Subharmonic vibration refers to the response of a dynamic system to excitation at a whole-number multiple (n) of its natural frequency by vibrating asynchronously at its natural frequency, that is, at (1/n) of the excitation. The phenomenon is generally associated with asymmetry in the stiffness vs. deflection characteristic of the system. It may be characterized as the “bouncing” of the rotor on the surface of the stiff support, energized by every nth unbalance impulse prior to contact. Second, third and fourth order subharmonic vibration responses have previously been observed in high speed rotating machinery with such an asymmetry in the bearing supports. An incident is reported where 8th and 9th order subharmonic vibration responses have been observed in a high speed rotor. A simple but exact computer model of the phenomenon has been evolved based on the numerical integration of a finite difference formulation. Response curves and wave forms of rotor deflection at individual speeds are computed. It is shown that the response is a series of pseudo-critical peaks at whole-number multiples of the rotational speed. Very high orders of subharmonic vibration are found to be possible for systems with low damping and extreme nonlinearity.

This is a report on work in progress on microelectrical and mechanical systems (MEMS)-based gas turbine engines, turbogenerators, and rocket engines currently under development at MIT. Fabricated in large numbers in parallel using semiconductor manufacturing techniques, these engines are based on micro-high speed rotating machinery with the same power density as that achieved in their more familiar, full-sized brethren. The micro-gas turbine is designed as a 1 cm diameter by 3 mm thick SiC heat engine producing 10-20 W of electric power or 0.05-0.

Abstract-A single-crystal silicon micromachined air turbine supported on gas-lubricated bearings has been operated in a controlled and sustained manner at rotational speeds greater than 1 million revolutions per minute, with mechanical power levels approaching 5 W. The device is formed from a fusion bonded stack of five silicon wafers individually patterned on both sides using deep reactive ion etching (DRIE). It consists of a single stage radial inflow turbine on a 4.2-mm diameter rotor that is supported on externally pressurized hydrostatic journal and thrust bearings. This paper presents the design, fabrication, and testing of the first microfabricated rotors to operate at circumferential tip speeds up to 300 m/s, on the order of conventional high performance turbomachinery. Successful operation of this device motivates the use of silicon micromachined high-speed rotating machinery for power microelectromechanical systems (MEMS) applications such as portable energy conversion, micropropulsion, and microfluidic pumping and cooling.[1161]

Subharmonic response in rotordynamics may be encountered when a rotor is operated with its rotational centerline eccentric to that of a close clearance static part, so that local contact can take place during each orbit when the rotor is excited by residual unbalance. The rotor will tend to bounce at or near its fundamental frequency when the rotor is operated at or near a speed which is a whole number [n] times that frequency. Using a simple numerical model of a Jeffcott rotor mounted on a nonlinear spring, it is found that the vibratory response in the transition zone midway between adjacent zones of subharmonic response has all the characteristics of chaotic behavior. The transition from subharmonic to chaotic response has a complex substructure which involves a sequence of bifurcations of the orbit with variations in speed. This class of rotordynamic behavior was confirmed and illustrated by experimental observations of the vibratory response of a high-speed turbomachine, operating at a speed between 8 and 9 times its fundamental rotor frequency when in local contact across a clearance in the support system. A narrow region between zones of 8th order and 9th order subharmonic response was identified where the response had all the characteristics of the chaotic motion identified in the numerical model.

When a rotor, excited by unbalance, is operating eccentrically within a clearance and in local contact with the stator it behaves as a bilinear oscillator with a natural periodic motion that resembles bouncing. When excited by unbalance at a subcritical rotative speed which is exactly or nearly 1/Nsuper times its natural frequency, the nonlinear system will respond by bouncing at or nearly at its natural frequency, or superharmonically at a frequency exactly Nsuper times the operating speed or forcing frequency. As in supercritical subharmonic response, there is a zone with characteristics of chaotic behavior in the transition zone between any order of superharmonic response and the next highest order of superharmonic response. There is also an intricate pattern of progressive bifurcations of the orbit on entry into this characteristically chaotic region and a reverse progression on exit from this region. The response is a mirror image or reciprocal set of the more thoughly studied supercritical subharmonic response of the same bilinear oscillator system which, when excited by unbalance at a supercritical rotative speed which is exactly or nearly a whole number Nsub times its natural frequency, the nonlinear system will respond by bouncing at exactly or nearly its natural frequency at a frequency exactly 1/Nsub times the operating speed or forcing frequency. Such supercritical subharmonic response is also characterized by the appearance of characteristically chaotic behavior in the transition zone between successive orders of subharmonic response and by patterns of progressive bifurcations of the orbit on entry into and exit from each region of characteristically chaotic response. Various aspects of subcritical superharmonic response are studied in a numerical model of the nonlinear system, and are compared to data taken on the core spool of an aircraft engine gas turbine. The engine data show many of the unique characteristics of response, wave form, and spectral content predicted by the numerical model of the bilinear oscillator when operating at subcritical rotative speed.