^
and q'm(x) represents the generalized displacement for the cyclic periodic system with K = 0 subjected to the same loading as that acting on the equivalent system. Substituting Eqs. (3.2.9) -(3.2.11) into the U-transformation (3.2.4a) yields wj(x) = w°j(x) + w'j(x)
(3.2.12)
in which W i(x)
°
=
K
2n 2N
e JJ
-^%T " ' '~{k~l)P^wJ^-nP(°^o
^mlx+cm2x2
3
+cmix
1 +±\tf)
(3.2.13a) and w'j(x) =
1
2N
* jy'"0""^^)
(3.2.13b)
-J2N ~ w'j(x) represents the deflection function of they'-th span for the equivalent system with K vanishing under the same loading, while w°j(x) account for effect of the elastic supports. The latter depends on the deflections at the elastic supports. Inserting j = j 0 + (s - \)p and x = 0 into Eqs. (3.2.12) and (3.2.13a) gives
w
> = "77IX*^* + w'>
s = l 2 2n
> --
(3214)
where i
2N
6 P>* = T^X '^""""^ 2N 'rl m—i
and
S
>k = 1A".,2».
(3.2.15)
120
Exact Analysis ofBi-periodic Structures
Ws = * W » , ( 0 ) ;
K = W'J.H.-I)P(P)
s = l,2,-..,2«
(3.2.16a,b)
It is obvious that J3sk (s,k = 1,2,...,2«) possesses the cyclic periodicity. The simultaneous equations (3.2.14) with unknown
Ws (j -1,2,...,2n)
can be
uncoupled by using the U-transformation. Let W,=-F=Yetl-l)"Q, V2/J
1
s=l,2,...,2n
(3.2.17a)
r = l,2,...,2n
(3.2.17b)
irr
2
"
Qr=-==Ye-i{s-x)r9Ws V2« j ^
with (p = n\n- py/. Eq. (3.2.14) can be expressed in terms of Qr (r = 1,2,...,2n) as Qr=~^ifiktle-,lk-i^Qr+br ^'*=,
r = l,2,...,2n
(3.2.18)
in which 1
2N
0 6 A^^Z^" "' -" 2W -1
(3219)
m
and 1
2
"
fef=-^ye-'(s-1)r^; V2n ,=i
(3.2.20)
Introducing Eq. (3.2.19) into Eq. (3.2.18) results in K 1 Qr = - ^ 7 - i > ^ 1 ) 2 „ i 0 a + br fiy
Z7 t=i
(3.2.21)
Bi-periodic Structures
where c r+(jfc _ 1)2n
0
121
has been defined in Eq. (3.2.11a), namely
v-co^-Kft-D&Jjr,
C
r+(*-l)2n,0 ~ 2S4
(3
^ 2 + cos[r + (A: - l)2n]^
The solution for Qr of Eq. (3.2.21) can be obtained as
r
r
/ [
384£7/>£f 2 + cos[r + (*-l)2wfo/j
in which br shown in Eq. (3.2.20) depends on the loading condition. When the specific load is given, br can be found without difficulty. 3.2.2
Example
Consider a concentrated load of magnitude P acting at the midpoint of the middle span, say kQ -th span [i.e., k0=(N +1)/2 and N is odd number], as shown in Fig. 3.2.1(a) where j = k0. For the equivalent system, equal and opposite concentrated loads must be applied to the k0 -th and (2N - k0 +1) -th spans with all other spans unloaded. The loading condition can be expressed as Fko(x) = PS(x); Fj(x) = 0
F2W(x)
j * k0,2N-k0
= -PS(x)
(3.2.24a)
+l,j = \,2,...,2N
(3.2.24b)
m = \,3,.-,2N-l
(3.2.25a)
where k0 =(N +1)/2 and N is odd number. Inserting Eq. (3.2.24) into Eq. (3.2.8) yields /J;(x) = - ^ f f i e - , ' ( V , ) m , , ' •J2N / ; ( * ) = <> with y/ = njN.
m = 2,4,..,2N
(3.2.25b)
122
Exact Analysis of Bi-periodic Structures
Noting that q'm(x) represents the solution for qm(x) f°(x)
of Eq. (3.2.5) with
vanishing subject to the boundary condition (3.2.6) and f'm(x)
is shown in
Eq. (3.2.25), q'm(x) can be found as -i(k -\)mw
2Pe 0 ^Ux) = ——j=^(cm0+cmix
+ cm2x2+cm3x3+—\x\
)
w = l,3,...,2iV-l q'm(x) = 0
(3.2.26a)
m = 2,4,...,2N
(3.2.26b)
where cm0~cmi have the same definition as those shown in Eq. (3.2.11). Introducing Eq. (3.2.26) into Eq. (3.2.13b) yields <
W
=
^
eiU-k°)mY(cm0+cmlx
£
+
cm2x2 +c M3 * 3 + ^ H 3 )
"•=1.3,5
j = \,2,...,2N Recalling
W's = Wh+(j.1)p(0),
k0=(N+
(3.2.27)
l)/2,
and
j0=(p
+ l)/2,
inserting
j = j 0 +(s- \)p and x = 0 into Eq. (3.2.27) gives n
2N-1
..
W! = — Y e-'m*/2e J EIN 4*.
1.
2
cm0
(3.2.28)
m
m=l,3,5
and then substituting Eq. (3.2.28) into Eq. (3.2.20) we have *' = | r ^ = e ' r ( ^ ) / 2 ^ c - ( ^ » . o £^P V2n T~f br=0
r=2,4,...,2«
r = \X.-,2n-\
(3.2.29a) (3.2.29b)
Since N, P are odd numbers, therefore, n(= N/p) is also odd. This property has been used in deriving Eq. (3.2.29a). Afterward inserting Eqs. (3.2.29) and (3.2.22) into Eq. (3.2.23) results in
Bi-periodic Structures
Qr=— ^=eir(
r = l,3,...,2«-l
r = 2,A,...,2n
123
(3.2.30a) V
;
(3.2.30b)
where
^ ^ 2 + cos[r + (A;-l)2nV K7 3
K 00 = — — 384£7
(3.2.32)
and ^ =^ , AT
p = ^n
(3.2.33)
K0 is a nondimensional parameter of the stiffness for the elastic support. Substituting Eq. (3.2.30) into Eq. (3.2.17a) yields
*Wo,
(3.2.34)
Now the deflection function for each span can be found by introducing Eqs. (3.2.27), (3.2.13a) and (3.2.34) into Eq. (3.2.12). The maximum deflection occurs at the midpoint of the loaded span. The maximum deflection can be obtained by inserting s = (n +1)/2 in Eq. (3.2.34) as
wmm = V . ) / 2 ( 0 ) ~ - I ^ / ( 1 + ^ o ^ ) ]
(3-2.35)
r=l,3
in which Hr shown in Eq. (3.2.31) depends on p and n or N. Noting N = np, Eq. (3.2.35) includes two independent parameters p and n or N besides K0 . Some
124
Exact A nalysis of Bi-periodic Structures
numerical results for Eq. (3.2.35) are given in Table 3.2.1 where p = 3 and total number of spans N and the nondimensional stiffness K0 take several values, respectively. Consider the particular case of K0 = 0. By substituting K0 = 0 and Eq. (3.2.31) into Eq. (3.2.35), the maximum deflection can be expressed as
w„
\j& ^pl-cos[r + {k-\)2n PI' ip £&k-(2 + cos[r + {k- l)2n 384£7 np PI' 1 ^^-i ' 7 /- -c ocos s m my/y 384£7 N ^-?,2 + cos mw m=l,3
(3.2.36)
'
The preceding result is in agreement with that obtained by Cheung et al [6]. Table 3.2.1
N(n)
Maximum Deflections of Continuous Beams with Bi-periodic Supports Subjected to Concentrated Load P at Midpoint (p = 3) 3(1)
9(3)
15(5)
21(7)
0.0
4.40000
4.19623
4.19615
4.19615
0.1
3.05556
2.95515
2.95514
2.95514
0.2
2.34043
2.28083
2.28083
2.28083
0.5
1.37500
1.35410
1.35410
1.35410
1.0
0.81481
0.80741
0.80741
0.80741
2.0
0.44898
0.44672
0.44672
0.44672
*o
Multiplier
PP/3S4EI
Chapter 4 STRUCTURES WITH BI-PERIODICITY IN TWO DIRECTIONS Cable and beam networks can be regarded as typical structures with periodicity in two directions. The static analyses of rectangular single and double layer grids were investigated by Chan et al [13][14] using the double U-transformation technique. The natural vibration and dynamic response analyses of cable and beam networks were studied by Cheung et al [15][16] employing the same method. A rectangular plate with uniform finite element meshes can also be regarded as a bi-directional periodic structure. Such a structure may be analyzed by means of the double U-transformation. The exact solution of the finite element equation for simply supported square plates was derived by Chan et al [17]. Recently the double U-transformation was successfully applied to the analysis of bi-directional bi-periodic structures. The static and vibration analyses of rectangular beam and cable networks with periodically distributed supports along two directions were performed by Chan et al [18] [19] using the double Utransformation twice. The static analysis of rectangular grids with periodic stiffening beams is investigated in this book applying the same technique. 4.1
Cable Networks with Periodic Supports
The network considered is made up of two sets of pretensioned straight cables orthogonal to each other with fixed ends, meeting at spot-welded nodes and supported by periodically distributed posts. For generality, consider an n,xn2 rectangular network with fixed ends at four edges as shown in Fig. 4.1.1 where the solid circles denote the nodes supported by posts. There are (m, -1) x (m2 -1) internal supports. The equivalent network with cyclic periodicity in x- and ^-directions can be produced by using image method [15][16]. At the outset, consider the extended network with 2«,x2n2 mesh shown in Fig. 4.1.2 where the loading pattern is antisymmetric with respect to two symmetric planes of the extended network. Moreover we regard the extended network as one having cyclic bi-periodicity in x- and ydirections, i.e., each pair of nodes (0, k) and (2«„ k) (h=0,l,2,...,2n2 ) and (j,0 ) and (j, 2n2) (j/=0,l,2,...,2n1 ) may be imaginarily put together and treated as one point in mathematics. The boundary conditions of die original system can be satisfied automatically in its equivalent system where the additional supports located at boundary and symmetric lines are necessary in order to form the cyclic bi-periodic
125
126
Exact Analysis ofBi-periodic Structures
system, but their supporting reactions are identically equal to zero.
0
1 (0,0)
p2
k
(m2-l)P2
M
2 (0,n2)
(0,k)
x-cable
n
y-cable (p„(m2-l)p2)
(P1.P2)
a.k)
0.0)
0.n2)
(m,-l)p, ((m,-l)p„p2) n,-l
(n„0)
(n„k)
wwwwww\ww WWWWWWWWY WWWWWVWWW «i
Figure 4.1.1
=mlpi
n2 = mlPl
«, x «2 network with (w, -1) x (m2 -1) supports
(ni,n2)
Structures with Bi-periodicity in Two Directions
127
-> y
I --. p . —
0 ft
t
k--(m -l)p.ft !
;m; p. - - - (m,*- I ) p . - 2 n j - k - - ( m.-Dpj
2
I 1 1
p, i
9
1 1 1
0.0)
j . 2n r k
(W
1
fi.2n,l
-p
p
1 1
|
t
1 1 1 1 1
*.«>-•
(m.214)
mi Pi
1
<
I
i i i i
11
i i
(2«rJ. 2n,-k)
2n,-j.k)
-P
i i i
P
i I
2n, ,
, .k>
n,=m, p,,
Figure 4.1.2
1 *
n,=m;p.
* t '<2n,.
n.)
1
V
(j.0MJ.2n,) (0, k)s(2n„ k)
(2n, 2tij)
j=0.l.2,--, 2n, k=0,l,2,-,2n,
Equivalent network with 2M, X 2n2 mesh and cyclic periodicity in x- and y-directions
128
Exact Analysis of Bi-periodic Structures
4.1.1
Static Solution
In order to use die periodicity, it is necessary that the supports should be replaced by the supporting reactions. The equilibrium equations for all nodes in the equivalent system can be expressed as (2KX + 2K2)w{Jk) -K x (w (j+ik) + w(J_xk)){j,k) = (jxpx,kxp2),
Jf,(w 0>ttl) + w (M _ 0 ) = F ( M ) + PUM
jx=l,2,...,2mx,
£,=l,2,...,2/n2
(4.1.1a)
(2KX + 2K2 )wUk) - Kx (w w+U) + w(,._U)) - K2 (w (M+1) + w{Jk_x)) = F (yjt) (j,k) * ti\P\, kiPz), 7i=l,2,...,2ffj„ Kx^,
kx=l,2,...,2m2
K2=\ a
(4.1.1b) (4.1.2)
b
where vv01) and F 0 A) denote the transverse displacement and loading of node (/, A) respectively; (/iPi» kxp2),jx=l,2,...,2mu kx=l,2,...,2m2represent the nodal numbers for the nodes supported by posts (see Fig. 4.1.2); P{hM represents the supporting reaction at node (jxpx, kj>2); Tx, T2 denote the pretensions of the cables in the * and y directions respectively and a, b denote the spacing of y- and ^-cables. The solution for w^ of Eq. (4.1.1) must be subjected to the restrained condition at supports, i.e., W
OAM>=0>
7=1.2,...,2w,;
*=l,2,...,2m2
(4.1.3)
In Eq. (4.1.1) die loading must be anti-symmetric and the supporting reaction can be determined by using Eq. (4.1.3). Eqs. (4.1.1) and (4.1.3) possess cyclic periodicity but the two periods are different. The cyclic periodic equation (4.1.1) subject to the cyclic periodic condition (4.1.3) is also cyclic bi-periodic. The solution for w(;-1) can be derived by applying the double U-transformation twice. Introducing die double U-transformation w
.k =
I V V g'O-O-r, eHk-\)s¥2 q ^ ^ y2n, iJ2n2 r=x s=x j=l,2,...,2nx;
k=\,2,...,2n2
(4.1.4a)
Structures with Bi-periodicity in Two Directions
129
and its inverse transformation
=
a
i
lih.
i
V V g-'U-Vrr, -Hk-i),y,,
2«i
2n,fin^M k=l r=l,2,...,2n,;
s=l,2,...,2«2
(4.1.4b)
with i//l = n/nx,
y/2=iiln2,
i=v-l,
nx=m^px,n2=m1p2.
(4.1.5)
in Eq. (4.1.1), the equilibrium equation (4.1.1) can be expressed in terms of the generalized displacement <7M as {2KX + 2K2 -2KX cosry/x -2K2 smry/2)q{rs) r=l,2,...,2nl;
= f*rs)
+f°s)
s=l,2,...,2n2
(4.1.6)
Y^e-^e-'^F^
(4.1.7a)
where /(;,s)
= ^
2m, 2w2
1
f
™
=
g
PTr-EI "" ""^"(Vr"',ipo-„»,) ^/2nj ,j2n2
M
y1=1
<4-1-7b>
t|= i
in which / / > and / ( V represent the generalized load and supporting reaction respectively. Noting the anti-symmetric properties for the loading, the supporting reaction is also anti-symmetric about two symmetric planes of the extended network. As a result 1
^2n - "t l
2n ^ " 22
> /2n 1> /2n 2 M k=l
130
Exact Analysis ofBi-periodic Structures
V 2 n i V 2 n 2 ;,-i *,-i
The solution for ^(M) of Eq. (4.1.6) may be formally expressed as 1(rj) = 9(%> + 4W)>
7=l,2,...,2n,; 5=l,2,...,2n2
(4.1.9)
where /"°
?£ ,i = (r,s)
— > 2/i: i (l-cosr^ 1 ) + 2 ^ 2 ( l - c o s s ^ 2 ) C)=0'
(r>s) * (2"i> 2 « 2 )
(r,s) = (2nlt2m)
(4.1.10a)
and /*, 9(*r 5)
=
~
)
( r > s ) * ( 2 « u 2n 2 )
2A", (1 - cos r y/x) + 2K2 (1 - coss y/2) l'(r,s)=0>
(r,s) = (2nu2n2)
(4.1.10b)
Substituting Eqs. (4.1.9), (4.1.10) and (4.1.7) into Eq. (4.1.4a) results in
and 2n,
a.*) U-
W
where
=
2n:
2-i 2-i PuMiPi,k,Pi)PUiA)
(4.1.12b)
Structures with Bi-periodicity in Two Directions
131
w°Jk) and w(*y>t) represent the displacements caused by the supporting reaction and loading for the equivalent network without supports. Noting that the loading pattern needs to satisfy the anti-symmetric condition, i.e., F
(2ni-j,k) = FU,2n2-k) = ~FU,k)
F
(2„l-j,2„1-k)=Fu,k)> 7=l>2,-,«i;
*=l,2,...,fi2
(4.1.14)
the Eq. (4.1.12a) becomes Hi-l n . - l n , - l « , - l
.
W
U,k) =
•
.
-
|
.
.
.
»
i i, i , i s m / r sm / y V" y yV~' sinyr^, ^ ^ i s msmksy/ " ^ 22s m v ismj ^liry/ ^ ylsi// 22 F 4 y \~'\~ lsmk
(4115)
Obviously, w'uk) shown in Eq. (4.1.15) satisfies the anti-symmetric condition and boundary condition of the original network, i.e.,
w
(V;,2n2-*) = w'u,k)>
/=1A-,»I;
(4.1.16)
*=I,2,...,B 2
and w'(Jk)=0,
j = 0,nl
or k = 0,n2
(4.1.17)
Introducing the notations W
U,k) s wO>,M) ' K,k)
s w
°UPlM)'
W
ik)
s W
*(JP,M)
•
(4-L18)
and substituting Eqs. (4.1.11), (4.1.12) into Eq. (4.1.3) yields w
u*)sWu*)+wu*)=0>
y'=l>2,..,2mi;
*=l,2,...,2iw2
(4.1.19)
132
Exact Analysis of Bi-periodic Structures
where 2m,
2m2
( 4 l - 2 °)
c=EZ^w^i and "(J'.*)(;'I>*I)
"(jpi.kpMPiAPi 2
1
^LZ'
=—— y 4rc,n2 7 ^
2
^iZ'
JU-hY9\
ei(k-kx)s
y ^ 2 A ' 1 ( l - c o s r ^ ' 1 ) + 2A' 2 (l-cos5^ 2 )
(4.1.21)
w i t h (px = n I rri\,
i=l,2,...,2m„ Ay,*X/„l)
=
P(jMWa)
-•-
=
/>(ux2 B i 1 -j+2,*,)
= ,,- =
^* 1 =U,...,2»ij
A;-I,*)(2«I.*I)
(4.1.22a)
=fi(j,2m1)(jl,2m1-k+\)= /*0\l)0„2m2-i+2) = • " = ^(y,*-l)(;,,2i»,)
j,j\ = l,2,-,2m»
k=l,2,-,2m2
In Eq. (4.1.19), the independent coefficients are /? 0jt)(U )
(4.1.22b) (/'=l,2,...,2m„
&=l,2,...,27n2). Eq. (4.1.22) indicates that Eq. (4.1.19) possesses cyclic periodicity and it can be uncoupled by using the double U-transformation. Let P r U.k)
=.
, v
y y c'"-1^ e«*-"'* a ,
\2m^2m2
r=i J=1
2mj
;=l,2,...,2/n„
2m2
*=l,2,...,2m2
(4.1.23a)
Structures with Bi-periodicity in Two Directions
133
or '(M)i-9i
yj2miyj2m2
0-Hk-l)'V
Pn U.*)
M k=x
r=l,2,...,2ml,
(4.1.23b)
s=l,2,...,2m2 1
Premultiplying Eq. (4.1.19) by the operator
/2/nlA/2m2
2mj 2m2 (J-Orw
-((t-l)ift
J=, t
and noting Eq. (4.1.22), we have 2BIJ 2m2 r V V p-'f"-')'-?', _-f(v-i)j0>j o -i^ LZjZj P(.,VX1,1)JW(M) -
L °(-v)
u=l v=l
=l,2,...,2iw,,
(4.1.24)
5 = 1 , 2 , ...,2m2
where
V,*) = and p.
v)(1,)
2wij
1 V
2w
2ffi2
-i(y-l)rp,
2m
-i(i-l)sp2 m > U,k)
i V 2 ;=i *=i
w,
(4.1.25)
can be obtained from Eq. (4.1.21) as 2n,-l
2n2-l
e i ( u - l ) r p , e i(v-l)sf!> z
A..VXM) - 44»i" „„ E 2-< 2 7 ^ •^f2A^ 1 (l-cosr^ 1 ) + 2AT 2 (l-cos5^ 2 )
(4.1.26)
Substituting the above equation into Eq. (4.1.24) results in
G(,„=-
•V.')
=l,2,...,2w,;5=l,2,...,2m2
(4.1.27)
where A(r,s) =-*—2$ii{2Kl P\Pl
+2K2 -2^,cos[r + 0 - l ) 2 W l M -2/: 2 cos[s + (A:-l)2m2]^2}-1
j=\ t=l
(4.1.28)
134
Exact Analysis of Bi-periodic Structures
Obviously, A^ is dependent on the structural parameters only and b{rs) IS dependent on the loading condition besides structural parameters. If the specific loads and structural parameters are given, the generalized reaction Q^ can be calculated and the supporting reaction P W ) can be found by substituting Q{rf) into Eq. (4.1.23a). Finally the transverse displacement for each node can be obtained from Eqs. (4.1.11), (4.1.12b) and (4.1.15) in which P0k) can be found as indicated above. The following example will show how this can be done. 4.1.1a
Example
Consider a uniform square network with 6x6 mesh and 2x2 supports subjected to a concentrated load of magnitude P acting at the center node as shown in Fig. 4.1.3. (0,0)
(0,2)
(0,1)
(0,3)
(0,4)
(0,5)
(0,6)
(1,0)
(2,2)
(2,0)
4
1
>
(2,4)
P (3,3)
(3,0)
(4,0)
1
)
(3,6)
1
(4,4)
(4,2)
a (5,0)
^
a _^
(6,0)
(6,6) (6,3)
Figure 4.1.3
6 x 6 network with 2 x 2 supports
Structures with Bi-periodicity in Two Directions
135
For the considered network, the specific parameters and loading may be written down as: «,=n2=6, pl=p2=2,
m,=m2=3,
y , = y 2 = —,
^=
6
a=b,
(4.1.29a) 5
TX=T2=T, Kt=K2=K
(4.1.29b)
and FM=P,
(/,*W3,3),
FW=0,
j,k = l,2,..,5
(4.1.30)
Inserting Eqs. (4.1.29) and (4.1.30) into Eq. (4.1.15) yields . n
s
. ,
.
K
7t .
7i
5 smjr — sinks — srnr — sins — P
2
WM,4SS
\
^•=»
2AT(2-cosr
S
>^ ^ A - . "
(4.1-31)
coss-) 6
6'
The nodal displacements caused by external loading without supporting reaction can be calculated from Eq. (4.1.31). The results are as follows: 3P w(u) = w(*Si5) = W.,5) = w' w) = j
^
(4.1.32a)
U W
W(*,,2) =
W
(2,l) =
*
W
0,3) = W(3,l) = ^3.5) = W(*5,3) = ^
(4.1.32b)
(M) = W(*4,l) = W(*2,5) = W(*5)2) = W*45) = W*j>4) = ^ p
*
*
W(2,3) = W(3,2) =
W
*
^.P
(4,3) = W(3i4) = —
* ,
23.P
W(3i3) = —
(4.1.32c)
.
..,,
(4.1.32d)
E>
W
(*2,2) = W(*4,4) = W(*2,4) = W(*4,2) = ~
(4.1.32e)
oA
w*, t ) =0,
7=6,12 (or 0) or A=6,12(or0)
(4.1.32f)
136
Exact Analysis of Bi-periodic Structures
and w
ln-m-k)=w'u,k)
W
(4.1.32g)
= w
0M2-t) = -wa,k)>
02-M>
>» *=1,2,...,6
(4.1.32h)
According to the definition shown in Eq. (4.1.18), the following can be written down directly < » = <,» = Ka) = »&.) = ^7
(4.1.33a)
fF ( * t ) =0,
(4.1.33b)
y=3,6
or £=3,6
and K-j,*-k)=K.k)
( 4 - 1 - 33c )
K-m = wL*-k) = -K» > J> *=!. 2
( 4 - 133d >
Noting that ^ ( * t ) has anti-symmetric property as shown in Eq. (4.1.16), Eq. (4.1.25) may be rewritten as
y2ffj, y2w 2 y=1 *=1 r=l,2,...,2m,;
s=l,2,...,2/n2
(4.1.34)
Substituting Eqs. (4.1.29) and (4.1.33) into the above equation results in
'(1,1) :"
(l-iS)P SK
, >
°(5.5)-
0(1,5) - 0(5,1) -
4 K
< I ± i ^
(4.1.35a)
(4.1.35b)
Structures with Bi-periodicity in Two Directions
b(r^=0,
m\,5
or s*l,5
137
(4.1.35c)
For the case under consideration as given in Eq. (4.1.29), then Eq. (4.1.28) becomes 2
4
2
6
"l£r
6
r,s=l,2,...,6
(4.1.36)
The useful results are =
-4(1,1)
-4(5,5) = -4(1,5) = -4(5,1) = —
(4.1.37)
Now the generalized reaction QM can be obtained by substituting Eqs. (4.1.35) and (4.1.37) into Eq. (4.1.27) as 2(,,,)=
l-j'VJ —^,
G(5.s)=
l + i'VJ — ^
e(,,5)=e(5,)=-|^ 0 M =O,
r*l,5 or s*l,5
(4.1.38a)
(4.i.38b) (4.1.38c)
For the present case, the double U-transformation (4.1.23a) becomes l ^ r - i
i(y-l)r-
i(i-l)j-
"r=l,5j=l,5
The supporting reaction P W) can be found by inserting Eq. (4.1.38) into Eq. (4.1.39) as [
and
(1,1) -
^ri.«=-Pan=^2.2)=~ ' (1,2) (2,1) ~ (2,2) 1
1
(4.1.40a)
138
Exact Analysis of Bi-periodic Structures
P{4,4) = ^(4,5) = ^(5,4) = ^5.5) = ~
(4.L40b)
P MM)
=
M4,l)
=
*0.5)
=
*(5,>)
=
-^(2,4)
=
M4,2)
=
'(2.5)
=
"(5,2)
=
T"
(4.1.4()C)
with the other P^ vanishing. The supporting reactions Pyk) (/', #=1,2) are actual ones for the network under consideration shown in Fig. 4.1.3. The nodal displacements w°jk) caused by the supporting reactions for the system without any supports can be calculated by substituting Eqs. (4.1.29), (4.1.40) and (4.1.13) into Eq. (4.1.12b). The results are summarized as follows:
0
0
<.,.) = <5,5) = <., 5) = < . ) = " —
(4.1.41a)
< 3 , = w<°.» = W
(4.1.41b)
0
0
0
0
^
0
0
W(l,2) = W(2,l) = W(l,4) = W(4.1) = W(2,5) = W(5,2) = ^(4,5) = ^(5,4) = ~
<2,3) = <3,2) = < 3 ) = <3,4) = "
— ,
^3,3) = "
/A i
~
(4.1.41c)
A*
\
(4.1.41d)
P
W0
( 2,2) = W(°4,4) = <2,4) = W(°4,2) = ~ 7 T 7
w.M ) = 0 ,
—
*
y=6,12 (or 0)
or *=6,12(or0)
(4.1.41e)
(4.1.41f)
and
<.2- M ) = < « - * ) = -<•,*). 7. A=l,2,..,6
(4.1.41h)
Finally the real displacements w ^ for the original system can be easily obtained by superimposing w°M) on w'UJc), giving
Structures with Bi-periodicity in Two Directions
W(..D = w ( W ) = w(1,5) = w(5,.) = ^
w(U) = w(3,i) = w(3,s) = w (5i3) = ^
w
7
(4-1 -42a)
7P :
(4.1.42b)
d,2) = w(2,i) = w(i,4) = W (4,D = w ( W ) = w(5>2) = w(4>5) = w (Ji4) = ^ ^ r 6P W(2.3) = W (3,2, = ^(4,3) = W (3i4) = —
W
139
(4.1.42c)
89P ,
W(33) = - ^ ^
(2,2) = W(4,4) = W(2,4) = W(4,2) = 0
.... (4.1.42d)
(4.1.42e)
and the displacements for the nodes on the boundary of the original network are equal to zero. Obviously the displacement solution shown in Eq. (4.1.42) satisfies the restrained condition shown in Eq. (4.1.3) besides the boundary condition. It can be verified that the displacements shown in Eq. (4.1.42) and supporting reactions shown in Eq. (4.1.40) satisfy the nodal equilibrium equations. The results also demonstrate the exactness of the formulas in this section. 4.1.2
Natural Vibration [19]
The rectangular network shown in Fig. 4.1.1 with lumped mass M at each node is considered. The natural vibration equation can be obtained from Eq. (4.1.1) where the nodal load F(jk) should be replaced by the inertial force Mo 2 w ( j , k ) , i.e., (2A-, + 2K2 -Mw2)wuk)
-Kx{wu+Vk)
U,k) = UiPi,k1p2) {IK, + 2K2 -Mco2)wUM
+ w(Mk))-K2(wUk+l)
jl=l,2,...,2ml,kl=l,2,...,2m2
- * , ( w ( y + l i t ) + w ( , _ U ) ) - K2(w(M+1)
U,k)*<JiPi,KPi)
+yvUJc_l)) = PUM
Ji=l,2,-,2mu
j = \,2,-,2nx,
k = l,2,...,2n2
(4.1.43a) + w(M_„) = 0
k1=\,2,...,2m2 (4.1.43b)
140
Exact Analysis of Bi-periodic Structures
where a
denotes the vibrational frequency;
w{jk)
and Py
k)
denote the
amplitude of transverse displacement and the support reaction respectively. Applying the double U-transformation (4.1.4) to Eq. (4.1.43), we have (2K{ + 2K2 -Ma>2 -2KX cosry/x -2K2 cossif/2)q{rj) r = l,2,...,2«,,
= /(°s)
s = l,2,...,2n2
(4.1.44)
where f°rs) is given in Eq. (4.1.7b). <7()v) can be expressed as a
(rs)
,o itfl 2Kl+2K2-Mo)2-2K{cosry/l-2K2cossy/2
=
(4 i 45)
Substituting Eqs. (4.1.45) and (4.1.7b) into the double U-transformation (4.1.4a) we have w
u.k) ~ 2-i2-iP(iMhPuKP2)PUxM
(4.1.46)
in which 2»H
0, ,( v)
* "'
2nj-l
e'(j-u)riin
eHk-v)sy/,
j^2K]+2K2-Mw2-2Klcosry/]-2K2cossy/2
4n,n2 j-f j,u=l,2,...,2nl,
k,v = l,2,...,2n2
(4.1.47)
Introducing Eq. (4.1.46) into the restrained condition (4.1.3) gives
2L2LPu*yUMPUM = ° J = 1 A...,2«„ k = l,2,...,2m2 and
(4.1.48)
Structures with Bi-periodicity in Two Directions
=—— y 4M M I 2 T\
y j-f2Kx+2K2-M
e
——
141
(4.1.49) -2Kxcosry/x-2K2cossy/2
with
s =l,2,...,2m2
(4.1.50)
where 2ffi|
\rA
2w2
e
= XX "' ( "" 1)r " e "''
(4-1-51)
and
/>(»,v)(U) -
4n^
2-t j-f2Kx
+ 2K2 -Mco2 - 2KX cosr^, - 2K2 c o s ^ 2 (4.1.52)
Inserting Eq. (4.1.52) into Eq. (4.1.51) results in A,r s, =
V y {2/:, + 2K2 -Ma2 - 2/sT, cos[r + (J - 1)2™,]^, - 2K2 cos[s + {k~ l)2m2]y2}~' (4.1.53)
Eq. (4.1.50) is made up of single degree of freedom equations. Let us consider the property of g ( r s ) (r = \,2,...,2mx;s = l,2,...,2m2). For the equivalent system, the supporting reactions P^jk) symmetric condition, i.e.,
(j = l,2,...,2mx; k = l,2,...,2m2) must satisfy the anti-
142
Exact Analysis OfBi-periodic Structures P{2m,-j,2«2-j) = pu,k) W M ) = Pua^-k)
7 = l,2,..,/n,-l;
= -p(m
* = 1,2,...,™2 - 1
J = 1 ' 2 >-> ffl > " I ; * = 1.2
w2 - 1
(4.1.54a) (4.1.54b)
and P(jh)
= 0 j = m1,2ml
or k = m2,2m2
(4.1.54c)
Substituting Eq. (4.1.54) into Eq. (4.1.23b) yields m,-l m2-l
4e'rip' e"92 -Jp ^ H 2 ( ^) = —; , > > sinyr^, sinfa^ 2 P ( y t ) y2/M, •yj2m2 J=1 t=1
(4.1.55)
That leads to 2(r,j>
=
0 r = mu2ml
or s=m2,2m2
(4.1.56)
and the other Q (r s) having the complex factor e'r9i e"92 . When the generalized supporting reactions QM are not identically equal to zero, the independent frequency equation is A
{r,s)=°
r = \,2,-,mx-\;s
= \,2,-,m2-\
(4.1.57)
and A ^ ^ is the function of CO as shown in Eq. (4.1.53). Consider n o w the natural mode. From the definition of A ( r s ) shown in Eq. (4.1.53), it can be verified that A ( r s ) = A(2mi_,tS) * A(ra„2_s) s A(2m<_ram2_s). Corresponding to each natural frequency satisfying AM=0, four generalized c a n b e e u a l to supporting reactions QM, Q(2m^s), Qir,2m2-s)> Q^-r.i^-,) q different constants for the extended network. We need to find the anti-symmetric mode. In view of Eq. (4.1.55), we must let Q(r,s\~Ce
*e
>
e (2m ,- r ,,) = -ce^-'-^'e^,
tf(2m,-r,2m2-j;
ce'
Q(n2m2_s) = -ceir«"e«2m2-^2
(4.1.58)
Structures with Bi-periodicity in Two Directions
143
with the other Q0Jc) vanishing, in which c denotes an arbitrary real constant. Substituting Eq. (4.1.58) into Eq. (4.1.23a) yields P(jk) =csin jr^ sinks
j = 1,2,...,2m,, k = \,2,...,2m2
(4.1.59)
where r and s represent the half wave numbers in x and y directions for the original network, respectively. The corresponding mode can be found by substituting Eqs. (4.1.59), (4.1.47) and the value of a into Eq. (4.1.46). When the supporting reactions are identically equal to zero, i.e., all of the supported nodes lie in the nodal lines of the mode for the network without the internal supports, the frequency equation can be obtained from Eq. (4.1.44) with /(^)=0,i.e., 2KX +2K2 -Mco2 -2Klcosriffl r=mu 2mu ..., (p,-l) ml
-2K2cossy/2
=0
or 5 = m2, 2m2,..., (p2-l) m2
(4.1.60)
Note that when r or/and s is replaced by 2n, - r or/and 2n2 -s in Eq. (4.1.60), the frequency equation has no change. Therefore corresponding to one natural frequency satisfying Eq. (4.1.60), there are four generalized displacements, %A - 1(2nx-r,S), lir.int-s) a n d 9(2n,-r,2n2-j) w h i c h are non-vanishing. In order to obtain the anti-symmetric mode, the corresponding generalized displacements should take the same form as that shown in Eq. (4.1.58), namely qirA = ce^'e1"",
q{2n^2ni_s)
= Ce«2"^
e ^ ^
<7(2„,-,„ = - c e ^ ^ e ^ , q{rani_s) = - c e '>
(4.1.61a) (4.1.61b)
with the other q^rs) vanishing. The corresponding mode can be Eq. (4.1.4a) as w
ii,k) = sin jry/x sinks iff2
obtained by introducing Eq. (4.1.61) into
j = 1,2,...,2«,; k = l,2,...,2n2
in which an arbitrary constant factor has been omitted.
(4.1.62)
144
Exact Analysis of Bi-periodic Structures
4.1.2a
Example
Consider a uniform square network with 6 x 6 mesh and 2 x 2 internal supports as shown in Fig. 4.1.3. The specific structural parameters can be written down as K
K
n\=n2=(>,mx=m2=l,px=p2=2,y/x=\i/1=
— ,
(4.1.63)
a = b, Tx = 7"2, Kx = K2 = K
When the supporting reactions are not identically equal to zero, the frequency equation is Eq. (4.1.57). Substituting Eqs. (4.1.63) and (4.1.53) into Eq. (4.1.57) gives
-Y^{4K-Mo)2-2Kcos[r
+ 6(k-l)]-yl
+ 6{j-\)]--2Kcos[s
=0
i=\ t=i
r=l,2;
(4.1.64)
5=1,2
The roots for co2 of the above frequency equation are summarized in Table 4.1.1.
Table 4.1.1 (M) 2
a,
Multiplier
Natural frequency (1)
(1,1)
(1,2)
(2,1)
(2,2)
4
4
4
4-V6
2
4 2
4-V2
4 + V6
6
6
4 + V2
KIM
When the supporting reactions are identically equal to zero, the corresponding frequency equation can be obtained from Eq. (4.1.60), where one of r and s must be equal to 3 for the present case, i.e.,
Structures with Bi-periodicity in Two Directions
4K-Mo)2
-2Kcosr—-2Kcoss— 6
6
145
=0
r = 3,5=1,2, —,5 and s = 3 , r = l,2,-,5
(4.1.65)
Because r and s on the left hand side of Eq. (4.1.65) are in agreement with the numbers of the half wave in x- and ^-directions for the original network. When r or j is equal to 3, all of the supported nodes must necessarily lie on the nodal lines of the corresponding mode, i.e., the supporting reactions are identically equal to zero. The corresponding frequencies can be expressed as 2 K ,. _ co =—(4-2cosr M
K
6
_ TZ\ 2coss—) 6
r = 3,5 = 1,2, —,5 and s =3,r = 1,2, —,5
(4.1.66)
The result is summarized in Table 4.1.2.
Table 4.1.2
M
co2
Natural frequency (2)
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(1,3)
(2,3)
(4,3)
(5,3)
4-VJ
3
4
5
4 + VJ
4-V3
3
5
4 + V3
Multiplier
KIM
The total number of natural frequencies is equal to 21 which is in agreement with the number of degrees of freedom for the original network. Next, consider the natural modes corresponding to the supporting reactions that do not vanish. The mode can be obtained by inserting the values of r, s, co and the supporting reactions shown in Eq. (4.1.59) into Eq. (4.1.46). Consider now the basic mode corresponding to the lowest natural frequency. The parameters r,s,a> can be obtained from Table 4.1.1 as r=l, and
s=l
(4.1.67a)
146
Exact Analysis of Bi-periodic Structures
2 CO =(4-V6)
K_ M
(4.1.67b)
Substituting Eqs. (4.1.63) and (4.1.67a) into Eq. (4.1.59) gives p
im = 4c> pw)
= pu»> p«.-m = puw
= -pu*)
J>k = V
<4-1-68)
while the other Pm vanished. Inserting Eqs. (4.1.47), (4.1.63), (4.1.67b) and (4.1.68) into Eq. (4.1.46), the mode can be found as shown in Table 4.1.3, in which the mode has been normalized according to the maximum amplitude taken as 1. Obviously, the mode shown in Table 4.1.3 satisfies the boundary condition and restrained condition at supported nodes. It can be verified that the free vibration equation shown in Eq. (4.1.43) is satisfied at the free nodes. Noting Tables 4.1.1 and 4.1.2, it is interesting that there are five independent modes corresponding to one natural frequency, i.e., co2 =AKIM , where four modes are corresponding to P(jk) * 0. They can be found by using the same procedure described in the above. The results are shown in Tables 4.1.4 - 4.1.7. Table 4.1.3
Natural mode w^
k
1
0
2
3
corresponding to < » 2 = ( 4 - v 6 ) — 4
5
6 p
J
m
0
0
0
0
0
0
0
0
1
0
1 4
£8
l 2
£8
l 4
0
2
0
0
£4
0
£8
0
1 2
0
0
£8
0 0 0
8 1 2
3
0
4
0
£8
5
0
6
0
£4
1
£4
0
£4
1 4
S
S
8
1 2
8
1 4
0
0
0
0
0
3
^"*—K j,k=\,24
P = MM)
Structures with Bi-periodicity in Two Directions
Table 4.1.4 Natural mode 1, w W) corresponding to a>2 =AKIM k
0
1
2
3
4
0
0
0
0
5
147
and (r,s)=( 1,1)
6
J
PM
0
0
0
0
1
0
0
-1/2
0
-1/2
0
0
P
2
0
1/2
0
1
0
1/2
0
=
3
0
0
-1
0
-1
0
0
4
0
1/2
0
1
0
1/2
0
5
0
0
-1/2
0
-1/2
0
0
6
0
0
0
0
0
0
0
* if
(l,\)=
P(l,2)
P(2.n=
M2.21=0
/T, * K2, the supporting reactions are non-zero.
Natural mode 2, w^k) corresponding to a2 = 4KIM
Table 4.1.5 k
0
1
2
3
4
5
and (r,s)=( 1,2)
6 p
m
j 0
0
0
0
0
0
0
0
1
0
0
_1 2
0
1 2
0
0
•* ( l , i ) = M 2 , l ) =
2
0
1 2
0
0
0
. 1 2
0
P(\,2)~ P{2,2)=~K
3
0
0
-1
0
1
0
0
4
0
1 2 0
0
0
0
_1
0
5
0
2 _ 1
0
1 2
0
0
0
0
0
0
2
6
0
0
0
148
Exact Analysis of Bi-periodic Structures
Natural mode 3, wuk) corresponding to co2=4KIM
Table 4.1.6 k
1
0
2
3
4
5
and (r,s)=(2,l)
6
^V)
j 0
0
0
0
0
0
0
0
1
0
0
1/2
0
1/2
0
0
^0,l)=
2
0
-1/2
0
-1
0
-1/2
0
M2,l) = P(2,2)~-K
3
0
0
0
0
0
0
0
4
0
1/2
0
1
0
1/2
0
5
0
0
-1/2
0
-1/2
0
0
6
0
0
0
0
0
0
0
Natural mode 4, w ^ corresponding to w2 = 4K/M
Table 4.1.7 k
0
1
2
3
4
5
P
{\,2)=K
and (r,5,)=(2,2)
6 p
w
j 0
0
0
0
0
0
0
0
1
0
0
-1
0
1
0
0
*(\,\)=P(2,2)=®
2
0
1
0
0
0
-1
0
P<2.n~ ^n.2> = "^n.n
3
0
0
0
0
0
0
0
4
0
-1
0
0
0
1
0
5
0
0
1
0
-1
0
0
6
0
0
0
0
0
0
0
if
Kx *K2, the supporting reactions are non-zero.
Structures with Bi-periodicity in Two Directions
Table 4.1.8 0
k
Natural mode 5, w^ corresponding to 1
2
3
4
5
149
a>2=4K/M
6 p
w
J 0
0
0
0
0
0
0
0
1
0
1
0
-1
0
1
0
P
2
0
0
0
0
0
0
0
=
3
0
-1
0
1
0
-1
0
4
0
0
0
0
0
0
0
5
0
1
0
-1
0
1
0
6
0
0
0
0
0
0
0
(l,\f
P
(\,2)
Pa.rT P(2.2\ = 0
The another mode where all the supported nodes lie in its nodal lines can be obtained by substituting Eq. (4.1.63) and r=s=3 into Eq. (4.1.62) as ™u*) = sin j y sin k y
(4.1.69)
The result is as shown in Table 4.1.8. In the same way as the above, the other modes can also be found without any difficulty. 4.1.3
Forced Vibration [19]
Consider the same network shown in Fig. 4.1.1 with lumped mass M at each node and subjected to harmonic load Fuk)e"" at node (j, k). The harmonic vibration equation for node (j, k) of equivalent system takes the form as (2AT, +2K2
-Ma>2)wuk)
-^,(>Vu)
U,k) = (JiPi,klp2)
+w
(j-w)-K2(.wUMD
jl=l,2,...,2ml;
+w
CM-»)
kt =l,2,...,2m2
= i
UA)
+F
U,»
(4.1.70a)
150
Exact Analysis ofBi-periodic Structures
(2Kl+2K2
-Mo)2)wUJl)-K](wij+ljl)+wu_iJ[))-K1(wUMl)+wijjl,i))
U,k)*{jxpx,k\p2)
J\
j = l,2
=1,2,...,2OT 1 ;
= F{jk)
k{ = l,2,-,2m2
2«,; k = l,2,...,2n2
(4.1.70b)
where F{jk) denotes the amplitude of harmonic force acting at node (j,k)
and the
other notations have the same meaning as those in Eq. (4.1.43). In Eq. (4.1.70), Ft ij* must satisfy the anti-symmetric condition as shown in Eq. (4.1.14). Applying the double U transformation (4.1.4) to Eq. (4.1.70), we have (2K, +2K2 -Ma2
-2K{ cosriy1-2K2cossy,2)q{rs) r = l,2
«/£,,+/£,,>
2n,; s = l,2,...,2n2
(4.1.71)
where / ( * s) and f°rs) areas shown in Eqs. (4.1.7a) and (4.1.7b) respectively. q(r s) in Eq. (4.1.71) can be written as 9(r.J)=9(°r,S)+9(*r,i)
r
= U,...,2n,; s = 1,2,...,2«2
(4.1.72)
where f°
Cs)= (I)
,
(r,l)
(r,5)^(2n„2n 2 )
2/C,+2A: 2 -Maj 2 -2A: i cosr^ 1 -2A: 2 cos5^ 2 (4.1.73a) ^.^,=0
(4.1.73b)
and q\r s) =
; 2Kx + 2K2 - Ma
— (r,s)* (2«, ,2n2) - 2Kx cos r \j/x - 2K2 cos s y/2 (4.1.73c)
Structures with Bi-periodicity in Two Directions
151
( 4 - L73d )
w»2>=°
Substituting Eqs. (4.1.72), (4.1.73) and (4.1.7) into Eq. (4.1.4a), we have w
(yjk)=w(°M)+w(M)
<4-174)
and
w
li,k) =L^PUMHP^)PUM
where fil/^xi
M
(4-L75a)
has the same definition as that shown in Eq. (4.1.47), i.e., 1
ZlhZ}
A
"2~l
2K2 - Ma2 - 2KX cosri//l-2K2
cos s y/2 (4.1.76)
w(;. k) a n d w(°. k) represent the amplitudes of the displacements caused by the harmonic loading and the supporting reactions acting on the equivalent network without the internal supports, respectively. Substituting Eqs. (4.1.76) and (4.1.14) into Eq. (4.1.75b), results in . '
W(i k)
_ 4 ^P ^k"1 ^T"1 ^ sinjr^sinfe^sin^r^sint,*^ ~ «, n2 2-f If la la 2KI + 2K2 - M a2 - 2K, cos r
°' '*'} (4.1.77)
It can be verified that w(*;t) shown in Eq. (4.1.77) satisfies the anti-symmetric condition and boundary condition of the original network with nx x n2 mesh, i.e., w
(*2»,-;,2n2-t) = wu,k); w(2n,-j,k) = wu,2»2-t) = -wlm>
j=h2,...,nu
A=l,2,...,n2
and
152
Exact Analysis of Bi-periodic Structures
W
\j,k) = ° J=0>
n
\
or
* = 0 ' n2-
The displacements at supported nodes must be equal to zero, i.e., W
(JP,^)
+
% * ) =°
7=l,2,...,2m„
A=l,2,...,2/n2
(4.1.78)
Introducing the notation
and Eq. (4.1.75a) into the restrained condition (4.1.78) yields 2ffl|
2ffl;
ZZAMXy,W /> (7,*,)=XM)'
;=l,2,...,2m„ fc=l,2,...,2m2 (4.1.80)
y,=i *,=i
where Pu,k)UM 1
s
PuPi,kpMPxMPi)
2^-1 2nj-l
=—— y 4n,n2 ^
JU-JiVn J(k-k,)sip2
y
^ —
(4.1.81)
~ 2/r, + 2/C2 - Af
with
(4.1.82)
JF(* _t) has been determined in Eqs. (4.1.79) and (4.1.77). It can be shown that the coefficients Pu,k)(hM
of the simultaneous equations
(4.1.80) have the cyclic periodicity, i.e., P(J,WM
~ Ay+L*)(2.*i) = • "
7=1,2,...,2m,; AJ.*X7I.1)
=
P<.jMW„i)
j,jx=l,2,-,2mx;
=
P(y-l,t)(2m1,t1)
k, &,=l,2,...,2m 2
(4.1.83a)
=••• = /^(;,*-i)(y,,2m2)
£=l,2,...,2m2
(4.1.83b)
Structures with Bi-periodicity in Two Directions
153
Eq. (4.1.80) takes the same form as that shown in Eq. (4.1.19), therefore it can be uncoupled by applying the double U-transformation. Applying the double U-transformation (4.1.23) to Eq. (4.1.80), we have 2/»! 2m2
J^-*-..^-*-.,,,^^^^^ = ^^
(4 L84)
where 1
2m, 2m,
^2ml A/2m2 "-f *=i and J
2^1
2^-1
eHu-i)rf,
' < - . « « >4«!« -^ E flf T fnf2Kt 2JC",++ 2^T - M
2
Hv-^sfi,
-2AT, cosr^, -2/f 2 cos.?^ 2
(4.1.86)
Substituting Eq. (4.1.86) into Eq. (4.1.84) results in Q{r,)=-^L
/-l,2,...,2m,;
s=l,2,...,2m2
(4.1.87)
where As) =
},y{2Kl
+2K2-Mco2-2Kxcos[r + (j-l)2ml]y/l -2K2cos[s+(k -1)2/^]^}"'
' fl Pi jJtf (4.1.88) If the specific load and structural parameters are given, the amplitudes of the nodal displacement and supporting reaction can be calculated from the relevant formulas given above. 4.1.3a
Example
Consider the same network shown in Fig. 4.1.3 subjected to the harmonic loading Pe,wl acting at its center, i.e.,
154
Exact Analysis of Si-periodic Structures
^(3,3)
=
^(9,9)
F(Jk) = 0 Noting that the definition
of
=
">
=
M3,9)
^(9,3)
=
—
"
j * 3,9 or * * 3,9
(4.1.89)
ff(*t) shown in Eq. (4.1.79) and substituting
Eqs. (4.1.63), (4.1.89) and (4.1.76) into Eq. (4.1.75b), results in W
=W
"(1,1)
W
(6-j,6-k)
=W
"(1,2)
=W
"(2,1)
= Wv
"(2,2)
A:(4-Q)(4-8Q + n ' )
W(6-y,*) = ^0,6-*)
W*m = 0
IP
-•
—
=
~W(j,k)
j,k=l,2
; = 3,6 or * = 3,6
(4.1.90)
where Q denotes the nondimensional parameter of frequency as shown in the following equation Q ^ ^ K
(4.1.91)
Introducing Eqs. (4.1.63) and (4.1.90) into Eq. (4.1.85) yields
Vi) =0-'^)^d.i) - *(«> =(1 + ''V3>(;1) V5)=V.,=2< 1 )
(4.1.92)
while the other b(rs) vanished. Inserting Eq. (4.1.63) into Eq. (4.1.88) gives A
-A -A ~ (,'5) "
W)
l 1Q 8 + Q2 -A ~ " (5 5) " ' " A: ( 4 - Q ) ( 4 - 8 Q + Q 2 )
Substituting Eqs. (4.1.92) and (4.1.93) into Eq. (4.1.87), we have
(4 193) (
}
Structures with Bi-periodicity in Two Directions
_
2(1-;V3>
155
__2{\ + iS)P
10-8Q + Q 2
w
e(,,5)= 6(5,.)=- 1 0 _ 8 4 ^ + Q 2 >
10-8D + Q 2
'
2M=0,
^1>5
or ^ 1 , 5
(4.1.94)
Substituting Eqs. (4.1.63) and (4.1.94) into Eq. (4.1.23a), the amplitudes of supporting reaction can be obtained as p
- p
_ p
(.,.)"'(..2)--(2,.)
^(6-./,6-*)
=
P(j*)'
Pm=0
P
U,6-k)
_ p
IP
_
--.(2,2)-
=
^(6-7,*)
y=3,6
1 0
=
_
g Q
~P(j,k)
or
+
Q2
J>
^~^'^
*=3,6
(4.1.95)
All of the nodal displacements can be found by substituting Eqs. (4.1.63), (4.1.89), (4.1.95) and (4.1.76) into Eqs. (4.1.75) and (4.1.74). Consider now the displacement of the loaded node, i.e., w(33). The final result can be expressed as w(X3)=H(Q)^
(4.1.96) A
where ( Q 2 - 8 Q ) 2 + 2 7 ( Q 2 - 8 Q ) + 178 ( 4 - O ) ( 1 0 - 8 Q + Q 2 )(13-8Q + Q 2 ) where Q has been defined as shown in Eq. (4.1.91). From Eq. (4.1.97), it can be shown that when i/(Q) approaches a finite value at a resonance frequency, the force is acting at a nodal point/line of the corresponding mode. The frequency response curve governed by Eqs. (4.1.96) and (4.1.97), w(33) versus Q , is plotted in Fig. 4.1.4.
Exact Analysis of Bi-periodic Structures
156
W ( 3 3 ) (Q)x
A:
1.5 1 0.5 0 -0.5 -1 -1.5 -2
Q =
Figure 4.1.4
4.2
Ma1 K
Frequency response curve,
w(33) versus Q
Grillwork with Periodic Supports [18]
The grillwork considered is made up of two orthogonal sets of beams, say x- and jy-beams, which are connected at the nodes so that no moments are transmitted from one set to the other (i.e., the torsional rigidity of the beams is neglected). For generality, consider now an MxN rectangular grillwork with simply supported ends at four edges as shown in Fig. 4.2.1, where EIX and EI denote the flexural rigidity of the beams in the x- and ^-directions respectively; a and b denote the spacing ofy- and x-beams; the solid circles denote the nodes supported by the posts. There are (m -1) x (w -1) internal supports.
Structures with Bi-periodicity in Two Directions
157
>y 1
2
p2+l
k
2p2+l
(n-1) p 2 +1
N
N+l
(1,1)
(1,2)
(I,*)
(1,N)
(2,1)
(2,2)
(2,k)
(2,N)
i 1 1
i
l»
F.
0.2)
(i,D
(j,k)
* <-
i
P
\
f»
i1
ik
b -*
t a
1 -f |P (M, 1)
1P
(M,2)
(M,N)
M=m p,, Figure 4.2.1
V
N=n p2
Grillwork with MxNmesh and (/n-l)x(n-l) internal supports
The equivalent grillwork with cyclic periodicity in x- and j-directions can be produced by using the image method as explained in Section 4.1. At the outset, consider an extended grillwork with 2M x 2N mesh as shown in Fig. 4.2.2 where the loading pattern is anti-symmetric with respect to two symmetric planes of the extended grillwork and the nodal loads acting at the nodes (r, s) (r = 2,3, • • •, M;
158
Exact Analysis of Bi-periodic Structures
s=2,7>,---N) are the real loads acting on the original grillwork. Moreover we regard the extended grillwork as having cyclic bi-periodicity in x- and ^-directions [i.e., each pair of nodes (1, k) and (2M+1, k) {k = 1,2,—,2N +1) and (/', 1) and (/', 2N+1) (j = 1,2, •••,2M +1) may be imaginarily regarded as the same node mathematically].
-* y
L ^ Symmetric plane p,+l-
k-
<(1.1)
d.k) a 1
P,+> i I
N
np2+l
2N-k+2
—
2N
2N+1
(I.2N
(I.N)
V \
fl.1) 2p,+l
2p,+l
-P
P
\
U-k) El
,
0 ' t 1
*EI,
M (M.1)
(M.H
• mp,+l
1
t Symmetnc plane
1| \t
-P
>
2M^+2
I i
{
P 2M-j+2
>
2M (2HI 2M»1 1 1
(2MN
CM* 2N-k*2
M = mp,, N = np2
Figure 4.2.2
0, I)=0,2N+1) (l,k)s(2M+l,k)
*
2N+I
j=l,2,-,2M+l k=1.2,-,2N+l
Equivalent grillwork with 2Mx2N mesh and cyclic periodicity in x- and ^-directions
Structures with Bi-periodicity in Two Directions
159
Such an extended grillwork is equivalent to the original one, because the simply supported boundary condition of the original grillwork can be satisfied automatically in its equivalent one with antisymmetric displacements. In the equivalent grillwork (see Fig. 4.2.2) the additional supports located at boundary and symmetric lines are necessary in order to form the cyclic bi-periodic system, but their supporting reactions are identically equal to zero. 4.2.1
Governing Equation
The principle of minimal potential energy will be applied to establish the equilibrium equation. The potential energy of the equivalent grillwork with 2M x 2N mesh may be expressed as 1M
IN
n=££n„
(4.2.i)
7=1 *=1
where I l ; t denotes the potential energy of substructure (/', k) and j , k denote the serial numbers of the substructure in the x- and^- directions, respectively. In general, the potential energy for the substructure can be written as Ujk=^{S}Tak)[K]sab{S}ilk)
-±({S}Tak){F}ak)
+{*}TUJI){F}U*))
(4-2-2)
in which [K]sub denotes the stiffness matrix of the substructure; W^t).^}(_,-,*) denote the displacement and loading vectors for the substructure (J, k), respectively and the overbar denotes complex conjugation. Here the loading vectors should include both the external load and supporting reaction. For the present case, the substructure is made up of two orthogonal segments of x- and j-beams as shown in Fig. 4.2.3. Each substructure includes three nodes. The substructural displacement vector is made up of three nodal displacement vectors; that is
s;
Wu,*,=k
where the nodal displacement vector can be expressed as
(4-2-3)
160
Exact Analysis ofBi-periodic Structures
(4.2.4)
i = 1,2,3
W(M> U,k)
in which w, 9x,9y (6xmdwl&,
denote the transverse displacement and two angular rotations
9y = dw I dy ). It is stipulated that wHjk) a w{jk),
9xHJk) = 9x(Jik),
Oy\(jJk) ~ ®y(j,k) •
X
^T
\
Figure 4.2.3
y EI..
Node (/', k)
(/.*)
EIr
Substructure and the ordinal number of nodes
By using the conventional stiffness superposition method, the stiffness matrix of the substructure can be obtained from two beam element stiffness matrices as
[*U = where
k
k
k
"•11
K
"-U
\2
k
k
k
"•21
"-22
""23
k31
ki2
k3i
(4.2.5)
Structures with Bi-periodicity in Two Directions
12
6a
[*..] = * ! 6a 4a 0 0
6a
0
[ka] = K} -6a
2a
2
0
0
0
12 [k21] = K, -6a
-6a 2
Aa
0
0
-12
0
6b
0
0
0
-6b
12 0
6b
0 + K2 0 0 0 0 6b 0 Ab2
-12 0
[*»] = * :
0"
2
0
(4.2.6a)
[k2l] = [kl2]7
0 [*33] = * 3
0
(4.2.6b)
12
0
-6b
0
0
0
-6b
[*3i] = [ * n ] r .
161
(4.2.6c)
2
0 Ab
[*23] = [*32l = [0]
(4-26d)
2
0 2fc
K2=EIy/b3
K^ElJa\
(4.2.7)
where the torsional rigidity of the beams is neglected. The continuity across the nodes requires the following conditions to be satisfied { * 2 } ( M ) = { * , W ) . {*3}(,, t ) ={*iW,). 7 = 1 , 2 - , 2 M ; * = 1,2-,27V (4.2.8) where {£,} (2M+U) = {£,} ( U ) , {<M(y,2jv+D = {<M(,u) d u e t 0 The restrained condition at supports can be written as r
[l+(7-l)Pl.l+(*-l)/> 2 ]
= 0,
j = 1,2 —,2m,
the c
y c l i c periodicity.
k = l,2---,2n
(4.2.9)
where p,,p2 and m, n are the geometric parameters of the grillwork (see Fig. 4.2.1). There are simple relations between the geometric parameters; that is pxm = M ,
p2n = N
(4.2.10)
For the present case, the double U-transformation and its inverse can be given as
162
Exact Analysis of Bi-periodic Structures
1M
«
w
(4.2.11a) and 1M
«
i
W
r-^
{}(M)
/,/,
J^ih
(4.2.11b)
yl2Myl2NJJ1-t in which \j/x = K/M , y/2 = n/N,
i — V - 1 and
i
(4.2.12)
{l\}(r,s) = (rj)
(r,s)
where {qj^jy is a vector while q{rs) is a scalar. Applying the double U-transformation (4.2.11) to Eqs. (4.2.1), (4.2.2) and (4.2.8) results in 2M
2N
2M
.
2N
,
(4.2.13) with .
{/}
M = r -
2M
2N
e iU X)me
YL ~ ~
'
WiW
(4-2- 14)
and {<72}<„> = «** {?.}<,,,).
{?,}(,.,> = ^
2
{?,}<,,,
(4-2.15)
The continuity condition (4.2.15) can be rewritten {?}^, = m ( , „ { ? , } ( , „
(4-2.16)
Structures with Bi-periodicity in Two Directions
163
where
m (riJ) =[/ M . e^IM,
e^I.J
(4.2.17)
and IM represents a unit matrix of order three. Substituting Eq. (4.2.16) into Eq. (4.2.13), we have 2M 2N
r=\
r1
1
n
J=l
in which [K]^
=[f]ls)[K]sab[T\ns)
(4.2.19)
{/C)=[nf,,, {/}(,,.,
(4-2-20)
and {?]}(,_,) (r = 1,2,---,2M;5 = l,2,---,2Af) are the independent variables. The necessary condition to make the potential energy minimum can be expressed as W M ^ I M M / } ; , , ) ,
r = l,2,-,2M;s
= l,2,-,2N
(4.2.21)
which is equivalent to the nodal equilibrium equation. Each equation in Eq. (4.2.21) with given r and 5 includes only three variables. Introducing Eqs. (4.2.5), (4.2.6) and (4.2.17) into Eq. (4.2.19) gives
24^' 1 (l-cosr^ 1 ) + 24A' 2 (l-cos5^ 2 ) -\2Kja sin riff,
\2Kj.a sin ry/l 4Kla2(2 + cosr^1)
-12A^ 2 i6sin5^ 2
0
\2K2ibsmsy/2 0 4AT2fe2(2 + cossi//'2) (4.2.22)
which is the Hermitian matrix. Without loss of generality, it is stipulated that the loading vector for substructure
164
Exact Analysis ofBi-periodic Structures
only includes the load acting at the first node of substructure. The loading vector can be expressed as
W w ) = L + i ( . , ) . °> °> •••> ° L j = l + (u-i)pl,
u = \,2-,2m;
k = l + (v-l)p2,
7'9tl + ( « - l ) p , ,
M = 1,2---,2W;
or
v = 1,2-,2«
k * \ + {y-\)p2,
(4.2.23a)
v = \,2 — ,2n (4.2.23b)
where F ( . k) denotes the external load acting at the node (/', k) (i.e., the first node of substructure (/, k) ) and P (uv) denotes the supporting reaction acting at the supported node [1 + (u - \)px, 1 + (v - \)p2 ] . The external loads of the equivalent grillwork must satisfy the anti-symmetric condition; that is F(2M-j+2,2N-k+2)
=
F(j,k) '
F(2M-j+2,k)
j = 2,3,-,M; F(M)=0,
=
~ ^j,2N-k+2)
k = 2,3,--,N
j=\,M+\
~^(j,k)
(4.2.24a)
or *=1,AH-1
(4.2.24b)
Introducing Eqs. (4.2.17), (4.2.14) and (4.2.23) into Eq. (4.2.20) results in f(r,s) +
{f)\r,s) = •
f(r,s)
(4.2.25)
0 0
in which 2M IN
Ar,s) ~~
X" 1 V 1 e-HJ-l)ry, e-Hk-l)sy,
2M4wj^i
p
(4.2.26a)
Structures with Bi-periodicity in Two Directions
1m
1
165
In
Noting the anti-symmetric condition (4.2.24), Eq. (4.2.26a) may be rewritten as M
4
f(r,s) =
N
XX S i l 1 ^ ~ 1)r^' ^ ^ ~ VS Wl FW
f— / yllMjlN
j=2
( 4 - 2 - 27 )
k=2
That leads to f(r s) = 0 ,
r = M,2M oi s=N, 2N
(4.2.28)
In the right side of Eq. (4.2.21) includes the unknown support reactions besides external loads which can be determined by using the restrained condition (4.2.9). 4.2.2
Static Solution
At the outset we need to derive the restrained condition in terms of the support reactions. Next we solve this equation by using the double U-transformation for the second time. Inserting Eqs. (4.2.22), (4.2.25) and (4.2.12) into Eq. (4.2.21), the solution of Eq. (4.2.21) can be expressed as *,., = (2 + C O S r ^ ) ( 2
+ C
°
S 5
^)(/^
+
O ,
r*2M,s*2N
q(rs) =0, r = 2M or s=2N
?X(M)=
3z'sin/•«/•. (2 +cos sy/,) 0 ^ —(/(,,,)+/(;,,)), r * 2 M , s * 2 W 0 X(JV)
9*r,)=
=0, r = 2Af or 5=2Af
3isinsy/2(2 + cosryO „ * -,,, „ Ar ^ — (/(,,,)+/(I,), r*2M, s*2N °^(r,,)
(4.2.29a) (4.2.29b) __ (4.2.30a) (4.2.30b) ^ ^ - M -> (4.2.31a)
166
Exact Analysis of Bi-periodic Structures
qy(rs)=0,
r = 2M or s=2N
(4.2.31b)
£>(rj) =12A:i(2 + c o s s ^ 2 ) ( l - c o s r ^ 1 ) 2 +12AT2(2 + c o s r ^ , ) ( l - c o s j ^ 2 ) 2
(4.2.32)
Substituting Eqs. (4.2.29a) and (4.2.26) into the first component equation of the double U-transformation (4.2.11a), we have w
a.k) =wu.k)+w(i.k)
(4.2.33)
%»=IZ4K./M
(4-2-34)
where
2m w
In
=
u,k) 2 J 2 j ^ M ) N » - i ) p , J * M f t ] ^ , » )
(4.2.35)
and
fi-
, _ ^ V r=l
V e . 0 - ^ , e - ( - ) ^ 2 (2 + cosr^)(2 + c o S ^ 2 ) j=l
(»Vl)
H e r e w(*JJt) a n d w ( °, t) denote the transverse displacements of the node (/", k) caused by the anti-symmetric external loads and supporting reactions, respectively. It is presupposed that F(jk) satisfies the anti-symmetric condition shown in Eqs. (4.2.24), otherwise Eq. (4.2.34) is not true. The term w^ A) is dependent on the unknown supporting reaction P(u v) which can be determined by means of the restrained condition (4.2.9). Inserting Eqs. (4.2.33) - (4.2.36) into Eq. (4.2.9) yields 2m
In
IIV/(»,)+C)=°> where
7=U.-,2m;* = lA-,2n (4.2.37)
Structures with Bi-periodicity in Two Directions
4M/V
r=)
167
D(rs)
j=1
Km
s w
(4-2-39)
[*+o-.)P„.+(*-i)p2]
Eq. (4.2.37) is the set of simultaneous linear algebraic equations with 4mn numbers of
variables.
Noting
that
the
coefficients
/?(;>i)(„,V)
(j,u =
l,2,---,2m;
k, v = 1,2, • • • ,2n )shown in Eq. (4.2.38) possess cyclic periodicity; that is AMXI.*)
=
=
Pu«**M
"" = A;-U)(2m,v)'
7 = l,2,-,2m;t,v = 1,2,-,2B
(4.2.40a)
and /W-,1) = /Wx«.2) = ••• = Put-Win)>
-/>
=
l A - > i ; * = l,2,-,2« (4.2.40b)
the deferent coefficients are y9(;i)(11) (y = l,2,---,2m; A: = 1,2,•••,2M). This property indicates that the simultaneous equation (4.2.37) can be uncoupled by using the double U-transformation. For the present case, the double U- and inverse double Utransformations can be given as 1
p
-
2m
2n
l
V V p'O-'t^ Jl*-*)'* n V2/wV2n " " y = l,2,-,2m;* = l,2,-,2n 1
_
(4.2.41a)
.
_ _ L _ V V -.-•"(/-D'-tt p-'(*-i)^2 p r = l,2,-,2i»; s = l , 2 , - , 2 » with
(4.2.41b)
p2y/2. *
2m
In
Premultiplying Eq. (4.2.37) by the operator _ _ _ Y V e -' (y-1), *
168
Exact Analysis of Bi-periodic
Structures
( 2m In
\
V u=l v=l
/
r = 1,2, •••,2m; 5 = 1,2,-,2/j
(4.2.42)
where 1
.
=
^
Y
2m
2n
™ 4di&%lrt ~*H**e~™*w™
A
h
e
e
^
^
(2 + c o S ^ l X 2
+
(4 2 43)
''
coS^2)
It can be proved that 6(2m 2n) = 0 leading to Q{2ma„) - 0 • Eq. (4.2.42) is made up of Amn numbers of single degree of freedom equations. The solution is e^)—-^-,
(r,5)*(2m,2n)
(4.2.45a)
A
(r,s)
e (2m ,2 n) =0
(4.2.45b)
where A
1 ^ ^ ( 2 + cos[r + 0--l)2w]y,X2 + cos[j + (*-l)2iiV 2 ) P\Vl
j=\ k=\
^
2
^
^[r+(y-l)2m,i+(i-l)2ii]
When the specific structure parameters and external loads are given, the generalized supporting reaction Q(rs) can be calculated from Eqs. (4.2.45), (4.2.46), (4.2.43), (4.2.39) and (4.2.34). Then the supporting reactions and all of the nodal displacements can be found from the related formulas derived above. Recalling the definitions of both generalized supporting reactions / ( ° s ) (r = l,2,---,2M; s = 1,2,---,2N)
and
Q{rs)
(r = l,2,---,2m; s = l,2,---,2n)
as
shown in Eqs. (4.2.26b) and (4.2.41b) respectively, there is a simple relation between both f°rs) and Q(r s); that is
Structures with Bi-periodicity in Two Directions
J [r+(y-l)2m,s+(*-l)2n]
—
j = l,2,-~,pl;k=l,2,-~,p2;
/
/
169
&(r,s) '
r = 1,2,-••,2m; s = l,2,---,2n
(4.2.47)
Consequently, if we are only interested in the nodal displacements, it is not necessary to find the supporting reactions. 4.2.3
Example
To explain the calculation procedure and to verify the exactness of the formulas derived in the above, a specific grillwork and loading is worked out as a numerical example. Consider a uniform square grillwork with 6x6 mesh simply supported along the four sides and by 2 x 2 internal supports subjected to a concentrated load with magnitude P acting at the centre node as shown in Fig. 4.2.4. The structural parameters are given as M = N = 6,
Kl = K2 = EI I a1 = K
m = n=3,Pl=p2=2,
(4.2.48a)
That leads to
W\=Wi= x/6»
(4.2.48b)
The anti-symmetric nodal loads for the equivalent grillwork can be expressed as ^,4) = / r (.o,.o, =P F^=FWQ)=-P
(4.2.49a) (4.2.49b)
with the other nodal loads vanishing. Substituting Eqs. (4.2.48) and (4.2.49) into Eq. (4.2.27) yields f{rs)
=
-|
s i
n^
s i n
^,
r,s = 1,2,-,12
Introducing Eqs. (4.2.48), (4.2.49) and (4.2.36) into Eq. (4.2.34) results in
(4.2.50)
170
Exact Analysis of Bi-periodic Structures
—>
y
(1,1)
0.2)
(1,3)
(1,4)
(1,5)
(1,6)
0,7)
X
(2,1)
«-
(3,1)
(4,1)
a
- > (3,3) i
ta \ EI 6r~ 1
*
4
1
(3,5)
1P >
P (4,7) (4,4)
i
(5,1)
(5,3)
* k (5, 5) %9
fP
(6,1)
(7,1)
... ... (7,4)
Figure 4.2.4
w
l*)
=
-Q- Z
(7,7)
Grillwork with 6x6 square mesh subjected to a concentrated load acting on the center node
Z ^
=1,3,5 s=l,3,5
r-\
s-\
2
2
H)
(2 + cos r — )(2 + cos 5 —) 6 6 sinO-l)r^sin(A-l)^ D,(r,s) (4.2.51)
Structures with Bi-periodicity in Two Directions
171
where D(rs) =l2K[(l-cosr—)2(2 6
+ coss-) + (l-coss—)2(2 6 6
+ cosr—)] 6
(4.2.52)
Obviously vv(*._t) satisfies the anti-symmetric condition shown in Eqs. (4.2.24) with w*y t ) instead of F(jk) and M = N = 6. Introducing Eq. (4.2.51) into Eq. (4.2.39) gives 529 P
w
K» = K» = ™ = K* = -^j K*)=0>
( 4 - 2 - 53a >
7 = M , * = U , - , 4 ; * = 1,4, 7=1.2,-,4
(4.2.53b)
f C - t ) =ȣ,),
(4-2-53c)
and
-A* = 2,3
ȣ**> = W'w_k) = X , * ) ,
j,k = 2,3
(4.2.53d)
which indicates the anti-symmetry for W*(.k). Substituting Eqs. (4.2.53) and (4.2.48) into Eq. (4.2.43) yields 529 P fyin = * « « = - — — (1.D
b (1.5)
(5,5)
4 g 0
529 P =b( 5 1 ) = — — (5.D
(4.2.54a)
K
(4.2.54b)
4 8 Q K
birs) = 0 r * 1,5 or s * 1,5 1613 in Inserting Eqs. (4.2.48) and (4.2.52) into Eq. (4.2.46) results (1,1) ~~ (5,5) _ ^(1,5) - ^(5,1) ~~ rsmrs 960/:
(4.2.54c)
(4.2.55)
By introducing Eqs. (4.2.54) and (4.2.55) into Eq. (4.2.45a), the generalized
172
Exact Analysis of Bi-periodic Structures
supporting reactions can be obtained as Q^=Q^~P
(4.2.56a)
10SR
e(1,5)=e(5,I)=-^/j
(4.2.56b)
Q(rs) =0, r * 1,5 or s * 1,5
(4.2.56c)
Inserting Eqs. (4.2.48a) and (4.2.56) into Eq. (4.2.47) yields /(i,i)
=
/(5,5>
=
/(7,7)
=
/(ii,ii)
/(i,5)
=
/(5,i)
=
/(7,5)
=
/(5,7)
=
=
J (n)
/(i,ii)
=
=
/(7,i)
=
/(5,ii)
=
/(ii,5)
=
/(ii,i)
=
/(7,ii)
=
/(n,7)
= _
.,..'
77TT
(4.2.57a)
(4.2.57b)
with the other components of f°rs) vanishing. Substituting Eqs. (4.2.56) into the double U-transformation (4.2.41a) gives 529 ^(2,2, = ^(3,3, = PW)
p(jk)
= ^(3,2) = ~ — P
(4.2.58a)
=0, j = 1,4 or k = 1,4
(4.2.58b)
and M8-A8-*)
=
(;'.*)'
(8-y.*)
=
(;,s-*)
=
(y.*)' 7
=
2 , 3 ; A; = 2,3
(4.2.58c)
Eqs. (4.2.58b) and (4.2.58c) indicate that the supporting reactions of the additional supports are equal to zero and the anti-symmetry of supporting reaction for the equivalent grillwork. The real supporting reaction for the original grillwork is given in Eq. (4.2.58a). Now all of the nodal displacements ( w, 0X, 9y ) can be calculated by introducing Eqs. (4.2.48), (4.2.50), (4.2.57) and (4.2.29) - (4.2.31) into the first three component equations in the vector equation (4.2.11a); that is
Structures with Bi-periodicity in Two Directions
HH)r-
e r~\
i(*-l)*T
6
j,k=\,2,-,n
6
e
173
(4.2.59)
s=\
(A*)
(',»)
The exact results for the original 6x6 grillwork are summarized in Tables 4.2.1-4.2.3. It can be proved that the results shown in Tables 4.2.1 - 4.2.3 satisfy all of the nodal equilibrium equations, the support conditions and the boundary conditions exactly.
Table 4.2.1
w,0\*)
1
2
3
4
5
6
7
1
0
0
0
0
0
0
0
2
0
5143
3841
1765
3841
5143
12
6 0
3
6 0
12
k j
3
0
3841 6
4
0
1765
7245
68143
3
4 0
12
7245 4
5
0
3841 6
6
0
5143
3841
1765
12 0
6 0
3 0
7
0
Multiplier
7245 4
P/tfSSUK)
0
3841 6
0
7245
1765
0
4 0
3 3841 6
0
3841
5143
0
6 0
12 0
0
174
Exact Analysis ofBi-periodic Structures
Table 4.2.2 k
6x(m 6
7
6601 8
1859 4
0
1989 4
1081 4
1425 4
0
15249 8
36607 8
15249 8
123 4
0
0
0
0
0
0
15249 8
123 4
0
1081 4
1425 4
0
6601 8
1859 4
0
1
2
3
4
1
0
1859 4
6601 8
2
0
1425 4
1081 4
3
0
123 4
4
0
0
5
0
123 4
15249 8
36607 8
6
0
1425 4
1081 4
1989 4
7
0
1859 4
6601 8
5
j
Multiplier
9049 8
9049 8 P/(75811aA:)
Structures with Bi-periodicity in Two Directions
Table 4.2.3
k
175
Oy(m
1
2
3
4
5
6
7
0
0
0
0
0
0
0
0
1425 4 1081
1859 4 6601
J 1 2 3
1859 4 6601
1425 4 1081
123 4 15249
8
4
8
0
123 4 15249
0
36607
0
15249
8
4
8
4
9049
5
6601
8
4
4
8
6
1859 4
1425 4
123 4
0
123 4
1425 4
1859 4
0
0
0
0
0
0
8
7
0
Multiplier
4.3
1989
4 1081
36607
8 15249
8
8
8
1989
4 1081
9049
8 6601
P/(J 5811 aK)
Grillwork with Periodic Stiffened Beams
The grid to be considered is made up of two orthogonal sets of beams, namely the x- and j-beams, which are connected at the nodes. Under the assumption that the torsional rigidity of the beams can be neglected, no moments are transmitted from one set to the other. For generality, consider now an MxN rectangular grid with simply supported ends and equidistant stiffening girders in one direction as shown in Fig. 4.3.1. The flexural rigidities of the regular beams and the stiffening girders in the x-direction are EIX and (1 + y)EIx respectively, whereas the flexural rigidity of the regular beams in the ^-direction is EIy. The spacing of the y- and x-beams are denoted by a and b respectively. Along the ^-direction, the («-l) internal stiffening girders divide the grid into n repetitive patterns, while there are p repetitive substructures within each pattern. In order to apply the U-transformation technique, we must define a cyclic bi-periodic system that is equivalent to the original one. Firstly, the grid is extended in the x- and _y-directions by its mirror images about two adjacent edges. Then die extended grid has a 2M x 2N mesh and its loading pattern must be
176
Exact A nalysis of Bi-periodic Structures
N=np 1
2
...
p+i
...
k
...
2/H-l
...
(n-\)p+\
...
N
N+\
(1,1)
(1,2)
(i.*)
O.AO
(2,1)
(2,2)
(2,*)
(2.A0
F
>f / (/,1)
0,k)
(/,2)
6
a
(Ml)
W2)
(M,N)
Figure 4.3.1 A grid with an AfxAf mesh and «-1 interior stiffening girders
anti-symmetric with respect to the two axes of symmetry as shown in Fig. 4.3.2. The nodal loads acting at the nodes (/', k) (j = 2,3,---,M;k = 2,3,---N) are real loads acting on the original grid while the other nodal loads can be determined by using the properties of anti-symmetry. Herey and k denote the ordinal numbers of the node (/', k) in the x- andy-directions, respectively. Secondly, such an extended system may be considered as cyclic bi-periodic in x- and ^-directions, when nodes (I, k) and
Structures with Bi-periodicity in Two Directions
177
(2M+1, k) (k = 1,2,•••,27V + 1) are hypothetically regarded as the same node mathematically, and nodes (/', 1) and (j, 2/V+l) (j = l,2,---,2Af + 1) are likewise regarded as the same node mathematically. The extended system with cyclic biperiodicity and anti-symmetric loading pattern is equivalent to the original one, as the simply supported boundary condition of the original grid can be satisfied automatically in its equivalent one having anti-symmetric displacements.
Axis of symmetry
1
...
P+ 1
0,0
P
(U)
4 (/.I)
2 +I
...
N
np+l
(/', 1) = (j, 2/V+l) (\,k) = (2M+l,k)
y=l,2, - , 2M+1 k=\,2,-,2N+l
2N-k+2 . . .
2N 2N+\ (1,2*0
(1/0
b
s
p
/
m
-P
r-
Eh
(MM
(M,l)
Axis of
symmetry
-P
2M-J+2
[2M,l)
r
•s
/
[2Mfl)
2M-J+2
(2MZN,
Figure 4.3.2 An equivalent grid with a 2Mx2Nmesh, 2n stiffening girders and cyclic periodicity in the*- andy-directions
178
Exact Analysis ofBi-periodic Structures
4.3.1
Governing Equation
t
Node (/, k) -o \
x
EL
\
,
62
Figure 4.3.3 A typical substructure showing the ordinal numbers of nodes Fig. 4.3.3 shows a typical substructure (J, k) where j and k denote the ordinal numbers of the substructure in the x- and y- directions, respectively. Each substructure is made up of three nodes and two orthogonal segments of x- and ybeams. The displacement vector {S}ijk) of substructure (/, k) is made up of three nodal displacement vectors, namely,
(4.3.1) (;.*)
where the nodal displacement vector can be expressed as
Wu*)
0y. and w,
i=l, 2 and 3
0.
(4.3.2)
(;.*)
0X and 0y denote, respectively, the transverse displacement and two
Structures with Bi-periodicity in Two Directions
179
angular rotations, i.e. 0X = dwl dx. and 0y = dwl dy . As the serial number of the first node of substructure (/', k) is also (/,fc),it can be stipulated that w1(;. k) = w(jk), d
x\(j,k) = 0x{j,k)
a
n
d
^i(M)
= ^(y,t) •
By using the conventional assembly process, the stiffness matrix [A^sub of a typical substructure can be obtained from the two stiffness matrices of the x- and ybeam elements lc
[*]s
lc
lc
A.,,
n.12 A.,3
lc
lc
"•21
"-22 "-23
lc
lc
"-31
^32
(4.3.3)
Ir lc " ^
where the sub-matrices [t ff ] (i,j=l, 2, 3) can be expressed as 12
6a 2 [ * „ ] = * , 6a 4a 0 0
0" 0
+
12 0 6b K2 0 0 0
0
6b 0 46
12 -6a 0
-6a 0 4a 2 0 0 0
[k33] = K.
-12 [*.2] = *"l - 6 a 0
6a 0 2a 2 0 0 0
-12 [k„] = K. 0 -6b
[k21] = Kx
[**,]= [*ul T ,
(4.3.4a)
12
0 -6b
0
0
-6b
[*3,]=[*,3]\
2
0
0 46
(4.3.4b)
2
0 6b 0 0 0 2b1
(4.3.4c)
[*23]=[*32l=[0]
(4.3.4d)
and
K^ElJa3,
K2=EIy/b3
(4.3.5)
When the substructure includes a segment of the stiffening girder, the stiffness parameter Kx in Eq. (4.3.4) must be replaced by (l + y)A^. The stiffness
180
Exact Analysis of Bi-periodic Structures
parameter (1 + y)Kl can be divided into Kx and yKl corresponding to the regular and additional stiffnesses, respectively. In order to make the structure cyclic monoperiodic, the opposite of the internal forces caused by the additional stiffness can be regarded as additional external loads. Note that the substructure (/', k) ( k = 1,1 + p , • • • ,1 + (2n - \)p) includes a segment of the stiffening girder. The additional load vector for substructure (/', k) can be expressed as *".
MU -
j=
l,2,--,2M;k=l,2,---,2N
(4.3.6a)
<M)
where ^
= -jKl[K]'
j = l,2,-,2M;k
*i
= l,l + p,-,l
= {0},
+ (2n-l)p
k*l,l + p,---,l +
(4.3.6b)
(2n-l)p
(4.3.6c)
<M>
{F,}°iM) = {0}, and yK\[KY
j = 1 , 2 , - , 2 M ; k = 1,2,• • • ,2N
(4.3.6d)
denotes the additional stiffness associated with the stiffening girder.
The matrix [K]e is given as
[K]e
[*.°.] =
12 6a 0" 6a 4a2 0 > 0 0 0
k"
k°
k°
k°
(4.3.7a)
' 12 1^22 J
—
-6a 0
-6a 4a2 0 0
0 0
(4.3.7b)
Structures with Bi-periodicity in Two Directions
-12
Kv
6a
0
2
- 6a 2a 0
181
0
0
[^M^r
(4.3.7c)
0
The additional loads associated with the stiffening girder depend on the nodal displacements w and 9X of the nodes at the stiffening girders. The total potential energy n of the equivalent system with 2M x 2N subsystems can be expressed as
n=EIn,,
(4.3.8a)
y=i t=i
where Yljk denotes the potential energy of substructure (/', k). In general, YlJk can be defined as n , , t =\{#}lk)[KU{S}w
-laS}l,k){F}W)+{S}Jm{F}(m)
where the superior bar denotes complex conjugation and {F}Uk) vector for substructure (/, k). The load vector {F}Uk)
(4.3.8b) denotes the load
should include both the
external loads and the additional loads associated with the stiffening girder, i.e. {F}(M)={F}lk)+{F}{
(4.3.9a)
(M)
in which {F}'(jk) denotes the external load vector for substructure (/', k). Without loss of generality, the load vector {F}[jk) only includes the load acting at the first node of substructure (/', k), i.e.
W\m=\Fu*)>
°> °> - '
°t
(4.3.9b)
where F(j k) denotes the transverse external load at node (/', k). The external loads for the equivalent system must satisfy the anti-symmetric condition, namely
182
Exact A nalysis of Bi-periodic Structures
f(2M-j+2,2N-k+2)
=
^(j,k)
>
^(2M-/+2,*)
j = 2,3,-,M; MM)=0'
=
**(j,2N-k+2) ~
k = 2,3,-,N
J=l,M+l;
or k=l,N+l
—
MM)
(4.3.9c) (4.3.9d)
The spatial relationship between adjacent substructures requires the following conditions to be satisfied iSi}u,k)
fcMuji) = (<M(y+i,*)' j = \,2--,2M;
=
W(M+»
k = \,2---,2N
(4.3.10)
where {SX}{2M+U) a {£,} ( l t ) and {^,}0,2A,+1) s {^1}(;,,) due to cyclic periodicity. For the present case, the double U-transformation and its inverse can be given as 2M 2N
j = l,2,-,2M;k = l,2,-,2N
(4.3.11a)
and 2M
2N
k=\
r = l,2, — ,2M;s = l,2,-,2N
(4.3.11b)
in which y/l = ;r/Af , y/2 = njN, i = V-1 , and
{?}(',,) S '
1^3
(4.3.12a, b)
{?,} ( M )
l*> (^)
By applying the double U-transformation (4.3.11) to Eqs. (4.3.8) and (4.3.10),
Structures with Bi-periodicity in Two Directions
183
we have 2M
2N i
2M 2N
.
(4.3.13)
f/>
«
2M 2W
'">-^^§§^"""*""" , F ! -"
(4314)
''
and {? 3 } (f „ = ^ 2 { < 7 , } „ „
{^}(r,s) = «*"{?.}<,,>,
(4.3.15a, b)
which indicate that fa2}(M) and {q3}{,tS) depend on {^,} (rj) ,and {?,}(r)S) are independent variables where r = 1,2,--,2M and s = 1,2,• • • ,2iV. The continuity condition (4.3.15) can be rewritten as {?}(,,,) = [*"](,,, {?,}(,,,)
(4.3.16a)
where the matrix [ r ] ( r j ) is made up of unit matrices / 3x3 of order three as
m ( ^,=[/ M . ^'4<» « fcri / M r
( 43 - 16b )
Substituting Eq. (4.3.16a) into Eq. (4.3.13), we have 2M w
n
r1
1
i
= i Z b { 9 , } L ) m ; . , { ? , } M --({?,c>{/>L)+{g,}L){/}^))j r=l »=1
Z
Z
(4.3.17) where m;v)=[f]^)m„b[7']M
(4.3.18)
{/}L) =[n^, s ) {/}(,„)
(4.3.19)
184
Exact Analysis of Bi-periodic Structures
The necessary condition to make the total potential energy a minimum can be expressed as r=\,2, — ,2M; s=l,2,-,2N
[^](r,.){?l}(r,i) -{/}(r,j)>
(4.3.20)
which is equivalent to the nodal equilibrium equation. Introducing Eqs. (4.3.3), (4.3.4) and (4.3.16b) into Eq. (4.3.18), we have
[*£,> = 24/T, (1 - cos r \f/x) + 2AK2 (1 - cos sy/2)
12/C, ai sin r y/l
\2K2bismsy/2
2
-\2Klaismry/l -\2K2bisms\f/2
4A^a (2 + cosr^,) 0 2 0 4A:2fe (2 + c o s i ^ 2 ) (4.3.21)
Obviously, [K]{r s) is a Hermitian matrix. Substituting Eqs. (4.3.14), (4.3.9), (4.3.6) and (4.3.16b) into Eq. (4.3.19), we have (4.3.22a)
{/};,,,={/}?,,,+{/}'(,,,, where lf\°
A 'YK iX
V V()-'1"-H(,-i|'-i|)«f1rn
6(1 - cos r ^ , ) = 4 -2>aismry/x
0 and
fj?\
l+(v-l)p)
3ai sin ry/t
0
a2(2 + cosr\f/])
0
0
0
(4.3.22b)
(4.3.22c)
Structures with Bi-periodicity in Two Directions
{/>'(,,,)=[/(,,,, 0,0] T
185
(4.3.22d)
_1M _ 2N _ _
1
Introducing the anti-symmetric condition (4.3.9) in Eq. (4.3.22e), we have M
4
f
N
™ = ~ /^7/^7 Z Z sin(" _ 1)r ^ sin(v ~1)j ^ ^(.,v) r=l,2, • • • ,2M; 5=1,2, • • • ,2W
(4.3.23a)
that leads to f(rs) = 0 ,
r = M, 2M or 5 = N, IN
(4.3.23b)
Because of the anti-symmetry of the nodal displacements, it can be shown that {/}(2W,2iv)={0}
(4.3.23c)
In Eq. (4.3.20), {/}(r?i) is dependent on the unknown nodal displacements associated with the stiffening girder besides external loads. The deformation compatibility condition of the stiffening girder needs to be satisfied. 4.3.2
Static Solution
At first, the deformation compatibility condition in terms of the nodal displacements associated with the stiffening girder must be established. Then the compatibility equation can be uncoupled by applying the double U-transformation again, which leads to the explicit solution. Inserting Eqs. (4.3.22a), (4.3.22d), (4.3.23b) and (4.3.23c) into Eq. (4.3.20), we have {
(4.3.24)
Substituting Eqs. (4.3.21) and (4.3.22) into Eq. (4.3.20), the solution for {9,}(r>i) can be expressed as
186
Exact Analysis ofBi-periodic Structures
(4.3.25a)
{
.qx.q y J(M)
^ G ](, S )££ e "' (u " 1)r " e "' (v " 1) ' B " !
r
r=\,2,-,2M;s=\,2,-~,2N
and (r,s)*(2M£N)
k,}^)
[G\ r,s)
e
Hr,S)
0
e
2(r,s)
1
*,
~2(r,s)
<-3(r,s)
A
3(r,,)
(4.3.25b)
( 4 - 3 - 25c )
=W
e
-Hr,*)
"J (u,l+(v-l)p)
(4.3.25d)
°
( l - c o s r ^ , ) 2 ( 2 + coss^ 2 )
3K2 -isinnf/^l-cossifj)2 aD,(r,*) 3A:, • ism s if/2 (1 - cos r\f/x ) 2 bD,0v)
Z)(M) = A", (1 - cos r y/\ )2 (2 + cos s y/2) + K2 (1 - cos s ^ 2 ) 2 (2 + cos r y/x)
(4.3.25e)
(4.3.25f)
(4.3.25g)
(4.3.25h)
and q (4.3.26a)
Wr* (^)
Structures with Bi-periodicity in Two Directions
, _ (2 + cosry/,,)(2 + cos.s^ 2 ) 4W) = 7Tr\ h'J)
(4.3.26b)
z'sin r ^, (2 + cos s ^ 2 ) Qx(r,s) ~
j'sini^ 2 (2 + cosr^ 1 ) 1y(r,s) ~
Aba(r,s)
187
f(r,s)
(4.3.26c)
A(r,s)
(4.3.26d)
Introducing Eq. (4.3.25a) into the double U-transformation (4.3.1 la), we have
w
w'
w°
o,
=.
M u,*) kJ 9V
+• 9'
01 9°
(4.3.27a)
X
<M)
9'
in which
w
1
1
2M
IN ^iv
~ y " y eiu-i)ryleKk-\)sW21
(4.3.27b)
6
°iu»
<-v)
and w
y 1 v 1 e'0-i>T, e-(*-o^
91
9 (4.3.27c)
•JlMflN^i: 1 < J (-v)
J (;,*)
where w ( '^ t ) , # ^ ] t )
and ^ ( y ) t ) denote the basic solution for the grid without
stiffening girders, while w(° t ) , 9°x^k)
and 9°Uk)
account for the effects of the
stiffening girders. The latter depends on the nodal displacements of the nodes at the stiffening girders. The deformation compatibility condition of the stiffening girders can be obtained from Eqs. (4.3.27) and (4.3.25). Substituting Eqs. (4.3.27b) and (4.3.25b) into Eq. (4.3.27a) and replacing the subscript (/', k) by (J, 1 + (k - \)p), we have
188
Exact Analysis of Bi-periodic Structures
jy
.2M ^n_
1
@f
-2u2JA;,*)(",v)>L \
(./,*)
+\&
"=1 v=l
7=1,2, •••,2M;A:=l,2,---,2n
(4.3.28)
where (4.3.29a) * J (,',*)
l-
*J(/,l+(*-l)/>)
ZM
(/,*)
(y,i+(*-i)p)
LIS
[A;,,)(U,v,] = -I^7£I^ 0 '-" ) r '^' ( i - v ) p s V 2 [Go] ( ,,o
(4.3.29b)
AMN
[Go](r,s) =
e
and 2(,,,)
1
(4.3.29c)
[G 0 ](2A,,2*)=[0]
In the compatibility equation (4.3.28), the unknown variables W(jk) and ®x(jyk) (j=\,2,---,2M)
denote the nodal displacements of the &-th stiffening girder. The
matrix [fiUyk)(uv) ] possesses cyclic periodicity, i.e. iPu.W.v) J ~ \-P(j+l,k)(2,v) J ~ ''" _ L/>(;-U)(2M,v) J
j=l,2,--,2M;k,v=l,2,---,2n
(4.3.30a)
and [/>U,*)(u,l) J = [ r(j,k+\)(u,2) J = ••' = lF(j,k-\)(u,2n)i j,u=l,2,---,2M;
(4.3.30b)
k=\,2,---,2n
The different matrices are [/?0i*)(u) ] (7=1,2,•••,2M; k-\,2,---,2n).
In
view of
Eq. (4.3.30), Eq. (4.3.28) can be uncoupled by using the double U-transformation. Let
Structures with Bi-periodicity in Two Directions
w,
=
6
' ^yye"^^-»>
189
(4.3.31a)
or n
1
e
=
,
2M 2„
!
yy e -'
with 97 = nIn = py/2 and
0'-l)Ti
-i(k-\)s
w
(4.3.31b) (y.t)
y/x=nlM. 2M
.
Premultiplying both sides of Eq. (4.3.28) by
In
yye-(;-')^,e-^V2MV2n ,=1 4=i
1)S(P
and applying the cyclic periodicity of [/3(yjt)(uv) ] , we have (2M
In
2
[
+
N
(4.3.32a)
with
6
2j ( r , s )
^£1.*-.*-^}
V2M
—>
It can be shown that {0}
(4.3.32c)
1} - »
(4.3.32d)
2 J (!M,1n)
and
I (2M,2n)
The vectors [Q,Qx]T(rs) and [fe,,ft2][rj) represent the generalized displacements of the stiffening girders with ^ * 0 and / = 0, respectively.
190
Exact Analysis of Bi-periodic Structures
Noting that
'. «
r = r a n d s =s
'
[0
' '
r'*rors'*s,
s+2n
'"''
S+
(P-V2»
s+2n,---, s+(p-\)2n (4.3.33)
with r=\,2,---,2M; s=\,2,••-,2n and substituting Eq. (4.3.29b) into Eq. (4.3.32a), we have
r=\,2,—,2M;s=\,2, — ,2n
(4.3.34)
Now the compatibility equation becomes a set of two-degree-of-freedom equations by using the double U-transformation. Introducing Eq. (4.3.29c) into Eq. (4.3.34), the solution for Q(rs) and Qx(rs) of Eq. (4.3.34) can be expressed as
0
•'l('-.s)
(4.3.35a)
l+ E
y Mr,s)
^2(r.s)
2x(r,j)
-
\ +y
"2(r,s)
"\(r,s)
(4.3.35b)
where
E E
2(r,s)
~
-If.
(4.3.36a)
e
(4.3.36b)
n^_t
2(r s+(*-l)2«)
When the structural parameters Kx, K2,
y, a, b , M , N (or n), p and
Structures with Bi-periodicity in Two Directions
191
the nodal loads are given, the generalized displacements Q{rs) and Qx(rs) can be calculated from the relevant formulas derived above. Noting the definitions of Wuk) and ®x(jk) given in Eq. (4.3.29a), and comparing Eqs. (4.3.25b) with (4.3.31b), we have
r,s+(*-l)2«) ~~
r,s+(k-l)2n) 1 n
r=l,2r",2M;s=l,2,-,2n; When g ( r s )
and Qx{rs)
f
k=\,2,-~,p
(4.3.37)
( r = l,2,---,2A/;s = l , 2 , - , 2 » ) are found,
{ q ^
{r=\,2,---,2M, s=l,2,---,2N) can be easily obtained from Eqs. (4.3.37) and (4.3.25). Therefore, substituting Eqs. (4.3.27b), (4.3.27c), (4.3.26) and (4.3.37) into Eq. (4.3.27a), the exact solution for w{jJc), 8x(jk)and 9y(jk) can be found. 4.3.3
Example
In order to explain the computational procedure and to verify the exactness of the derived formulas, a specific example of rectangular grid is chosen for detailed discussion. Consider a uniform square grid with a 6x6 mesh and two internal stiffening girders as shown in Fig. 4.3.4. The grid is simply supported at the four sides and subjected to a concentrated load with magnitude P acting at the center node. The structural parameters are given as M = N = 6,
Ki=K2=EIIal
p = 2,
=K,
a=b
(4.3.38a)
and y/x = y/2 = n/6,
(p = K/3 ,
n=3
(4.3.38b)
The external nodal loads for the original grid can be expressed as F{4A=P FUM=0,
(y,£)*(4,4);
(4.3.39a) j,k = l,2,-,7
(4.3.39b)
192
Exact Analysis ofBi-periodic Structures
y (1,2)
d.l)
(1,3)
(1,4)
(1,5)
(1,6)
(1,7)
t X (2,1)
(\+y)EI
^ a ^ (3,1)
a (4,1)
\ EI
\ P ) (4,4)
(4,7)
(5,1)
(6,1)
(7,1)
Figure 4.3.4
(7,7)
(7,4)
A grid with a 6 x 6 square mesh and two interior stiffening girders subjected to a concentrated load acting on the center node
Introducing Eqs. (4.3.38) and (4.3.39) into Eq. (4.3.23a), we have . P . m . sn J,.,, = sin — sin — Hr,s) 3 2 2
r,s = 1,2,-,12
(4.3.40)
Substituting Eqs. (4.3.38), (4.3.40), (4.3.26) and (4.3.25h) into Eq. (4.3.27c), all nodal displacements for the equivalent system with y - 0, namely w[. k), 9'x(J k) and 0'yUk), can be found in explicit form. The results for the original grid are summarized in Table 4.3.1. The other nodal displacements for the extended grid can
Structures with Bi-periodicity in Two Directions
193
be obtained from anti-symmetry, namely -%4-j,*) = -w('y,i4-*> = W ( M ) '
w,(14-;,14-t)
J'k
=1'2>">6
(4.3.41a)
^(14-y,14-t) - 0*0,14-*) = -0x(14-j,t) ~ ~^'xU,k) >
7> * = 1>2," " " ,6
^(i4-;,M-t) =03>(M-7,*)
7>* = 1 > 2 , " " , 6
= _
0yu,i4-*)
=-
0>(y,*)>
(4.3.41b)
(4.3.41c)
Substituting Eq. (4.3.38) and the results shown in Table 4.3.1 and Eq. (4.3.41) into Eq. (4.3.32b), we have
-'1(1,0
23 A{l + -±fi)^, 40 K J
U\2-r,6-s)
V
2(1,1)
23 bx<.x)=-A(\~£)£40
r
K
r=l,3,5; 5=1
\(\2-r,s\
\r s) ~ 0.
7
2A P ^ , 47 A
, =-V,6-,)=V,)>
u
~
=
is e v e n or 5 ^ 1,5
" _10 n f ( 74 + ^ ) 7aA F ' *w)=-pr-77. 47 aA
(4.3.42a) (4.3.42b) (4.3.24c)
ft
2(5,,)=—77T T->/3 . aK 10 1^4 (4.3.42d)
b
=-b2(r,6-s)
2(l2-r,s)
=-b2(l2-r,6-s)
^2(r,j)=0>
=
b
2(r,s)
>
^ =1,3,5;
S=l
r is even or 5^1,5
(4.3.42e)
(4.3.42f) (4.3.42g)
72
Inserting Eqs. (4.3.42), (4.3.36), (4.3.25e), (4.3.25f) and p=2 into Eq. (4.3.35), results in . ^0.1)
4^(40 + 23 V3-) , ™
o«
„-,
P
/T t- '
160 + 807-23y>/3 K
_
AA P_ .., „ 94 + 53y AT
^(3,1) - „ .
(.t.J.tOd;
194
Exact Analysis of Bi-periodic Structures
Table 4.3.1 k
[w[m/{P/K),
0'x(m/(P/aK),
0'y(j,k)/(P/aK)f
1
2
3
4
5
6
7
0
0
0
0
0
0
0
0
343 1880
2369 7520
681 1880
2369 7520
343 1880
0
0
0
0
0
0
0
0
0
2633 15040
115 376
7979 22560
115 376
2633 15040
0
0
2411 15040
23 80
2531 7520
23 80
2411 15040
0
343 1880
2411 15040
353 3760
0
0
115 376
529 960
0
353 3760
2369 7520
J
1
2
3
4
5
353 3760
2411 15040
92 141
529 960
115 376
0
2829 15040
234 940
2829 15040
353 3760
0
23 80
2829 15040
0
2829 15040
23 ~80
2369 ~7520
0
7979 22560
92 141
8999 11280
92 141
7979 22560
0
0
0
0
0
0
0
0
681 1880
2531 7520
234 940
0
234 ~940
2531 ~7520
681 ~1880
0
115 376
529 960
92 141
529 960
115 376
0
0
353 ~3760
2369 7520
23 80
2829 15040 2829 15040
343 ~ 1880
234 ~940
2829 15040
353 ~3760
0
0
2829 15040
23 ~80
2369 ~7520
(To be continued)
Structures with Bi-periodicity in Two Directions
195
(Continued)
k
1
2
3
4
5
6
7
0
2633 15040
115 376
7979 22560
115 376
2633 15040
0
23 ~80
2531 ~7520
23 ~80
2411 15040
353 "3760
2411 15040
J 6
2411 15040
0
7
0
343 1880
2411 15040
353 3760
0
0
0
0
0
0
0
0
343
2369 7520
681
2369 7520
343
0
1880
1880
0
1880
0
0
0
0
0
4^(40-23V3)
e ( 5 ,„
2(r,s)=0>
*»~
P
r =1,3,5; *=1
r is even or j * l , 5
12A-(7 + W3) P \60 + S0y-23ySaK'
. U
™
Q«n-r,s) = -QX(r,6-s) = -QX(u-rfi-s) = Q*(r,s), Qx(r,S)
=
0,
0
(4.3.43c) (4.3.43d)
™ P 94 + 53r aK
UAi(7-4S) P 160+ 80/+ 23^73 aK
2*(5,1)
343 1880
(4.3.43b)
160 + 80f + 23/73 K
fi(i2-,,6^)=-fi(i2-M)=-fi(r,6-,)=e(^,,
ti
0
_
^
^
(4.3.43Q
r =1,3,5; 5=1 (4.3.43g)
''is even or s * 1,5
(4.3.43h)
196
Exact A nalysis of Bi-periodic Structures
Finally, introducing Eqs. (4.3.37), (4.3.43), (4.3.25) and (4.3.38) into Eq. (4.3.27b), the effects of the stiffening girders, namely w(° A) , 0°xUk) a n d 0°yUk), c a n b e found. The nodal displacements for the original grid are symmetric, i.e., 7,A=1,2,3 a0 a
xU,t-k)
°y(*-],k)
ft" ' °*(j,k)
~ °y(j,k)
(4.3.44a)
&x(S-j,k) ~
"x(Z-j,i-k)>
j> * 1>2,3
(4.3.44b)
°y(ifi-k)
-el, >(8-;,8-4) :
j , *=1,2,3
(4.3.44c)
~
The results for nodes (j, k) (j, £=1,2,3,4) of a quarter of the grid are summarized in Table 4.3.2. The solution for the nodal displacements of the original grid can therefore be obtained from Eq. (4.3.27a) using the results given in Tables 4.3.1 and 4.3.2 together with Eq. (4.3.44). It can be proved that the results satisfy all of the nodal equilibrium equations and simply supported boundary condition without any error. Therefore, the solution is shown to be exact. Consider now the effect of y on the nodal displacements. When y approaches zero, the effects of the stiffening girders w(°.it), d°Uk)
and 0°y(jk) as shown in
Table 4.3.2 approach zero. Obviously the solution with y = 0 is applicable to the grid without stiffening girders.
When y approaches infinity, w(°y k), 0°xUk) a n d
Q°yU,k) approach finite values as their limits respectively. given in Table 4.3.3. W
These limiting values are
Comparing Table 4.3.1 and Table 4.3.3, it is obvious that
( M ) = ~WU,k) ••
°xU,k)
-6'*U.k) '
7 = l , 2 , - , 7 ; * = 3,5
(4.3.45)
which indicates that the stiffening girders have no flexural deformations. The solution with y approaching infinity is therefore also applicable to a continuous grid with equidistant line-supports. Finally, let us consider the deflections of the nodes on the axis of symmetry orthogonal to the stiffening girders. These nodal deflections can be expressed as
w(4>1) = 0 ,
w(4>2) =
529/
72
6400 + 4787/ 2 202?(/) COO.
7979 — 22560 K
(4.3.46a)
Structures with Bi-periodicity in Two Directions
"(1,3)
"(4,4)
23y_ 6452 + 4813^ 36 B(y) 529^ 6400+ 4787^ 36 20B(y) w(4.8-t) = w(4,t).
53 ) 92 -\ C(y) J 141
CM *=1,2,3
8999 11280
197
(4.3.46b)
(4.3.46c)
(4.3.46d)
where 5(y) = 25600 + 25600^ + 4813^ 2
(4.3.46e)
C{y) = 47(94 + 53^)
(4.3.46f)
The numerical results for w ( 4 t ) (A: = 1,2,•••,7) with increasing values of y are given in Table 4.3.4.
198
Exact Analysis of Bi-periodic Structures
Table 4.3.2 k
(a)
[wlk)/(P/K),0°xak)/(P/aK),0°yUk)/(P/aK)]T 1
(A=l, 2) 2
J 1
0 0
23/6440+4807/ 24 5B(r)a
0
0
0
529/6400 + 4787/ 72 40S(/)
0
4_ C(rY
23/45440 + 33829/ 24 40*00
0
22 C(/)
12167/(4 + 3/) 24*00
0
5 2 9 / 2 8 0 + 209/ 24 10*00
529/(640+479/) 160*(/)
+
_J_» C(/)
529/(280 + 211/) 80*(/)
°
529/6400 + 4787/ 72 205(y)
0
Notes:
2_ C(/)
529/(640 + 481/) 320fl(/)
2 3 / 2 1 2 0 + 1589/ 8 SB(r)
2 3 / 2 1 2 0 + 1589/ 4 5*(/)
23 C(r)b
|
2 CO')
0 (
2 C(/)
" * ( / ) = 25600 + 25600/ + 4 8 1 3 / 2 ;
23/45440+33829/ 24 20*(/) h
C(/) = 47(94 + 53/)
; +
22 C(/)
Structures with Bi-periodicity in Two Directions
Table 4.3.2
[w°(j,k)/(P/K),0°M)/(P/aK),0°yUk)/(P/aK)]J
(b)
53
23/45440+33691/ 24
20B(/)"
c(rY 53
23/6452 + 4813/ " 36 25(/)
c(r)
529/(640+481/) 1605(/)
Sir)
0
23/2120+1589/ 8 105(/)
C(r)
529/(6400 + 4813/) 960fl(/) 23/45440+33691/ 24 40B(y)
12167/(4 + 3/) l2B(y)
53 C(r)
+
5 2 9 / 2 8 0 + 209/ 24 5B(r)
529/(640 + 479/) 3205(/) 23/6452 + 4813/
fi(/)
53 +
23/2120 + 1589/ 55(/)
+_
c(rY
0
529/6400 + 4787/ 36 205(/)
C(rY
0
Notes:
c(r)
529/6400 + 4787/ __ ) 36 405(/) C(/)
)
23/(6360 + 4813/)
" 36
5*00
1_,
0
0
80
(A=3, 4)
529/ 280+209/ 12
0 0
2 C(/)'
" £ ( / ) = 25600+25600/ + 4813/ 2 ;
fc
199
C(/) s 47(94 + 53/)
{
2 C(/)'
200
Exact Analysis of Bi-periodic Structures
Table 4.3.3
[w^/iP/^^^/iP/a^^^/iP/aK)]1
(a)
when y —> oo , &=1,2 k
1
2
7 1 529 209 1 ' 24 24065 2491 0 529 4787 2_ ' 72 192520 2491 529 481 -x320 4813 23 1589 4 ~ Y 24065 ~ 2491 0 0 529 479 -x 160 4813
23 33829 22 ~ 24 192520 ~~ 2491 12167 38504 529 209 1 ~ 24 (48130 + 2491 529 211 x80 4813
529 4787 2 " 72 (96260 + 2491
0 23 1589 2 " 4 24065+ 2491
23 33829 22 ~ 24 96260+ 2491
Structures with Bi-periodicity in Two Directions
Table4.3.3 (b) [ w ' J f f / D . O j . J t J ' M l . O ; , , / ^ ] ' when y -> oo , k
k=3, 4
3
4
0
0
J 1
2369 ~7520 0
115 376 23 80
529 209 _ 1 ~ 12 24065 ~ 2491 0
529 4787 2_ ~ 36 192520 ~ 2491 529 481 • x 160 4813
23 1589 2_ 8 48130^2491
0
529
12167
960
19252
2829 15040
529 209 2 ~ 24 24065 + 2491
529 479 320 4813
0
92 141
529 4787 2 ) 36 96260 + 2491
0 23 1589 2 8 24065 ' 2491
0 0
201
202
Exact Analysis ofBi-periodic Structures
Table 4.3.4
The effect of y on
w(4k)/(P/K)
r 0
0
0.35367908
0.65248227
0.79778369
0.65248227
0.35367908
0
i
0
0.27818231
0.51909174
0.64679016
0.51909174
0.27818231
0
2
0
0.22836407
0.43139247
0.54715367
0.43139247
0.22836407
0
5
0
0.14591315
0.28656658
0.38225183
0.28656658
0.14591315
0
10
0
0.087341033
0.18384088
0.26510760
0.18384088
0.087341033
0
50
0
0.0095569068
0.047550913
0.10953935
0.047550913
0.0095569068
0
100
0 -0.0035018672 0.024681044
0.083421798
0.024681044 -0.0035018672
0
103
0
-0.016136958 0.0025557357
0.058151617
0.0025557357
-0.016136958
0
104
0
-0.017450069 0.00025648434
0.055525394
0.00025648434 -0.017450069
0
00
0
-0.017596549
0
0.055232433
0
-0.017596549
0
Chapter 5 NEARLY PERIODIC SYSTEMS WITH NONLINEAR DISORDERS The mode localization phenomenon in infinite periodic mass-spring systems with one disorder was investigated by Cai et al [20] using the U-transformation technique. The localized mode in mono-coupled periodic mass-spring systems having two nonlinear disorders was studied [21] by means of the U-transformation and L-P method. Recently the same method was applied to analyze the localized modes in a twodegree-coupling periodic system with a nonlinear disordered subsystem [22] and the forced vibration for the damped periodic system having one nonlinear disorder [23]. 5.1
Periodic System with Nonlinear Disorders - Mono-coupled System [21]
Consider the system shown in Fig. 5.1.1. It consists of infinite number of subsystems connected to each other by means of the linear springs having stiffness ekc. Each subsystem consists of a mass M connected to a rigid foundation by a spring with linear stiffness (for ordered ones) or nonlinear stiffness (for disordered ones). In Fig. 5.1.1, j \ and j 2 denote the ordinal number of the disordered subsystem.
h ~ 1 -A
J, +1
Figure 5.1.1
h ~1
J\
j 7 +1
Periodic system with nonlinear disorders
203
204
Exact Analysis of Bi-periodic Structures
It is assumed that the system energy is large enough in order to realize the localized motion mode and both the coupling stiffness ekc and the coefficient sy of the cubic term of the nonlinear stiffness in the disordered subsystems are weak, i.e., £ is a positive nondimensional small parameter. This section is aimed at analyzing the mode localization phenomena. The localized modes of the system with infinite number of subsystems are hardly affected by the conditions at infinity. Consequently the system under consideration may be regarded as a cyclic periodic one. 5.1.1
Governing Equation
At the outset a cyclic periodic system with N number of subsystems is considered. Then by adopting a limiting process with N approaching infinity, the governing equation will be applicable for a cyclic periodic system with infinite number of subsystems. Applying Newton's second law to every mass M, we can write the differential equations of motion as follows: Afxk+Kxk-ekc(xk+1+xk_l-2xk) Mxk+Kxk-£kc(xk+l+xk_i-2xk) where
xk
=0 + £)xl=0
k*ji,j2 k = j„j2
(5.1.1a) (5.1.1b)
denotes the longitudinal displacement of the &-th mass and
xN+i = x,, x0 = xN due to cyclic periodicity. The superior double dot denotes the second derivative with respect to the time variable t. K and M denote the stiffness and mass for the ordered subsystems. The nonlinear terms play the role of disorder. Introducing the time substitution x = cot
(5.1.2)
into Eq. (5.1.1) results in k 0}2xl+o)lxk=£-f-(xk+i+xk_]-2xk)
co2x"k+colxk = e-f-(xk+l+xk_,-2xk)-e M
k*jx,j2
—x\ M
k = j„j2
where the prime sign designates differentiation with respect to T and
(5.1.3a)
(5.1.3b)
Nearly Periodic Systems with Nonlinear Disorders
205
M The co in Eqs. (5.1.2) and (5.1.3) is the fundamental frequency to be determined. The periodicity condition may be expressed as xk(T + 2n) = xk(r)
k = \,2,-,N
(5.1.5)
According to the L-P method, the solution of Eq. (5.1.3) is assumed to have the form xk(r) = xk0(T) + exk](r) + s2xk2(T) + -
k = l,2,-,N
(5.1.6)
and co is given as w = a>0+£G)l+£2co2+---
(5.1.7)
Substituting Eqs. (5.1.6) and (5.1.7) into Eq. (5.1.3), the coefficients of equal powers of S on both sides of Eq. (5.1.3) must be equal, i.e., o\x\\ +
k = l,2,-,N
(5.1.8a)
co2x"k, + co2xkl = -2fi>0<w,x;0 + T H * * + I , O + *t-i.o - 2xk,o) M
* * 7i. Ji
k 0)2xnk] +co2xkl = -2fiv»X 0 +-£-(* M
y _2JC 4 ) - - — j£ 0 M
+x
* =j,J2
(5.1.8b)
(5.1.8c)
Without loss of generality, we can assume that the initial velocity for each subsystem is zero, i.e., <;(0) = 0
y = 0 , l , - ; k=l,2,-,N
Then the solution of Eq. (5.1.8a) may be expressed as
(5.1.9)
206
Exact Analysis of Bi-periodic Structures
xko=Akocosr
(5.1.10)
k = l,2,---,N
Inserting Eq. (5.1.10) into Eqs. (5.1.8b) and (5.1.8c) results in
®o(*ti+**i) = 2eo0a,Ak0
+ -77(^+1,0 + 4M,O - 2 A k , o ) M
(5.1.11a)
k*ji,j2
and
<»o(*M+**l) =
k i \ 3v 2a)0(olAk0 + 7 7 ^ + 1 , 0 + ^t-1,0 - 2 / 4 t 0 ) - — — Ak0 COST M AM Y 4
M
^.3O cos3r *,0
k = j.,j2
(5.1.11b)
Jt>J2
In the second equation the identical relation cos3 r = —(3cosx + cos3r) has been used. In order to eliminate the secular term from xkx (k = l,2,---,N ) we must set the coefficients of COST on the right hand sides of Eqs. (5.1.11a) and (5.1.11b) equal to zero, i.e., 2(o0wlAk0+-£-{Ak+l0 M 1
2o>o<Mto
+1±(A+I,O
M
+ Ak_i0-2Akt0)
=0
k^j„j2
(5.1.12a)
* =J\Ji
(5.1.12b)
-y
+^-,,o - 2 A , o ) - T T 7 < o = 0 4M
One can now apply the U-transformation technique to Eq. (5.1.12). The U and inverse U transformations may be expressed as ,N
(5.1.13a)
Nearly Periodic Systems with Nonlinear Disorders
-l)mlf/
*m,o
m = \X--,N
207
(5.1.13b)
JN{
with y/ = — , i = v - 1
and a^.,, = aOT in which the superior bar denotes
complex conjugation. By using the U-transformation, Eq. (5.1.12) becomes 2co0colam0+^(cosmW-\)am0-^--L £ M 4M y/N kt£h
^ ^ , = 0
m = l,2,-,N
(5.1.14)
Consequently am0 takes the form as
=
a„n
at.
• L ^ - l + cosm J
- j LV
e^-»-"^0
(5.1.15)
Substituting Eq. (5.1.15) into Eq. (5.1.13a) yields
•"to
_LV
—
N4^
8*.
eHk-j)my
M (Da(Ox
. 1 + COS/M^
4,
(5.1.16a)
Noting that the imaginary part of the summation of the series in the square brackets is equal to zero, Eq. (5.1.16a) may be rewritten as
A o ~~
Zy 8£
£ t
-.
— 7
cos(£ - j)m w\ a>na>,
1 + cos m w
A%
(5.1.16b)
By letting N approach infinity, the limit of the series summation in the square brackets on the right hand side of Eq. (5.1.16b) becomes the definite integral [5], i.e.,
208
Exact Analysis of Bi-periodic Structures
lim — /
cos(& - j)m y/
n->oo A/ *-^ ' y m=\
. 1 + cos m y/
c
/
\
1 r*
=-
M G)0a>,
(
\ cos{k - j)6 coQ(o,
M
l + cos<9
d6 = p{k:,
(5.1.17)
If M ^0^1
(5.1.18)
>1
the definite integral shown in Eq. (5.1.17) is in existence. The condition (5.1.18) is equivalent to 2
o>
2
or
>(o„
(5.1.19)
where w1 =a\ +2CO0CO1£ + O(E1) and cou, coL denote the upper and lower limits of the pass band for the ordered system (y = 0), i.e.,
co„
K M
4afc„ M
K_ M
(5.1.20)
Eq. (5.1.19) indicates that a lies in the stop band of the ordered periodic system. Because the frequency corresponding to localized mode must lie in the stop band, we have no interest for the case that the integral shown in Eq. (5.1.17) is not in existence. Consider now the system having infinite number of subsystems. Consequently Eq. (5.1.16b) becomes A
"=jr\p«-MAl*+p«-hA\
*=u,-,«>
(5.1.2D
where /? (t _ ;) has been defined by Eq. (5.1.17). Setting £ = _/', and ^respectively, yield
Nearly Periodic Systems with Nonlinear Disorders
209
^,o~k4o+4,w2)4o]
(5.122a)
^=|k-/,A+Hl
< 5 - i - 22b )
and
The unknown amplitudes Ak0 can be obtained by substituting the nontrivial solution of Eq. (5.1.22) into Eq. (5.1.21). Before we discuss the solutions of Eqs. (5.1.22a) and (5.1.22b), let us calculate the definite integral A shown in Eq. (5.1.17). l A =- f —Ed0 = s&n(")-r== n Jo a + cos0 Ja* _l
Pi=P-i=-\*-^-de n Jo a + cos0
= \-apo
H >1 ''
(5-1-23)
\a\>\
SW d0 fi2=fi-2=~[n Jo aC+° cos0 = ~ 2 a + (2«2 ~ 1)A a
A = A-3 = ~ f C O S 3 ^ d9 = 4a2 - 1 - a(4a2 -3) A ^ Jo a + cos0
(5.1.24)
H > 1 (5-1-25) |a|>l (5.1.26)
where a=co0col
1
(5.1.27)
It can be verified that
A =A =A =£ A A A where
( 5.i.2 8)
210
Exact A nalysis ofBi-periodic Structures
% = -a + sgn(a)Va 2 -l
\a\ > 1
(5.1.29)
It is obvious that £ and a are of opposite sign and |^|<15.1.2
Localized Modes in the System with One Nonlinear Disorder
The periodic system with infinite number of subsystems in which only one subsystem having nonlinear stiffness is considered. The governing equation (5.1.22) can be reduced to
^o=^-fio4o
(5-130)
There is a nontrivial solution of the above equation if and only if ^>M;,O=1
8*.
(5.1.31)
Inserting Eqs. (5.1.23) and (5.1.27) into Eq. (5.1.31) results in
K M
(5.1.32) *7
where >7 = f ^ 2 o Noting that co2 = col + ^eco^ spring) co2 > — + MM
+ 0(s2),
(5-1.33)
it can be proved that if r > 0 (hardening
and r < 0 (softening spring) a2 < — . Consequently M
we can come to the conclusion that if y > 0 there is one localized mode with a greater than
Nearly Periodic Systems with Nonlinear Disorders
^ o ^ ^ o ^ / M l o ^ o
211
(5.134a)
4 - w = V2.0 = ^ " / M j o = £ 2 4„o
(5.L34b)
4,+*,o = 4,-M = | f - ^ 4 o = tkAj,.o
(5.1-34C)
It indicates that the amplitudes decay exponentially on either side of the nonlinear disorder. The attenuation rate £ of localized mode may be found by substituting Eqs. (5.1.27) and (5.1.32) into Eq. (5.1.29) as 4 = ^-yl^n)/rj
(5.1.35)
4 is the odd function of nondimensional parameter TJ . When TJ approaches zero, £ also approaches zero. It indicates that if kc decreases or y increases or Aj
0
increases, the mode will approach strongly localized state. The localized level of the modes is dependent on not only the structural parameters kc and y but also the amplitude Aj
0
related to the initial condition or the total energy of system. Aj
0
can be determined by applying the energy conservation to the considered system. Ignoring the 0(e) term it can be proved that the following equation A
lo+
Z f c w
+
i w l = ^
(constant)
(5.1.36)
*=1,2,...
is a necessary condition for the energy conservation, p1 is an energy-like quantity which is related to the total energy E of the system by
—Kp2+ 0(e) = E .
Inserting Eqs. (5.1.34), (5.1.35) and (5.1.33) into Eq. (5.1.36) results in
212
Exact Analysis of Bi-periodic Structures
8^
3r
(5.1.37)
which indicates that when kc/y is given, in order that a localized mode would occur, however small the amplitude, the energy constant p2 must be greater than |8Arc /3x| • If the system energy is below the required level it is impossible to realize any localized mode. This property is different from that for the linear periodic system. 5.1.3
Localized Modes in the System with Two Nonlinear Disorders
For the system under consideration, the governing equation (5.1.22) becomes
+ ^o=^^k» ^J 8k
(5.1.38a)
^0=1^0(^,0+40)
(5.1.38b)
in which n = |_/, - j21 and 4 has been defined by Eqs. (5.1.29) and (5.1.27). Eq. (5.1.38a) minus and plus Eq. (5.1.38b) yield
4o-4o~/?o(i-r)fe-4o) 8k,
(5.1.39a)
4o+4o~/?o(i+r)(4,o+4o)
(5.1.39b)
The above equations are equivalent to 4o=4o
4o-"4o
or
^-^o(l-r)(4,o + 4 o 4 o + 4 o ) = 1 8*.
or ^ ^ 0 ( l + ^ ) ( ^ 0 - ^ i 0 ^ 0
+
40)=l
(5.1.40a)
(5.1.40b)
Nearly Periodic Systems with Nonlinear Disorders
213
respectively. The simultaneous equations (5.1.40) may be divided into three sets of equations, i.e, (i)
A
(5.1.41a)
A
J,o -
h*
^&(l-4")(40+AJi0AJ20+Al0)=l (ii)
A
j,o ~
(5.1.42a)
A
j2o
r)(4,o-^,o^o+4o)=l
(5.1.42b)
+ =1 ^Ai-r)fe+v A » 4.) 8*
(5.1.43a)
fA(i+ffc.-V.o+4o)= 1
(5.1.43b)
^^0(l 8* (iii)
(5.1.41b)
+
Let us discuss in detail the case of j 2 = y, + 1 , i.e., n = 1. (i) The first set of equations, i.e., Eq. (5.1.41), describes the symmetric localized modes. Inserting Eq. (5.1.41a) and « = 1 into Eq. (5.1.41b) yields
If-AO-*;.-! Recalling Eq. (5.1.33), substituting Eq. (5.1.44) results in
2K 1 + - 1
conw, = •
M
77(77 + 2 )
(5.1.44)
Eqs. (5.1.23), (5.1.27) and (5.1.29)
AM
77 +
1 ?7 + 2
A)fi,
77>-l
into
(5.1.45)
where 77 is the nondimensional parameter as shown in Eq. (5.1.33). It is obvious that if 77 > 0 (i.e., y>0),
Ik (o0col >—- (i.e., a> xou), M
there is a symmetric
214
Exact A nalysis of Bi-periodic Structures
localized mode with higher frequency (i.e., co > a>u) and if —1 < 77 < 0, co
the symmetric localized mode can
be obtained from Eq. (5.1.21) as Aj^0=-Ah+k,0
=^-(fc -A+,Ko = m , 0 ,
k = 0,1,2,...,co
(5.1.46)
which indicates that the amplitudes on either side decay exponentially away from the nonlinear disorders. The attenuation rate £ of the amplitudes of symmetric mode may be found by substituting Eqs. (5.1.27) and (5.1.45) into Eq. (5.1.29), as
!/>-!
£ = - 77 + 2
(5.1.47)
The sum of the squares of the amplitudes should be equal to constant p2, i.e.,
24 | 0 (l + « f + £ 4 +
) = p2
+
or (5.1.48)
Inserting Eqs. (5.1.47) and (5.1.33) into the above equation results in
A* - i
p2>-—^
16*
-+jp4 +
3r p2>
m
<~ ly
2
(for y>0) (5.1.49) 2y 3y In order that the symmetric localized mode may occur, p2 must be greater than max
32*, 4*, Zy 1y
(for y<0),
Nearly Periodic Systems with Nonlinear Disorders
215
The stability analysis of the symmetric localized mode shown in Eqs. (5.1.46) and (5.1.47) is investigated by superposing the small perturbations on the mode solution and applying the Floquet theory [24]. It is concluded that when 77 lies in the region [0, 2], the symmetric localized mode is unstable and when 77 > 2 or —1 < 77 < 0 the symmetric localized mode is stable. If 77S-I, the symmetric localized mode does not exist. (ii) The second set of equations shown in Eq. (5.1.42) describes the antisymmetric localized modes. Inserting Eq. (5.1.42a) and n-\ into Eq. (5.1.42b) yields ^yS0(l +^
i 0
= l
(5.1.50)
Substituting Eqs. (5.1.23), (5.1.27) and (5.1.29) into Eq. (5.1.50) results in 2k
c 1 fi\>a>, = - 7 7 - —
3f 1 r = T777
.2 ^,,0.
7<1
... (5.1.51)
Recalling 77 = —-A'X, it is concluded that if 77 < 0 (/ <0), ft)0t», < 0 (a> < atL) 3>y there is an antisymmetric localized mode with lower frequency (co < coL ); and if 2k 0 < 77 < 1, o)0ct)l > — - (fflxa,,) there is an antisymmetric localized mode with M higher frequency (co>o}u). The amplitudes for the antisymmetric localized mode can be found by inserting j 2 = jx +1 and AJi0 = AJi0 into Eq. (5.1.21) as 4,-*.o = AJ,«.o = | f (&
+
A + iKo = <^/,o •
k
= 0,l,2,...,co
(5.1.52)
Substituting Eqs. (5.1.27) and (5.1.51) into Eq. (5.1.29) results in Z = -TL-
n
(5-1.53)
2-77 For the present case Eq. (5.1.48) is also applicable in which £ should be defined as
216
Exact Analysis of Bi-periodic Structures
in Eq. (5.1.53) instead of Eq. (5.1.47). Substituting Eqs. (5.1.53) and (5.1.33) into Eq. (5.1.48) results in
A2
32*,
2
-I
2
16fc c
3/
3y
p2 >
(fory>0),
3r
32kc
I 4
2
P
Ah
£3r
(for ^ < 0 )
(5.1.54)
In order that the antisymmetric localized mode may occur p2 must be greater than ~32k max
c_
3/
4k, c_
3y
By applying the Floquet theory to the stability analysis of the antisymmetric localized mode shown in Eqs. (5.1.52) and (5.1.53), it is concluded that when 77 lies in the region [-2, 0], the antisymmetric localized mode is unstable, and when 77 < - 2 or 0 < 77 < 1 the antisymmetric localized mode is stable. If 77 > 1, it is impossible for the antisymmetric localized mode to occur. (iii) The third set of equations shown in Eq. (5.1.43) describes the nonsymmetric localized modes. By inserting Eqs. (5.1.23), (5.1.29) and w = l into Eq. (5.1.43), it can be proved that Eq. (5.1.43) is equivalent to
h°AhO
or
A
3r
A1 + 4Z
-
4^ 3y
A,_ =-±A,. V 2 0n
fl + sgn(a)Va 2 -lj
where a is defined as Eq. (5.1.27). Without loss of generality, we assume U . 0 > U . J Ah0
;,o
(5.1.55a)
(5.1.55b)
(|T7|<2).
Eliminating
from Eq. (5.1.55b) and substituting Eq. (5.1.27) into it results in 2
2K 1 + (77 -277 + 4) 2 M
877(77 +4)
2
|T7|<2
(5.1.56)
Nearly Periodic Systems with Nonlinear Disorders
where 77 is given in Eq. (5.1.33). It is concluded that if y>0 ( (o>(o{J
I
217
(77 > 0) then
2k } coaw{ > — - , therefore the non-symmetric localized mode with higher
M)
frequency ( a >
* = 0,1A...,«
(5.1.57a)
Ah+kfi
* = 0,1,2,.., co
(5.1.57b)
and = | f (flw^o + A 4 o ) .
Recalling Eqs. (5.1.28) and (5.1.38) with n -1, Eq. (5.1.57) may be rewritten as Ah_kfi=l;kAha,
fc
= 0,1,2,..., co
(5.1.58a)
k = 0,l,2,...,co
(5.1.58b)
and Ah+kfi
= SkAha,
where AJ0 and ^ ; z 0 must satisfy Eq. (5.1.55a). Substituting Eqs. (5.1.56) and (5.1.27) into Eq. (5.1.29) results in
? = --¥-,
\ri\<2
7+4 Noting Eq. (5.1.58), the sum of the amplitude squares may be expressed as
Inserting Eqs. (5.1.59) and (5.1.55a) in the above equation results in
(5.1.59)
218
Exact Analysis of Bi-periodic Structures
2kc
(>r + 4) 3 4
3/
_
2
ij(Tj +4r] +\6)
A
mode3
2
H**
(5.1.61)
Ajzo/Aj,o
mode2
mode2
mode3
£> 7) model
model
Figure 5.1.2
Localized modes \A^QJAjQ)
mode3
versus //
8*,
^ T
mode 1 — symmetric mode mode 2 — antisymmetric mode mode 3 — non-symmetric mode — stable mode; unstable mode
1
17
^
\early Periodic Systems with Nonlinear Disorders
219
Sk
Recalling ;/ = — - A ~ ~ 0 > >l ls impossible to find the closed form solution for A2J0 of 32£
Eq. (5.1.61).
But it can be proved that if and only if p~ >
, the solution of 9y
Eq. (5.1.61) exists, i.e., there are two non-symmetric localized modes \^;,o/^ ;i o and
Aji0/Ah0=-rj/2).
It can be concluded that when U < 2 in which t] s — - A^ there are two 3/ non-symmetric stable localized modes where one mode is the symmetric image of the other. The condition for each type of localized mode to occur is shown in Fig. 5.1.2. 5.2
Periodic System with One Nonlinear Disorder — Two-degree-coupling System [22]
Consider the two-degree-coupling periodic system with infinite number of subsystems as shown in Fig. 5.2.1, where K and M denote the linear stiffness and mass for the ordered subsystem respectively; skc and 2dcc denote the stiffness for two kinds of coupling springs in longitudinal and inclined directions respectively; s denotes the ordinal number of the disordered subsystem with nonlinear stiffness.
Figure 5.2.1
Two-coupling periodic system with a = n 14
220
Exact Analysis of Bi-periodic Structures
It is assumed that the system energy is large enough to realize the localized motion mode and that the coupling stiffness and the coefficient ey0 of the cubic term of the nonlinear stiffness in the disordered subsystem are weak, i.e., s is a positive nondimensional small parameter. This section is aimed at analyzing the localized modes. The localized modes in the system with infinite number of subsystems are hardly affected by the conditions at infinity. Consequently the system under consideration may be regarded as a cyclic periodic one. At the outset a cyclic periodic system with TV number of subsystems is considered. Then by adopting a limiting process with N approaching infinity, the limiting solution will be applicable to the system having infinite number of subsystems. 5.2.1
Governing Equation
Applying Newton's second law to every mass M, the differential equations of motion can be expressed as MxXj + KxXj + skc (4xw - x1J+1 - *„•_, - x2J+l - x2M) = 0 j = l,2,—,N
and j*s
(5.2.1a)
Mx\j + Kx2j + d.c {4x2j - x2J+i - x2M - xlJ+l - xlM) = 0 j = \,2,---,N
and j*s
(5.2.1b)
A^i, + Kxu + ^c ( 4 *i, - *i,+! ~ *i,-i - ^2s+i - *2*-i) = -£7^i
(5.2.1 c)
Mx2s + Kx2s + £kc(4x2s -x2s+x - x j , . , -xu+1 -xt,_t) = -ey0xls
(5.2.1d)
where xXj and x2j denote the longitudinal displacements of two masses in the y'-th subsystem and xkJJ+l=xhl,
xk0=xkN
(^=1,2) due to cyclic periodicity. The
superior double dot denotes the second derivative with respect to the time variable t. The nonlinear terms play the role of "load" in the above equation. In order to apply the L-P method, it is necessary to transfer Eq. (5.2.1) into the standard form with the uncoupled linear terms. One can now apply the U-transformation to Eq. (5.2.1). The U- and inverse Utransformations may be expressed as
Nearly Periodic Systems with Nonlinear Disorders
1
221
JV
jy^""^
k = l,2;j = l,2,-,N
(5.2.2a)
Jfc = l,2;iH = l , 2 , - , t f
(5.2.2b)
and N
1km
* JVu-U-i^
1K_
with w = — and i = V-T . Y N N Applying the U-transformation to Eq. (5.2.1), i.e., premultiplying both sides of 1
Eq. (5.2.1) by the operator - = Y e'iU~r)mw, we obtain
2
2
*
M =
—e-'l'-^rfLLrUa
a
••• a 1
m = l,2,~,N
(5.2.3a)
, 2k Vim +(oaR2m +e—f-[-(cosmyr)qlm M r-~
e
, .
+(2-cosmy/)q2m]
•*2JW21>(/22>
'llN)
m = l,2,---,N
(5.2.3b)
where <w0 denotes the natural frequency of the single subsystem, i.e., a»o2=7T M
(5-2-4)
Conveniently the frequency equation for the linear system corresponding to the differential equation (5.2.3) with y0 vanishing can be obtained as
222
Exact Analysis ofBi-periodic Structures
0)1 -CO2 +
Ilk,
( 2 - c o s my/-)
M 2sfc
M
co\-co2
-cosmtf/
2A, -cosmy/ M =0 2skc + (2-cosm^) M
(5.2.5)
where a> denotes the natural frequency of the ordered system. The solutions for co2 of Eq. (5.2.5) are
co2=(o20+
4st
c
-,
co2=col+
M
4fife ^(1-cosmy) M
m =l,2,---,Af (5.2.6)
The lower and upper bound (coL and cov ) of the pass band can be obtained from Eq. (5.2.6), as 2 2 co J =a> L~UJrs>
col=co2 +-
<"u ~ •"<>n
Sek^ c M
(5.2.7)
Introducing the time substitution (5.2.8)
T = COt
into Eq. (5.2.3) results in °> iL
+
+co
o
1 e-^""
?±xl(qu,qn,-,qw)\
2k. 0 qlm + °>0 llm = ~£ j ~J7"t" (C0S m ^llm
+ ^e^-^^xi(q2],q22,.-.,q2N)\
+
m=l,2,-,N
+( 2
(5.2.9a)
~ C ° S m ^2m 1 +
m = l,2,-,N
(5.2.9b)
in which co denotes the fundamental frequency to be determined and the prime
Nearly Periodic Systems with Nonlinear Disorders
223
symbol designates differentiation with respect to the new time variable x . According to the L-P method, the solution of Eq. (5.2.9) is assumed to have the form 9 fen =9t ra .o+ fi ?*m,i + ^ 2 9 tm ,2+-"
* = l,2;i» = l , 2 , - , t f
(5.2.10)
and a is given as
(5.2.11)
Substituting Eqs. (5.2.10) and (5.2.11) into Eq. (5.2.9), the coefficients of equal powers of 8 on both sides of Eq. (5.2.9) must be equal, i.e., ®o?iVo+«»o?ta.o=0
m = \,2,--,N
(5.2.12a)
«»o?L,o+«»o?ta,o=0
m=\,2,-,N
(5.2.12b)
2k <»o9iVi + a,o?L..i = -2flv»i0ta,o - — H ( 2 - c o s m y O ? l l M M }_Zo_e-H'-l)my
1 /
.
-(cosmy/)q2m0]
x
m = \,2,-,N ®o?L.i
+fi,
(5.2.13a)
2k o?2«,i = - ^ o ^ m . o — r r [ - ( c o s w i ^ l m , 0 + ( 2 - c o s m ^ 2 m 0 ] --^—Z°-e'i(s'i)mvxr' e
/-— , ,
(a
a
•*2s,0W21,0>l/22,0>
-a
1 >H2Nfi)
•sJN M
m = \£ — ,N
(5.2.13b)
where xUfi and x2sfi denote zero-order approximation for xu and x2s which can
224
Exact Analysis of Bi-periodic Structures
be expressed in terms of the generalized displacements with zero-order approximation as =_
1
N
^I,\0 •" T= Ee'0",)"'^.o 4N •m=\ V
r
,o = -j^e^'q^
J = h2,-,N
(5.2.14a)
j = \,2-,N
(5.2.14b)
Without loss of generality, we can assume that the initial velocity for each mass is equal to zero, i.e.,
5.2.2
<7,Vo(0) = 0, <7L,o(0) = 0
m = l,2,-,N
C ( ° ) = °> lLA°) = °
m = \,2,-,N
(5.2.15a)
(5.2.15b)
Perturbation Solution
The solution for q]nQ and q2mfi of Eq. (5.2.12) with initial conditions shown in Eq. (5.2.15a) can be expressed as 9)m,o =«i»,,ocosz-
m = \,2, — ,N
(5.2.16a)
tit = 1,2, —,N
(5.2.16b)
*i;,o=4,-,ocosr
7 = 1,2," •,N
(5.2.17a)
X
7 = 1,2,--,N
(5.2.17b)
l2m,0 =a2m,0COST
Inserting Eq. (5.2.16) in Eq. (5.2.14) yields
2j,0
= ^2j,0 C
0 S T
where
V=^i>' ( '~ , ) " r f l .»,o
(5-2-18a>
Nearly Periodic Systems with Nonlinear Disorders
A
u,o=-)=1te>U~l)m¥aw
225
(5.2.18b)
Substituting Eq. (5.2.17) into the right hand sides of Eq. (5.2.13) results in ©oV-.i + ®foi*.i = J2ffl0<»i
3^0
1
g„-i(s-l)m - l - D¥- r j^i
1„„„
Xo 1 „-i(j-l)m|y L C 0 S T __L2__^ = r e -i'-i)»r j i
cos3r
m = l,2,-,tf
(5.2.19a)
r 2)t fi'o^L.i +fl>o92m.i = J2o06J,a2m>0 —jjj-[-(cos»i^)a lMi0 +(2-cosm^)a 2 M j 0 ]
-^-J= e-'W'Al 0 cos r - - ^ - - J = e-' ( '- ,) "M2, 0 cos 3r 4A/ VJV J 4M Jfi m=l,2,-,tf
(5.2.19b)
In order to eliminate the secular terms from qlm, and q2ml, the coefficients of COST ontheright sides of Eq. (5.2.19) must be equal to zero, i.e., [2^,
2* c / „ .,-, ,2*e, . 3/0 1 3 - - M 2 - c o s ^ ) K , , 0 + - ^ ( c o s m ^ ) a 2 m , 0 = - i 2 - _ e - < i(s-l)m(C ' - « » ^A.ls,0 M M 4M T]N m = l,2, — ,N
—^-(cos/«^)a, m ,o+[2«o«i—— (2-cos>w^)]fl2M0 = _-e Af M 4M y/N m=\X
— ,N
The unknowns alm 0 and a2m 0 can be expressed formally as
(5.2.20) M2
0
(5.2.21)
226
Exact Analysis of Bi-periodic Structures
n
-
lw
a
A„
= 2m A.
(5 2 21)
where 4k A m = ( — c - ) 2 C l ( n + cosmif/) M Alm =^^e-^l^(^)[(n 4M -JN M
+
(5.2.23a)
^cosmW)Al0-UcosmW)Al0] 2 I
A2m =^^e-i^^^)[-Ucosm¥)Al0+(Q 4M -JN M 2
+
\cosm¥)Ai0] 2
(5.2.23b)
(5.2.23c)
Here me nondimensional parameter Q is defined as Q =«„<»,— -1
(5.2.24)
Substituting Eqs. (5.2.22) and (5.2.23) into Eq. (5.2.18) results in
4y,o=/VC+«>
(5.2.25b)
in which a
. ;I
=
i ^ l l V [ e - 0 - ) - 20 + c o s ^ 32)tcQA^^ Q + cosro^
'* =- | ^ i l > ' 32kc Q N j ^ (
0
^ ^ ^ - ] fi + cosm^
(5.2.26b)
Consider now the system with infinite number of subsystems. By letting TV approach infinity, the limits of the series summation on the right sides of Eq. (5.2.26) become the definite integral, i.e.,
Nearly Periodic Systems with Nonlinear Disorders
limite^)-^±cosj^ = _ L | ^ °° N j ^ i Q + cosm^ In Jo
Ar_>
20 + c o . f l ^ Q + cos# (5.2.27a)
= SJ,+QBJ_
N
^"° N ~
fi
+ cos/n^
227
2;r Jo
Q + cos0
= Sj,-nBj_
(5.2.27b)
where
**=L y s
~
2* Jo
(5-2-28)
. Q + cos6»
The definite integral Bk can be expressed in terms of the elementary functions as
B
° = T " P r , ! /»rfg 2;r Jo Q + cos#
=
sgn(")-r= M >! vft - 1
1 f2ff cos $ 5., = 5 , = — f -«W = l - n * 0 2n Jo Q + cos0 B2=B2=—
1 c2n cos 2/9 d& = -2Q + (2Q2-l)B0 2;r Jo Q + cos#
(5.2.30a)
Q>1
(5.2.30b)
Q >1
(5.2.30c)
|Q| > 1 is necessary and sufficient condition for that the definite integral Bt is in existence. Let us consider the physical meaning of the condition \Q\ > 1. Recalling the definition of Q shown in Eq. (5.2.24), |Q| > 1 is equivalent to
228
Exact Analysis of Bi-periodic Structures
col +2«o0
(5.2.31a)
or col + 2ecoacax > col +
H^u)
(5.2.31b)
M which indicate co (co0 +scol) lies in the stop band of the ordered periodic system. It is well known that the frequency corresponding to localized mode must lie in the stop band. It can be proved that Bk has the property [21] ^ - = E, Bk
£ = 0,1,2,-
(5.2.32a)
* = 1,2,-
(5.2.32b)
or Bk=4kBQ with £ = - Q + sgn(fi)VQ 2 -l
|fi|>l
(5.2.33)
|<*| is always less than one. Inserting Eq. (5.2.27) in Eq. (5.2.26) with Napproaching infinity yields
a =s
* lut ( H*> + *'- J
; = 1,2
'"'
(5 234a)
"
Setting j = s in Eq. (5.2.25) gives
4,.o=«»
(5-2-35a)
Nearly Periodic Systems with Nonlinear Disorders
4,,o=A,
229
(5.235b)
where ass and Pa can be obtained from Eq. (5.2.34) as a
"=^r(i+jB°)' 32&c Q
^- = ^r ( -?r + B «»> 32«:c
(5 236)
-
D,
The localized modes in the system under consideration must satisfy Eqs. (5.2.35) and (5.2.36), where Q, and B0 are as shown in Eqs. (5.2.24) and (5.2.30a) respectively. 5.2.3
Localized Modes
The periodic system with infinite number of subsystems and having one nonlinear disorder is considered. The governing equation for the localized modes has been given as shown in Eqs. (5.2.35) and (5.2.36). Eq. (5.2.35a) minus or plus Eq. (5.2.35b) yields 4,,o - 4 , o = («„ - fin X < o " 4 , o )
(5.237a)
K* + As,o = («„ + A , ) « o + < o )
(5.2.37b)
The above equations are equivalent to 4,o=4,,o
or ( a „ - / ? „ X < o + 4 . . o 4 , . o + < o ) = l
(5.2.38a)
and 4,o=-4,o
or ( a „ + / ? „ ) « 0 - ^ . o ^ , o + < o ) = l
(5.2.38b)
respectively. The simultaneous equations (5.2.38a,b) may be divided into three sets of equations; that is (I) 4 , , = 4 , o
(5.2.39a)
230
Exact Analysis ofBi-periodic Structures
l
4,o ~ 4,o4,o + 4 i 0 =
-—
a
ss
(5.2.39b)
+ Pss
(II) Alsfi=-A2sfi
(5.2.40a)
4 , o + AsfiA2sfi
+<
0
=—
^
a
ss
(III) <
0
+ AufiA2sfi
1 ^—
+ 4_0 = «M
4,o - 4 , o 4 , o + 4 , o =
(5.2.40b)
Pss
(5.2.41a)
H ss
1
—
(5.2.41b)
There are three types of localized modes corresponding to the above three sets of equations, respectively. Let us discuss each type of localized modes in detail. (I) The first set of Eqs. (5.2.39a,b) describes the symmetric localized mode about the longitudinal centre line. Inserting Eq. (5.2.39a) in Eq. (5.2.39b) gives
AUfl
= A2sfl
1 —
(5.2.42a)
.
Substituting Eq. (5.2.36) into the right side of Eq. (5.2.42a) yields 4 , o = ^ 4 3Xo *o
(5.2.42b)
Inserting Eq. (5.2.30a) into the above equation gives s g n ( n ) - = L = = 77 VQ2-1 where
|Q|>1
(5.2.43)
Nearly Periodic Systems with Nonlinear Disorders
16*. ,_j
vsT^AZ°
231
(5 2 44)
--
is a nondimensional parameter. Substituting Eq. (5.2.24) into Eq. (5.2.43) results in JT72+1
2k
,»„„,,=-Hl +Xi
Noting
<s>2 =a>l +2so}0a>l + 0(s2),
(hardening spring), co2 xo2 +
8afc
)
(5.2.45)
it can be proved that if
y0 > 0
and if y0 <0 (softening spring), a2 < co\ .
M It can be concluded that if y0 > 0 , there is one symmetric localized mode with co greater than au
and if y0 < 0, there is one symmetric localized mode with co
less than coL . For the symmetric localized mode, the amplitudes of all masses can be found by substituting Eqs. (5.2.34) and (5.2.39a) into Eq. (5.2.25) as 3r ° R A3 16K,.
-
A
A
-
3r
°
R A3
k = 0,±1,±2,
(5.2.46a)
A; = 0,±l,±2,-
(5.2.46b)
Inserting Eq. (5.2.32b) into the right sides of the above equations yields A,+k,o = 4,-*.o = £ kA,o
k = 0,1,2,-
(5.2.47a)
^ + t , o = ^-*,o = £ *4,,o
k = 0,1,2,-
(5.2.47b)
which indicates that the amplitudes decay exponentially on either side of the nonlinear disorder. The attenuation constant t, of localized mode can be found by inserting Eqs. (5.2.24) and (5.2.45) into Eq. (5.2.33) as
232
Exact Analysis ofBi-periodic Structures
tJ-zJlE+L
(5.2.48)
£ is the odd function of rj as shown in Fig.5.2.2. When TJ approaches zero, £ also approaches zero. Recalling Eq. (5.2.44), it is concluded that if kc decreases or y0 increases or Auo increases, the mode will approach a strongly localized state and if y0 is greater/less than zero, then £ is less/greater than zero; i.e., the displacements of two adjacent subsystems are of opposite/same signs in the corresponding localized motion mode. The localized level of the modes is dependent on not only the structural parameters kc and/ 0 but also the amplitude Auo related to the initial condition or the total energy of the system. Auo can be determined by applying the energy conservation to the considered system. Ignoring the 0(f) term, the conservation of energy can be expressed as IS"
T
QO
00
K o + £ ( 4 + * , o + 4 U o ) + < o + I > L * , o + 4s-k,o)] = E
(5-2.49)
where E denotes the total energy of system. Inserting Eq. (5.2.47) into Eq. (5.2.49) yields KAl0(l + -±-2T) = E
(5.2.50)
w
Substituting Eq. (5.2.48) into the above equation results in KAlfi^fT\=E
(5.2.51)
Recalling Eq. (5.2.44), the solution for A*sfi of Eq. (5.2.51) can be found as \E2 J 6 t As,n - J „ 2 (^r - ^) ' A" 3y,
l6k„K
<E
(5.2.52)
3^o
which indicates that when the structural parameters K, kc and y0 are given, in
Nearly Periodic Systems with Nonlinear Disorders
2Vs
order that a symmetric localized mode would occur, however small amplitude, the energy constant E must be greater than
If the system energy is below the 3r0 required level, it is impossible to realize any localized mode with symmetry. The stability analysis of the symmetric localized mode shown in Eq. (5.2.47) is investigated by superposing the small perturbation on the mode solution and applying the Floquet theory [24]. It is concluded that when \rj\ < 2v2 , the symmetric localized mode is unstable and when W > 2V2 , the mode is stable. (II) The second set of equations shown in Eqs. (5.2.40a,b) describes the antisymmetric localized mode about the longitudinal centre line. Inserting Eqs. (5.2.40a) and (5.2.36) into Eq. (5.2.40b) yields <
0
= - ^ Q
|Q|>1
(5.2.53)
-vo Recalling the definition of Q shown in Eq. (5.2.24), Eq. (5.2.53) becomes Ik 1 co0o}1=-f-(l + -) M r / It is concluded that if -l
|77|<1 <0(axo)L),
(5.2.54) there is an anti-
localized
mode with lower frequency (axa>L) and if 4k 0
=
^\s-kfi ~ 0
^2s*kfi ~ A2s-k,0 ~ 0
^U,0
=
_
^ 2 J , 0
*o
A: = l , 2 , - - -
(5.2.55a)
*=u-
(5.2.55b) (5.2.55c)
which indicates that all of the subsystems are motionless except for the disordered
234
Exact Analysis ofBi-periodic Structures
one. This is a strongly localized mode. The amplitudes for the disordered subsystem can also be determined by using the conservation law of energy shown in Eq. (5.2.49), namely < o = f
(5-2-56)
which indicates that any amount of energy E, however small, is sufficient to cause one anti-symmetric localized mode. By applying the Floquet theory to stability analysis for the anti-symmetric localized mode shown in Eq. (5.2.55), one comes to the conclusion that when |^| < 1, the anti-symmetric localized mode is stable. If \T^ > 1, the anti-symmetric localized mode does not exist. (Ill) The third set of equations, i.e., Eqs. (5.2.41a,b), describes the nonsymmetric localized mode. Eq. (5.2.41a) minus and plus Eq. (5.2.41b) yields 4,.o4,,o=
/" ®ss
2
(5.2.57a)
Pss
and < o +4,o=
""
(5.2.57b)
respectively. Substituting Eq. (5.2.36) into Eqs. (5.2.57a,b) results in 4,,o4,,o=^(n-^-)
(5.2.58a)
< o + < o = ^ M " +-^)
(5.2.58b)
Recalling Eq. (5.2.30a) and multiplying the left (right) side of Eq. (5.2.58b) by the left (right) of Eq. (5.2.58a), we have A,fiKfi
« o + < o ) = (|^-)2
(5.2.59)
Nearly Periodic Systems with Nonlinear Disorders
235
Introducing the nondimensional mode parameter R = ^2-
(5.2.60)
and eliminating A2s0 from Eqs. (5.2.59) and (5.2.60) results in
R>0
%—
(5.2.61a)
Recalling Eq. (5.2.44), the above equation can be rewritten as Tj^sgniy^lJR
+ R1
R>0
(5.2.61b)
which indicates that the amplitude ratio R must be greater than zero, i.e., Au „ and A2sQ are of the same signs. Considering Eqs. (5.2.30a) and (5.2.44), dividing both sides of Eq. (5.2.58a) by 4 l o gives R = ^ [ Q _ S gn(Q)VQ 2 -l]
(5.2.62)
The solution for Q of Eq. (5.2.62) can be expressed as Q = sSn(r0)~(~f=L=r+"'R 2 ^R + Ri
+ Ri
)
(5.2.63)
R
which leads to 3 2kcn 1 R <JR + ,R\_. « o « , = - r f [l + s g n 0 ' 0 ) - ( - r = = + )] M 2 R iJR + R3
,r^<:A, (5.2.64)
Noting that the sum of two terms in the round brackets on the right side of Eq. (5.2.64) is always greater than two, it is concluded that if x 0 < 0 ,
236
Exact Analysis of Bi-periodic Structures
4k co0o)l < 0(co < coL) and if y0 >0,ca0col >—c-{co>au), i.e., a lies in the stop M band. For arbitrary positive value of R, the nonsymmetric localized mode is in existence. Replacing R in Eq. (5.2.64) by R~x, coQa)\ is invariable. Therefore when all of the structural parameters are given, there are two nonsymmetric localized modes with the same frequency and different amplitude ratio. All of the amplitudes for the nonsymmetric localized mode can be found by substituting Eqs. (5.2.34) and (5.2.32b) into Eq. (5.2.25) as 4,.os
(5.2.65a)
4,.o = - < < , + < „
(5.2.65b)
and 4,-*.o = 4 , + W = 4 * < o
k = 1,2,-
(5.2.65c)
4,-*.o = 4,+*.o = £ * < o
*=1A-
(5.2.65d)
where < o = ^ , o - ^.o) = ^ r i « o 2
-
(5.2.66a)
+
(5.2.66b)
32KC Q
< o = ^(4,.o + A2sfi) = ^ 2 - S 2 32A:C
0
«
0
The attenuation constant E, has been defined as shown in Eq. (5.2.33), but Q in. Eq. (5.2.33) must correspond to nonsymmetric localized mode. Substituting Eq. (5.2.63) into Eq. (5.2.33) results in £ = -sgn(r 0 )
R
JR + R3
It is obvious that |^| is always less than one.
R>0
(5.2.67)
Nearly Periodic Systems with Nonlinear Disorders
237
Eqs. (5.2.65a-d) indicate that the nonsymmetric localized mode may be divided into symmetric and anti-symmetric components, the amplitudes of the symmetric component decay exponentially on either side of the nonlinear subsystem but the anti-symmetric mode component is confined to the disordered one with nonzero amplitude. According to the parameter equation shown in Eqs. (5.2.67) and (5.2.61b), the £ - 77 curve can be obtained as shown in Fig. 5.2.2. Substituting Eqs. (5.2.65a-d) into the equation of energy conservation, i.e., Eq. (5.2.49), results in *[(
2
] =£
(5.2.68)
•mode 1 -mode 3
0.5
-> V
•0.5
Figure 5.2.2
S, versus 77 curves
238
Exact Analysis of Bi-periodic Structures
Recalling Eqs. (5.2.66), (5.2.60), (5.2.61a) and (5.2.67), the above equation can be expressed in terms of R as 1 + 4R2+R*
4Kk,
3|r0| (l-R + R2)ylR + R*
(5.2.69)
The nonsymmetric localized mode for arbitrary value of R is stable. The conditions for each type of localized mode to occur are shown in Fig. 5.2.3 where the longitudinal coordinate TJ is redefined as
^21,0
> ri
Figure 5.2.3
—— versus r\ curves
Nearly Periodic Systems with Nonlinear Disorders
n=^ with Amm = maxttAU0\,\A2s0u, localized modes with A2s0 / Als0
^
239
(5-2.70)
instead of Eq. (5.2.44). A pair of nonsymmetric being equal to R and R~l, are corresponding to
the same value of TJ defined in Eq. (5.2.70). According to the new definition of TJ shown in Eq. (5.2.70), R should be less than or equal to one in Eq. (5.2.61b) and ^
= 2V2 . When \TJ\> 2^2,
the
nonsymmetric localized mode does not exist. 5.3
Damped Periodic Systems with One Nonlinear Disorder [23]
Consider the system shown in Fig. 5.3.1(a) which consists of n number of subsystems connected to each other by means of a linear spring having stiffness sfec. Each subsystem is made up of a mass M connected to both a dashpot with a nondimensional damping coefficient e£0 and a spring with linear stiffness K (for ordered subsystems) or nonlinear stiffness K + ey0x2 (for disordered one), where f is a positive small parameter. In Fig. 5.3.1(a), s denotes the ordinal number of the disordered subsystem and Xj denotes the longitudinal displacement of the j-th mass. In order to apply the U-transformation to uncouple the linear terms of the governing equation, an equivalent system with cyclic periodicity must be created. It is necessary to extend the original system by its symmetrical image and apply the antisymmetric loading on the corresponding extended part as shown in Fig. 5.3.1(b) in which the first and last (2n-th) masses are imaginarily jointed by a spring with stiffness skc. This imaginary spring is not subjected to any load for antisymmetric vibration. If and only if the dynamic response of the extended system is antisymmetric, two extreme end conditions of the original system are satisfied in the extended one, i.e., the extended system is equivalent to the original one. The response of the first half (i.e., substructures 1 ~ n) of the equivalent system is the same as that of the original system. 5.3.1
Forced Vibration Equation
Applying Newton's second law to every mass in the equivalent system, one can write the differential equations of motion as follows
240
Exact Analysis of Bi-periodic Structures
£Fi0cosQt
ekr
J'
Ph^ ...
*,
-W*-
iKx + er^x
4-^-r M
i
eF,„cosQt
/
...
(a)
>^t&A-J>s+i
t
...
M „
Original system
sFJ0cosQt
eF,„cos£lt
J' M i
...
,-
Centre line eFt0 cosSit
*.
eFjB cosiit
J*
A'
Kx + ey^x*
M
/—E—L_
A*Lm'A
Z
Wtwm 2/i - s
(b)
Figure 5.3.1
2n-s + \
"
2« - j +1
M
A a A'
In
Equivalent system
Damped periodic system with a nonlinear disorder
Nearly Periodic Systems with Nonlinear Disorders
241
Mxj + 2Mco0£^0Xj +(K + 2ekc )Xj - dcc (xJ+l + *._,) = eFj j = l,2,-,2n
(5.3.1)
and Fj = FJ0 cos Qt Fj =FjfjcosQ.t-y0x)
j*s,2n-s
+l
j = s,2n-s
(5.3.2a) +l
(5.3.2b)
where the superior dot denotes the derivative with respect to the time variable t, a>0 denotes the natural frequency for the single ordered subsystem and *2n+i s x\ > xo = x2n ^ u e t o cyclic periodicity, eFJ0 denotes the amplitude of the harmonic force acting on the 7-th mass and £2 denotes the driving frequency, ey0 is the coefficient of the cubic term of the nonlinear stiffness in the disordered subsystem. The external excitation for the equivalent system must satisfy the antisymmetry condition, i.e., F
2»-y+i.o = FJ,O
J = 1,2, • • •, n
(5.3.3)
where F, 0 ~ FnQ indicate the real excitation acting on the original system. If the initial conditions are antisymmetric, then the dynamic displacements are also antisymmetric, i.e., *2*-, + i=*,
y = l , 2 , - , fi
(5.3.4)
One can now apply the U-transformation to the governing equation (5.3.1). The U- and inverse U- transformations may be expressed as
x
=*Ye>lJ~^qm V2«f^
y = l,2,-,2«
(5.3.5a)
m = l,2,-,2n
(5.3.5b)
and 1~=-F='Ze~'U~1)""'xJ
V2n ~t
242
Exact Analysis ofBi-periodic Structures
with y/ = — and i = , where In denotes the total number of subsystems for n the equivalent system. Noting that the displacements are always real variables, it can be proved that the generalized displacements qm{m = l,2,--,2«) have the following property Hln-m
m = l,2,--*,n
Hm
(5.3.6)
and qn, q2n must be real variables, in which the superior bar denotes complex conjugation. By using the U-transformation, i.e., premultiplying both sides of Eq. (5.3.1) by 1
2
"
the operator —=Ye-'u-l)mr, V2« ^
Eq. (5.3.1) becomes
qm+2co0£
(5.3.7)
where i—mw
/ „
2e2 •sfln
( »
I^cos(j--)my
Fj0 cos Qt - cos(s
--)my/r0xis(qx,q2,-",q2„)
(5.3.8) , K+ a* =
2ek,(l-cosmw) — M
(5.3.9)
co, =con
(5.3.10a)
Nearly Periodic Systems with Nonlinear Disorders
^ = ^ ^ = " 4 + 4^-
243
(5.3.10b)
Introducing the time substitution Q.t = r + (p
(5.3.11)
into Eq. (5.3.7) results in ? ; + ^ „ = f ( - ^ r2 MQ
- 2 ^ ) Q
m = l,2, —,2«
(5.3.12)
and
m
Q2
Q2
MQ2
V
'
where the prime symbol designates differentiation with respect to the new time variable r and tp is an unknown phase angle. Consider now the case of primary resonance, i.e., Q « a0. By letting (5.3.14)
Eq. (5.3.13) can be written as v
l =l + £J7m
(5.3.15)
where
nm
2k(l-cosmw) MQ2
(5.3.16)
Inserting Eq. (5.3.15) in Eq. (5.3.12), gives ll+qm=£Gm
/n = l , 2 , - , 2 n
(5.3.17)
244
Exact Analysis of Bi-periodic Structures
in which .1 i—mw
G =
2e 2 2
M2 V2n
" 1 ]TCOS(/'-->K^F,0
cos(r +
(quq2,--,q2„)
V;=i
(5.3.18)
7T— 9™ - ^ ^ m
According to the perturbation method, we seek a solution of Eq. (5.3.17) in the form of power series in £ not only for qm(j), but also for
(5.3.19) (5.3.20)
Eq. (5.3.19) is equivalent to Xj (r) = xJ0 (r) + £ xjX (r) + £2xJ2 (r) + • •
(5.3.21)
with W =4-|]e'(y-,)-r?„ V2^ m= i
r = 0,1,2,-
(5.3.22)
Substituting Eqs. (5.3.19) and (5.3.20) into Eqs. (5.3.17) and (5.3.18), the coefficients of equal powers of £ on both sides of Eq. (5.3.17) must be equal, i.e.,
(5.3.23a)
?™i + 9mi = i—mw
2e2
MQ.24ln
^ F ; o c o s ( 7 - - ) 7 n ^ cos(r + ^ 0 ) - cos(s - - ) w ^ r0^3o V7=i
2<»oCo . p. 9m0
VmVmO
(5.3.23b)
Nearly Periodic Systems with Nonlinear Disorders
245
1m2 + 9 ra 2 =
i—mu/
1 1 ^ V ; 0 c o s O - - > ^ ) {-q>x sin(T +
2e2
2
MQ. 4ln 2tt>
W=i
o£"o , rt
5.3.2
1 (p0j)-cos(s--)miff(3y0xlxsX)
"ml
(5.3.23c)
/nif/ml
Perturbation Solution
The solution of Eq. (5.3.23a) may be expressed as qmo=amoc°ST + bmosinT
m=l,2,-,2n
(5.3.24)
with «2n-m,o=flm,o an< i *2.-«.o = *-,o d u e t o ?a.-m.o = 9«,o, where am0 and 6m0 ( /M = 1,2, • • • ,2n) are complex constants to be determined. The physical displacements corresponding to qm0 shown in Eq. (5.3.24) can be obtained from Eq. (5.3.22) with r=0 as xJ0 = Aj0 cos T + Bj0 sin r
j = l,2, — ,2n
(5.3.25a)
where A
B
i
2
"
—i-Ve'O-""'/!
V
i(M)mv e
h
(5.3.25b)
(5.3.25c)
i4 .„ and Bj0 are real numbers and A2n_J0 = ^ . 0 , B2n_j0 = Bj0, which lead to Without loss of generality, we can assume that the initial velocity for the
246
Exact Analysis of Bi-periodic Structures
disordered subsystem is zero besides the antisymmetry for both initial displacement and velocity, which leads to (5.3.26)
5,0=0 and X
(5.3.27)
A
S0=
SOCOST
In order to prevent secular terms, the coefficients of COS T and sin T on the right side of Eq. (5.3.23b) must be zero. Introducing Eqs. (5.3.24) and (5.3.27) into Eq. (5.3.23b), letting the coefficients of cos r and sin r be equal to zero, give i—mw
2e2
( "
1
'm0
n'fm"m0 m<*mo = o
Mtffln 2ft>0Co Q
^
1 3y cos cp0 - cos(j — ) m y/ — - A3s0
(5.3.28a)
m = l,2,--,2n
t—mtf
2e2
MQ2V2n
sin^ 0
+
a Q
mo-nmi>mo = o
7=1
m = l,2,--,2n
(5.3.28b)
Consider now a specific loading condition as that there is no excitation acting on each subsystem except the disordered one, i.e., F,0=0
j*s
and ^o"0
j = \,2, — ,n
(5.3.29a) (5.3.29b)
Inserting Eq. (5.3.29) into Eq. (5.3.28), the solution for am0 and bm0 of simultaneous equations (5.3.28a,b) can be expressed as
Nearly Periodic Systems with Nonlinear Disorders
1 2 ™
2
4«f
;=e
.
1.
+CL +C2
C/.-fD + l-cosm^)/,
cos(s—)»!«/•—•—i
2Kfbi
Y
2'
-^
247
(5.3.30a)
(5.3.30b)
\-cosm¥)2+C2
(D +
in which Fs0coscp0--y0A3s0
/, =
I2 =FsQsin
D=
MQa>0£0
K-MQ.2 ,. 2afc
(5.3.31a) (5.3.31b)
Q. <0 co0(£kc/K)
Q , at -[l-(—)2]/(2-r) «n ^T
(5.3.31c)
(5.3.3 Id)
C and D are two nondimensional parameters. They are dependent on the Q ek nondimensional frequency, stiffness and damping constant, i.e., — , — - and a0 K eg0. In Eqs. (5.3.31a) and (5.3.31b), As0 and
4o=^(«/.+/W
j = l,2,--,2n
(5.3.32a)
*,o=—
j = l,2,--,2n
(5.3.32b)
iPA-cCjh)
where 1
'
2
"
„ ,. K , 1, D +1 - cos m w 2 cosO - -)m y/ cos(s - ~)m yr — — J 2n*-f m=l L 2 2 (D + \-cosmy/y +C
(5.3.33a)
248
Exact Analysis of Bi-periodic Structures
i
2
"
2 cos( /
)m w cos( s 2
)mw 2
; (Z) + l - c o s » n / ) 2 + C 2
(5.3.33b)
Consider now the s-th set of simultaneous equations in Eq. (5.3.32). Inserting j=s and Eq. (5.3.26) in Eq. (5.3.32), yields j
=2kcasAs0
(5.3.34a)
*>fi _ 2kJsAs0
(5.3.34b)
2
~ al+ti
Noting the definitions of /, and I2 shown in Eqs. (5.3.3la,b), Eqs. (5.3.34a,b) may be rewritten as 3 ,3 2kca.A.0 Fs0 cos(p0 = -y0A;0 + f s '? s 4/0 <*/+# „ . Fs0sm
(5.3.35a)
2kJsAs0 c '°
(5.3.35b)
From the above equation, we can find the phase angle with zero-order approximation as
UJs
(pQ = t a n
(5.3.36)
2
2kea,+-r0A Mtf+fi) and the frequency response curve as t'.n^-y
4/C„
a*+#
4ft Ct
«;+A
5 2
2\
/3
.? ..2
4'
in which as and /?s are dependent on Q . They can be expressed as
(5.3.37)
Nearly Periodic Systems with Nonlinear Disorders
249
a = — > 2cos 2 (^—)my/ £—r 2nj^ 2 (Z) + l - c o s m ^ ) 2 + C 2
(5.3.38a)
P, = — Y 2 c o s 2 ( i - - ) m ^ 2n^ 2 (£> + l - c o s m ^ ) 2 + C 2
(5.3.38b)
where C and Z) are dependent on Q/a>0 besides the structural parameters as shown in Eqs. (5.3.31c) and (5.3.31d). If the parameters of the system and loading are given, the response As0 for the loaded subsystem can be calculated from Eq. (5.3.37), and the other AJ0 and BJ0 can be obtained by substituting Eq. (5.3.34) into Eq. (5.3.32) as aja
+0,0, «, + P,
B
J° =
Pias-aiBs
ni
j = l,2,.-.,2n
(5.3.39a)
J = !>2,-,2»
(5.3.39b)
A
.°
a. + P.
i i The characteristic of the frequency response (\A s0\
Q
) curve is similar to
that for the single nonlinear subsystem, i.e., the jump phenomenon may occur. For the specific case of ey0 = 0.2 , e£Q = 0.2, ekc = 0.25, K = 2.5 and eFs0 = 30 with n approaching infinity, the frequency response curve is as shown in Fig. 5.3.2. Introducing Eqs. (5.3.27) and (5.3.24) into Eq. (5.3.23b), noting the coefficients of COST and sinr on the right side of Eq. (5.3.23b) vanishing, yields i i—mu/
2
2e 1 r <7», +?„, = - , # , i— cos(s - -)m y/ -f As0 cos 3r MQ 2 V2« 2 4 The solution for qmX of Eq. (5.3.40) can be expressed as V
J±
i—mw
Qm\ = am\ c o s r + bm< sinT + ° s0.— e2 16MQ2V2«
(5.3.40)
1
cos(s—)my/cos?>T 2
(5.3.41)
250
Exact Analysis of Bi-periodic Structures
£7 0 = 0.2 K=2.5
££o=0.2
ekc =0.25
EFSO=30
n' w: Figure 5.3.2
The frequency response (\As0\ v e r s u s — ) curve 0)n
Substituting Eq. (5.3.41) into Eq. (5.3.22) with r = 1 results in j x,.=A j.cosT Jl '
n • Yd^la 1 V^ Hi-^liV + B,.sm.T + — f "2 — > e 2 ' 16A/Q 2n±i
y = l,2,-",2n
, 1 . cos(s—)mu/cos3r 2 (5.3.42)
where 1
A
2
"
=—Ve^''"*'a
(5.3.43a)
Nearly Periodic Systems with Nonlinear Disorders
B
n=-ir1Lei(H)mVb^ V2«
Noting
the initial velocity
In
y cos2(s
251
(53 43b)
-
"
vanishing
for
the nonlinear
subsystem
and
j
)m y/ = n, inserting j = s into Eq. (5.3.42) gives JC„ = A. cos T + r°As\
cos 3r
(5.3.44)
^=-?==Z e '' ( " , ) m ^' = 0
(53 45)
and
-
Substituting Eqs. (5.3.41), (5.3.44) and (5.3.27) into Eq. (5.3.23c) and letting the coefficients of COST and sinr on the right side of Eq. (5.3.23c) be equal to zero gives am.=
p=2e2 2kcyfa
, 6m1 =
1 „ 4mr , j= 2e 2 cos(s Ikfhi
cos(s
)my/2' Y (D +
^-^ r~ l-cosmysf+C1
*\ C7*-(D + l-cosmy)/ 2 * )mw —•— , V 2 (D + l - c o s myff+C2
(5.3.46a)
, „ . m (5.3.46b)
in which /,- = -
+ j ^ )
(5.3.47a) (5.3.47b)
252
Exact Analysis of Bi-periodic Structures
4l=^-(«/.*+/V2*)
(53-48a)
Bj^jj-i/tJ-ajO
(5.3.48b)
in which the definitions of at and /?. are as shown in Eq. (5.3.33). Recalling Eq. (5.3.45), inserting j - s into Eq. (5.3.48) yields I* =
. 2
=
2keCa,A,, 2 ' 2'
(5.3.49a)
2kJsA
(5349b)
a
"«. +tf
Introducing Eq. (5.3.47) into Eq. (5.3.49) gives + 4. = -T^^-AT^O^O ^ 7 K 1 2 8 M Q 2 / V ° *° a , 2 + #
+
&
tan
?o)l
(5.3.50a)
(5.3.50b)
Inserting Eq. (5.3.49) into Eq. (5.3.48) yields a
A
n-
j<*.
a] Pj<*s
Bfl-
„ 2
+
M , + tf "
-«,A{ , /?2
Jl
(5.3.51a)
(5.3.51b)
The forced response with first-order-approximation can be obtained by substituting Eqs. (5.3.25a), (5.3.39), (5.3.42) and (5.3.51) into Eq. (5.3.21), as
Nearly Periodic Systems with Nonlinear Disorders
Xj
a ,as + = -J—i—tr~
+£
253
ft,fts ftias - a jfts (4o +s4,i)cosr +-L-2—-7— (i4l0 +£4 j l )sinr
Yo^so — 2_, c °s(7 - - ) w V cos(5 - - ) m ^ cos3r 16MQ2 2w 7 = 1,2,—,2«
(5.3.52)
with r = Q f - (^>0 + £(px). Consider now the sum of series in the square brackets as follows — 2^ cos(y 2n-
)m y/ cos(s
)m yr = — / f c o s ( . / + s- \)m y/ + cos(j - s)m y/]
0
j *s,2n-s
1
.
—
1 =s,2n-s
2
+\ (5.3.53)
„
+l
J
Introducing the above result into Eq. (5.3.52), noting a2n_J+l =(Xj and A»-y+i = ftj. y i e l d s x
m-j+\ =xj =
ajas+pjps ft 2—TJ— K o +a4 s l )cosr + <*s +fts j = \,2,---,n
and
as
-ajfts -———(A s 0 +£A s l )smr Ct, + ft,
j*s
*2„-s+, =x, ={A,0 + £Asi)cosT + ^^cos3r
(5.3.54a)
(5.3.54b)
It is obvious that the forced vibration shown in Eqs. (5.3.54a,b) satisfies the antisymmetric condition shown in Eq. (5.3.4), i.e., the solution x. (j = l,2,---,n) is applicable to the original system. In Eq. (5.3.54a), a . and /?, are dependent on the total number of subsystems besides the parameters C and D. Consider now the periodic system with infinite number of subsystems. By letting n approach infinity, the limit of the series
254
Exact Analysis of Bi-periodic Structures
summation on the right sides of Eqs. (5.3.33a) and (5.3.33b) become the definite integral [5] respectively, i.e., 1 ?\ ,• lx« / K „ £> + l - c o s 0 ,. a,=— 2cos(;—)6cos(s—)6 ; -d0 1 J In Jo 2 2 (£> + l-cos
1 r2"
=
J_[[cos(j+s-l)e In Jo
+c o s U
-
r> +1—cos o c m
d9
(5.3.55a)
(£> + l - c o s # ) + C
and C Pj = T " f[cosC/ + 5 - l ) « + c o s ( y - ^ ] — — rfg ' 2;r Jo (D + l-cos0) +C2
(5.3.55b)
When a finite periodic system is considered, i.e., n is a finite number, a . and /?, can be expressed exactly as the series forms shown in Eq. (5.3.33). The series form can be regarded as the rectangular integration formula for the definite integral shown in Eqs. (5.3.55a,b), where the integration interval [0,2n ] is divided into 2w subintervals, i.e., each subinterval is y/. If the integral form is adopted instead of the series form, the error is in agreement with that for rectangular integral formula, i.e., 0(«"'). Generally, there are infinite subsystems between the disordered subsystem and the extreme one at infinity, i.e., (j + s-l) is an infinite number. Introducing the Riemann lemma into Eqs. (5.3.55a,b) yields 1
r2"
D + l— cos f)
«,+*=r-J ^ ^ — rt-7Td0> In Jo (Z> + l - c o s 0 ) + C C fi^—fcoske In Jo (D + 1-cosOy
* = °,±1.±2,- (5-3.56a)
dO, * = 0,±1,±2,- (5.3.56b) +C
The above definite integrals can be expressed in terms of elementary functions, such as as=(D + l)E0 - £ , ,
fi,
= CE0
(5.3.57a)
Nearly Periodic Systems with Nonlinear Disorders
255
a_, =a J + 1 ^KD + l ^ + C ^ o - ^ + l ^ - l
(5.3.57b)
A-i=A+i=CBi
( 5 - 3 - 57c )
where
0
IK JO (D + l - c o s ( 9 ) 2 + C 2
= p(o 2 + 2Z> + C 2 + V[(2 + Z)) 2 +C 2 ](Z) 2 +C 2 ))j
H
1 2
2
>lD +C
r+ -
1
J(2 + D)2+C2
/
(5.3.58a)
1
_ J_ f cos VdU ~2;r Jo (D + 1 - c o s 0 ) 2 + C 2
= |2(D2+2D + C2+A/[(2 + Z))2+C2](JD2+C2)J|2
4D2~7C2
J(2 + D)2+C (5.3.58b)
All of **s±k an<^ y^s±* (^ = ^'2>''') c*m be expressed as the linear combination of E0 and £ , . 5.3.3
Localized Property of the Forced Vibration Mode
The periodic response with zero-order approximation shown in Eq. (5.3.25a) can be written as Xj = AJOCOST +BJ0siar = XJCOS(T-$J)
j = l,2,---,oo
(5.3.59)
256
Exact Analysis ofBi-periodic Structures
where
(5.3.60a)
(5.3.60b)
It is clear that X,=A,0
and
0s=O
(5.3.61)
Because of as_j = as+j and Ps_} = fis+j, we have and
X,_J=X„J
0,.j=0,+J
(5.3.62)
which indicate the symmetry of the forced vibration about the nonlinear subsystem. The localized level of the mode is dependent on the attenuation rate of the amplitudes. Let
^ • - ^ - =J
U +k+
J
^:k
k = l,2,-
(5.3.63a)
and Mk=°.+k-0,*t-x
k = \,2,-
(5.3.63b)
A0k indicates the phase difference between the corresponding displacements in (s + £)-th and (s + &-l)-th subsystems. £k and A0k are only dependent on three nondimensional parameters, i.e., Q/ft>0 , skjK and eg0, and independent from the nonlinear parameter eya. The numerical results are given as shown in Tables 5.3.1 and 5.3.2. The accurate numerical results show that W ,
=&=-
(5.3.64a)
Nearly Periodic Systems with Nonlinear Disorders
A0,=A02=-
257
(5.3.64b)
which are in agreement with those obtained from the linear periodic system. By using the results shown in Eqs. (5.3.64a) and (5.3.64b), Eq. (5.3.59) can be written as x,_t=x,+t=£*4ocos(r-*A01)
* = 0,1,2,-
(5.3.65)
Lff| is always less than one except that eg0 = 0 and Q lies in the pass band as shown in Table 5.3.1. The case of |^| = 1 represents that the corresponding mode is not localized. Moreover let us consider the forced vibration with first-order approximation. By using the above results, Eqs. (5.3.54a,b) can be expressed as *,-*=*.•* =£*(4o +sAsl)cos(T-kA0x)
+ ^^-cos3T 32MQ2
Sk0
(5.3.66)
where x = Q t - (
2,-
conclusion that (3) when O =a>0{\ +
2fit
- ) , A6X is identically equal to 90°
K v2
2~
which can be proved mathematically as follows: Inserting Q = co0 (1 + Eqs. (5.3.3Id) and (5.3.58b) yield D = -\
2fifc„
-) into
and £, = 0 , leading to as = 0 and
Ps+X = 0, then introducing these results into Eq. (5.3.60b) gives A0, = 0J+1 = 90°,
258
Exact Analysis of Bi-periodic Structures
w h i c h is independent o f t h e damping; (4) w h e n Q <eOg(l + with increasing increasing
e£0;
and
when
Q 2 >a>l{\ +
2at
-),
2£fc
- ) , A 0l increases
A0,
decreases
K
e£0.
Table 5.3.1
(a)
Attenuation constant £,
(£ 2 ) for — ^ = 0.1 K
0.00001
0.00100
0.01000
0.8
0.2679491920 (0.2679491920)1
0.2679450672 (0.2679450672)
0.2675379778 (0.2675379778)
0.9
0.3819660094 (0.3819660094)
0.3819475651 (0.3819475651)
0.3801450670 (0.3801450670)
1.0"
0.9900497512 (0.9900497512)
0.9047621882 (0.9047621882)
0.7270194833 (0.7270194833)
1.1
0.9998789013 (0.9998789013)
0.9879630216 (0.9879630216)
0.8864552245 (0.8864552245)
1.2
0.9998904637 (0.9998904593)
0.9891055471 (0.9891055471)
0.8964375957 (0.8964375957)
1.3
0.9998683525 (0.9998683525)
0.9869214264 (0.9869214264)
0.8772952844 (0.8772952844)
1.4 C
0.9891812677 (0.9891812677)
0.8968356116 (0.8968356116)
0.7066092868 (0.7066092868)
1.5
0.3819660082 (0.3819660082)
0.3819352703 (0.3819352703)
0.3789568507 (0.3789568507)
1.6
0.2679491916 (0.2679491916)
0.2679409423 (0.2679409423)
0.2671293807 (0.2671293807)
e£o 2
n7*>„
" The numerical results in the round brackets denote £ 2 ; b c
Case of Q = coL; Case of Q = w,,.
with
Nearly Periodic Systems with Nonlinear Disorders
Table 5.3.1
(b) eC»
at Attenuation constant £, (£ 2 ) for —£- = 0.1 0.10000
0.20000
0.2360679775
0.1867233484 a
a
(0.2360679775)
(0.1867233484)
0.2892626024
0.2042325086
(0.2892626024)
(0.2042325086)
0.3460143392
0.2168453354
(0.3460143392)
(0.2168453354)
0.3833046482
0.2215530221
(0.3833046482)
(0.2215530221)
0.3877945824
0.2174286857
(0.3877945824)
(0.2174286857)
0.3612922949
0.2058395544
(0.3612922949)
(0.2058395544)
0.3130958493
0.1895643223
(0.3130958493)
(0.1895643223)
0.2599562179
0.1715728753
(0.2599562179)
(0.1715728753)
0.2149005549
0.1540711857
(0.2149005549)
(0.1540711857)
The numerical results in the round brackets denote £ 2 ; Case of Q. = coL\ c Case of Q = co,,.
b
259
260
Exact Analysis ofBi-periodic Structures
Table 5.3.2
(a)
Difference of phase angles A#,
ek (A02) (degree) for — - = 0.1 K
0.00001
0.00100
0.01000
0.0029587412
0.2958701883
2.954809549
(0.0029587414)"
(0.2958701883)
(2.954809549)
0.0048617080
0.4861451448
4.836423380
(0.0048617081)
(0.4861451448)
(4.836423380)
0.5729530204
5.724792592
17.96423592
(0.5729530204)
(5.724792592)
(17.96423592)
60.00000024
60.00242543
60.23852969
(60.00000025)
(60.00242543)
(60.23852969)
90.00000000
90.00000000
90.00000000
(90.00000000)
(90.00000000)
(90.00000000)
119.9999997
119.9971337
119.7189490
(119.9999995)
(119.9971337)
(119.7189490)
179.3767671
173.7737712
160.4907647
(179.3767671)
(173.7737712)
(160.4907647)
179.9937236
179.3724114
173.7774321
(179.9937236)
(179.3724114)
(173.7774321)
179.9958157
179.5815819
175.8267916
(179.9958157)
(179.5815819)
(175.8267916)
< 0
&K 0.8
0.9
1.0b
1.1
1.2
1.3
1.4C
1.5
1.6
a
The numerical results in the round brackets denote
b
Case of Q = a)L;
c
Case of Q. = a>u .
A02;
Nearly Periodic Systems with Nonlinear Disorders
Table 5.3.2
(b)
Difference of phase angles A0, (A0 2 ) (degree) for
< 0
2
nK
0.10000
0.20000
2
0.8
26.56505118 (26.56505118)
0.9
1.0"
1.1
1.2
1.3
1.4C
1.5
1.6
43.80251522 a
(43.80251522)
36.79497241
53.97312668
(36.79497241)
(53.97312668)
51.82729237
65.53019948
(51.82729237)
(65.53019948)
70.47593457
77.80809134
(70.47593457)
(77.80809139)
90.00000000
90.00000000
(90.00000000)
(90.00000000)
108.6373579
101.3892055
(108.6373579)
(101.3892055)
124.7702433
111.4677448
(124.7702433)
(111.4677448)
136.9286010
120.0000000
(136.9286010)
(120.0000000)
145.2509396
127.0128869
(145.2509396)
(127.0128869)
a
The numerical results in the round brackets denote
b
Case of Q = co, ;
c
Case of Q = co,,
A02;
K
261
0.1
This page is intentionally left blank
REFERENCES 1. 2. 3. 4. 5.
6.
7.
8. 9. 10.
11.
12.
13.
14.
15.
H.C. Chan, C.W. Cai and Y.K. Cheung, Exact analysis of structures with periodicity using U-transformation. World Scientific, Singapore, 1998. C.W.Cai, Y.K.Cheung and H.C.Chan, Uncoupling of dynamic equations for periodic structures, Journal of Sound and Vibration, 139(2), 253-263, 1990. C.W.Cai and F.G.Wu, On the vibration of rotationally periodic structures, Acta Scientiarum Naturalium Universitatis Sunyatseni, 22(3), 1-9,1983. C.W.Cai, On analytical solution of rotationally periodic structures, Acta Scientiarum Naturalium Universitatis Sunyatseni, 25(2), 64-67,1986. C.W.Cai, Y.K.Cheung and H.C.Chan, Dynamic response of infinite continuous beams subjected to a moving force — an exact method, Journal of Sound and Vibration, 123(3), 461-472, 1988. Y.K.Cheung, H.C.Chan and C.W.Cai, Exact method for static analysis of periodic structures, Journal of Engineering Mechanics, ASCE, 115(2), 415434,1989. C.W.Cai, Y.K.Cheung and H.C. Chan, Transverse vibration analysis of plane trusses by analytical method, Journal of Sound and Vibration, 133(1), 139-150, 1989. H.C.Chan, C.W.Cai and Y.K.Cheung, A static solution of stiffened plates, Thin-Walled Structures, 11,291-303,1991. H.C. Chan and C.W. Cai, Dynamics of nearly periodic structures, Journal of Sound and Vibration, 213(1), 89-106,1998. C.W. Cai, H.C. Chan and Y.K. Cheung, Exact method for static and natural vibration analyses of bi-periodic structures, Journal of Engineering Mechanics, ASCE, 124(8), 836-841, 1998. C.W. Cai, H.C. Chan and J.K. Liu, Analytical solution for plane trusses with equidistant supports, Journal of Engineering Mechanics, ASCE, 126(4), 333339,2000. C.W. Cai, J.K. Liu, F.T.K. Au and L.G. Tham, Dynamic analysis of continuous plane trusses with equidistant supports, Journal of Sound and Vibration, 246(1), 157-174,2001. H.C.Chan, C.W.Cai and Y.K.Cheung, Moments and deflections of simply supported rectangular grids — an exact method, International Journal of Space Structures, 4(3), 163-173, 1989. H.C.Chan, C.W.Cai and Y.K.Cheung, An analytical method for static analysis of double layer grids, International Journal of Space Structures, 4(2), 107-116, 1989. YKCheung, H.C.Chan and C.W.Cai, Natural vibration analysis of rectangular
263
264
Exact Analysis ofBi-periodic Structures
networks, International Journal of Space Structures, 3(3), 139-152,1988. 16. Y.K.Cheung, H.C.Chan and C.W.Cai, Dynamic response of orthogonal cable networks subjected to a moving force, Journal of Sound and Vibration, 156(2), 337-347, 1992. 17. H.C.Chan, C.W.Cai and Y.K.Cheung, Convergence studies of dynamic analysis by using the finite element method with lumped mass matrix, Journal of Sound and Vibration, 165(2), 193-207, 1993. 18. H.C. Chan, C.W. Cai and J.K. Liu, Exact static solution of grillwork with periodic supports, Journal of Engineering Mechanics, ASCE, 126(5), 480-487, 2000. 19. H.C. Chan, Y.K. Cheung and C.W. Cai, Exact solution for vibration analysis of rectangular cable networks with periodically distributed supports, Journal of Sound and Vibration, 218(1), 29-44, 1998. 20. C.W.Cai, Y.K.Cheung and H.C.Chan, Mode localization phenomena in nearly periodic systems, Journal of Applied Mechanics, ASME, 62(1), 141-149, 1995. 21. C.W. Cai, H.C. Chan and Y.K. Cheung, Localized modes in periodic systems with nonlinear disorders, Journal of Applied Mechanics, ASME, 64(4), 940945, 1997. 22. C.W. Cai, H.C. Chan and Y.K. Cheung, Localized modes in a two-degreecoupling periodic system with a nonlinear disordered subsystem, Chaos, Solitons and Fractals, 11(10), 1481-1492,2000. 23. H.C. Chan, C.W. Cai and Y.K. Cheung, Forced vibration analysis for damped periodic systems with one nonlinear disorder, Journal of Applied Mechanics, ASME, 67(1), 2000. 24. A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations, John Wiley and Sons, New York, 1979.
NOMENCLATURE loading vector unit matrix stiffness matrix mass matrix U-matrix [see Eqs. (1.1.8a, b)] displacement vector elementary cyclic matrix [see Eq. (1.1.12)]
F I K M U X e. j
EI Wj Jm'
F
•'
{ \],k)
f(r,s)
generalized loads for systems with cyclic periodicity in one and
4(r,s)
two directions, respectively imaginary unit generalized displacements for systems with cyclic periodicity in
i qm,
{S}j >
Wu*)
V, 9 $ 03
n n
;>
[]
T
flexural rigidity of beam loading vectors for subsystems j and (j, k), respectively
n,,t
one and two directions, respectively displacement vectors for subsystems
j
and
(j, k),
respectively periods of cyclic bi-periodic system attenuation constant for localized modes vibration frequency lower and upper bounds of pass band, respectively potential energy of whole system potential energy of subsystems j and (j, k), respectively transposed matrix of [ ]
[]"'
inverse matrix of [ ]
(")
second derivative of ( ) with respect to time variable t
(")
complex conjugate of ( )
265
This page is intentionally left blank
INDEX
A attenuation constant, 231, 236 attenuation rate, 211,214, 256
B bi-periodic structure (s), 15, 27, 81, 125 bi-periodic system (s), 30,44, 59 bi-periodicity, 27,45, 69
c cable network (s), 125 complex conjugation, 3, 84, 159, 181, 207, 242 concentrated load, 94, 121, 169, 170, 191, 192 continuity condition, 86, 87, 162, 183 continuous beam, 115 continuous truss, 81, 99, 103, 104, 106, 107 cyclic bi-periodic equation, 15, 18, 20, 26 cyclic bi-periodic system, 34, 45, 69, 80, 82, 126, 159, 175 cyclic bi-periodicity, 27, 45, 60, 71, 80,83, 115, 125, 158, 177,204 cyclic matrix, 4 cyclic periodic equation, 2, 16, 128 cyclic periodicity, 2-5, 11-15, 18, 21, 24, 27, 30, 35, 40, 44,48, 49, 84, 86,91,92,101,117,120,125, 127, 128, 132, 152, 157, 158, 161, 167, 182, 188, 189, 220, 239, 241
D damped periodic system, 203 diagonal matrix, 5 Dirac delta function, 117 disordered subsystem, 203, 204, 219, 220,234,239,241,246,254 double U-transformation, 13, 14,21, 23, 24, 125, 128, 132, 137, 140, 141, 153, 161, 162, 165-167, 172, 182, 185, 187, 188, 190
E eigenvalue, 9 eigenvectors, 9 energy conservation, 211, 232, 237 equilibrium equation, 14, 28, 29, 30, 34, 88, 96, 99, 116, 128, 129, 139, 159, 163, 173, 184, 196 equivalent system, 27, 44, 45, 60, 64, 67, 69, 72, 91, 108, 109, 115-117, 119, 121, 125, 128, 141, 149, 181, 192, 239, 241, 242
F forced vibration, 39, 107, 203, 253, 256, 257 frequency equation, 36, 37, 51-53, 55, 63, 75-77, 80, 100, 102, 103, 106, 142-144, 221 frequency response, 58, 69, 113, 155, 248-250, 257
268
Exact Analysis ofBi-periodic Structures
G
M
generalized displacement, 7, 9, 29, 38, 39, 53, 64, 67, 77, 87, 100, 108, 119, 129, 143, 189,191, 224, 242 governing equation, 11, 12, 15, 27, 31, 45, 90, 92, 204, 210, 212, 229, 239, 241 grid, 175, 176, 187, 191, 192, 196 grillwork, 156-159, 161, 164, 169, 172, 173
mass-spring system (s), 27, 45, 58, 203 mode subspace, 10, 27
H harmonic load (s), 40, 107, 112, 149, 151,153 Hermiltian matrix, 6
I influence coefficient (s), 30, 35 inverse U-transformation, 7, 12, 18, 28,86
N natural frequency, 9-11, 34, 39, 44, 58, 67, 76, 79, 100, 115, 142, 143, 145,146,221,222,241,242 natural mode, 9, 10, 37-39,44, 53-55, 64, 65, 77, 79, 80, 102,106, 142, 145 natural vibration, 9, 34, 44, 99, 125, 139 nonlinear disorder, 203, 211, 214, 229,231,240,257
o orthogonal basis, 3
P J jump phenomenon, 249
L linear periodic structure, 27 linear periodic system, 44, 212, 257 localized mode (s), 203, 204, 208, 210-217, 219, 220, 228-231, 233, 234, 236-239 L-P method, 203, 205, 220, 223
pass band, 11, 104-106,208, 222, 242, 257 periodic structure, 1, 2, 3, 7, 9, 11, 27, 81, 84, 125 periodicity, 4, 15, 20, 27, 86, 125, 128, 205 phase difference, 3, 10, 11, 256 potential energy, 84, 87, 159, 163, 181,184
Index
Q quasi-diagonal matrix, 4
R resonance, 58, 113, 115, 155, 243 rotating mode, 3, 10, 11
s stability analysis, 215, 216, 233,234 stable mode, 218 static problem, 115 stiffening girder (s), 175, 179-181, 185, 187-189, 191, 192, 196 stiffness matrix, 84, 85, 159, 160, 179 stop band, 208, 228, 236
269
u U-matrix, 2, 3 unit matrix, 3,4, 20, 87, 163 unitary space, 3 unstable, 215, 216, 218, 233 U-transformation, 6-8, 11-14, 16-18, 20, 21, 23, 24, 26-28, 30-32, 34, 35, 40, 41, 44, 45, 48-51, 61, 62, 69, 73, 74, 81, 84, 86, 91, 92, 96, 100, 101, 109, 110, 115, 117, 119, 120, 141, 167, 175, 203, 206, 207, 220,221,239,241,242
w Warren truss, 81, 94, 102, 112
By using the U-transformation method, it is possible to uncouple linear simultaneous equations, either algebraic or differential, with cyclic periodicity. This book presents a procedure for applying the U-transformation technique twice to uncouple the two sets of unknown variables in a doubly periodic structure to achieve an analytical exact solution. Explicit exact solutions for the static and dynamic analyses for certain engineering structures with doubly periodic properties — such as a continuous truss with any number of spans, cable network and grillwork on supports with periodicity, and grillwork with periodic stiffening members or equidistant line supports — can be found in the book. The availability of these exact solutions not only helps with the checking of the convergence and accuracy of numerical solutions, but also provides a basis for optimization design for these types of structures. The study of the force vibration and mode shape of periodic systems with nonlinear disorder is yet another research area which has attained considerable success by the U-transformation method. This book illustrates the analytical approach and procedure for the problems of localization of the mode shape of nearly periodic systems together with the results. B
World Scientific www. worldscientific.com 4940 he
Exact Analysis of
Bi-Periodic
