On the first frequency of reinforced partially hinged plates

Abstract

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve the stability of the plate, it seems reasonable to place a certain amount of stiffening material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In the present paper we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions.

About Symmetry in Partially Hinged Composite Plates

“…The proof that the first eigenfunction is of one sign, and hence simple, follows by exploiting the so-called dual cone decomposition technique which relies on the positivity preserving property stated in Corollary 2.3. Since the proof is standard we omit it and we refer the interested reader to [9,Lemma 7.2] where the same issue, together with the simplicity of the first eigenvalue, was proved for a related fourth order eigenvalue problem in dimension 1. As concerns the regularity of u p , it follows by combining elliptic regularity and embedding arguments.…”

“…[3,[18][19][20][21][22]29]. As far as we are aware, the partially hinged composite plate problem (1.1)-(1.3) has only been studied in [9], see also [7] for results about higher eigenvalues; in [9] it is proved that the infimum in (1.3) is achieved by the piecewise constant density:…”

“…by considering Navier boundary conditions on sufficiently smooth domains or Dirichlet boundary conditions on balls). Regarding problem (1.1), the above mentioned difficulties are further increased by the unusual boundary conditions and only few results about qualitative properties of the first eigenfunction were proved in [9] for p = p(y). An important step forward in the study of (1.1) has been recently done in [8] by computing explicitly the Fourier expansion of the Green function of the operator in (1.1) and by showing its positivity, see Proposition 2.1 below.…”

A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions

“…As already remarked, the validity of the PPP for problem (2) is by no means an obvious fact; recall that it does not hold for problem (1) on rectangular planar domains; by numerical observations, we do not even expect its validity for the partially clamped plate problem, i.e., (2) with Dirichlet conditions instead of Navier, see Figure 1 on the right. Furthermore, we believe that the validity of the PPP will help in making a significant step forward in the spectral analysis of the operator in (2) and in the related stability analysis for partially hinged plates, especially in the nonhomogeneous case, see, e.g., [2] and [3].…”

“…Let m ⩾ 3 be an integer and set a m ∶= 1 m 3∕2 The function vanishes at t = 0 and t = ±t 1 with t 1 ∈ 2 2m+1 ,3 2m . Furthermore,m (t) > 0 in [− 2 m+1 , −t 1 ) and (0, t 1 ) while m (t) < 0 in (−t 1 , 0) and (t 1 , 2m+1 .Proof of Theorem 2.2 completedSince m is odd, it is sufficient to study its behavior in [0, 2 m+1 .…”