Semiclassical analysis of systems of Schrödinger equations

The scalar Schrödinger equation models the probability density distribution for a particle to be found in a point x given a certain potential V(x) forming a well with respect to a fixed energy level E_0. Formally two real inversion points a,b exist such that V(a)=V(b)=E_0, V(x)<0 in (a,b) and V(x)>0 for xb. Following the work made by D.Yafaev and performing a WKB approximation we obtain solutions defined on specific intervals. The aim of the first part of the thesis is to find a condition on E, which belongs to a neighbourhood of E_0, such that it is an eigenvalue of the Schrödinger operator, obtaining in this way global and linear dependent solutions in L2. In quantum mechanics this condition is known as Bohr-Sommerfeld quantization. In the second part we define a Schrödinger operator referred to two potential wells and we study the quantization conditions on E in order to have a global solution in L2xL2 with respect to the mutual position of the potentials. In particular their wells can be disjoint,can have an intersection, can be included one into the other and can have a single point intersection. For these cases we refer to the works of A.Martinez, S. Fujiié, T. Watanabe, S. Ashida.

Prediction and measurement of forced response of a composite blade undergoing nonlinear vibration

The main objective of this project is to experimentally demonstrate geometrical nonlinear phenomena due to large displacements during resonant vibration of composite materials and to explain the problem associated with fatigue prediction at resonant conditions. Three different composite blades to be tested were designed and manufactured, being their difference in the composite layup (i.e. unidirectional, cross-ply, and angle-ply layups). Manual envelope bagging technique is explained as applied to the actual manufacturing of the components; problems encountered and their solutions are detailed. Forced response tests of the first flexural, first torsional, and second flexural modes were performed by means of a uniquely contactless excitation system which induced vibration by using a pulsed airflow. Vibration intensity was acquired by means of Polytec LDV system. The first flexural mode is found to be completely linear irrespective of the vibration amplitude. The first torsional mode exhibits a general nonlinear softening behaviour which is interestingly coupled with a hardening behaviour for the unidirectional layup. The second flexural mode has a hardening nonlinear behaviour for either the unidirectional and angle-ply blade, whereas it is slightly softening for the cross-ply layup. By using the same equipment as that used for forced response analyses, free decay tests were performed at different airflow intensities. Discrete Fourier Trasform over the entire decay and Sliding DFT were computed so as to visualise the presence of nonlinear superharmonics in the decay signal and when they were damped out from the vibration over the decay time. Linear modes exhibit an exponential decay, while nonlinearities are associated with a dry-friction damping phenomenon which tends to increase with increasing amplitude. Damping ratio is derived from logarithmic decrement for the exponential branch of the decay.