This study investigates the dynamics and vibration of high-speed planetary gears, spinning cantilevered beams, and gear pairs. High-speed planetary gear dynamics and vibration are analyzed using a lumped-parameter model. A continuous model is used to study spinning cantilever beam vibration. A finite element/contact mechanics model is used for the dynamics of gear pairs.
Chapter 2 investigates the modal property structure of high-speed planetary gears with gyroscopic effects. The vibration modes of these systems are complex-valued and speed-dependent. Equally-spaced and diametrically-opposed planet spacing are considered. Three mode types exist, and these are classified as planet, rotational, and translational modes. The properties of each mode type and that these three types are the only possible types are mathematically proven. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds.
In Chapter 3, the structured properties of the critical speeds and associated critical speed eigenvectors of high-speed planetary gears are identified and mathematically proven. Planetary gears have only planet, rotational, and translational mode critical speeds. Divergence instability is possible at speeds adjacent to critical speeds, and whether or not it occurs is determined using a perturbation method. Numerical results verify the critical speed locations and the stability near these critical speeds. Flutter instabilities occur at extremely high speeds, and these are investigated numerically for each mode type.
Chapter 4 demonstrates unusual gyroscopic system eigenvalue behavior observed in a lumped-parameter planetary gear model. While the model has been used for dynamic analyses in industrial applications, the focus is on the eigenvalue phenomena that occur at especially high speeds rather than practical planetary gear behavior. The behaviors include calculation of exact trajectories across critical speeds, uncommon stability features near degenerate critical speeds, and unique stability transitions. These eigenvalue behaviors are not evident in the vast literature on gyroscopic systems.
Chapter 5 investigates eigenvalue sensitivity to model parameters and eigenvalue veering in high-speed planetary gears. The eigenvalue perturbation approach is formulated such that the results apply to discrete, continuous, and hybrid discrete-continuous gyroscopic systems. Third-order perturbation approximations for the eigenvalues are determined. From the second-order perturbation approximation an eigenvalue veering parameter is defined and used to analyze veering in high-speed planetary gears. The sensitivity of the eigenvalues to model parameters are written in terms of modal kinetic and potential energies. Eigenvalue veering is prominent in planetary gears that have disrupted cyclic symmetry.
In Chapter 6, the single-mode vibrations of high-speed planetary gears are investigated in the rotating carrier-fixed and the stationary inertial reference frames. The properties of the structured planetary gear modes result in gear motions with interesting geometry. The frequency content of the motion differs between the rotating carrier-fixed and stationary inertial bases. The results from this work assist the analysis of experimental planetary gear measurements.
A linear model for the bending-bending-torsional-axial vibration of a spinning cantilever beam with a rigid body attached at its free end is derived in Chapter 7 using Hamilton's Principle. The rotation axis is perpendicular to the beam (like a helicopter blade). The equations split into two uncoupled groups: coupled bending in the direction of the rotation axis with torsional motions, and coupled bending in the plane of rotation with axial motions. The practically important first case above is examined in detail. The governing equations of motion are cast in a structured way using extended variables and extended operators. With this structure the equations represent a classical gyroscopic system. Using the extended operator structure, the equations are discretized using Galerkin's method, and subsequently the eigenvalues and mode shapes are calculated for varying rotation speeds.
In Chapter 8, the general Euler-Lagrange equations for gyroscopic continuum are derived from Hamilton's Principle using kinetic, potential, and virtual work expressions with specific functional dependencies typical of gyroscopic continua. These equations are useful in problems with multiple variables, where directly taking variations of the Lagrangian is cumbersome. The equations can be used to derive linear and nonlinear governing equations. The formulation is in a form that is suitable for programming in computer algebra software. The resulting Euler-Lagrange equations are applied to axially moving media, rotating shafts, and spinning beams to determine governing equations of motion.
In Chapter 9, a finite element formulation for the dynamic response of gear pairs is proposed. Following an established approach in lumped parameter gear dynamic models, the static solution is used as the excitation in a frequency domain solution of the finite element vibration model. The nonlinear finite element/contact mechanics formulation provides accurate calculation of the static solution and average mesh stiffness that are used in the dynamic simulation. The frequency domain finite element calculation of dynamic response compares well with numerically integrated (time domain) finite element dynamic results and previously published experimental results. Simulation time with the proposed formulation is two orders of magnitude lower than numerically integrated dynamic results. This formulation admits system level dynamic gearbox response, which may include multiple gear meshes, flexible shafts, rolling element bearings, housing structures, and other deformable components.