Harmonic Balance Analysis of General Squeeze Film Damped Multidegree-of-Freedom Rotor Bearing Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 499-507 ◽  
Author(s):  
E. J. Hahn ◽  
P. Y. P. Chen
Keyword(s):  
Harmonic Balance ◽  
Reference Frames ◽  
Squeeze Film ◽  
Equilibrium Solutions ◽  
Rotor Bearing ◽  
Balance Analysis ◽  
Force Components ◽  
Lift Off ◽  
Rotating Reference Frames ◽  
The Stability

Squeeze film dampers introduce nonlinear motion dependent damper forces into otherwise linear rotor bearing systems, thereby considerably complicating their analysis. Noncircular orbit type dampers, such as unsupported or uncentralized dampers, have generally necessitated transient solutions, which are computationally prohibitive for design studies of large order systems, particularly for systems with low damping. By utilizing harmonic balance with appropriate condensation, it is possible to considerably reduce the number of simultaneous nonlinear equations inherent to this approach. The stability (linear) of the equilibrium solutions may be conveniently evaluated using Floquet theory, particularly if the damper force components are evaluated in fixed, rather than rotating, reference frames. The versatility of this technique is illustrated on systems of increasing complexity with and without damper centralizing springs. Of particular interest, is its applicability to unsupported systems illustrating how such systems can lift off and, with further increase in speed, the damper forces can be linearized about the orbit center.

scholarly journals Analytical and Experimental Investigation of the Stability of the Rotor-Bearing System of a New Small Turbocharger

1987 ◽  
Author(s):  
H. R. Born
Keyword(s):  
Experimental Investigation ◽  
Dynamic Stability ◽  
Squeeze Film ◽  
Test Results ◽  
Flow Capacity ◽  
Squeeze Film Dampers ◽  
Rotor Bearing ◽  
The Stability ◽  
Turbine Design ◽  
Bearing System

This paper presents an overview of the development of a reliable bearing system for a new line of small turbochargers where the bearing system has to be compatible with a new compressor and turbine design. The first part demonstrates how the increased weight of the turbine, due to a 40 % increase in flow capacity, influences the dynamic stability of the rotor-bearing system. The second part shows how stability can be improved by optimizing important floating ring parameters and by applying different bearing designs, such as profiled bore bearings supported on squeeze film dampers. Test results and stability analyses are included as well as the criteria which led to the decision to choose a squeeze film backed symmetrical 3-lobe bearing for this new turbocharger design.

scholarly journals Subharmonic Oscillations in Squeeze Film Damped Rotor Bearing Systems Without Centralizing Springs

1993 ◽  
Author(s):  
Peter Y. P. Chen ◽  
E. J. Hahn ◽  
G. Y. Wang
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Excitation Frequency ◽  
Squeeze Film ◽  
Equilibrium Solutions ◽  
System Parameters ◽  
Fundamental Frequencies ◽  
Damped Systems ◽  
Subharmonic Orbits ◽  
Spring Support

This paper investigates the nonsynchronous motion of a rigid rotor in squeeze film damped systems without spring support subjected to both unbalance and unidirectional loading. Both harmonic balance and 4th order Runge–Kutta integration are used to obtain the unbalance response, the asymptotic stability of the harmonic balance equilibrium solutions being determined by Floquet theory. Depending on the system parameters stable subharmonic orbits with fundamental frequencies of half, one third and one quarter of excitation frequency were obtained. Also, bistable, tristable and quadristable solution possibilities were found. The effect of these subharmonic orbits on maximum transmissibilities is included.

Stability of Squeeze Film Damped Multi-Mass Flexible Rotor Bearing Systems

1985 ◽  
Vol 107 (3) ◽  
pp. 402-409 ◽  
Author(s):  
L. J. McLean ◽  
E. J. Hahn
Keyword(s):  
Performance Monitoring ◽  
Operating Conditions ◽  
Squeeze Film ◽  
Flexible Rotor ◽  
Equilibrium Solutions ◽  
Rotor Bearing ◽  
Stability Maps ◽  
Damper Design ◽  
Synchronous Orbit ◽  
And Performance

A technique is presented for investigating the stability of and the degree of damping in the circular synchronous orbit equilibrium solutions pertaining to radially symmetric multi-mass flexible rotor bearing systems. It involves the analysis of appropriate linearized perturbation equations about the equilibrium solutions and is applicable to systems with several squeeze film dampers. For a system with a single damper, stability threshold maps, independent of unbalance distribution, may be found in terms of the same damper parameters and operating conditions as the equilibrium solutions, thereby allowing for damper design and performance monitoring. The technique is illustrated for a simple symmetric four degree of freedom flexible rotor with an unpressurized damper. This example shows the utility of zero frequency stability maps for delineating multiple solution possibilities and that for low (in this case of the order of 0.06 or lower) bearing parameters, the introduction of an unpressurized squeeze film damper may promote instability in an otherwise stable system.

scholarly journals Transition curves in a parametrically excited pendulum with a force of elliptic type

2012 ◽  
Vol 468 (2148) ◽  
pp. 3995-4007 ◽  
Author(s):  
Si Mohamed Sah ◽  
Brian Mann
Keyword(s):  
Numerical Integration ◽  
Elliptic Function ◽  
Harmonic Balance ◽  
Floquet Theory ◽  
Parameter Plane ◽  
Elliptic Type ◽  
Mathieu’S Equation ◽  
Balance Analysis ◽  
The Stability ◽  
Transition Curves

This article investigates the equilibria and stability of a pendulum when the support has a prescribed motion defined by an elliptic function. Stability charts are generated in the parameter plane for different values of the elliptic function modulus. Numerical integration and Floquet theory are used to generate stability charts that are later obtained through harmonic balance analysis. It is shown that the size and location of the instability tongues is directly linked to the elliptic function modulus. Comparisons are also made between the stability charts of Mathieu's equation and those of the pendulum when the prescribed motion is defined by an elliptic function.

scholarly journals Comprehensive Reduced-Order Models of Electrostatically Actuated MEMS Switches and Their Dynamics Including Impact and Bounce

2010 ◽  
Author(s):  
Michael G. Snow ◽  
Anil K. Bajaj
Keyword(s):  
Galerkin Method ◽  
Rarefied Gas ◽  
Dielectric Substrate ◽  
Contact Dynamics ◽  
Mode Shapes ◽  
Squeeze Film ◽  
Equilibrium Solutions ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Lift Off

As MEMS technology develops it is becoming better understood that MEMS designers must account for the large uncertainties characteristic of the relevant manufacturing processes. Uncertainty quantification tasks the designer with evaluating many different possible outcomes from the manufacturing process which creates a demand for models that are accurate and comprehensive, yet fast to evaluate. This work presents a comprehensive reduced-order model of electrostatically actuated switches incorporating a range of effects that are typically included only in FE modeling codes. Specifically, the model accounts for variable electrode geometry, stretching of centerline or large displacement effects, fringing field, squeeze film and rarefied gas damping, and allows for elastic contact with the dielectric substrate. Individual compact models for each of these effects are taken from literature and included in the model for the system. The dielectric substrate is modeled as an elastic foundation. The resulting partial differential equation for the switch modeled as a beam is discritized via a Galerkin method into ordinary differential equations for modal amplitudes. The Galerkin method uses the linear un-damped mode shapes of the beam to approximate the solution. Both cantilever and fixed-fixed type switches are analyzed. Static equilibrium solutions as a function of the applied voltage are developed along with their stability. Static pull-in voltages, first time of switch closure, and voltage for lift-off are studied with the model. To capture the contact dynamics, the contact condition is evaluated with the substrate divided into a large number of elements and the contact force is projected on to the beam basis functions. In the case of cantilever geometry and slow voltage variations, three stable regimes of contact configuration and hysteresis between them are demonstrated.

Periodic Solutions in Rotor Dynamic Systems With Nonlinear Supports: A General Approach

1989 ◽  
Vol 111 (2) ◽  
pp. 187-193 ◽  
Author(s):  
C. Nataraj ◽  
H. D. Nelson
Keyword(s):  
Dynamic Systems ◽  
Original Problem ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Trigonometric Collocation ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Future Work ◽  
Rotor Dynamic ◽  
Large Rotor

A new quantitative method of estimating steady state periodic behavior in nonlinear systems, based on the trigonometric collocation method, is outlined. A procedure is developed to analyze large rotor dynamic systems with nonlinear supports by the use of the above method in conjunction with Component Mode Synthesis. The algorithm discussed is seen to reduce the original problem to solving nonlinear algebraic equations in terms of only the coordinates associated with the nonlinear supports and is a big improvement over commonly used integration methods. The feasibility and advantages of the procedure so developed are illustrated with the help of an example of a typical rotor dynamic system with an uncentered squeeze film damper. Future work on the investigation of the stability of the periodic response so obtained is outlined.

Nonlinear Analysis of Rotor-Bearing Systems Using Component Mode Synthesis

1983 ◽  
Vol 105 (3) ◽  
pp. 606-614 ◽  
Author(s):  
H. D. Nelson ◽  
W. L. Meacham ◽  
D. P. Fleming ◽  
A. F. Kascak
Keyword(s):  
Forced Response ◽  
Component Mode Synthesis ◽  
Squeeze Film ◽  
System Response ◽  
Reduced Order ◽  
Squeeze Film Dampers ◽  
Rotor Bearing ◽  
System Equations ◽  
Transient System ◽  
Blade Loss

The method of component mode synthesis is developed to determine the forced response of nonlinear, multishaft, rotor-bearing systems. The formulation allows for simulation of system response due to blade loss, distributed unbalance, base shock, maneuver loads, and specified fixed frame forces. The motion of each rotating component of the system is described by superposing constraint modes associated with boundary coordinates and constrained precessional modes associated with internal coordinates. The precessional modes are truncated for each component and the reduced component equations are assembled with the nonlinear supports and interconnections to form a set of nonlinear system equations of reduced order. These equations are then numerically integrated to obtain the system response. A computer program, which is presently restricted to single shaft systems has been written and results are presented for transient system response associated with blade loss dynamics, with squeeze film dampers, and with interference rubs.

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