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DETERMINATION OF NATURAL OSCILLATION FREQUENCIES OF A HOMOGENEOUS VISCOELASTIC PLATE BY THE METHOD OF G. I. PSHENICHNOV
Seitmuratov A.
Doctor of Philosophy, Professor of Korkyt Ata KSU
Mukeyeva G.
doctoral student of KSU named after Korkyt Ata
Nurgaliyeva A.
Master student of KSU named after Korkyt Ata
Abstract
When solving applied problems of oscillation of rectangular planar elements, a wide class of oscillation problems arises related to various boundary value problems: approximate equations of oscillation, various boundary conditions at the edges of a plane element, and initial conditions. In the theory of oscillations, an important point is the determination of the natural oscillations frequencies, the solution of problem-induced oscillations of a plane element, and the study of the propagation of harmonic waves. Depending on the particular types of planar element under consideration, in the general solutions of the three-dimensional problem, the main unknown functions are selected: displacements or deformations at points of a fixed plane of the planar element, in particular, in the median plane of a plate of constant thickness. Displacements and stresses at an arbitrary point of a plane element are expressed in terms of the basic unknown functions, which are determined from the boundary conditions on the surfaces of the plane element. The obtained equations for the basic unknown functions are the general equations of oscillation of a plane element, containing derivatives of functions with respect to coordinates and time of any arbitrarily large order.
Keywords: Displacement, stress, oscillations, frequency oscillations,, viscoelastic plate.
General solutions are given as power series according to flat element thickness. The general solution relates to an equation of hyperbolic type, which describes the oscillatory and wave process in a flat element. When defining a finite number of first members
of sum in series of general equation, we obtain approximate equations of oscillation of one or another plane element.
Turn to the problem of natural vibrations of a rectangular plate of elastic material based on an approxi-
based on the decomposition method developed in the works of Professor G.I. Pshenichnyi [1.2] for static
mate sixth-order equation. The problem is limited by a problems.
conditions when all four edges of the plate is rigidly fixed. If all four edges are arbitrarily fixed, then it is not possible to obtain exact frequency equations.
For such problems, one can successfully apply the approximate method of obtaining frequency equations
The problem includes solution of the approximate equation of a homogeneous isotropic viscoelastic plate [ 3-5]
P (W)+ — P3 (n+10N-1 + 5M-2 - 4p2 (3N- + 9M- - 4MN)x
120 dt
xA^^- +16 p(4 +3MN-1 - M2 N-2 )a2 - 32M (l - MN-1 )a3 W ] = )
(1)
(1) with the right-hand side equal to zero, with that, the viscoelastic operators are replaced by elastic constants. The problem solution is as follows:
W(x, y, t) = expl i b *t W (x, y)
(2)
To define the constant W0 see the equation:
a3w + D a2w + D AW + DW = 0
(3)
Where factors D0, D1, D2 equal to:
_ 24v 2 - 46v + 21 ^ 5 Do = 4h2(1 - v) * - h2;
_(14v2 - 37v +19) 4 5(2 - v) D 8h4 (1 - v) * h4
*2;
(4)
D2 =
(64v2 - 104v + 41) 6 5(7-8v) 4 15(1 -v) g2.
* * + fi * ;
64h6 (1 - v) * 8h6
v - Poisson's constant is determined by formula [6]
H _ b2 _ 1 - 2v 1 + 2^ a2 2(1 - v)
2h6
(5)
Boundary data of the problem with rigid fixing are as follows:
dv d2 v
v = — = —v = 0; (n = x, y; x = 0) dr dr2 n , y; '
d4 v
+ 3A2-^X + 3A4 -4X + 1« 5 2v
da6
da4 dp2
da2 dp4
dp6 +a 14
+Q0I
da
+21 d 4 v. +14 d 4 v
da2 dp2
+ Q2I v = 0
d2v .2 d2v
da2
■ + 12
dp2
dp4
+
(6)
Where factors Q0, Q1, Q2 are equal to
Q0 = D h2; Q1 =Dxh4;
Q2 =D2 h6.
Following the decomposition method, let's formulate auxiliary problems: [7] to determine V, that satisfies the equation and boundary conditions
da
define V2, that satisfies the conditions
^ = f «(a, ß) v, = ^ = = 0
da da
(a = 0; x);
(8)
X /-(2)
dßk
+ f (2)(a,ß); v2 =
_ d 2 V2 = Q
dß dß2 (ß = 0;x)
(9)
In the rest of equation (6) let's use unknown data V3, that satisfies the conditions
3X2
d V + ;l2 d6 V3
+ QX
da4 dß2 d2 v,
da2 dß4
d 2v
dß2
da2
+ X
+ Q0X2
^ + 2X2^X + Ä2 d 4 v3
da4 " " "
da 2dß2
dß4
+
(10)
+ Q2X6 V3 + f (1)(a, ß)+ f (2)(a, ß)= 0;
f{J)(a, 0) = £ ^ a^ sin(«a)sin(m0).
«=1 m=1
General solutions of the auxiliary problems (8) and (9) are as follows
5 3 2
vi = f1 (a, 0) + Op <p, (0) + Of + Of ^ 0) + a ^5 (0) + (A), A6 V2 = f2 (a, 0) + 05 ^i (0) +04 ^ (0) + 03 ^ (0) +02 (A) + 0^5 (0) + ^ (0);
(11)
Where functions f(a, ß) are [8]
œ œ
(i)
fi(a, ß)=-E E -T sin(na)sin(mß);
n=i m=1 n
œ œ a (2)
f2(a, ß)=-E E "Tsin(na)sin(mß);
riä m6
(12)
Based of boundary conditions [9] to define ty
{ß\¥} (at
and expressions (12) , we find the following
360
Pi =T4~
x
dfi
da
24
P2 = —
x
a=0
f da
+
f a
,+7
f a
(13)
12
P3 = — x
dfi
da
,+2
f a
P4 = 0; p= f .. P6 = 0
da
a=0 5
And
360
Wi =Xr
x
f dß
ß=0
+
dß,
ß
24
W2 =—3 x3
12
W3 = — x
8
dß,
ß
ß=0
+7
ß=x
dß1
ß
ß=x
dß1
ß
+2 dß
ß=0 +2 dß
ß=x
dß I
W4 = 0;Ps =-~^ß I ß=0^6 = 0
a=x
8
a=x
a-x
With that
dfl
da
dfu
dfl
d//=o
a
(i)
6=1 m=1 n
-=-£ S (- 1)
sin (/m),
a(1)
n n,m
n
sin (/m),
(15)
=-s s
a
(2 )
n=1 m=1 ^^
-sin (an),
a (2 )
m n,m
(a |a=0 = S S ( 1) m 5
da n=1 m=1 m
sin( an);
If in series (12) we are limited only by first sum members, [10] then under condition V V at (a, p) = —
~(1) _ 7-6,-,(2) receive a11 = 1 a11.
Using condition (9) and equation (10) regarding 3 and set
V3 = 1[V1 + V2 ].
We receive characteristic equation
a,/ =
2
2Л611 -1 к-л4 (1)(2 -f - 5f]ô1
2(1 + Л6 )+3Я2 (1+ Я2 )2 - - - ^
V f f
= 0
(1+ * (
4 v. 24 5ж]. о2 L 3
(1+ Г ) 2+ 4Я2|1 --
ôo -
(16)
I
Related to frequency ~ of the 6th series, that approximately describes the first three frequencies of natural vibrations. [11]
Thus, the approximate decomposition method allows one to find the natural frequencies of flat elements. The problems for the viscoelastic material of flat element are solved in a similar way.
REFERENCES:
1. Pshenichnov G.I. The decomposition method of the solution. - М. DAN USSR, 1985, V.182, № 4, p.792-794.
2. Fillippov I.G. Cheban V.G. The mathematical theory of oscillations of elastic and viscoelastic plates and rods. -Kishinev: Shtiintsa, 1988,-p.190
3. Pshenichnov G.I. Solutions for some problems of structural mechanics using the decomposition
method. / - Structural mechanics and estimation for buildings, 1986, № 4, p.12-17.
4. Fillippov I.G. Egorychev O.A. Wave processes in linear viscoelastic medium. - М: Mechanical engineering, 1983. - p. 272
5. Fillippov I.G. An approximate method for solution of dynamic problems for viscoelastic medium. - PMM, V.43, № 1, 1979, p.133-137.
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7. Seitmuratov A., ZharmenovaB., Dauitba-yevaA., Bekmuratova A. K., TulegenovaE., Ussenova G. Numerical analysis of the solution of some oscillation problems by the decomposition method //News of NAS RK. Series of physico-mathematical.20191(323): 28 - 37. ISSN 2518-1726 https://doi.org/10.32014/2019.2518-1726.4
n=1 m=1
n,m
2
V
