Bolt-nut fasteners are widely used in mechanical structures due to the systems' ease of disassembly for maintenance and their relatively low cost. However, vibration-induced loosening has remained problematic. In this paper, we investigated the mechanisms of the loosening process due to micro bearing-surface slip within the framework of the three-dimensional finite element method (FEM). The results show close agreement with Kasei's experimental results. It is found that the early-stage nut rotation observed experimentally originates from simultaneous bolt-nut rotation induced by the tightening torsion of the bolt and does not correspond to loosening rotation. Therefore, loosening rotation should be defined by the relative rotation angle of the nut with respect to the bolt. It is also found that small loosening is initiated when the vibration force reaches about 50 to 60% of the critical loading necessary for bearing-surface slip. Attention should be paid to the contact state of both bearing and thread surfaces when considering the loosening of bolt-nut tightening systems. Contact states can be classified into three types: complete slip involving no sticking region, micro slip involving no constant-sticking region over a vibration cycle, and localized slip involving a constant-sticking region over a vibration cycle. It is also found that loosening rotation can proceed when either micro slip or complete slip occurs at both the thread and bearing contact surfaces.

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System of Dirac fermions with random-varying mass is studied in detail. We reformulate the system by transfer-matrix formalism. Eigenvalues and wave functions are obtained numerically for various configurations of random telegraphic mass m(x).Localized and extended states are identified. For quasi-periodic m(x), low-energy wave functions are also quasi-periodic and extended, though we are not imposing the periodic boundary condition on wave function. On increasing the randomness of the varying mass, states lose periodicity and most of them tend to localize. At the band center or the low-energy limit, there exist extended states which have more than one peak spatially separate with each other comparatively large distance. Numerical calculations of the density of states and ensemble averaged Green's functions are explicitly given. They are in good agreement with analytical calculations by using the supersymmetric methods and exact form of the zero-energy wave functions.

We study delocalization transition in a one-dimensional electron system with quenched disorder by using supersymmetric (SUSY) methods. Especially we focus on effects of nonlocal correlation of disorder, for most of studies given so far considered δ-function type white noise disorder. We obtain wave function of the "lowest-energy" state which dominates partition function in the limit of large system size. Density of states is calculated in the scaling region. The result shows that delocalization transition is stable against nonlocal short-ranged correlation of disorder. Especially states near the band center are enhanced by the correlation of disorder which partially suppresses random fluctuation of disorder. Physical picture of the localization and the delocalization transition is discussed. Quenched disorder plays important roles in various physical phenomena. Anderson localization is one of these examples[1]. Especially in two and lower dimensions almost all states are localized, and extended delocalized states are isolated points in physical parameter region. The transition between quantum Hall plateaus is such an example. For the transition between integer quantum Hall plateaus, useful model, named network model, has been proposed[2], which incorporates effects of localization and quantum tunneling in a strong magnetic field. Numerical studies indicate that the network model belongs to the same universality class of the transition between integer quantum Hall plateaus. However it is rather difficult to solve the network model analytically because there exists no controllable parameter for perturbative expansion and also because of nature of quenched disorder itself. Compared with the twodimensional (2D) systems, one dimensional (1D) systems are more tractable. These 1D systems include Dyson's study on random strength harmonic springs[3], random Ising model by McCoy and Wu[4], and more recently random exchange spin chains[5]and random hopping tight-binding (RHTB) model [6,7,8].Recently supersymmetric (SUSY) methods appear useful for handling the quenched disorder. SUSY methods are applied to the network model [9], the 1D RHTB model [7] etc. In this paper we shall revisit the 1D RHTB model by applying the SUSY methods. Especially we shall consider nonlocally correlated quenched disorder and study stability of the delocalization transition which exists in the case of the δ-function-type white noise disorder. The model contains two parameters which control magnitude of fluctuation and correlation length of disorder. We expect that we can get detailed physical picture of (de)localization transition from calculations of density of states, Green's functions, etc. as a function of the above parameters. Another motivation of the present work is rather technical, i.e., we show how to use the SUSY methods for nonlocally correlated disorder systems.

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