We examine the non-stationary response of a one-degree-of-freedom non-linear system to a non-periodic parametric excitation with varying frequency. We use the method of multiple scales to obtain equations governing the stationary and non-stationary responses of the system, and we analyze the stability of the stationary responses. The response displays several phenomena, including penetration of the trivial response into the unstable trivial region, oscillation of the response about the non-trivial stationary solution, convergence of the non-stationary response to the stationary solution, lingering of the non-trivial response into the stable trivial region, and rebounding of the non-trivial response. These phenomena are affected by the sweep rate, the initial conditions, and the system parameters. Digital and analog computers are used to solve the original governing differential equation. The results of the simulations agree with each other and with those obtained by using the method of multiple scales.

Response of a single-degree-of-freedom system to a non-stationary principal parametric excitation

There are many good references for classical balance problems; however, specific ways of handling non-ideal conditions are rarely presented. Therefore, this tutorial focuses on six case histories, which illustrate actual problems that were encountered while trying to field balance different types of rotating machinery. Coupling lockup, thermal bows, eccentricity, looseness, and structural resonances can prevent successful balancing if not properly identified and corrected. This tutorial shows how these real world pitfalls can complicate balance procedures and provides some ways of dealing with these issues such as: indexing coupling parts, using inference fits, adding structural bracing, and acquiring vibration measurements at additional locations. This tutorial also gives a brief description of the common types of unbalance. INTRODUCTION Generally, when machinery experiences high vibration at 1× running speed, the first course of action that a person may take is to improve the balance of the rotor. This tutorial provides a brief overview of different unbalance conditions, theories, and criteria used when balancing rotating equipment. There are many good books, papers, and articles that explain balancing procedures in great detail as listed in the Bibliography. These works may also cover the limitations of various methods and provide warnings which state that vibration problems due to misalignment or instability can not be “balanced out.” A literature search uncovered papers by Rieger (1984) and Baxter and Whaley (1995) containing case histories where balancing proved difficult. It is the intent of this tutorial to show how instances where balancing was not initially successful can be addressed. The six case histories demonstrate many of the pitfalls that can occur when trying to balance rotating equipment. The purpose of balancing a rotor is to help ensure that the machinery is safe and reliable. This is achieved when the rotor mass and rotational centerlines are as close to equal as possible. Excessive unbalance can cause vibration and stress in the shaft or attached pieces. For example, a generator with high shaft vibration can cause the overhung exciter to loosen and fail. Dynamic forces can also be transmitted to supporting structures, such as the bearing housing or equipment case. An unbalanced centrifuge on a common deck can cause vibration in adjacent pieces of equipment. If operators have to walk on vibrating grating, this can cause annoyance and fatigue. Excessive vibration can cause wear in bearings, seals, gears, etc. and reduce the life expectancy of these parts. As a rotor spins, centrifugal forces act upon it. The surface around the periphery is stressed as particles are pulled outward from the axis of rotation. If all of these radial forces are equal, the rotor is said to be balanced and should not vibrate. However, if the rotor contains a heavy spot to one side, these radial forces will not cancel and the unbalanced force will tend to pull the rotor from the center of rotation. Figure 1 shows that the centrifugal force produced by unbalance depends on the mass and radius as well as the rotational speed of the rotor.

Balance This! Case Histories From Difficult Balance Jobs.

Analysis of deceleration phenomenon of high speed rotor systems

Response of a cracked rotating shaft with a disk during passage through a critical speed

The problem of a single-degree-of-freedom rotor decelerating slowly through its critica speed is solved by an energy approach; a closed solution is obtained. A small discontinuous downward jump of rotor speed across the critical speed is shown to be required, either with or without damping in the system. The maximum increment of deflection, hence bending stress, in the rotor shaft is shown to be small, provided the rotor is carefully balanced and/or the system is sufficiently damped.

W. K. Bodger

Papers

Deceleration of an Unbalanced Rotor Through a Critical Speed