Vibrations of Spinning Rings With Radial and Circumferential Displacement Constraints

The frequencies and mode shapes of rolling rings with radial and circumferential displacement constraints are investigated. The displacement constraints practically come from the point contact, e.g., rolling tire on the road, or other applications. The proposed approach to analysis is calculating the natural frequencies and modes of a non-contacted spinning ring, then employing the receptance method for displacement constraints. The frequency equation for the constrained system is hence obtained, and it can be solved numerically or graphically. The receptance matrix developed for the spinning ring is surprisingly found not symmetric as usual. Moreover, the cross receptances are discovered to form complex conjugate pairs. That is a feature that has never been described in literature. The results show that the natural frequencies for the spinning ring in contact, as expected, higher than those for the non-contacted ring. The variance of frequencies to rotational speeds are then illustrated. The analytic forms of mode shapes are also derived and sketched. The traveling modes are then shown for cases.

Determination of Instability Boundaries for a Highly Flexible Prismatic-Jointed Robot Performing Repetitive Maneuvers

Presented in this work are the equations of motion governing the behavior of a simple, highly flexible, prismatic-jointed robotic manipulator performing repetitive maneuvers. The robot is modeled as a uniform cantilever beam that is subject to harmonic axial motions over a single bilateral support. To conveniently and accurately predict motions that lead to unstable behavior, three methods are investigated for determining the boundaries of unstable regions in the parameter space defined by the amplitude and frequency of axial motion. The first method is based on a straightforward application of Floquet theory; the second makes use of the results of a perturbation analysis; and the third employs Bolotin’s infinite determinate method. Results indicate that both perturbation techniques and Bolotin’s method yield acceptably accurate results for only very small amplitudes of axial motion and that a direct application of Floquet theory, while computational expensive, is the most reliable way to ensure that all instability boundaries are correctly represented. These results are particularly relevant to the study of prismatic-jointed robotic devices that experience amplitudes of periodic motion that are a significant percentage of the length of the axially moving member.

Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials As Assumed Shape Functions

The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in Cartesian coordinate system are transformed into the modified polar coordinate system. Boundary characteristic orthogonal polynomials in the radial direction, and trigonometric functions in the angular direction are used to express the deflection of the plate. These deflection shapes are classified into four basic categories, depending on its symmetrical or antisymmetrical property about the major and minor axes of the ellipse. The first six natural modes in each of the above categories are presented in the form of contour plots.

Suppression of Subsynchronous Vibration in the SSME HPFTP

SSME HPFTP hot-fire dynamic data evaluation and rotordynamic analysis both confirm that two of the most significant turbopump attributes in determining susceptibility to subsynchronous vibration are impeller interstage seal configuration and rotor sideload resulting from turbine turnaround duct configuration and hot gas manifold. Recent hot-fire testing has provided promising indications that the incorporation of roughened “damping” seals at the impeller interstages may further increase the stability margin of this machine. A summary of the analysis which led to the conclusion that roughened seals would enhance the stability margin is presented herein, along with a correlation of the analysis with recent test data.

Axisymmetric Free Vibration of Thick Orthotropic Hemispherical Shells Under Various Edge Conditions

Axisymmetric free vibration of moderately thick polar orthotropic hemispherical shells are studied under the various boundary conditions of sliding, guided pin, clamped and hinged edges. Based on the improved linear elastic shell theory with the transverse shear strain and rotatory inertia taken into account, the dynamic equilibrium equations are formulated and transformed into the displacement form in terms of mid-surface meridian and radial displacements and parallel circle cross-section rotation. These partial differential equations are solved by the Galerkin method using proper Legendre polynomials as admissible displacement functions with the aid of the orthogonality and a number of special integral relations. Numerical results of the present theory compare well with existing data, which is available only in the isotropic theories. Good convergence is obtained for natural frequencies and mode shapes. Study of the effects of thickness and modulus ratio reveals higher frequencies for the thicker and/or stiffer shells with E\ oriented parallel to the meridians. Ranking of the natural frequencies descends in the order of guided pins, sliding, clamped and hinged edges in general. Also seen are the effects of transverse shear strain from the mode shapes with clamped and sliding edges on the slant. For the guided pin and sliding edges, frequencies increase fast as thickness increases so that new fundamental modes are generated in filling up the “frequency gap”. These are the new discoveries in the field of anisotropic shells, as a result of polar orthotropy of shell material and construction.

Dynamic and Static Design of Engine Test Benches

The elastic mechanical strength of the large and medium reciprocating engine test benches, is calculated for vertical and horizontal bendings and for torsion. The dynamic loads - horizontal vibrating forces and vertical vibrating couples (bending and torsion) and horizontal vibrating couple,- and the static loads, - distributed weight of the bench itself and the machinery and the motor torque (torsion), - are analyzed. It is pointed out that the bench body of reinforced concrete is submitted to bending moments and shearing forces and to torsion couples. This paper is the continuation of the paper “Dynamic Loads in Engine Test Benches” in 12th Biennial ASME Conference on Mechanical Vibration and Noise.

Dynamics of Flexible Beams and Plates Under Large Overall Motions

The dynamic response of flexible beams, plates, and solids undergoing arbitrary spatial motions are systematically derived via a unified approach. This formulation is capable of incorporating arbitrary representation of the kinematics of deformation, phenomenon of dynamic stiffening, and complete nonlinear interaction between elastic- and rigid-body dynamics encountered in constrained multi-body systems. It is shown that the present theory captures the phenomenon of dynamic stiffening due to the transfer of the axial and membrane forces to the bending equations of beams and plates, respectively. Examples are presented to illustrate the proposed formulations.

Numerical Analysis of the Transient Responses of Bladed Disks

A numerical model for the dynamic analysis of a rotationally periodic structure under rotation subjected to non-stationary excitation system of forces is presented on the basis of principles of finite elements and cyclic symmetry. Based on this model, the transient response characteristics of a representative practical bladed disk system during run-up subjected to nozzle-excitation are studied taking into account many important practical aspects of the problem. Some fundamental effects concerning the response maxima and resonance shift of these responses with respect to the cantilever blade response, angular acceleration and the order of excitation are discussed.

Dynamic Response of an Axially Moving Beam Coupled to Hydrodynamic Bearings

The vibration response of a distributed axially moving beam, controlled through distributed hydrodynamic bearing forces, is analyzed by the transfer function method. In the axially moving beam, all modes are affected by the distributed bearing coupling. The bearing force is described in terms of impedance functions. An approximate closed-form transfer function for the coupled beam-bearing system is derived. The transfer function is applicable to all linear, one-dimensional, coupled systems, and it is exact for localized, linear constraint forces. The derivation requires knowledge of the transfer function of the axially moving beam which is not available in the literature. A method for determining the beam transfer function is also presented. The frequency response of the coupled system is illustrated for several beam and bearing design parameters.

A 3-D Laminated Plate Theory and Nonlinear Impact Modal Analysis

A three-dimensional theory of cross-ply laminated thick plates is developed via an elasticity approach for nonlinear impact modal analysis. The simply supported plate is strictly stress free over all four edges and both lateral surfaces, in addition to stress-loading equilibrium over the patch loading area. The 3-D dynamic displacement fields are expressed in a mixed mode of double Fourier series and cubic polynomials. A system of modified Lagrange’s equations is derived incorporating all surface conditions and interface displacement and stress continuities. Validity of the present 3-D theory is proved in comparison to the only existing exact solutions in the basic cases of statics and free vibration. The nonlinear impact modal analysis is performed using the Hertz contact law in patch loading and Green’s function for small time linearization. The 3-D displacements and stresses are found to predict a tensile crack at the unimpacted side and probably, a delamination at the interface. They are always unsymmetric with respect to the mid-plane in all cases of unidirectional, symmetric and antisymmetric cross-ply laminates due to the one-sided loading.