Journal of Applied Mechanics, Transactions ASME

The orthogonality properties among the eigenfunctions for a gyroscopic system are derived for a stiffness operator that is not positive definite. The derivatives of the eigenvalues with respect to certain parameters in the system are then obtained. The results are applied to a spinning disk in contact with a stationary load system, which contains such parameters as friction force, transverse mass, damping, stiffness, and the analogous pitching elements, to predict the effects of these parameters and the stiffening of the disk due to the centrifugal force on the natural frequencies and stability of the spinning disk. The results obtained provide a theoretical understanding for previously reported observations based on numerical solutions.

In a previous paper (Chen and Bogy, 1992) we studied the effects of various load parameters, such as friction force, transverse mass, damping, stiffness and the analogous pitching parameters, of a stationary load system in contact with the spinning disk on the natural frequencies and stability of the system when the original eigenvalues of interest are well separated. This paper is a follow-up investigation to deal with the situations in which two eigenvalues of the freely spinning disk are almost equal (degenerate) and strong modal interactions occur when the load parameters are introduced. After comparing an eigenfunction expansion with the finite element numerical results, we find that for each of the transverse and pitching load parameters, a properly chosen two-mode approximation can exhibit all the important features of the eigenvalue changes. Based on this two-mode approximation we study the mathematical structure of the eigenvalues in the neighborhood of degenerate points in the natural frequency-rotation speed plane. In the case of friction force, however, it is found that at least a four-mode approximation is required to reproduce the eigenvalue structure. The observations and analyses presented provide physical insight into the modal interactions induced by various load parameters in a spinning disk-stationary load system.

A generalized approach to predict the physical instability mechanisms that are involved in the interaction between a rotating flexible disk and a stationary constraining system is developed. Based upon equations derived for an energy flux analysis, unified instability conditions for various lateral interactive forces are presented. These developments lead to a clear understanding of the physical mechanisms involved in the development of vibrational instabilities. New developments also involve the stability analysis of a rotating disk subjected to multiple moving concentrated regenerative and follower interactive forces that act over a space-fixed sector. The lateral regenerative interactive forces that are responsible for self-excited vibrations in saw-blade cutting are identified and modeled. The generalized Fourier series method is proposed to develop a characteristic equation for time-varying dynamic systems with or without time lag. The resulting equation can be solved efficiently by using Mu¨ller’s algorithm with deflation.

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

Using the von Karman field equations, large amplitude vibrations of a spinning disk are analysed. For definiteness, the disk is assumed to be free, and its deflection is represented by a two-term polynomial. A vibration mode associated with two nodal diameters is studied in more detail. A familiar phenomenon of a decreasing period of vibration with an increasing amplitude is corroborated. The results specialized to the linear case show a close agreement with the classical results of Lamb and Southwell. The dependence of the membrane stresses on the amplitude of vibration and the velocity of spin is discussed.

This paper presents results of an investigation on the effect of a transverse load on the stability of a spinning elastic disk. The disk rotates at constant angular velocity and the load consists of a mass distributed over a small area of the disk, a spring, and a dashpot. The equation of motion for the transverse vibration of the disk is written as a system of linear ordinary differential equations with constant coefficients. The analysis indicates that the disk system is unstable for speeds in a region above the critical speeds of vibration of the spinning disk due to the effects of load stiffness. The mass and damping of the load system cause a terminal instability and other instabilities occur as a result of modal interaction.

This paper describes the modeling, theoretical formulation, and eigenvalue analysis for a combined system of a spinning flexible disk and a pair of head and suspension systems that contact the disk at opposing points on its two sides. In the analytical model a constant friction force between the sliders and disk and the slider pitch motion, as well as its transverse motion, are taken into account. From the eigenvalue analysis it is found that pitch stiffness and moment of inertia of the heads induce instability above the critical rotation speed similarly to the transverse stiffness and mass. This instability can be effectively stabilized by increasing the external damping which is spinning with the disk. It is also found that the friction force makes all forward modes unstable over the entire rotational speed range. The friction induced instability can be effectively suppressed by increasing the transverse stiffness and mass and it can be stabilized by the pitch damping and the external damping. The characteristics of instability due to the friction force qualitatively agree well with experimental results reported previously.

An investigation of the dynamic response of a circular elastic disk excited by a moving mass-spring-dashpot system is presented. Three distinct types of unstable response are observed. A stiffness instability region occurs immediately above each critical speed of the disk. Other regions of instability arise as a consequence of modal interaction. A terminal instability region exists for all load speeds above a certain limiting value. The presence of viscous dissipation in the loading system may destabilize the total system.

A novel design of a two-stage nonlinear vibration isolation system, with each stage having a high-static-low-dynamic stiffness (HSLDS), is studied experimentally in this paper. The positive stiffness in each stage is realized by a metallic plate, and the corresponding negative stiffness is realized by a bistable carbon fiber–metal (CF) composite plate. An analytical model is developed as an aid to design a bistable composite plate with the required negative stiffness, and a static test of the plate is conducted to measure the actual stiffness of the plate. Dynamic tests of the two-stage isolator are carried out to determine the effectiveness of the isolator. Two tests are conducted, one with the bistable composite plates removed so that the isolator behaves as a linear device and one with the bistable composite plates fitted. An improvement in the isolator transmissibility of about 13 dB at frequencies greater than about 100 Hz is achieved when the bistable composite plates are added.

Problems in aircraft dynamics such as stability and response of the rigid airplane may be affected by fuel motion in the tanks. Such problems also might arise in connection with missiles. In this paper the response of the fuel to simple harmonic motions of a rectangular tank in translation, pitching, and yawing is studied. Using the force and moment expressions, simple mechanical systems equivalent to the fuel are constructed. These systems respond to motions of the tank walls in the same fashion as the fuel, producing identical forces and moments. The use of such mechanical analogies should simplify in many cases the analysis of the complete dynamic system.