Goncharenko D.S., Pasechnik V.A., Atanova M.Y.
Oles Honchar Dnipropetrovsk National University, Ukraine
RESEARCH ON FORCED OSCILLATIONS OF PLATES WITH MIXED BOUNDARY CONDITIONS
The usage of thin plates as supporting elements for constructions in many fields of modern industry makes the problem of analyzing and solving boundary problems of plate theory of great importance.
The boundary conditions of reallife constructions are often complicated and combine different types of support, clamping and hinging. These conditions may occur due to either project or the change of material’s properties during its lifetime.
The study of plate oscillations is reduced to solving fourthorder partial differential equations which satisfy corresponding boundary conditions. The exact solution of these problems can only be obtained for the several linear cases, when the boundary conditions do not prevent separation from spatial variables. Otherwise the problem may be solved approximately using numerical methods of which variations methods are mostly used. Unfortunately, the efficiency of variations methods declines in the case of mixed boundary conditions because of difficulties in building coordinate functions [1].
Therefore the development of approximate analytical methods for solving and analyzing these boundary value problems is a rather actual problem.
The problem of forced oscillations of thin homogeneous isotropic rectangular plate was considered. The plate is hinged on sides and on sides and has mixed boundary conditions of hingedclamped type, symmetrical about the axis (Figure 1). Forcing pressure is applied to the plate:
The differential equation for this problem is:
(1)
We will seek for its solution in the form of:
(2) 
After dimension removal and substitution of (2) to (1) we get:
(3) 
where
Figure 1. Plate with mixed boundary conditions
The parameter is inserted into the boundary conditions in a way that provides fully hinged boundaries when and considered boundary conditions when :
(4) 

(5) 
where with as Heaviside function.
Presenting the form and eigenvalue by based expansion (6) we can apply the pertrubation method to boundary value problem (3), (4), (5). [2]
(6) 
A recurrent system of boundary value problems is obtained after asymptotic expansion of the original boundary value problem. Its solution is to be found in the form of:
(7) 
where is a partial solution of nonhomogeneous equation like (3) with homogeneous boundary conditions:
(8) 
while describes solution of the following boundary value problem:
(9) 
.
This system can be reduced to infinite system of algebraic equations and due to the necessity to satisfy boundary conditions (5), the next system is obtained:
(10) 
where
Presence of the parameter in (10) allows us to simplify it by using the small parameter method and obtain recurrent formulas for determining the approximation of unknown coefficients:
(11) 

(12) 

(13) 
The Pade approximation for which is: [3]
. 
(14) 
Limits and then give exact solutions for the (3), (4), (5) problem for corresponding values.
Plate forced oscillation problem for mixed boundary conditions of hingedclamped type with nonsymmetrical clamped area location can be solved analogically to the above shown method.
Considering the results obtained from the proposed method and numerical methods, the combination of boundary perturbation method and Pade approximation is applicable to plate forced oscillations problems with mixed boundary conditions of various types.
Literature:
1. Тимошенко С.П. Статические и динамические проблемы теории упругости / С.П. Тимошенко. – К.: Наук. думка, 1975. – 576 с.
2. Коул Д. Методы возмущений в прикладной математике / Д. Коул. – М.: Мир, 1972. – 227 с.
3. Бейкер Дж. Аппроксимации Паде / Дж. Бейкер, П. Грейвс–Моррис. – М.: Мир, 1986. – 502 с.