General Beam Kinematics

With regard to the dynamics of rotating structures on the one hand, nonlinear influences in the beam kinematics have to be taken into account, while analytical formulation of the constitutive relation of beams with complicated cross-sections on the other hand, is only possible for thin walls and linear kinematics. This gives rise to a combined procedure with a linear analysis to determine the beam properties and a succeeding non-linear analysis to investigate the global beam behavior. For the latter, a general beam with adequate kinematics will be examined first and subsequently transcribed into the intended thin-walled beams. 

---

Based upon the adaptive shell description given in the previous chapter, a thin-walled beam formulation for general anisotropic cross-sections with arbitrary open branches and/or closed cells will be derived in this chapter. After the deduction of non-linear kinematic relations for the general beam and the linear kinematic relations for the thin-walled beam, the torsional warping effects of the latter are examined. Subsequently, the constitutive relation and the equations of equilibrium are established. 

One of the early hopes of heavy ion induced transfer reactions was that new states in nuclei would be preferentially populated The fact that this hope diminished was due primarily to insufficient understanding of the reaction mechanism, but also to the generally poorer energy resolution obtained with heavy ions as compared to light ions. In this paper, I hope to demonstrate that with the proper kinematical conditions there is a remarkable selectivity which can be. obtained with a proper. choice of the reaction and that these reactions can be valuable spectroscopic tools The data in this talk have been taken using beams from the Brookhaven National Laboratory double MP tandem facility with particles identified in the focal plane of a QDDD spectrometer ... 

In 1912, when M. Lane suggested to W. Friedrich and P. Knipping the irradiation of a crystal with an X-ray beam in order to see if the interaction between this beam and the internal atomic arrangement of the crystal could lead to interferences, it was mainly meant to prove the undulatoiy character of this X-ray discovered by W.C. Rontgen 17 years earher. The experiment was a success, and in 1914 M. Laue received the Nobel Prize for Physics for the discovery of X-ray diffraction by crystals. In 1916, this phenomenon was used for the first time to study the structure of polycrystalhne samples. Throughout the 20 century. X-ray diffraction was, on the one hand, studied as a physical phenomenon arrd explained in its kinematic approximation or in the more general context of the dynamic theory, and on the other, implemented to study material that is mainly solid. 

There are two general theories that describe observed intensities in x-ray diffraction these are the kinematical and dynamical theories. The kinematical approach, which is the better known and most commonly employed, treats the scattering from each volume element independently of each other except for the incoherent power losses associated with the beam reaching and leaving a particular volume element. 

The theory presented in Section 3.3.1 is called the kinematic theory of scattering. Diffraction by a three-dimensional body is, however, more complex than suggested by Fig. 3.11. On the one hand, the primary radiation is attenuated by diffraction and the secondary beams may be rediffracted. Hence, the different volume elements do not all receive the same primary intensity for this reason, the kinematic theory does not obey the law of conservation of energy. On the other hand, the interference between the primary wave and the divers diffracted waves has been neglected. All these effects generally lead to diffracted intensities that are weaker than those predicted by the kinematic theory. These... 

To determine the equations of equilibrium as well as the constitutive relations of the beam, the principle of virtual work may be applied and its individual contributions be examined, respectively. Thus, the foundations for an analytic solution with regard to the statics of the non-rotating structure can be provided. Furthermore, the principle of virtual work will serve to set up the equations of motion in consideration of the dynamics of the rotating structure. This, in addition, requires the study of inertia effects and the inclusion of stiffening effects due to kinematic non-linearity with reference to relatively slender and flexible beams. The derivation of the principle of virtual work for the general case is presented in Section 3.4, and it will now be adapted and extended to depict adaptive thin-walled beams. Therefore, the various virtual work contributions will be discussed individually. 

---