Bending oscillations of a rectangular orthotropic plate with massive inclusion
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BENDING OSCILLATIONS OF A RECTANGULAR ORTHOTROPIC PLATE WITH MASSIVE INCLUSION M. A. Sukhorol’s’kyi and T. V. Shopa
UDC 539.3
We study the problem of natural and forced oscillations of a hinged rectangular plate with massive elliptic inclusion. The process of bending of the plate is described by the modified equations of the Tymoshenko theory of plates. The numerical solution of the problem is obtained by the indirect method of boundary elements based on the sequential representation of distributions and the collocation method.
The problems of vibration of plates with various types of stiffeners and concentrated masses prove to be quite urgent because structural elements of this sort are encountered in engineering and it is necessary to know the frequency range in which these elements operate outside the resonance zone. Free oscillations of rectangular plates containing concentrated masses were investigated by the Rayleigh – Ritz method with various boundary conditions in [1, 2]. The vibration of the plates with arbitrarily oriented reinforcing ribs was studied in [3]. The finite and invariable jumps of mass in a rectangular domain were analyzed with the help of Chebyshev polynomials and the Galerkin – Ritz method in [4]. The process of vibration of a rectangular plate with stiffeners with distributed mass along the diagonals was investigated in [5] by modeling the analyzed system by a plate with masses concentrated at many points. The solution of the problem of isotropic plate with a jump of mass along a straight line is known [6]. We now consider the problem of oscillations of a plate with massive absolutely rigid inclusion. The Tymoshenko mathematical model of orthotropic plates taking into account the normal component of the inertial force is based on the equilibrium equations [7] ∂Mi1 ∂Mi2 + − Qi = – mi ∂α1 ∂α 2
( i = 1, 2 ), (1)
∂Q1 ∂Q2 ∂u + − 2hδ 2 = – q, ∂α1 ∂α 2 ∂t 2
and the physical equations ∂γ j ⎞ ⎛ ∂γ Mii = Di ⎜ i + νij ⎟, ⎝ ∂αi ∂α j ⎠ ∂w ⎞ ⎛ Q = Λi γ i + ⎝ ∂αi ⎠
⎛ ∂γ j ∂γ i ⎞ Mij = Mji = Dij ⎜ + ⎟, ⎝ ∂αi ∂α j ⎠ ( i, j = 1, 2, i ≠ j ),
(2)
“L’vivs’ka Politekhnika” National University, Lviv, Ukraine. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 6, pp. 41–46, November–December, 2008. Original article submitted October 2, 2007. 1068–820X/08/4406–0783
© 2008
Springer Science+Business Media, Inc.
783
784
M. A. SUKHOROL’S’KYI
AND
T. V. SHOPA
where Di =
2h 3Ei , 3(1 − ν12 ν21)
2h 3E12 , 3
D12 =
Λi =
5hGi 3 , 3
Ei , E12 , Gij , νij are the elasticity constants ( E1 ν12 = E2 ν21 , G12 = G21 ), α 1 , α2 , and α 3 are Cartesian coordinates (α 3 = 0 is the equation of the median plane), 2h is the thickness of the plate, δ is the density of the material, w is a deflection, γ1 and γ2 are the angles of rotation of the normal to the median plane, Qi and M i j are internal forces, and q and mi are external loads. The displacement of any point of the plate is given by the formulas ui = γ i α 3
( i = 1, 2 )
and
u3 = 0.
(3)
The modified equations of bending of the plate
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