<p>In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case.</p><p>Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series.</p> / Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.

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In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case. Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series. / <p>Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.</p>

The vibration of complex mechanical systems that include coupled rotating and stationary bodies motivates this work. A semi-analytical model is developed for high-speed, compliant, rotating bodies. Exploiting the axisymmetry of the rotating body, the developed semi-analytical model only discretizes the two-dimensional radial cross-section; Fourier series are used in the circumferential direction. The corresponding formulation for thin-walled, axisymmetric shells is given. Even though the body is axisymmetric, its deflection as well as external forces, constraints, and supports acting on the body are allowed to be asymmetric. These asymmetric elements can be stationary or rotating. The model includes Coriolis and centripetal effects. The prestress (or geometric) stiffness matrix that arises from external forces and constant centripetal acceleration has additional terms compared to the literature, and these terms can significantly change the natural frequencies. Discrete stiffness-damper elements, elastic foundations, and constraint equations are used to couple the rotating body to other rotating and stationary bodies. The model is developed in a stationary reference frame to avoid time-dependent coefficients in the equations of motion when coupled to stationary components. Surface constraints are developed using equivalent force relations between multiple points on the surface and a reference node. Discrete stiffness-dampers, asymmetric elastic foundation, and asymmetric constraints introduce non-axisymmetry in the system. The speed-dependent natural frequencies and complex-valued vibration modes, presence of multiple Fourier harmonics in each mode, changes to critical speeds, divergence and flutter instability phenomena, and eigenvalue veering are investigated for spinning systems with asymmetric features. The developed semi-analytical model is used for rotationally periodic systems, for example, planetary gears. Rotationally periodic systems consist of multiple vibrating, rotating central components and substructures. The model is developed in a reference frame rotating with the central component that supports the substructures. Structured modal properties of the cyclically symmetric systems and diametrically opposed systems are investigated. The modes of the spinning system are categorized into translational-tilting, rotational-axial, and substructure modes. Time-varying coupling elements act as parametric excitation in the system. Large strain energy in the coupling elements lead to large parametric instability regions. The analytical closed-form expression of the parametric instability bandwidth obtained using a perturbation method compares well with numerical results from Floquet theory. / Doctor of Philosophy / Complex mechanical systems, for example, mechanical transmission, consist of coupled rotating and stationary bodies. The vibrations of rotating bodies are transmitted to the other bodies through coupling elements. To reduce weight of the system, the rotating bodies are made thin-walled resulting in increased flexibility of the body. The existing lumped parameter/rigid body models do not account for the flexibility of these rotating bodies. Conventional three-dimensional finite element models lead to a large number of degrees of freedom in the system, increasing the computational cost. We aim to develop a computationally efficient model to analyze the dynamics and vibration of complex mechanical systems. Most rotating bodies can be approximated as axisymmetric. The axisymmetric property of the rotating body is harnessed to reduce the three-dimensional model of the body to a two-dimensional radial cross-section using Fourier series in the circumferential direction. This reduces the system degrees of freedom. Coriolis, centripetal, and prestress effects are included in the model. Discrete stiffness-dampers, elastic foundations, and constraint equations couple the rotating body to other rotating and stationary bodies. Non-axisymmetric coupling elements and forces introduce asymmetry in the system. The system model for these asymmetric systems are developed in a stationary reference frame to avoid time-dependent coefficient equations of motion. Flexible stationary bodies alter the natural frequencies and vibration modes of the system. Instabilities, critical speeds, effects of asymmetry on the natural frequencies and vibration modes of the system are investigated. The model is extended for rotationally periodic systems, for examples, planetary gears and bearings. This model is developed in the reference frame that rotates with the central component that supports substructures. Structured modal characteristics are observed for the rotationally periodic systems. Changing contact conditions act as a source of parametric excitation in systems. Parametric resonances occur when natural frequencies of vibration with large strain energy in the coupling elements sum to the excitation frequency. Parametric instability regions obtained using an analytical equation compare well with numerical results.