ISSN: 2690-5779
Doo Sung Lee*
Received: October 28, 2020 Published: January 29, 2021
Corresponding author: Doo-Sung Lee, Department of Mathematics College of Education, Konkuk University Seoul, Korea
DOI: 10.32474/JOMME.2021.01.000115
The existence and uniqueness of the solutions to an infinite system of nonlinear equations arising in the dynamic analysis of large deflection of a circular plate are discussed. Under the condition that the rim of the plate is prevented from inplane motions, explicit equation for the coupling parameter is given.
Keywords: Existence; uniqueness; Berger’s method;1991 Mathematics Subject Classification; 46(Functional analyss);73(Mechanics of solids)
In the analysis of classical mechanics problems, there are
cases where linear mathematical model can not fully describe the
phenomena. If the deflection of the plate is of order of magnitude
of its thickness, the differential equations for the deflection and
displacements can be written in terms of nonlinear equations. These
nonlinear equations are usually difficult to obtain the solution.
Thus, several attempts have been tried to obviate the difficulties.
Among these attempts, it was Berger’s method which drew
much attention. Berger[1] derived as implified nonlinear equations
for a plate with large deflections by assuming that the strain energy
due to the second invariants of the middle surface strains can be
neglected when deriving the differential equations by energy
method. Berger restricted his analysis to static and isotropic cases.
Later, his procedure was generalized to dynamics of isotropic
plates by Nash and Modeer [2] and to dynamic phenomena
in anisotropic plates and shallow shells by Nowinski [3].
Berger’methods is dealt in recent books [4] and [5]. In the research
paper[6], Banerjee and Mazumdar review various approximate
methods including Berger’s in relation to the investigation of
geometrically nonlinear problems. In Sathymoorthy and Chia
[7], a nonlinear vibration theory is formulated for rectilinearly
orthotropic circular plates using Berger’s method. On the other
hand Han and Petyt[8] report that the large vibration of in-plane
membrane forces over the plate span for some of the laminated
plates has been observed which will definitely affect the application
of Berger’s hypothesis to the geometrically nonlinear analysis of
these laminated plates.
There are many other papers giving explicit solutions to various
cases, however the search for the existence and uniqueness of the
solution is rare, thus it is the purpose of this paper to discuss this
matter. We now briefly go over the Berger’s method for the circular
plate. The deformation of the middle surface pertinent to the large
transverse deflections is described by the equations
In the above equations w = deflection of plate in the normal direction. u, v =displacement in plane
The strain energy due to the bending can be written as
where S denotes the surface of the circular plate. We can write the strain energy due to the stretching of the middle plane as
The variation of the work by the external force is now
By virtue of d’Alembert’s principle the motion of the structure is replaced by a state of static equilibrium governed by by the equation of minimum potential energy of the system,
Burger’s method is to set e2 = 0 in (2) as it is relatively negligible compared to other terms, and equation (3) then gives following equations,
from which integration of (4) gives the interesting result the
first strain invariant is a constant. The
governing equations are
We consider a clamped circular plate so that boundary conditions are
The coupling parameter is now determined from the equation
(6) and boundary conditions (7)
as follows
We attempt to find Fourier series solutions to (5) when q = 0.
Here ( ), ( ) n n J γ I γ are the Bessel functions of the first kind and modified Bessel function of the first kind, respectively. When this solution is substituted in (5) and (8) we find
We discuss the existence and uniqueness of the solutions to above infinite system of nonlinear equations. The initial conditions on (11) will be taken as
If we multiply (11) by T0jand sum j from 0 to infinity, we can show that
At first glance it would appear that if the initial conditions (12) and (13) satisfy a finite energy condition, i.e.,
then (11) should have a solution for all t > 0. Indeed this is the case for finite system of the form (11) since the finite system
has associated with it a Lipschitz constant. Therefore, successive approximation method may be applied to prove the existence of solution to (16). However, the infinite system of equations (11) is not Lipschitz continuous since the coefficients of Tj is unbounded as j →∞. Thus the method of successive approximation fails and an alternative procedure is necessary.
In section 2, it will be shown that under the initial conditions (12) and (13) solution of the finite system (16) converge to a solution (11) as N → ∞. In section 3 it will be shown that the solution of (11) satisfying initial conditions (12) and (13) is unique.
To prove solution existence of (11), we define a set of functions T_{j>}, N in the following way: for j ≤ N, T_{j>}, N is a solution of the finite system of equations (16) satisfying the initial conditions (12) and (13) for j = 0, 1, 2, · · ·, N and for j > N set T_{j>}, N = 0. The functions T_{j>}, N are solutions of the infinite system (11),i.e.
If in addition the initial data (12) and (13) satisfy the finite energy condition (15) it follows That
Thus there exist constants M1, M2 and M3 independent of N
Lemma 1. |A0N | is uniformly bounded independent of N where prime indicates differentiation with respect to t. Proof. After differentiating the function AN, if we employ Schwarz inequality we obtain
in view of the relations (20) and (21).
Lemma 2. |AN | is uniformly bounded independent of N
Proof. |AN | is uniformly bounded independent of N from the
relation (22).
Thus AN is uniformly bounded and equicontinuous; by
Arzela’s lemma, there exists a subsequence {ANi} which converges
uniformly to a continuous function A(t). Let Tj be the solution of
the(linear) equation
satisfying the initial conditions (12) and (13). The existence of solutions to (11) is settled by the following theorem.
Theorem 1. The infinite system of (11) have a solution satisfying the initial data (12) and (13). Proof. It is only necessary to show that the solutions of linear system (24) furnish a solution of system (11). For this purpose it suffices to show that
The series which occurs in (25) converges since(cf.(22))
The equality in (25) follows from the estimate
The right side of (27) can be made arbitrarily small by first choosing n, then choosing N_{i}.
In this section it will be shown that the infinite system (11) has at most one solution satisfying the initial conditions (12) and (13). We write (11) in the following way.
Let T_{j} and S_{j} be solutions of (11) satisfying the initial conditions (12) and (13) i.e. T_{j} is the solution of (28) with q(t) being given by (30) and S_{j} is the solution of
If we multiply (28) by Sj and from the resulting equation, we substract(31) multiplied by Tj and integrate it from zero to infinity, we get, after integrating by parts
where we have used the initial conditions (12),(13),(33),(34). According to Gel’fand and Levitan[9] there exists function K(t, x) having continuous partial derivatives of first and second order such that
If we substitute (36) into (28),we find that the partial differential equation
and the boundary conditions
are satisfied. Similarly,
and the boundary conditions
Thus if we substitute (36) and (40) into (35), we find that
where c(t) = q(t) − p(t). Making change of variables and changing the order of integration in (44) we get
The left hand side of (45) is a function of λ,whereas right hand side is a constant. The equality holds only when both sides are equal to zero. Thus
and from Gronwall’s Lemma we get c(t) = 0, i.e.,
Let U_{j} = K_{j} − S_{j} . We show U_{j} = 0 in the following. From (28) and (31) we have
because of (49). Let us denote
Then Vj (t) will be the solution of
With the initial condition (54), the solution of (52) is Vj = 0 from
the semi-group theory. So we have the following theorem.
Theorem 2.The system of equations (11) have at most one
solution satisfying the initial conditions (12) and (13).
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