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Classical mechanics is the branch of physics concerned with describing the motion of bodies. The subject is based on three simple axioms relating forces and movement. These axioms were first postulated by Newton in the 17th century and are known as his three laws of motion. Lagrangian mechanics is a restatement of the Newtonian formulation. It deals with energy quantities and paths-of-motion instead of forces. This often makes it simpler to use when working with non-trivial mechanical systems. In this thesis, we use the Lagrangian method to model two such systems; A rotating torus and a variant of the classical double pendulum. It soon becomes clear that the complexity of these systems make them difficult to attack by hand. For this reason, we take a computer-based approach. We use a software-package called Sophia which is a plug-in to the computer algebra system Maple. Sofia was developed at the Department of Mechanics at KTH for the specific purpose of modeling mechanical problems using Lagrange’s method. We demonstrate that this method can be successfully applied to the analysis of motion of complex mechanical systems. The complete equations of motion are derived in a symbolic form and then integrated numerically. The motion of the system is finally visualized by means of 3D graphics software Blender.

Digitized by Kansas Correctional Industries

Digitized by Kansas Correctional Industries

Digitized by Kansas Correctional Industries

Photocopy of typescript. / Digitized by Kansas State University Libraries

Call number: LD2668 .T4 1951 E36 / Master of Science

Modelling of elastically coupled rigid bodies is an important research topic in multibody dynamics. We consider the problem of modelling what can be called flexural joints, where two essentially rigid bodies are coupled by a substantially more elastic body. For modelling general elastic couplings one would like to have generically applicable, lumped parameter, intuitive, Euclidean geometric, accurate models with desirable physical symmetries. The model constitutive equations should be easily and quickly computable. For purely elastic coupling the constitutive equations should be truly energy conservative: the configuration-wrench equations should be derivable from a potential function. Linear and angular momentum should be conserved. Quaternion-based and twist-based modelling methods are presented. The constitutive equations to calculate the configuration-wrench behavior are derived via geometric potential energy functions. Wrenches are computable given the configurations of the rigid bodies and all the stiffness matrices of the compliant element. For an arbitrary pair of elastically coupled rigid bodies there exist coincident, unique points on the bodies known as centers of stiffness at which translation and rotation are minimally coupled. At the center of stiffness there exist three sets of orthonormal principal axes and corresponding principal stiffnesses. These parameters are useful in both analysis and numerical simulation. A finite-element-based method for computing canonical stiffness parameters of elastically coupled rigid bodies is presented. The method is applied to notch hinges and Remote Center of Compliance (RCC) hinges. Standard procedures are presented on how to determine canonical stiffness parameters at the center of stiffness of a spatial compliance. Series of canonical stiffness parameters can be generated automatically using the methods provided. Key program listings are provided which can be used to duplicate the results presented.

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In Part I the nature of the electron correlation problem is briefly discussed and some approaches to its solution and analysis are described. In addition, some previous work concerning the 2p2 3P state of helium-like systems is reviewed. In Part II, position space electron correlation effects in the 2p2 3P state of H-, He, Li+ and Be++ are investigated. This study is conducted by examining the effects of correlation on a variety of radial, angular and interparticle distribution functions and expectation values. A number of different correlated and uncorrelated wavefunctions are employed for this purpose, allowing the merits of different approaches to be assessed. In particular, a natural orbital analysis performed on a configuration interaction wave- function is used to gauge the relative importance of the angular and radial components of correlation, which are found to act in unison. It is seen that the correlation effects are similar in kind to those found in the ground state systems, though they are of a greater magnitude in this doubly excited state. The variation of the correlation effects with the atomic number, Z, is also examined. To complement the results for position space, a parallel study of electron correlation effects in momentum space is presented in Part III. In contrast to the position space investigation, the radial and angular components of correlation are found to have opposing effects, producing a rich and informative total correlation effect. As in position space, the Z-dependent correlation behaviour is studied and it is seen to be more informative in momentum space. The correlation effects in momentum space are similar to the effects in ground state systems, but are of a greater magnitude. Some techniques used in this research are described in three appendices in Part IV.