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DUCTING IN EXTENDED PLATES OF VARIABLE THICKNESS
Safarov I.I.
Bukhara Technological-InMitute of Engineering Bukhara, Republic of UzbekiMan
Akhmedov M.S.
Bukhara Technological-InMitute of Engineering Bukhara, Republic of UzbekiMan
Boltaev Z.I.
Bukhara Technological-InMitute of Engineering Bukhara, Republic of UzbekiMan
ABSTRACT
The paper deals with the spread of its own waves on the visco elaflic plate with variable thickness. Basic relations of the classical theory of plates of variable thickness obtained on the basis of the principles of virtual displacements. The spectral problem, which is not self ad joint. Built for the task biorthogonality conditions, based on the Lagrange formula. The numerical solu-tion of the spectral tasks performed on the computer software syflem based on the method of orthogonal shooting S.K. Godunov in combination with the method of Muller.
Keywords: waveguide, spectral problem, plane wave biorthogonality, plaflic, dual problem.
Introduction. Known [1,2,3] that normal wave deformable layer (Lamb wave) is not or-thogonal thickness, i.e. the integral of the scalar product of vectors of displacements of two different waves, considered as a function of position perpendicular to the surface layer is not zero. They are also not orthogonal conjugate wave, which is obtained from a consideration of the
dual problem. This introduces additional difficulties in solving practical problems. This article is based conjugate spectral biorthogonality objectives and conditions for the problem.
1. The mathematical formulation of the problem.
We consider the visco elaflic waveguide as an infinite axial x1 variable thickness (Figure 1).
Figure 1. Design scheme: plates of variable thickness.
Basic relations of the classical theory of plates of variable thickness can be obtained on the basis of the principles of virtual displacements. The varia-tion equation problem visco elaflicity theory in three-dimensional flatement has the form
(<r. § . + pu.<Su. )d ..d , = 0 / "J j j ^ > > i> 3 21 (i =1,2,3; j = 1,2,3) (1}
where p- material density; i - displacement components; cij and eij - components of the flress tensor and flrain; h- plate thickness; V - the volume occupied by the body. In accordance with the hypotheses of Kirchhoff - Love
ai (2) Neglecting in (1) the members of which take into account the inertia of rotation normal to the middle plane, will have the following variation equation:
h 2
J d S : + 2r2 S j +r2£ 2 y 3 + J p—j- Swdx3 = 0
dt2
(3)
1 (du, du ) d
s, = - —- +—- -x.
21 dxJ dx, I dx, dxJ
, ,, j = 1,2;
=-—(s + ^ 2 );
1 - V
E l \ E
a2 =--(s2 + s j ); ct, = --st
1-v 1+ V
(4)
EnP(t ) = E0n
f(t)-JR„ (t-r)ç(t)dT
(5,a)
E„v = Eo j [l -rf ((Or )- ,r / (WR )]p
(5,6)
C(wR)= jR (v)coswFTdv T„s(aR ) = JR (T)srna)RTdT
where 0 , 0
, respectively, cosine and sine Fourier transforms relaxation kernel material. As an example, the visco elaflic material take
Ri (t) = Ane-ß /f1"*
three parametric relaxation nucleus
A a B
Here " ' n,Hn - parameters relaxation nucleus. On the effect
of the function R (t t) superimposed usual requirements
inerrability, continuity (except f = T). signs - certainty and monotony:
a
Re >0 , —L < 0, 0<j Rs (t)ti <1. " 0
Introducing the notation for points
fd2 w d2 w ) -fd2 w d2 w )
I , = DI—- + v—- I M, = DI—r + v—r I 1 3x2, ) ^ dx2 J
Based on the geometric relationships and relations of the generalized Hooke's law, taking into account the kinematic hypotheses (2), the expressions for the components of the flrain and flress tensor has the form
ME = D (1 - v)
d2 w dx: dx2
D = Eh3
D =
Here E is the
When R (t t) =0, then modulus of elaflici-ty.
Integrating (3) in the flrip thickness leads to the following form
M,
d Sw dx 2
+ 2M,
d 2Sw dx:dx2
+M„
d 2Sw dx2
Jph d-W Swds = 0.
dt
where ) - arbitrary function of time; v Poisson's ratio;
rb (t - T)- the core of relaxation; E<i - inflantaneous modulus of elaflicity; We accept the in-tegral terms in (5, a) small, then
the function 9>(t) = ^(t)e R , where v(t) - slowly varying
function of time, Wr - real conflant. The [7], we replace of (5,a) approximate species
(6)
Integrating twice by parts and alignment to zero, the coefficients of varia-tion w inside the body and on its boundary, and we obtain the following differential equation
d2 M, d M d 2M, i „2 , 2n
^ + 2 _ _ 1 + _ + phw = 0, (w = d2 w/dt2 )
dx2 5x15x2 dx 2
with natural boundary conditions:
dw
dx„
= 0
w = 0; x2 = 0 :12 (8)
w
^ = 0
ÔTj xl = 0; l1
w = 0 ; (9)
The main alternative boundary conditions to them
fM, = 0
dM_ „ dM
2 + 2- 1
M = 0
dx
= 0; x2 = 0 ; l2
(10)
dM, ncM,
1 + 2-^ = 0 ; x1 = 0 : l1.
dx,
dx^
w=W;
p =
3x2
_ (d 2W d W M =1 —- +
dx,2
5x2
dM, dM.
Q = +
5x„
ôx,
(12)
3Q 32M N 32p , 32W „
- +-- + D '(1 -v)—y +ph—- = 0
dx2 3x,2
3x,
dM - Q - D"(1-v)dW = 0 D^-M + Dd-W = 0^--p = 0
öx2 v ' dx2 . dx2
dx12
dW öx 2
and alternative boundary conditions x2=0: x2=l2:
M - D (1 - vf-M- = 0;
p = 0 or dxJ
Q + D (1 - v)d2pp = 0.
w = 0 v X2
(14)
or
and x1=0, x1=l1,
M - D (1 - = 0
p = 0
or
dx 2
(Q, M ,p,W ) =(q , M ,p,W )e'(ax1-
a )
(16)
In both cases, the imaginary part kI or CI characterized by the intensity of the dissipative processes. Subflituting (16) in (17), we obtain a syflem of firfl order differential equations solved for the derivative
Q '-a2~-a2D'(l -v)p-pfia2W = 0;
~'- Q +a2 D '(l - v)W = 0;
p ' -— M-a2W = 0; _ D W'-p = 0
(17)
with boundary conditions at the ends of the band x2=0, l2, one of the four types
a.
swivel bearing: W = M = 0; (18)
6. sliding clamp:
: Q=p = 0
(19)
1 (11) For, we conflruct a spectral problem by entering the following change of variables
dw
b. anchorage:
r. free edge:
: W =p = 0
(20)
M + a2D (1 -v)W = 0 Q-a1 (1 -v) Dp = 0
(21)
Subflituting (12) into (7) we obtain the differential equation of the sys-tem relatively sparse on the firfl derivatives x2 :
Thus, the spectral formulated task (17) and (21) the parameter a2, de-scribes the propagation of flexural waves in planar waveguide made as a band with an arbitrary coordinate on the thickness change x2. It is shown that the spectral parameter a2 It takes complex values (in the case of ^0) If =0, whereas the spectral parameter a2 It takes only real values. Transform this syflem (17). We have
Q ' = M' + D '(1 -v)aW + D '(1 - v)a2V From whence
M"+ D'(1 - v)a2W -a2M-pk W = 0 Moreover
W -1M-aW = 0.
D
Thus, the conversion syflem is of the form
\M ' -a2 M - {phrn2 - D ' (1 - v)a2 ) W = 0
|W ' -a2W - = M = 0 I D
(22)
Q + D (1 - v)^ = 0 W = 0 or dx1 (15)
Now consider the infinite along the axis x1 band with an arbitrary thick-ness changes h=h(x2). We seek a solution of problem (13) - (15) in the for
Describing the harmonic plane waves propagating along the
axis x1. Here (q , M ,^,W) - complex amplitude - function;
- wave number; C (C = CR +iCi )- complex phase velocity; ra
- complex frequency.
To clarify their physical meaning, consider two cases:
1) k = kR; C = CR +iCi, ( =m' +im' )then the solution of differential equations (13) has the form of a sine wave at x1, whose amplitude decays over time;
2) k = kR +ikI; C = CR, Then at each point x1 fluctuations eflablished, but x1 attenuated.
The boundary conditions (18) - (21) in alternating it has the form:
a. swivel bearing: W=M = 0;(23)
6. sliding clamp: W' = M '-a2 D ' (1 -v)W = 0; (24)
b. anchorage: W = W' = 0 (25)
M ' + a2 D (1 -v)W = 0
r. free edge: M ' - a(1 -v)(DW ) ' = 0 (26)
at x2=0 or x2=+l2
Let and some own functions of the syflem (22) - (26) may
have a complex meaning. Multiply the equation syflem (22) to
function and , complex conjugate to and . Identical converting
the firfl equation, we in-tegrate the resulting equality x2 and
composed of the following linear combina-tion
i-i ^ i-i ^ i-i -fMWti 2 -a2 (1 — v)J (DW) "Wd 2 +a2 (1 -v) J (DWfWd 2 -
"2 — "2 — "2 — -a2jMWd 2 -a2JphWWi 2 -a2(1 -v)JD"WWi 2 +
r — -y'r — rMM
-JW "Mrf 2 -a2 JWMrf 2 d 2 = 0
Integrating (27) by parts,
M '-a1 (1 - V DW 'W - J [M W ' + M W ''
'2 '2 — + a2(1 -v)JDW'W'd 2 -a1 J [MW' + MW]j 2
-2a2 (l-v)J D'WWd 2-a2 J phWWd 2 -J MMd 2 + WM
-2a2(I-v)JdW Wd 2 + a2(1 -v)J D'WWd 2 + a2(l-v)J DW"Wi 2
or
Im'-a2(l-v)DW) Iw 2 + \M + a2(l-v)DWIw'I
J^WM + WM
valid at
=0. The expressing a2 (28) We find that
J (MW '+M W ') d 2 +J MMd 2 + ( JphW Wd
D
1
D
a
a .
-a
J (30)
on the left-hand side of equation (29) will be as follows
JQ'Q' -a'MQ'-a'D' (1 -v)pQ' - phm'WQ' +M'M' - QM' +
0
+ a2 D '(1 -v)WM " + -1 Mf -aWf + W ' W " -fW 2 = 0
(31)
or, integrating parts
QQ ' + rr' + WW ']|- j\(Q ' + M')Q + (M•' + a2Q ' -0 0 L
+ )M + (V + W' + a2D'(1 -v)Q' )v+ (W +a2y'-
-a2 D' (1 -v)M' + phofQ ' )W^ 2 = 0 (32)
Thus the conjugate (30) - (32), the syflem has the form
' Q + M' = 0
M'' +a2Q • +—m' = 0
^ D
f + W '+a2 D '(1 -v) Q * = 0 W '-a1 D ' (1 -v)M ' +af + pha2Q ' = 0
(33)
Moreover, we get the conjugate boundary conditions of equality to zero is integral members expression in (32):
-JM
_f^Li 2phWWi 2 -2a2(1-v)fD"WWi 2 + a'(1-v)fD'¡WW 1 à 2 = 0.
o D o i Jo V ^ (28)
It is easy to make sure that is the integral terms of (28) vanish at any combination of the boundary conditions (23) - (26). It should also be noted that all the functions under the integral
RÂ (t-t) =r
a. swivel bearing:
: r= Q •= 0,
x2 = 0, l2 (34)
6. sliding clamp:
. W • = M • = 0,
2(35)
B.
anchorage: M Q 0,
x2 = 0, l2 (36)
r+a2 D (1 - v)Q * = 0,
[W*-a2D(1 -v)M' = 0, x2 = 0, l2
r. free edge: For conditions biorthogonality solutions once again use the Lagrange formula (29) in the form
J [l(u )V •+ L' (V • )U = Z (U ,V • )|2
(37)
»2 — — *2 — *2 — j(MW + MW)d 2 -2(1 -v)jD"WWd 2 -(1 -v)jD'(WW)'d 2 0 0 0 -
real number.
Thus (with R (t -t) =0), It is shown that the square of the wave number for own endless flrip of varying thickness is valid
for any combination of boundary conditions. If Rj (t t) ^ 0 , then It is a complex value for any combination of boundary conditions.
2. Adjoin spectral problem, orthogonality condition. The resulting spectral problem (17) - (21) is not self-adjoin. Built for her adjoin problem using this Lagrange formula [16]
i i i
j L(U) • V * si = Z(U,VJ |-jL * (VJ •Udx,
0 0 0 (29) where L and L* - direct and adjoin linear differential operators; U and V* - arbitrary decisions of relevant boundary value problems. In our case
-a2D'(l - v) - pha2 0 -a2 D '(l - v)
that leads to the consideration of the following integral
l2V____ _ _ ___,_
j\Qi' Qj'- a2MiQ; - a2D' (1 - v)vQ - phrn2WlQ] + Mh Mj -0
- QM'j +a2D'(1 -v)WlM'J +W] - -=MivJ - a2i W vj +
+ ■=Mfj + ff + W'f + a^D'Çl - v)Qjf + WW '+ + a2Wf' -a2D'(1 - vWM • + phrn2QW]d 2 = 0,
(38)
(Q , M,f, W,, )T
where
- own form, corresponding to the
Eigen value ai original spectral problem; Qj, j )
own form, corresponding to the Eigen value aj adjoin. Integrating (38) by parts
k -aj2)
J [- M Qj - D '(1 - v)Q]f + D '(1 - v)W M] - W, f ^ 2 +
(39)
+
D (1 - v)Q-f - D (1 - v)W M'j ] 0 = 0,
where to i=j we have the condition biorthogonality forms:
i2
j[Mi + D'(1 -v)v)Q] + W(Vj'= D'(1 -v)M)2 +
0
+m-vpiM--qV]\;2=sj (40)
The expression W,Mj - Qj v zero, if the border is set to any of the condi-tions (18) - (21) in addition to the conditions of the free edge.
3. Fixed problem for a semi-infinite flrip of variable thickness.
Consider a semi-infinite axial x1 lane variable section, wherein at the end (x1 =0) harmonic set time exposure of one of two types of:
+
+
0
0
+ w;W -fW + QjQ, + MQ + M'Ml + a jMQ] +
-a
L
0
W = fw (X2 )e", M, = fw (X2 )e", or
, = f, (x ", Q. = fQ ( X2 y, Xj = 0 where
xi = 0 (41)
(42)
Biorthogonality ratio (30) gives
i l — ak = -J fQ ( X2 )Qid
dW n n = , Qi = D dx.
d 3W „„ d W
—r + 2(1 -v)-2
dx, dx, dx -,
(43)
Transform the boundary conditions (41) so that they contain only select-ed our variables W,9, M and
W = fw (X2 )e", D
-I d 2W d2W
dx ,2
- + v-
dx2
= fM (X2 )e", Xi = 0,
W = fw (X2 )e", D
d 2W d 2W
dx2
dx2
— d 2W -D(1 -v) — = fM(X2)e", x, = 0, dx2
(49)
4. Tefling software syflem and fludy the properties of propagation of flexural waves in a band of variable thickness.
Tefling program was carried out on the task of diflributing the flexur-al waves in a plate of conflant thickness. Consider the floor plate of conflant thickness infinitely satisfying Kirchhoff-Love hypotheses, with supported long edges (Figure 1). at end face x1=0 specified: w=f1(x2)eiwt, M11=f2(x2)eiwt (50)
Spread along the axis x1 flexural wave is described by the differential control syflem (13)
Q' + + phw' = 0 (w' = d2 w / dt2)
dx1
or
dW dx,
= fr(x2 Y" ,-
d2W d 2W
+ D (1 -v)-
d2 I dW
= fq (x2 )e", x, = 0,
Of finally
W = fw (x2 )e", M = [fM (x2 ) + D (1 - v) fW(x2 )]e", x. = 0
(44)
dW dx.
= f,(x2)e"
dM dx.
= [fq (x2 ) - D (1 -v) f,( x2 )]e", x1 = 0.
(45)
Assume that the desired solution of the no flationary problem can be expanded in a series in Eigen functions of the solution of the spectral problem. In the case of conflant thickness it is evident, and in general, the queflion re-mains open.
The solution of the flationary problem (17) - (21) (41) - (42) will seek a
= 2 a
,k (x2 ) Mt (x2 )
Qk ( x2 ),
(46)
Wk, , eK Mk
where k' k, k, k - biorthonormal own forms of the spectral prob-lem (17) - (21).
The representation (46) gives us the solution to the problem of non-Sationary wave in the far field, i.e., where it has faded not propagating modes. The number of propagating modes used N course for each specific frequency w, since the cutoff frequency is greater than the other w.
Consider two cases of excitation of Sationary waves in the band:
а) fw=0 - antisymmetric relative х1;
б) fw=0 - symmetric.
In the case of antisymmetric excitation, sub^ituting (46) into (44) and expressing fM(x2), obtain
N _
fM (x2 ) = YakMk (x2 )
k=i (47)
Value biorthogonality (40) gives expression to determine the
«2 _
ak = J fM (x2 Qk'(x2 У 2
unknown coefficients 0 .
f (x )
In the case of a symmetrical excitation J Q v 2 7 rearranging (46) to (45) in the following form
We obtain
M'-в = 0 ,
1 ,, d2
■=M + - 2 D dx,2
w'-,= 0, h = h0
(51)
with boundary conditions of the form (15)
— d2 w
M - D (1 - v)—— = 0, x2 = 0, n
w=0, dx 1 . (52)
Introducing the desired motion vector in the form of
M
M Ф
W
К У
-i ( akx1 -")
(53)
Go to the spectral problem
Q ' -a2M - m2 phW = 0, M ' - Q = 0, _, 1
=M -aW = 0, D
W '-, = 0,
(54)
with the boundary conditions
W = 0, M + D(1 -v)a2W = 0, x2 = 0, n or
W = 0, M = 0, x2 = 0, n(55) Rewrite the syflem (54) as follows
W" --Lm-a2W = 0, D
M ''-a2M - m2 phW = 0,
(56)
and W = 0, M = 0, x2 = 0, n. We seek the solution of (56) in the form W = aw sin nx2,
M = aM sinnx2,( n = 1,2,...) (57)
satisfying the boundary conditions (55). We obtain an algebraic homogeneous syflem
- n2aW - a2aW - -= aM = 0
fQ (x2 ) = 2 (-—kakMk (x2 )
(48)
= 0 (58)
+ v
d
0
D
+
e
w
К У
IW Л I Wk (x2 ) 1
e
2
2
-n ам~а a m — со
W
For the exigence of a nontrivial solution, which is necessary to require the vanishing of its determinant
det
or
D
a2 ph
= 0,
2 , 2 , Ph + n = ±w 4=t- , 1 " D
(59,a)
where, when
R-a (t-t) =
=0
2 2 , ph a,~ =-n ±a.\L—.
1,2 V D
(59,b)
Ownership of conflant thickness flrip bending vibrations are of the form
(60)
S7 1,2 •
Wn = sin n x2 .
M= ± ph D sin
1,2
pn = n cos nx 2,
Qn1'2 = +0phD cosnx2.
We conflruct the solution of the problem adjoin to (54)-(55)
W * + a2p' + phm2Q * = 0,
p' + W' = 0
— 1 2 — M *' +=p* + a Q • = 0, D
(61)
Q • + M * = 0 and
f* = 0, Q* = 0, x2 = 0, n(
(64)
f1'2 = ±œ^D
sin »
a"'2 = Sin nx2 ,
W/1'2 = ±n( fi D cos » M j1'2 = n cos nx„.
(65)
J Q.'Qj* - a,2 M, Qj - OphWQj + Qj Q, + MjQ, + M, M j - QM j +
+m j mt +D%;mi +u]q;mi +f fj - d mt fj'-a2w, f1+f1 % + +w;% + w' w; - %w; + w' w+afw+phoq'w ik 2 = s.
(66)
where
Qt, Mt, , Wt -
own form for the direct
problem, the corre-sponding Eigen values
q: , mj, fj, wj
, and - own form of the dual problem, the corresponding Eigen value aj. Integrating by parts in (66), using the boundary conditions (55) and (62) we obtain the desired condition:
JM,
Qj + w%:
iVj JT 2 ~Si
0 (67)
We now verify biorthogonality received their own forms (60) and (65) using the condition biorthogonality (67)
| [j phD Sin(k 2 ) • Sinj 2 ) + phDSin(k 2 )Sin(jk 2 2 =
0
^_n _
= 2^JphD | Sin(k 2 )Sin(jk 2 )d 2 = n^jphD Sf
0
The normalized adjoin eigenvector on , we have a
syflem of ei-genvectors satisfying the condition (67).
We now obtain the solution of the problem of the diflribution of the fla-tionary wave in the semi-infinite flrip of conflant thickness. Suppose that at the border x1=0 set the following flationary diflurbance:
w = Weica = bW sin(» 2)eiœt,
M = Meim = bM sin(» 2)eia, Xi = 0 (68)
We seek a solution of a problem
*2 " (62) Transforming (61) - (62) we obtain the following syflem of firfl order differential equations
\p' -a2p'-phm2Q '= 0
Q•" -a2Q•-DP'= 0, x2 = 0,^: p' = 0,Q' = 0.
2 (63)
The solution of (63) in the form
p' = a'p sin nx2, Q' = aQ sin n x2
w(xi, x2, t) = £ akWk M(x, x2, t) = atMt k=1 , k=1
where
Wk = W (xa )e Wk è Mk - ,
-,(akxi -a)
Mk = M ( x 2 )e
(69)
-i ( akxi-at )
From whence It has the same form (59a), own forms of vibrations are of the form:
For biorthogonality conditions direct solutions and the adjoin
problem is necessary to consider the following equation
a -k - —k - own form (60), corresponding to k It is evident from the band at the end face at x1=0 decision (69) mufl sat-isfy the boundary conditions (68)
bW sin(K 2 )e>°* = aW (x2 )eM,
K=1
TO
bM sm(M 2 )eia =£ aM (x2 )eia,
K=1
or go to the amplitude values
to
bw sin(* 2 Y0* =E aW,
bM sin(« 2 )aia =£ atMk,
k=i (70)
Consider the following integral
jMQ/+ W%;V 2 =J
£ atMtQ-+£ aWfj
= jj at J^MtQ:+ Wj 2 = aj.
2 2 n2 +a
22 n2 +a
2
k=1
2
On the other hand on the edge x1 = 0 the same integral as follows
n r
J\bM sin(* 2)Qj + K sin(» 2W
obtain
n 1 q
J \bu sin(n x2 ) —==sin(p 2 ) ± bw sin (nx2 ) — sin (x 2 0 n^phD n
J sin (nx2 ) ■ sin( f 2 2
nJphD n
(73)
a ± = £.
b„
2-J phD
bwa ' 2
(74)
W = (a+W+ + aW -Vi(axi -Qt)
\ n n n n /
M = (a-U+ + aM )e(a-Qt )
\ n n n n /
where
Wn± =±sinnx2, Mn = ±(phD sinnx2, ratio (74).
Now suppose that the fleady influence on the border of semiinfinite flrip x1=0 it has the form
w = fw x )e(, M = Jm (x2 )e"
. u - 2 , ^ j + L
0 (72)
Subflituting in (72) from the normalized own form (65) we
(76)
Let us expand the function and Fourier series of sinus in the in-terval [0,n]
fw (X2 ) = Z BW sin (k X2 X fM (X2 ) = Z bM sin (k X2 )
k=i k=i (77)
Using the results of the previous problem, we find that the solution can be represented as a Fourier series:
W
==±a
,a+W ++ a-W -) e
- a-at)
From a comparison of the formulas (71) and (73) it is clear that under such boundary conditions is excited only «n» - Single private form:
M = X(a+M++ a-M-) e
i (ax;-at )
(78)
determined from the
Thus, the solution of the no flationary problem for a half-flrip of con-flant thickness has the form
(75)
a - determined from the
, aWk1, Mk
where 2VphD ratio (60).
5. Numerical results and analysis. The numerical solution of spectral problems carried out by computer software syflem based on the method of or-thogonal shooting S.K. Godunov [4] combined with the method of Muller. The results obtained in tefling with the same software package analytically up to 4-5 mark frequency range from 0.01 to 100. Hereinafter, the entire analysis is conducted in dimensionless variables, in which the density of the material, half the width of the waveguide l2 and E modulus taken to be unity, and the parameters of relaxation
kernel A = 0,048; P = 0,0 ; a = 0,1 .
2,5 2,0 1,5 1,0
C
D.5
4
6
0
2
8
w
Figure 2. The dependence of the phase velocity on frequency
Figure 3. The dependence of the frequency of the wave
The calculation results are obtained when A = 0. Figure 2 shows the spectral curves of the lower modes of oscillation of conflant thickness plate, the corresponding n=0, 1, 2, 3, 4, 5 for Poisson's ratio v=0,25. Analysis of the data shows that the range of applicability of the theory of Kirchhoff-Love to a plate of conflant thickness is limited by the low frequency range. For example, for the firfl mode (h = 0), the range of application of the theory 03 because of the unlimited growth of the phase propagation velocity with increasing fre-quency, for high frequencies Cf Cs .
At high frequencies, where the wavelength is comparable or less than the fashion of flrip thickness, there is, as is well known, localized in the faces of the Rayleigh wave band at a speed slower speed Cs, however, as is obvious, this formulation of the problem, in principle, does not allow to obtain this re-sult. However, it should be noted that in the application of the theory of Kirchhoff - Love platinum conflant thickness is obtained the correct conclu-sion about the growth of the number of propagating modes with increasing frequency that is well seen from the spectral curves of Figure 2 and Figure 3, which shows the dependence of the wave number the frequency for the same modes of waves.
Figure 4 shows the obtained numerical form for the above modes of os-cillations coincided with the same accuracy (4-5 decimal places) in the division of bandwidth by 90 equal segments.
Figure 5 illuflrates the solution of the flationary problem: the amplitude of the excited oscillation modes linearly depends on the frequency.
We proceed to the propagation of flexural waves in a symmetric band Kirchhoff - Love of variable thickness. Let us firfl consider a waveguide with a linear thickness change, presented in Figure 6 and 7 which are free edges. Fig-ure 8 shows the dispersion curves for the firfl mode, depending on the verge of tilt angle 9/2. Curve I corresponds to a flrip of conflant thickness h0=h1. Curve 2 corresponds to a waveguide with an angle of inclination of faces 9/2 =/4 or tg 9/2=1 and curve 3 corresponds to a waveguide tg 9/2=0,2. The fig-ure shows that, unlike the bands in the case of conflant cross-section of the waveguide with a small tapered angle at the base of the wedge (Curve 3) there exifls a finite limit of the phase velocity of the fashion spread, and
lim C f = 2Cs
P 2
where Cs -. The speed of shear waves, which coincides with the results of other fludies [5,6,14,15] Thus, it is shown that -Lava Kirchhoff theory pro-vides a wave propagating in the waveguide is tapered with a sufficiently small angle at the base of the wedge-speed, lower shear wave velocity and different from the Rayleigh wave velocity. Moreover, these waves from a frequency dis-tributed without dispersion. This wave is called «wave Troyanovskiy - Safarov» [10, 12,13].
Figure 9 shows the waveform of the same frequency for ra=10, from which it follows that the flrip of conflant thickness behaves like a rod while at the wedge-shaped flrip there is a significant localization of waves in the area of acute viburnum, and the more, the smaller the angle 9. The above fact explains the Kirchhoff theory -Lava applicability for fludying wave propagation in waveguides is tapered, as the frequency increases with decreasing length of one side of the wave modes, with different wave localizes with the sharp edge of the wedge so
that the ratio of the wavelength and the effective thickness of the material is in the field of applicability of the theory. This flatement is true, the smaller the angle at the base of the wedge.
It should also be noted that the numerical analysis of the dispersion equation (33) does not allow to show the presence of flrictly limit the speed of wave propagation modes, since the computer cannot handle infinitely large quantities. We can only speak about the numerical flability result in a large frequency range, which is confirmed by research. For example, when tg^/2=0,2 value of the phase velocity of a measured without shear wave veloc-ity at ra=3 and ra=40 It differs fifth sign that corresponds to the accuracy of calculations, resulting in tefl problem.
In the example h0 = 0,0001, it certainly gives an increase ofthe phase ve-locity when the frequency increases further, since such a flrong localization of the wave to the thin edge of the wedge, flarts to affect the characteriflic dimen-sion - the thickness of the truncated wedge, and Kirchhoff hypothesis -Lava flops working. To solve the problem of acute wedge numerically is not possible, since the dispersion equation contains a term D-1, and the thickness tends to zero flexural rigidity D behaves as a cube and the thickness goes to zero. This significantly increases the «rigidity» (ie the ratio between the small and large coefficient) syflem, increases dramatically the computing time and decreases the accuracy of the results. However, it is clear that you can trufl the results obtained where the agreed parameters h0 and a. We note also that the numerical experiment showed no significant dependence of the phase velocity of the firfl mode of the Poisson's ratio , and the fact that a family of disper-sion curves with different apex angles of the wedge have a similarity property: the ratio of the phase velocity to the limit does not depend on the angle of the wedge 9. On the modes, flarting from the second, the speed limit dependence on Poisson's ratio becomes noticeable - about 8.5% for the second mode when changing
0 < < 0,5. Generally, the limit speed increases with the flronger and the more the mode number.
Figure 10 shows the dispersion curves for the firfl modal wedge tg^/2=0,2. The figure shows that the speed of the firfl mode (curve I) is equal to zero for ra=0 and since the frequency ra=1, virtually unchanged. The speed of the second mode (curve 2) is nonzero and finite for ra=0 and flabilized at ra=3. The refl of the modes (curve number matches the number of fashion) have a cut-off frequency, which can be easily determined from Figure 11 (the dependence of the wave number a of the frequency), and decreasing, flabilized (seen 3 and 4 modes) at top speed.
Figure 12 shows the evolution of the firfl waveform with the frequency ra for frequencies ra=0,5; 1; 5 h 20. Pronounced localized form with increasing frequency. Figure 13-16 shows the own forms respectively for 2-4 modes of vibration for different frequencies: ra = 1, 2, 3 and 4 (the number of grid points corresponds to the number form). And here there are localized forms in the ar-ea of thin wedge edge. Figure 16 gives an idea of the degree of localization of the forms at the frequency ra = 1, obviously, the lower the number of forms, the flronger it is localized at the edge of the wedge.
Figures 17-19 shows the spectral curves of the firfl three events in the case of the nonlinear dependence of the thickness of the flrip from the coordi-nates x2.
h(X2 ) = ho + k 2 P,
0< X2 < 1,
where the parameter p It was assumed to be 1.5; 2; 2.5; 3 (curves 1, 2, 3, 4, respectively, curve «0» corresponds to p = 1 -linear relationship).
Figure 4. Form for the higher oscillation modes
From the equation of «0» with the remaining curve shows that they are located on the horizontal high-frequency asymptote, monotonically to zero. The midrange is observed a characteriflic peak which is shifted to lower fre-quencies with an increase in «p». In accordance with the charts of waveforms at ris.20-22 quicker and localization of motion near the edge of the waveguide.
Thus, it can be concluded that the phase velocity of the wave in the local-ized waveguide edge is defined as the frequency
increases the rate of change of thickness in the vicinity of the sharp edge.
Figures 23-28 illuflrate the solution of the flationary problem for a wedge-shaped waveguide with a linear change in the thickness of the coordi-nates x2 depending on the location of the excitation zone, from which it is clear that the main contribution to the resulting solution brings a sharp edge excited waveguides. Analysis of figures 23-25 shows that, if aroused sharp edge of the wedge is raised moflly firfl oscillation mode, and ratio a1 increases with in-creasing frequency.
Figure 5. The amplitude of the excited mode depending on the fre-quency
The amplitude of the remaining modes is not more than 5% from the firfl (ra = 10). Upon excitation of the central waveguide portion (fig.26 and fig.27) the amplitude of oscillation is 20-50 times lower than when excited sharp edge and decreases with increasing frequency. On Figure 28 shows the factors driv-ing modes when the excitation zone does not capture any region, the
center of the waveguide. The oscillation amplitude is also here oscillations 20-50 times less than in the firfl case. 26-28 of the drawings can be made and another con-clusion that in this case the entire frequency range can be divided into zones, in which one of the modes propagates mainly. For example, in the case of Figure 25:
0 < œ < 2 I fashion; 2 < œ < 5 II fashion; 5 < œ < 10 III fashion, t. i.
x
X3
Figure 7. The settlement scheme
Figure 8. The dependence of the real and imaginary parts of the phase velocity on frequency.
Figure 11. Dependence of the frequency of the wave
Figure 12. Changing the shape of the coordinate fluctuations X1.
Figure 13. Changing the shape of the coordinate fluctuations X2
Figure 15. Changing the shape of the coordinate fluctuations
Figure 16. Changing the shape of the coordinate fluctuations
Figure 17. Changing the phase velocity as a function of frequency
C
0,4 0,3 0,2 0,1
1 2 3 4 W
Figure 18. Changing the phase velocity as a function of frequency
Figure 19. Changing the phase velocity as a function of frequency
w
0,8
0,6
0,4
0,2
I форма
0
2
1
3
4
w
0 0,2 0,4 0,6 0,8 X2
Figure 22. Changing the shape of the coordinate fluctuations
Figure 23. The change factor a depending on the frequency
Figure 26. The change factor a depending on the frequency.
-1
-2
-3
-4
Xl0_i
0.9 1
Figure 27. The change factor a depending on the frequency.
a xl u 4
3 2
1 0 -1 -2 -3
Figure 28. The change factor a depending on the frequency.
On the basis of these results the following conclusions:
- On the basis of the variation equations of elaflicity theory, the mathe-matical formulation of the problem of wave propagation in the extended plates of variable thickness. A syflem of differential equations with the appropriate boundary conditions.
- Showing that the square of the wave number for own endless bands of variable thickness in any combination of the action of the boundary conditions.
- Obtained spectral problem is not self-adjoin. Built conjugate problem for her. Coupling syflem consifls of ordinary differential equations with the appropriate boundary conditions. With the help of the Lagrange formula ob-tained conditions biorthogonality forms. The problem is solved numerically by the method of orthogonal shooting S.K. Godunov in conjunction with the method of Muller.
- Analysis of the data shows that the region with the imaginary theory of Kirchhoff-Love to the plate of conflant thickness is limited by the low frequen-cy range. At high frequencies, when
wavelength comparable to fashion or less than the thickness of the plate theory Kirchhoff -Love does not yield reliable results.
- For the phase velocity of propagation modes in the band of variable thickness, there is final repartition unlike the contant cross-section flrip.
References
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a
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